NUMERICAL ANALYSIS OF CONDITIONS NECESSARY FOR
NEAR-SURFACE SNOW METAMORPHISM
by
Andrew Edward Slaughter
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Engineering
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 2010
© Copyright
by
Andrew Edward Slaughter
2010
All Rights Reserved
ii
APPROVAL
of a dissertation submitted by
Andrew Edward Slaughter
This dissertation has been read by each member of the dissertation committee and
has been found to be satisfactory regarding content, English usage, format, citations,
bibliographic style, and consistency, and is ready for submission to the Division of
Graduate Education.
Dr. Edward E. Adams
Approved for the Department of Civil Engineering
Dr. Brett W. Gunnink
Approved for the Division of Graduate Education
Dr. Carl Fox
iii
STATEMENT OF PERMISSION TO USE
In presenting this dissertation in partial fulfillment of the requirements for a doc-
toral degree at Montana State University, I agree that the Library shall make it
available to borrowers under rules of the Library. I further agree that copying of this
dissertation is allowable only for scholarly purposes, consistent with “fair use” as pre-
scribed in the U.S. Copyright Law. Requests for extensive copying or reproduction of
this dissertation should be referred to ProQuest Information and Learning, 300 North
Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the exclusive right
to reproduce and distribute my dissertation in and from microform along with the
non-exclusive right to reproduce and distribute my abstract in any format in whole
or in part.”
Andrew Edward Slaughter
April 2010
iv
ACKNOWLEDGMENTS
First, I would like to thank my advisor, Dr. Adams, for all of his assistance
through the years, for securing various sources of funding to keep me going, and
mostly for allowing me to discover my own path. Thanks to Dr. McKittrick for his
guidance with numerical analysis and setting up servers for all my projects. And,
thank you to my entire committee for their time and patience.
This dissertation would not be possible without the dedicated Yellowstone Club
Ski Patrol that helped produce a world-class data set, specifically Tom Leonard,
Doug McCabe, Irene Henninger, Doug Catherine, and Henry Munter. I would also
like to thank my fellow graduate students for their assistance in gathering data, the
conversations, and the encouragement. A special thanks to those I shared an office
with: Peter Gammelgard, Patrick Staron, and Rich Schertzer. Thank you to Dave
Neal for my experience in his 5th grade class. Thank you to my parents as well
as all of my family and friends for their support. Thank you to my beautiful wife,
Deanne, for her patience, emotional and financial support, editing skills, and mostly
for her love. Lastly, I am thankful for the fortitude granted by our heavenly Father
to continue the pursuit of this achievement.
vTABLE OF CONTENTS
1. INTRODUCTION ........................................................................................1
2. SIGNIFICANCE AND BACKGROUND OF SURFACE WEAK LAYERS .......5
2.1 Introduction ............................................................................................5
2.2 Significance of the Near-Surface Layers .....................................................5
2.2.1 Avalanches........................................................................................5
2.2.2 Climate ............................................................................................6
2.2.3 Synopsis ...........................................................................................8
2.3 A Review of Surface Hoar ........................................................................9
2.4 A Review of Near-Surface Facets ............................................................ 17
2.5 Conclusions ........................................................................................... 21
3. FIELD INVESTIGATION OF SURFACE HOAR ......................................... 24
3.1 Introduction .......................................................................................... 24
3.2 Methods................................................................................................ 25
3.2.1 Weather stations ............................................................................. 25
3.2.2 Instrumentation .............................................................................. 25
3.2.3 Snow observations ........................................................................... 26
3.3 Results.................................................................................................. 27
3.4 2007/2008 Surface Hoar Events .............................................................. 29
3.4.1 Event A-1: January 24, 2008............................................................ 29
3.4.2 Event A-2: February 15, 2008 .......................................................... 31
3.4.3 Event A-3: February 19–21, 2008 ..................................................... 32
3.4.4 Event A-4: February 22, 2008 .......................................................... 36
3.4.5 Event A-5: February 26, 2008 .......................................................... 38
3.4.6 Event A-6: March 10, 2008 .............................................................. 40
3.4.7 Event A-7: March 30, 2008 .............................................................. 42
3.5 2008/2009 surface hoar events ................................................................ 44
3.5.1 Event B-1: January 23, 2009 ............................................................ 44
3.5.2 Event B-2: January 30–31, 2009 ....................................................... 45
3.5.3 Event B-3: February 4, 2009 ............................................................ 47
3.5.4 Event B-4: February 7–8, 2009 ......................................................... 49
3.5.5 Event B-5: February 13–14, 2009...................................................... 51
3.5.6 Event B-6: February 28, 2009........................................................... 52
3.5.7 Event B-7: March 13, 2009 .............................................................. 53
3.6 Analysis ................................................................................................ 55
3.7 Future Considerations ............................................................................ 59
3.8 Conclusion ............................................................................................ 60
vi
TABLE OF CONTENTS – CONTINUED
4. FIELD INVESTIGATION OF NEAR-SURFACE FACETS ........................... 61
4.1 Introduction .......................................................................................... 61
4.2 Methods................................................................................................ 61
4.3 Results.................................................................................................. 63
4.4 2007/2008 Near-surface Facet Events ...................................................... 66
4.4.1 Event C-1: January 21, 2008 ............................................................ 66
4.4.2 Event C-2: February 14–16, 2008...................................................... 66
4.4.3 Event C-3: February 18–20, 2008...................................................... 69
4.4.4 Event C-4: February 26–27, 2008...................................................... 71
4.4.5 Event C-5: March 6, 2008 ................................................................ 75
4.4.6 Event C-6: March 10, 2008 .............................................................. 75
4.4.7 Event C-7: March 13, 2008 .............................................................. 77
4.4.8 Event C-8: March 15, 2008 .............................................................. 78
4.4.9 Event C-9: March 19, 2008 .............................................................. 79
4.4.10 Event C-10: March 22, 2008 ........................................................... 81
4.4.11 Event C-11: March 28, 2008 ........................................................... 82
4.4.12 Event C-12: March 30, 2008 ........................................................... 82
4.4.13 Event C-13: April 2–4, 2008 ........................................................... 84
4.4.14 Event C-14: April 6, 2008 .............................................................. 85
4.4.15 Event C-15: April 8, 2008 .............................................................. 87
4.5 2008/2009 Near-surface Facet Events ...................................................... 89
4.5.1 Event D-1: February 4, 2009 ............................................................ 89
4.5.2 Event D-2: February 8, 2009 ............................................................ 90
4.5.3 Event D-3: February 12–14, 2008 ..................................................... 92
4.5.4 Event D-4: February 19, 2009 .......................................................... 93
4.5.5 Event D-5: February 21, 2009 .......................................................... 95
4.5.6 Event D-6: February 27–28, 2009 ..................................................... 97
4.5.7 Event D-7: March 7, 2009 ................................................................ 99
4.5.8 Event D-8: March 12–14, 2009 ....................................................... 101
4.5.9 Event D-9: March 20, 2009 ............................................................ 104
4.5.10 Event D-10: March 30, 2009 ......................................................... 105
4.5.11 Event D-11: April 6, 2009 ............................................................ 107
4.6 Analysis .............................................................................................. 108
4.7 Conclusions ......................................................................................... 111
5. SNOW THERMAL MODEL ..................................................................... 114
5.1 Introduction ........................................................................................ 114
vii
TABLE OF CONTENTS – CONTINUED
5.2 Background......................................................................................... 115
5.3 Model Development ............................................................................. 117
5.3.1 Conservation of Energy.................................................................. 117
5.3.2 Application ................................................................................... 120
5.3.3 Numerical Solution ........................................................................ 121
General Numeric Equation ................................................................. 121
Boundary Conditions ......................................................................... 123
Matrix Solution ................................................................................. 124
5.3.4 Short-wave Radiation .................................................................... 125
5.3.5 Surface Flux Terms ....................................................................... 127
5.3.6 Boundary Layer Application .......................................................... 130
5.3.7 Material Properties........................................................................ 132
5.4 Analysis with VIS/NIR Components..................................................... 133
5.5 Reliability of Model ............................................................................. 136
5.6 Closing Remarks.................................................................................. 137
6. SOBOL SENSITIVITY ANALYSIS: THEORY AND EXAMPLES .............. 139
6.1 Introduction ........................................................................................ 139
6.2 Sensitivity Defined............................................................................... 142
6.3 Decomposition of Variance ................................................................... 144
6.3.1 Closed Variance............................................................................. 145
6.3.2 Total-effect Variance...................................................................... 145
6.4 SOBOL Method .................................................................................. 146
6.4.1 Basic Premise................................................................................ 147
6.4.2 Improved SOBOL Method ............................................................. 149
6.4.3 A “Less Expensive” SOBOL .......................................................... 153
6.5 Confidence Levels and Bias Correction .................................................. 153
6.5.1 BCa Confidence Level Intervals ...................................................... 155
6.5.2 Bias Correction ............................................................................. 157
6.6 Example 1: SOBOL ............................................................................ 157
6.7 Example 2: Temporal Analysis ............................................................. 160
6.8 Closing Remarks.................................................................................. 161
7. IMPLEMENTATION OF NUMERICAL ANALYSIS TECHNIQUES........... 162
7.1 Introduction ........................................................................................ 162
7.2 Thermal Model Input Distributions ...................................................... 162
7.3 Model Evaluations ............................................................................... 168
viii
TABLE OF CONTENTS – CONTINUED
7.4 Sensitivity Analysis.............................................................................. 168
7.5 Monte Carlo Analysis .......................................................................... 170
7.6 Highest Density Regions....................................................................... 170
7.7 Empirical Probability Density Functions ............................................... 173
7.8 Goodness-of-fit Hypothesis Test............................................................ 174
7.9 Closing Remarks.................................................................................. 175
8. NUMERICAL ANALYSIS OF SURFACE HOAR ....................................... 176
8.1 Introduction ........................................................................................ 176
8.2 Methods.............................................................................................. 177
8.3 Results: Sensitivity Analysis ................................................................ 178
8.3.1 Mean Mass-Flux, Φ ....................................................................... 178
8.3.2 Minimum and Maximum Mass-flux (Φmin and Φmax) ....................... 180
8.3.3 Positive and Negative Mean Mass-flux (Φpos and Φneg) .................... 182
8.4 Discussion: Sensitivity Analysis ............................................................ 186
8.5 Results and Discussion: Monte Carlo Simulations .................................. 188
8.6 Analysis: Comparison with Field Observations ...................................... 191
8.7 Closing Remarks.................................................................................. 199
9. SENSITIVITY ANALYSIS OF NEAR-SURFACE FACETS ........................ 202
9.1 Introduction ........................................................................................ 202
9.2 Methods.............................................................................................. 202
9.3 Results and Discussion......................................................................... 205
9.3.1 Snow Temperatures ....................................................................... 205
Snow Surface Temperature ................................................................. 205
Snow Temperatures at Depth ............................................................. 211
9.3.2 Temperature Gradient ................................................................... 216
Gradient Computed at 2 cm............................................................... 216
“Knee” Temperature Gradient............................................................ 219
9.4 Closing Remarks.................................................................................. 223
10. MONTE CARLO SIMULATIONS OF NEAR-SURFACE FACETS ............. 225
10.1 Introduction ...................................................................................... 225
10.2 Methods ............................................................................................ 225
10.3 Results .............................................................................................. 228
10.4 Discussion ......................................................................................... 232
10.5 Analysis ............................................................................................ 234
ix
TABLE OF CONTENTS – CONTINUED
Control Location ............................................................................... 237
North Location.................................................................................. 238
South Location .................................................................................. 239
10.6 Closing Remarks ................................................................................ 241
11. CONCLUSIONS ....................................................................................... 243
REFERENCES CITED.................................................................................. 247
APPENDICES .............................................................................................. 258
APPENDIX A: Yellowstone Club Weather Stations .................................. 259
APPENDIX B: YCweather User Manual ................................................. 305
APPENDIX C: Thermal Model Software User Manual ............................. 343
APPENDIX D: Sensitivity Analysis Software User Manual ....................... 366
APPENDIX E: Sensitivity Analysis Results for Surface Hoar .................... 391
APPENDIX F: Sensitivity Analysis Results for Near-Surface Facets .......... 413
APPENDIX G: Yellowstone Club Daily Logs ........................................... 468
vLIST OF TABLES
Table Page
2.1 A summary of the conditions necessary for surface hoar growth as re-
ported in the literature reviewed. ............................................................. 16
2.2 A summary of the conditions necessary for near-surface facet growth as
reported in the literature reviewed. .......................................................... 22
2.3 Summary of quantifiable parameters shown to lead to the formation of
(a) surface hoar and (b) near-surface facets as presented in the available
literature................................................................................................ 23
3.1 Detailed information on each of the three weather stations situated on
Pioneer Mountain. .................................................................................. 26
3.2 Summary of mean nightly weather conditions for all days recorded as
surface hoar events. ................................................................................ 28
3.3 Summary of the snow conditions for the layer underlying the surface hoar,
as recorded in the field notes. .................................................................. 28
3.4 Kolmogorov-Smirnov test results comparing the distributions set shown
in Figures 3.31 and 3.30; the null hypothesis (H0) was that the data are
from the same distribution. ..................................................................... 57
3.5 Percentiles of environmental variables coupled to the formation of surface
hoar. ...................................................................................................... 58
4.1 Summary of snow conditions prior to the near-surface facet events as
recorded in the field notes. Events tagged with an asterisk (*) indicate
events, as noted in the field notes, that were likely dominated by non-
radiation processes.................................................................................. 64
4.2 Summary of mean daily weather conditions for all days recorded as near-
surface facets events................................................................................ 65
4.3 Kolmogorov-Smirnov test results comparing the distribution sets shown
in Figures 4.52 and 4.53; the null hypothesis (H0) was that the data were
from the same distribution. ................................................................... 109
4.4 Percentiles of environmental variables coupled to the observed formation
of near-surface facets............................................................................. 110
5.1 List of constant variables utilized for computing the heat source term of
Equation (5.30). ................................................................................... 128
xi
LIST OF TABLES – CONTINUED
Table Page
6.1 Matrix detailing the output vectors (~a) used to compute the necessary
sensitivity parameters. This table was adapted from Saltelli (2002) and
should be used in conjunction with Equations (6.33) through (6.40).
Note, the j superscript is omitted for simplicity. ..................................... 152
6.2 Improved SOBOL sensitivity indices, expresses as percentages, of Equa-
tion (6.51). ........................................................................................... 158
7.1 List of input parameters, their associated symbol, and index (i) refer-
enced in the analysis throughout Chapters 8–10...................................... 163
7.2 Snow property uniform distribution parameters used for sensitivity anal-
ysis and Monte Carlo simulations........................................................... 165
7.3 Environmental input parameter distribution sets used for sensitivity anal-
ysis and Monte Carlo simulations; the coefficients (a, b, and c) correspond
to the parameters provided in Equations (7.1)–(7.4)................................ 166
8.1 List of input parameters, associated symbol, and reference index used in
the analysis throughout this chapter. ..................................................... 178
8.2 Table summarizing the sensitivity analysis parameters for the Control
location calculated from Φ..................................................................... 180
8.3 Summary of crystal size, long-wave radiation (LW ), and Π observed sur-
face hoar events at the North and South Stations. .................................. 192
8.4 Regions of mass-flux and expected surface hoar crystal size. .................... 198
9.1 First-, second-, total-, and higher-order sensitivity indices for the
South/KTGmid results (see Table 7.1 for reference)................................. 222
10.1 Summary of results from laboratory experiments conducted by Morstad
et al. (2007) and Slaughter et al. (2009). ................................................ 227
A.1 Detailed location information on each of the three weather stations situ-
ated on Pioneer Mountain. .................................................................... 260
A.2 List of output data from North and South weather stations. .................... 263
A.3 Summary of the instrumentation utilized at each weather station during
each winter season. ............................................................................... 265
xii
LIST OF TABLES – CONTINUED
Table Page
A.4 Summary of calibration constants of weather station sensors. The values
inside the brackets give the serial number of the sensor and all calibration
numbers are given as W/m2/mV. .......................................................... 268
A.5 Tabular wiring layout for North and South weather stations. ................... 269
E.1 Control / Mass Flux with Time (Total-effect)......................................... 393
E.2 Control / Mass Flux / Mid-day ............................................................. 393
E.3 South / Mass Flux with Time (Total-effect) ........................................... 394
E.4 South / Mass Flux / Mid-day................................................................ 394
E.5 North / Mass Flux with Time (Total-effect) ........................................... 395
E.6 North / Mass Flux / Mid-day................................................................ 395
E.7 Control / Mass Flux / Mean ................................................................. 396
E.8 South / Mass Flux / Mean.................................................................... 396
E.9 North / Mass Flux / Mean.................................................................... 396
E.10 Control / Positive Mass Flux with Time (Total-effect) ............................ 397
E.11 Control / Mass Flux / Positive Mid-day................................................. 397
E.12 South / Positive Mass Flux with Time (Total-effect)............................... 398
E.13 South / Mass Flux / Positive Mid-day ................................................... 398
E.14 North / Positive Mass Flux with Time (Total-effect)............................... 399
E.15 North / Mass Flux / Positive Mid-day ................................................... 399
E.16 Control / Mass Flux / Positive Mean..................................................... 400
E.17 South / Mass Flux / Positive Mean ....................................................... 400
E.18 North / Mass Flux / Positive Mean ....................................................... 400
E.19 Control / Negative Mass Flux with Time (Total-effect) ........................... 401
E.20 Control / Mass Flux / Negaitve Mid-Day............................................... 401
E.21 South / Negative Mass Flux with Time (Total-effect) ............................. 402
E.22 South / Mass Flux / Negaitve Mid-Day ................................................. 402
xiii
LIST OF TABLES – CONTINUED
Table Page
E.23 North / Negative Mass Flux with Time (Total-effect) ............................. 403
E.24 North / Mass Flux / Negaitve Mid-Day ................................................. 403
E.25 Control / Mass Flux / Negative Mean ................................................... 404
E.26 South / Mass Flux / Negative Mean ...................................................... 404
E.27 North / Mass Flux / Negative Mean...................................................... 404
E.28 Control / Mass Flux / Maximum........................................................... 405
E.29 South / Mass Flux / Maximum ............................................................. 405
E.30 North / Mass Flux / Maximum ............................................................. 405
E.31 Control / Mass Flux / Minimum ........................................................... 406
E.32 South / Mass Flux / Minimum.............................................................. 406
E.33 North / Mass Flux / Minimum.............................................................. 406
E.34 Control / Temp. at 0cm with Time (Total-effect) ................................... 407
E.35 Control / Temp. at 0cm / Mid-day........................................................ 407
E.36 South / Temp. at 0cm with Time (Total-effect)...................................... 408
E.37 South / Temp. at 0cm / Mid-day .......................................................... 408
E.38 North / Temp. at 0cm with Time (Total-effect)...................................... 409
E.39 North / Temp. at 0cm / Mid-day .......................................................... 409
E.40 Control / Temp. at 0cm / Mean............................................................ 410
E.41 South / Temp. at 0cm / Mean .............................................................. 410
E.42 North / Temp. at 0cm / Mean .............................................................. 410
E.43 Control / Temp. at 0cm / Maximum ..................................................... 411
E.44 South / Temp. at 0cm / Maximum........................................................ 411
E.45 North / Temp. at 0cm / Maximum........................................................ 411
E.46 Control / Temp. at 0cm / Minimum...................................................... 412
E.47 South / Temp. at 0cm / Minimum ........................................................ 412
xiv
LIST OF TABLES – CONTINUED
Table Page
E.48 North / Temp. at 0cm / Minimum ........................................................ 412
F.1 Control / Temp. at 0cm with Time (Total-effect) ................................... 415
F.2 Control / Temp. at 0cm / Mid-day........................................................ 415
F.3 South / Temp. at 0cm with Time (Total-effect)...................................... 416
F.4 South / Temp. at 0cm / Mid-day .......................................................... 416
F.5 North / Temp. at 0cm with Time (Total-effect)...................................... 417
F.6 North / Temp. at 0cm / Mid-day .......................................................... 417
F.7 Control / Temp. at 0cm / Mean............................................................ 418
F.8 South / Temp. at 0cm / Mean .............................................................. 418
F.9 North / Temp. at 0cm / Mean .............................................................. 418
F.10 Control / Temp. at 0cm / Maximum ..................................................... 419
F.11 South / Temp. at 0cm / Maximum........................................................ 419
F.12 North / Temp. at 0cm / Maximum........................................................ 419
F.13 Control / Temp. at 0cm / Minimum...................................................... 420
F.14 South / Temp. at 0cm / Minimum ........................................................ 420
F.15 North / Temp. at 0cm / Minimum ........................................................ 420
F.16 Control / Temp. at 2cm with Time (Total-effect) ................................... 421
F.17 Control / Temp. at 2cm / Mid-day........................................................ 421
F.18 South / Temp. at 2cm with Time (Total-effect)...................................... 422
F.19 South / Temp. at 2cm / Mid-day .......................................................... 422
F.20 North / Temp. at 2cm with Time (Total-effect)...................................... 423
F.21 North / Temp. at 2cm / Mid-day .......................................................... 423
F.22 Control / Temp. at 2cm / Mean............................................................ 424
F.23 South / Temp. at 2cm / Mean .............................................................. 424
F.24 North / Temp. at 2cm / Mean .............................................................. 424
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LIST OF TABLES – CONTINUED
Table Page
F.25 Control / Temp. at 2cm / Maximum ..................................................... 425
F.26 South / Temp. at 2cm / Maximum........................................................ 425
F.27 North / Temp. at 2cm / Maximum........................................................ 425
F.28 Control / Temp. at 2cm / Minimum...................................................... 426
F.29 South / Temp. at 2cm / Minimum ........................................................ 426
F.30 North / Temp. at 2cm / Minimum ........................................................ 426
F.31 Control / Temp. at 5cm with Time (Total-effect) ................................... 427
F.32 Control / Temp. at 5cm / Mid-day........................................................ 427
F.33 South / Temp. at 5cm with Time (Total-effect)...................................... 428
F.34 South / Temp. at 5cm / Mid-day .......................................................... 428
F.35 North / Temp. at 5cm with Time (Total-effect)...................................... 429
F.36 North / Temp. at 5cm / Mid-day .......................................................... 429
F.37 Control / Temp. at 5cm / Mean............................................................ 430
F.38 South / Temp. at 5cm / Mean .............................................................. 430
F.39 North / Temp. at 5cm / Mean .............................................................. 430
F.40 Control / Temp. at 5cm / Maximum ..................................................... 431
F.41 South / Temp. at 5cm / Maximum........................................................ 431
F.42 North / Temp. at 5cm / Maximum........................................................ 431
F.43 Control / Temp. at 5cm / Minimum...................................................... 432
F.44 South / Temp. at 5cm / Minimum ........................................................ 432
F.45 North / Temp. at 5cm / Minimum ........................................................ 432
F.46 Control / Temp. at 8cm with Time (Total-effect) ................................... 433
F.47 Control / Temp. at 8cm / Mid-day........................................................ 433
F.48 South / Temp. at 8cm with Time (Total-effect)...................................... 434
F.49 South / Temp. at 8cm / Mid-day .......................................................... 434
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LIST OF TABLES – CONTINUED
Table Page
F.50 North / Temp. at 8cm with Time (Total-effect)...................................... 435
F.51 North / Temp. at 8cm / Mid-day .......................................................... 435
F.52 Control / Temp. at 8cm / Mean............................................................ 436
F.53 South / Temp. at 8cm / Mean .............................................................. 436
F.54 North / Temp. at 8cm / Mean .............................................................. 436
F.55 Control / Temp. at 8cm / Maximum ..................................................... 437
F.56 South / Temp. at 8cm / Maximum........................................................ 437
F.57 North / Temp. at 8cm / Maximum........................................................ 437
F.58 Control / Temp. at 8cm / Minimum...................................................... 438
F.59 South / Temp. at 8cm / Minimum ........................................................ 438
F.60 North / Temp. at 8cm / Minimum ........................................................ 438
F.61 Control / “Knee” Temp. with Time (Total-effect)................................... 439
F.62 Control / “Knee” Temp. / Mid-day ....................................................... 439
F.63 South / “Knee” Temp. with Time (Total-effect) ..................................... 440
F.64 South / “Knee” Temp. / Mid-day ......................................................... 440
F.65 North / “Knee” Temp. with Time (Total-effect) ..................................... 441
F.66 North / “Knee” Temp. / Mid-day ......................................................... 441
F.67 Control / “Knee” Temp. / Mean ........................................................... 442
F.68 South / “Knee” Temp. / Mean.............................................................. 442
F.69 North / “Knee” Temp. / Mean ............................................................. 442
F.70 Control / “Knee” Temp. / Maximum .................................................... 443
F.71 South / “Knee” Temp. / Maximum....................................................... 443
F.72 North / “Knee” Temp. / Maximum....................................................... 443
F.73 Control / “Knee” Temp. / Minimum ..................................................... 444
F.74 South / “Knee” Temp. / Minimum........................................................ 444
xvii
LIST OF TABLES – CONTINUED
Table Page
F.75 North / “Knee” Temp. / Minimum ....................................................... 444
F.76 Control / Temp. Gradient at 2cm with Time (Total-effect) ..................... 445
F.77 Control / Temp. Gradient at 2cm / Mid-day.......................................... 445
F.78 South / Temp. Gradient at 2cm with Time (Total-effect)........................ 446
F.79 South / Temp. Gradient at 2cm / Mid-day ............................................ 446
F.80 North / Temp. Gradient at 2cm with Time (Total-effect)........................ 447
F.81 North / Temp. Gradient at 2cm / Mid-day ............................................ 447
F.82 Control / Temp. Gradient at 2cm / Mean.............................................. 448
F.83 South / Temp. Gradient at 2cm / Mean ................................................ 448
F.84 North / Temp. Gradient at 2cm / Mean ................................................ 448
F.85 Control / Temp. Gradient at 2cm / Maximum ....................................... 449
F.86 South / Temp. Gradient at 2cm / Maximum.......................................... 449
F.87 North / Temp. Gradient at 2cm / Maximum.......................................... 449
F.88 Control / Temp. Gradient at 2cm / Minimum........................................ 450
F.89 South / Temp. Gradient at 2cm / Minimum .......................................... 450
F.90 North / Temp. Gradient at 2cm / Minimum .......................................... 450
F.91 Control / Temp. Gradient at 5cm with Time (Total-effect) ..................... 451
F.92 Control / Temp. Gradient at 5cm / Mid-day.......................................... 451
F.93 South / Temp. Gradient at 5cm with Time (Total-effect)........................ 452
F.94 South / Temp. Gradient at 5cm / Mid-day ............................................ 452
F.95 North / Temp. Gradient at 5cm with Time (Total-effect)........................ 453
F.96 North / Temp. Gradient at 5cm / Mid-day ............................................ 453
F.97 Control / Temp. Gradient at 5cm / Mean.............................................. 454
F.98 South / Temp. Gradient at 5cm / Mean ................................................ 454
F.99 North / Temp. Gradient at 5cm / Mean ................................................ 454
xviii
LIST OF TABLES – CONTINUED
Table Page
F.100 Control / Temp. Gradient at 5cm / Maximum ....................................... 455
F.101 South / Temp. Gradient at 5cm / Maximum.......................................... 455
F.102 North / Temp. Gradient at 5cm / Maximum.......................................... 455
F.103 Control / Temp. Gradient at 5cm / Minimum........................................ 456
F.104 South / Temp. Gradient at 5cm / Minimum .......................................... 456
F.105 North / Temp. Gradient at 5cm / Minimum .......................................... 456
F.106 Control / Temp. Gradient at 8cm with Time (Total-effect) ..................... 457
F.107 Control / Temp. Gradient at 8cm / Mid-day.......................................... 457
F.108 South / Temp. Gradient at 8cm with Time (Total-effect)........................ 458
F.109 South / Temp. Gradient at 8cm / Mid-day ............................................ 458
F.110 North / Temp. Gradient at 8cm with Time (Total-effect)........................ 459
F.111 North / Temp. Gradient at 8cm / Mid-day ............................................ 459
F.112 Control / Temp. Gradient at 8cm / Mean.............................................. 460
F.113 South / Temp. Gradient at 8cm / Mean ................................................ 460
F.114 North / Temp. Gradient at 8cm / Mean ................................................ 460
F.115 Control / Temp. Gradient at 8cm / Maximum ....................................... 461
F.116 South / Temp. Gradient at 8cm / Maximum.......................................... 461
F.117 North / Temp. Gradient at 8cm / Maximum.......................................... 461
F.118 Control / Temp. Gradient at 8cm / Minimum........................................ 462
F.119 South / Temp. Gradient at 8cm / Minimum .......................................... 462
F.120 North / Temp. Gradient at 8cm / Minimum .......................................... 462
F.121 Control / “Knee” Temp. Gradient with Time (Total-effect)..................... 463
F.122 Control / “Knee” Temp. Gradient / Mid-day ......................................... 463
F.123 South / “Knee” Temp. Gradient with Time (Total-effect) ....................... 464
F.124 South / “Knee” Temp. Gradient / Mid-day ........................................... 464
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LIST OF TABLES – CONTINUED
Table Page
F.125 North / “Knee” Temp. Gradient with Time (Total-effect) ....................... 465
F.126 North / “Knee” Temp. Gradient / Mid-day ........................................... 465
F.127 Control / “Knee” Temp. Gradient / Mean ............................................. 466
F.128 South / “Knee” Temp. Gradient / Mean................................................ 466
F.129 North / “Knee” Temp. Gradient / Mean ............................................... 466
F.130 Control / “Knee” Temp. Gradient / Maximum ...................................... 467
F.131 South / “Knee” Temp. Gradient / Maximum......................................... 467
F.132 North / “Knee” Temp. Gradient / Maximum......................................... 467
vLIST OF FIGURES
Figure Page
2.1 Example images of (a) surface hoar (Cooperstein et al., 2004) and (b)
near-surface faceted snow crystals (Morstad, 2004). ....................................9
3.1 Image from event A-1 of surface hoar (1 mm grid) captured from the
North Station on January 24, 2008. ......................................................... 29
3.2 Weather data for event A-1 (January 24, 2008) recorded for both the
(a,b) North and (c,d) South weather stations............................................ 30
3.3 Image from event A-2 of surface hoar (1 mm grid) captured from the
North Station on February 15, 2008. ........................................................ 31
3.4 Weather data for event A-2 (February 15, 2008) recorded for a the North
weather Station. ..................................................................................... 32
3.5 Images from event A-3 of surface hoar captured from the (a) North and
(b) South Stations on February 19, 2008 and surface hoar from the (c)
South Station that was recorded on February 20....................................... 33
3.6 North Station weather data for event A-3 (February 19–21, 2008). ............. 34
3.7 South Station weather data for event A-3 (February 19–21, 2008). ............. 35
3.8 Images from event A-4 of surface hoar captured from the (a) North and
(b) South Stations on February 22, 2008 and at the (c) North station the
following day after the surface hoar was buried by new snow. .................... 36
3.9 Weather data for event A-4 (February 22, 2008) for both the (a,b) North
and (c,d) South weather stations. ............................................................ 37
3.10 Images from event A-5 of surface hoar captured from the (a) North and
(b) South Stations on February 26, 2008. ................................................. 38
3.11 Weather data for event A-4 (February 26, 2008) for both the (a,b) North
and (c,d) South weather stations. ............................................................ 39
3.12 Images from event A-6 of surface hoar captured from the (a) North and
(b) South Stations on March 10, 2008. ..................................................... 40
3.13 Weather data for event A-6 (March 10, 2008) for both the (a,b) North
and (c,d) South weather stations. ............................................................ 41
3.14 Images from event A-7 of surface hoar captured from the (a) North and
(b) South Stations on March 30, 2008. ..................................................... 42
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LIST OF FIGURES – CONTINUED
Figure Page
3.15 Weather data for event A-7 (March 30, 2008) for both the (a,b) North
and (c,d) South weather stations. ............................................................ 43
3.16 Images from event B-1 of surface hoar (2 mm grid) captured from the
North Station on January 23, 2009. ......................................................... 44
3.17 North Station weather data for event B-1 (January 23, 2009). ................... 45
3.18 Images from event B-2 of surface hoar captured from the (a) North and
(b) South Stations on January 30, 2009 and the (c) North Station on
January 31, 2009. ................................................................................... 46
3.19 North Station weather data for event B-2 (January 30–31, 2009). .............. 47
3.20 Images from event B-3 of surface hoar captured from the (a) North and
(b) South Stations on February 4 and (c) the North Station on February
5, 2009. .................................................................................................. 48
3.21 North Station weather data for event B-3 (February 4, 2008)..................... 49
3.22 Images from event B-4 of surface hoar captured from the North and South
Stations on (a) February 7 and (b,c) 8, 2009............................................. 50
3.23 South Station weather data for event B-4 (February 7, 2009)..................... 50
3.24 Images from event B-5 of surface hoar captured from the North Station
on (a) February 13 and (b) 14, 2009......................................................... 51
3.25 North Station weather data for event B-5 (February 13–14, 2009). ............. 52
3.26 Images from event B-6 of surface hoar captured from the North Station
on February 28, 2009. ............................................................................. 53
3.27 North Station weather data for event B-6 (February 28, 2009). .................. 53
3.28 Images from event B-7 of surface hoar captured from the North Station
on March 13, 2009. ................................................................................. 54
3.29 Weather data for event B-7 (March 13, 2009) for the North Station. .......... 54
3.30 Histogram comparing the daily average air/snow temperature difference
for the entire season (all data), along with the days associated with sur-
face hoar events at either the North or South Stations. ............................. 55
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LIST OF FIGURES – CONTINUED
Figure Page
3.31 Histograms comparing the frequency of recorded daily average weather
conditions at both the North and South Stations (all data), along with
the days associated with surface hoar events observed at the North or
South Stations. ....................................................................................... 56
4.1 Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station for January
21–22, 2008 (C-1). .................................................................................. 67
4.2 Four images of a near-surface facet event at the South Station on Febru-
ary 14, 2008 (C-2): (a) initial observation (1100) at the South Station,
(b) second observation (1400) at the South Station, (c) observations at a
near-by south facing slope, and (d) following day (Feb. 15) South Station
observation (1100). ................................................................................. 68
4.3 Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station for February
14–16, 2008 (C-3). .................................................................................. 69
4.4 Facets formed on February 14, 2008 at the South Station that persisted
through warmer temperatures and new snow until February 16th. .............. 69
4.5 Images of near-surface facets at the South station described as diurnal
recrystallization., that formed on February 18–20, 2008 (C-3). ................... 70
4.6 Graph of air temperature and snow surface temperature at the South
Station on February 17–20, 2008 (C-3). .................................................... 71
4.7 Images taken at the South Station of (a) surface hoar formed the day
prior to the (b) near-surface facets that formed on February 27, 2008
(C-4) and persisted through (c) the following day. .................................... 73
4.8 Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station on February
26–28, 2008 (C-4). .................................................................................. 74
4.9 Snow temperature profiles from February 26, 2008 (C-4) at the South
Station................................................................................................... 74
4.10 Images from March 6, 2008 (C-5) near-surface facet event at the South
Station: (a) initial observation at 1100 and (b) second observation at 1330. 75
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LIST OF FIGURES – CONTINUED
Figure Page
4.11 Image of near-surface facets observed on March 10, 2008 (C-6) at the
South Station. ........................................................................................ 76
4.12 Recorded short- and long-wave radiation at the Aspirit station (C-6) as
well as the air and snow surface temperatures at the South station for
March 10, 2008. ...................................................................................... 77
4.13 Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March
12–13, 2008 (C-7). .................................................................................. 78
4.14 Images taken at the South Station during the (a) March 15, 2008 (C-8)
near-surface facet event; this layer persisted the following two days (b
and c). ................................................................................................... 79
4.15 Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March
15–17, 2008 (C-8). .................................................................................. 80
4.16 Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March
19, 2008 (C-9). ....................................................................................... 80
4.17 Images from the (a) March 22, 2008 (C-10) near-surface facet event at
the South Station and (b) facets that persisted through the following day. . 81
4.18 Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station for March
22–24, 2008 (C-10).................................................................................. 82
4.19 Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station, on March
28, 2008 (C-11)....................................................................................... 83
4.20 Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March
30, 2008 (C-12)....................................................................................... 84
4.21 Image of near-surface facets formed on March 30, 2008 (C-12) at the
South Station. ........................................................................................ 84
4.22 Images of surface snow at (a) 1130 and (b) 1430 at the South Station on
April 3, 2008 (C-13) showing formation of near-surface facets . .................. 85
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LIST OF FIGURES – CONTINUED
Figure Page
4.23 Recorded short- and long-wave radiation at the Aspirit Station, as well
as the air and snow surface temperatures at the South Station, on April
2–4, 2008 (C-13). .................................................................................... 86
4.24 Images of surface snow at (a) 1030 and (b) 1430 on April 6, 2008 (C-14)
showing formation of near-surface facets at the South Station.................... 87
4.25 Recorded short- and long-wave radiation at the Aspirit station, as well
as the air and snow surface temperatures at the South Station, on April
6, 2008 (C-14). ....................................................................................... 87
4.26 Recorded short- and long-wave radiation at the Aspirit station as well as
the air and snow surface temperatures at the South station for April 8–9,
2008....................................................................................................... 88
4.27 Image near-surface facets formed on April 8, 2008..................................... 89
4.28 Image of near-surface facets formed (a) at the South Station on February
4, 2009 (D-1) ; the facets persisted through the night and were also
observed (b) on Feb. 5. ........................................................................... 90
4.29 Recorded weather data from the South Station on February 4, 2009 (D-
1); due to instrumentation malfunctions the data prior to 1200 on Feb.
4 was not recorded.................................................................................. 90
4.30 Images of a near-surface facet event that occurred on February 8, 2009 (D-
2) at the South Station. Images include facets observed (a) at the initial
observation, (b and c) at the second observation, and (d) the following
day despite being buried underneath new snow......................................... 91
4.31 Recorded weather data from the South Station on February 8, 2009 (D-2).. 92
4.32 Images of near-surface facet event that occurred at the South Station on
(a,b) February 12, 2009 (D-3) and that continued on (c) Feb. 13 and (d) 14.93
4.33 Recorded weather data from the South Station on February 12–14, 2009
(D-3). .................................................................................................... 94
4.34 Images of a near-surface facet event (D-3) that occurred at the South
Station and persisted beneath 9 cm of new snow falling on the night of
Feb. 14. ................................................................................................. 94
4.35 Image of near-surface facets observed at the South Station on February
19, 2009 (D-4). ....................................................................................... 95
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LIST OF FIGURES – CONTINUED
Figure Page
4.36 Recorded weather data from the South Station on February 19, 2009 (D-4).95
4.37 Recorded weather data from the South and North stations on February
21, 2009 (D-5). ....................................................................................... 96
4.38 Images of near-surface facets on February 21, 2009 (D-5) that formed
small-faceted crystals at both the South and North Stations...................... 97
4.39 Recorded weather data from the South Station on February 27–28, 2009
(D-6). .................................................................................................... 98
4.40 Images of radiation-recrystallized near-surface facets from Event D-6 that
formed at the South Station on (a,b) February 27 and (c,d) 28, 2009. ........ 99
4.41 Images of a near-surface facet event that occurred at the South Station
on March, 7 2009 (D-7). ........................................................................ 100
4.42 Recorded weather data from the South Station on March 7, 2009 (D-7). .. 100
4.43 Recorded weather data from the South Station on March 12–14, 2009
(D-8). .................................................................................................. 101
4.44 Images of event a near-surface facet event (D-8) that occurred on three
consecutive days (March 12–14, 2009) at the South Station. .................... 103
4.45 Images of large near-surface facets captured at the South Station during
the second observation (1440) on March 14, 2009 (D-8)........................... 104
4.46 Recorded weather data from the South Station on March 20, 2009 (D-9).. 105
4.47 Image of slight faceting that occurred at the South Station on March, 20
2009 (D-9)............................................................................................ 105
4.48 Images from two observations—(a) 1130 and (b) 1315—of a near-surface
facet event that occurred at the South Station on March, 30 2009 (D-9)... 106
4.49 Recorded weather data from the South Station on March 30, 2009 (D-10).106
4.50 Image from near-surface facet event that occurred at the South Station
on April 6, 2009 (D-11). ........................................................................ 107
4.51 Recorded weather data from the South Station on April 6, 2009 (D-11). .. 108
4.52 Histograms comparing daily average radiation conditions for the entire
data set (2007/2008 and 2008/2009 seasons) against the days associated
with near-surface facet events. ............................................................... 112
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LIST OF FIGURES – CONTINUED
Figure Page
4.53 Histograms comparing the daily average weather conditions—(a) air tem-
perature, (b) snow surface temperature, (c) wind speed, (d) wind direc-
tion, and (e) relative humidity—for the entire data set (2007/2008 and
2008/2009 seasons ) against the days associated with near-surface facet
events at the South Station. .................................................................. 113
5.1 Schematic of the arbitrary control volume (CV ) enclosed by the control
surface (CS)......................................................................................... 119
5.2 Schematic of snowpack layering utilized for numerical solution of snow
temperatures with time. The superscript j represents the j-th time step
and the subscript i represents the layer number. ..................................... 122
5.3 Schematic that demonstrates the application of short-wave attenuation
in a layered snowpack. .......................................................................... 127
5.4 Example of temperature differences observed by differing application of
the Neumann boundary condition. ......................................................... 131
5.5 Resulting output distributions—(a) snow surface temperature and (b)
temperature gradient—from the Monte Carlo simulations........................ 132
5.6 Comparison between six model evaluations with varying irradiation inputs.136
5.7 Graphs demonstrating the model behavior with respect to measurement
error including (a) a contour plot of the largest deviation from the input
evaluation and (b) 90% confidence intervals with input evaluation and
measured values.................................................................................... 138
6.1 Results from the SOBOL sensitivity analysis of Equation (6.51), includ-
ing the first-order (Si) and total-effect sensitivity (STi) terms. (The error
bars reflect the 90% confidence intervals.) .............................................. 158
6.2 First- and second-order indices for the first input parameter (x1) from
analysis of Equation (6.51). ................................................................... 160
6.3 Stacked area plot of the time-dependent total-effect indices resulting from
the analysis of Equation (6.52). ............................................................. 161
7.1 Probability distribution functions for input data based on measured data. 167
7.2 Comparison of 3-D representations of (a) the raw data as a scatter plot
and (b) the data encapsulated by 5% (inner), 50% (middle), and 95%
(outer) HDRs. ...................................................................................... 172
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LIST OF FIGURES – CONTINUED
Figure Page
7.3 The (a) bi-variate probability density function was constructed from the
raw data points shown in sub-figure b; the probability distribution was
then sliced such that 95% of raw data had a probability density greater
than this value resulting in a highest density region trace also shown in
sub-figure b. ......................................................................................... 172
7.4 Example of tri-variate data analysis including (a) a 3-D scatter plot of
raw data, (b) a 3-D 95% HDR, (c) 2-D HDRs encapsulating specific bands
of Ψ, and (d) the 10%, 50%, 90%, and 95% HDRs of complete data set
(the number of data points used to construct each region is included in
the parenthesis). ................................................................................... 174
8.1 Total-effect indices for Φ for each of the three locations considered, see
Table 8.1 for reference. .......................................................................... 179
8.2 Total-effect indices for (a) Φmin and (b) Φmax for each of the three loca-
tions considered (see Table 8.1 for reference). ......................................... 181
8.3 Total-effect indices for (a) Φneg and (b) Φpos for each of the three locations
considered (see Table 8.1 for reference)................................................... 182
8.4 First-, second- and higher-order indices for Φpos for control location and
each of the four important inputs: LW (6), Vw(9), Ta(10), and RH(11)
(see Table 8.1 for reference). The higher-order interactions for these terms
are provided in the Sh grouping. ............................................................ 184
8.5 First-, second- and higher-order indices for Φpos for North location and
each of the four important inputs: ρ(1), T ints (4), LW (6), Vw(9), Ta(10),
and RH(11) (see Table 8.1 for reference). The higher-order interactions
for these terms are provided in the Sh grouping. ..................................... 185
8.6 First-, second- and higher-order indices for Φpos for South location and
each of the six important inputs: ρ(1), T ints (4), LW (6), Vw(9), Ta(10),
and RH(11) (see Table 8.1 for reference). The higher-order interactions
for these terms are provided in the Sh grouping. ..................................... 186
8.7 Highest density regions (95%) comparing Monte Carlo simulation results
for LW (6) and Π for (a) all values with Ta < 0 and (b) ΦSH . .................. 190
8.8 Highest density regions (95%) comparing the complete set (All) of Monte
Carlo simulation results to the data limited to surface hoar formation
(SH) for the (a) Control, (b) North, and (c) South locations.................... 191
xxviii
LIST OF FIGURES – CONTINUED
Figure Page
8.9 Comparison of Control/ΦSH highest density regions with field observa-
tions from the North- and South-facing stations...................................... 193
8.10 Comparison of 99% (outer) and 50% (inner) HDRs for the Control/ΦSH
results and all recorded field data with the field observations from the
North- and South-facing stations. .......................................................... 194
8.11 Comparison of 99% (outer) and 50% (inner) HDRs for the ΦSH results
and all recorded field data with the field observations from the (a) South-
and (b) North-facing stations................................................................. 196
8.12 Comparison of North/ΦSH highest density regions with field observations
from the North-facing station. ............................................................... 197
8.13 Highest density regions based on the North location and various mass-flux
rates. ................................................................................................... 199
9.1 Schematic of “knee” temperature profile and related sensitivity analysis
output parameters. ............................................................................... 204
9.2 Stacked area charts of normalized total-effect sensitivity (S∗T ) for T0(t)
for the (a) Control, (b) North, and (c) South locations. The regions are
stacked from bottom to top in order as listed in Table 7.1. ...................... 206
9.3 Total-effect sensitivity indices for the Control, South, and North locations
for (a) T0, (b) T0max, and (c) T0mid (see Table 7.1 for reference)............. 208
9.4 First-, second-, and higher-order indices for (a) the South/T0mid sensi-
tivity analysis and (b) a zoomed view focusing on α(8) (see Table 7.1 for
reference). ............................................................................................ 209
9.5 First-, second-, and higher-order indices for the (a) Control/T0mid and
(b) North/T0mid sensitivity analysis (see Table 7.1 for reference). ............ 211
9.6 Stacked area charts of normalized total-effect sensitivity as a function of
model evaluation time at the (a) “knee”, (b) 2 cm, (c) 5 cm, and (d) 8
cm depth for the South locations. The regions are stacked from bottom
to top in order as listed in Table 7.1. ..................................................... 212
9.7 Stacked area charts of normalized total-effect sensitivity as a function
of model evaluation time at the 2 cm depth for the (a) Control and (b)
North locations. The regions are stacked from top to bottom in order as
listed in Table 7.1 (see Table 7.1 for reference)........................................ 213
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LIST OF FIGURES – CONTINUED
Figure Page
9.8 Total-effect sensitivity indices for the Control, South, and North locations
for TKmid (see Table 7.1 for reference). .................................................. 214
9.9 First-, second-, and higher-order indices for TKmid sensitivity analysis
for the (a) Control, (b) North, and (c) South locations (see Table 7.1 for
reference). ............................................................................................ 215
9.10 Stacked area charts of normalized total-effect sensitivity of TG2(t) for
the (a) Control, (b) North, and (c) South locations and (d) the total-
effect indices computed from TG2mid output. The regions are stacked
from bottom to top in order as listed in Table 7.1. .................................. 217
9.11 Contour plots of snow temperature with incoming short-wave radiation
of (a) 300 W/m2 and (b) 400 W/m2. (see Table 7.1 for reference) ........... 218
9.12 First-, second-, and higher-order indices for TG2mid sensitivity analysis
for the South location, which highlights the overwhelming importance of
higher-order interactions. (see Table 7.1 for reference)............................. 219
9.13 Stacked area charts of normalized total-effect sensitivity of KTG(t) for
the (a) Control, (b) North, and (c) South locations. The regions are
stacked from bottom to top in order as listed in Table 7.1. ...................... 220
9.14 Grouped bar charts of total-effect sensitivity for KTGmid for all three
locations. (see Table 7.1 for reference) ................................................... 221
9.15 First-, second-, and higher-order indices for KTGmid sensitivity analysis
for the (a) Control and (b) North locations. (see Table 7.1 for reference) . 221
10.1 Comparison between the two seasons (2007/2008 and 2008/2009) of
recorded long-wave radiation values for near-surface facet events.............. 229
10.2 Probability distribution functions for snow properties including the com-
plete (all) input distribution and data limited by KTGmid from 200–600
◦C/m (limited) . Table 7.1 (p. 163) defines the variables and the units
for each graph. ..................................................................................... 231
10.3 Probability distribution functions for radiation inputs including the com-
plete (all) input distribution and data limited by KTGmid from 200–600
◦C/m (limited) . Table 7.1 (p. 163) defines the variables and the units
for each graph. ..................................................................................... 232
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LIST OF FIGURES – CONTINUED
Figure Page
10.4 Comparison of the complete input (A) with the input limited by KTGmid
from 200–600◦C/m (B) for the (a) Control, (b) North, and (c) South
locations. ............................................................................................. 235
10.5 Comparison of tri-variate PDF of all input (A) with the input limited by
KTGmid from 200–600◦C/m (B) for the South location. .......................... 236
10.6 Contour plot of HDRs for the Control results including the field observa-
tions from Chapter 4 and laboratory data of Morstad et al. (2007) and
Slaughter et al. (2009)........................................................................... 238
10.7 Contour plot of HDRs for the North location including the field observa-
tions from Chapter 4 and laboratory data of Morstad et al. (2007) and
Slaughter et al. (2009)........................................................................... 239
10.8 Contour plot of HDRs for the South location including the field observa-
tions from Chapter 4 and laboratory data of Morstad et al. (2007) and
Slaughter et al. (2009)........................................................................... 240
10.9 Chart showing the relationship of ρ, k, and γ as well as two commonly
utilized ρ and k relationships as presented by Sturm et al. (1997). ........... 241
A.1 Google Earth images of Pioneer Mountain showing the locations of (a)
the South and (b) the North and American Spirit weather stations. ......... 261
A.2 Wiring schematic for North and South weather stations. ......................... 267
A.3 Comparison of incoming short-wave radiation at the Yellow Mule RAWS
and Aspirit weather stations. ................................................................. 279
B.1 YCweather Program Control window. .................................................... 308
B.2 Example of the Data List window. ......................................................... 310
B.3 Example graph showing dual-axis capabilities. ........................................ 311
B.4 Example of a YCweather workspace....................................................... 317
B.5 Prompt that appears by default when opening a workspace. .................... 318
B.6 YCweather preferences window.............................................................. 320
B.7 Example of the image viewer for YCweather........................................... 323
B.8 Examples of the daily log options available in YCweather........................ 325
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LIST OF FIGURES – CONTINUED
Figure Page
B.9 Program Control with all side panels showing......................................... 326
B.10 Example workspace showing a graph of thermocouple data...................... 327
B.11 (a) RadTherm/RT file exporter and (b) an example output file. .............. 330
B.12 Example of file structure of the database directory used by YCweather. ... 334
B.13 Example format file utilzed by YCweather. ............................................ 335
B.14 Example format file utilized by YCweather that includes the thermo-
couple ID for plotting temperature profile data (this is not a complete
file)...................................................................................................... 337
B.15 Example entries for prescribing units within the units.txt file, which is
utilized by the getunit.m function. ....................................................... 340
C.1 Flow chart demonstrating how the various functions discussed in Section
C.2 and C.3 interact. ............................................................................ 345
C.2 Example of the (a) “SnowProperties” and (b)“AtmosphericSettings”
worksheets for Excel file read by xls input.m. ....................................... 348
C.3 Example of the “Constants” worksheets for Excel file read by xls input.m.349
C.4 MATLAB implementation of runmodel.m and the resulting data structure.350
C.5 Required data structure of albedo.mat.................................................. 352
C.6 Graphical user interface for implementing the snow thermal model. ......... 354
C.7 Example graphs of snowpack temperatures demonstrated the two graph-
ing options available: (a) profiles and (b) contours. ................................. 356
C.8 Example graphs of snowpack temperature demonstrated the two graphing
options available for displaying confidence level intervals: (a) C.I. profiles
and (b) C.I. contours. ........................................................................... 357
D.1 Flow chart of the main sensitivity analysis function sobol.m and associ-
ated sub functions................................................................................. 370
D.2 MATLAB code for saFANG.m function.................................................... 371
D.3 MATLAB code for saG.m function. ........................................................ 371
D.4 Syntax for implementation of saMODEL2.m.............................................. 374
xxxii
LIST OF FIGURES – CONTINUED
Figure Page
D.5 MATLAB code demonstrating the definition of the input *.mat files for
the saMODEL2.m function. ...................................................................... 375
xxxiii
NOMENCLATURE
Variables
Variable(s) Description
~aj Sensitivity analysis vectors of for the jth model output vectors, Eq.
(6.32)
âcc Acceleration of bootstrap confidence level intervals, Eq. (6.48)
a, b, c, d Coefficients for numeral solution of heat equation (Eq. (5.18));
coefficients of probability density functions (Sec. 7.2)
b Arbitrary vector used in Gauss’ theorem, Eq. (5.6)
B Number of bootstrap re-samplings (Sec. 6.5)
biasB Bootstrap-computed estimate of bias, Eq. (6.50a)
C Number of computations required for SOBOL method (Chap. 6)
cp, cpa Specific heat capacity of snow and air, respectively
CS Control surface, Fig. 5.1
CV Control volume, Fig. 5.1
ea, es Water-vapor pressure above and at the snow surface, respectively
(Sec. 5.3.5)
e0 Reference vapor pressure, Tab. 5.1
E Internal energy, Eq. (5.1)
E(yj) Expected value of the jth output parameter (Chap. 6)
Ec Eckert number (Sec. 8.4)
h Specific heat supply (Sec. 5.3.1)
xxxiv
Variable(s) Description
k Thermal conductivity tensor (Chap. 6)
k Thermal conductivity scalar (Chapter 5); shape parameter for gen-
eralized extreme value and generalized Pareto distributions (Sec.
7.2)
Ke, Kh Transfer coefficients for latent and sensible heat equations, Tab. 5.1
KE Kinetic energy, Eq. (5.1)
K Number of Monte Carlo simulations for SOBOL method (Chap. 6)
Ls Latent heat of sublimation phase change, Tab. 5.1
LW,LW a Incoming long-wave radiation, the superscript a (Tab. 4.2) denotes
the American Spirit Station
Ma,Mv Dry-air and water-vapor molecular weights, respectively, Tab. 5.1
m Number of output parameters for SOBOL analysis (Chap. 6)
n Number of input parameters for SOBOL analysis (Chap. 6)
nˆ Outward normal vector of the control surface, Fig. 5.1
N Matrix of inputs for computing SOBOL output vectors (Chap. 6)
Nb Number of bootstrap estimates greater than the test statistic (Sec.
6.5)
p The p-value test statistic for goodness-of-fit (Sec. 7.8)
pi Probability density function of ith SOBOL input parameter, Eq.
(6.3)
P Composite probability density function of all inputs, Eq. (6.3)
xxxv
Variable(s) Description
Patm Atmospheric pressure, Eq. (5.34)
~q Weighting function vector for use in the g-function for sensitivity
analysis example (Sec. 6.6)
q Heat flux vector across the control surface, Fig. 5.1
qi Short-wave heat flux (Chap. 5)
qV ISi Short-wave heat flux in visible wavebands (Chap. 5)
qNIRi Short-wave heat flux in near-infrared wavebands (Chap. 5)
qs Heat flux at the snow surface (Chap. 5)
qLW Long-wave heat flux, Eq. (5.31)
qe, qh Latent and sensible heat flux, respectively (Sec. 5.3.5)
R Rate of heat input, Eq. (5.1)
Ra Specific gas constant for air, Tab. 5.1
Rv Specific gas constant for water vapor, Tab. 5.1
RH Relative humidity
Sh Higher-order sensitivity analysis index
Si, Si,l First- and second-order sensitivity indices (Chaps. 8 and 9), an
alternate representation of the following term
Sji , S
j
il First- and second-order sensitivity indics with respect to the jth
output parameter (Chap. 6)
STi Total-effect sensitivity index (Chaps. 7–10), an alternate represen-
tation of the following term
xxxvi
Variable(s) Description
SjTi Total-effect sensitivity with respect to the jth output parameter
(Chap. 6)
S∗T Normalized total-effect sensitivity index (Sec. 6.7 and Chap. 9)
Sj∗Ti Normalized total-effect sensitivity index with respect to jth output
parameter (Sec. 6.7)
SW Incoming short-wave radiation
t Time
T Temperature
Ta, Ts Air and snow surface temperature, respectively
T ints Initial snow temperature at the start of thermal model evaluations
(Chaps. 7–10)
T0 Reference temperature, Tab. 5.1
T0, T2, T5, T8 Modeled snow temperatures at 0, 2, 5, and 8 cm depths, respectively
(Chap. 10)
TK Modeled snow temperature at the “knee” location (Chap. 10)
TG2, TG5, TG8 Modeled snow temperature gradients between the snow surface and
2, 5, and 8 cm depths, respectively (Chap. 10)
KTG Modeled snow temperature gradient between the surface and the
“knee” temperature (Chap. 10)
~u A subsample vector of the input vector ~x (Sec. 6.4)
U ji A parameters used to defined the sensitivityS
j
i (Chap. 6)
Û j
c
i , Û
jc
−i, Û
jc
il Estimates of the “closed” U -terms for computation of sensitivity
indices (Chap. 6)
xxxvii
Variable(s) Description
~v A subsample vector of the input vector ~x (Sec. 6.4)
V j Total variance of jth output parameter (Chap. 6)
V ji , V
j
il First- and second-order partial variance of ith input parameter with
respect to the jth output parameter (Chap. 6)
V j
c
i , V
jc
il “Closed” first- and second-order partial variance with respect to the
jth output parameter (Chap. 6)
V jTi Total-effect variance with respect to the jth output parameter
(Chap. 6)
Vw Wind velocity
W Rate of work acting on the system, Eq. (5.1)
W,W
′
Replicates of the Monte Carlo input matrices used for computing
N (Eq. (6.29))
~x Vector of input parameters, where ~x = xi | i = 1, 2, . . . , n (Chap.
6)
~x∗b The bth bootstrap re-sampling of input parameters of ~x (Sec. 6.5)
~y Vector of output parameters, where ~y = yj | j = 1, 2, . . . ,m (Chap.
6)
z Distance (Chap. 5)
zˆ0 An estimate of bias for computing bootstrap confidence levels (Sec.
6.5)
zα, z1−α Standard normal percentiles for computing bootstrap confidence
levels (Sec. 6.5)
α Snow albedo; shape parameter (Eq. (7.3)) for Weibull distribution
xxxviii
Variable(s) Description
αhi, αlo Upper and lower percentiles for computation of bootstrap confi-
dence intervals (Sec. 6.5)
β Scale parameter for Weibull distribution, Eq. (7.3)
γ location parameter for Weibull and lognormal distribution (Eq.
(7.3) and (7.4)); thermal diffusivity (Chap. 10)
∆T Difference between the air and snow temperature (Chap. 3)
ε Emissivity of snow
θˆ Test statistic (Sec. 6.5)
θˆ∗b Test statistic computed from bootstrap re-samplings (Sec. 6.5)
θˆ(r) The rth jackknife statistic (Sec. 6.5)
θˆ(·) The sum of the jackknife statistics (Sec. 6.5)
θˆ∗b(·) Mean of the bootstrap estimates of the test statistic (Sec. 6.5)
ϑ Specific internal energy (Sec. 5.3.1)
κ Extinction coefficient for snow
µ Location parameter for generalized extreme value and generalized
Pareto distributions; continuous shape parameter of lognormal dis-
tribution (Sec. 7.2)
Π Dimensionless parameter, Eq. (8.2)
Π1,Π2 Arbitrary terms utilized in highest density region examples (Sec.
7.6)
ρ, ρa Density of snow and air, respectively
xxxix
Variable(s) Description
σ Scale parameter for generalized extreme value and generalize Pareto
distributions (Sec. 7.2); continuous shape parameter for lognormal
distribution (Sec. 7.2); standard deviation (Chap. 10)
Φ Standard normal distribution (Sec. 6.5); mass-flux at the snow
surface (Chap. 8)
Ψ Arbitrary term utilized in highest density region examples (Sec.
7.6)
Ω Dimensionless parameter, Eq. (10.3)
~∇ Del vector operator (Sec. 5.3.1)
Indices
Variable(s) Description
b Index for bootstrap re-samplings, b = 1, 2, . . . , B (Sec. 6.5)
i Index for depth in numerical solution of heat equation (Chap. 5);
index for SOBOL input parameters, i = 1, 2, . . . , n (Chap. 6)
−i Short-hand notation used to represent “all except i” (Chap. 6)
i(−i) Short-hand notation used to represent the coupling of the ith value
with all values except i (Chap. 6)
j Index for time in numerical solution (Chap. 5); index for SOBOL
output parameters, j = 1, 2, . . . ,m (Chap. 6)
l Index of input parameters, l = 1, 2, . . . , n and l 6= i (Chap. 6)
r Index for sensitivity analysis Monte Carlo replicates, r = 1, 2, . . . , K
(Chap. 6)
xl
ABSTRACT
Faceted snow crystals develop at or near the snow surface due to temperature
gradients. After burial, snow avalanches regularly fail on these layers. Generally, sur-
face hoar deposits when the snow surface is cooler than the surrounding environment;
near-surface facets form when the subsurface is warmed by solar radiation and the
surface is cooled by radiative, convective, and latent heat exchange.
Field research stations were established that included daily observations and me-
teorological data. In two seasons, 14 surface hoar and 26 near-surface facet events
were recorded. Statistical analysis of the surface hoar events indicated three factors
that were related to surface hoar growth: incoming long-wave radiation, snow surface
temperature, and relative humidity. The ideal conditions for each of these parameters
were 190–270 W/m2, -22 to -11 ◦C, and 45–80%, respectively. For near-surface facet
formation, long- and short-wave radiation and relative humidity were statistically
linked to the events. The ideal conditions for these parameters ranged from 380–710
W/m2, 210–240 W/m2, and 23–67%, respectively.
Using a thermal model, sensitivity analysis, and Monte Carlo simulations the
conditions that lead to facet formation were explored. Based on computed mass-flux,
the formation of surface hoar was mainly driven by changes in long-wave radiation, air
temperature, wind velocity, and relative humidity. From these terms graphical tools
were developed to predict surface hoar; the numerical results matched reasonably well
with the field observations. Based on the presence of a specific temperature gradient
understood to lead to near-surface facets, three terms were determined to be the
most influential: density, thermal conductivity, and incoming long-wave radiation.
Using these terms, albedo, and incoming short-wave radiation—a requirement for
radiation-recrystallization—a means for predicting the presence of near-surface facets
was presented.
The physical and analytical data presented indicates that incoming long-wave
radiation is the most influential parameter governing the conditions that lead to
surface hoar and near-surface facet growth. The analysis suggests that snow with
low density and high thermal conductivity may be conducive to the formation of
near-surface facets.
1CHAPTER 1
INTRODUCTION
Within the avalanche community, surface hoar and near-surface facets are es-
tablished areas of importance and have been reported to account for up to 73% of
human-triggered avalanches (Schweizer and Jamieson, 2001; Tremper, 2001). More-
over, the influence of these surface layers on the environment extends well beyond the
scope of avalanche prediction into the domain of global climatology. Seasonally, snow
covers 46,000,000 km2 of the earth, which is about 31% of the Earth’s land surface
(Weast, 1981; Frei et al., 1999). Consequently, the extent of snow cover affects the
global temperature, principally due to the reflectivity of snow—in contrast to the
reflectivity of soil or vegetation. However, the crystal structure of snow itself can
alter its reflectivity, since the optical properties of particles are controlled by the size
and shape of the particle. The presence of a layer such as surface hoar (crystals that
form via vapor deposition on the snow surface) could have global effects, especially
when considering the vast extent of snow cover that exists. Leathers et al. (1995)
stated that “a thorough knowledge of the dynamics of snow cover to atmosphere
interactions is needed if an understanding of present climate variations and long-term
climate change is desired.”
Qualitative data regarding the formation of both surface hoar and near-surface
facets is readily available, and the processes for both are driven by radiation processes.
Surface hoar is known to form when the snow surface cools due to a radiation loss and
water vapor condenses on the surface. Near-surface facets form when radiation heats
the snow just below the surface and the surface is cooled because of radiation losses.
In both cases, large temperature and vapor pressure gradients develop resulting in
the formation of facets.
2While conceptual knowledge of conditions leading to the morphology of such a
faceted layer is important, a quantitative definition of these conditions is the ulti-
mate goal of this dissertation. There has only been a modicum of prior success in
this effort. The work that coined the term “near-surface facets” focused heavily on
the temperature gradient near the surface, which was generally 100–300 ◦C/m (Birke-
land et al., 1998). A study in the Bolivian Andes reinforced these findings with the
meteorological and snow profile data collected after two avalanches claimed several
lives. Prior to the fatal avalanche the average nighttime near-surface temperature
gradient was determined to be 160 ◦C/m (Hardy et al., 2001). Perhaps the most con-
clusive quantitative study to date regarding facet formation was a laboratory study
of radiation-recrystallization (Morstad, 2004; Morstad et al., 2004, 2007). Thirteen
experiments were performed where environmental conditions including short-wave
and long-wave radiation, snow density, and humidity were controlled . Ten of these
experiments produced 1/4–1 mm faceted crystals near the surface.
The study of the near-surface snow environment is well-founded. However, a
solid qualitative comprehension of the environmental parameters necessary to yield
facet formation has not been established. Considering that the micro-structure of
snow is a critical component of avalanche research and climatology, an antecedent
need is an understanding of the environmental and micro-structural factors that drive
snow morphology—the overriding objective for this dissertation. To meet this general
objective two specific tasks were completed, the details of which are the basis for this
dissertation:
1. Two extensive weather stations, at north- and south-facing locations, were cou-
pled with rigorous daily observations and grain scale images of the snow surface.
The data collected from this investigation contains two complete winter seasons
3of observations and comprises the most extensive and detailed field study of
near-surface facets and surface hoar.
2. Both field and laboratory research of snow metamorphism are limited by time
and nature, thus a numerical exploration was conducted that provided quan-
titative results in two capacities. A sensitivity analysis defines the relative
importance of the various driving factors known to influence the snowpack and
using Monte Carlo simulations the specific quantities for the factors deemed
important were defined.
The work presented in this dissertation is divided into nine chapters, plus this
introduction and a short conclusion. These chapters generally were written to be self
supporting. The dissertation is also divided into four parts. Part I includes a single
chapter (2) that begins with providing additional details regarding the relevance of the
research and then provides a literature review of the two near-surface morphologies
of snow considered: near-surface facets and radiation-recrystallization. This review
focuses mainly on research that investigated the environmental conditions leading to
the formation of these two types of crystals.
Part II, which includes two chapters (3 and 4), details the first accomplishment
listed above, the field investigation of surface hoar and near-surface facets respec-
tively. Within each chapter the recorded events were summarized with images of
the snow crystals (before and after metamorphism) and graphs of the weather condi-
tions surrounding the event. Also, based on the recorded events, the environmental
conditions were analyzed providing physically based evidence of the most influential
environmental parameters.
The next portion of the dissertation, Part III, includes the methods. In Chapter 5
the thermal model used for the numerical investigation is derived. Additional analysis
4in this chapter examines the importance of various wave lengths for incoming short-
wave radiation and the influence of measurement error on the model output. Chapter
6 presents the theory and application of a sensitivity analysis methodology used for
the numerical investigation portion of this dissertation.
Part IV of the dissertation includes four chapters (7–10). Chapter 7 summarizes
all the methods defined in Part III, which includes the development of input distri-
butions for the thermal model, a discussion of sensitivity analysis, methods used to
develop the Monte Carlo simulations, and various data analysis techniques employed
for analyzing the results. Chapter 8 includes both the sensitivity analysis and Monte
Carlo simulation results for surface hoar formation.
Part IV continues with an analysis of near-surface facet formation separated into
two chapters. Chapter 9 focuses on the sensitivity analysis and quantifies the most
influential inputs on a variety of model outputs. Chapter 10 continues by applying
the results of the sensitivity analysis in conjunction with Monte Carlo simulation
data to define specific regions of inputs that are likely to result in near-surface facet
development.
5CHAPTER 2
SIGNIFICANCE AND BACKGROUND OF SURFACE WEAK LAYERS
2.1 Introduction
The introduction (Chapter 1) of this dissertation provided a brief overview of the
importance of examining the near-surface layer of a snowpack and sets the stage for
the main objective of the entire body of work presented. The objective of which is to
improve the current understanding of near-surface facet and surface hoar formation.
Along these lines, the first step to obtaining such an understanding is to compile the
preexisting data, which is the purpose of this chapter. That is, to define the state
of the art concerning the driving factors associated with the development of surface
hoar and near-surface facets.
2.2 Significance of the Near-Surface Layers
2.2.1 Avalanches
On the regional scale the importance of studying the near-surface layer of snow
lies with predicting avalanche hazards. Tremper (2001) provided myriad statistics
regarding avalanche fatalities within the United States and Western Europe from
1953 through 2000. It is reported that the average fatalities in the U.S. increased from
approximately 5 annually in the 1960s to near 25 in the late 1990s. The Colorado
Avalanche Information Center (CAIC, 2010) provided more recent statistics up to
2008, and reported that the current U.S. average is now approaching 30 fatalities
per year. The avalanche concern is not limited to the U.S. For example, according
to McClung and Schaerer (2006) Switzerland alone averaged nearly 30 fatalities per
year between 1983 and 2003.
6In addition to fatalities, property damage due to avalanches is a serious concern In
alpine countries (Switzerland, Austria, and France) property damage is often 20 times
as great as in the U.S. (McClung and Schaerer, 2006). McClung and Schaerer (1993,
2006) pointed out that the true cost of avalanches to society is difficult to define, but
the collateral costs such as the cost of avalanche protection—typically four times that
of annual property damage—as well as indirect costs that result from highway and rail
closures drive up the financial liability of avalanche activity. Inarguably avalanches
are a deadly threat and have a significant associated cost.
The most common weak layer associated with avalanches develops in the surface
layer as near-surface facets or surface hoar. Schweizer and Jamieson (2001) reported
that in approximately 73% of 103 investigated avalanches in Canada and Switzerland,
the weak layer was composed of facets (excluding depth hoar) and surface hoar.
Furthermore, of the 45 avalanches investigated by Schweizer and Lutschg (2001) the
weak layer or interface layer (layers adjacent to the failure plane) of 29 were composed
of facets or surface hoar. In studying the processes associated with near-surface facets
in southwest Montana, Birkeland (1998) reported that of 51 avalanches investigated
90% had a weak layer composed of near-surface facets (59%) or surface hoar (31%).
Additionally, research had indicated that such layers are persistent over time and can
be hazardous well after the layer initially formed (Lang et al., 1984; Davis et al., 1996;
Hachikubo and Akitaya, 1996).
2.2.2 Climate
On a global scale the surface layer of snow has perhaps more significant impli-
cations than avalanche formation, that is, the link between this uppermost layer of
snow and the global climate. Globally, snow covers approximately 46,000,000 km2
annually during January and February (Frei et al., 1999), which is nearly one third
7of the Earth’s total land surface (Weast, 1981). Consequentially, snow can have a
vast effect on the climate, specifically on air temperature (Karl et al., 1993; Leathers
and Robinson, 1993; Groisman et al., 1994a). For example, Groisman et al. (1994b)
explained that snow cover increases planetary albedo and reduces outgoing long-wave
radiation. It was shown that the changing radiation balance from decreasing snow
cover in the northern hemisphere between 1979 and 1990 may account for 0.5 ◦C of
the recorded 0.98 ◦C temperature increase (Groisman et al., 1994b).
Leathers et al. (1995) examined temperature depressions associated with snow
cover in the Northeastern United States and concluded that a snow cover of greater
than 2.5 cm caused temperature depressions near 5 ◦C during early and late portions
of the season. Utilizing a snow pack model and data from four cold air masses
moving across the United States Great Plains, Ellis and Leathers (1999) reported that
decreasing the albedo affects the air temperature significantly. Changing the uniform
albedo for the region from 0.9 to 0.5 increased mean daytime air temperature 3–6 ◦C
and maximum day temperatures by 7–12 ◦C (Ellis and Leathers, 1999). Additionally,
snow depth variations between 30 cm, 15 cm, and 2.5 cm yielded little difference in
the change of air temperature (Ellis and Leathers, 1999). Ellis and Leathers (1999)
explained that the temperature differences developed from the model agree with those
of the measured data and stated that the difference in temperature may be due to
sensible and possibly latent heat fluxes.
The snow albedo is an important factor when considering climatic changes in areas
that are covered in snow. Typically, snow albedo is reported as a singular value or
all-wave albedo that includes the solar (short-wave) electromagnetic spectrum, but
the albedo for each wavelength varies. The solar reflectance of radiation is highest
in the ultraviolet and visible range (0.2–0.8 µm) and decreases to nearly zero in the
8near-infrared range (0.7–5.0 µm); hence, a majority of the radiation absorbed is in
the near-infrared spectra (Oke, 1978; Warren and Wiscombe, 1980; Warren, 1984).
The albedo of snow also varies on snow age and grain size; Oke (1978) indicated
that snow albedo ranges from 0.4 to 0.95 for old and new snow, respectively. Albedo
has been shown to change dramatically due to aging and contamination of the snow,
but albedo also varies due to physical changes in the surface morphology (Oke, 1978;
Armstrong and Brun, 2008). Additionally, it is reported that the albedo of snow
reduces upon aging due to its increasing grain size as well as the introduction of
impurities such as carbon soot, particularly in the near-infrared (Warren, 1982, 1984).
As mentioned previously, grain size is a critical component when discussing the
optical or solar adsorption properties of snow (Mellor, 1965; Bohren and Barkstrom,
1974; Warren, 1982). Moreover, Kokhanovsky and Zege (2004) indicated that current
reflectivity studies of snow are limited to spherical grains, but pointed out that the
shape of the grains has a profound effect on the reflectivity of snow. These researchers
provided a “new approach to snow optics with a more realistic model of snow as a
medium with non-spherical and close-packed snow grains.”
2.2.3 Synopsis
On the surface of a snow pack, the grain size and shape impact the albedo and
consequently the surrounding climate. In this respect, near-surface facets and surface
hoar are critical, since both develop either at the surface, as in the case of surface
hoar, or in the upper few centimeters of a snowpack (Birkeland et al., 1996; Birkeland,
1998; Morstad et al., 2007). Additionally, these layers are known to be associated
with avalanche release. Surface hoar and near-surface faceted crystals associated with
skier-triggered avalanches were typically around 2 mm in size (Schweizer and Lutschg,
2001). However, surface hoar crystals have been observed to grow up to 10 mm in
9the mountains of southwest Montana (Tremper, 2001; Cooperstein et al., 2004), but
typically range from 1–5 mm (McClung and Schaerer, 2006). Near-surface facets are
smaller in size. Tremper (2001) reported that near-surface facets range from 0.5–2.0
mm. While Cooperstein et al. (2004) indicated that a majority of crystals observed
were less than 0.5 mm. In a laboratory study, artificially grown near-surface facets
ranged from 0.1–1.0 mm (Morstad et al., 2007). Figure 2.1 provides examples of
near-surface faceted and surface hoar crystals.
(a) Surface Hoar
 
(b) Near-surface Facet
Figure 2.1: Example images of (a) surface hoar (Cooperstein et al., 2004) and (b)
near-surface faceted snow crystals (Morstad, 2004).
2.3 A Review of Surface Hoar
Surface hoar has been studied for nearly 75 years (Seligman, 1936) and in a discus-
sion of observations collected regarding the accumulation and stratification of snow,
Gow (1965) observed “spike-like crystals” now referred to as surface hoar. These
10
crystals were reported to form during periods of still weather and clear skies, and
were more prominent on snow dunes and sastrugi (sharp irregular ridges on the snow
surface formed by wind erosion). Gow (1965) theorized that this hoar frost may orig-
inate from the condensing of water originating in the dune itself. Furthermore, Gow
(1965) emphasized that hoar frost only forms during “exceptionally” calm weather.
As with the aforementioned research, work regarding the accumulation of moisture
on Antarctic ice has also been examined by Linkletter and Warburton (1976). Their
work took into consideration the accumulation due to surface hoar and rime (crystals
formed from liquid water condensation as opposed to vapor deposition that forms
surface hoar) and determined that these processes may contribute 5–10% of the annual
accumulation on the Ross Ice Shelf. Their observations indicated that both rime and
surface hoar develop during super-cooled fog events. The fog events typically occurred
with a light wind from 0–5 m/s and frequently demonstrated temperature oscillations
as rapid as 0.5 ◦C/min with fluctuations as large as 5 ◦C (Linkletter and Warburton,
1976). However, surface hoar was also shown to develop on three occasions when no
visible fog was present (Linkletter and Warburton, 1976).
Lang et al. (1984) proclaimed that their research was the first “thorough quanti-
tative study of surface hoar.” The study utilized a thermocouple stack that recorded
temperatures of the air above and within first the few centimeters of the snowpack.
In an overnight experiment, Lang et al. (1984) observed that surface hoar formation
was associated with significant differences between the snow surface temperature and
the air temperature, with the major growth occurring when the gradient in the air
was largest at 380 ◦C/m. Additionally, a temperature gradient reversal was observed
between the -2 cm temperature in the snow and the snow surface; the gradient re-
versed from -80 ◦C/m to 70 ◦C/m during the growth of the surface hoar (Lang et al.,
1984). Snow surface temperatures were observed to range between -21◦C and -12.5
11
◦C when surface hoar crystals developed. These results agree with work conducted by
Mason et al. (1963), who reported the growth of snow crystals from the vapor phase.
This study indicated the dendritic, feather-like crystals grew with the air temperature
between -12 ◦C and -16 ◦C. Lang et al. (1984) concluded that “a large near-surface
air temperature gradient, due to nocturnal clear sky conditions, is insufficient in itself
for significant condensation onto the snow surface to occur.” It also explained that
any horizontal air movement close to the surface was observed to prohibit surface
hoar growth—a result that has been disputed (Colbeck, 1988).
In maritime climates, Breyfogle (1986) determined that two dominant scenarios
exist for the development of surface hoar. The first of the two formation scenarios
outlined by Breyfogle (1986) occurred when a cold air front moved into the region
causing a“highly saturated environment at the snow/air interface;” this was associ-
ated with radiation cooling of the snow surface to the dew point, hence deposition
occurred. Generally, surface hoar tended to develop when the air temperature was
between -6 ◦C and -12 ◦C at the snow surface, relative humidity was at least 70%,
and the formation occurred in low-wind conditions. The second observed formation
conditions occurred in an under-saturated environment, but a secondary source of
water vapor must be present (Breyfogle, 1986). This secondary source was similar to
that of the aforementioned research by Linkletter and Warburton (1976); Breyfogle
(1986) indicated that the source of vapor was supercooled cloud decks that established
a large vapor flux when clear sky and low humidity conditions prevailed above the
cloud deck; in these conditions the cloud deck acted as a radiator and this layer was
termed the “peripheral zone.”
Breyfogle (1986) provided two examples of the latter scenario that occurred during
a four day period where air temperature and relative humidity were monitored con-
tinuously. During two nights the relative humidity dropped from near 100% during
12
the day to near 60% at night. Additionally, the air temperatures increased during the
night by approximately 2–3 ◦C. These environmental observations we observed at a
station located 300 m above the study plot, directly in the peripheral zone. During
both of these events surface hoar was shown to form precisely when the observation
station was within the peripheral zone.
Colbeck (1988) calculated the importance of temperature profiles and humidity
to form surface hoar by employing a theory for temperature decay of a surface being
cooled by outgoing radiation explained by Sutton (1953). The theory is an exponential
function based on environmental conditions such as long-wave radiation, thermal
conductivity, temperature gradient within the snow, latent heat, and heat capacity,
among others. Using a number of assumptions from the literature, Colbeck (1988)
considered a no-wind condition which was described as necessary by Lang et al. (1984).
The study concluded that “some wind” is a necessity to supply vapor to the snow
surface causing surface hoar growth and such wind could be due to “near-surface, low-
speed gravity drainage.” These winds are common in mountainous areas undergoing
radiation cooling, which causes pools of cool air to accumulate in constricted areas
that suddenly release (Colbeck, 1988). Thus, no detectable wind may be present for
a majority of the time, but surges could supply vapor necessary for the growth of
surface hoar.
Surface hoar crystals were found to persist on the surface of a snow pack for many
days despite evaporation during the day. During two multi-day case studies, surface
hoar formed during the clear, humid nights with air temperatures between -6 and
-10 ◦C and snow surface temperatures around -15 ◦C (Hachikubo and Akitaya, 1996,
1998). Hachikubo and Akitaya (1996) indicated that the surface hoar did not degrade
as expected during the daytime hours. Data indicated that hoar crystals developed
during a positive latent heat flux with condensation of 74 gm/m2. Evaporation of 62
13
gm/m2 occurred during the day, indicating that the crystals should have been reduced;
however, they persisted. One possible explanation provided by the researchers was
that the vapor lost due to evaporation may have been from the underlying layer of
snow, a few centimeters below the surface. Hachikubo and Akitaya (1996) suggested
that “the surface hoar crystals were cooled by outgoing radiation even in the daytime
and kept their size, while the snow grains underneath the surface were warmed by
solar radiation and evaporated.” These are precisely the near-surface faceted crystals
that shall be discussed in Section 2.4.
As previously discussed, wind has been presented as both necessary (Colbeck,
1988) and destructive (Lang et al., 1984) for surface hoar formation. Hachikubo and
Akitaya (1997) provided sufficient evidence that the presence of wind is significant in
the formation of surface hoar crystals and made the general statement that surface
hoar grew “when the snow surface temperature was 5 ◦C (or more) lower than the
air temperature, the humidity was higher than 90%, and the wind speed was 1 to
2 m/s at 0.1 m high.” Additionally, Hachikubo and Akitaya (1997) stated that the
net radiation (sum of sensible, latent, and conductive heat fluxes) slowly decreased
during formation, as did the bulk transfer coefficient. The bulk transfer coefficient
is utilized as an estimate for condensation rate, which was shown to increase as the
surface hoar forms. Hence, the formation of surface hoar itself increases the surface
roughness of snow thus increasing the bulk transfer rate. In other words, a feedback
process exists for the means of condensation and surface hoar formation (Hachikubo
and Akitaya, 1997). An investigation by Ho¨ller (1998) and values inferred from the
work of Hachikubo and Akitaya (1997) suggest that surface hoar is far more likely
to form in open areas than in forested or partially forested areas. This conclusion
was based upon a preliminary study that examined three study plots in open and
14
forested areas by comparing the air temperature, relative humidity, wind speed, and
total radiation balance at each site (Ho¨ller, 1998).
Hachikubo (2001) conducted a comparison between field data and two numerical
models—the “Simple” and the “Crocus.” Crocus is a snow model originally developed
by Brun et al. (1992). Hachikubo (2001) concluded that when relative humidity
was less than 80% there existed a wind speed at which the sublimation rate was
maximized. For example, at 60% the sublimation rate reached its maximum around
2 m/s, while at 70% it was greatest near 3.5 m/s (Hachikubo, 2001). For conditions
with a relative humidity greater than 90% sublimation increased with wind speed in
the range examined from 0–6 m/s. The two models explored both disagreed with
field experiments in some capacity. The Crocus Model underestimated snow surface
temperature and the Simple Model disagreed with results when clouds were present
(Hachikubo, 2001).
A more recent study of surface hoar and near-surface facets compared the differ-
ences between aspect (north versus south) with respect to crystal formation. In two
examples given by Cooperstein et al. (2004), surface hoar formed larger crystals on
the north-facing site when compared to the south-facing site. The conditions were
similar in both examples: the temperature gradient at the snow/air interface was
larger on the north site, the snowpack absorbed less short-wave radiation, and the
snow surface temperatures were lower. Interestingly, the second example discussed
also exhibited growth of near-surface faceted crystals, which formed better on the
south-facing slope.
In a study examining the spatial variability of surface hoar formation, Feick et al.
(2007) reported three surface hoar events that occurred at three weather stations.
Feick et al. (2007) provided exhaustive details of these events. To summarize, the
events occurred with air temperatures of approximately −10 to −5 ◦C, with snow
15
surface temperatures approximately 10 ◦C colder, and with wind speeds less than 2.5
m/s. The study concluded that wind and short-wave radiation contributed to the
destruction of the surface hoar and that wind speed was key to predicting formation
and destruction. This conclusion suggests that forecasting is nearly impossible due
to the difficulties of modeling wind speeds across complex terrain.
Finally, Colbeck et al. (2008) detailed an investigation into a specific mechanism
for surface hoar formation due to valley clouds. This work explained that in order to
achieve growth rates observed, the surface hoar likely formed as the cloud expanded
up-slope where the wind speeds were between 1 and 2 m/s and the snow was still
exposed to the cold sky allowing for radiative losses at the snow surface.
Overall, the study of surface hoar has progressed during the years from simple
observations to complex numerical modeling. Throughout all the literature presented,
three major variables are consistently explored: wind speed, air/snow temperature
gradients, and relative humidity. Wind speed has been shown to be critical for forming
surface hoar by providing a source of moisture for condensation on the surface. The
research presented indicated that winds in the range of a few meters per second are
optimum. A temperature difference between the snow surface and the air must be
present; this difference was shown to be near 5 ◦C, with air temperature being the
higher value. Finally, humidity varies greatly depending on the conditions. Although
not mentioned in a majority of the literature, another critical factor is the radiation
balance and heat exchange, which was shown to be slightly negative but changed
little during surface hoar formation. Table 2.1 summarizes the work presented within
this section, highlighting the chief focus of each article reviewed.
16
Table 2.1: A summary of the conditions necessary for surface hoar growth as reported
in the literature reviewed.
Author Surface Hoar Observations
Mason et al.
(1963)
Dentritic feather-like crystals grew from vapor phase between -12 ◦C
and -16 ◦C with supersaturation with respect to ice greater than 20%.
Gow (1965) Surface hoar formed during clear and still weather; on dunes and
sastrugi; theorized that condensed moisture may have originated from
dune feature.
Linkletter and
Warburton
(1976)
Crystals formed in association with fog events (in addition to three
periods without fog) and with light winds of 0–5 m/s; fog had rapid
(0.5 ◦C/min) and large (5 ◦C) temperature variations.
Lang et al.
(1984)
Surface hoar formed with snow surface to air temperature gradients in
excess of 300 ◦C/m, a gradient reversal of snow to snow-surface (-80 to
70 ◦C/m), snow surface temperatures between -12 and -16 ◦C, and no
horizontal wind movement.
Breyfogle
(1986)
Formation tended to occur with air temperatures between -6 and -12
◦Cwith humidity of 70%; two dominant processes occurred: 1) a highly
saturated condition with low wind and 2) an undersaturated condition
with the presence of a secondary vapor source.
Colbeck
(1988)
Formation required low wind conditions such as provided by gravity
drainage winds; pure diffusion of water vapor was unable to account
for growth rates observed.
Hachikubo
and Akitaya
(1996, 1998)
Surface hoar growth occurred during clear humid nights; air
temperature was between -6 and -10 ◦C; snow surface temperature was
near -15 ◦C.
Hachikubo
and Akitaya
(1997)
Growth occurred with a surface temperature 5 ◦Clower than the air
temperature, humidity greater than 90%, wind speed between 1–2 m/s
at 0.1 m above snow, and as net radiation decreased from -85 to -50
W/m2.
Ho¨ller (1998) Environment for surface hoar growth suggested to be open terrain
compared to forested areas with 20% and 80% canopy coverage.
Hachikubo
(2001)
Sublimation was maximized between wind speeds from 0–6 m/s when
relative humidity was less than 90%; when humidity was over 90%,
sublimation increased throughout the range tested.
Cooperstein
et al. (2004)
Surface hoar crystals up to 10 mm formed with temperature gradients
up to 92 ◦C/m at the snow/air interface, net short-wave of 181 W/m2,
average surface temperature of -10 ◦C, and average wind speed of 2.1
m/s.
Feick et al.
(2007)
Various extents of surface hoar was observed at three weather stations
in a single basin. The surface hoar developed with air temperatures of
-5 to -10 ◦C, wind speeds of less than 2.5 m/s, and a snow surface
temperature approximately 10 ◦C less than the air temperature.
17
2.4 A Review of Near-Surface Facets
Colbeck (1989) conducted one of the first studies specifically examining near-
surface faceted crystal formation, although previous research exists (LaChapelle, 1970;
LaChapelle and Armstrong, 1977; Armstrong, 1985; Akitaya and Shimizu, 1987). Col-
beck (1989) developed a “theory of spatial and temporal variations in temperature
with sinusoidally varying surface temperature and periodic solar radiation at the sur-
face.” Colbeck (1989) examined three scenarios using the theory developed: seasonal,
high-altitude, and polar ice-sheet snow covers. The seasonal snow cover was assumed
to be 1 m thick, have a 30 ◦C air temperature swing, and have a peak solar radiation
absorption of 70 W/m2. Under these conditions Colbeck (1989) explained that the
conditions necessary for increased crystal growth just beneath the surface existed,
especially when the diurnal effects were coupled with penetration solar radiation.
Field observations and laboratory experiments were conducted by Fukuzawa and
Akitaya (1993) with regard to the formation of faceted crystals near the snow surface.
A January case study consisted of 3 cm of new snow (70 kg/m3) on top of a lightly
compacted layer (170 kg/m3). The snow surface metamorphosed into faceted grains
over two nights of clear skies (Fukuzawa and Akitaya, 1993). These grains formed due
to temperature gradients that averaged 159 ◦C/m, occurred at about 1 cm depth, were
subjected to high solar radiation during the day and radiative cooling at night, and
experienced low wind speeds averaging 1.0 m/s (Fukuzawa and Akitaya, 1993). Sim-
ilar results were found during March, except that the growth rates were double that
of January, which was likely due to the increased solar radiation during this month.
Fukuzawa and Akitaya (1993) constructed an experimental setup that established a
temperature gradient across a snow sample and were able to explore the effects of
high temperature gradients on low density snow as observed in the field. During these
18
experiments faceted crystals grew from fine ice particles into 0.2 mm facets within
16 hr and into 0.4 mm facets in 48 hr. Fukuzawa and Akitaya (1993) concluded that
three conditions were necessary to form faceted crystals near the surface: (1) a low
density layer (less than 3 cm) must overlay older, denser snow, (2) the subsurface
snow temperature must increase due to diurnal solar radiation, and (3) the surface
temperature must decrease rapidly due to upward long-wave radiation.
The process of diurnal heating of the snow beneath the surface applies to more
than just the formation of faceted crystals. Oke (1978) explains such a process and
its importance to understanding the boundary layer climate, and Ozeki and Akitaya
(1996) explained a similar process that lead to the formation of ice crusts.
Fierz (1998) explained the formation of a near-surface faceted layer similarly to
Fukuzawa and Akitaya (1993), including the three key factors pointed out. Fierz
(1998) reported that 11 cm of new low-density snow (70–110 kg/m3) accumulated on
a denser pack (260 kg/m3) that was followed by clear weather. The result was a crust
5 mm thick on the surface with 1.5–2 mm facets underneath.
The establishment of near-surface faceted crystals as an specific area of expanded
study is partially due to work conducted by Birkeland et al. (1996), Birkeland (1998),
and Birkeland et al. (1998), in which the term “near-surface facets” was coined to
describe “snow formed by near-surface vapor pressure gradients resulting from tem-
perature gradients near the snow surface.” Birkeland (1998) defined three specific
processes that lead to the formation of near-surface facets:
In a case study in the mountains of Montana, diurnal recrystallization was shown
to develop 1 mm facets within 36 hours during a period characterized by fresh snow-
fall (5–10 cm) followed by clear weather (Birkeland et al., 1996; Birkeland, 1998).
Temperatures during the day were near -3 ◦C, which was proceeded by nighttime
temperatures near -15 ◦C. This shift in temperature caused a large temperature gra-
19
dient in the upper 5 cm of the snow pack. A gradient of -250 ◦C/m (in this paper
a negative gradient is associated with warmer temperatures at depth) was recorded
during the night followed by a 100 ◦C/m gradient during the day, resulting in faceted
crystals up to 1 mm in size.
In an investigation of two avalanches, one of which claimed the lives of two
climbers, Hardy et al. (2001) suggested that the conditions existed in high-altitude
tropical mountains that led to the formation of a thick (10 cm), well-developed layer
of large-faceted grains (3–5 mm). Although specific measurements regarding the
formation of the faceted layer were not available, using data from a nearby weather
station Hardy et al. (2001) concluded that a layer of buried facets (27–37 cm deep)
was once at the surface and exposed to clear and cold conditions. Additionally, high
amounts of incoming short-wave solar radiation are typical of these high-altitude areas
(950 W/m2) and temperature gradients of 161 ◦C/m were recorded, both of which
have been shown to be sufficient for the formation of near-surface faceted crystals
(Hardy et al., 2001).
Aspect was also shown to affect the formation of near-surface faceted crystals.
Cooperstein et al. (2004) reported that faceted crystals were better developed on
south-facing slopes compared to a north-facing aspect with similar elevations. During
a 24 hour period in which near-surface faceted crystals were observed to form, the
southern exposure had a larger temperature gradient and exhibited a reversal while
the opposing site did not show a reversal (Cooperstein et al., 2004). As mentioned in
the previous section, this near-surface facet event was coupled with the formation of
surface hoar, which grew larger on the north site.
To re-examine the effects of canopy coverage on facet formation Ho¨ller (2004)
conducted a study similar to that summarized in the preceding section on surface
hoar. The effects of tree cover were examined between an open area, a clearing (30%
20
canopy coverage), and a forest (75% canopy coverage). Temperature gradients were
shown to be as high as 130 ◦C/m for the open area, 42 ◦C/m in the clearing, and less
than 25 ◦C/m for the forest region. Using snow profiles during three different winter
seasons, Ho¨ller (2004) did not observe any near-surface faceted crystals forming in
the forest areas, but observed faceting in the open area and the clearing.
The most complete quantitative study of near-surface faceting was conducted un-
der laboratory conditions, utilizing an environmental chamber that allows for the con-
trol of incoming long-wave and short-wave radiation, among other variables. Morstad
et al. (2007) successfully grew radiation recrystallized near-surface facets in a lab set-
ting, whence facets ranged from 1/8 to 1 mm in size. The complete results from this
endeavor are provided in Morstad (2004). Of the thirteen experiments performed, 10
produced 1/4–1 mm faceted crystals near the surface. The environmental conditions
were controlled as follows: short-wave radiation ranged from 595–1180 W/m2, long-
wave was relatively constant near 280 W/m2, density varied between 175–10 kg/m3,
and humidity was between 15% and 40%. The temperature gradients found during
the facet formation were between 100–550 ◦C/m.
Slaughter et al. (2009) detailed the work originally presented in McCabe et al.
(2008) and Slaughter et al. (2008) that included field observations of near-surface
facets and laboratory simulations of three events. McCabe et al. (2008) summarized
six radiation-recrystallization events that formed facets under similar conditions: a
warm, melting subsurface overlain by a frozen surface. The temperature gradient
between the melt-layer and surface was estimated to range from 240–400 ◦C/m and
short-wave input ranged from 575–840 W/m2 at a weather station with a clear view
of the sky. The facets formed in new snow in each event. Slaughter et al. (2009, 2008)
mimicked these three observed events in a laboratory environment. Despite that the
21
experiments utilized old, rounded snow, facets developed, albeit not to the extent
observed in the field.
In summary, according to the literature reviewed the formation of near-surface
faceted crystals was driven by a temperature gradient that forms in the upper few
centimeters of the snowpack. This gradient may be established in any number of ways,
including radiation penetration into the snow surface, long-wave radiative losses, or
because of the presence of a buried layer of warmer snow from a melt cycle. This
gradient was also shown to be both positive and negative. Generally speaking, a
temperature gradient of equal to or greater than 100 ◦C/m was required for facets to
form. Table 2.2 summarizes each article reviewed regarding the formation of near-
surface faceted crystals.
2.5 Conclusions
The near-surface layer is important with respect to global climatology and
avalanche safety. A significant portion of the Earth’s surface is coated by a layer of
snow seasonally. Thus, the changing and aging of a snowpack can have a large effect
on the environment due to the drastic changes in albedo that may occur in snow.
Additionally, avalanches are a significant societal hazard accounting for numerous
deaths and high cost to highway departments in mountainous regions.
Avalanche research with respect to snow crystal metamorphism has been ongoing
for a number of years, a portion of which has focused on two specific formations at the
snow surface: near-surface facets and surface hoar. However, minimal quantitative
work has been completed regarding the driving environmental conditions that lead
to the formation of each of these crystals. Tables 2.3a and 2.3b summarize the
quantifiable parameters found in the research reviewed. Researchers indicate that
22
Table 2.2: A summary of the conditions necessary for near-surface facet growth as
reported in the literature reviewed.
Author Near-Surface Facet Observations
Colbeck (1989) Radiation recrystallization formed crystals at 0.1 m below the
surface with a peak solar radiation of 126 W/m2 and a
temperature swing of 20 ◦C.
Fukuzawa and
Akitaya (1993)
Formation required temperature gradients between 100–300
◦C/m and a low density layer (3 cm) overlaying an old denser
layer.
Fierz (1998) Facets (1.5–2 mm) formed under a 5 mm crust when 11 cm of
new low-density snow (70–110 kg/m3) accumulated on a
denser pack (260 kg/m3) followed by days of clear weather.
Birkeland et al.
(1996); Birkeland
(1998); Birkeland
et al. (1998)
Diurnal recrystallization occurred on a layer of new snow 5 to
10 cm thick with temperature gradients shifting from -250 to
100 ◦C/m between the -20 ◦Cnight and -3 ◦Cday.
Hardy et al. (2001) Temperature gradients of 161 ◦C/m and incoming short-wave
radiation of 950 W/m2, were shown to account for a 10 cm
layer of large-faceted crystals using dust and chloride
concentration evidence.
Cooperstein et al.
(2004)
Well-developed facets formed with temperature gradients
between 126 and -60 ◦C/m just below the surface, with net
short-wave of 587 W/m2 and an average surface temperature
of -7 ◦C.
Ho¨ller (1998) Near-surface facet growth was observed in an open region and
forest clearing ( 30% canopy coverage) with maximum
temperature gradients of 130 ◦C/m and 42 ◦C/m, respectively.
Morstad (2004);
Morstad et al. (2004,
2007)
Ten experiments produced near surface faceted crystals with
1/4–1 mm grains and temperature gradients between 100–550
◦C/m. The environmental conditions were controlled:
short-wave ranged from 595 1180 W/m2, long-wave was near
280 W/m2, snow density varied between 175–410 kg/m3, and
humidity was between 15% and 40%.
McCabe et al. (2008);
Slaughter et al. (2008,
2009)
Six radiation-recrystallization events were observed in the field
with temperature gradients in the surface layer ranging from
240–400 ◦C/m with incoming short-wave radiation ranging
from 575–840 W/m2 at a station with a clear view of the ski.
Three of these events were reproduced in laboratory
simulations, producing facets despite drastically different
snowpacks from those observed in the field.
23
surface hoar forms due to large temperature gradients between the snow and the
overlying subfreezing air. Near-surface facets tend to form in the top few centimeters
of the snowpack, with high solar radiation, and with various snow densities. As with
surface hoar, a temperature gradient is necessary to induce growth of near-surface
faceted crystals, however in the case of the latter phenomenon this gradient exists
within the snowpack.
Table 2.3: Summary of quantifiable parameters shown to lead to the formation of (a)
surface hoar and (b) near-surface facets as presented in the available literature.
(a) Surface Hoar
Quantitative Parameter Range Adapted From Literature
Air temperature -6 to -16 ◦C
Difference in surface and air temperature 5 ◦C
Humidity 60 to 100%
Net radiation -85 to -50 W/m2
Net shortwave radiation 181 to 587 W/m2
Temperature gradient (snow/air) 0 to 300 ◦C/m
(b) Near-surface Facets
Quantitative Parameter Range Adapted From Literature
Changes in air temperature 17 to 20 ◦C
Humidity 15 to 40%
Snow density 0 to 410 kg/m3
Short-wave radiation 587 to 1180 W/m2
Long-wave radiation 280 W/m2
Temperature gradient -250 to 550 ◦C/m
Overall, the research that has been performed to date marks a significant step
in developing an understanding of the surface environment of a snowpack. A solid
conceptual understanding of the formation process exists, but further work is required
to quantify these parameters. Quantification is a requirement for greater comprehen-
sion of the processes that alter the snow surface, which affects the environment and
presents a risk to the life of people who work and recreate in mountainous regions.
24
CHAPTER 3
FIELD INVESTIGATION OF SURFACE HOAR
3.1 Introduction
Surface hoar is a snow crystal that forms on a surface, typically on seasonal snow,
via vapor deposition from the surrounding environment. When buried by additional
snowfall, this layer is a particularly dangerous weak-layer that leads to a significant
number of avalanches. Approximately 30% of skier-triggered avalanches have been
associated with surface hoar (Birkeland, 1998; Schweizer and Lutschg, 2001). This
layer is known to be persistent (McClung and Schaerer, 2006), thus understanding the
conditions surrounding the formation of surface hoar has been the topic of numerous
research projects. A complete review of surface hoar is provided in Chapter 2.
Despite the body of research that exists for surface hoar formation, minimal re-
search exists that defines specific environmental conditions that surround surface
hoar formation. To this end, two weather stations were established and daily obser-
vations and crystal-scale photographs of the snow surface were taken. The goal of
these stations and extensive observations was to quantify the conditions necessary to
yield surface hoar formation for use in subsequent research. The following chapter
summarizes the results obtained from a field investigation of surface hoar, which is
only one portion of an ongoing investigation resulting from the collaborative efforts
of researchers at the Subzero Science and Engineering Research Facility at Montana
State University and the Yellowstone Club (YC) Ski Patrol.
25
3.2 Methods
3.2.1 Weather stations
Two study plots were chosen, one on a north-facing and another on a south-facing
slope. Data collection began during the 2005/2006 season; however, snow observa-
tions did not begin until the following season (2006/2007) and images were added
to the observations during the 2007/2008 season. All three forms of data (weather,
observations, and images) continued through 2008/2009, and will continue as the
project is ongoing. Both sites were used in previous research: Cooperstein et al.
(2004) used these locations in a study of surface hoar and near-surface facet devel-
opment; Staples et al. (2006) utilized the weather data when modeling snow surface
temperatures; Slaughter et al. (2009) performed laboratory experiments mimicking
conditions observed at the south location; Adams et al. (2009) used recorded data
as a basis for spatial modeling of weak-layers; and Slaughter and Adams (2009) used
the weather data for the basis of a sensitivity analysis of the conditions leading to
the formation of near-surface facets and surface hoar.
The North and South study plots are located on Pioneer Mountain at the Yellow-
stone Club near Big Sky, Montana. A third station (Aspirit), which is maintained
by the YC Ski Patrol, is positioned near the top of a ridge at the American Spirit
chair lift. This location has a nearly clear view of the sky for gathering unobstructed
radiation data. The locations and elevations for each station are detailed in Table
3.1. Both the North and South study plots have a slope angle of approximately 30◦.
3.2.2 Instrumentation
The North and South sites were similarly instrumented to measure air tempera-
ture and humidity (Campbell Scientific, Inc. CS215 with 41303-5 naturally aspirated
26
Table 3.1: Detailed information on each of the three weather stations situated on
Pioneer Mountain.
Station Latitude Longitude Elevation Aspect
Aspirit 45◦14′23.0′′N 111◦26′34.5′′W 2690 m n/a
North 45◦14′52.3′′N 111◦27′21.8′′W 2530 m 0◦
South 45◦13′47.7′′N 111◦26′33.0′′W 2740 m 187◦
radiation shield), snow depth (NovaLynx Corp.), snow surface temperature (Everest
Interscience, Inc. 4000.4ZL), incoming long-wave radiation (Eppley Lab., Inc. PIR
and Kipp & Zonen CGR3), slope-parallel incoming and reflected shortwave radiation
(LI-COR, Inc. Li200 and Kipp & Zonen CGR3), wind speed and direction (Met One
Instruments, Inc. 034B-L), and subsurface snow temperature taken at 2 cm inter-
vals (Omega Eng., Inc. type T thermocouples). Data was recorded with Campbell
Scientific, Inc. (CSI) CR10(x) dataloggers. The third location, Aspirit, measured
unobstructed incoming short- and long-wave radiation (Eppley Lab., Inc. PSP and
PIR). Appendix A includes a completed discussion of the instrumentation.
3.2.3 Snow observations
Each season, from mid-January to early April each season the YC Ski Patrol daily
maintained visual observations and images describing the upper 5 cm of the snowpack.
Snow crystal images were captured using a Panasonic PV-500, Olympus SP-510 UZ, or
a Nikon Coolpix fitted with a 10× loupe from a Brunel 8×30 ocular. The same Brunel
scope was utilized for the crystal classification. The daily records and images were
cataloged in a custom weather software package, YCweather, designed for use in this
project. The software also includes regional weather data and extensive graph creation
capabilities (see Appendix B). Complete copies of all the daily logs are included in
Appendix G.
27
3.3 Results
During the 2007/2008 (A) and 2008/2009 (B) winter seasons, 14 significant events
of surface hoar were recorded. These events occurred at both the South and North
weather stations, but not always both. Table 3.2 includes the mean values of the mea-
sured weather data for each event at the station(s) that crystals were observed. The
means were computed using the data from the night prior to event date shown, from
dusk to dawn. The difference between the air (Ta) and snow surface (Ts) temperature
(∆T = Ta − Ts) was also computed.
Throughout the field notes more references to surface hoar exist than those dis-
cussed here; however,some of this data was omitted as the event was minimal (i.e.,
the field notes and images did not demonstrate that the surface hoar was widespread
or the surface hoar crystals only accounted for a small portion of the surface layer).
The surface hoar often developed in newer snow at the North site while it develop on
melt-freeze crusts at the South location. Table 3.3 summarizes the snow underlying
the surface hoar on the day of observation.
In all but a few events, complete weather data as well as grain-scale images of
the crystals exist. The following sections summarize each of the events recorded.
These summaries provide a brief overview of the event and include the weather data
surrounding the events and images of the surface hoar observed. Images included in
this chapter are un-cropped and are representative of all the images captured for that
event.
28
Table 3.2: Summary of mean daily weather conditions for all days recorded as surface
hoar events, including long-wave (LW ) radiation, air (Ta) and snow surface (Ts)
temperature, wind speed (Vw) and direction (Dir), relative humidity (RH), and the
difference between the air and snow temperatures (∆T ). Blank regions indicate that
surface hoar was not observed at that location.
Event Date LW Ta Ts Vw Dir RH ∆T LW Ta Ts Vw Dir RH ∆T
W/m2 W/m2 ◦C ◦C m/s deg. ◦C W/m2 W/m2 ◦C ◦C m/s deg. ◦C
A-1 01/24/2008 252 -13.1 -22.1 1.0 155 71 9.0 354 -10.5 -19.7 1.3 82 40 9.2
A-2 02/15/2008 225 -11.1 -16.3 1.1 117 78 5.2
A-3a 02/19/2008 217 -4.8 -15.3 1.2 142 54 10.5 376 -4.7 -11.5 1.4 76 51 6.8
A-3b 02/20/2008 417 -0.9 -9.7 1.6 45 18 8.8
A-4 02/22/2008 206 -8.2 -18.0 1.4 151 64 9.9 371 -6.6 -10.5 1.1 53 57 3.9
A-5 02/26/2008 274 -8.8 -15.0 1.1 132 81 6.2 277 -9.8 -15.8 0.9 93 84 5.9
A-6 03/10/2008 206 -7.6 -17.6 1.2 165 57 10.0 369 -7.5 -15.0 1.4 122 57 7.4
A-7 03/30/2008 199 -15.9 -22.4 1.3 152 79 6.5 267 -15.9 -22.4 1.1 89 74 6.5
B-1 01/23/2009 263 -5.7 -9.0 0.5 165 83 3.4
B-2a 01/30/2009 226 -5.8 -12.4 1.5 105 72 6.6
B-2b 01/31/2009 210 -6.4 -15.1 1.4 119 61 8.7
B-3 02/04/2009 202 -6.2 -15.4 1.3 159 63 9.2
B-4a 02/07/2009 Surface hoar observed, data missing
B-4b 02/08/2009 Surface hoar observed, data missing 190 -7.1 -14.9 1.2 246 55 7.8
B-5a 02/13/2009 188 -14.0 -21.3 1.4 159 72 7.3
B-5b 02/14/2009 193 -14.1 -20.8 1.2 155 69 6.7
B-6 02/28/2009 175 -14.7 -22.4 1.3 159 75 7.7
B-7 03/13/2009 317 -8.3 -17.5 1.3 134 50 9.2
Table 3.3: Summary of the snow conditions for the layer underlying the surface hoar,
as recorded in the field notes. Blank regions indicate that surface hoar was not
observed at that location.
Event Event Date North: Description of Snow South: Description of Snow
A-1 1/24/2008 1–3 mm stellars, plates 1 mm decomposing new snow
A-2 2/15/2008 stellars 2 mm, facets 0.5 mm
A-3a 2/19/2008 0.5–1 mm decomposing stellar crystals 0.25x1 mm spaghetti chains
A-3b 2/20/2008 melt freeze/sun crust
A-4 2/22/2008 0.5–1 mm decomposing snow moist melt freeze crust
A-5 2/26/2008 broken stellars 2–3 mm rimed new snow, 2mm
A-6 3/10/2008 partly decomposed 1 mm melt freeze crust (frozen/dry) 1.5 mm
A-7 3/30/2008 highly broken 0.25 mm melt freeze crust
B-1 1/23/2009 0.5 mm decomposing
B-2 1/30/2009 0.5–1 mm decomposing
B-3 2/4/2009 1 mm rounds melt freeze crust
B-4a 2/7/2009 2 mm new snow, 0.25 mm decomposing
B-4b 2/8/2009 0.25 mm highly decomposed melt-freeze (moist)
B-5a 2/13/2009 Decomposing stellars 0.5–1 mm
B-5b 2/14/2009 0.5 mm decomposing and rounds
B-6 2/28/2009 0.25 mm highly decomposed
B-7 3/13/2009 1 mm highly decomposed particles
29
3.4 2007/2008 Surface Hoar Events
3.4.1 Event A-1: January 24, 2008
Surface hoar measuring 2–3 mm in height formed the between January 23 and
24, 2008 at the North Station; “decomposing surface hoar” 1–2 mm in size was
recorded at the South location. Figure 3.1 is an image of the surface hoar from
the North location; no images showing the surface hoar at the South station were
captured. Figure 3.2 contains graphs of the weather data surrounding the event. At
both locations the surface hoar formed on new snow. At the South the short-wave
radiation was beginning to break down the crystals at the time of observation.
Figure 3.1: Image from event A-1 of surface hoar (1 mm grid) captured from the
North Station on January 24, 2008.
30
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
60
65
70
75
80
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
240
245
250
255
260
265
270
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.3
0.6
0.9
1.2
1.5
1.8
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
25
31
37
43
49
55
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(c) South
1200 1800 2400 0600 1200
300
350
400
450
500
550
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.5
1
1.5
2
2.5
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(d) South
Figure 3.2: Weather data for event A-1 (January 24, 2008) recorded for both the
(a,b) North and (c,d) South weather stations.
31
3.4.2 Event A-2: February 15, 2008
The second significant surface hoar event occurred on February 15, 2008 primarily
at the North Station. The field notes from the South Station indicated that surface
hoar was present, but only constituted “a very small percent of the surface snow,”
thus it was assumed that the significant growth only occurred at the North Station.
Figure 3.3 is an image showing the faceted surface hoar that developed on the stellar
arms of new snow. The weather data from the North Station (Figure 3.4) indicates
that the surface hoar likely began forming just after midnight, which is marked by a
rapid decrease in incoming long-wave radiation and subsequent snow surface cooling.
Figure 3.3: Image from event A-2 of surface hoar (1 mm grid) captured from the
North Station on February 15, 2008.
32
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
50
60
70
80
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
190
200
210
220
230
240
250
260
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.4: Weather data for event A-2 (February 15, 2008) recorded for a the North
weather Station.
3.4.3 Event A-3: February 19–21, 2008
The field notes from the North Station indicated that small (0.5 mm) surface hoar
formed in the night prior to the February 19, 2008 and persisted nearly unaltered
until the following day, at which point it degraded. However, the surface hoar was
still visible on the February 21. The field notes stated that the “snow appeared very
similar to [the] previous two days, but had signs of having dried out.” The notes
also indicated that surface hoar was observed at the South Station, but these crystals
were intermingled with “spaghetti chains” of near-surface facets. Images from both
stations on February 19 are provided in Figure 3.5 and the weather data from both
stations is provided in Figures 3.6 and 3.7.
Interestingly, the field notes from the South Station on the February 20 stated that
“surface hoar was barely attached to the crust below, suggesting [that] it [formed]
33
last night” since the surface hoar observed on the previous day was “well-linked to
the crystal below it.”
(a) North: Feb. 19 (b) South: Feb. 19 (1 mm grid)
(c) South: Feb. 20 (1 mm grid)
Figure 3.5: Images from event A-3 of surface hoar captured from the (a) North and
(b) South Stations on February 19, 2008 and surface hoar from the (c) South Station
that was recorded on February 20.
34
1200 1800 02/19 0600 1200 1800 02/20 0600 1200 1800 02/21 0600 1200
−20
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
10
22
34
46
58
70
R
el
at
iv
e 
hu
m
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 02/19 0600 1200 1800 02/20 0600 1200 1800 02/21 0600 1200
200
210
220
230
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
0
0.8
1.6
2.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.6: North Station weather data for event A-3 (February 19–21, 2008).
35
1200 1800 02/19 0600 1200 1800 02/20 0600 1200 1800 02/21 0600 1200
−15
−10
−5
0
5
10
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
10
20
30
40
50
60
R
el
at
iv
e 
hu
m
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) South
1200 1800 02/19 0600 1200 1800 02/20 0600 1200 1800 02/21 0600 1200
300
400
500
600
700
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
0
0.6
1.2
1.8
2.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) South
Figure 3.7: South Station weather data for event A-3 (February 19–21, 2008).
36
3.4.4 Event A-4: February 22, 2008
A widespread surface hoar event occurred on the night prior to February 22, 2008.
Well-defined 1–2 mm surface hoar was observed at the North site. Smaller crystals
were found at the South Station, which were growing on a “moist melt-freeze crust.”
Images from February 22 for both sites are included in Figure 3.8 as well as an image
of the surface hoar at the North Station the following day, after being buried under
a few centimeters of new snow. The weather data for the night surrounding the
formation is included in Figure 3.9.
(a) North: Feb. 22 (1 mm grid) (b) South: Feb. 22 (1 mm grid)
(c) North: Feb. 23 (3 mm grid)
Figure 3.8: Images from event A-4 of surface hoar captured from the (a) North and
(b) South Stations on February 22, 2008 and at the (c) North station the following
day after the surface hoar was buried by new snow.
37
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
40
50
60
70
80
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
200
205
210
215
220
225
230
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.3
0.6
0.9
1.2
1.5
1.8
2.1
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
1200 1800 2400 0600 1200
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
20
35
50
65
80
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(c) South
1200 1800 2400 0600 1200
300
350
400
450
500
550
600
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.4
0.8
1.2
1.6
2
2.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(d) South
Figure 3.9: Weather data for event A-4 (February 22, 2008) for both the (a,b) North
and (c,d) South weather stations.
38
3.4.5 Event A-5: February 26, 2008
Large surface hoar formed at both the North (4–8 mm) and South (2–4 mm)
Stations the night prior to February 26, 2008. According to the field notes the surface
hoar persisted at the North Station through the following few days, despite being
buried by a few centimeters of new snow. Examples of these crystals are shown in
Figure 3.10 and the weather data is provided in Figure 3.11.
(a) North: Feb. 26 (1 mm grid) (b) South: Feb. 26 (3 mm grid)
Figure 3.10: Images from event A-5 of surface hoar captured from the (a) North and
(b) South Stations on February 26, 2008.
39
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
60
68
76
84
92
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
200
220
240
260
280
300
320
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.4
0.6
0.8
1
1.2
1.4
1.6
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
50
60
70
80
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(c) South
1200 1800 2400 0600 1200
250
300
350
400
450
500
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.5
0.9
1.3
1.7
2.1
2.5
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(d) South
Figure 3.11: Weather data for event A-4 (February 26, 2008) for both the (a,b) North
and (c,d) South weather stations.
40
3.4.6 Event A-6: March 10, 2008
On March 10, 2008 small (1 mm) surface hoar, as shown in Figure 3.12, was
observed at both the North and South Stations. The weather conditions around
this event were similar to many of the prior events and characterized by the snow
temperature dropping upwards of 10 ◦C below the air temperature, see Figure 3.13.
The surface hoar on the North site persisted to the following day, as the field log
noted “0.5 mm decomposing surface hoar” on March 11.
(a) North (1 mm grid) (b) South (1 mm grid)
Figure 3.12: Images from event A-6 of surface hoar captured from the (a) North and
(b) South Stations on March 10, 2008.
41
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
45
50
55
60
65
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
200
220
240
260
280
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.6
0.9
1.2
1.5
1.8
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
35
41
47
53
59
65
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(c) South
1200 1800 2400 0600 1200
300
350
400
450
500
550
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
1
1.3
1.6
1.9
2.2
2.5
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(d) South
Figure 3.13: Weather data for event A-6 (March 10, 2008) for both the (a,b) North
and (c,d) South weather stations.
42
3.4.7 Event A-7: March 30, 2008
The final significant surface hoar event of the 2007/2008 season occurred on the
night prior to March 30, 2008, resulting in 1 mm and 0.3–0.5 mm crystals at the North
and South stations, respectively. Images from both locations are included in Figure
3.14. The weather data surrounding the event is included in Figure 3.15. The field
notes indicated that the crystals did not persist beyond the initial day of observation.
(a) North (1 mm grid) (b) South (1 mm grid)
Figure 3.14: Images from event A-7 of surface hoar captured from the (a) North and
(b) South Stations on March 30, 2008.
43
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
45
60
75
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
180
200
220
240
260
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.6
1.2
1.8
2.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
45
60
75
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(c) South
1200 1800 2400 0600 1200
250
300
350
400
450
500
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.5
1
1.5
2
2.5
3
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(d) South
Figure 3.15: Weather data for event A-7 (March 30, 2008) for both the (a,b) North
and (c,d) South weather stations.
44
3.5 2008/2009 surface hoar events
3.5.1 Event B-1: January 23, 2009
The first surface hoar event of the 2008/2009 season resulted in 0.5–4 mm crystals
developing the night between January 22 and 23, 2009, as shown in Figure 3.16.
The field notes indicated that the surface hoar was 0.5 mm in size. However, the
images indicated crystals as large as 4 mm. It is likely that these larger crystals were
not as widespread, but photographed preferentially due to their size. The weather
conditions from the North Station surrounding this event are provided in Figure 3.17.
Note, this event occurred on the first full day of fully operational weather stations,
thus the weather data that begins the previous day was not recorded.
Figure 3.16: Images from event B-1 of surface hoar (2 mm grid) captured from the
North Station on January 23, 2009.
45
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
70
75
80
85
90
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
200
220
240
260
280
300
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.28
0.56
0.84
1.12
1.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.17: North Station weather data for event B-1 (January 23, 2009).
3.5.2 Event B-2: January 30–31, 2009
Event B-2 was a wide-spread surface hoar event that resulted in crystals forming
at both the North and South Stations the night between January 29 and 30, 2009.
The weather data from the North Station surrounding the event is included in Figure
3.19. The field notes from the North Station indicated 0.5 mm facets; however, the
image in Figure 3.18 indicates that larger crystals existed. Interestingly, the images
from the South Station show much larger and more pronounced surface hoar crystals
than from the North, but only images from the South station were taken on this day.
Also, the weather data from the South Station surrounding the event was absent due
to a technical difficulty, thus the observation at this station cannot not be confirmed.
An example of the well-defined surface hoar crystals is provided in Figure 3.18.
The surface hoar on the North site persisted and likely became larger the following
night. The field notes for January 31 indicated 1 mm surface hoar. As shown in
Figure 3.18, the crystals were more pronounced than previous day. The images from
46
the South Station indicated that the surface hoar degraded, as it was not evident
from in the images on the January 31.
(a) North: Jan. 30 (2 mm grid) (b) South: Jan. 30 (1 mm grid)
(c) North: Jan. 31 (2 mm grid)
Figure 3.18: Images from event B-2 of surface hoar captured from the (a) North and
(b) South Stations on January 30, 2009 and the (c) North Station on January 31,
2009.
47
1200 1800 01/30 0600 1200 1800 01/31 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
45
60
75
90
R
el
at
iv
e 
hu
m
id
ity
 (%
)
Air temp. Snow temp. Humidity
1200 1800 01/30 0600 1200 1800 01/31 0600 1200
180
200
220
240
260
280
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
0
0.6
1.2
1.8
2.4
3
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
Figure 3.19: North Station weather data for event B-2 (January 30–31, 2009).
3.5.3 Event B-3: February 4, 2009
Large (5 mm) surface hoar developed at both the North and South Stations on
February 4, 2009 as shown in Figure 3.20. At the North Station the surface hoar
persisted to the following day (February 5), but was noticeably decomposing. No
evidence of the crystals were reported in the notes after the February 5. At the
South Station, the field notes indicated that facets were present, but it was unclear
if these crystals were near-surface facets or decomposed surface hoar. Figure 3.21
includes the weather data for both the North and South Stations surrounding this
48
event. Unfortunately, as with the previous event, the weather data for the South
Station was unavailable due to technical difficulties.
(a) North: Feb. 4 (2 mm grid) (b) South: Feb. 4 (2 mm grid)
(c) North: Feb. 5 (2 mm grid)
Figure 3.20: Images from event B-3 of surface hoar captured from the (a) North and
(b) South Stations on February 4 and (c) the North Station on February 5, 2009.
49
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
42.5
55
67.5
80
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
195
200
205
210
215
220
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.6
0.9
1.2
1.5
1.8
2.1
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.21: North Station weather data for event B-3 (February 4, 2008).
3.5.4 Event B-4: February 7–8, 2009
On February 7, 2009 1 mm surface hoar developed on new snow at the North
station (Figure 3.22). No evidence of surface hoar was reported at the South station.
On the following day, surface hoar was reported at both stations, 1 mm at the South
and 5 mm at the North (Figure 3.22). The surface hoar from the North Station
persisted through the February 8 and was visible underneath a few centimeters of
new snow on February 9. On February 10, after 14 cm of new snow, no evidence of
the layer was found in the snowpack. The weather data surrounding this event at the
South Station is included in Figure 3.23. The weather data from the North Station
is unavailable due to a battery failure during the event.
50
(a) North: Feb. 7 (1 mm grid) (b) South: Feb. 8 (1 mm grid)
(c) North: Feb. 8 (2 mm grid)
Figure 3.22: Images from event B-4 of surface hoar captured from the North and
South Stations on (a) February 7 and (b,c) 8, 2009.
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
30
38
46
54
62
70
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) South
1200 1800 2400 0600 1200
180
190
200
210
220
230
240
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.6
0.9
1.2
1.5
1.8
2.1
2.4
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) South
Figure 3.23: South Station weather data for event B-4 (February 7, 2009).
51
3.5.5 Event B-5: February 13–14, 2009
Surface hoar was reported at the North site on two consecutive days: February
13 and 14, 2009. On February 13, the surface hoar was reported as “[half] surface
hoar and [half] decomposing stellars” 1 mm in size. The following day, February 14,
the surface hoar was more pronounced and reported as 1.5 mm in size. Images from
both days are included in Figure 3.24 and the weather data from the North site is
included in Figure 3.25.
(a) North: Feb. 13 (1 mm grid) (b) North: Feb. 14 (1 mm grid)
Figure 3.24: Images from event B-5 of surface hoar captured from the North Station
on (a) February 13 and (b) 14, 2009.
52
1200 1800 02/13 0600 1200 1800 02/14 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
40
50
60
70
80
R
el
at
iv
e 
hu
m
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 02/13 0600 1200 1800 02/14 0600 1200
150
175
200
225
250
275
300
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
0
0.3
0.6
0.9
1.2
1.5
1.8
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.25: North Station weather data for event B-5 (February 13–14, 2009).
3.5.6 Event B-6: February 28, 2009
Surface hoar crystals 1.5 mm in size were reported at the North Station on Febru-
ary 28, 2009. These crystals persisted to the following day, but were reported to
be “decomposing” on March 1. Images of crystals from these events are included in
Figure 3.26 and the weather data surrounding the event is included in Figure 3.27.
53
(a) North: Feb. 28 (1 mm grid) (b) North: Mar. 01 (1 mm grid)
Figure 3.26: Images from event B-6 of surface hoar captured from the North Station
on February 28, 2009.
1200 1800 2400 0600 1200
−25
−20
−15
−10
−5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
60
65
70
75
80
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
160
180
200
220
240
260
280
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0
0.3
0.6
0.9
1.2
1.5
1.8
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.27: North Station weather data for event B-6 (February 28, 2009).
3.5.7 Event B-7: March 13, 2009
Small (0.5–1 mm) surface hoar, as shown in Figure 3.28, was reported at the
North weather station on March 13, 2009. This was the final surface hoar event for
the 2008/2009 season. The event only occurred at the North station and the surface
54
hoar did not persist beyond the observation date. The weather data surrounding this
event is provided in Figure 3.15.
Figure 3.28: Images from event B-7 of surface hoar captured from the North Station
on March 13, 2009.
1200 1800 2400 0600 1200
−20
−15
−10
−5
0
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
20
30
40
50
60
Re
la
tiv
e h
um
id
ity
 (%
)
Air temp. Snow temp. Humidity
(a) North
1200 1800 2400 0600 1200
200
250
300
350
400
450
500
Time
Irr
ad
ian
ce
 (W
/m
2 )
 
 
0.3
0.6
0.9
1.2
1.5
1.8
2.1
W
in
d 
sp
ee
d 
(m
/s)
Long−wave Wind speed
(b) North
Figure 3.29: Weather data for event B-7 (March 13, 2009) for the North Station.
55
3.6 Analysis
The data presented in this chapter was used to assess the weather conditions that
lead to the formation of surface hoar. First, the data from both stations was combined
into a single set. This was performed primarily due to the small number of surface
hoar events that occurred, which would make statistical analysis difficult if the values
were not combined. Next, the mean nightly averages from each event (Table 3.2) were
compared to the nightly averages from each day recorded during the two seasons at
both stations.
Histograms showing all the measured data, with the surface hoar events superim-
posed, are included Figure 3.30 and 3.31. The thickness of the histogram bar for the
North and South sites provides the frequency. For example, the bar above the 200
W/m2 tick mark in Figure 3.31 indicates that the North site includes 8 events and
the South 1 event. The height of the all data provides the actual frequency.
0 5 10
0
10
20
30
40
50
60
Air/snow temp. difference (◦C)
Fre
que
ncy
 
 
North
South
All data
Figure 3.30: Histogram comparing the daily average air/snow temperature difference
for the entire season (all data), along with the days associated with surface hoar
events at either the North or South Stations.
56
100 200 300 400 500
0
20
40
60
80
100
Long-wave (W/m2)
Fre
que
ncy
 
 North
South
All data
(a)
−25 −20 −15 −10 −5 0
0
10
20
30
40
50
60
70
Air temperature ( ◦ C)
Fre
que
ncy
 
 
North
South
All data
(b)
−30 −25 −20 −15 −10 −5
0
10
20
30
40
50
60
Snow temperature ( ◦ C)
Fre
que
ncy
 
 
North
South
All data
(c)
0 1 2 3 4
0
20
40
60
80
100
Wind speed (m/s)
Fre
que
ncy
 
 
North
South
All data
(d)
50 100 150 200 250 300
0
20
40
60
80
100
Wind direction (deg.)
Fre
que
ncy
 
 
North
South
All data
(e)
20 40 60 80
0
20
40
60
80
100
Relative humidi ty (%)
Fre
que
ncy
 
 
North
South
All data
(f)
Figure 3.31: Histograms comparing the frequency of recorded daily average weather
conditions at both the North and South Stations (all data), along with the days
associated with surface hoar events observed at the North or South Stations.
57
Using a Kolmogorov-Smirnov test (KS-test), the two distributions—all days and
event-only days (i.e., days when surface hoar was observed)—were compared (Massey,
1951). The test provided a means for determining if the two data sets were from the
same distribution. The results of these comparisons are provided in Table 3.4, where
the null hypothesis (H0) was that the two data sets were from the same distribution at
the 5% significance level. The hypothesis test stated that p-values less than 0.05 (i.e.,
5% significance level) would result in a failure to reject the null hypothesis, meaning
that the distributions were likely different.
Table 3.4: Kolmogorov-Smirnov test results comparing the distributions set shown in
Figures 3.31 and 3.30; the null hypothesis (H0) was that the data are from the same
distribution.
Figure Variable H0 result P -value
3.31a Long-wave reject 0.01
3.31b Air temperature fail to reject 0.84
3.31c Snow temperature reject 2.17× 10−3
3.31d Wind speed fail to reject 0.19
3.31e Wind direction fail to reject 0.94
3.31f Relative humidity reject 0.02
3.30 Air/snow temp. difference reject 1.28× 10−7
The results from the K-S test indicated that incoming long-wave radiation, snow
surface temperature, relative humidity, and air/snow temperature difference likely
originate from different distributions. Thus, it is assumed that these factors are
related to the formation of surface hoar. Surprisingly, considering the body of research
discussing its importance, wind speed was not one of the these factors. This is likely
an artifact of the weather station locations, which typically had wind speeds from
1–2 m/s (see Figure 3.31d). These values are within the range typically reported
as necessary for surface hoar formation (Linkletter and Warburton, 1976; Hachikubo
and Akitaya, 1997; Feick et al., 2007).
58
The event-only distributions deemed significant via the KS-test were used to assign
a range of “optimum” conditions for surface hoar development. Using the Bootstrap
Method (Efron and Tibshirani, 1993), the percentiles of the empirical distribution
function for each event-only result were developed and presented in Table 3.5. The
percentiles are a simple quantification of the distribution. For example, referring to
Table 3.5, the 10% percentile of long-wave radiation is 190 W/m2, which means that
90% of the observed events had a higher value of incoming long-wave radiation. If
a normal distribution is assumed, the percentiles are proportional to the probability,
i.e., the 50% percentile would be the most probable for the formation of surface hoar.
Also, assuming a normal distribution, 68.2% of the data fits between the 15.9 and
84.1 percentiles, which is one standard deviation.
Table 3.5: Percentiles of environmental variables coupled to the formation of surface
hoar.
Variable Units 10% 20% 30% 40% 50% 60% 70% 80% 90%
Long-wave W/m2 190 199 208 220 238 261 291 332 369
Snow temperature ◦C -22.0 -20.8 -19.1 -17.4 -16.2 -15.3 -14.5 -13.0 -10.9
Relative humidity % 45 53 57 61 65 70 73 77 81
Air/snow temp. difference ◦C 5.0 6.1 6.6 7.0 7.5 8.1 8.7 9.2 9.7
Cooperstein et al. (2004) detailed two surface hoar events that occurred at the
same locations used for the data presented in this chapter, where minimum snow
surface temperatures were reported as -15.1 and -14.2 ◦C for the North site and -11.1
and -12.5 ◦C at the South Station. Mean values for the night were not reported by
Cooperstein et al. (2004), but the minimum value in many of the events reported
in this chapter was representative of the mean (i.e., the snow surface temperature
remained relatively constant through the night). The values from the North fell near
the 70th and 80th percentile, which is within one standard deviation. Thus these
values match well with the events presented in this chapter. The South temperatures
59
lie between the 80th and 90th percentiles, which indicate that surface hoar formation
is less probable at the South site than at the North. These results are expected
for two reasons. First, if Figure 3.31c is examined, the snow temperatures from the
South events tended to occur at slightly warmer temperatures. Secondly, Cooperstein
et al. (2004) concluded that the surface hoar at the South Station was less developed
than at the North station, which indicated that the conditions are less favorable for
development.
A similar analysis to the above may be performed using two surface hoar events
recorded in Japan (Hachikubo and Akitaya, 1996), which identified two multi-day sur-
face hoar events. The average snow surface temperatures for the five nights reported
were one of three temperatures: -12, -14, and -16 ◦C. These values are within the
range defined in Table 3.5 and the -16 ◦C value reported corresponds with the most
probable temperature for surface hoar formation defined by the data set presented in
this chapter.
3.7 Future Considerations
The results presented in both this chapter as well as in Chapter 4, which covers
near-surface facets, show that surface hoar and near-surface facets often occur in con-
junction with each other. Nearly 60% of the dates summarized in Table 3.2 occurred
the night before or after a near-surface event reported in Table 4.2. For example, on
February 14, 2008 near-surface facets were observed (C-2) the day following 1 mm
surface hoar (A-2). Similarly, on March 10, 2009 surface hoar was observed (C-6)
followed by the observation of near-surface facets on the snow surface (A-6) that
same day. This relationship is also evident in one of the surface hoar events described
by Cooperstein et al. (2004) at the same locations. To the author’s knowledge, no
60
other investigation has shown this relationship. As such, it should be the topic of
future investigations.
3.8 Conclusion
Throughout two seasons, 2007/2008 and 2008/2009, 14 surface hoar events were
observed at north- and south-facing weather stations. Four parameters—incoming
long-wave radiation, snow surface temperature, relative humidity, and the air/snow
surface temperature difference—were shown to be the weather factors important to
the formation of surface hoar. The analysis of the data defined the following opti-
mum conditions for surface hoar formation for each of the aforementioned factors,
respectively: 190–270 W/m2, -22 to -11◦C, 45–80%, and 5-10 ◦C.
The ranges developed in this chapter may be used as one tool among many for
determining if surface hoar growth is likely based on weather data. However, the
data presented was developed from only two locations on the same mountain and
only for two seasons of data, thus increasing the data set would likely provide a more
reliable tool. This project is ongoing therefore the data set will be expanding. The
methodology presented herein may be easily applied, thus forecasting agencies with
reliable weather data and daily observations of the snow surface could perform a
similar analysis and build probability charts for their own region.
61
CHAPTER 4
FIELD INVESTIGATION OF NEAR-SURFACE FACETS
4.1 Introduction
Dry slab avalanches cause extensive property damage and fatalities each year
throughout the world. A majority of these avalanches slide on a weak layer that
was formed at or near the snow surface and were subsequently buried (Schweizer
and Lutschg, 2001; Birkeland, 1998). Specifically, near-surface facets—a layer that
forms at or near the snow surface due to temperature gradients—accounted for 59%
of avalanches in a case study in Southwest Montana (Birkeland, 1998), which is also
the location of the field investigation presented herein. Chapter 2 includes a review
of the of the body of literature discussing the conditions leading to near-surface facet
development.
4.2 Methods
A discussion of the methods used for this investigation are provided in the methods
section of Chapter 3 and a complete discussion of the instrumentation is included in
Appendix A. for the sake of brevity, the details are not repeated. However, it should
be noted that Chapter 3 is a discussion of surface hoar that almost exclusively develops
at night. Thus, the short-wave radiation was not considered. As short-wave radiation
is crucial to the formation of near-surface facets, the focus of this chapter, two items
related to the short-wave radiation need to be mentioned: solar contamination of
long-wave radiation and short-wave radiation sensor orientation.
Long-wave radiation data from the 2007/2008 season at the South Station is con-
sidered unreliable due to solar contamination—a problem discussed by Albrecht and
62
Cox (1977)—associated with the instruments used, the Eppley Lab Inc., PIR. The
PIR sensor measures the incoming long-wave radiation, but the protective dome and
the case of the instrument also contribute to this value when their temperatures
differ from that of the sensor itself. Generally, correcting for the case temperature, as
was done, is adequate (PIR, 2007). However, in certain applications of intense solar
(short-wave) radiation the dome temperature must also be adjusted, this is the situa-
tion at the South station (Albrecht and Cox, 1977). The sensors were mounted slope
parallel, hence nearly in-line with the solar zenith angle. This problem was corrected
during the 2008/2009 season by upgrading to Kipp and Zonen CGR3 instruments
that account for this problem.
Slope parallel orientation is also used for the upward- and downward-facing short-
wave sensors. Therefore, the sensors are measuring both incoming as well as reflected
radiation from the surrounding environment (i.e., albedo), which includes snow from
the slope below the research site. Slope parallel orientation is desired at the stations
as the sensor is intended to measure the radiation actually impacting the snow, but
causes the recorded values to be higher than expected. For comparison, peak irra-
diation values at the South Station often exceed over 1200 W/m2. Values of this
magnitude are within the range of values expected based on ASTM G-173, which
reports an average value of 1000 W/m2 for terrestrial direct radiation (ASTM G-173,
2003) at a 37◦ latitude (i.e., average for the continental United States) as well as the
solar standard of 1367 W/m2.
63
4.3 Results
The 2007/2008 (C) and 2008/2009 (D)1 seasons include detailed weather data,
observations, and images of 26 near-surface facet events. Though weather data was
collected prior to 2007, the results are excluded here due to the lack of snow crystal
images. All except three of these events (C-3, C-7, and D-5), as noted in the field
notes, were likely dominated by radiation processes. The results presented here does
not differentiate these events since it was not possible to determine if these events
were not due to radiation. In many cases, the events have before and after images and
observations showing facet development in a manner of hours. Table 4.1 summarizes
the type of snow that existed prior to the formation of facets, as reported in the field
notes.
Each event of near-surface facets that occurred is summarized in Sections 4.4 and
4.5. The summaries presented attempt to provide a brief but thorough overview
of each event, with minimal interpretation. Images included in this document are
uncropped and selected as a representative of all the images taken on each day.
The event summaries for the 2007/2008 season (C) utilize long- and short-wave
data from the Aspirit Station. As mentioned in Section 4.2, the long-wave sensors
at the station were unreliable during this season. Thus, for consistency among the
reported long- and short-wave radiation values, unless otherwise noted data from the
Aspirit Station was used throughout the summaries for the 2007/2008 season.
The summaries for the 2008/2009 season (D) utilize the radiation data at the
station itself, thus the values mentioned in the summaries are not comparable between
the seasons. However, Table 4.2 provides the complete data set for both stations and
1The C and D notations are used to differentiate event references from Chapter 4.
64
seasons, allowing for a comparison to be made. This table includes daily mean values
at both the South and Aspirit Stations for each event including short- and long-wave
radiation, snow surface and air temperature, wind speed and direction, and relative
humidity. The mean values for all parameters were calculated for the duration in
which short-wave radiation was greater than zero.
Table 4.1: Summary of snow conditions prior to the near-surface facet events as
recorded in the field notes. Events tagged with an asterisk (*) indicate events, as
noted in the field notes, that were likely dominated by non-radiation processes.
Event Event Date Description of Snow
C-1 1/21/2008 rimed stellars and plates 0.2–0.5 mm
C-2 2/14/2008 new snow, stellars rimed 2 mm, plates 1 mm
C-3* 2/18/2008 rimed stellars
C-4 2/26/2008 2–3 mm stellar dendrites, some heavily rimed, some not at all
C-5 3/6/2008 1 mm rimed stellars, 2 mm stellars
C-6 3/10/2008 highly broken 0.25 mm (dry)
C-7* 3/13/2008 2 mm stellars, 1 mm decomposing stellars, 1 mm facets
C-8 3/15/2008 stellars rimed 1–2 mm
C-9 3/19/2008 rimed new snow, 1–2 mm
C-10 3/22/2008 1 mm plates, columns, capped columns, stellars
C-11 3/28/2008 decomposing rimed new snow
C-12 3/30/2008 highly broken new snow, 0.5 mm
C-13 4/2/2008 rimed stellars, 1 mm
C-14 4/6/2008 new snow, rimed irregular grains, 1 mm
C-15 4/8/2008 new snow, 1mm
D-1 2/4/2009 1–2 mm graupel
D-2 2/8/2009 1.5 mm new snow
D-3 2/12/2009 1 mm new snow
D-4 2/19/2009 1–2 mm new snow
D-5* 2/21/2009 1 mm stellars
D-6 2/27/2009 0.5–3 mm new snow
D-7 3/7/2009 1–2 mm new snow
D-8 3/12/2009 0.5–1 mm decomposing snow and 0.1–0.3 mm surface hoar
D-9 3/20/2009 1 mm graupel
D-10 3/30/2009 1–2 mm new snow
D-11 4/6/2009 0.5–3 mm new snow and some surface hoar
65
Table 4.2: Summary of mean daily weather conditions for all days recorded as near-
surface facets events, including short-wave (SW ) and long-wave (LW ) radiation, air
(Ta) and snow surface (Ts) temperature, relative humidity (RH), and wind speed (Vw)
and direction (Dir). The superscript a denotes Aspirit station. Events tagged with
an asterisk (*) indicate events that were likely dominated by non-radiation processes.
Event Date SW LW Ta Ts Vw Dir RH LWSW SW a LW a LW
a
SWaW/m2 W/m2 ◦C ◦C m/s deg. % W/m2 W/m2
C-1 01/21/2008 607 340 -16.1 -16.6 0.7 157 68 1.79 170 166 1.03
C-2 02/14/2008 547 414 -7.2 -9.2 1.0 187 58 1.32 292 190 1.54
C-3a* 02/18/2008 593 479 -2.0 -5.4 1.4 232 45 1.24 291 202 1.44
C-3b* 02/19/2008 680 558 4.3 -3.2 1.3 151 20 1.22 318 192 1.65
C-3c* 02/20/2008 664 549 3.4 -3.2 1.2 209 21 1.21 310 197 1.57
C-4 02/26/2008 675 440 -4.0 -6.5 1.9 150 60 1.53 383 196 1.95
C-5 03/06/2008 626 432 -6.6 -7.9 1.6 216 63 1.45 413 194 2.13
C-6 03/10/2008 619 484 1.0 -4.2 1.6 154 43 1.28 404 218 1.85
C-7* 03/13/2008 262 348 -4.9 -5.8 1.1 207 68 0.75 221 265 0.84
C-8 03/15/2008 506 403 -6.3 -7.8 1.5 165 60 1.25 375 221 1.70
C-9 03/19/2008 416 388 -3.9 -7.8 2.1 146 52 1.07 306 230 1.33
C-10 03/22/2008 669 458 -6.7 -10.2 1.4 177 47 1.46 423 171 2.47
C-11 03/28/2008 499 380 -9.5 -12.8 1.9 171 50 1.31 365 207 1.77
C-12 03/30/2008 614 434 -8.6 -11.7 1.5 198 43 1.41 441 185 2.39
C-13a 04/02/2008 508 400 -6.8 -10.2 1.9 163 48 1.27 383 213 1.80
C-13b 04/03/2008 647 455 -4.0 -7.8 1.9 154 44 1.42 487 185 2.64
C-13c 04/04/2008 356 393 -2.1 -6.1 2.3 150 38 0.90 286 243 1.17
C-14 04/06/2008 506 386 -5.2 -7.3 1.9 156 58 1.31 399 229 1.75
C-15 04/08/2008 564 396 -5.4 -7.4 1.6 136 60 1.43 448 220 2.03
D-1 02/04/2009 706 230 5.7 -0.9 2.0 122 24 3.07 306 – –
D-2 02/08/2009 504 240 -1.0 -3.9 1.2 156 51 2.10 211 224 0.94
D-3 02/12/2009 691 224 -7.6 -8.9 1.2 121 56 3.08 208 172 1.21
D-4 02/19/2009 608 220 -5.8 -7.5 2.0 101 64 2.76 313 202 1.55
D-5* 02/21/2009 687 215 -0.2 -5.2 1.9 107 37 3.20 334 201 1.67
D-6a 02/27/2009 516 229 -9.8 -9.7 1.5 111 65 2.26 363 190 1.92
D-6b 02/28/2009 711 197 -2.1 -7.0 1.2 131 31 3.61 372 180 2.07
D-7 03/07/2009 685 221 -7.2 -9.3 2.0 106 57 3.09 399 197 2.02
D-8a 03/12/2009 704 194 -4.6 -8.6 1.3 163 33 3.62 438 180 2.43
D-8b 03/13/2009 640 222 0.0 -5.7 1.8 126 35 2.89 416 211 1.97
D-8c 03/14/2009 661 222 -1.0 -5.5 2.3 124 34 2.98 422 211 2.00
D-9 03/20/2009 690 238 3.7 -2.1 1.9 107 48 2.90 337 213 1.58
D-10 03/30/2009 488 247 -8.5 -9.2 1.2 150 66 1.98 436 221 1.97
D-11 04/06/2009 739 218 4.4 -3.8 1.7 97 24 3.39 578 204 2.83
66
4.4 2007/2008 Near-surface Facet Events
4.4.1 Event C-1: January 21, 2008
The first event (C-1) of the 2007/2008 season was reported as both a surface
hoar and near-surface facet event on January 22, 2008. The surface was described as
“surface hoar 4–6 mm” and the subsurface at 1, 2, and 3 cm depth was described as
facets with broken stellars still slightly visible with a grain size of less than 0.5 mm.
The observations occurred at 0900, thus it is likely that the facets formed the day
prior (Jan. 21). No images were taken of this event.
The air and snow surface temperatures as well as the incoming short- and long-
wave radiation from Aspirit for this event are included in Figure 4.1. New snow was
reported on Jan. 20 and then the weather cleared. On the 21st, short-wave peaked
at 340 W/m2 and long-wave was approximately 170 W/m2 throughout the daylight
hours. During daylight on Jan. 21 the snow was also warmer than the air, confirming
the absorption of significant short-wave radiation. It is assumed that facets formed
during the sunny conditions and persisted, and perhaps grew, until the observation
the following day.
4.4.2 Event C-2: February 14–16, 2008
Event C-2 was a considerable and wide-spread near-surface facet event and has
been the subject of additional analysis (McCabe et al., 2008; Slaughter et al., 2008,
2009). Initial observations occurred at 1100 on the South Station. It was reported
that evidence of minimal amounts of surface hoar existed. The YC ski patrol made
additional observations at 1400 showing additional needle-like growth. Figures 4.2a–
4.2c include before and after images taken at the South Station as well as images
67
01/21 0600 1200 1800 01/22 0600 1200 1800 01/23
0
100
200
300
400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−40
−30
−20
−10
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.1: Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station for January 21–22, 2008
(C-1).
taken from the Pinnacles area, which is a near-by slope with an elevation of 2800 m,
aspect of 182◦, and slope angle of 28–38◦.
The formation of these crystals occurred with air temperatures that rose between
600 and 1400 from -17 ◦C to -4 ◦C and snow surface temperatures that increased
from -25 ◦C to -4 ◦C, as shown in Figure 4.3. The facets formed in new snow that
fell the previous day; the density was reported as 20 kg/m3. This warming was more
pronounced in the subsurface, and it was reported the snow between 1 cm and 5
cm was moist. Thus, the temperature gradient in the upper centimeter of snow was
approximately 400 ◦C/m. Figure 4.3 confirms that the sky was clear: long-wave
radiation was only 160 W/m2 and short-wave peaked at 575 W/m2.
On Feb. 15 facets were observed again in the surface layer at 1100, as shown
in Figure 4.4. The radiative conditions were nearly identical on Feb. 14 and 15;
however, the air temperature on Feb. 15 increased to 2 ◦C (see Figure 4.3). No
68
secondary observations were made on Feb. 15. The facets persisted despite being
buried on Feb. 16 by 4–5 cm of new snow, as recorded in the field notes: “facets are
still visible on the upper crust interface” (see Figure 4.4). No mention of the layer of
facets was recorded after this day.
(a) South at 1100 (2 mm grid) (b) South at 1400 (2 mm grid)
(c) Pinnacles (2 mm grid) (d) South at 1100 on Feb. 15 (2 mm
grid)
Figure 4.2: Four images of a near-surface facet event at the South Station on Febru-
ary 14, 2008 (C-2): (a) initial observation (1100) at the South Station, (b) second
observation (1400) at the South Station, (c) observations at a near-by south facing
slope, and (d) following day (Feb. 15) South Station observation (1100).
69
02/14 0800 1600 02/15 0800 1600 02/16 0800 1600 02/17
0
100
200
300
400
500
600
700
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−30
−25
−20
−15
−10
−5
0
5
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.3: Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station for February 14–16, 2008
(C-3).
Figure 4.4: Facets formed on February 14, 2008 at the South Station that persisted
through warmer temperatures and new snow until February 16th.
4.4.3 Event C-3: February 18–20, 2008
Event C-3 consisted of an observation of small (0.5 mm) facets in the surface
layer at 0924 on February 18, 2008 followed by two days (Feb. 19 and 20) in which
the field notes indicated the presence of facets underlying surface hoar crystals. The
70
facets were described as “spaghetti of diurnal-recrystallization chains.” Figure 4.5
includes an image from each of these three days.
The recorded weather conditions indicated drastic changes in the snow surface
temperature (see Figure 4.6). During these three days the snow surface temperature
changed an average of 13 ◦C between daylight and night, with the biggest change
being a 17 ◦C increase from night to daylight on Feb. 18. The changes occurred in a
matter of hours.
(a) Feb. 18 (2 mm grid) (b) Feb. 19 (1 mm grid)
(c) Feb. 20 (1 mm grid)
Figure 4.5: Images of near-surface facets at the South station described as diurnal
recrystallization., that formed on February 18–20, 2008 (C-3).
71
02/17 1200 02/18 1200 02/19 1200 02/20 1200 02/21
−25
−20
−15
−10
−5
0
5
10
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 
Surface Temperature
Air Temperature
Figure 4.6: Graph of air temperature and snow surface temperature at the South
Station on February 17–20, 2008 (C-3).
4.4.4 Event C-4: February 26–27, 2008
The field notes explained that “1 mm columns [and] needles” formed on February
27, 2008 (C-4). New snow (10 inches) fell two days prior (Feb. 25), surface hoar
formed the following day, and distinct facets were then found on Feb. 27. These
facets persisted for the following two days. Figure 4.7 includes images of the preceding
surface hoar and subsequent near-surface facets.
Figure 4.8 is a graph of the radiation and temperature data for the days surround-
ing the C-4 event. The near-surface facets likely formed during the day on Feb. 26,
were reported on Feb. 27, and persisted for several overcast days after formation.
The observation on Feb. 26 was made at 0830 and showed significant surface hoar;
however, the short-wave was high (640 W/m2 at Aspirit at mid-day) and the incoming
long-wave was less than 200 W/m2 throughout the day. The following day (Feb. 27)
short-wave was much lower (490 W/m2) and long-wave was higher (250–290 W/m2).
Thus, it is possible that the near-surface facets formed on Feb. 26.
72
An examination of the snow temperature data from the thermocouple array placed
in the snow provided further evidence that the facets may have formed on Feb. 26.
Figure 4.9 includes measured temperature profiles during the daylight hours on Feb.
26, which indicate subsurface heating and surface cooling. Note, the data was suspect
due to solar contamination of the thermocouples (above freezing snow temperature),
however the general trend of the temperature measurements indicated a temperature
gradient near the snow surface with the subsurface warmer than the surface. On Feb.
27 temperature profiles indicate that the snow surface was the warmest portion of
the snowpack; however these measurements were invalid due to melting around the
thermocouple array. On Feb. 26 three thermocouples were exposed to the air and
the following day 10 were exposed. Thus, melting of the snow caused 14 cm of the
array to become exposed over 24 hours.
73
(a) Feb. 26 (3 mm grid) (b) Feb. 27 (1 mm grid)
(c) Feb. 28 (1 mm grid)
Figure 4.7: Images taken at the South Station of (a) surface hoar formed the day prior
to the (b) near-surface facets that formed on February 27, 2008 (C-4) and persisted
through (c) the following day.
74
02/26 0800 1600 02/27 0800 1600 02/28 0800 1600 02/29
0
100
200
300
400
500
600
700
800
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−24
−20
−16
−12
−8
−4
0
4
8
Te
m
pe
ra
tu
re
 (°  
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.8: Recorded short- and long-wave radiation at the Aspirit Station as well as
the air and snow surface temperatures at the South Station on February 26–28, 2008
(C-4).
−10 −8 −6 −4 −2 0 2 4 6 8
0
5
10
15
20
25
30
35
40
Temperature (°C)
D
ep
th
 (c
m)
 
 
0800
1800 1600
1400 1000
1200
Figure 4.9: Snow temperature profiles from February 26, 2008 (C-4) at the South
Station. The horizontal line represents the snow surface.
75
4.4.5 Event C-5: March 6, 2008
Observations at the South Station were made at 1100 and 1330 on March 6, 2008.
The initial observation reported the “surface snow [was] composed of 2–3 mm stellars
and stellar fragments throughout the top ten centimeters. . . no surface hoar or other
faceting was present.” In the second observation the field report stated that “small
cups and needles were observed in the cold snow on the surface.” Figure 4.10 includes
images from both observations. This event has been discussed and analyzed in prior
research (McCabe et al., 2008; Slaughter et al., 2008, 2009).
On Mar. 6 the short-wave radiation at Aspirit peaked at 690 W/m2 and long-wave
radiation remained constant at approximately 200 W/m2 throughout the day. Air
temperature increased from -15 ◦C at sunrise to -4 ◦C during the daylight. This event
occurred after five consecutive days of snow totaling 25 mm of snow water equivalent
and a depth of 30 cm.
(a) 1100 (2 mm grid) (b) 1330 (2 mm grid)
Figure 4.10: Images from March 6, 2008 (C-5) near-surface facet event at the South
Station: (a) initial observation at 1100 and (b) second observation at 1330.
4.4.6 Event C-6: March 10, 2008
Near-surface facets and surface hoar were observed on March 10, 2008 (C-6). The
faceted crystals were “0.5 mm to 1 mm in size and mostly rectangular in shape; some
76
chaining [was] observed.” Chaining refers to the facets being arranged in long, narrow
chains. The field notes also reported that distinct surface hoar grains were visible.
Figure 4.11 is an image of near-surface facets located at the snow surface. The facets
were reported to persist until Mar. 12 on the surface; the field notes indicated facets
on the surface “from [Mar. 10] surface hoar event.”
The air temperature was mild on Mar. 10, rising to 5 ◦C. The snow surface
temperature reached 0 ◦C for a few hours at mid-day. The short- and long-wave
radiation at Aspirit were similar to the previous event: short-wave peaked at 660
W/m2 and long-wave was approximately 220 W/m2 throughout the day.
Figure 4.11: Image of near-surface facets observed on March 10, 2008 (C-6) at the
South Station.
77
04/02 0800 1600 04/03 0800 1600 04/04 0800 1600 04/05
0
150
300
450
600
750
900
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
5
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.12: Recorded short- and long-wave radiation at the Aspirit station (C-6)
as well as the air and snow surface temperatures at the South station for March 10,
2008.
4.4.7 Event C-7: March 13, 2008
A layer of 0.5 mm facets was observed 4–5 cm below the surface on March 13,
2008 (C-7). This layer was between two crusts and the facets were described as
“small and relatively round, but very [non-cohesive].” The weather data, Figure 4.13,
was not as conducive to radiation-recrystallization when compared to the previous
event described. Long-wave radiation was 280 W/m2 during the day and short-wave
peaked at only 350 W/m2. The prior days conditions were similar to other events,
but no indication of facets was recorded at the 1245 observation. Thus, the resulting
facets may have formed due to the diurnal temperature change (the snow surface
temperature changed over 10 ◦C between day and night) or melt-layer recrystallization
as indicated by the presence of crusts surrounding the layer.
78
03/12 0600 1200 1800 03/13 0600 1200 1800 03/14
0
200
400
600
800
1000
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−15
−12
−9
−6
−3
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.13: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March 12–13, 2008
(C-7).
4.4.8 Event C-8: March 15, 2008
Event C-8 was marked by an observation of small (0.5 mm) facets (Figure 4.14a)
above a crust that was 2 cm below the surface at 1045 and 1400 on March 15,
2008. Approximately 8 cm of new snow fell the day prior (Mar. 14). The short-
wave radiation at Aspirit peaked at 770 W/m2 and long-wave was approximately 210
W/m2 throughout the day. The facets persisted (Figures 4.14b and 4.14c) through
the following two days despite being covered by approximately 3 cm of new snow.
The radiation and temperature data for March 15–17 is included in Figure 4.15.
79
(a) Mar. 15 (3 mm grid) (b) Mar. 16 (2 mm grid)
(c) Mar. 17 (2 mm grid)
Figure 4.14: Images taken at the South Station during the (a) March 15, 2008 (C-8)
near-surface facet event; this layer persisted the following two days (b and c).
4.4.9 Event C-9: March 19, 2008
Small (0.25–0.5 mm) facets were observed at 1330 on March 19, 2008 (C-9). These
facets formed in new snow under cool, clear conditions. Short-wave radiation at
Aspirit peaked at 550 W/m2 and long-wave radiation increased from 166 W/m2 at
sunrise to 275 W/m2 at sunset. Air temperatures reached -2 ◦C at mid-day; the
snow surface peaked at -4 ◦C. No images were taken of this event. The radiation and
temperature data for the event are included in Figure 4.16.
80
0
200
400
600
800
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
03/15 0800 1600 03/16 0800 1600 03/17 0800 1600 03/18
−20
−15
−10
−5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.15: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March 15–17, 2008
(C-8).
03/19 0300 0600 0900 1200 1500 1800 2100 03/20
0
150
300
450
600
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−20
−15
−10
−5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.16: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March 19, 2008
(C-9).
81
4.4.10 Event C-10: March 22, 2008
The conditions of Event C-10 on March 22, 2008 were very similar to Event C-2 on
February 14–16, 2008 (Section 4.4.2). Widespread 0.3–1.5 mm facets were found at
the South Station on March 22, 2008 and persisted until Mar. 25. The facets formed
in new snow that fell two days prior. Images of the facets observed are included in
Figure 4.17.
On Mar. 22 both the snow surface and air temperature reached -4 ◦C, short-
wave at Aspirit peaked at 800 W/m2, and long-wave was between 160 W/m2 and
190 W/m2. On subsequent days following the event, air temperatures and long-wave
radiation increased, while short-wave decreased compared to the event day. The
field notes on Mar. 25 indicated that new snow had fallen but the facets were still
observable beneath the new snow. The temperature and radiation data for Mar.
22–24 is included in Figure 4.18.
(a) Mar. 22 (2 mm grid) (b) Mar. 23 (2 mm grid)
Figure 4.17: Images from the (a) March 22, 2008 (C-10) near-surface facet event at
the South Station and (b) facets that persisted through the following day.
82
03/22 0800 1600 03/23 0800 1600 03/24 0800 1600 03/25
0
200
400
600
800
1000
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−20
−15
−10
−5
0
5
Te
m
pe
ra
tu
re
 (°  
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.18: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station for March 22–24, 2008
(C-10).
4.4.11 Event C-11: March 28, 2008
Event C-11 occurred on March 28, 2008, facets measuring 0.5 mm were observed.
The field notes indicated difficulty deciphering if the observed crystals were surface
hoar or near-surface facets; however, the field notes also stated “it did appear that the
[facets were] slightly subsurface.” No images were taken on this day. The weather
data, Figure 4.19, showed that the conditions were similar to the previously men-
tioned events: short-wave at Aspirit peaked at 630 W/m2 and long-wave ranged from
150 W/m2 to 250 W/m2 throughout the day. The increase in long-wave radiation
throughout the day may be why only a “small amount” of faceting was observed.
4.4.12 Event C-12: March 30, 2008
Event C-12 on March 30, 2008 was similar to many of the previously mentioned
events: short-wave at Aspirit peaked at 760 W/m2 and long-wave remained below
83
03/28 0300 0600 0900 1200 1500 1800 2100 03/29
0
150
300
450
600
750
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−30
−25
−20
−15
−10
−5
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.19: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station, on March 28, 2008
(C-11).
200 W/m2 during the entire day. Air temperature increased from -15 ◦C at night to
-5 ◦C at midday. The radiation and temperature data for Mar. 30 are included in
Figure 4.20 and an image of the crystals observed is shown in Figure 4.21. The facets
formed in 3 cm of new snow that fell on the night between Mar. 29 and 30. The field
notes stated that the surface snow “was mostly 1 mm facets with some forms that
appeared to be surface hoar.”
84
03/30 0300 0600 0900 1200 1500 1800 2100 03/31
0
200
400
600
800
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−30
−22.5
−15
−7.5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.20: Recorded short- and long-wave radiation at the Aspirit Station as well
as the air and snow surface temperatures at the South Station on March 30, 2008
(C-12).
Figure 4.21: Image of near-surface facets formed on March 30, 2008 (C-12) at the
South Station.
4.4.13 Event C-13: April 2–4, 2008
Facets were observed at the South Station on three consecutive days in early April,
2008 (C-13). At 1200 on Apr. 2 facets measuring 0.5–1 mm were observed in the
upper 1 cm of the snowpack. The following day (Apr. 3) small facets were observed
85
at 1130 and at 1430; the field notes indicated that “additional facet growth” had
occurred. Also, only the top 1 cm of the snow remained frozen. Figure 4.22 includes
images from the two observations made on Apr. 3. On Apr. 4, well-developed facets
were observed at the surface; they were described as “needles and sheath, striated
cups.” Additional faceting was observed 3 cm below the surface between melt-freeze
crusts.
The radiation and temperature weather data for April 2–4, 2008 is displayed in
Figure 4.23. The conditions for both Apr. 2 and 3 are similar to many of the events
discussed in this chapter: high short-wave peaks and long-wave radiation near 200
W/m2. On Apr. 4 conditions change: short-wave radiation decreased to a peak of
600 W/m2 and long-wave radiation increased to 280 W/m2 by sunset.
(a) 1130 (1 mm grid) (b) 1430 (1 mm grid)
Figure 4.22: Images of surface snow at (a) 1130 and (b) 1430 at the South Station
on April 3, 2008 (C-13) showing formation of near-surface facets .
4.4.14 Event C-14: April 6, 2008
Event C-14 occurred April 6, 2008. The initial (1030) observation at the South
Station reported 1 mm surface hoar and 2–3 mm stellars; at 1430 facets measuring 1
mm were reported. The field notes state that “it was clear that the observed snow
86
04/02 0800 1600 04/03 0800 1600 04/04 0800 1600 04/05
0
150
300
450
600
750
900
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
5
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.23: Recorded short- and long-wave radiation at the Aspirit Station, as well
as the air and snow surface temperatures at the South Station, on April 2–4, 2008
(C-13).
was more faceted in the [afternoon] than in the [morning].” The facets that formed
during the day were described as “champagne glass facets.” Figure 4.24 includes
images from both observations.
The temperature and radiation data are provided in Figure 4.25. The day prior
was overcast as indicated by the low incoming short-wave radiation and light snow
was reported in the field notes. The night prior to the event, the skies cleared and
some surface hoar formed, this is indicative of the large gradient between air and snow
temperatures. The near-surface facets formed during the day under similar conditions
as many of the other events described: incoming short-wave radiation peaked at 837
W/m2 and long-wave radiation remained near 230 W/m2 throughout the day.
87
(a) 1030 (1 mm grid) (b) 1430 (1 mm grid)
Figure 4.24: Images of surface snow at (a) 1030 and (b) 1430 on April 6, 2008 (C-14)
showing formation of near-surface facets at the South Station.
04/05 0600 1200 1800 04/06 0600 1200 1800 04/07
0
200
400
600
800
1000
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.25: Recorded short- and long-wave radiation at the Aspirit station, as well as
the air and snow surface temperatures at the South Station, on April 6, 2008 (C-14).
4.4.15 Event C-15: April 8, 2008
The final event (C-15) recorded during the 2007/2008 occurred on April 8, 2008 at
the South Station. Again the conditions were similar to other events described: days
were characterized by high incoming short-wave radiation (peaked at 800 W/m2) and
88
long-wave radiation (average around 220 W/m2 throughout the day). The field notes
indicated that 10 cm of new snow fell the night prior to the event; this is noticeable in
the weather data presented in Figure 4.26, which shows a jump in incoming long-wave
radiation on Apr. 9.
The actual event was not recorded in the notes until Apr. 9, the entry stated that
“at [0930] pictures were taken of facets in [the] 3 mm layer on top of [a] melt freeze
crust; presumably, these facets formed during yesterday’s clear skies in a radiation
process.” These facets persisted through the following day and were buried by new
snow that began falling at 1200, as stated in the notes: “at [1400] 5 mm of new snow
had fallen on this [faceted] layer, which appears to have survived today’s radiation,
which lasted until noon or so. In the photos, the facets are still observable underneath
the newly fallen stellar crystals.” Photographs of the facets taken on Apr. 9 are
included in Figure 4.27.
04/08 0600 1200 1800 04/09 0600 1200 1800 04/10
0
200
400
600
800
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−20
−15
−10
−5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave Short−wave Surface Temp. Air Temp.
Figure 4.26: Recorded short- and long-wave radiation at the Aspirit station as well
as the air and snow surface temperatures at the South station for April 8–9, 2008.
89
Figure 4.27: Image near-surface facets formed on April 8, 2008 (C-15). Photo was
taken on April 9 after buried by new snow, the scale is unknown.
4.5 2008/2009 Near-surface Facet Events
4.5.1 Event D-1: February 4, 2009
The first recorded event (D-1) of the 2008/2009 season occurred on February 4,
2009 at the South station. The field notes reported 0.5 mm facets at the surface,
as shown in Figure 4.28a. Due to a malfunction with the weather station instru-
mentation, the weather data for this event does not begin until 1200 on Feb. 4, see
Figure 4.29. Nonetheless, the data showed that short-wave radiation peaked near
1090 W/m2 at the South Station and 520 W/m2 at Aspirit. The long-wave radiation
averaged 230 W/m2 throughout the day at the South Station. Air temperatures were
well above freezing (6.7 ◦C) and the snow surface was 0◦C at midday.
The facets persisted through the night and were observed the following day (Feb.
5). However, as shown in Figure 4.28b, they appeared to be decomposing. The field
notes indicated that subsurface melting occurred beneath the thin layer of facets on
the snow surface.
90
(a) Feb. 4 (2 mm grid) (b) Feb. 5 (2 mm grid)
Figure 4.28: Image of near-surface facets formed (a) at the South Station on February
4, 2009 (D-1) ; the facets persisted through the night and were also observed (b) on
Feb. 5.
1200 1400 1600 1800 2000 2200 02/05
0
200
400
600
800
1000
1200
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−20
−15
−10
−5
0
5
10
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.29: Recorded weather data from the South Station on February 4, 2009
(D-1); due to instrumentation malfunctions the data prior to 1200 on Feb. 4 was not
recorded.
4.5.2 Event D-2: February 8, 2009
An initial observation of the surface snow was conducted at 1045 on February
8, 2009 (D-2) at the South Station, which indicated widespread faceting. A second
91
observation was made at 1300 that “found additional faceting.” The facets persisted
until the following day (Feb. 9), despite being buried by 2–3 cm of new snow. Images
from both observations on Feb. 8 as well as those taken on the following day, are
included in Figure 4.30. The weather conditions for Event D-2 were similar to the
previous event, only slightly cooler as shown in Figure 4.31. This event was per-
haps one of the most obvious of all events, the images show facets that were easily
distinguishable and were present in a variety of forms.
(a) 1045 Feb. 8 (b) 1300 Feb. 8 (#1)
(c) 1300 Feb. 8 (#2) (d) 1130 Feb. 9
Figure 4.30: Images of a near-surface facet event that occurred on February 8, 2009
(D-2) at the South Station. Images include facets observed (a) at the initial obser-
vation, (b and c) at the second observation, and (d) the following day despite being
buried underneath new snow.
92
02/08 0300 0600 0900 1200 1500 1800 2100 02/09
0
250
500
750
1000
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−20
−15
−10
−5
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.31: Recorded weather data from the South Station on February 8, 2009
(D-2).
4.5.3 Event D-3: February 12–14, 2008
Event D-3 occurred on February 12, 2009, observations showed that new snow
became faceted in a manner of hours. The initial observation at 1015 reported 1–2 mm
of new snow (Figure 4.32a). At 1245 facets measuring 1 mm were observed (Figure
4.32b) at the snow surface with a moist layer between 2 and 5 cm. Facets at the surface
were observed for the following two days (Figures 4.32c and 4.32d). The field notes
on Feb. 14 stated, “[the facets] look larger than yesterday, but not as many striations
are noted [and they] don’t seem to be standard [radiation recrystallized near-surface
facets].” Hence, it is unknown if the facets observed on the days following the initial
event formed during the day, night, or both.
Weather data, including snow and air temperatures as well as long- and short-
wave radiation, is provided in Figure 4.33. The conditions for each day were similar
to many other events described herein: high incoming short-wave radiation and long-
wave radiation near 200 W/m2.
93
On the night of Feb. 14 new snow (9 cm) was recorded. The following day
(Feb. 15) at 0845 the facets detailed in Event D-3 were intact underneath this layer,
as shown in Figure 4.34a. At 1300 the buried facets were also observed, although
significant decomposition (Figure 4.34b).
(a) 1015 Feb. 12 (1 mm grid) (b) 1245 Feb.12(1 mm grid)
(c) Feb. 13 (1 mm grid) (d) 1245 Feb.14(2 mm grid)
Figure 4.32: Images of near-surface facet event that occurred at the South Station on
(a,b) February 12, 2009 (D-3) and that continued on (c) Feb. 13 and (d) 14.
4.5.4 Event D-4: February 19, 2009
Small 0.3 mm facets were observed on February 19, 2008 (D-4) at 1245 at the
South Station, as shown in Figure 4.35. The facets appeared intermixed with the 8
94
02/12 0800 1600 02/13 0800 1600 02/14 0800 1600 02/15
0
200
400
600
800
1000
1200
1400
1600
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−24
−21
−18
−15
−12
−9
−6
−3
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.33: Recorded weather data from the South Station on February 12–14, 2009
(D-3).
(a) 0845 (2 mm grid) (b) 1300 (1 mm grid)
Figure 4.34: Images of a near-surface facet event (D-3) that occurred at the South
Station and persisted beneath 9 cm of new snow falling on the night of Feb. 14.
cm of new snow that was recorded at 0700. The weather conditions (Figure 4.36)
were typical of many of the other events described throughout this chapter.
95
Figure 4.35: Image of near-surface facets observed at the South Station on February
19, 2009 (D-4).
02/19 0300 0600 0900 1200 1500 1800 2100 02/20
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−14
−12
−10
−8
−6
−4
−2
0
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.36: Recorded weather data from the South Station on February 19, 2009
(D-4).
4.5.5 Event D-5: February 21, 2009
Small 0.5 mm facets were observed at the South Station in the snow surface at
1045 on February 21, 2009 (D-5), the weather data surrounding the event is provided
in Figure 4.37. Similar facets were observed at the North Station at 0900. The field
notes from the South Station stated that “as with the north plot, there seems to be
96
some small facets at the surface. Not that many advanced forms and it doesn’t look
like surface hoar.” The field notes form the North Station also stated that “a few
[of the facets] look like the classic [radiation-recrystallized near-surface facets] we saw
last year at the South plot.” Figures 4.38a and 4.38b are images from the South and
North Stations, respectively.
02/20 0800 1600 02/21 0800 1600 02/22 0800 1600 02/23
−20
−15
−10
−5
0
5
Time
Te
m
pe
ra
tu
re
 (°
C)
 
 Surface Temp. (South)
Air Temp. (South)
Surface Temp. (North)
Air Temp. (North)
Figure 4.37: Recorded weather data from the South and North stations on February
21, 2009 (D-5).
The following day, facets were reported at both the South and North sites once
again. At the North Station “0.5 mm mixed facets” were reported between 1 cm and
3 cm. At the South Station 0.5 mm facets were reported at the snow surface. Images
of the facets from the South are included in Figure 4.38c; no images were taken from
the North Station.
Since these crystals were observed at both locations, it is assumed that they
formed due to diurnal temperature fluctuations, or possibly they were the beginnings
of surface hoar crystals. Figure 4.37 shows the snow surface and air temperatures
surrounding the event for both the North and South locations. The night prior to
97
the event, the conditions at both sites were nearly identical: air temperatures near
-9 ◦C and snow surface temperature near -18 ◦C.
(a) 1045 Feb. 21 at South Station (2 mm grid) (b) 0900 Feb. 21 at North Station (2 mm grid)
(c) 1300 Feb. 22 at South Station (1 mm grid)
Figure 4.38: Images of near-surface facets on February 21, 2009 (D-5) that formed
small-faceted crystals at both the South and North Stations.
4.5.6 Event D-6: February 27–28, 2009
Event D-6 consisted of an occurrence of radiation-recrystallization at the South
Station on consecutive days: February 27 and 28, 2009. The daily weather conditions
were typical of the events discussed throughout this chapter, as shown in Figure 4.39.
98
The two-day conditions where day two was warmer than day one were similar to
Events C-2 and C-4.
At 1200 on Feb. 27 the field notes stated that the snow surface was composed
of “some small facets (0.25 mm) mixed with new snow.” A second observation was
made at 1400; the notes explained that the “surface had changed to 0.5–1 mm facets.”
Images of these two observations are included in Figure 4.40.
02/27 0800 1600 02/28 0800 1600 03/01 0800 1600 03/02
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
5
10
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.39: Recorded weather data from the South Station on February 27–28, 2009
(D-6).
The following day (Feb. 28) an observation was made at 1000; the notes explained
that the facets “from yesterday [were] still visible, but a bit more rounded.” A second
observation at 1430 detailed that the “facets in the surface snow appear to have grown
some amount since [the] morning [observation].” Images of the two observations from
this day are also included in Figure 4.40. The field notes also explained that the
“pictures don’t really do it justice” and that “some facet forms look like surface hoar;
this was not observed in the [morning].” The facets were also observed on March 1,
2009 but melted during the day due to warm temperatures.
99
(a) 1200 Feb. 27 (2 mm grid) (b) 1400 Feb. 27 (2 mm grid)
(c) 1000 Feb. 28 (2 mm grid) (d) 1430 Feb. 28 (2 mm grid)
Figure 4.40: Images of radiation-recrystallized near-surface facets from Event D-6
that formed at the South Station on (a,b) February 27 and (c,d) 28, 2009.
4.5.7 Event D-7: March 7, 2009
Event D-7 occurred on March 7, 2009. In the initial observation at 0945 new snow
was reported at the South Station, as shown in Figure 4.41a. A second observation
was made at 1315 in which “small 0.5 mm facets” were found on the surface layer
at the South Station, as pictured in Figure 4.41b. The subsurface at the second
observation was moist between 1 cm and 3 cm deep.
Air and snow surface temperatures as well as long- and short-wave radiation for
Event D-7 are shown in Figure 4.42. Short-wave radiation peaked near 1200 W/m2
100
and long-wave averaged 200 W/m2. The drop in temperatures and short-wave radia-
tion at 1200 is was due to a “brief period of mostly cloudy sky” that occurred around
1145.
(a) 0945 (1 mm grid) (b) 1315 (1 mm grid)
Figure 4.41: Images of a near-surface facet event that occurred at the South Station
on March, 7 2009 (D-7).
03/07 0300 0600 0900 1200 1500 1800 2100 03/08
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−24
−21
−18
−15
−12
−9
−6
−3
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.42: Recorded weather data from the South Station on March 7, 2009 (D-7).
101
4.5.8 Event D-8: March 12–14, 2009
Event D-8 was composed of three consecutive days (March 12–14, 2009) of near-
surface facet formation at the South Station. The temperature and radiation data
for all three days is presented in Figure 4.43. The prior day (Mar. 11) small 0.1–0.3
mm facets were observed. These small facets persisted throughout the day without
change, and at the North Station 1–3 mm surface hoar was reported. The night prior
to Mar. 11 the air temperatures (-18 ◦C) and snow surface temperatures (-26 ◦C)
were the same at both weather stations. Thus, these facets were assumed to be small
surface hoar.
03/12 0800 1600 03/13 0800 1600 03/14 0800 1600 03/15
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
5
10
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.43: Recorded weather data from the South Station on March 12–14, 2009
(D-8).
Facets 0.5 mm in size were reported at the South Station during the initial ob-
servation (0945) on Mar. 12. Later that day (at 1345) near-surface facets 1 mm in
size were reported on the surface. Images of both observations are shown in Figures
4.44a and 4.44b.
102
On Mar. 13, the field notes explained that facets observed on the surface at the
South Station were composed of “easily visible needles [seen with] the naked eye.”
Two sets of images were taken: an AM and PM. However, the field notes did not
distinguish the times—only a single time was given of 1220. Figures 4.44c and 4.44d
include images from each of the observations made on the second day.
The third day of Event D-8 included two observations at the South Station made
at 1120 and 1440. Images of from the snow surface for these observations are included
in Figures 4.44e and 4.44f. The field notes described that facets on the snow surface
ranged in size from 0.5–2 mm and that the snow was melting to a depth of 15 cm.
Examining the images taken, the facets observed clearly changed between the two
observations on Mar. 14. The initial images show crisp rectangular facets, while the
images show facets that seem to be mixing with melting snow and are more hexag-
onal. These hexagonal crystals also tended to be the some of the largest observed
(measuring over 4 mm) for any of the events over the two seasons reported herein.
Figure 4.45 includes two examples of these large hexagonal crystals.
The facets formed during Event D-8 were still visible on Mar. 15 and 16. The field
notes stated that the crystals were melted into the top of the melt-freeze crust that
developed due to a slight decrease in incoming short-wave radiation and increased
wind speed on Mar. 15.
103
(a) 0945 Mar. 12 (1 mm grid) (b) 1345 Mar. 12 (1 mm grid)
(c) Mar. 13 AM (1 mm grid) (d) Mar. 13 PM (1 mm grid)
(e) 1120 Mar. 14 (2 mm grid) (f) 1440 Mar. 14 (1 mm grid)
Figure 4.44: Images of event a near-surface facet event (D-8) that occurred on three
consecutive days (March 12–14, 2009) at the South Station.
104
Figure 4.45: Images of large near-surface facets captured at the South Station during
the second observation (1440) on March 14, 2009 (D-8).
4.5.9 Event D-9: March 20, 2009
An occurrence of radiation-recrystallized “forms” was recorded on March 20, 2009
(D-9) at the South Station. The weather data shown in Figure 4.46 corresponded well
with many of the other events discussed: short-wave radiation peaked at 1200 W/m2
and long-wave averaged 240 W/m2. However, the images from this event showed only
a few hints of faceted crystals, see Figure 4.47. Observations were made at 1120 when
air temperatures were reaching nearly 7 ◦C; thus, this event was likely on the cusp
between melting and near-surface faceting.
105
03/20 0300 0600 0900 1200 1500 1800 2100 03/21
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−16
−12
−8
−4
0
4
8
12
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.46: Recorded weather data from the South Station on March 20, 2009 (D-9).
Figure 4.47: Image of slight faceting that occurred at the South Station on March,
20 2009 (D-9).
4.5.10 Event D-10: March 30, 2009
Event D-10 occurred on March 30, 2009 at the South Station. Initial observations
at 1130 indicated that the snow surface was composed of 1–2 mm new snow, see
Figure 4.48a. A second observation at 1415 reported that stellars had 0.3 mm facets
attached, as shown in Figure 4.48b. The weather data, see Figure 4.49, shows that
106
the skies likely cleared between 1000 and 1100, just prior to the initial observations.
At this time the long-wave radiation decreased from 300 W/m2 to 200 W/m2.
(a) 1130 (1 mm grid) (b) 1315 (1 mm grid)
Figure 4.48: Images from two observations—(a) 1130 and (b) 1315—of a near-surface
facet event that occurred at the South Station on March, 30 2009 (D-9).
03/30 0300 0600 0900 1200 1500 1800 2100 03/31
0
200
400
600
800
1000
1200
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−25
−20
−15
−10
−5
0
5
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.49: Recorded weather data from the South Station on March 30, 2009 (D-10).
107
4.5.11 Event D-11: April 6, 2009
The final near-surface event (D-11) of the 2008/2009 winter season occurred on
April 6, 2009 at the South Station. This event was very similar to Event D-9, though
the snow surface temperature remained a few degrees cooler resulting in more iden-
tifiable facets, as shown in Figure 4.50. The temperature and radiation data for this
event is included in Figure 4.51.
Figure 4.50: Image from near-surface facet event that occurred at the South Station
on April 6, 2009 (D-11).
108
04/06 0300 0600 0900 1200 1500 1800 2100 04/07
0
200
400
600
800
1000
1200
1400
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
−18
−14
−10
−6
−2
2
6
10
Te
m
pe
ra
tu
re
 (°
C)
Long−wave
Short−wave
Surface Temp.
Air Temp.
Figure 4.51: Recorded weather data from the South Station on April 6, 2009 (D-11).
4.6 Analysis
The data collected throughout both seasons presented herein may be utilized to
help pinpoint the conditions favorable for near-surface facet development, particularly
due to radiation recrystallization. This was accomplished by comparing daily averages
of all recorded weather data over the two seasons with the specific days associated
with the near-surface facet events.
Figures 4.52 and 4.53 include histograms showing the frequency of all data ob-
served superimposed with the data on each day with a near-surface facet event. Figure
4.52 contains the radiation data, including short- and long-wave, as well as the ra-
tio of the two for both the South and Aspirit Stations. The long-wave radiation,
and consequently the ratio with short-wave radiation from the South Station only
contains data from the 2008/2009 (B) season due to unreliable short-wave radiation
data (see Section 3.2.2). Figure 4.53 contains the histograms of air and snow surface
109
temperature, wind speed and direction, and relative humidity from the South Station
for both seasons.
For each of these histograms, the complete daily averages were compared to the
event-only data. This was accomplished via the two-sample Kolmogorov-Smirnov
test (KS-test), which determines if the two data sets are from the same distribution.
The results of these comparisons are included in Table 4.3, where the null hypothesis
(H0) was that two data sets were from the same distribution with a 5% significance
level. The hypothesis test stated that p-values less than 0.05 (i.e., 5% significance
level) would result in a failure to reject the null hypothesis, that is the distributions
are likely different.
Table 4.3: Kolmogorov-Smirnov test results comparing the distribution sets shown
in Figures 4.52 and 4.53; the null hypothesis (H0) was that the data were from the
same distribution.
Figure Variable H0 Result p-value
4.52a Short-wave (South) Reject 5.32× 10−9
4.52b Short-wave (Aspirit) Reject 3.19× 10−3
4.52c Long-wave (South 08/09) Reject 8.23× 10−7
4.52d Long-wave (Aspirit) Reject 3.27× 10−8
4.52e SW:LW (South 08/09) Reject 7.14× 10−4
4.52f SW:LW (Aspirit) reject 1.73× 10−7
4.53a Air Temp. Fail to reject 0.12
4.53b Snow Temp Fail to reject 0.35
4.53c Wind Speed Fail to reject 0.18
4.53d Wind Dir. Fail to reject 1.00
4.53e Relative Humidity Reject 2.97× 10−5
The KS-test results showed that the entire weather data set from both seasons
(2007/2008 and 2008/2009) and the event-only weather data are different for both
long- and short-wave radiation as well as relative humidity. Thus, the ranges observed
could be linked to the formation of the near-surface facet crystals. Using the Boot-
strap method, percentiles for the daily mean values of each environmental variable
110
were determined (Efron and Tibshirani, 1993). Table 4.4 includes the percentiles
for the environmental variables of interest that were coupled to the near-surface facet
events observed in the field (these variables resulted in a rejection of the null hypothe-
sis). The percentiles presented in the table provide a tool for predicting environmental
conditions conducive to near-surface facet formation.
Table 4.4: Percentiles of environmental variables coupled to the observed formation
of near-surface facets.
Variable Units 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95%
Short-wave (South) W/m2 376 450 508 542 587 621 648 670 685 701 714
Short-wave (Aspirit) W/m2 199 203 212 218 221 223 225 229 234 240 243
Long-wave (South 08/09) W/m2 209 242 292 316 341 368 389 407 425 450 489
Long-wave (Aspirit) W/m2 172 178 186 192 198 203 208 214 220 230 242
SW:LW (South 08/09) 2.09 2.20 2.48 2.74 2.90 3.00 3.08 3.18 3.33 3.50 3.56
SW:LW (Aspirit) 0.99 1.14 1.41 1.57 1.68 1.80 1.91 2.00 2.14 2.41 2.59
Relative humidity % 22.9 26.7 34.1 39.9 44.8 48.9 53.3 57.6 61.1 64.7 66.5
Morstad et al. (2007) successfully formed radiation-recrystallized near-surface
facets in ten laboratory experiments. The mean short-wave radiation for these ex-
periments was one of three values: 595, 755, or 1180 W/m2. Long-wave radiation
ranged from 270–320 W/m2 and relative humidity ranged between 15 and 40%. A
comparison of these values with the tabulated data in Table 4.4 indicated that only
the experiment conducted with short-wave at 595 W/m2 fell within the range of av-
erage values observed in the field at the South Station. Interestingly, this experiment
resulted in the largest facets (1 mm) of any experiment conducted by Morstad et al.
(2007, Table 1). The long-wave radiation and relative humidity from the Morstad
et al. (2007) experiments fell in the lower portion of the values observed in the field
observation discussed in this Chapter. Finally, a computation of the ratio of short-
to long-wave radiation from the laboratory experiments yielded four values: 2.2, 2.4,
2.7, and 3.9. These values, with exception of the 3.9, fit between the 10th and 30th
111
percentiles. Observations made at the South Station by Cooperstein et al. (2004)
reported near-surface facet growth during 587 W/m2 of short-wave radiation, which
falls at the 40th percentile of the data presented in this work.
4.7 Conclusions
Throughout two seasons of observations, 26 near-surface events were observed on
a south-facing slope with no conclusive events recorded at the north-facing slope. The
events reported at the South station in this work typically formed under clear skies.
A comparison of the daily mean environmental conditions from the near-surface facet
events with that of the daily means entire data set support that only short- and
long-wave radiation as well as relative humidity demonstrate statistically significant
differences. Slope parallel incident short-wave radiation ranged from 380–710 W/m2,
long-wave ranged from 210–240 W/m2, and relative humidity between 23% and 67%
for all near-surface facet events. However, these results are only based on a small set of
data from two locations. Further research documenting near-surface facet formation
is needed from various locations for multiple seasons.
The formation of near-surface facets seemed to be dominated by the interaction
of short-wave radiation gain just below the surface and cooling at the snow surface
due to long-wave radiation loss. This was evident through field notes that indicated
that a majority of the events showed crystals which appeared to develop during the
daylight hours and that the facets were often diminished the day following the event.
However, this was not the case for all events. This finding emphasizes that the tradi-
tional separation of diurnal- and radiation-recrystallized facet formation as separate
processes may not be appropriate. These processes likely occur simultaneously and
radiation-recrystallization may be more prevalent than previous research suggests.
112
200 400 600 8000
5
10
15
20
25
Short−wave (W/m2): South Station
Fr
eq
ue
nc
y
 
 
Events
All data
(a)
0 200 400 600 800 10000
10
20
30
40
50
Short−wave(W/m2): Aspirit Station
Fr
eq
ue
nc
y
 
 
Events
All data
(b)
180 200 220 240 260 280 3000
2
4
6
8
10
12
14
16
Long−wave, 08/09 (W/m2): South Station
Fr
eq
ue
nc
y
 
 
Events All data
(c)
140 200 260 320 380 440 500 560 6200
10
20
30
40
50
60
Long−wave (W/m2): Aspirit Station
Fr
eq
ue
nc
y
 
 
Events
All data
(d)
0 1 2 3 40
2
4
6
8
10
12
SW:LW, 08/09: South Station
Fr
eq
ue
nc
y
 
 
Events All data
(e)
0 0.5 1 1.5 2 2.5 30
10
20
30
40
SW:LW: Aspirit Station
Fr
eq
ue
nc
y
 
 
Events
All data
(f)
Figure 4.52: Histograms comparing daily average radiation conditions for the en-
tire data set (2007/2008 and 2008/2009 seasons) against the days associated with
near-surface facet events: (a) short-wave radiation at the South Station, (b) short-
wave radiation at the Apsirit station, (c) long-wave radiation at South Station for
2008/2009 season, (d) long-wave radiation at the Aspirit Station, (e) the ratio of
short- to long-wave radiation (SW:LW) at South Station for 2008/2009 season, and
(f) the ratio between short- and long-wave radiation at the Aspirit station.
113
−15 −10 −5 0 50
5
10
15
20
25
30
35
Air temperature (°C)
Fr
eq
ue
nc
y
 
 Events
All data
(a)
−15 −10 −5 00
5
10
15
20
25
30
Snow temperature (°C)
Fr
eq
ue
nc
y
 
 
Events
All data
(b)
0.5 1 1.5 2 2.5 3 3.50
5
10
15
20
25
30
35
Wind speed (m/s)
Fr
eq
ue
nc
y
 
 
Events
All data
(c)
100 150 200 2500
10
20
30
40
Wind direction (deg.)
Fr
eq
ue
nc
y
 
 
Events
All data
(d)
20 30 40 50 60 70 800
5
10
15
20
25
30
35
Relative humidity (%)
Fr
eq
ue
nc
y
 
 
Events
All data
(e)
Figure 4.53: Histograms comparing the daily average weather conditions—(a) air
temperature, (b) snow surface temperature, (c) wind speed, (d) wind direction, and
(e) relative humidity—for the entire data set (2007/2008 and 2008/2009 seasons )
against the days associated with near-surface facet events at the South Station.
114
CHAPTER 5
SNOW THERMAL MODEL
5.1 Introduction
Various models exist for snow that range from simple 1-D conduction to complete
3-D finite element constructs. Section 5.2 provides a broad overview of various mod-
eling endeavors, including the model discussed in this chapter. The model presented
is a simple 1-D heat equation-based energy balance model, written in MATLAB (The
Mathworks, Inc.), which includes attenuation of short-wave radiation for computing
snowpack temperatures.
This chapter serves to highlight the details surrounding the model development,
including the theoretical derivation, numerical representation, example usage, and the
reliability of the output. Details regarding the usage of the model, including the source
code, are included in the user manual in Appendix C. The model presented herein was
originally implemented by Morstad et al. (2007) and was subsequently re-developed,
as presented here, to improve the application of the boundary conditions, improve
computational efficiency, and enhance usability for future researchers. Increasing the
computational efficiency was necessary to perform the analysis presented in Chapters
7–10, which required millions of model evaluations. Additionally, visible (VIS) and
near-infrared (NIR) components were added to the short-wave radiation attenuation
to allow snow material properties related to radiation to vary between these two
wavebands.
115
5.2 Background
Modeling the thermal behavior of snow is not a new endeavor. LaChapelle (1960)
cited papers from as early as 1892 that examined temperature profiles of snow. In
his critique, experimental and theoretical means were explored regarding thermal
conductivity and vapor transport, all of which show a high degree of associated un-
certainty. This uncertainty is expected considering that snow is a complex system
that consists of a porous, phase-changing material that is subject to atmospheric
radiation. Adams and Sato (1993) explored modeling thermal conductivity using an
idealized snowpack; the findings indicated that, under certain geometric conditions,
the theoretical results were in close agreement with empirical data and, as expected,
the geometry of the snow grains was critical to determining the thermal conductiv-
ity. A significant amount of work has examined snow using a continuum mechanics
theory of mixtures (Adams and Brown, 1989, 1990; Morland et al., 1990; Bader and
Weilenmann, 1992; Brown et al., 1994, 1999). However, Boone and Etchevers (2001)
indicated that the application of such models can be unattractive due to computation
time; therefore a detailed comparison with simpler models was explored that yielded
comparable results.
A non-dimensional approach was utilized by Gray and Morland (1994) that re-
duced a mixture theory analysis (Morland et al., 1990) to a set of four differential
equations using a variety of simplifications. Then, contour plots were developed for
varying non-dimensional time and depth using non-dimensional parameters for snow
temperature, snow density, and surface air velocity. However, the results obtained
were developed for long time scales (winter season) but neglected to account for the
effects of solar penetration. Additional non-dimensional work was later conducted to
assess the snow for shorter time scales (15 min) and small layers (mm scale) (Bartelt
116
et al., 2004). Using a thermal non-equilibrium approach, this study also indicated
that temperature differences between the pore air and ice particles in the uppermost
layer of snow (within 0.2 m of the surface) were on the order of ±5 ◦C and that inter-
facial heat exchange between snow crystals played a significant role in determining
the temperature profile.
Perhaps the most comprehensive model developed to date is the SNOWPACK
model, which was designed to improve avalanche warnings (Lehning et al., 1999).
The model was intended to provide snowpack information using data from a number
of automated weather stations. Initial results examining the accuracy of the model
were considered a “reasonable representation” (Lehning et al., 1999). SNOWPACK
has been explained in detail in a series of papers (Bartelt and Lehning, 2002; Lehning
et al., 2002a,b). The model was explained as a one-dimensional three-phase (ice, wa-
ter, and air) model that accounts for heat transfer, water transport, vapor diffusion,
and mechanical deformation with special conditions for wind drifting and snow abla-
tion (Bartelt and Lehning, 2002). Research conducted in an attempt to validate the
SNOWPACK model yielded reasonable results: The predicted temperature profiles
were stated to be “fairly accurate” by Lundy et al. (2001), but Fierz and Lehning
(2001) encouraged additional work regarding the initial stage of snow metamorphism,
specifically the processes involving particles changing to small faceted or rounded
crystals.
Attempts to model the metamorphism of snow due to temperature gradients using
a heat-transfer approach has been attempted by many authors; some of the earliest
work was presented by Adams and Brown (1982a,b, 1983). Adams and Brown (1982a)
initially examined vapor transport through a pore space of two snow crystals as es-
tablished by the presence of a temperature gradient. The results agreed with exper-
imental work conducted by other researchers. Adams and Brown (1983) expanded
117
upon this effort by including a heat-conduction equation that considered internal
heat generation. This was the basis of the model utilized by Morstad et al. (2007)
as well as the model presented in this chapter. The aforementioned models were
based on general heat transfer principles, the basic equation of which may be found
in introductory heat transfer textbooks (e.g., Incropera et al., 2007). However, the
model has been refined significantly with respect to the input terms, specifically the
supply term. Morstad et al. (2007) included the surface effects of long-wave radiation
exchange, latent heat, and sensible heat as well as the internal heat generation of
absorbed shortwave radiation. This augmentation is similar to many other models
including SNOWPACK (Lehning et al., 2002b) and is common when assessing the
energy balance of a snowpack (Armstrong and Brun, 2008). The governing equations
for each of these terms were adopted from a variety of sources and a detailed descrip-
tion of each is contained in Morstad (2004) as well as the following sections. It is
important to note that other similar models have been developed, but with varying
supply terms. Singh and Gan (2005) compared three models that vary the input
terms and determined that a model that approximates heat flow into the snow using
a periodic boundary (diurnal) temperature forcing at the surface was shown to be the
most statistically accurate for determining snow surface temperature.
5.3 Model Development
5.3.1 Conservation of Energy
The First Law of Thermodynamics, also known as conservation of energy, states
(e.g., Narashimhan, 1993):
The time rate of change of the sum total of the kinetic energy and the
internal energy in the body is equal to the sum of the rates of work done by
118
the surface and body loads in producing the deformation (or flow) together
with the heat energy that may leave or enter the body at a certain rate.
As a mathematical expression, this principle may be written as
d
dt
(KE + E) = W +R, (5.1)
which, as stated above, is a function of macroscopic kinetic energy (KE), internal
energy (E), rate of work acting on the system (W ), and rate of heat input (R).
The system is assumed to remain at rest, thus macroscopic kinetic energy is ne-
glected. Additionally, it shall be assumed that no mechanical work is being performed
on the system, so the work rate term is also neglected. Finally, the rate of heat in-
put may be broken into two parts: the heat added to the system across its surface
boundary and the heat generated internally or supplied to the volume, that is
R =
∮
CS
−ξ · nˆdA+
∫
CV
ρhdV. (5.2)
This relationship is best described using a control volume as shown in Figure 5.1,
which is an illustration detailing that the rate of heat input (R) is equivalent to the
sum flux of heat (ξ), across the control surface (CS) and the internally generated
heat (ρh). The outward normal vector (nˆ) defines the surface and the internal heat
is defined as the product of the material density (ρ) and the specific heat supply
(h). The negative sign preceding the flux vector defines flux of heat into the control
volume as positive (i.e., the dot product of the flux and outward normal vector is
negative when the flux is entering the control volume).
The internal energy may be described using specific internal energy (ϑ) as
d
dt
E =
d
dt
∫
CV
ρϑdV. (5.3)
119
ρh
ξ
nˆ
CV
CS
R
Figure 5.1: Schematic of the arbitrary control volume (CV ) enclosed by the control
surface (CS). The rate of heat generation (R) is a result of the heat flux across the
surface (ξ) and the heat supply (ρh); nˆ is the outward normal vector.
Therefore, Equation (5.1), with the aforementioned assumptions, may be rewritten
as
d
dt
∫
CV
ρϑdV =
∮
CS
−ξ · nˆdA+
∫
CV
ρhdV. (5.4)
Using the Reynolds Transport Theorem and the continuity equation, assuming
the specific internal energy is a continuous and differentiable function, the left-side of
this relationship may be rewritten as (Reddy, 2008, Eq. 5.2.28)
d
dt
∫
CV
ρϑdV =
∫
CV
d
dt
ρϑdV. (5.5)
Gauss’ Theorem (Liu, 2002) is defined as
∮
CS
b · nˆdA =
∫
CV
~∇ · bdV, (5.6)
where b is an arbitrary vector. This expression allows the area integral of Equation
(5.4) to be represented as a volume integral. By applying both Equations (5.5) and
(5.6), Equation (5.4) may again be rewritten as
∫
CV
[
d
dt
ρϑ+ ~∇ · ξ − ρh
]
dV = 0. (5.7)
120
For Equation (5.7) to be valid for an arbitrary control volume, the integrand must be
zero. The resulting relationship is the differential form of the First Law of Thermo-
dynamics, commonly referred to as the local form of conservation of energy with the
absence of mechanical processes:
d
dt
ρϑ+ ~∇ · ξ − ρh = 0. (5.8)
This relationship was the basis of the thermal model for snow presented in this chap-
ter.
5.3.2 Application
The following section details the application of Equation (5.8) to the form used
for developing the thermal model. The internal energy component of Equation (5.8)
is expressed in terms of the specific heat capacity (cp) at a constant pressure, density
(ρ), and the time rate of change of the material temperature (T ) as follows,
d
dt
ρϑ = ρcp
∂T
∂t
. (5.9)
The heat flux across the control surface (ξ) is separated into two components such
that
ξ = qk + q, (5.10)
where qk is the heat flux due to conduction and q is an additional heat flux component.
The latter is detailed in Section 5.3.4, which includes heat flux due to short-wave
radiation. This radiation term may be considered a volumetric heat source that would
be accounted for in the ρh term of Equation (5.8). However, due to the method used
to compute and measure this term as a heat flux (W/m2); radiation was implemented
here as an additional flux term (q) As such, the ρh term of Equation (5.8) is zero.
121
Fourier’s Law of Conduction is defined as
qk = −k~∇T, (5.11)
where k is the thermal conductivity tensor (Narashimhan, 1993). Using this rela-
tionship, Equation (5.10), and the aforementioned assumption that ρh = 0, Equation
(5.8) may be written as
ρcp
∂T
∂t
= ~∇ · (k~∇T )− ~∇ · q. (5.12)
Finally, the material in question is assumed to be thermally isotropic (i.e., thermal
conductivity is a scalar, k) and reduced to one-dimensional heat flow in the vertical
direction, z. Thus, Equation (5.12) becomes
ρcp
∂T
∂t
= k
∂2T
∂z2
−
∂q
∂z
. (5.13)
Each additive term in this equation has units of W/m3, which may be described as
the rate of change of energy per unit volume within the system. This equation is an
adaptation of the 1-D heat diffusion equation (Incropera et al., 2007) and used for
the basis of the thermal model presented in this chapter.
5.3.3 Numerical Solution
General Numeric Equation: Equation (5.13) may be solved numerically for the
temperature (T ) of a layered system throughout time (t). Referring to Equation
(5.13), the Crank-Nicolson Method may be applied as (Chapra and Canale, 2002, p.
849)
ρcp
∂T
∂t
∼= ρicpi
[
T j+1i − T
j
i
∆t
]
(5.14)
and
k
∂2T
∂z2
∼=
ki
2
[
T ji+1 − 2T
j
i + T
j
i−1
(∆z)2
+
T j+1i+1 − 2T
j+1
i + T
j+1
i−1
(∆z)2
]
, (5.15)
122
where the j index represents the j-th time step and the layer index is i = 1, 2, . . . , n,
where n is the number of layers.
Additionally, the heat flux (q) from Equation (5.13) may be rewritten as a numer-
ical representation using the forward difference approximation (Chapra, 2005):
−
∂q
∂z
∼=
qji − q
j
i+1
∆z
. (5.16)
Throughout this chapter the layer thickness is assumed constant, thus the i index
is dropped from ∆z term; this is done for simplicity and is not a requirement of the
method. Figure 5.2 is a schematic showing the layered snowpack for the numerical
solution presented here, which includes the temperatures at each node (T ji ) and heat
flux across the layer (qji ), for the j-th time step. The node above the surface is
considered a “phantom” node, which is necessary for the application of the upper
boundary condition.
Snow Surface
Ground
...
T j
1
T j
2
T j
n-1
T j
n
T j
n+1
T j
0
q j
1
q j
2
q j
n-1
q j
n
Figure 5.2: Schematic of snowpack layering utilized for numerical solution of snow
temperatures with time. The superscript j represents the j-th time step and the
subscript i represents the layer number.
123
The numerical representation of Equations (5.14)–(5.16), may be substituted into
Equation (5.13). The result of this substitution yields
ρicpi
∆t
T j+1i −
ki
2(∆z)2
(
T j+1i+1 − 2T
j+1
i + T
j+1
i−1
)
= . . .
ρicpi
∆t
T ji −
ki
2(∆z)2
(
T ji+1 − 2T
j
i + T
j
i−1
)
+
qji − q
j
i+1
∆z
, (5.17)
where all of the j + 1 and j temperature terms are relocated to the left and right
sides, respectively.
The constant terms of Equation (5.17) may be grouped together as
ai =
ki
(∆z)2
, (5.18a)
bi =
ρcpi
∆t
, (5.18b)
ci = bi + ai, and (5.18c)
di = bi − ai. (5.18d)
In conjunction with the coefficients defined in Equation (5.18), the general numerical
representation of the heat equation is written as
−ai
2
T j+1i−1 + ciT
j+i
i +
−ai
2
T j+1i+1 =
ai
2
T ji−1 + diT
j
i +
ai
2
T ji+1 +
qji − q
j
i+1
∆z
. (5.19)
Boundary Conditions: The bottom temperature of the snowpack is assumed to
be constant, thus the bottom boundary condition may be defined as
T jn+1 = Tbottom, (5.20)
where Tbottom is a constant temperature.
The top boundary condition is defined as
k
∂T
∂z
∣
∣
∣
z=0
= qs, (5.21)
124
where qs is the heat flux across the surface layer at node i = 1. This flux boundary
condition is known as the Neumann condition (Incropera et al., 2007), which states
that the flux entering the system at the surface is conducted into the uppermost
layer of the system. Section 5.3.5 details the components of qs. It may be writ-
ten numerically—using the central difference approximation (Chapra, 2005)—for the
current (j) and future (j + 1) time steps as
qjs = k1
T j0 − T
j
2
2∆z
and (5.22a)
qj+1s = k1
T j+10 − T
j+1
2
2∆z
. (5.22b)
Next, it is assumed the heat flux at the surface in the present time step may be
applied to the future time step. This allows Equation (5.22) to be solved for T j0 and
T j+10 , which results in
T j0 =
2qjs∆z
k1
+ T j2 and (5.23a)
T j+10 =
2qjs∆z
k1
+ T j+12 . (5.23b)
Equations (5.23a) and (5.23b) are then substituted into Equation (5.19) to produce
the upper boundary condition as
c1T
j+1
1 − a1T
j+1
2 = d1T
j
1 + a1T
j
2 + 2
qjs
∆z
+
qj1 − q
j
2
∆z
. (5.24)
Matrix Solution: Equations (5.19), (5.20), and (5.24) may be represented in
matrix form as shown in Equation (5.25),
125



















c1 −a1 0 0 · · · 0 0 0
−a2
2 c2
−a2
2 0 · · · 0 0 0
0 −a32 c3
−a3
2 · · · 0 0 0
0 0 −a42 c4 · · · 0 0 0
...
...
...
...
. . .
...
...
...
0 0 0 0 · · · −an2 cn
−an
2
0 0 0 0 · · · 0 0 1



















︸ ︷︷ ︸
[A]



















T j+11
T j+12
T j+13
T j+14
...
T j+1n
T j+1n+1



















︸ ︷︷ ︸
x
=



















d1T
j
1 + a1T
j
2 + 2
qjs
∆z +
qj1−q
j
2
∆z
a2
2 T
j
1 + d2T
j
2 +
a2
2 T
j
3 +
qj2−q
j
1
∆z
a3
2 T
j
2 + d3T
j
3 +
a3
2 T
j
4 +
qj3−q
j
2
∆z
a4
2 T
j
3 + d4T
j
4 +
a4
2 T
j
5 +
qj4−q
j
3
∆z
...
an
2 T
j
n−1 + dnT
j
n +
an
2 T
j
n+1 +
qjn−q
j
n−1
∆z
T jbottom



















︸ ︷︷ ︸
b
, (5.25)
which is in the form [A]x = b. Thus, the temperatures given in x may be solved using
the temperatures from the previous time step, such that x = [A]−1b. Hence, the
model must be initialized with a temperature profile and it is assumed that material
properties (k, ρ, and cp; see Section 5.3.7) as well as the heat flux terms (q and qs)
are known model inputs.
5.3.4 Short-wave Radiation
The heat flux term (q) of Equation (5.13) comprises the short-wave radiative
(0.28–2.8 µm) heat flux that penetrates the snow surface and is absorbed throughout
126
the snow; the amount absorbed is a strong function of wavelength (Armstrong and
Brun, 2008). The total radiative flux absorbed by the snow (SW in) is computed
using snow all-wave albedo (α) in this range. Albedo is the ratio of reflected to
incident irradiance. It is assumed that both SW in and α are known model inputs.
Numerically, for the j-the time step, the incoming short-wave radiation flux in the
first layer (i = 1), that is qj1, may be described as
qj1 = SW
in(1− α). (5.26)
The remaining short-wave radiation penetrates the snowpack and is assumed to
be absorbed following an exponential decay function, as presented by Gray and Male
(1981). This decay function is applied to the layered snowpack, for i > 1, with the
following relationship:
qji+1 = q
j
i · exp(−κi∆z). (5.27)
The extinction coefficient, κ, has units of 1/m and is multiplied by the layer thickness,
∆z. Figure 5.3 graphically shows the application of these relationships to a layered
snowpack. Notice that the extinction coefficient may differ for each layer, this allows
for the inclusion of different types of snow such as contaminated snow or an ice layer.
In application, i.e., Equation (5.25), the following relationship is useful:
qabs
j
i = q
j
i − q
j
i+1 = q
j
i · (1− exp(−κi∆z)). (5.28)
As mentioned previously, short-wave absorption is a strong function of wavelength.
Thus, the model presented in this chapter divides the short-wave flux term into two
components: a visible (VIS) and near-infrared (NIR). This allows the short-wave flux
term to be written as
qji = q
j
V ISi
+ qjNIRi . (5.29)
127
i = 1
i = 2
i = 3
qj1 = SW
in(1− α)
qj2 = q
j
1 exp(−κ1∆z)
qabs
j
1 = q
j
1 − q
j
2 = q
j
1(1− exp(−κ1∆z))
qabs
j
2 = q
j
2 − q
j
3 = q
j
2(1− exp(−κ2∆z))
qabs
j
3 = q
j
3 − q
j
4 = q
j
3(1− exp(−κ3∆z))
qj3 = q
j
2 exp(−κ2∆z)
...
Figure 5.3: Schematic that demonstrates the application of short-wave attenuation
in a layered snowpack.
Both of these components behave as shown in Figure 5.3, but allow for different
albedo and extinction coefficients to be defined for the two wavebands: 380–700 nm
(VIS) and 700–1400 nm (NIR); see Appendix C for additional details.
5.3.5 Surface Flux Terms
For application to snow, the heat flux at the surface (qs) includes three energy
balance inputs: the heat flux input due to long-wave radiation (qLW ), sensible heat
(qe), and latent heat (qh), which may be written as
qs = qLW + qe + qh. (5.30)
This may be directly substituted into Equation (5.25).
As mentioned in Section 5.2, the energy balance factors have been used in vari-
ous forms in many models. Morstad (2004) detailed the factors summarized in the
following sections, and Armstrong and Brun (2008) provided a detailed overview of
each energy balance component. Table 5.1 summarizes the various constant values
used throughout this section, specifically Equations (5.31) through (5.35).
128
Table 5.1: List of constant variables utilized for computing the heat source term of
Equation (5.30).
Variable Description Value
Ls Latent heat of sublimation phase change [kJ/kg] 2833
Ke Transfer coefficient for water vapor 0.0023
Kh Transfer coefficient 0.0023
Mv/Ma Ratio of dry-air to water-vapor molecular weights 0.622
Ra Gas constant for air [kJ/(kg· K)] 0.287
Rv Gas constant for water vapor [kJ/(kg· K)] 0.462
T0 Reference temperature for vapor pressure [◦C] -5
e0 Reference vapor pressure [kPa] 0.402
ε Emissivity of snow 0.988
Long-wave radiation (3.5–50 µm) is simply another name for thermal radiation.
The heat flux due to long-wave radiation is a balance between the incoming and
outgoing radiation. The incoming radiation (qinLW ) is assumed to be a known or
measured value. The outgoing radiation is governed by the Stefan-Boltzman Law
(Wetly et al., 2008, p. 365), which is a function of the snow emissivity (ε) the snow
temperature (Ts) and the Stefan-Boltzman constant (σ = 5.670× 10−8W/(m2 ·K4)).
The emissivity is assumed constant at 0.988, the same as for pure ice in the spectrum
defined for long-wave radiation. The net long-wave radiation may be written as
qLW = q
in
LW − εσT
4
s . (5.31)
If qLW is computed as a negative value, then the surface is cooling or loosing heat.
The same convention applies to the qe and qh parameters.
Latent heat is the result of energy associated with phase-change and is driven by
the water-vapor pressure gradients at the snow surface, i.e., changes in the energy
state of the snow cause phase changes rather than temperature changes. Latent heat
may be estimated as(Martin and Lejeune, 1998; Ishikawa et al., 1999)
qe =
(Mv/Ma)ρaLsKeVw
(
ea RH100% − es
)
Patm
. (5.32)
129
Based on this equation, latent heat (qe) is a function of the latent heat of sublimation
(Ls), the transfer coefficient for water-vapor (Ke), the water-vapor pressures above
the snow surface (ea) and at the snow surface (es), the ratio of dry-air to water-
vapor molecular weights (Mv/Ma), the density of air (ρa), wind velocity (Vw), and
the atmospheric pressure (assumed to be a known model input, Patm).
The saturation water-vapor pressures above and at the snow surface may be cal-
culated with the the Clausius-Clapeyron Equation for water-vapor (Gray and Male,
1981; Bejan, 1997):
ei = e0 · exp
[
Ls
Rv
(
1
T0
−
1
Ti
)]
. (5.33)
The variables e0 and T0 are reference values, Rv is the gas constant for water vapor,
and the temperature, Ti, represents either the air (i = a) or snow temperature (i = s).
The air density is calculated via the ideal gas law, which uses the gas constant for air
(Ra), atmospheric pressure, and air temperature:
ρa =
Patm
RaTa
. (5.34)
At the snow surface, saturation is assumed, but above the snow surface (i.e., in
the air) the calculated partial pressure of water-vapor (ea) must be adjusted for
undersaturated conditions. Therefore, the relative humidity (RH) is multiplied by ea
in Equation (5.32).
Sensible heat, qh, is calculated using Equation (5.35),
qh = ρacpaKhVw(Ta − Ts), (5.35)
and is a function of the convection between the air and snow surface. Sensible heat is
associated with a change in energy state of the material that results in a temperature
change. Sensible heat is dependent on the density of air (ρa), specific heat capacity
130
of air (cpa), the transfer coefficient (Kh), wind speed (Vw), and the snow and air
temperatures (Ts and Ta, respectively).
Both the latent and sensible heat relationships are based on simple bulk transfer
formulations (Armstrong and Brun, 2008); qh is a direct application of the convective
heat flux equation: qh = h(T1 − T2) (Wetly et al., 2008, p. 302). The transfer
coefficients Ke and Kh are based on melting snow (Martin and Lejeune, 1998; Ishikawa
et al., 1999). Armstrong and Brun (2008) detailed another common method for
determining transfer coefficients based on surface roughness parameterization. These
methods were used in the SNOWPACK model (Lehning et al., 2002b), which was also
based on experiments examining wet snow (Calanca, 2001). Hence, for application to
dry snow these coefficients do not directly apply, but are utilized nonetheless since a
suitable alternative does not exist.
5.3.6 Boundary Layer Application
The implementation of surface heat flux (qs) from Equation (5.30) and the appli-
cation of short-wave radiation (Section 5.3.4) differ from that presented by Morstad
et al. (2007). This research assumed that the surface heat flux applied to the Neu-
mann boundary condition was composed of the three components defined in Equation
(5.30) as well as a fourth component that includes the amount of short-wave radiation
absorbed between the surface and mid-point of the first layer (Morstad et al., 2007).
In general, the difference in the computed temperatures between the two methods
were on the order of tenths of a degree, as shown in Figure 5.4, which is a comparison
of Experiment #1 from Morstad et al. (2007) evaluated using the two methods. Ad-
ditionally, Monte Carlo simulations with 500 replicates of the model were conducted
with the two different model setups. For both models, the inputs were varied as de-
tailed in Chapter 7 (“Control” location). Two of the resulting output distributions, as
131
shown in Figure 5.5, were compared: the snow surface temperature with the “night”
(Figure 5.5a) and the temperature gradient computed between the surface and 2 cm
(Figure 5.5b) with the “day-light” configuration (see Chapter 7). Statistically, at
the 5% confidence level interval using the Kolmogorov-Smirnov (Massey, 1951) and
Ansari and Bradley (1960) tests, the distributions between the two models do not
differ.
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
30
35
40
Time (hrs)
Dept
h(cm
)
Tem
p.
Diffe
ren
ce
(
◦ C
)
−0.6
−0.4
−0.2
0
0.2
0.4
Figure 5.4: Example of temperature differences observed by differing application of
the Neumann boundary condition.
The discrepancy in the boundary condition application of Morstad et al. (2007)
was not identified until after the analysis in this dissertation was complete, as such the
application of the Neumann boundary condition as conducted by Morstad et al. (2007)
was utilized throughout this dissertation. Since statistically the resulting distributions
do not differ between the two versions of the model, the results presented in the
subsequent chapters should agree with the analysis if it were performed with the
derivation presented here. However, the derivation presented in this chapter is more
rigorous and is recommended for future applications of the thermal model.
132
−50 −40 −30 −20 −10 0 10
0
0.02
0.04
0.06
0.08
0.1
Temp. [◦C]
Pr
ob
ab
ilit
yD
ens
ity
Morstad ‘07
Slaughter
(a)
−500 0 500
0
1
2
3
4
5
x 10−3
Temp. Gradient [ ◦C/m]
Pr
ob
ab
ilit
yD
ens
ity
Morstad ‘07
Slaughter
(b)
Figure 5.5: Resulting output distributions—(a) snow surface temperature and (b)
temperature gradient—from the Monte Carlo simulations.
5.3.7 Material Properties
Equation (5.8) requires three material properties of snow: density (ρ), specific
heat capacity (cp), and thermal conductivity (k). The density is assumed to be a
measured or known value, thus is not discussed. The specific heat of snow is assumed
to be only a function of its temperature (T in ◦C), according to the relationship in
Equation (5.36) as was utilized by Morstad et al. (2007),
cp = 1000 · (2.115 + 0.00779 · T ), (5.36)
which is a relationship for ice (Gray and Male, 1981).
The thermal conductivity, k, relationship used by Gray and Male (1981) is given
in Equation (5.37),
k = 0.021 + 2.5·
(
ρ
1000
)2
, (5.37)
which is a strict function of density, ρ (kg/m3). The conductivity is expressed as
effective thermal conductivity because it is assumed to account for various aspects
of heat-transfer including conduction through the air and ice matrix as well as heat
133
transfer across the pore-space from vapor diffusion. This is only one of many rela-
tionships that exist for modeling thermal conductivity; Sturm et al. (1997) compiled
an extensive list of experimentally attained relationships.
In Chapters 9–10 the thermal conductivity and specific heat were assumed to be
known, thus the relationships of Equations (5.36) and (5.37) were not utilized.
5.4 Analysis with VIS/NIR Components
The equations defined in the previous sections were used to build a thermal model
which is solved using MATLAB (The Mathworks, Inc.). The complete program is
detailed in Appendix C, including instructions for operating the model via the MAT-
LAB command-line or via a graphical interface. This section highlights the inclusion
of the NIR and VIS short-wave radiation components. Besides the code being written
more efficiently, the inclusion of these radiation components is the only substantial
difference between the model presented herein and that used by Morstad et al. (2007).
This detail was added to improve the model behavior with respect to attenuating
radiation.
The sun emits radiation primarily in the visible (VIS, 0.3–0.8 µm) and near-
infrared (NIR, 0.8–1.5 µm) wavelengths. ASTM G-173 (2003) provided a standard
reference for direct incident short-wave radiation. According to this standard over
the entire electromagnetic spectrum that reaches the Earth’s surface (0.28–4 µm) the
average total irradiation is 1000 W/m2, at a latitude of 37◦. Of this value, 54.5%
is in the visible range, 27.4% is in the near-infrared, and 8.7% is in the short-wave
infrared range (SWIR, 1.5–2.8 µm). The remaining 9.4% of the incident irradiation
is for wavelengths greater than 2.8 µm. A negligible amount of incident irradiation is
due to the bands omitted from this analysis: 0.28–0.3 µm. The wavebands presented
134
here differ slightly from the formal definitions, but were defined to align with bands
presented by Armstrong and Brun (2008) for analysis purposes.
Experiment two in Morstad et al. (2007) resulted in 1 mm near-surface facets
due to radiation recrystallization; this experiment was utilized here to demonstrate
the VIS/NIR component added to the model. Morstad (2004) provided complete
details on this experiment, which used a constant value of 650 W/m2 for short-wave
irradiance. This value was measured using an Eppley PSP radiation sensor, which
measures between 0.3 and 1.5 µm. Thus, this value may be divided into VIS, NIR, and
SWIR components based on the aforementioned divisions. The resulting components
are 391, 196, and 62 W/m2, respectively. This division is reasonable, even for the
laboratory experiments, because the solar simulation system is within 2.6% of the
CIE (1989) standard (Scott, 2001).
Morstad et al. (2007) measured the albedo for this experiment to be 0.81. This
value is similar to the albedo of 0.78 reported by Armstrong and Brun (2008, p. 57)
for a Class 1 snow type. This class of snow has albedo values for the VIS, NIR, and
SWIR of 0.94, 0.80, and 0.59, respectively, and extinction coefficient, κ, values of 40
m-1 and 110 m-1 for the VIS and NIR spectral ranges, respectively, which average to
a value of 75 m-1. Morstad et al. (2007) used 82 m-1 for the extinction coefficient. In
the SWIR range, κ is reported as infinite, therefore it acts only at the snow surface.
In addition to the SWIR irradiance, a small band between 2.8 µm and 3.5 µm
was unmeasured. Wavelengths between 3.5 µm and 50 µm were measured by a long-
wave sensor (Eppley Lab., Inc. PIR) and applied to the snow surface, since in these
wavebands snow acts nearly as a blackbody (Warren, 1982; Armstrong and Brun,
2008). Using the ASTM G-173 (2003), 3.2% of the total incoming radiation is in
this “missing” range. This value may be estimated using the measured value of
650 W/m2, which is measured over the wavelengths that comprise 91.3% of radiation
135
emitted by the sun. Therefore, the missing portion of the spectrum may be estimated
as (0.032)(650 W/m
2
0.913 ) = 0.035 · 650 W/m
2 = 23 W/m2. A missing portion on the
order of 3.5% may seem insignificant, but once the albedo values are applied to the
incident radiation the value yielded—23 W/m2—becomes a significant contributor to
the energy balance.
Using the irradiance, albedo, and extinction coefficient values defined here, the
thermal model presented in this chapter was executed for six scenarios defined below:
1. “AS-IS” was executed as in Morstad et al. (2007) but with α = 0.78 and κ =
75 m−1.
2. “VIS” was executed with only visible irradiation.
3. “NIR” was run with only near-infrared irradiation.
4. “VIS-NIR” used both the visible and near-infrared values.
5. “SWIR(1)” was the same as the previous simulation, except the 25 W/m2 from
the SWIR range was added to the long-wave radiation component that acts at
the snow surface (62 W/m2(1− 0.59) = 25 W/m2).
6. “SWIR(2)” was the same as the “SWIR(1)” simulation, except the “missing”
23 W/m2 was also added to the long-wave radiation component.
Figure 5.6 compares the model evaluations with the measured values after eight
hours, which was when the largest facets were observed in the experiment. This
figure also shows the importance of including each of the radiation components. As
expected, the “AS-IS” model evaluation behaves almost identically to the evaluations
presented in Morstad (2004). Notice, this evaluation tended towards a melt-layer
beneath the snow surface, which was not as prominent in the measured data. The
136
−14 −12 −10 −8 −6 −4 −2 0
0
2
4
6
8
10
12
14
16
18
20
Temperature (°C)
D
ep
th
 (c
m)
 
 
AS−IS
VIS
NIR
VIS−NIR
SWIR(1)
SWIR(2)
Measured
Figure 5.6: Comparison between six model evaluations with varying irradiation in-
puts.
melt-layer in the model was common (9 of 13 experiments) in the model/experiment
comparisons in Morstad (2004). Additionally, the “SWIR(2)” evaluation matches
the measured data the closest, particularly in the inflection point region. This single
example highlights the importance of considering, with as much detail as available,
the various components of radiation that impact the energy balance.
5.5 Reliability of Model
Often, the thermal model presented here is used in comparison with measured
temperature data using various environmental sensors for input. Thus, each input
factor has an associated measurement error. Using 1,000 re-samplings, the 95% con-
fidence level intervals were computed using the bootstrap percentile method (Press
137
et al., 1986; Efron, 1987). Based on the “SWIR(2)” evaluation from the previous
section, confidence intervals were calculated assuming that all input parameters have
an associated measurement error that is ±5% of the desired value and which may be
described by a normal distribution such that the 1% tails of the distribution occur at
the 5% values.
Figure 5.7a contains a contour plot showing the maximum deviation from the
mean value of the 1,000 samplings over a 10-hour period, which indicates that the
largest error is approximately 2 ◦C and occurs just below the snow surface. Figure
5.7b depicts a single profile at the 8-hour mark that includes the confidence level
intervals, the “SWIR(2)” model evaluation, and the measured values from Morstad
et al. (2007) Experiment Two. Note, the analysis presented here was simply an
example. Confidence level intervals, a feature available in the thermal model software
presented in Appendix C, need to be computed for any model evaluation. Nonetheless,
both Figures 5.7a and 5.7b show that accurate measurements are crucial when using
the model to compare modeled and measured data.
5.6 Closing Remarks
This chapter summarized the theoretical and numerical development of a 1-D
model for computing snowpack temperatures, which was utilized for additional nu-
merical computation in Chapters 7–10. In addition to the model development, an
example was presented that highlights the importance of the short-wave radiation
attenuation. This example indicated that using both visible and near-infrared com-
ponents may provide more accurate results than using the all-wave component alone.
Finally, in another example, the computed temperature profiles were shown to be
138
0 2 4 6 8 10
0
5
10
15
20
25
30
35
40
Time ( hr)
Depth(
cm
)
Maxde
via
tion
(
◦ C
)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(a)
−12 −10 −8 −6 −4 −2 0
0
5
10
15
20
25
30
35
40
Temperature (°C)
Depth(c
m)
08:30 5% C.I.
08:30 95% C.I.
Measured Data
(b)
Figure 5.7: Graphs demonstrating the model behavior with respect to measurement
error including (a) a contour plot of the largest deviation from the input evaluation
and (b) 90% confidence intervals with input evaluation and measured values.
affected by measurement error when measured data was used for the input terms.
Thus, it is critical that care is taken when using measured data.
139
CHAPTER 6
SOBOL SENSITIVITY ANALYSIS: THEORY AND EXAMPLES
6.1 Introduction
Sensitivity analysis is used to examine the output of a model and how the variation
of this output can be apportioned to the various input factors. The typical purpose for
performing such an analysis is to determine how important the input parameters are
to the overall outcome. Chan et al. (2000) elaborated on the importance of sensitivity
analysis stating that it “is a prerequisite for model building in any setting. . . .”
Two methods of global, variance-based sensitivity analysis (see Saltelli et al., 2008)
shall be briefly discussed: the Fourier Amplitude Sensitivity Test (FAST) and an
extension of the SOBOL method, which was named after I.M. Sobol (1993). FAST
relies on transforming the input parameters into the frequency domain and then
analyzing the variance via the Fourier coefficients. The improved SOBOL method
assesses the variance via Monte Carlo samplings. Cukier et al. (1977) explained the
advantages of using the FAST method, as detailed in the excerpt below.
The sensitivity analysis presented here is nonlinear so that it permits us to
examine large deviations from the nominal parameter values. In addition,
since all parameters are varied simultaneously, one explores regions of
parameter space where more than one parameter is far from its nominal
value. Because of this thorough exploration of the parameter space, it
often turns out that sensitivities of an unexpected nature are revealed.
A careful study of the model will then reveal some complex coupling be-
tween variables, unexpected prior to the analysis, which leads to observed
sensitivity. . . Another frequent and important finding is that a number of
140
sensitivity coefficients corresponding to a large set of parameters turn out
to be negligible. This permits one to focus one’s attention on a greatly
reduced set of Equations.
Since that publication, the SOBOL method has become an equally, if not more,
effective method of sensitivity analysis. FAST was developed throughout a series of
papers to analyze chemical rate equations (Cukier et al., 1973; Schaibly and Shuler,
1973; Cukier et al., 1975) and is centered around theories presented by Weyl (1938).
However, as stated by the creator, the method is not limited to chemical rate equa-
tions. In fact, in a later comprehensive review Cukier et al. (1978) stated that FAST
was developed for sensitivity analysis of large systems of coupled nonlinear equations.
Since the creation of FAST it has been implemented in a variety of applications
(McRae et al., 1982; Uliasz, 1988; Collins and Avissar, 1994; Colonna et al., 1994).
Additionally, variations and improvements have been applied to the method (Smith
and Ginsburg, 1977; Saltelli and Bolado, 1998; Saltelli et al., 1999; Fang et al., 2003).
These papers are not an exhaustive list of the applications and modifications of FAST;
additional references exist for each of the references given here, and Frey and Patil
(2002) provided even more references as well as a review of many other sensitivity
analysis procedures.
Both FAST and the SOBOL method provide sensitivity indices that communicate
the relative importance of each input parameter. The power of the methods is that
both are capable of determining the importance of each factor independent from the
others as well as the importance of the interactions between the inputs. The results
are typically reported as first-order, second-order, and/or total-effect sensitivity in-
dices; first-order yields sensitivity without any interactions, second-order yields the
interactions between pairs of inputs, and total-effect results in the combined effect of
141
first-order and all interactions. The originally FAST was only capable of computing
first-order, but Saltelli et al. (1999) developed an extended FAST method capable of
computing the total-effect indices.
Saltelli (2002) developed an improvement of the SOBOL method that advanced
the computational efficiency beyond that of the extended FAST, which was previously
the most efficient method for computing “total-effect” indices. The SOBOL method
was originally introduced by Sobol (1990, 1993) and been reported and improved by
various authors (Saltelli et al., 1993; Chan et al., 1997; Sobol, 2001; Saltelli, 2002;
Saisana et al., 2005). The improvements made by Saltelli (2002) allow for the calcu-
lation of first- and second-order sensitivity indices as well as the total-effect indices
that reduces computational cost by nearly 50% compared to other methods.
Due to the superior computational efficiency offered by the SOBOL method, it
was selected for the analysis performed in Chapters 8 and 9. This chapter focuses
solely on the theory and application of the SOBOL method. Appendix D includes
the complete program code capable of implementing both methods, which was used
for the examples in this chapter as well as the analysis detailed in later chapters.
This chapter was designed to be a generic, model-independent explanation of the
improved SOBOL method of sensitivity analysis. The chapter begins with a general
discussion of variance and variance-based sensitivity parameters. Then, theoreti-
cal development and stepwise instructions for implementing the SOBOL method are
given. A method for computing confidence levels and adjusting for bias that does
not require any additional model evaluations is then presented. Finally, examples are
included that illustrate the usage and interpretation of the results.
142
6.2 Sensitivity Defined
The following derivation of sensitivity was gathered primarily from Saltelli (2002),
and is one of many derivations presented in the literature (see also Ishigami and
Homma, 1990; Chan et al., 1997, 2000; Homma and Saltelli, 1996). The SOBOL
method is applicable to any function that has a discrete input and output. The only
stipulation is that the input parameters must be independent. Consider the generic
mathematical function
~y = f(~x), (6.1)
where ~y = yj | j = 1, 2, . . . ,m are the model outputs if the function f is evaluated
for the model input parameters, ~x = xi | i = 1, 2, . . . , n.
The mean of the output parameters, E(yj), may be represented as an ensemble,
as:
E(yj) =
∫ ∫
· · ·
∫
yj(x1, x2, . . . , xn)P (x1, x2, . . . , xn)dx1dx2 . . . dxn, (6.2)
where P (x1, x2, . . . , xn) is the combined probability density function of all the input
parameters in ~x.
Equation (6.2) is also known as the expected output for all possible inputs (Cacuci,
2003). For further discussion of the mean ensemble, including a simplified two-
parameter example, review McRae et al. (1982). The expected value computation
for sensitivity analysis relies on the knowledge of each input parameters’ probability
density function, pi(xi), and that each is independent of the others. Thus the total
probability becomes
P (x1, x2, . . . , xn) =
n∏
i=1
pi(xi). (6.3)
143
Using the total probability as defined in Equation (6.3), the expected output may
be re-written as (Saltelli, 2002)
E(yj) =
∫ ∫
· · ·
∫
yj(x1, x2, . . . , xn)
n∏
i=1
pi(xi)dxi, (6.4)
where the dxi is inserted into the product results in the dx1dx2 . . . dxn in Equation
(6.2). In similar fashion the total variance of the output may be expressed as (Saltelli,
2002)
V (yj) =
∫ ∫
· · ·
∫
(yj(x1, x2, . . . , xn))
2
n∏
i=1
pi(xi)dxi − E(y
j)2. (6.5)
Next, one of the input values, xk, is fixed to an arbitrary value, x˜k, where k is an
arbitrary value of the index i representing xk. The resulting variance of the desired
function is then re-written as
V (yj|xk = x˜k) =
∫ ∫
· · ·
∫
(yj(x1, x2, . . . , xk, . . . , xn))
2
n∏
i=1
i6=k
pi(xi)dxi
− E(yj|xk = x˜k)
2. (6.6)
The main purpose of a sensitivity analysis is to remove the necessity for fixing
values. Thus Equation (6.6) is integrated over the probability distribution of the
fixed term x˜k, resulting in the expected value of the variance for the kth input,
E(V (yj|xk)) =
∫ ∫
· · ·
∫
(yj(x1, x2, . . . , xn))
2
n∏
i=1
pi(xi)dxi
−
∫
E(yj|xk = x˜k)
2pk(x˜k)dxk. (6.7)
Subtracting Equation (6.7) from Equation (6.5) results in
V (yj)− E(V (yj|xk)) =
∫
(E(yj|xk = x˜k))
2pk(x˜k)dxk − (E(y
j))2. (6.8)
The left side of Equation (6.8) is equivalent to the variance of the expected value
of the jth output of the function y for the factor xk, which is written as V (E(yj|xk))
144
(Saltelli, 2002). This is the fundamental quantity of variance-based sensitivity anal-
ysis. When normalized with respect to the total variance, V (yj), it is exactly the
first-order sensitivity index Sjk, where
Sjk =
V j(E(yj|xk))
V (yj)
. (6.9)
Recalling that k is an arbitrary value of the i index, this relationship is redefined as
Sji =
V j(E(yj|xi))
V (yj)
. (6.10)
This basic relationship, Equation (6.10), is also commonly referred to as the cor-
relation ratio (Chan et al., 1997). An estimation of this parameter is the foundation
of the SOBOL method.
6.3 Decomposition of Variance
Before presenting the details specific to the SOBOL method of sensitivity analysis,
an understanding of how the variance may be separated into parts is necessary. This
section defines various ways in which variance may be separated into components as
well as some notational conventions that will be used in Section 6.4 when the specific
method of SOBOL is detailed.
As discussed previously, Equation (6.10) is equivalent to the first-order sensitivity
index Sji (Chan et al., 2000). This measure of sensitivity yields the portion of the
total variance that may be contributed to the ith input parameter. Sji refers to the
ith parameter only, uncoupled with any other factors. However, each input factor
may be coupled with each of the other input parameters, thus higher order terms
exist. For example, Sjil refers to the ith second-order indices, where l = 1, 2, ..., n and
l 6= i. The second-order indices give the portion of the total variance due to the ith
and lth inputs interacting.
145
The denominator of Equation (6.10) is the total-variance, which is renamed here
for simplicity as V ji . The total variance may also be expressed as a summation of the
various components and interactions as follows (Chan et al., 2000):
V j =
n∑
i=1
V ji +
n∑
i=1
n∑
g=1
g 6=i
V jil +
n∑
i=1
n∑
l=1
l 6=i
n∑
h=1
h6=l∨i
V jilh + . . . . (6.11)
For example, in a three-input parameter model the total-variance would break down
into three components: first-, second-, and third-order components, namely
V j = V j1 + V
j
2 + V
j
3︸ ︷︷ ︸
1st Order
+ V j12 + V
j
13 + V
j
23︸ ︷︷ ︸
2nd Order
+ V j123︸︷︷︸
3rd Order
. (6.12)
6.3.1 Closed Variance
As done for a single parameter in Section 6.2, the variance of the expected value
for multiple input factors may also be determined, i.e., V (E(yj|xi, xl)). In sensitivity
analysis, this is called a “closed” variance (Saltelli et al., 2004). A closed variance is
the variance associated with respect to specific input parameters, namely
V (E(yj|xi)) = V
j
i
c
= V ji , (6.13a)
V (E(yj|xi, xl)) = V
j
il
c
= V ji + V
j
l + V
j
il , and (6.13b)
V (E(yj|xi, xl, xg)) = V
j
ilg
c
= V ji + V
j
l + V
j
g + V
j
il + V
j
ig + V
j
lg + V
j
ilg. (6.13c)
Notice, for the three parameter case (n = 3) the variance V jilg
c
equals V j because it
contains all the possible variance in the function.
6.3.2 Total-effect Variance
The total-effect variance ,V jTi , is introduced as
V jTi = V
j
i + V
j
i(−i) (6.14)
146
where the −i indicates “all except i” (Homma and Saltelli, 1996; Chan et al., 1997;
Saltelli and Bolado, 1998; Saltelli et al., 2000; Chan et al., 2000). The subscript
i(−i) represents the coupled interactions of the ith input parameter with all other
parameters. For example, referring to the three parameter model, the total-effect
index for the first (i = 1) parameter may be written as
V jT1 = V
j
1 + V
j
12 + V
j
13 + V
j
123︸ ︷︷ ︸
V j1(−1)
. (6.15)
As such, the total variance of Equation (6.11) may be reduced to a summation of
three terms:
V ji =
n∑
i=1
[V ji + V
j
i(−i) + V
j
−i]. (6.16)
This introduces a third term, V j−i, which is the variance not coupled to the ith pa-
rameter. For the first parameter of a three parameter model this term would be
V j−1 = V
j
2 + V
j
3 + V
j
23. (6.17)
6.4 SOBOL Method
In the preceding sections (6.2 and 6.3), no applications specific to the SOBOL
method were defined. This section details the application of the SOBOL method
developed by Saltelli (2002) and further summarized in Saltelli et al. (2004) and
Saltelli et al. (2008). This method is an adaptation of the original SOBOL method
introduced by Sobol (1993).
147
6.4.1 Basic Premise
Equation (6.10) was defined as the basic relationship for computing sensitivity
parameters, which is redefined by breaking the numerator into two components:
Sji =
U ji − E(y
j)2
V j
. (6.18)
Referring to Equation (6.8),
U ji =
∫
E(yj|xi)
2pi(xi)dxi. (6.19)
The SOBOL method relies on the estimation of U ji in Equation (6.19);
Û ji =
∫ ∫
· · ·
∫
yj(x1, x2, . . . , xi, . . . , xn)y
j(x
′
1, x
′
2, . . . , xi, . . . , x
′
n)
n∏
i=1
pi(xi)dxi
n∏
l=1
l 6=i
pi(xl)dxl.
(6.20)
The theory of this transformation is beyond the scope of this summary, but a de-
tailed derivation is presented in Ishigami and Homma (1990) as well as a summary
in Saltelli et al. (1993) and Saltelli (2002). This transformation may be considered a
representation of the square of the expected value, E(yj|xi)2, of a new function that
is defined as the product of yj evaluated with two different input sets, ~x and ~x
′
. The
sets are produced via uniform Monte Carlo samplings, both with K replicates of each
input parameter, i.e., xr,i | r = 1, 2, . . . , K.
The use of these Monte Carlo input parameters allows Equation (6.20) to be
estimated as a summation:
Û ji =
1
K
K∑
r=1
yj(xr,1, xr,2, . . . , xr,i, . . . , xr,n)y
j(x
′
r,1, x
′
r,2, . . . , xr,i, . . . , x
′
r,n). (6.21)
This simplification is only representative provided that the Monte Carlo sample size
is adequately large; Saltelli (2002) utilized K = 1024. The multiplier prior to the
148
summation differs slightly from that of Saltelli (2002), who used 1K−1 instead of
1
K .
The value of 1K was used here for simplicity. With respect to variance, both methods
can be utilized, however for large K, 1K ≈
1
K−1 (Freund and Simon, 1995).
Next, the integral function that defines the expected value, E(yj), and total vari-
ance, V j, that is Equations (6.4) and (6.5), may be estimated in a similar fashion:
Ê(yj) =
1
K
K∑
r=1
yj(xr,1, xr,2, . . . , xr,n) (6.22)
and
V̂ j =
1
K
K∑
r=1
[yj(xr,1, xr,2, . . . , xr,n)]
2 − E(yj)2. (6.23)
As alluded to in Section 6.3, a further extension of SOBOL involves the computa-
tion of variance subsets. For example, consider a function with four input parameters
(n = 4), where ~u = {xr,2, xr,3} and ~v = {xr,1, xr,4}. Recalling the definition of closed
variance in Equation (6.13), the effect of ~v on the total variance may be estimated as
V̂ (E(yj|~v)) = V̂ j
c
~v = Û
jc
~v − Ê(y
j)2, (6.24)
where,
Û j
c
~v =
1
K
K∑
r=1
f(xr,1, xr,2, xr,3, xr,4)f(xr,1, x
′
r,2, x
′
r,3, xr,4). (6.25)
Hence, Û ji , Û
jc
−i, and Û
jc
il may be written as
Û ji =
1
K
K∑
r=1
f(xr,1, xr,2, . . . , xr,i, . . . , xr,n)f(x
′
r,1, x
′
r,2, . . . , xr,i, . . . , x
′
r,n), (6.26)
Û j
c
−i =
1
K
K∑
r=1
f(xr,1, xr,2, . . . , xr,i, . . . , xr,n)f(xr,1, xr,2, . . . , x
′
r,i, . . . , xr,n), (6.27)
and
Û j
c
il =
1
K
K∑
r=1
f(xr,1, xr,2, . . . , xr,i, . . . , xr,n)f(x
′
r,1, x
′
r,2, . . . , xr,i, . . . , xr,l, . . . , x
′
r,n).
(6.28)
149
The ability to estimate the variance of subsets, as done here, provides the basis
for the improved SOBOL method which is detailed in the following section (Saltelli,
2002). This method is capable of computing first-order, second-order, and total-effect
sensitivity parameters.
6.4.2 Improved SOBOL Method
The improved method of SOBOL derived by Saltelli (2002) relies on two Monte
Carlo sampling matrices, each with K replicates of the input variables. These two
matrices are considered the “sample” (W ) and “re-sample” (W ′) matrices:
W =









x1,1 x1,2 · · · x1,n
x2,1 x2,2 · · · x2,n
...
...
. . .
...
xK,1 xK,2 · · · xK,n









and W
′
=









x
′
1,1 x
′
1,2 · · · x
′
1,n
x
′
2,1 x
′
2,2 · · · x
′
2,n
...
...
. . .
...
x
′
K,1 x
′
K,2 · · · x
′
K,n









. (6.29)
These two matrices are used to develop the Ni and N−i matrices:
Ni =









x
′
1,1 x
′
1,2 · · · x
′
1,i−1 x1,i x
′
1,i+1 · · · x
′
1,n
x
′
2,1 x
′
2,2 · · · x
′
2,i−1 x2,i x
′
2,i+1 · · · x
′
2,n
...
...
...
...
...
...
. . .
...
x
′
K,1 x
′
K,2 · · · x
′
K,i−1 xK,i x
′
K,i+1 · · · x
′
K,n









(6.30)
and
N−i =









x1,1 x1,2 · · · x1,i−1 x
′
1,i x1,i+1 · · · x1,n
x2,1 x2,2 · · · x2,i−1 x
′
2,i x2,i+1 · · · x2,n
...
...
...
...
...
...
. . .
...
xK,1 xK,2 · · · xK,i−1 x
′
K,i xK,i+1 · · · xK,n









. (6.31)
150
In Equations (6.32)a–(6.32)d a set of vectors of length K is introduced. Recalling
that i = 1, 2, . . . , n and l = 1, 2, . . . , n|l 6= i, these vectors are defined as follows:
~aj0 = y
j(W ), (6.32a)
~aji = y
j(Ni), (6.32b)
~aj−i = y
j(N−i), and (6.32c)
~ajK = y
j(W
′
). (6.32d)
These vectors refer to the output of the function evaluated with the various input
matrices defined in Equations (6.29) through (6.31). The output vectors aj0 and a
j
K
are j × K dimensional and aji and a
j
−i are j × K × n dimensional. Therefore, the
necessary evaluations of the function in question results in C = K(2n + 2) model
evaluations.
Saltelli (2002) presented a table to aid with calculating the sensitivity analysis
parameters, which assumed a five-parameter model with a single output variable.
Table 6.1 was developed from this table, but was simplified to contain only values
pertinent to the discussion at hand. Saltelli (2002) demonstrated that results when
n < 5 are a special case of this table. However, the methodology presented here
may be applied to all cases; the differences arise in the off-diagonal terms that were
excluded in Table 6.1.
Using the results from Equations in (6.32)a–d and Table 6.1 as a guide, the first-
order, second-order, and total-effect sensitivity indices may be computed. Recogniz-
ing that the scalar products of two ~ai output vectors ~a are proportional to Ê(yj),
151
V̂ (yj), Û j, Û j
c
−i, and Û
jc
il these estimates may be redefined as
Ê(yj)2 =
1
K
~aj0 · ~a
j
K =
1
K
~aji · ~a
j
−i, (6.33)
V̂ j =
1
K
~aji · ~a
j
i − Ê(y
j)2 =
1
K
~aj0 · ~a
j
0 − Ê(y
j)2 =
1
K
~ajK · ~a
j
K − Ê(y
j)2, (6.34)
Û ji =
1
K
~aj0 · ~a
j
−i =
1
K
~aji · ~a
j
K , (6.35)
Û j
c
−i =
1
K
~aj0 · ~a
j
i =
1
K
~aj−i · ~a
j
K , (6.36)
and
Û j
c
il =
1
K
~aji · ~a
j
−l =
1
K
~aj−i · ~a
j
l . (6.37)
Notice each of the above equations has multiple relationships that may be used
for estimation; this results in double estimates of the first, second, and total-effect
indices (Saltelli, 2002). Recalling the break-down of the closed variance in Equation
(6.13), the desired sensitivity parameters are calculated as follows:
Sji =
Û ji − Ê(y
j)2
V̂ j
, (6.38)
Sjil =
Û j
c
il − Ê(y
j)2 − V̂ ji − V̂
j
l
V j
=
V̂ j
c
il − V̂
j
i − V̂
j
l
V j
, (6.39)
and
SjTi = 1−
Û j−i − Ê(y
j)2
V̂ j
. (6.40)
Saltelli (2002) provided the following restrictions on the usage of the total vari-
ance, V j, and expected value, E(yj), estimates, which ensured that all the estimates
available were applied:
1. In computing the first-order indices, Sji , ~a
j
0, and ~a
j
K should be used for Ê(y
j)2
and ~ajK should be used for the computation of V̂
j.
152
2. The total-effect indices should be computed using ~aj0 only for Ê(y
j)2, while ~aj0
and ~ajK should be used for V̂
j.
3. Computation of the second-order indices is completed using one of the output
vectors in the same row or column, i.e., V̂ j
c
il should be calculated using ~a
j
−i and
~aji for Ê(y
j)2 and using ~aj−l and ~a
j
−l for V̂
j.
Finally, the higher-order interaction sensitivity indices may be computed via Equa-
tion (6.41),
Sji(−il) = S
j
Ti
− Sji −
n∑
l=1
l 6=i
Sjil (6.41)
which accounts for any higher-order variance not accounted for by the first- or second-
order indices.
Table 6.1: Matrix detailing the output vectors (~a) used to compute the necessary
sensitivity parameters. This table was adapted from Saltelli (2002) and should be
used in conjunction with Equations (6.33) through (6.40). Note, the j superscript is
omitted for simplicity.
~a0 ~a1 ~a2 ~a3 ~a4 ~a5 ~a−1 ~a−2 ~a−3 ~a−4 ~a−5 ~aK
~a0 V̂ (y)
~a1 ST1 V̂ (y)
~a2 ST2 V̂ (y)
~a3 ST3 V̂ (y)
~a4 ST4 V̂ (y)
~a5 ST5 V̂ (y)
~a−1 S1 Ê(y)2 V̂ c12 V̂ c13 V̂ c14 V̂ c15 V̂ (y)
~a−2 S2 V̂ c12 Ê(y)2 V̂ c23 V̂ c24 V̂ c24 V̂ (y)
~a−3 S3 V̂ c13 V̂ c23 Ê(y)2 V̂ c34 V̂ c35 V̂ (y)
~a−4 S4 V̂ c14 V̂ c24 V̂ c34 Ê(y)2 V̂ c45 V̂ (y)
~a−5 S5 V̂ c15 V̂ c24 V̂ c35 V̂ c45 Ê(y)2 V̂ (y)
~aK Ê(y)2 S1 S2 S3 S4 S5 ST1 ST2 ST3 ST4 ST5 V̂ (y)
153
6.4.3 A “Less Expensive” SOBOL
Saltelli (2002) presented a second approach that is computationally less expensive
(C = K(n+ 2)) than the aforementioned method. Two disadvantages to this method
are that only single estimates are produced and the second-order indices may not
be computed. To perform this analysis, solve the desired relationship to obtain the
output vectors listed in Equations (6.32)a–d, except omit (6.32)c. The first-order
and total-effect indices can be computed as previously described using the Equations
(6.33)–(6.40), but without including any −i vectors. Table 6.1 and the steps listed
on page 151 may still be utilized to perform the calculations; simply overlook the ~a−i
terms and the third step.
6.5 Confidence Levels and Bias Correction
The SOBOL method for computing sensitivity indices, as shown in the previous
sections, is based on estimates of variance. Therefore, confidence should be applied
to the results. When computing the confidence intervals, traditional statistical tech-
niques may not be appropriate, especially for models that require significant computa-
tion time. For example, using traditional statistics, the sensitivity analysis would be
repeated to develop a set of sensitivity indices from which confidence intervals may be
computed. This may be unreasonable for models that require significant computation
time. To circumvent this issue the bootstrap method is presented for calculating the
confidence levels of the SOBOL analysis. The bootstrap method allows for these
calculations without any additional model evaluations. Additionally, a bootstrap
method for estimating the bias is presented.
The theory behind the computations presented in this section is beyond the scope
of this chapter. The information presented is meant to be an overview as required for
154
implementation. For further details discussion of bootstrap analysis, refer to Efron
and Tibshirani (1993). Additional information may also be found in Efron (1987),
DiCiccio and Efron (1996), Hesterberg et al. (2005), and Manly (2007).
Bootstrap calculations are based on re-samplings of the input parameters to create
replicate, or bootstrap, samples. Consider, for example, the data set ~x = {1, 5, 6, 8}.
Bootstrap data sets are created by randomly selecting replacement parameters from
the original data set to create another of the same size, which might be ~x∗1 =
{8, 5, 1, 8}. This process is repeated B times, resulting in B bootstrap samples:
~x∗1, ~x∗2, . . . , ~x∗B. Each set of bootstrap samples is then used to calculate a new
bootstrap estimate of the statistic of interest, for example
θˆ = f(~x) (6.42a)
and
θˆ∗b = f(~x∗b) | b = 1, 2, . . . , B (6.42b)
where θˆ is the value computed using the original data set (~x) and θˆ∗b is computed
with the bootstrap data sets (~x∗b).
With the SOBOL method, the samplings are generated from output vectors de-
fined in Equation (6.32). The re-sampling procedure is performed for each of the
output vectors in a fashion that is consistent across the vectors, such that model
evaluations are not disordered. As expected, the statistics of interest are the sen-
sitivity indices: Sji , S
j
il, and S
j
Ti
. For example, referring to Equation (6.38) and
(6.33)–(6.35), the first-order index would be computed as follows, the superscript j
and the double estimates were omitted for simplicity:
S∗bi =
Û∗bi − (Ê(y)
∗b)2
V̂ ∗b
(6.43)
155
where,
(Ê(y)∗b)2 =
1
K
~a∗b · ~a∗bK , (6.44)
V̂ ∗b =
1
K
~a∗bi · ~a
∗b
i − (Ê(y)
∗b)2, (6.45)
and
Û∗bi =
1
K
~a∗b0 · ~a
∗b
−i. (6.46)
Performing re-sampling and computing B bootstrap sensitivity analysis parame-
ters results in data sets from which confidence intervals are computed following the
general procedure detailed in the following section.
6.5.1 BCa Confidence Level Intervals
Efron and Tibshirani (1993) explained that the BCa method—an abbreviation
for bias-corrected and accelerated—is a “good” method for automatic computation
of confidence level intervals. The bias correction accounts for the difference in ex-
pectation of the original statistic, θˆ, and bootstrap estimates, θˆ∗b. The acceleration
accounts for the rate of change between the standard error of θˆ and the true value. For
details regarding calculation of the BCa confidence levels refer to Efron and Tibshirani
(1993).
The BCa method begins by computing an estimate of bias, zˆ0 that is
zˆ0 = Φ
−1
(
Nb
B
)
, (6.47)
which is a function of the number of bootstrap estimates, θˆ∗b, that are less than the
measured statistic θˆ, which is defined here as Nb. This value is normalized against
the total number of bootstrap samples, B, and then applied to the inverse of the
standard normal cumulative distribution function, Φ−1.
156
The acceleration, âcc, is computed based on the jackknife values of the statistic
θˆ(r). The subscript r is used here because the computation of the jackknife statistic
in the SOBOL method is based on the k Monte Carlo re-sampling. These values
are determined by computing the value of the statistics with the ith input parameter
removed,for example θˆ(r) = f(a−r). The function f(a−r) in the SOBOL method refers
to the computation of the sensitivity analysis parameters using all except the rth
Monte Carlo re-samplings. Equation (6.48) utilizes these jackknife values to compute
acceleration,
âcc =
∑K
r=1(θˆ(·) − θˆ(r))
3
6
[∑K
i=r(θˆ(·) − θˆ(r))
2
]3/2 (6.48)
where θˆ(·) =
∑K
r=1
θˆ(r)
K .
The BCa adjusted percentiles are computed using the relationships in Equation
(6.49),
αlo = Φ
(
zˆ0 +
zˆ0 + z(α/2)
1− âcc(zˆ0 + z(α/2))
)
(6.49a)
αhi = Φ
(
zˆ0 +
zˆ0 + z(1−α/2)
1− âcc(zˆ0 + z(1−α/2))
)
(6.49b)
where z(α) and z(1−α) are computed using the standard normal cumulative distribu-
tion. The α-value corresponds to the percentile interval desired, e.g., α = 0.1 results
in confidence intervals between the 5% and 95% values of the bootstrap estimates of
θ̂∗b. For example, if α = 0.1 then z(0.95) = Φ−1(0.95) = 1.645.
The confidence level intervals are then determined using the αlo and αhi values.
Take for example, a BCa bootstrap sample size of B = 2000 and calculated intervals
αlo = 0.110 and αhi = 0.985. The confidence levels would be the 220th and 1970th
ordered values of θˆ∗b.
157
6.5.2 Bias Correction
An unbiased result is defined as E(θˆ) = θ, i.e., the expected value is equal to the
true value. Using the bootstrap replicates, an estimate of the bias may be computed.
Note, this estimate is different from the bias calculation in the previous section. The
bias estimate method presented here should be applied to the computed value of the
SOBOL sensitivity indices, whereas the previous estimate should be used with the
confidence intervals. The bootstrap bias, biasB, is easily estimated from the mean of
the bootstrap estimates:
θˆ∗b(·) =
B∑
b=1
θˆ∗b/B and (6.50a)
biasB = θˆ
∗b
(·) − θˆ. (6.50b)
6.6 Example 1: SOBOL
The test case presented here is based on the “g function,” which is commonly
used throughout the literature. This function, as defined below, is used here as an
example of the SOBOL technique.
g(x1, x2, . . . , xn) =
n∏
i=1
|4xi − 2|+ qi
1 + qi
. (6.51)
The vector, ~q is defined as ~q = {0, 0.5, 3, 9, 99, 99} and acts as a weighting parameter
for the xi inputs. Small values cause the associated input to become more impor-
tant. Each input parameter, xi, is uniformly distributed between 0 and 1, where
i = 1, 2, . . . , 6 (i.e., n = 6). A SOBOL sensitivity analysis was performed on this
function where K = 10, 000 replicates of the input parameters. Bootstrap confidence
level intervals were computed using B = 10, 000 bootstrap re-samplings. The results
indicate the importance of each xi.
158
Figure 6.1 provides the results of this analysis for the first-order and total-effect
indices. This figure can be directly compared to Saltelli (2002, Fig. 3), which shows
similar results. Additionally, bias-corrected first-order, second-order, and total-effect
indices are provided in Table 6.2.
1 2 3 4 5 6
0
10
20
30
40
50
60
Sensitiv
ity(
%)
S 1
S 2
S 3 S 4 S 5 S 6
(a) First-Order Sensitivity
1 2 3 4 5 6
0
20
40
60
Sensitiv
ity(
%)
S T1
S T2
S T3 S T4 S T5 S T6
(b) Total-effect Sensitivity
Figure 6.1: Results from the SOBOL sensitivity analysis of Equation (6.51), including
the first-order (Si) and total-effect sensitivity (STi) terms. (The error bars reflect the
90% confidence intervals.)
Table 6.2: Improved SOBOL sensitivity indices, in percent, of Equation (6.51); the
values on the diagonal are the first-order indices, the off-diagonal terms are the second-
order indices (e.g., S12 = 14.19%), and the bottom row reflects the total-effect indices.
The tales is a symmetric matrix, but the upper triangular values were omitted for
readability.
i,l 1 2 3 4 5 6
1 57.26
2 14.19 25.73
3 3.61 1.61 3.51
4 -0.67 -0.29 0.04 0.58
5 0.03 0.02 0.00 0.00 0.00
6 0.02 -0.02 0.00 0.00 0.00 0.00
ST 68.30 34.73 3.51 0.00 0.00 0.00
159
This example allows for an analysis that is indicative of what is desired from a
sensitivity of a given function. The first two parameters are the most significant, as
both the first- and second-order indices reveal. It may also be possible to state that the
x3, . . . , x6 terms are negligible and may be omitted in future analyses. The importance
of computing the total-effect and/or the second-order indices is also illustrated. S12
accounts for 14% of the total variance. Without computing the total-effect or second-
order indices this would remain undetected.
This example also highlights the importance of computing confidence intervals.
Recall the total-effect index includes individual sensitivity and the sensitivity of all
interactions. Thus, the sum of the columns in Table 6.2 should comprise a value less
than the total-effect. Performing this computation for the first column yields a value
of approximately 75%, which differs from the reported total-effect index in the table
(68.3%). This difference is likely due to the uncertainty present in the calculation of
the first- and second-order indices, which when added compounds; Figure 6.2 shows
the uncertainty in these terms. This uncertainty is not compounded in the calculation
of the indices themselves, since each is determined from different estimates according
to the criteria listed in Section 6.4.2.
160
1 2 3 4 5 6
0
20
40
60
Sensitiv
ity(
%)
S 1
S 1 ,2
S 1 , 3 S 1 , 4 S 1 , 5 S 1 , 6
Figure 6.2: First- and second-order indices for the first input parameter (x1) from
analysis of Equation (6.51).
6.7 Example 2: Temporal Analysis
In situations that are time dependent SOBOL is particularly useful. The following
example considers an arbitrary function that is dependent on time t:
y(t) =
1
t2
sin(x1) + 7t sin
2(x2) + 0.1t
2x43 sin(x1). (6.52)
For any given set of input parameters this function may be evaluated at any time
t. For example, SOBOL is implemented as usual, but instead of incorporating a
single output as in the previous example, the function will output m values, i.e.,
yj = y(tj) | j = 1, 2, ...,m. Thus, when SOBOL is performed m sets of sensitivity
indices are computed. This allows the sensitivity indices to be plotted with time,
as in Figure 6.3, which depicts the aforementioned time dependent relationship for
t between 1 and 5. Using this figure, it is possible to examine the contribution of
each parameter to the total of all values. For example, at t = 4 the normalized
total-effect S∗Ti indices are 0.58, 0.05, and 0.37, respectively. The asterisk indicates
that the total-effect indices are normalized to the sum of all the total-effect indices,
161
thus guaranteeing that the individual indices are between 0 and 1. For additional
examples and illustrations refer to Saltelli et al. (2000).
1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
T ime (t )
S∗ T
i
x 1x 2x 3
Figure 6.3: Stacked area plot of the time-dependent total-effect indices resulting from
the analysis of Equation (6.52).
6.8 Closing Remarks
This chapter summarizes the theory behind the SOBOL method of sensitivity
analysis, which is a variance-based method capable of computing first-order, second-
order, and total-effect sensitivity indices. The methods defined here have been utilized
by a variety of researchers to analyze chemical rate equations and climate energy
balance models, among others. The tools defined here, including a set of MATLAB
(The Mathworks, Inc.) functions (see Appendix D), were used for analyzing the
critical parameters of the formation weak-layers on the snow surface that often lead
to subsequent avalanches, Chapters 8–10.
162
CHAPTER 7
IMPLEMENTATION OF NUMERICAL ANALYSIS TECHNIQUES
7.1 Introduction
The main objective of the research presented throughout this dissertation was
to define the conditions favorable for surface hoar and near-surface facet develop-
ment. To achieve this goal, two numerical analysis techniques—sensitivity analysis
(see Chapter 6 for details) and Monte Carlo simulation—were employed, these provide
complementary products. The sensitivity analysis quantified the amount of variance
in the model output that was due to the variance of the input parameters, simply
stated, it quantifies the importance of each input on the output. The Monte Carlo
simulations supplemented this analysis by providing a means for determining the set
of inputs that led to a certain output. That is, this technique allowed for a range of
inputs to be associated with a range of outputs. This chapter provides the information
necessary to understand how these methods, as well as two additional data analysis
techniques, were implemented for the problem at hand. This chapter sets the stage
for the work detailed in Chapters 8–10.
7.2 Thermal Model Input Distributions
The analysis presented in Chapters 8–10 is based on a snowpack model derived
from the heat equation (Equation (5.13)). For details regarding the model used, refer
to Chapter 5. This model was chosen for two reasons. First, a nearly identical model
is implemented in RadTherm/RT (ThermoAnalytics, Inc.1) that has been shown to
1http://www.thermoanalytics.com/
163
be successful in predicting snow surface temperatures and mass-flux over spatially
complex terrain (Staples et al., 2006; Adams et al., 2009). Secondly, the analysis
presented in this chapter is computationally expensive. The method used required
hundreds of thousands of model evaluations. Therefore, a computationally efficient
thermal model was advantageous. The evaluation time of the model presented in
Chapter 5 is on the order of a few tenths of a second (Hewlett Packer dv9000; Windows
Vista x64; Intel T9300 at 2.50 GHz; 4 GB RAM).
The model used in the analysis required either 8 or 11 input parameters which are
listed in Table 7.1 along with its respective index reference (i) and assigned symbol
(Sym.) that are referenced throughout the remainder of this chapter and Chapters
8–10. The parameters listed are divided into two groups: snow properties—ρ(1), k(2),
cp(3), κ(5), and α(8)—and environmental conditions.
Table 7.1: List of input parameters, their associated symbol, and index (i) referenced
in the analysis throughout Chapters 8–10.
i Sym. Units Name
1 ρ kg/m3 Snow density
2 k W/(m K) Thermal conductivity
3 cp kJ/(kg K) Specific heat capacity
4 T ints
◦C Initial snow temperature
5 κ m−1 Extinction coefficient
6 LW W/m2 Incoming long-wave radiation
7 SW W/m2 Incoming short-wave radiation
8 α Albedo
9 Vw m/s Wind speed
10 Ta ◦C Air temperature
11 RH % Relative humidity
Both the sensitivity analysis and Monte Carlo simulations required that each
input be assigned a continuous distribution function. The distributions were then
sampled randomly so that all possible values and combinations for each input were
evaluated. Two scenarios were considered in the analysis: “day-light” and “night.”
164
The day-light sets considered all the input parameters including solar input (SW (7))
and the related snow properties, albedo (α(8)) and extinction coefficient (κ(5)). The
night sets were executed in absence of these three “solar” parameters, which explains
the difference between the number of input parameters considered. Within each of
these two scenarios, three locations were developed: a Control set that used uniform
distributions, a South set based on weather data from the South-facing weather sta-
tion, and a North set based on the North-facing weather station. The term “control”
is used loosely to refer to the synthetic location created based only on reasonable
values of each of the input parameters; i.e., if no location specific weather conditions
existed, the distributions defined by Control location may be reasonable estimates of
the input parameters. From this point forward these data sets will be referred to as
scenario/location, e.g., night/North or day-light/South.
Due to limited information regarding the snow properties, uniform distributions
consistent with data published by Armstrong and Brun (2008) were used in all
cases for the five terms listed in Table 7.2. The remaining terms—the environmen-
tal conditions—were fit to distributions based on two seasons of recorded weather
data from two weather stations at the Yellowstone Club ski area located near Big
Sky, Montana. Using a distribution-fitting software package, EasyFit 5.0 (Mathwave
Technologies), these distributions were determined based on mean values of the input
parameters measured at the weather stations for the day-light or night scenarios, with
one exception. Namely, the initial snow temperature T ints (4) for the entire snowpack
was assumed to be the temperature of the snow prior to the onset of day-light or
night.
All the Control input sets were composed of uniform distributions that spanned
reasonable values, these values also served as limits for the South and North data set
distribution functions. For each of the North and South input data sets, the best-
165
Table 7.2: Snow property uniform distribution parameters used for sensitivity analysis
and Monte Carlo simulations.
min. max.
ρ 50 500
k 0.01 0.7
cp 1795 2115
α 0.4 0.95
κ 40 200
fitting distributions functions, based on a Kolomogorov-Smirnov test, were selected
that were also available in MATLAB (The Mathworks, Inc.). Four different distribu-
tion functions were utilized: generalized extreme value (gev), generalized Pareto (gp),
Weibull (wbl), and lognormal (logn), as defined in the probability density functions
(EasyFit 5.0, 2009) which follow. The functions chosen were the
The resulting distributions for each parameter are tabulated in Table 7.3, with
the relevant distribution functions provided in Equations (7.1)–(7.4). Each input
distribution utilized is graphed in Figure 7.1. The distributions presented indicate
the probability that the inputs equal the mean for any given day or night during
the two seasons. Figure 7.1 includes the probability density functions for the input
parameters based on measured data.
fgev(x) =



1
σ exp(−(1 + kz)
−1/k)(1 + kz)−1−1/k k 6= 0,
1
σ exp(−z − exp(−z)) k = 0,
(7.1)
where z = x−µσ and k, σ, and µ are the shape, scale, and location parameters, respec-
tively, which correspond to a, b, and c, respectively, in Table 7.3.
fgp(x) =



1
σ
(
1 + k x−µσ
)−1−1/k
k 6= 0,
1
σ exp
(
− x−µσ
)
k = 0,
(7.2)
166
where k, σ, and µ are the shape, scale, and location parameters, respectively, which
correspond to a, b, and c, respectively, in Table 7.3.
fwbl(x) =
α
β
(
x− γ
β
)α−1
exp
(
−
(
x− γ
β
)α)
, (7.3)
where α, β, and γ are the shape, scale, and location parameters, respectively, which
correspond to a, b, and c, respectively, in Table 7.3.
flogn(x) =
exp
(
− 12
( ln(x−γ)−µ
σ
)2
)
(x− γ)σ
√
2pi
, (7.4)
where σ and µ are the continuous shape parameters and γ is the location parameter,
which correspond to a, b, and c, respectively, in Table 7.3.
Table 7.3: Environmental input parameter distribution sets used for sensitivity anal-
ysis and Monte Carlo simulations; the coefficients (a, b, and c) correspond to the
parameters provided in Equations (7.1)–(7.4).
South North Control
Type* a b c Type* a b c Min. Max.
D
a y
-l
ig
ht
T ints gev -0.39 5.80 -16.34 gev -0.36 6.15 -16.13 -40 0
LW gev -0.09 63.62 287.97 gev -0.04 33.85 245.29 100 600
SW gp -0.89 575.79 39.09 wbl 128.08 2.20 0.00 50 800
Vw logn 0.52 0.33 0.00 gev -0.09 0.28 1.05 0 4
Ta gev -0.24 4.47 -8.19 gev -0.39 4.68 -7.59 -30 10
RH gev -0.66 15.92 60.43 gev -0.73 13.33 62.99 0 100
N
ig
ht
T ints gev -0.45 5.32 -12.65 gev -0.40 5.19 -13.15 -40 0
LW gev -0.25 42.46 262.03 gev 0.07 35.84 236.83 100 600
Vw gev 0.00 0.54 1.17 gev -0.17 0.35 1.02 0 10
Ta gev -0.28 4.51 -11.33 gev -0.41 4.68 -10.11 -30 10
RH gev -0.99 13.72 72.14 gev -0.80 11.26 70.90 0 100
*gev = generalized extreme value; gp = generalized Pareto; wbl = Weibull; logn = lognormal
167
−40 −30 −20 −10 0
0
0.02
0.04
0.06
0.08
Ini tial snow temp. ( ◦ C)
Proba
bilityDe
nsity
(a)
0 200 400 600 800
0
2
4
6
8
x 10−3
Short-wave irradiance (W/m 2)
Proba
bilityDe
nsity
(b)
0 200 400 600
0
0.005
0.01
0.015
Long-wave irradiance (W/m 2)
Proba
bilityDe
nsity
(c)
0 1 2 3 4
0
0.5
1
1.5
Wind speed (m/s)
Proba
bilityDe
nsity
(d)
−40 −20 0 20
0
0.02
0.04
0.06
0.08
0.1
Air temp. ( ◦ C)
Proba
bilityDe
nsity
(e)
0 20 40 60 80 100
0
0.02
0.04
0.06
0.08
Relative humidi ty
Proba
bilityDe
nsity
(f)
Control Day-light/North Night/North Day-light/South Night/South
Figure 7.1: Probability distribution functions for input data based on measured data.
168
7.3 Model Evaluations
The sensitivity analysis and Monte Carlo simulations rely on a multitude of model
evaluations. In each model evaluation, with the exception of short-wave radiation,
all of the input parameters remained constant through time. For each evaluation the
inputs were selected from the input distributions randomly, i.e., random numbers that
follow the assigned distributions defined in Table 7.3. Initially, the entire snowpack
was assumed to begin at the same temperature defined by T ints (4), and the model
was evaluated for a 10-hour period. Data for use in computation was exported in
20 minute intervals. Short-wave radiation was defined as a sine function that had a
mean value equivalent to the mean value recorded in the field measurements.
7.4 Sensitivity Analysis
The extended SOBOL variance-based sensitivity method (Saltelli, 2002) quantifies
the contribution of the variance of each input parameter (see Table 7.1) and the
interactions between input parameters to the total variance of the model output.
The theory behind the method used here, as well as a detailed discussion of variance
with respect to sensitivity, is provided in Chapter 6.
The SOBOL method was performed using the thermal model and a sampling size
of 10,000 replicates (K, see Section 6.4.2) to compute output based on the input
parameters and associated statistical distributions. . The 90% confidence intervals
were calculated using the bootstrap BCa method (Efron and Tibshirani, 1993) with
10,000 re-samplings (B, see Section 6.5). The basic methodology is summarized by
the following steps:
169
1. The input distributions previously defined in Section 7.2 were re-sampled twice
each, 10,000 times in this case, to form two input matrices.
2. These two matrices are used systematically, as defined by the improved SOBOL
method in Section 6.4.2, to construct sets of input parameters for evaluation.
3. These sets of input data are analyzed with the thermal model thus producing
sets of output vectors associated with each of the input parameters, see Chapter
5 and Section 7.3.
4. These vectors are combined using the improved SOBOL methodology to esti-
mate the various sensitivity parameters detailed below, see Section 6.4.2.
5. Confidence level intervals for each parameter are obtained by re-sampling the
output vectors, in this case 10,000 times, creating 10,000 values for each sensi-
tivity parameter from which the 90% confidence intervals were computed, see
Section 6.5.
The results discussed include four terms: the first-order index (Si) gives the con-
tribution of the ith input parameter; the second-order index Si,l gives the contribution
due to interaction between ith and lth terms; the higher-order index (Sh) that includes
all interactions greater than second-order; and the total-effect index (STi ) provides the
contribution of the ith parameter and all associated interactions to the kth order (e.g.,
ST1 = S1 + S1,2 + S1,3 + . . .+ S1,k + Sh), where k is the number of input factors (i.e.,
8 or 11 here), see Section 6.3.2. The i and l subscripts refer to the variable numbers
in Table 7.1. The term “residual” is also used throughout this chapter and refers to
all variance not associated with the parameter under consideration.
170
7.5 Monte Carlo Analysis
While the SOBOL method quantifies the significance of the input parameters,
its limitation is that it does not directly link inputs to a particular output. The
specific outputs utilized are defined in Chapters 8 and 9 and include mass flux, snow
temperatures, and temperature gradients. Thus, Monte Carlo simulations (Press
et al., 1986) were utilized to further quantify the environmental conditions and snow
properties by separating the portion of critical input parameters that led to a specific
output, e.g., the levels of long-wave radiation that are associated with mass-flux
rates typical of surface hoar formation (see Section 8.2). To perform this analysis,
no additional model evaluations were necessary; it was sufficient to rearrange—as
described below—the input and output from the SOBOL method to create a large
set of Monte Carlo simulated data.
The use of this reordered data is a natural extension, as the SOBOL method
is based on Monte Carlo simulations organized in a certain manner. Referring to
Chapter 6 (Section 6.4.2), this is accomplished by gathering the input matrices (W ,
W ′, Ni and N−i; see Equations (6.29)–(6.31)) with the corresponding output vectors
(~aj0, ~a
j
K , ~a
j
−i, and ~a
j
K ; see Equation (6.32)). Using the results from the SOBOL a
Monte Carlo data set was produced with 240,000 and 180,000 replicates for the day-
light and night data sets, respectively.
7.6 Highest Density Regions
A methodology for analyzing the Monte Carlo simulation data is presented; the
data in this section is hypothetical and used simply as an example of how this analyt-
ical tool was applied in the following chapters. Using the Monte Carlo simulations it
171
was possible to separate the inputs responsible for a specified output. Thus, subsets of
the complete Monte Carlo data were defined. These subsets may contain any number
of data points, so it was necessary to define a region that surrounded the data. The
information in this section demonstrates the usage of highest density regions (HDRs),
as defined by Hyndman (1996).
An HDR is defined by cropping the probability density function (PDF) such that
the desired amount of data remains, 95% for example. Imagine slicing a plane through
a normal distribution such that 95% of the data has a probability density greater
than that of the plane. Hence, it may be stated that an observation from within the
population (i.e., a model evaluation) has a 95% chance of falling within this range
(Hyndman, 1996). The HDR region is defined by the slice of the PDF function that
encapsulates the data. By definition this region is the smallest possible region that
satisfies this condition (Hyndman, 1996). Additional discussions of the usage of HDRs
may be found in Scott (1992) and Martinez (2008).
Consider the data presented in Figure 7.2a, which includes an arbitrary output
value (Φ) that is a function of two additional variables (Π1 and Π2). A 3-D HDR
was used to enclose the data in a region that contained a certain amount of the data.
To compute the HDR, first a tri-variate PDF was defined. This is done using the
normal Product Kernel (Martinez, 2008) that resulted in an empirical multi-variate
distribution function. Based on the distribution, HDRs were defined. Figure 7.2b
shows the 5%, 50%, and 90% HDRs of the Monte Carlo simulations based on the
data points in Figure 7.2a.
The 3-D HDRs were not a practical graphic for gathering useful information re-
garding the data presented. Therefore, the 3-D HDR shown in Figure 7.2 is simplified
into a 2-D plot, ignoring the vertical dimension. As was done with the 3-D data, a
Kernel Product estimation of the probability density function (PDF) was defined,
172
Π2
Π1
Φ
(a) (b)
Figure 7.2: Comparison of 3-D representations of (a) the raw data as a scatter plot
and (b) the data encapsulated by 5% (inner), 50% (middle), and 95% (outer) HDRs.
resulting in the bi-variate distribution shown in Figure 7.3a. This PDF was then
used to define the region that encapsulates 95% of the data, as shown in Figure 7.3b.
Π2Π1
Pr
ob
ab
ili
ty
 d
en
sit
y
(a)
Π2
Π
1
(b)
Figure 7.3: The (a) bi-variate probability density function was constructed from the
raw data points shown in sub-figure b; the probability distribution was then sliced
such that 95% of raw data had a probability density greater than this value resulting
in a highest density region trace also shown in sub-figure b.
173
A second example of data set was then considered, where Ψ = f(Π1,Π2). First, for
comparison, the raw data as shown in Figure 7.4a was encapsulated by a 95% HDR,
as was performed in the previous example. Again, utilizing the 3-D representation
is problematic so the data was reduced to a 2-D representation. In Figure 7.4c the
vertical dimension, represented by Ψ, was not ignored but separated into three bands.
The bands only included inputs resulting in the output for the prescribed bands (i.e.,
inputs resulting in values of Ψ of 100-200, 200-300, and 300-400 in Figure 7.4c). For
each band the 95% HDR are shown in Figure 7.4c. For the data in this example, as
demonstrated in Figure 7.4c, Π2 exhibited a much larger range for values of Ψ above
200. Figure 7.4d considers all the data without banding, but includes four different
HDRs: 95%, 90%, 50%, and 10%. This example demonstrates that 50% of the data is
expected to have Π1 ≈ 0.25–0.7 and Π2 ≈ −0.2–0.2. Both of these analysis techniques
were utilized throughout Chapters 8–10.
7.7 Empirical Probability Density Functions
The highest density regions discussed in the previous section required the compu-
tation of the multi-variate probability distribution functions. This was accomplished,
as mentioned, using a product kernel estimate (Scott, 1992; Martinez, 2008) with
a Gaussian kernel. In addition to the multivariate PDFs used in the HDR com-
putations, one-dimensional PDFs were also used for displaying data. In this case,
the kernel estimate was also used, but with an Epanechnikov kernel (Scott, 1992;
Martinez, 2008).
174
−1
0
1
2
3
0
2
4
6
0
100
200
300
400
Π1Π2
Ψ
(a)
−0.5 0
0.5 1
−10
12
3
0
100
200
300
400
500
Ψ
Π2Π1
(b)
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
2
Π 1
Π2
 
 
100−200 (9579)
200−300 (2444)
300−400 (803)
(c)
−0.5 0 0.5 1 1.5
−0.5
0
0.5
1
1.5
2
2.5
3
Π 1
Π2
95%
90%
10%
50%
(d)
Figure 7.4: Example of tri-variate data analysis including (a) a 3-D scatter plot of
raw data, (b) a 3-D 95% HDR, (c) 2-D HDRs encapsulating specific bands of Ψ, and
(d) the 10%, 50%, 90%, and 95% HDRs of complete data set (the number of data
points used to construct each region is included in the parenthesis).
7.8 Goodness-of-fit Hypothesis Test
Throughout Chapters 8–10 distributions of data were compared using the
Kolmogorov-Smirnov Test (Massey, 1951). In all cases the null hypothesis was that
the two distributions being compared were from the same distribution. The test
175
returns a p-value; if the p-value is greater than the level of significance desired then
the null hypothesis is rejected. For example, if two distributions returned p = 0.074,
then at the 5% significance level the null hypothesis would be rejected and it may be
concluded that the two distributions are likely from different populations. However,
at a 10% significance level the test would fail to reject the null hypothesis, meaning
the distributions may be from the same population.
7.9 Closing Remarks
The methods presented in this chapter summarize the tools utilized throughout
the following analytical chapters. The sensitivity analysis quantifies the important
factors influencing the thermal model output. The Monte Carlo simulations allow
for the identification of the input parameters responsible for the desired conditions.
To aid in the visualization of the data, highest density regions were defined. Fi-
nally, throughout the analysis, the Kolmogorov-Smirnov test for goodness-of-fit was
employed to quantify the similarity or difference between various distributions.
176
CHAPTER 8
NUMERICAL ANALYSIS OF SURFACE HOAR
8.1 Introduction
Surface hoar is a particular morphology of faceted snow crystals that forms on
the snow surface. When buried, it is often a contributor to snow avalanches. The
environmental and snow material properties that are conducive to surface hoar forma-
tion have been investigated by a number of authors (Mason et al., 1963; Lang et al.,
1984; Hachikubo and Akitaya, 1997; Cooperstein et al., 2004; Feick et al., 2007). A
review of the literature provided in Chapter 2 reports that surface hoar forms when
air temperature is between -5 ◦C and -15 ◦C, relative humidity is between 60% and
100%, and when the snow surface is about 5 ◦C cooler than the air temperature.
Despite the research that exists, a minimal amount of quantifiable data is available
to firmly determine the necessary conditions for surface hoar formation.
The information presented in this chapter expands on the current understanding
of the conditions surrounding surface hoar formation. Using numerical methods,
namely the SOBOL method of sensitivity analysis (Saltelli, 2002) as well as Monte
Carlo simulations (Press et al., 1986), augmented with observed surface hoar events,
the conditions were explored with a simple 1-D snow thermal model similar to that
used by Morstad et al. (2007). The main objectives of the analysis presented were
two-fold:
1. To identify the most important environmental conditions and snow properties
causing surface hoar formation.
2. To provide a tool for determining when surface hoar forms based on environ-
mental and snow conditions.
177
8.2 Methods
Chapter 7 describes the methods used throughout this chapter, including the
input data set development. The variables considered for this analysis are repeated
here, Table 8.1, for convenience. Since surface hoar typically forms at night, only the
night input scenario was considered. However, all three locations—Control, North,
and South—were considered. The Control location was designed to be generic in
nature and was developed assuming uniform distributions for each of the 11 input
parameters. As the names suggest, the North and South locations were developd
based on site specific weather conditions. Refer to Section 7.2 for a description of the
locations and details of the input distributions.
In the case of surface hoar, one output “class”1 was analyzed: mass-flux (Φ) at
the snow surface. The mass-flux was computed as
Φ =
qe
Ls
, (8.1)
where qe is the latent heat flux computed from Equation (5.35) on page 129 and Ls
is the latent heat of sublimation of ice (see Table 5.1).
Various calculations, referred to as “types,” were considered including the mean,
minimum, and maximum for the 10-hour model evaluations. For reference the mean,
minimum, and maximum of Φ are respectively named as Φ, Φmin, and Φmax. In
addition to considering mass-flux in general, output considering only positive (Φpos)
and negative (Φneg) values was also considered. This was accomplished in each case
by setting mass-flux values equal to zero for Φ < 0 and Φ > 0 for the positive and
negative cases, respectively. Only the mean values were considered for the positive
and negative Φ outputs, referred respectively to as Φpos and Φneg.
1The term class is used here to remain consistent with Chapter 9, which uses multiple classes.
178
The consideration of three locations, one class, and five types generated 15 dif-
ferent outputs data sets. The results shall be referred to by the convention of
location/class-type, e.g., North/Φmin. The complete tabulated results are provided
in Appendix E. Also, in all cases the senstivity indices are listed as percentages, that
indicate the percent of the total variance that may be attributed to the paramter
under considerstion (e.g., S1 = 15% indicates that 15% of the total variance is due to
the ρ acting independently).
Table 8.1: List of input parameters, associated symbol, and reference index used in
the analysis throughout this chapter.
i Sym. Units Name
1 ρ kg/m3 Snow density
2 k W/(m K) Thermal conductivity
3 cp kJ/(kg K) Specific heat capacity
4 T ints
◦C Initial snow temperature
6 LW W/m2 Incoming long-wave radiation
9 Vw m/s Wind speed
10 Ta ◦C Air temperature
11 RH % Relative humidity
8.3 Results: Sensitivity Analysis
8.3.1 Mean Mass-Flux, Φ
The total-effect indices (STi ) result from analyzing Φ for each of the three locations
are presented in Figure 8.1. The figure is a grouped bar chart, with a group of bars
for each of the model inputs. Within each group a bar exists that represents STi for
the specified location, ordered from left to right for the Control, North, and South
input locations. For example, the total-effect index for LW (6) was approximately
40% for the Control and 75% for the North input. The error bars provide the 90%
confidence levels for each index. In all cases, k(2) and cp(3) were irrelevant to the
179
mass-flux at the snow surface. The North results demonstrated the most dramatic
results, where LW (6) dominated the other factors and all of the contributing factors
interacted with each other.
Table 8.2 includes the first-, second-, and higher-order as well as the total-effect
sensitivity indices computed for Φ from the Control/Φ results. The second-order
interactions included ρ(1) interacting with T ints (4) and Vw(9); both of these inter-
actions were of similar magnitude to the first-order index, indicating that the in-
teraction of these parameters interacting was as important as ρ(1) itself. A similar
relationship existed between Ta(10) and RH(11). The first-order indices for these
parameters were approximately 9.0% and 10.7%, respectively, and the second-order
index S10,11 ≈ 7.9%. The similarity of these values also indicated that the interaction
of these two parameters was as important as either parameter individually.
0
20
40
60
80
Sensitiv
ity(
%)
 
 
S T1
S T2 S T3
S T4
S T6
S T9 S T10 S T11
Control
North
South
Figure 8.1: Total-effect indices for Φ for each of the three locations considered, see
Table 8.1 for reference.
The second-order interactions of the parameters observed in the Control/Φ results
were nearly non-existent when the North and South inputs were considered. In fact,
only the interaction of ρ(1) with the T ints (4) (S1,4) was non-zero: 4.5% ≤ S1,4 ≤
8.6% and 4.1% ≤ S1,4 ≤ 9.4% for the North and South, respectively. However, the
results from all locations included significant higher-order interactions, see the tables
provided in Appendix E for the specific values.
180
Table 8.2: Table summarizing the sensitivity analysis parameters (in percent) for the
Control location calculated from Φ. The italic ranges indicate the confidence levels of
each parameter; the first-order indices (Si) are along the diagonal, the second-order
indices (Si,j) are on the off-diagonal (e.g., S1,2 = 0.6), the total-effects (STi ) are listed
in the bottom row, and the higher-order interactions (Sh) in the row labeled “Higher.”
@
@@i
j 1 2 3 4 6 9 10 11
1 1.7 0.6 0.6 2.3 0.2 1.6 0.3 0.9
0.8–2.5 -0.8–2.1 -0.8–2.1 0.8–3.7 -1.7–2.2 0.1–3.0 -1.3–1.9 -0.6–2.5
2 0.6 0.1 -0.2 -0.1 -0.2 -0.5 -0.3 0.0
-0.8–2.1 0.0–0.2 -0.4–0.0 -0.3–0.1 -1.2–0.8 -1.1–0.1 -0.7–0.1 -0.4–0.4
3 0.6 -0.2 0.0 -0.1 -0.2 -0.4 -0.3 -0.0
-0.8–2.1 -0.4–0.0 -0.1–0.2 -0.4–0.3 -1.2–0.8 -1.0–0.2 -0.7–0.2 -0.5–0.4
4 2.3 -0.1 -0.1 -0.3 0.2 0.2 0.4 0.4
0.8–3.7 -0.3–0.1 -0.4–0.3 -1.0–0.5 -1.6–1.9 -1.1–1.5 -1.1–1.9 -1.0–1.9
6 0.2 -0.2 -0.2 0.2 24.6 3.0 2.2 -0.1
-1.7–2.2 -1.2–0.8 -1.2–0.8 -1.6–1.9 23.2–26.0 0.7–5.3 0.1–4.3 -2.0–1.8
9 1.6 -0.5 -0.4 0.2 3.0 11.5 1.6 4.2
0.1–3.0 -1.1–0.1 -1.0–0.2 -1.1–1.5 0.7–5.3 10.3–12.6 0.2–2.9 2.9–5.5
10 0.3 -0.3 -0.3 0.4 2.2 1.6 9.0 7.9
-1.3–1.9 -0.7–0.1 -0.7–0.2 -1.1–1.9 0.1–4.3 0.2–2.9 7.7–10.3 5.6–10.3
11 0.9 0.0 -0.0 0.4 -0.1 4.2 7.9 10.7
-0.6–2.5 -0.4–0.4 -0.5–0.4 -1.0–1.9 -2.0–1.8 2.9–5.5 5.6–10.3 9.7–11.8
Higher 9.2 1.0 1.4 8.7 10.5 14.9 12.3 1.9
Total 17.5 0.3 0.9 11.6 40.1 36.1 33.1 26.0
14.6–20.3 -2.8–3.5 -2.3–4.0 8.6–14.7 37.7–42.6 33.5–38.6 30.4–35.8 23.4–28.7
An interesting result was evident in the behavior of the RH(11). As shown in
Figure 8.1, RH(11) was only significant for the Control/Φ data. Therefore, relative
humidity at the North- and South-facing location had little effect on mass-flux, at
least with respect to the model used here and Φ. This result may find reason in the
range of relative humidity recorded in the field, which was typically between 40%
and 80%; refer to Figure 7.1f (p.167). Field observations (see Chapter 3) at the same
locations demonstrated that surface hoar formed with relative humidities across this
entire range.
8.3.2 Minimum and Maximum Mass-flux (Φmin and Φmax)
The sensitivity analysis results based on the minimum and maximum mass-flux
over the 10-hours simulation provided information regarding the dominant direction
181
of mass-flux. The results of the Φmin analysis are presented in Figure 8.2a. When
compared, the Φ and Φmin results were similar. For example, statistically speaking
the value of ST6 (LW ) did not differ between the mean and minimum results for all
three locations (see Appendix E). This behavior was not observed when comparing
Φ and Φmin.
0
20
40
60
80
Sensitiv
ity(
%)
 
 
S T1
S T2 S T3
S T4
S T6
S T9
S T10 S T11
Control
North
South
(a) Φmin
−20
0
20
40
60
80
Sensitiv
ity(
%)
 
 
S T1
S T2 S T3
S T4
S T6
S T9
S T10
S T11ControlNorth
South
(b) Φmax
Figure 8.2: Total-effect indices for (a) Φmin and (b) Φmax for each of the three locations
considered (see Table 8.1 for reference).
The Φmin results (Figure 8.2b) show a diminished importance of LW (6) and Vw(9)
and increased importance of Ta(10) and RH(11) of the Control location. For the
Control/Φmin results, Ta(10) and RH(11) overshadowed all other inputs. The dif-
ference between the Φmin and Φ results likely indicates that negative mass-flux (i.e.,
smaller values) are dominant. However, making this conclusion definitively is difficult
182
since the Φmin and Φ output used to compute the result does not distinguish the sign,
just the largest or smallest value observed for the model evaluations.
8.3.3 Positive and Negative Mean Mass-flux (Φpos and Φneg)
The dominance of negative mass-flux values was confirmed by considering the
positive and negative mass-flux results separately; the total-effect results are shown in
Figure 8.3. These results did not statistically differ from the minimum and maximum
data shown in Figure 8.2, as the confidence levels for all terms overlapped. Making
this distinction is critical for applying these results to surface hoar formation, which
forms with a mass-flux onto the surface.
0
20
40
60
80
Sensitiv
ity(
%)
 
 
S T1
S T2 S T3
S T4
S T6
S T9
S T10 S T11
Control
North
South
(a) Φneg
0
20
40
60
80
Sensitiv
ity(
%)
 
 
S T1
S T2 S T3
S T4
S T6
S T9
S T10
S T11
Control
North
South
(b) Φpos
Figure 8.3: Total-effect indices for (a) Φneg and (b) Φpos for each of the three locations
considered (see Table 8.1 for reference).
183
For the model used here a positive mass-flux was defined as mass being added
to the snow surface. Therefore, the use of the Φ results shown in Figure 8.1 is
not appropriate, since the results presented in Figures 8.2 and 8.3 indicate that the
negative terms dominate the sensitivity of Φ. For the remainder of the analysis, the
Φpos results will be utilized.
The main objective of the sensitivity analysis was to identify as well as quantify the
most important model inputs for surface hoar formation. The results of Φpos (Figure
8.3b) indicated that the important factors differ depending on the input location
being considered. Also, the total-effect index (STi ) included interaction terms, so it
is important to examine the role these play to truly determine the critical terms.
First the Control location was considered for Φpos; An examination of Figure 8.3b
indicates that only LW (6), Vw(9), Ta(10), and RH(11) contributed to the variance of
the output Φpos. As such were the only parameters that must be considered for this
location.
Figure 8.4 is a grouped bar chart that differs from the charts already displayed
therefore requires some explanation. The groups include four bars, one for each of the
parameters listed as important for Control/Φpos: LW (6), Vw(9), Ta(10), and RH(11).
If the corresponding bars in each group were summed it would equal the total-effect
value provided in 8.3b. The height of the bars provides either the first-, second-, or
higher-order index for the parameter associated with the bar. Consider the following
example, the left-most bar in each group of four refers to LW (6). The first-order
index (S6) for this term is located in the sixth group, which is labeled S6,i. The i
is replaced by the index of the parameter being considered, in this case i = 6, thus
represents S6,6 = S6—the first-order index of LW (6). All other locations in the same
group yield the second-order index for this term, e.g., the first column in the S10,i
group represents the interaction of LW (6) with Ta(10), since the i is replaced with
184
the corresponding index for the column reference it becomes S10,6. This is the second-
order index representing the second-order interaction of LW (6) with Ta(10). Finally,
the higher-order index is provided in the last group labeled Sh. For LW (6) this value
was near zero. Error bars were excluded from this value because it was computed
post-analysis directly from STi ; as such the error is similar in magnitude from the
error associated with the total-effect index.
0
10
20
30
40
Sen
sitivity
(%)
S 1 ,i S 2,i S 3 ,i S 4 ,i
S 6 ,i S 9 ,i
S 10 ,i S 11 ,i
S h
 
 
RH (11)
T a (10)
V w (9)
LW (6)
Figure 8.4: First-, second- and higher-order indices for Φpos for control location and
each of the four important inputs: LW (6), Vw(9), Ta(10), and RH(11) (see Table 8.1
for reference). The higher-order interactions for these terms are provided in the Sh
grouping.
The sensitivity results presented in Figure 8.4 indicate that the most important
parameter for causing changes in Φpos was the interaction of Ta(10) and RH(11):
23.6% ≤ S10,11 ≤ 34.6%. Note, the S10,11 and S11,10 terms are exactly equal. The
first-order index, Ta(10) acting independently, was second in importance: 17.0% ≤
S10 ≤ 22.5%. The total of the first- and second-order indices for Ta(10) and RH(11)
(S10 + S11 + S10,11) is approximately 59.4%. This means that nearly 60% of the
variance observed in the output parameter Φpos for the Control location was due to
changes in these two parameters.
Next, the sensitivity terms for Φpos based on the North location were considered
in a similar fashion. The total-effect results (shown in Figure 8.3b) indicated that six
terms are non-zero: ρ(1), T ints (4), LW (6), Vw(9), Ta(10), and RH(11). Each of these
185
terms was grouped and displayed in Figure 8.5. The North location showed results
that differed greatly from those of the Control location. The most dominant term in
this case was S6, i.e., the effect of LW (6) acting alone, where 27.1% ≤ S6 ≤ 31.5%.
The next most important term was the effect of Ta(1) acting alone, 16.1% ≤ S10 ≤
19.3%, followed by the higher-order terms except for Sh associated with RH(11). Also
significant was the interaction between LW (6) and Ta(10): 4.7% ≤ S6,10 ≤ 11.9%.
The significance of the higher-order terms indicated that the North location was more
interactive than the Control.
0
10
20
30
40
Sen
sitivity
(%)
S 1 ,i
S2,i S 3 ,i
S 4 ,i
S 6 ,i
S 9 ,i
S 10 ,i
S 11 ,i
Sh
 
 
RH(11)
T a (10)
V w (9)
LW(6)
T i nts (4)
ρ(1)
Figure 8.5: First-, second- and higher-order indices for Φpos for North location and
each of the four important inputs: ρ(1), T ints (4), LW (6), Vw(9), Ta(10), and RH(11)
(see Table 8.1 for reference). The higher-order interactions for these terms are pro-
vided in the Sh grouping.
Despite the larger role of interactions, a majority of the variance observed in the
North/Φ results may be attributed to only two parameters: LW (6) and Ta(10). If
the mean values of the first- and second-order terms are considered (S6 +S10 +S6,10),
55.3% of the variance is accounted for by these two parameters alone. And, if the
higher-order terms are included then it may be stated that approximately 79% of the
variance of Φ is in some way attributed to changes in LW (6) and Ta(10) for the North
location.
186
The total-effect results from the south/Φpos shown in Figure 8.3b indicate that the
same six terms as the south/Φpos data were non-zero: ρ(1), T ints (4), LW (6), Vw(9),
Ta(10), and RH(11). However, as shown in Figure 8.6, the sensitivity results differ.
For the South location, mass-flux is highly interactive, since the higher-order terms
dominate. Only the first-order index for LW (6) (18.8% ≤ S6 ≤ 24.0%) was on the
same scale as the higher-order terms, with the exception of RH(11). The highly
interactive results shown here indicated that only in certain situations, when many
terms work together, were the conditions able to influence the positive mass-flux
output.
0
10
20
30
Sen
sitivity
(%) S 1 ,i
S2,i S 3 ,i
S 4 ,i
S 6 ,i
S 9 ,i
S 10 ,i
S 11 ,i
Sh
 
 
RH(11)
T a (10)
V w (9)
LW(6)
T i nts (4)
ρ(1)
Figure 8.6: First-, second- and higher-order indices for Φpos for South location and
each of the six important inputs: ρ(1), T ints (4), LW (6), Vw(9), Ta(10), and RH(11)
(see Table 8.1 for reference). The higher-order interactions for these terms are pro-
vided in the Sh grouping.
8.4 Discussion: Sensitivity Analysis
The sensitivity analysis presented in the previous section demonstrated that at all
locations four parameters—LW (6), Vw(9), Ta(10), and RH(11)—were important to
positive values of mass-flux at the snow surface. The results from South/Φpos were
shown to be highly interactive. As such specific critical parameters were difficult to
187
identify and therefore these results were not considered here. The North/Φpos data
also indicated that both ρ(1) and T ints (4) influenced Φpos to some respect. Examining
the mean values of the sensitivity indices indicated that only 7.7% of the variance of
Φpos may be attributed to these terms alone (S1, S4, and S1,4). Due to this relatively
small influence, these terms were assumed to be secondary influences. Therefore, the
sensitivity analysis indicated that Φpos may be approximated as a function of four
terms: Φpos ≈ f(LW, Vw, Ta, RH). Through the use of a dimensionless parameter Π,
defined as
Π =
−V 2w
CpairTa
·RH, (8.2)
the five-dimensional function is re-written as Φpos ≈ f(LW,Π).
The relationship in Equation 8.2 is analogous the Eckert number (Ec), which is
defined as Ec = U
2
cpT0
and is important for dissipation problems, where U is the free-
stream velocity, T0 the fluid reference temperature (White, 1999, 2006), cp the specific
heat of the fluid. The fluid for the problem at hand is air, thus the specific heat of
air is assumed to be a constant of 1001 kJ/(kg◦K)(Armstrong and Brun, 2008). The
negative sign is applied such that the result becomes positive (this is discussed further
in Section 8.5). The appearance of the Eckert number is reasonable, since dissipation
is defined as a system losing energy resulting in heat generation due to friction or
turbulence—a process that is likely occurring at the snow surface to some extent.
Hence, the dimensionless term in Equation 8.2 is analogous to Ec ·RH, but not equal
to this parameter, since the thermal model is simplistic and assumes nothing regarding
a boundary layer. The velocity (Vw) and temperature (Ta) discussed throughout this
chapter were assumed to act precisely at the snow surface. Finally, RH—being a
dimensionless term itself—is simply applied to make the mass-flux relationship as a
function of two variables.
188
8.5 Results and Discussion: Monte Carlo Simulations
Before delving into the Monte Carlo simulation results, two problems with relating
surface hoar formation to Φpos must be addressed. The issues are rooted in how Φpos
is computed from Equation (8.1) on page 177. The basic premise of the problem is
that Φpos alone does not lead to surface hoar formation, other conditions must be
satisfied, namely the wind speed and air temperature.
In the latent heat equation, Vw(9) is a simple linear relationship, thus the higher
the wind speed, when the air temperature is warmer than the snow, the higher the
mass-flux to the snow. With respect to surface hoar formation this behavior is not
accurate; surface hoar formation becomes inhibited by the high wind (Colbeck, 1988).
However, the wind speeds used here were limited to 4 m/s, which is within the realm
observed by other researchers (see Chapter 2). So, this problem was assumed to be
irrelevant for the results presented here.
The second problem that must be addressed is that of air temperature, which may
be as high as 10 ◦C. The thermal model simply returns a mass-flux and says nothing
of the phase. Based on the literature reviewed in Chapter 2, when air temperature is
above freezing surface hoar would likely not form, even though the mass-flux is well
within the range defined previously. To account for this problem, an assumption was
made that surface hoar only forms with below-freezing air temperatures. Therefore, a
negative value assigned to the Π-term of Equation 8.2 will always result in a positive
quantity.
Knowing that Φpos is a function of two parameters—LW (6) and Π—it was possible
to separate the inputs that lead to a specific output through the use of Monte Carlo
simulations. For the results presented in this section, only the inputs from the Monte
Carlo simulations resulting in a positive mass-flux onto the snow surface over a range
189
from 1.5×10−4–3.0×10−3 gm/(m2s) were considered. This range was defined by a
mass-flux known to be reasonable for surface hoar formation based on recorded data
published by Feick et al. (2007) that reported average mass-flux rates of 1.3×10−3
gm/(m2s) over a 24-hour period resulting in approximately 1 cm surface hoar. For
comparison, Hachikubo and Akitaya (1997) reported mass-flux values of similar mag-
nitude. Using this value, the time frame was shifted to 10 hours and surface hoar
from 0.5–10 mm was assumed possible. This yielded the range of mass-flux defined.
The size of the surface hoar considered here was consistent with the sizes observed in
the field events discussed in Chapter 4, which are used in the following section. The
mass-flux limited as such is defined referred to as ΦSH herein.
As mentioned previously, for the analysis presented here Ta(10) was limited to
sub-freezing values. Making this assumption reduced the number of simulations being
assessed from 180,000 to 135,333, 179,028, and 177,172 for the Control, North, and
South locations respectively. From these data sets, the 95% highest density region
(HDR) was computed for each location. These regions surround the data such that
95% is contained within the area traced. Chapter 7 Section 7.6 provides details
regarding HDRs.
An HDR comparison between the locations of all possible values of LW (6) and
Π with with Ta > 0 is included in Figure 8.7a. Limiting mass-flux to the range
previously defined resulted in Figure 8.7b. These regions are composed of 10.0%,
35.0%, and 17.6% of the simulations considered for the Control, North, and South
locations, respectively.
The simulations shown in Figure 8.7b indicate that in the most general case,
based on the Control location LW (6) ranges from approximately 100–400 W/m2
and Π ranges from approximately 10-4–100. However at both the North and South
locations, the ranges of LW (6) is reduced to 200–300 W/m2 and 10-3–100, respectively.
190
10−7 10−5 10−3 10−1 101
0
200
400
600
800
Π
LW
(6)[W/m
2 ]
 
 Control North South
(a) All Data
10−5 10−3 10−1 101
100
200
300
400
500
Π
LW
(6)[W/m
2 ]
 
 Control
North
South
(b) ΦSH
Figure 8.7: Highest density regions (95%) comparing Monte Carlo simulation results
for LW (6) and Π for (a) all values with Ta < 0 and (b) ΦSH .
This difference may be attributed to the differences in the input distributions; values
of less than 200 W/m2 are improbable (see Chapter 7 Section 7.2). Comparing these
results with the entire data set presented in Figure 8.7a showed that the values of
LW (6) and Π for ΦSH were concentrated to a region in the lower-right corner.
All the Monte Carlo simulations were compared with the ΦSH simulations for
each location in Figure 8.8. Aspect seemed to have a minimal affect on the regions,
which should be expected since the conditions at night should be similar irrespective
of aspect: short-wave radiation is not contributing. The similarity of the North and
South input data sets was evident in Figure 7.1 (p. 167) of Chapter 7. Despite the
similarities in the input distributions, the number of data points resulting in each
region differs by about 50%; the South location is composed of about one-half the
number of points as the North location. This difference, along with the vastly different
and interactive results obtained from the sensitivity analysis, make drawing specific
conclusions on the regions difficult.
191
10−7 10−5 10−3 10−1 101
0
200
400
600
800
Π
LW
(6)[W/m
2 ]
 
 All SH
(a) Control
10−5 10−3 10−1 101
100
200
300
400
500
600
Π
LW
(6)[W/m
2 ]
 
 
All
SH
(b) North
10−4 10−2 100
100
200
300
400
500
Π
LW
(6)[W/m
2 ]
 
 
All
SH
(c) South
Figure 8.8: Highest density regions (95%) comparing the complete set (All) of Monte
Carlo simulation results to the data limited to surface hoar formation (SH) for the
(a) Control, (b) North, and (c) South locations.
8.6 Analysis: Comparison with Field Observations
The sensitivity analysis (Section 8.3) defined the critical parameters governing
mass-flux onto the snow surface, while the Monte Carlo simulations (Section 8.5)
provided a means for identifying values of these parameters that may be conducive
to surface hoar formation. In this section, the Monte Carlo simulation results are
compared with observed events.
192
Chapter 3 details 14 surface hoar events that occurred on the North- and South-
facing slopes—the same locations that the weather station data utilized for the nu-
merical examination presented in this chapter was collected. Based on the weather
data recorded, as well as the field notes describing the surface hoar crystals, values
for crystal size were determined. Table 8.3 includes the average crystal size as well as
LW (6) and Π for each event. In total, 23 observations were made: 15 at the North
Station and 8 at the South Station. The average crystal size observed ranged from
0.5–8 mm. The mass-flux rate of 0.0015 gm/(m2s) defined in the previous section
results in a 0.5 cm crystal in 10 hours.
North South
Event Size LW Π Size LW Π
(mm) W/m2 (mm) W/m2
A-1 2–3 252 5.87E-03 1–2 354 6.87E-03
A-2 0.5 225 9.13E-03
A-3a 0.5 217 1.58E-02 1 376 2.07E-02
A-3b 0.5 417 5.02E-02
A-4 1–2 206 1.51E-02 0.5–1 371 9.93E-03
A-5 4–8 274 1.18E-02 2–4 277 7.55E-03
A-6 1 206 1.01E-02 1 369 1.57E-02
A-7 1 199 8.96E-03 <0.5 267 5.57E-03
B-1 0.5–4 263 4.08E-03
B-2a 0.5 226 2.80E-02
B-2b 2–4 210 1.79E-02
B-3 5 202 1.85E-02
B-4b 1 190 1.06E-02
B-5a 1 188 9.43E-03
B-5b 1.5 193 7.56E-03
B-6 1.5 175 8.18E-03
B-7 0.5–1 317 1.03E-02
Table 8.3: Summary of crystal size, long-wave radiation (LW ), and Π observed surface
hoar events at the North and South Stations.
The field observations were first compared with the numerical results from the
Control location. Figure 8.9 includes HDRs for Control/ΦSH results as well as the
field observations for both the South- and North-facing weather stations listed in
Table 8.3. To some respect the observations fit the regions: 43% (10/23) of the
193
observations fell within the 50% HDR and all the data points were within the 99%
HDR. The later results was particularly interesting considering the 99% HDR contains
an abnormality that the encompassed five points that may otherwise have not fit in
the region. On the other hand, 26% (6/23) of the data points fell outside the 90%
HDR, where only two or three should have been in this region. The discrepancy
between the observations and numerical results was likely caused by two factors:
1. Very small values of Π (on the order of 10-3 or less) resulted due to decimal
values of Vw(9) that were raised to the second power and to a lesser extent large
values of Π resulted from similarly small decimal values of Ta(10).
2. The observations of surface hoar were limited by the conditions observed in the
field.
10−5 10−4 10−3 10−2 10−1 100 101
0
100
200
300
400
500
Π
LW
(6)
[W
/m
2 ]
 
 
Field Data
Control /ΦSH
99%
90% 50% 10%
Figure 8.9: Comparison of Control/ΦSH highest density regions with field observa-
tions from the North- and South-facing stations.
To account for the first of the two problems defined, two adjustments to the nu-
merical data were made. First, Figure 8.10 must be introduced. This figure includes
194
two sets of HDRs. The first set was generated from the Monte Carlo simulations
as discussed above. The second set was generated from daily means from the entire
weather data set—the same data used to generate the location-specific input distribu-
tions, as detailed in Section 7.2. Two restrictions to the Monte Carlo simulation data
were implemented to minimize the small-number issues introduced: Vw(9) was limited
to values greater than 0.25 m/s and Ta(10) to values less than -0.1 ◦C. These values
were selected such that the extent of the Π values in Figure 8.10 were approximately
the same magnitude for the two region sets displayed. These adjustments are included
in the following analysis.
10−5 10−4 10−3 10−2 10−1 100 101
0
100
200
300
400
500
Π
LW
(6)
[W
/m
2 ]
 
 
Control / ΦS H
All Field
SH Observations
Figure 8.10: Comparison of 99% (outer) and 50% (inner) HDRs for the Control/ΦSH
results and all recorded field data with the field observations from the North- and
South-facing stations.
The Control results yielded promising results. Referring to Figure 8.10, all of the
observed surface hoar events fell within the 99% HDR from the field data. Although,
if the observed events were a purely randomly selected subset, half of the points
would lie inside the 50% HDR, but only 30% (7/23) were within this region. So, it is
195
reasonable to state that the observed surface hoar events were not a random sample
of the complete data set. Next, if the observed surface hoar events were a sample
from the numerically generated data, then half of the points would lie within the 50%
HDR of the Control/ΦSH results. As stated earlier, 10 of the 23 events (43%) were
within this region; this is promising considering a major portion of this 50% HDR is
outside the conditions attainable in the field on any given day.
A similar analysis was performed for the North and South locations exclusively,
as shown in Figure 8.11, but the HDRs were limited to the numerical Monte Carlo
simulations results only. Hence, the North/ΦSH and South/ΦSH regions were com-
puted from the data limited to mass-flux rates defined as conducive for surface hoar
development. And the regions labeled at “All South Sim.” and “All North Sim.”
were developed from all values of the inputs (i.e., not limited). The “Sim.” identifier
is added to decifer this data from that of Figure 8.10 that displays regions computed
from the measured field data, whereas the regions in 8.11 were computed entirely
form simulated data (i.e., “Sim.”).
In Figure 8.11a, the results from the South indicated that the numerical results
did not correlate with the observed field data. Due to this lack of correlation, as well
as the sensitivity analysis results that yielded little in the way of definitive results,
the data from the South was excluded from further analysis.
The most promising results were obtained from the data based on the North-facing
station, as shown in Figure 8.11b. First, as expected, all of the observed events were
within the field data 99% HDR. Also, only 27% (4/15) of the data was within the
50% HDR of the field data, which indicated that the observed events at the North
Station were not a random sample of the complete data set itself. With respect to
the numerically generated 50% HDR from the North/ΦSH results, 53% (8/15) of the
observed events were within this region. To illustrate this result further, Figure 8.12
196
10−4 10−3 10−2 10−1 100
100
150
200
250
300
350
400
450
Π
LW
(6)
[W
/m
2 ]
 
 
South / ΦS H
All South Sim.
SH Observations
(a) South
10−4 10−3 10−2 10−1 100
100
150
200
250
300
350
400
450
500
Π
LW
(6)
[W
/m
2 ]
 
 
North / ΦS H
All North Sim.
SH Observations
(b) North
Figure 8.11: Comparison of 99% (outer) and 50% (inner) HDRs for the ΦSH results
and all recorded field data with the field observations from the (a) South- and (b)
North-facing stations.
was generated to compare various HDRs from the numerical results with the observed
events.
197
10−3 10−2 10−1 100
160
180
200
220
240
260
280
300
320
A-1
A-2
A-3a
A-4
A-5
A-6A-7
B-1
B-2a
B-2b
B-3
B-5aB-5b
B-6
B-7
Π
LW
(6)
[W
/m
2 ]
 
 
Field Data
North/Φ SH
99%
90%
50%
25%
10%
Figure 8.12: Comparison of North/ΦSH highest density regions with field observations
from the North-facing station.
Comparing the HDRs and observations in Figure 8.12 demonstrated that the
observations fit reasonably well within the numerically generated data. For example,
1 of 15 (7%) points were within the 10% HDR, 3 of 15 (20%) were within the 25%
HDR, 8 of 15 (53%) were within the 50% HDR, and 14 of 15 (93%) were within the
99% HDR.
This apparent fit was confirmed using the Kolmogorov-Smirnov goodness-of-fit
test (KS-test) for both LW and Π. These KS-tests were performed independently on
the two parameters comparing the Monte Carlo simulation data, North/ΦSH , with
the observed events. The results were p-values of 0.198 and 0.094, for LW (6) and
Π respectively. The null hypothesis was that the observed events and the Monte
Carlo simulation data were from the same distribution. Hence, for both terms, the
hypothesis fails to be rejected at the 5% confidence level indicating that the two data
sets were likely from the same distribution. Interestingly, the Π data fit better than
the LW (6) data (larger p-value for Π), which does not seem to be the case visually.
198
However, the B-7 event likely caused this result for the LW (6) data since it was well
outside the North/ΦSH data.
Finally, the surface hoar size was examined. Table 8.4 defines four ranges of mass-
flux and the expected size of the surface hoar crystal based on the previously defined
rate. Figure 8.13 includes 95% HDRs for each of these. Upon examining the size
of the observed events reported in Table 8.3 as well as the over lap of the regions,
it was obvious that the regions did not correspond to surface hoar size. This figure
does however demonstrate the bias in the model towards large values of Π for large
mass-flux values. This bias was likely, as previously discussed, due to the nature of
the latent heat equation utilized that unequivocally increased as wind velocity and
air temperature increased. This result confirmed that mass-flux alone is insufficient
for predicting the size of surface hoar formation.
Since surface hoar size and mass-flux were uncorrelated in Figure 8.13, a KS-test
was performed in similar fashion as done previously for the North/ΦSH , but here the
field data was compared to all positive values of mass-flux for the North location:
North/Φ > 0. The goodness-of-fit results obtained in this differed significantly from
the North/ΦSH results. In the case of LW (6) the fit actually improved significantly
(p = 0.84), but the Π goodness-of-fit decreased markedly (p ≈ 0). Therefore, with
respect to Π simply considering a positive mass-flux is not appropriate. So, Figures
8.9 and 8.12 may be the best tools for determining if a positive mass-flux exists
conducive to surface hoar formation.
Table 8.4: Regions of mass-flux and expected surface hoar crystal size.
Mass-flux rate Crystal size
10-3gm/(m2s)
0–0.15 <0.5 mm
0.15–3 0.5–10 mm
3–6 1–2 cm
6–30 2–10 cm
199
10−4 10−3 10−2 10−1 100
140
160
180
200
220
240
260
280
300
320
Π
LW
(6)[W/m
2 ]
 
 
 A−1
 A−2
 A−3a
 A−4
 A−5
 A−6
 A−7
 B−1
 B−2a
 B−2b
 B−3
 B−5a B−5b
 B−6
 B−70−0.15 (5988)
0.15−3 (63053)
3−6 (7677)
6−30 (687)
Figure 8.13: Highest density regions based on the North location and various mass-
flux rates.
8.7 Closing Remarks
Using SOBOL sensitivity analysis, four parameters—incoming long-wave radia-
tion, air temperature, wind velocity, and relative humidity—were defined to be the
most important based on their effect on mass-flux rate at the snow surface. An input
data set composed of conditions that varied uniformly demonstrated that air temper-
ature and relative humidity were the most important, with long-wave radiation and
wind velocity playing a secondary role. However, data sets derived from measured
data indicated the long-wave radiation was the most important parameter, accounting
for 20–30% of the total variance observed in mass-flux by itself. When interactions
were considered, approximately 60% of the mass-flux observed was in some way re-
lated to incoming long-wave radiation. This finding may be the most significant
finding to result from the work presented in this chapter. To the author’s knowledge
previous field studies—excluding the data presented in Chapter 4—typically neglect
200
to consider incoming long-wave radiation, but this study indicates that such data
may be vital for identifying the conditions leading to surface hoar development.
The shortfall of this study lies in trying to relate surface hoar formation to mass-
flux rates. Thus, the results obtained which indicated that incoming long-wave ra-
diation is a critical parameter was limited by the use of mass-flux as the indicator
variable. The Monte Carlo simulation data demonstrated that mass-flux alone was
insufficient for predicting surface hoar.
In Section 8.6 the results from the Monte Carlo simulations were explored and
compared with observed surface hoar events. The results indicated, based on the
model evaluation presented here, that mass-flux was insufficient for predicting surface
hoar particularly in characterizing the size of the surface hoar. Out of the three data
sets explored—Control, South, and North—the North input set resulted in the best
correlation. However, this correlation was limited strictly to the presence of a positive
mass-flux. As such, Figures 8.9 and 8.12 were presented as possible tools for assessing
if a positive mass-flux exists that would be conducive to surface hoar formation. Based
on this data, long-wave radiation of 100–200 W/m2 and a value of Π from 10−2.9–
10−1.5 may be considered the optimum conditions for positive mass-flux to occur,
which is necessary of surface hoar formation. This range was defined from a 50%
HDR of the most general case (the Control location). If the North location is used
this range narrows to long-wave of 200–250 W/m2 and Π from 102.5–10−1.7 (see Figure
8.12). These charts demonstrate the ultimate goal for the work presented throughout
this chapter: to develop a methodology for assessing surface hoar formation based
on modeled data that may then be applied spatially via the RadTherm/RT software
package. The results presented here require additional validation for such application,
but are a stepping stone to this end.
201
The work presented shows the potential for sensitivity analysis, Monte Carlo sim-
ulations, and similar numerical methods to aid in exploring the behavior of the snow-
pack to an extent not possible with laboratory or field experimentation alone. But, in
the case of surface hoar formation, the application of these methods is restricted by
the limited knowledge that exists of surface hoar formation on the micro-structural
and micro-meteorological scale. This knowledge is critical for taking the next step in
spatial modeling.
202
CHAPTER 9
SENSITIVITY ANALYSIS OF NEAR-SURFACE FACETS
9.1 Introduction
Faceted snow crystals form at or near the snow surface due to temperature gradi-
ents that are induced by a variety of sources including diurnal temperature changes,
melt-layers, and solar radiation (Fukuzawa and Akitaya, 1993; Hardy et al., 2001;
Morstad et al., 2007; Slaughter et al., 2009). However, the specific environmental
conditions required to form near-surface facets are not well understood. A review of
the literature investigating these conditions is presented in Chapter 2.
The research presented in this chapter used a numerical methodology to expand
the knowledge surrounding the formation of near-surface facets, particularly focused
on the process of radiation-recrystallization. The main objective was to quantify
the most critical environmental and snow micro-structure parameters that lead to
near-surface metamorphism, which was accomplished using variance-based sensitivity
analysis. The data presented sets the stage for Chapter 10 that further quantified—
through the use of Monte Carlo simulations—the specific conditions necessary for
facet formations.
9.2 Methods
This chapter employed the variance-based sensitivity analysis method of SOBOL
(Saltelli, 2002), which was summarized in Section 7.4 and described in detail in Chap-
ter 6. The input parameters used in this chapter are defined in Table 7.1, which
includes the index (i), symbol (Sym.), and description that are used for referencing
203
the inputs. The short-wave radiation input (SW (7)) was assumed to vary temporally
according to a sine-wave (see Section 7.3).
For this analysis three input data sets were considered based on the “location”
from which the distributions were developed: Control, South, and North. The de-
velopment of these data sets was explained in Section 7.2. For each location, nine
different output “classes” were considered:
ˆ the snow temperature at the surface, 2, 5, and 8 cm deep (T0, T2, T5, and T8
respectively);
ˆ the snow temperature at the depth of the “knee” temperature gradient (TK);
ˆ the temperature gradient between the snow surface and 2, 5, and 8 cm deep
(TG2, TG5, and TG8 respectively); and
ˆ a “knee” temperature gradient (KTG).
The “knee” related outputs were defined according to the temperature profile that
is characteristic of solar penetration and surface radiative cooling (Figure 9.1), which
is typically associated with radiation-recrystallization. In this case, the gradient was
calculated between the surface and the inflection point. If the “knee” shape was not
present a value of zero was assigned to the output, otherwise the magnitude of the
gradient was utilized.
For each class mentioned above various model output calculations were preformed.
These various outputs of the thermal model were used in the SOBOL sensitivity
analyses. First, a sensitivity analysis was performed temporally at 20 minute intervals
for each of the classes mentioned, resulting in 30 sets of indices for each class. Next,
the temporal data from the thermal model was reduced to a single output in the
204
Temperature
Snow Surface
Depth
“Knee” Temp. (TK)
Surface Temp. (T0)
“Knee” Depth (d K ) }
“ K
ne
e”
 T e
m p
.  P
rof
i l e
KTG =
TK − T0
dK
Figure 9.1: Schematic of “knee” temperature profile and related sensitivity analysis
output parameters.
form of the mean, maximum, minimum, and mid-day values from the entire 10-
hour simulations. The results from all of these sensitivity analyses are denoted by
modifying the class previously defined as:
ˆ The temporal results list the symbols as a function of time, e.g., T0(t).
ˆ The mean results utilizes the bar, e.g., T0.
ˆ The minimum (min) and maximum (max) for each evaluation and mid-day
utilize a superscript, e.g., T0min, T0max, T0mid.
The input and output combinations will henceforth be referred to as loca-
tion/symbol, where symbol is the modified class symbols listed above. For example,
South/KTG was used when considering the mean “knee” temperature gradient for
the South input location and Control/T5max when the maximum temperature at a
depth of 5 cm was considered based on the Control input location.
205
9.3 Results and Discussion
The sensitivity analyses results presented use the various combinations of input
locations (i.e., Control, South, and North) and outputs (e.g., T5max or TG2min) to
provide a better understanding of what inputs are the most important to the forma-
tion of near-surface facets. Due to the enormity of the data produced, considering
all the results is impossible and it is easy to become enveloped in the subtle nuances
of the analyses conducted. To make the data presented in this chapter accessible, a
majority of the results are presented graphically and specific quantities are kept to a
minimum, only being listed when deemed to be a significant finding with respect to
radiation-recrystallization. Also, when quantities are reported only the mean values
are given, however each parameter has defined confidence levels associated. Appendix
F includes the complete results from the sensitivity analyses presented here, with the
exception of the temporal data, which only includes the total-effect results with time
and the complete results at mid-day.
9.3.1 Snow Temperatures
All of the gradient computations are based on the gradient between the snow
surface and a temperature at depth. Hence, analyzing the results of the snow surface
temperature (T0) is an obvious starting point. The critical input parameters affecting
the snow surface temperature, the temperatures at 2 cm, 5 cm, and 8 cm depth (T2,
T5, and T8, respectively), and the temperature at the “knee” (TK) were considered.
Snow Surface Temperature: The total-effect indices as a function of model
evaluation time for the three locations are included in Figure 9.2. The indices in
these figures are expressed as the normalized total-effect indices (S∗T ; see Chapter 6
206
Section 6.7). Two obvious results arise: (1) LW (6) is the most influential input at
all locations and (2) the importance of SW (7) and α(8) differ with location.
Time, hr
S∗ T
(%)
LW(6)
T a (10)
T i nts (4)
ρ(1)
V w (9)
0 2 4 6 8 10
0
20
40
60
80
100
(a) Control
Time, hr
S∗ T
(%) LW(6)
T a (10)
ρ(1)
T i nts (4)
α(8)
0 2 4 6 8 10
0
20
40
60
80
100
(b) North
Time, hr
S∗ T
(%)
0 2 4 6 8 10
0
20
40
60
80
100
α(8)
T a (10)
ρ(1)
SW(7) V w (9)
T i nts (4) LW(6)
(c) South
Figure 9.2: Stacked area charts of normalized total-effect sensitivity (S∗T ) for T0(t)
for the (a) Control, (b) North, and (c) South locations. The regions are stacked from
bottom to top in order as listed in Table 7.1.
The area charts shown are useful for monitoring the progression of the sensitivity
parameters over time, but two difficulties arise when trying to gather specific quanti-
ties. Figure 9.2 only includes the total-effect results, which incorporate interactions,
207
but do not decipher distinctly the components of the interactions. To determine what
interactions are important, the first- and second-order indices must be considered,
but doing so would require 11 area charts (one for each input parameter) for every
output under consideration, which is not practical. Secondly, the area charts do not
display the confidence bounds. Therefore, it is necessary to establish a suitable single
parameter capable of capturing the important information contained in the temporal
data.
First, the mean surface temperature is considered, as presented in Figure 9.3a. For
the T0 analysis LW (6) is the most influential parameter—two-thirds of the output
variance is due to some respect with changes in LW (6). The only other parameters
that influence the snow surface temperature (i.e., a non-zero total-effect index) are
ρ(1), T ints , and Ta(10). Additionally, the results did not differ with location. However,
the lack of significance of the SW (7) and α(8) indicated that the mean does not reflect
the conditions during the day.
The results obtained by considering the maximum value of the surface tempera-
ture (i.e., T0max) were considered first, see Figure 9.3b. These results yielded total-
effect indices consistent with the temporal data, that is the SW (7) and α(8) inputs
contributed to the output variance. Upon further scrutiny the T0max results were
problematic in practice, since the sensitivity analysis does not decipher when the
maximum occurs, the maximum used for computation could then be from any point
during the simulations. With respect to radiation-recrystallization the fluctuation
due to solar radiation may be missed. Therefore, the mid-day results were computed
(see Figure 9.3c) and assumed to be the most relevant to the problem at hand, which
is explored in further detail in the following sections.
The T0mid results, by definition, align exactly with the five hour point on each
of the graphs in Figure 9.2. Overwhelmingly, LW (6) was the most influential input.
208
Also, ρ(1), T ints (4), SW (7), α(8), Vw(9), and Ta(10) each influence the snow surface
to some extent, depending on the location. Before specific conclusions may be stated
regarding the most influential terms on snow surface temperature the interactions
should be considered, as shown in Figure 9.4 for the South location.
−20
0
20
40
60
80
Sensitiv
ity(
%) S T1
S T2 S T3
S T4
S T5
S T6
S T7 S T8
S T9
S T10
S T11
 
 
South
North
Control
(a) T0
0
20
40
60
80
Sensitiv
ity(
%)
S T1
S T2 S
T
3
S T4
S T5
S T6
S T7 S T8 S T9
S T10
S T11
 
 
South
North
Control
(b) T0max
0
20
40
60
80
Sensitiv
ity(
%) S T1
S T2 S T3
S T4
S T5
S T6
S T7
S T8 S T9
S T10
S T11
 
 
South
North
Control
(c) T0mid
Figure 9.3: Total-effect sensitivity indices for the Control, South, and North locations
for (a) T0, (b) T0max, and (c) T0mid (see Table 7.1 for reference).
The first-, second-, and higher-order interactions for the South location are pre-
sented in Figure 9.4 and for the Control and North locations in Figure 9.5. These fig-
ures only include data for the values listed previously—ρ(1), T ints (4), LW (6), SW (7),
α(8), Vw(9), and Ta(10)—as important from the total-effect results. These grouped
bar charts require explanation, they differ from the charts already presented, but
209
−5
0
5
10
15
20
25
30
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i S4 ,i S5 ,i
S6 ,i
S7 ,i
S8 ,i
S9 ,i
S10 ,i
S11 ,i
Sh
 
 
T a (10)
V w (9)
α(8)
SW(7)
LW(6)
T i nts (4)
ρ(1)
see inset (b)
(a) South/T0mid
 
 S 8
S 8 ,9 S 8 ,10S 8 ,7
S 8 ,4
S 8 ,1
(b) Zoom inset of α(8)
Figure 9.4: First-, second-, and higher-order indices for (a) the South/T0mid sensitiv-
ity analysis and (b) a zoomed view focusing on α(8) (see Table 7.1 for reference).
are similar to the charts used in Chapter 8 (see p. 183). For each chart, the bars
are grouped. Each group includes seven bars, one for each of the parameters listed
above as important. The height of the bars provides either the first- or second-order
index for the parameter associated with the bar. Consider the inset in Figure 9.4b,
it provides a zoomed view of the group associated with α(8) from Figure 9.4a. For
each bar in the group the i is replaced with the corresponding input term in the
legend. For example, the first bar corresponds with ρ(1), thus becomes S8,1, which is
the interaction of α(8) and ρ(1). In this example, the first-order index occurs with
i = 8 (S8,8 = S8). The higher-order interactions—labeled as Sh—for each term are
210
provided in similar fashion. Based upon the data presented in Figures 9.4 and 9.5
the following may be stated:
1. Control/T0mid (Figure 9.5a): The output variance is mainly related to LW (6),
which accounts for approximately 60% of the variance, if the total-effect is
considered (ST6 ≈ 60%). The next most important term is S
T
10, approximately
25% of the output variance may be attributed to this term.
2. North/T0mid (Figure 9.5b): A vast majority of the output variance (80%) was
due to the first-order indices. Over one-third (35%) of the output is attributed
to LW (6) alone, 21% to Ta(10), 12% to α(8), 9% to SW (7), and 2% to T ints (4).
3. South/T0mid (9.4): A majority (59%) of the total variance may be attributed
to the first- and second-order results for LW (6), SW (7), and α(8) (i.e., S6 +
S7 +S8 +S6,7 +S6,8 +S7,8 ≈ 59%). The remainder of the variance is associated
with Ta(10) or the higher-order interactions.
211
−10
−5
0
5
10
15
20
25
30
35
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i S4 ,i S5 ,i
S6 ,i
S7 ,i
S8 ,i S9 ,i
S10 ,i
S11 ,i
Sh
 
 
T a (10)
V w (9)
α(8)
SW(7)
LW(6)
T i nts (4)
ρ(1)
(a) Control/T0mid
−5
0
5
10
15
20
25
30
35
40
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i
S4 ,i
S5 ,i
S6 ,i
S7 ,i
S8 ,i
S9 ,i
S10 ,i
S11 ,i Sh
 
 
T a (10)
V w (9)
α(8)
SW(7)
LW(6)
T i nts (4)
ρ(1)
(b) North/T0mid
Figure 9.5: First-, second-, and higher-order indices for the (a) Control/T0mid and
(b) North/T0mid sensitivity analysis (see Table 7.1 for reference).
Snow Temperatures at Depth: The time-dependent sensitivity analysis results
for snow temperature at various depths—TK(t), T2(t), T5(t), T8(t)—for the South
location are provided in Figure 9.6. The most obvious result gained was the similarity
between TK(T ) and T2(t). In fact, at mid-day the TK(t) values do not statistically
differ from the T2(t) results; for all the inputs parameters the confidence intervals
overlapped. This result was also true for the Control and North locations. Therefore,
212
it is reasonable to conclude that the “knee” is located at approximately 2 cm deep in
the snowpack. For comparison, Figure 9.7 includes the TK(t) results for the Control
and North locations.
Time, hr
S∗ T
(%)
LW(6)
SW(7)
α(8)
T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(a) South/TK(t)
Time, hr
S∗ T
(%)
0 2 4 6 8 10
0
20
40
60
80
100
LW(6)
T i nts (4)
ρ(1)
SW(7)
α(8)
T a (10)
k (2)
(b) South/T2(t)
Time, hr
S∗ T
(%)
0 2 4 6 8 10
0
20
40
60
80
100
ρ(1)
k (2)
SW(7)
α(8)
T a (10)
LW(6)T i nts (4)
(c) South/T5(t)
Time, hr
S∗ T
(%)
0 2 4 6 8 10
0
20
40
60
80
100
ρ(1)
k (2)
LW(6)
SW(7)
α(8)
T i nts (4)
(d) South/T8(t)
Figure 9.6: Stacked area charts of normalized total-effect sensitivity as a function of
model evaluation time at the (a) “knee”, (b) 2 cm, (c) 5 cm, and (d) 8 cm depth for
the South locations. The regions are stacked from bottom to top in order as listed in
Table 7.1.
213
Conceptually, the T2(t), T5(t), and T8(t) results in Figures 9.6b–d demonstrate
the attenuation of short-wave radiation expected. At 8 cm the role of SW (7) and
α(8) are greatly diminished. Interestingly, the extinction coefficient (κ(5)) appears to
be negligible with respect to effecting the snow temperature at depth. As discussed,
the time-dependent results are not practical for providing quantitative results. Thus,
considering the focus of this chapter on the radiation-recrystallization process, only
the TK results are analyzed further. Additionally, given the findings for the snow
surface temperature (T0) only the mid-day values are considered. The total-effect
results for TKmid are presented in Figure 9.8.
Time, hr
S∗ T
(%)
LW(6)
SW(7)
α(8)
T a (10)
T i nts (4)
V w (9)
0 2 4 6 8 10
0
20
40
60
80
100
(a) Control/TK(t)
Time, hr
S∗ T
(%)
ρ(1)
LW(6)
SW(7)
α(8)
T a (10)
T i nts (4)
k (2)
0 2 4 6 8 10
0
20
40
60
80
100
(b) North/TK(t)
Figure 9.7: Stacked area charts of normalized total-effect sensitivity as a function of
model evaluation time at the 2 cm depth for the (a) Control and (b) North locations.
The regions are stacked from top to bottom in order as listed in Table 7.1 (see Table
7.1 for reference).
Based on the total-effect results in Figure 9.8 only four parameters influenced
the “knee” temperature: LW (6), SW (7), α(8), and Ta(10). For all three locations
LW (6) was the most influential term. However, both SW (7) and α(8) approach the
total-effect of LW (6) for the South location. Again, to truly quantify the important
214
0
20
40
60
80
Sensitiv
ity(
%)
S T1 S T2 S T3
S T4 S T5
S T6
S T7
S T8
S T9
S T10
S T11
 
 
South
North
Control
Figure 9.8: Total-effect sensitivity indices for the Control, South, and North locations
for TKmid (see Table 7.1 for reference).
parameters the interactions must also be considered. The first-, second-, and higher-
order sensitivity indices are provided in Figure 9.9. This figure only includes the four
terms listed above as influential on the TKmid output.
In similar fashion as the snow surface temperature, the following statements are
provided based on the sensitivity analysis results of the “knee” temperature at mid-
day (TKmid):
1. Control/TKmid (9.9a): Approximately 44% of the output variance is due solely
to three terms—LW (6), SW (7), α(8), and Ta(10)—without any interactions,
with 23% from LW (6) alone. The second-order interactions of these four terms
account for an additional 29% of the total variance, the remainder is associated
with higher-order interactions.
2. North/T0mid (9.9b): A vast majority of the output variance (72%) was due to
the first-order indices: 31% from LW (6), 13% from SW (7), 18% from α(8), and
10% from Ta(10). The remainder of the output variance is due to a variety of
second- and higher-order interactions.
215
3. South/T0mid (9.9c): A majority (69%) of the total variance may be attributed
to the first- and second-order results for LW (6), SW (7), and α(8) (i.e., S6+S7+
S8 +S6,7 +S7,8 +S7,8 ≈ 69%). An additional 11% of the output variance is due
to the total-effect of Ta(10) and the remainder from higher-order interactions.
0
5
10
15
20
25
30
35
40
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i S4 ,i S5 ,i
S6 ,i
S7 ,i
S8 ,i
S9 ,i
S10 ,i
S11 ,i
Sh
 
 
T a (10)
α(8)
SW(7)
LW(6)
(a) Control/TKmid
0
5
10
15
20
25
30
35
40
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i S4 ,i S5 ,i
S6 ,i
S7 ,i
S8 ,i
S9 ,i
S10 ,i
S11 ,i Sh
 
 
T a (10)
α(8)
SW(7)
LW(6)
(b) North/TKmid
0
5
10
15
20
25
30
35
40
Sensitiv
ity(
%)
S1 ,i S2,i S3 ,i S4 ,i S5 ,i
S6 ,i
S7 ,i
S8 ,i
S9 ,i
S10 ,i
S11 ,i Sh
 
 
T a (10)
α(8)
SW(7)
LW(6)
(c) South/TKmid
Figure 9.9: First-, second-, and higher-order indices for TKmid sensitivity analysis for
the (a) Control, (b) North, and (c) South locations (see Table 7.1 for reference).
216
9.3.2 Temperature Gradient
As shown in the previous section, the sensitivity results for snow temperature at
a depth of 2 cm were statistically identical to the results for the “knee” temperature.
Therefore, only the TG2 and KTG results are considered in this section.
Gradient Computed at 2 cm: The TG2(t) results for each location are included
in Figure 9.10a–c. The results shown in this figure indicated that sensitivity indices
for the input parameters did not change significantly with time. The exception was
SW (7) and α(8) that became evident after about 2 hours of model evaluation time.
Generally speaking, three terms—ρ(1), T ints , and LW (6)—dominate the charts. Ad-
ditionally, as was done for temperature, the mid-day value (TG2mid) was used to
simplify the temporal results. Figure 9.10d shows the total-effect results for TG2max,
which indicates six terms as influencing the gradient: ρ(1), k(2), T ints (4), LW (6),
Vw(9), and Ta(10). Interestingly, in both the TG2(t) and TG2mid neither SW (7) or
α(8) appear to influence the gradient to a significant extent.
The reason behind this behavior is due to subsurface melting.1 Consider the
scenario were the only parameter altered is short-wave radiation. For example, two
different temperature contour plots are shown in Figure 9.11. In these figures, all
inputs —except for SW (7)—were held constant: ρ(1) = 150, k(2) = 0.08, cp(3) =
2030, T ints (4) = −10, κ(5) = 60, LW (6) = 250, α(8) = 0.8, Vw(9) = 1, Ta(10) = −10,
and RH(11) = 50 (units are consistent with values in Table 7.1). The value of SW (7)
was changed from 300 W/m2 to 400 W/m2, but despite this change the temperature
gradient computed at 2 cm for both scenarios is approximately 600 ◦C/m, since the
subsurface is at 0◦C and the surface at -12◦C. Hence, despite changes in SW (7), the
1The thermal model presented in Chapter 5 is constrained such that the temperature of the snow
remains at or below 0◦C and it does not model melting snow.
217
Time, hr
S∗ T
(%)
ρ(1)
T i nts (4)
LW(6)
V w (9)
T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(a) Control/TG2(t)
Time, hr
S∗ T
(%)
ρ(1)
T i nts (4)
LW(6)
T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(b) North/TG2(t)
Time, hr
S∗ T
(%)
ρ(1)
T i nts (4)
LW(6)
T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(c) South/TG2(t)
0
20
40
60
80
Sensitiv
ity(
%)
S T1
S T2
S T3
S T4
S T5
S T6
S T7
S T8
S T9
S T10
S T11
 
 
South
North
Control
(d) TG2mid
Figure 9.10: Stacked area charts of normalized total-effect sensitivity of TG2(t) for
the (a) Control, (b) North, and (c) South locations and (d) the total-effect indices
computed from TG2mid output. The regions are stacked from bottom to top in order
as listed in Table 7.1.
gradient remains the same. This behavior is the culprit behind the low values for the
sensitivity indices reported previously.
This result coincides with observed near-surface facet events described in Chapter
4, which often were reported to occur with subsurface melting. Typically radiation-
218
0 2 4 6 8 10
0
2
4
6
8
10
Time ( hr)
Depth(
cm
)
Tem
pera
ture
(
◦ C
)
−12
−10
−8
−6
−4
−2
0
(a) SW (7) = 300 W/m2
0 2 4 6 8 10
0
2
4
6
8
10
Time ( hr)
Depth(
cm
)
Tem
pera
ture
(
◦ C
)
−12
−10
−8
−6
−4
−2
0
(b) SW (7) = 400 W/m2
Figure 9.11: Contour plots of snow temperature with incoming short-wave radiation
of (a) 300 W/m2 and (b) 400 W/m2. (see Table 7.1 for reference)
recrystallization is explained as requiring incoming short-wave radiation, which is
true, but the intensity is less important than the intensity of the incoming long-wave
radiation.
Next, the interactions should be considered before providing specific results. The
total-effect results for TG2mid indicate that only three parameters—cp(3), κ(4), and
RH(11)—may be neglected at all locations. Hence, Figure 9.12 is presented that
includes the first-, second-, and higher-order indices for each of the other parameters.
This figures contains far too many factors to be easily used, but is presented to illus-
trate one point—the overwhelming importance of the higher-order interactions. This
219
result indicates that temperature gradient is only affected when certain combinations
exist of the various input parameters.
−5
0
5
10
15
20
25
30
35
40
Sensitiv
ity(
%)
S1 ,i
S2,i
S3 ,i
S4 ,i
S5 ,i
S6 ,i
S7 ,i S8 ,i S9 ,i S10 ,i S11 ,i
Sh
 
 
T a (10)
V w (9)
α(8)
SW(7)
LW(6)
T i nts (4)
k (2)
ρ(1)
Figure 9.12: First-, second-, and higher-order indices for TG2mid sensitivity analysis
for the South location, which highlights the overwhelming importance of higher-order
interactions. (see Table 7.1 for reference)
“Knee” Temperature Gradient: Unlike the temperature results, the KTG data
differed slightly from TG2 results. The temporal results for each location for the
“knee” temperature gradient (KTG(t)) are included in Figure 9.13. The KTG(t)
results were less uniform than observed for TG2(t) (see Figure 9.10), k(2) became
the dominant input around mid-day, and T ints (4) and LW (6) had a diminished im-
portance.
The total-effect indices for KTGmid are provided in Figure 9.14. Two characteris-
tics of the KTG(t) (Figure 9.13) data were captured by the mid-day results: (1) the
importance of k(2) is evident and (2) the SW (7) and α(8) inputs also show some level
of importance depending on the location being considered. The large error associated
with the total-effect may be attributed to the nature of the KTG calculation, which
sets all input scenarios that did not yield a “knee” temperature profile to a value
220
Time, hr
S∗ T
(%)
ρ(1)
k (2)
LW(6)
T a (10)
V w (9)
T i nts (4)
0 2 4 6 8 10
0
20
40
60
80
100
(a) Control/KTG(t)
Time, hr
S∗ T
(%)
ρ(1)
k (2)
T i nts (4)
LW(6)
V w (9) T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(b) North/KTG(t)
Time, hr
S∗ T
(%)
ρ(1)
k (2)
LW(6)
V w (9)
T a (10)
0 2 4 6 8 10
0
20
40
60
80
100
(c) South/KTG(t)
Figure 9.13: Stacked area charts of normalized total-effect sensitivity of KTG(t) for
the (a) Control, (b) North, and (c) South locations. The regions are stacked from
bottom to top in order as listed in Table 7.1.
of zero, this effectively reduces the number of simulations from which the sensitivity
parameters were computed, inducing greater error.
The total-effect results also differed significantly with locations. Figure 9.15a
presents the indices from the Control location, which include ρ(1), k(2), LW (6),
Vw(9), and Ta(10). The total-effect results for the North/KTGmid indicated that
five terms—ρ(1), k(2), T ints , LW (6), and α(8)—influenced the temperature gradient,
thus Figure 9.15b is presented. Finally, considering that the South/KTGmid were
influenced by seven terms, a complete table of these results is provided in Table 9.1.
221
−20
0
20
40
60
80
Sensitiv
ity(
%)
S T1
S T2
S T3
S T4 S T5
S T6
S T7 S T8
S T9
S T10
S T11
 
 
South
North
Control
Figure 9.14: Grouped bar charts of total-effect sensitivity for KTGmid for all three
locations. (see Table 7.1 for reference)
0
10
20
30
40
Sensitiv
ity(
%)
S1 ,i
S2,i
S3 ,i
S4 ,i S5 ,i
S6 ,i
S7 ,i S8 ,i
S9 ,i
S10 ,i
S11 ,i
Sh
 
 
T a (10)
V w (9)
LW(6)
k (2)
ρ(1)
(a) Control/KTGmid
−10
0
10
20
30
40
Sensitiv
ity(
%) S1 ,i
S2,i
S3 ,i
S4 ,i
S5 ,i
S6 ,i
S7 ,i
S8 ,i S9 ,i S10 ,i S11 ,i
Sh
 
 
α(8)
LW(6)
T i nts (4)
k (2)
ρ(1)
(b) North/KTGmid
Figure 9.15: First-, second-, and higher-order indices for KTGmid sensitivity analysis
for the (a) Control and (b) North locations. (see Table 7.1 for reference)
222
Table 9.1: First-, second-, total-, and higher-order sensitivity indices for the
South/KTGmid results (see Table 7.1 for reference).
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.8 3.3 1.2 3.1 1.8 0.1 2.5 2.6 3.4 1.9 1.4
0.1–3.5 -0.7–7.4 -0.5–2.8 1.2–5.0 -0.3–3.8 -2.6–2.9 0.2–4.8 0.5–4.7 1.3–5.6 -0.6–4.4 -0.3–3.1
2 3.3 13.2 1.2 2.3 1.8 5.4 -0.7 0.4 3.7 4.4 1.1
-0.7–7.4 10.9–15.4 0.0–2.4 0.1–4.4 0.1–3.5 1.4–9.4 -3.6–2.1 -1.9–2.8 1.2–6.1 1.4–7.4 -0.1–2.2
3 1.2 1.2 -0.2 0.5 0.3 -0.2 0.3 0.5 0.5 0.4 0.3
-0.5–2.8 0.0–2.4 -0.4–0.1 -0.1–1.0 -0.3–0.8 -1.6–1.2 -0.3–0.9 -0.1–1.1 -0.0–1.0 -0.2–1.1 -0.1–0.8
4 3.1 2.3 0.5 -0.2 0.3 -0.7 0.5 0.5 0.8 0.4 0.3
1.2–5.0 0.1–4.4 -0.1–1.0 -0.7–0.2 -0.7–1.4 -2.4–0.9 -0.5–1.4 -0.4–1.5 -0.2–1.7 -0.6–1.4 -0.6–1.2
5 1.8 1.8 0.3 0.3 0.1 -0.4 0.1 0.3 0.5 -0.1 0.3
-0.3–3.8 0.1–3.5 -0.3–0.8 -0.7–1.4 -0.7–0.9 -2.4–1.6 -1.5–1.6 -1.2–1.9 -1.0–2.1 -1.8–1.7 -1.1–1.7
6 0.1 5.4 -0.2 -0.7 -0.4 9.3 -0.1 0.3 1.4 0.6 -0.3
-2.6–2.9 1.4–9.4 -1.6–1.2 -2.4–0.9 -2.4–1.6 7.4–11.1 -2.7–2.5 -2.0–2.7 -1.0–3.9 -2.1–3.3 -2.2–1.6
7 2.5 -0.7 0.3 0.5 0.1 -0.1 -0.5 0.7 0.4 0.3 -0.0
0.2–4.8 -3.6–2.1 -0.3–0.9 -0.5–1.4 -1.5–1.6 -2.7–2.5 -1.8–0.9 -1.4–2.9 -1.9–2.6 -2.1–2.6 -2.0–2.0
8 2.6 0.4 0.5 0.5 0.3 0.3 0.7 -0.1 1.0 0.8 0.9
0.5–4.7 -1.9–2.8 -0.1–1.1 -0.4–1.5 -1.2–1.9 -2.0–2.7 -1.4–2.9 -1.1–1.0 -0.8–2.8 -1.2–2.7 -0.7–2.5
9 3.4 3.7 0.5 0.8 0.5 1.4 0.4 1.0 1.6 1.4 0.5
1.3–5.6 1.2–6.1 -0.0–1.0 -0.2–1.7 -1.0–2.1 -1.0–3.9 -1.9–2.6 -0.8–2.8 0.8–2.3 0.0–2.8 -0.5–1.6
10 1.9 4.4 0.4 0.4 -0.1 0.6 0.3 0.8 1.4 2.0 -0.2
-0.6–4.4 1.4–7.4 -0.2–1.1 -0.6–1.4 -1.8–1.7 -2.1–3.3 -2.1–2.6 -1.2–2.7 0.0–2.8 0.7–3.3 -2.1–1.7
11 1.4 1.1 0.3 0.3 0.3 -0.3 -0.0 0.9 0.5 -0.2 0.0
-0.3–3.1 -0.1–2.2 -0.1–0.8 -0.6–1.2 -1.1–1.7 -2.2–1.6 -2.0–2.0 -0.7–2.5 -0.5–1.6 -2.1–1.7 -0.2–0.2
Higher 5.7 2.9 -7.5 -1.5 2.0 27.5 11.1 5.3 -3.1 4.6 -7.3
Total 28.9 38.9 -2.6 6.3 7.0 43.1 14.5 13.4 12.0 16.6 -2.9
19.7–38.2 32.4–45.5 -13.9–8.6 -4.8–17.3 -6.2–20.2 31.3–54.8 4.1–24.9 3.9–22.9 3.0–21.0 7.7–25.4 -13.8–8.0
Given the data presented in this section, the following statements are provided
based on the sensitivity analysis results of the “knee” temperature gradient at mid-
day (KTGmid):
1. Control/KTGmid (9.15a): A majority of the variance observed in the output was
due to three terms and their associated interactions. The variance due to ρ(1)
and LW (6) was primarily from higher-order interactions, while the variance due
to k(2) was primarily from the first-order index and second-order interactions
with ρ(1) and Ta(10).
2. North/KTGmid (9.15b): Two terms—ρ(1) and k(2)—were the most influential
on the output variance, with LW (6) having having a secondary effect. The
first-order index for k(2) was the most significant accounting for approximately
27% of the total variance observed.
223
3. South/KTGmid (9.1): A majority of the observed variance was due to ρ(1),
k(2), and LW (6), which had total-effect indices of approximately 29%, 39%, and
43%, respectively. Both ρ(1) and k(2) were composed of a significant number
of second-order interactions. SW (7) and α(8) were also shown to affect the
output, but this effect was composed of interactions with ρ(1) directly or high-
order terms, the first-order indices were approximately zero.
9.4 Closing Remarks
As stated the goal of this chapter is to identify the inputs that influence the
temperature gradient, specifically gradients induced by short-wave radiation gains
that are known to lead to radiation-recrystallization. The snow temperatures were
affected by a number of inputs and the relative importance shifted with evaluation
time. However, considering the mid-day temperatures the snow temperatures were
mainly affected by five parameters: T ints (4), LW (6), SW (7), α(8), and Ta(10).
The broadest conclusion that may be drawn from the results presented in this
chapter is that incoming long-wave radiation is the most influential parameter that
effects snow temperature, temperature gradient, and “knee” related outputs. Also,
that cp(3), κ(5), and RH(11) are negligible in this regard.
Another conclusion that may be drawn from the data presented in this chap-
ter is that incoming short-wave radiation is a necessary, but secondary influence
on the “knee” temperature gradient assumed to be associated with radiation-
recrystallization. In general, three terms governed changes in the temperature gra-
dient: ρ(1), k(2), and LW (6). And, only when interacting with one or more other
parameters did SW (7) influence the gradient. This indicated that only in specific
224
situations did the “knee” gradient develop, defining these situations is the topic of
Chapter 10.
The minimal influence of SW (7) was due to subsurface melting. Changes in
SW (7) caused alterations in the extent of subsurface melting that occurred. This
was evident in the sensitivity analyses focused on snow temperature that showed
both SW (7) and α(8) as dominant. However, the snow surface temperature, at least
with respect to the model used here, was not affected by short-wave radiation to the
extent of other parameters, namely LW (6). Thus, when the gradient is considered
both the subsurface temperature (near melting) and the surface temperature are not
influenced significantly by changes in short-wave radiation. This finding indicates
that the “knee” temperature gradient is often associated with subsurface melting.
Based on the snow temperature results this melting occurs near a depth of 2 cm. The
observed near-surface facets events provided in Chapter 4 present physical evidence
of this behavior, as many of the events occurred under these conditions.
225
CHAPTER 10
MONTE CARLO SIMULATIONS OF NEAR-SURFACE FACETS
10.1 Introduction
Conceptually the process of near-surface faceting of snow is well understood, how-
ever the quantitative data detailing the conditions under which these crystals form
is based on limited field and laboratory data. Using a simple thermal model, in
Chapter 9 sensitivity analysis was employed to quantify the most influential model
inputs on the snow temperatures and temperature gradients. The inputs included
both snow properties and environmental conditions. Based on these results, through
the use of Monte Carlo simulations, it is possible to examine specific quantities of each
input that resulted in a specific output. This type of numerical analysis allows for an
infinite number of input parameter combinations to be explored, which is impossible
with physical experiments.
The results and analysis presented throughout this chapter aim to meet a sin-
gle objective: to provide a graphical tool for assessing the likelihood of radiation-
recrystallization based on snow and environmental conditions.
10.2 Methods
Details of the methods used in this section are provided in Chapter 7. The Monte
Carlo simulations (Section 7.5) preformed were constructed from the evaluations from
the sensitivity analysis discussed in Chapter 9. The reference scheme defined in
Section 9.2 is also used throughout this chapter. The input parameters discussed
remain the same as in Chapters 7–9, these values are listed in Table 7.1. In total,
240,000 model evaluations were preformed for each of the three locations: Control,
226
North, and South. Examining the data from such a large number of simulations
required the use of highest density regions (see Section 7.6), which simply offer a
means to encompass a percentage of the simulations by a region or contour.
In addition to the data generated from the simulations, three physically based data
sets were used for comparison: (1) the near-surface facet events detailed in Chapter 3,
(2) the laboratory experiments conducted by Morstad et al. (2007), and (3) laboratory
experiments conducted by Slaughter et al. (2009). The laboratory data from Morstad
et al. (2007) and Slaughter et al. (2009) is provided in Table 10.1. The bottom four
entries in the table were conducted by Slaughter et al. (2009), which included six
total experiments, but two were missing long-wave radiation data and thus excluded.
The thermal conductivity (k) values reported for this work were estimated from the
measured density using the relationship proposed by Sturm et al. (1997). The work
by Morstad et al. (2007) was the result of 13 laboratory experiments, 10 of which
resulted in near-surface facet formation. The data in this table includes only the
parameters used in this chapter, for complete results refer to Morstad (2004). The
thermal diffusivity (γ) was used here for convenience, where
γ =
k
ρcp
. (10.1)
A constant value of cp = 2030 J/(kg ·K) of ice was used for the calculations presented
in Table 10.1. Diffusivity is presented with units of m2/s were used throughout this
chapter. The Ω term is introduced in Section 10.4 as the ratio of SW (1 − α) and
LW .
Chapter 4 detailed 26 near-surface facet events observed at the South-facing
weather station. The observations did not include micro-structural parameters, as
such estimates for ρ, k, and α were required. Rather than use a single estimate
227
Table 10.1: Summary of results from laboratory experiments conducted by Morstad
et al. (2007) and Slaughter et al. (2009).
Exp. Size TG SW LW α ρ k log(γ) Ω
# mm ◦C/m W/m2 W/m2 kg/m3 W/(m· K)
1 200 330 254 0.75 195 0.20 -6.30 0.32
2 1.0 350 595 273 0.81 174 0.10 -6.55 0.41
3 3/4 550 755 280 0.81 175 0.10 -6.55 0.51
4 1/2 400 1180 300 0.78 200 0.12 -6.53 0.87
5 1/4 400 755 280 0.76 250 0.18 -6.45 0.65
6 1/2 300 755 280 0.84 187 0.11 -6.54 0.43
7 1/2 150 755 280 0.85 270 0.20 -6.44 0.40
8 100 208 242 0.73 170 0.15 -6.36 0.23
9 1/4 170 755 280 0.79 257 0.40 -6.12 0.57
10 1/4 200 755 320 0.78 540 0.17 -6.81 0.52
11 1/8 200 755 280 0.78 410 0.75 -6.05 0.59
12 20 0 207 303 0.25 -6.39
13 1/2 200 755 280 0.75 300 0.25 -6.39 0.67
Feb14#3 443 315 0.92 284 0.11 -6.71 0.11
Mar6 701 272 0.87 350 0.18 -6.59 0.34
Apr#1 679 286 0.91 284 0.11 -6.71 0.21
Apr3#2 638 330 0.89 320 0.15 -6.65 0.21
of these properties a range of values was assigned based on published data, thus a
confidence region was defined to encapsulate the observed events.
Nearly all the measured near-surface facet events reported in Chapter 4 occurred
after recent snowfall events, thus a density range was assigned as such. Armstrong and
Brun (2008, p. 59) state that newly fallen snow typically ranges between 60 kg/m3
and 120 kg/m3. Using the raw data from the entire body of thermal conductivity
measurements presented by Sturm et al. (1997, Fig. 4), k was assumed to vary from
0.04–0.25 W/(m·K). The specific heat capacity is assumed to be a constant value of
cp = 2030 J/(kg ·K) . For the estimated range of γ, both ρ and k were assumed to
vary according to a normal distribution such that the aforementioned limits were at
the 95% tails (i.e., two standard deviations from the mean). This resulted in log(γ)
having an estimated range of -6.57 to -5.81.
228
To determine the range of α, first it was assumed the new snow metamorphosed
into facets of class 1, 2, or 3 (Armstrong and Brun, 2008, p. 28). This assumption
allowed for tabulated values (see Armstrong and Brun (2008, p. 57)) of α to be
utilized for determining the possible extent of α. However, the tabulated data alone
is insufficient due to the wavelength dependence of albedo. Using a weighted average
determined from the various wavebands defined by ASTM G-173 (see Section 5.4)
and the tabulated values of Armstrong and Brun (2008, p. 57), α is assumed to
range from 0.8–0.87.
As discussed in Chapter 4, the long-wave radiation sensors used during the
2007/2008 winter season were influenced by preferential heating from incoming short-
wave radiation, particularly at the south-facing slope. This problem was corrected
in the 2008/2009 data. The histogram of Figure 10.1 is an illustration of the long-
wave radiation data that highlights the difference in the recorded values between the
two seasons. To account for this problem a correction is applied to the 2007/2008
data based on a comparison with the American Spirit (Aspirit) radiation data (see
Chapter 4). The mean daily incoming long-wave radiation values of the events at the
South Station were 2.07 times that of Aspirit during the 2007/2008 season and 1.11
for the 2008/2009 season (see Table 4.2). Assuming the 2008/2009 ratio applies to
the previous season, the long-wave values reported in Table 4.2 were reduced by a
factor of 0.54 (1.11/2.07) for usage in the analysis presented here.
10.3 Results
Near-surface facets are known to form with significant temperature gradients, the
values reported in the literature range from approximately 100–600 ◦C/m (Fukuzawa
and Akitaya, 1993; Hardy et al., 2001; Morstad et al., 2007; Slaughter et al., 2009).
229
200 250 300 350 400 450 500 550
0
5
10
15
20
Incoming Long-wave Radiation [W/m 2 ]
Fre
qu
enc
y
 
 
2 0 0 7/20 0 8
2 0 0 8/20 0 9
Figure 10.1: Comparison between the two seasons (2007/2008 and 2008/2009) of
recorded long-wave radiation values for near-surface facet events.
The data presented from Morstad et al. (2007) in Table 10.1 indicates that gradients
on the order of 100 ◦C/m may be inadequate (Exp. #8). One experiment is of little
value for making such a claim, but this statement gains traction considering the work
of Pinzer and Schneebeli (2009). Their work provides additional evidence that on
time scales of less than a day that “temperature gradients on the order of 100 ◦C/m
do not lead necessarily to faceting. . . ” Considering that the simulations presented
here only spanned 10 hours, a lower limit of 200 ◦C/m was assumed to be necessary
for facet formation to occur. This is in agreement wtih observations in Chapter 4
as well as in Slaughter et al. (2009). The upper limit was assumed to remain at 600
◦C/m. Values larger than this were observed in the simulations but assumed to be
unrealistic.
In Chapter 10, the temperature gradient (both TG2 and KTG) was demonstrated
to be primarily affected by ρ, k, and LW . Additionally, relying on the results from
sensitivity analysis of KTGmid, four other terms influence the gradient for the Control
and South locations: SW , α, Vw, and Ta. However, as discussed in the analysis in
Section 10.4, the Vw and Ta were not utilized. Therefore, the remainder of the results
230
focused on five parameters listed—ρ, k, LW , SW , and α—as well as the above
temperature gradient range.
The input parameters under investigation were divided into two groups: snow
properties (ρ, k, and α) and radiation input (SW and LW ). Based on the KTGmid
values these five inputs were limited to values with a KTGmid from 200–600 ◦C/m
resulting from the simulations. Limiting the data as such reduced the simulations
considered to 27,893 (11.6%), 24,251 (10.1%), and 22,257 (9.3%) for the Control,
North, and South locations, respectively. Estimated probability distribution functions
(PDFs) for the snow properties are provided in Figure 10.2 and for the radiation inputs
in Figure 10.3. Each graph includes the PDFs for the complete (all) input data set
and the limited data (limited). For the snow properties the “all” input distributions
do not differ between the locations. The PDFs were computed via the kernel estimate
method, see Section 7.7.
Negative values appeared in the PDFs of ρ, k, and SW , which is impossible
considering the parameters. However, this was strictly a graphical issue due to the
computation method used, which was only employed to visualize the results, no nu-
merical computations used these distributions.
As shown in Figures 10.2 and 10.3, the PDFs had various differences, which were
compared in two ways: (1) for each location and input parameter the complete input
was compared with the KTGmid limited data and (2) the limited data for each pa-
rameter was compared across the locations (e.g., the limited data from the Control
locations was compared with the limited data from the South location). In all cases,
despite apparent similarities—such as for ρ in Figure 10.2a—statistically the distri-
butions are different. Using the Kolmogorov-Smirnov test (KS-test; see Section 7.8),
a p-value of approximately zero for each comparison was computed.
231
0 200 400 600
0
1
2
3
4
x 10−3
ρ(1)
Prob.
Density
(a)
0 0.2 0.4 0.6 0.8
0
2
4
6
8
k (2)
Prob.
Density
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
2
3
α(8)
Prob.
Density
(c)
all North/limited South/limited Control/limited
Figure 10.2: Probability distribution functions for snow properties including the com-
plete (all) input distribution and data limited by KTGmid from 200–600 ◦C/m (lim-
ited) . Table 7.1 (p. 163) defines the variables and the units for each graph.
Visually, substantial changes in the distributions occurred. The most notable
change was the spike in probability density for values of ρ near 100 kg/m3 for the North
and South locations. This spike was not observed in the Control location data and the
probability density actually became lower in that same region. Thermal conductivity
(k) yielded drastically different distributions for all locations and incoming short-wave
radiation showed little change.
232
0 100 200 300 400 500 600 700 800
0
0.005
0.01
0.015
LW (6)
Prob.
Density
(a)
0 100 200 300 400 500 600 700 800
0
2
4
6
8
x 10−3
S W (7)
Prob.
Density
(b)
North/all North/limited South/all South/limited Control/all Control/limited
Figure 10.3: Probability distribution functions for radiation inputs including the
complete (all) input distribution and data limited by KTGmid from 200–600 ◦C/m
(limited) . Table 7.1 (p. 163) defines the variables and the units for each graph.
10.4 Discussion
Based on the sensitivity analysis of Chapter 9, ρ, k, and LW were determined
to be the influential parameters on temperature gradient, particularly the “knee”
gradient typical of radiation-recrystallization. Hence, a simple function was defined
using the thermal diffusivity (γ) as
TG ≈ f(γ, LW ), (10.2)
233
where TG is used as a generic temperature gradient not associated with any specific
computation method such as KTG or TG2. Including the specific heat, cp, was
simply a natural selection when grouping ρ and k.
Conceptually, Equation 10.2 seems lacking, considering the focus of this research is
radiation-recrystallization, so naturally the incoming short-wave should be included.
Excluding short-wave radiation, as well as the albedo negates, the North and South-
data sets for practical application, since there is nothing to distinguish the locations.
An unexpected result was the number of resulting simulations with KTGmid limited
from 200–600 ◦C/m: 10.1% and 9.3% for the North and South locations, respectively.
Intuitively, the South location should yield a larger portion of “knee” gradients. The
likely culprit behind this result was the range of α assumed, which was 0.4–0.9. Ex-
amining Figure 10.2c demonstrates that the North location favors low values and the
South high values of α. Therefore, it is assumed here that the important parameter to
consider is the absorbed short-wave radiation. When combined with LW a convenient
dimensionless term arises:
Ω =
SW
LW
(1− α). (10.3)
Therefore, Equation 10.2 is redefined as
TG ≈ f(γ,Ω). (10.4)
Including the SW and α terms is not only natural but offers some statistical
advantage—although only a small advantage considering the sensitivity analysis—
since the variability associated with these terms is included.
Similarly, if Vw and Ta were included all of the variance in the system would be
accounted for since no other input terms would remain that influenced the variance of
KTGmid, refer to Figure 9.1. In this case, TG = f(ρ, k, LW, SW,α, Vw, Ta), notice the
approximation sign used in Equation (10.4) would be inappropriate for this complete
234
equation. The sensitivity analysis demonstrated, at least with respect to the model
and KTGmid output considered, that only these seven terms alone caused changes
in the temperature gradient. An attempt was made to analyze all the parameters,
but it was deemed impractical and made an already complex analysis even more so.
Therefore, the relationship defined in Equation 10.4 was used for the remainder of
the analysis presented.
10.5 Analysis
Using Equation 10.4, the Monte Carlo simulation inputs and temperature gradient
output were simplified to a three-dimensional data set. First, the inputs of γ and LW
are compared in similar fashion as done in the previous section. However, here the
variables are considered together using two-dimensional PDFs, as shown in Figure
10.4.
Visually these 2-D PDFs illustrate the most profound difference in the complete
data set and the data limited by KTGmid from 200–600 ◦C/m, the bi-modal behavior.
The low point or saddle of the bi-modal distributions coincided with the peak of the
complete data set. This indicates that the most common value of γ in the simulations
is unfavorable for the developing a strong temperature gradient. The 2-D PDFs in
Figure 10.4 were the basis of the end result of the work presented in this chapter that
includes comparison with field and laboratory measurements of near-surface facets.
However, before continuing it is important to explain that the full analysis of Equation
10.4 requires a tri-variate PDF. An example of which is presented in Figure 10.5.
The tri-variate, 38% HDR for the South location is provided in Figure 10.5. The
38% HDR was selected because it is akin to a confidence level of ±12σ for a normal
distribution (i.e., a probable outcome), where σ is the standard deviation. In this
235
−7.5
−7
−6.5
−6
−5.5
0
1
2
3
0.5
1
1.5
P
ro
b.
D
en
si
ty
Ω
log(γ)
B
A
(a) Control
−7.5
−7
−6.5
−6
−5.5
0
0.2
0.4
1
2
3
4
5
P
ro
b.
D
en
si
ty
Ω
log(γ)
B
A
(b) North
−7.5−7
−6.5−6
−5.5
0
0.5
1
0.5
1
1.5
2
P
ro
b.
D
en
si
ty
log(γ)
Ω
B
A
(c) South
Figure 10.4: Comparison of the complete input (A) with the input limited by KTGmid
from 200–600◦C/m (B) for the (a) Control, (b) North, and (c) South locations.
figure the TG2mid output comprises the vertical axis because it produced output for
the complete data set. The limited data was still based on the KTGmid criteria, but
the associated TG2mid quantities were displayed for consistency. The two regions of
236
the limited data, clearly shows the bi-modal behavior of the data, which indcated
that there are two regions of Ω and γ likely to induce significant “knee” temperature
gradients.
−7.5 −7
−6.5 −6
−5.5 −5
−0.20
0.20.4
0.60.8
−200
−100
0
100
200
300
T
em
p
.
G
ra
d
ie
nt
,
T
G
2m
id
[◦
C
/m
]
log(γ)Ω
Figure 10.5: Comparison of tri-variate PDF of all input (A) with the input limited
by KTGmid from 200–600◦C/m (B) for the South location.
In any dimension, one, two, or three, the PDFs presented throughout this chapter
may be thought of in the same fashion. Consider the KTGmid limited distribution for
the South location presented in Figure 10.4 and the following hypothetical situation.
If γ was determined to be approximately 10-6, then the most likely scenario to lead
to gradients of 200–600 ◦C/m would be an Ω of approximately 0.17, which is located
in the depression between the two peaks. Based on a comparison with the other
probability densities it is then possible to assess how likely it is that a gradient
will develop. Even at the peak of the distributions it is never possible to state, with
certainty, that the gradient will develop because the data includes additional variance.
Sources of additional variance included: known variance in terms not considered such
237
as Ta and Vw, unknown variance due to uncertainty associated with the sensitivity
analyses (i.e., the error bars shown throughout Chapter 9), and errors associated with
the non-infinite sample size in the Monte Carlo simulations.
Control Location: Contour plots computed from the 2-D probability distri-
butions in Figure 10.4 were computed. First ,the Control results are considered, as
presented in Figure 10.6. The contours are composed of the 95%, 68%, 38%, 20%, and
10% HDRs. These ranges were selected because they were approximately proportional
to confidence levels typically associated with a normal distribution. And, if the data
was normally distributed these HDRs would be equivalent to 2σ, 1σ, 12σ,
1
4σ, and
1
8σ
confidence regions.
The region labeled “field” estimates the location of all 26 observations from the
field data presented in Chapter 4. The regions were defined using the ranges for γ
and α defined in Section 10.2. Assuming a uniform distribution for these terms, these
distributions were sampled 10,000 times for each observed event, thus constructing a
synthetic set of data of likely values from which the 95% HDR was computed. Simply
stated, the field observations likely fall somewhere within this region.
Statistically comparing the Monte Carlo simulation computed HDRs with the ob-
served laboratory and field near-surface facets is not appropriate. The laboratory data
of Morstad et al. (2007) was exploratory in nature so the conditions were explicitly
selected based on success of facet formation. However, nearly all of the laboratory
experiments shown are within the 38% HDR. On the other hand, two of these points
(Exp. #1 and #8) did not result in near-surface facet formation, but considering how
tight the contours are in this region, a small error in Ω could shift these parameters
outside of the most probable regions.
238
log(γ)
Ω
 
 
−8 −7.5 −7 −6.5 −6 −5.5 −5
0
0.5
1
1.5
2
2.5
3
HDRs
Morstad (2007)
Slaughter (2009)
Field
95%
68%
38%20%
10%
Figure 10.6: Contour plot of HDRs for the Control results including the field obser-
vations from Chapter 4 and laboratory data of Morstad et al. (2007) and Slaughter
et al. (2009).
Although only composed of four experiments, the data presented by Slaughter
et al. (2009) were all near the center of the regions. Since these laboratory experiments
were based on observed events from the South-facing weather station (see Chapter
4) it is reasonable to expect these values to be well predicted by the Monte Carlo
simulations. However, such is not the case for the field estimate region presented,
which did not seem to align well with the regions.
North Location: Figure 10.7 includes the Monte Carlo simulations results com-
puted form the North location. Generally, the observed field and laboratory data
match poorly with the results from the North location, which is expected considering
that much of the physical data was developed by examining data from the South-
facing weather station. The North HDR regions are skewed to lower values of Ω
when compared to the Control results of Figure 10.6, this is expected considering the
lower values of α observed in Figure 10.2c.
239
log(γ)
Ω
 
 
−8 −7.5 −7 −6.5 −6 −5.5 −5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
HDRs
Morstad (2007)
Slaughter (2009)
Field
95%
68%
38% 20% 10%
Figure 10.7: Contour plot of HDRs for the North location including the field obser-
vations from Chapter 4 and laboratory data of Morstad et al. (2007) and Slaughter
et al. (2009).
South Location: The results from the South location were the most intriguing.
As shown in Figure 10.8, the bi-modal behavior observed is the most prominent in
this data set. This is evident by the 38% and 20% HDRs located with γ ≈ 10−5.6.
Unfortunately, the correlation with the laboratory data and field data in this region is
weak, so it is not possible to make any definitive statements. Visually, the laboratory
experiments particularly those conducted by Morstad et al. (2007) do not seem to
correlate.
It is important to point out that the most probable regions for both the North
and South locations (Figures 10.7 and 10.8) were similar to the Control set. In fact,
in direct comparison the North and South data appear to be subsets of the Control
location. This is expected considering the Control input was designed to a generic
unbiased approach with the North and South being oriented, as their name suggests,
with different aspects.
240
log(γ)
Ω
 
 
−8 −7.5 −7 −6.5 −6 −5.5 −5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2 HDRs
Morstad (2007)
Slaughter (2009)
Field
68%
95%
38%
10%
38%
20%
20%
Figure 10.8: Contour plot of HDRs for the South location including the field obser-
vations from Chapter 4 and laboratory data of Morstad et al. (2007) and Slaughter
et al. (2009).
Returning to the bi-modality of the data observed. Conceptually the reality of
this behavior may be explored. Figure 10.9 is a reconstruction of a figure presented by
Sturm et al. (1997) relating ρ and k, but here it also includes the value of γ associated
as well as two relationships commonly used for relating ρ and k. Consider the larger
of the two spikes in the probability distribution function, which occurred with values
of γ approximately from 10−7.25–10−6.25. The other peak occurred with γ ≈ 10−5.6.
Using Figure 10.9, the first range is in the realm of reasonable values of γ, especially
when compared to the raw data presented by Sturm et al. (1997, Fig. 4).
The second peak was associated with low density snow with high k values, which is
counter intuitive, i.e., based on the typical relationships used for relating conductivity
and density. The data presented in Chapter 4 indicated facets commonly form in low-
density snow. The only observed data to approach this region in the numerical results
was the field observations. And, since this field data region was constructed assuming
241
typical ρ-k relationship this misalignment is expected. Hence, this second region may
be physical evidence that supports the existence of the low-density, high-conductivity
region. This conclusion is not unfounded, theoretically k used here can be assumed
to be due to any mode of heat transfer such as conduction or vapor diffusion. In fact,
this deviation from the typical ρ-k relationship is discussed to some extent by Sturm
et al. (1997).
Densi ty [ kg/m 3 ]
Therm
alCo
ndu
ctivi
ty[W
/(m
K)]
−8
−7.5
−
7
−
6.5
−
6
−
5.
5
−
5
100 200 300 400 500 600 700
0.01
0.1
1
log
(γ
)
−8
−7.5
−7
−6.5
−6
−5.5
−5
−4.5
Abel Sturm
Figure 10.9: Chart showing the relationship of ρ, k, and γ as well as two commonly
utilized ρ and k relationships as presented by Sturm et al. (1997).
10.6 Closing Remarks
The main objective of this chapter was to present a tool for assessing near-surface
faceting due to radiation-recrystallization, which was accomplished with Figures 10.6–
10.8. These figures related the non-dimensional term Ω to thermal diffusivity (γ).
Based on the numerical simulations preformed it was possible to define the ideal
conditions that lead to strong temperature gradients (200–600 ◦C/m) that have a
temperature profile conducive to radiation-recrystallization. In the most general case,
242
irrespective of aspect, the optimum region for facet formation is Ω from 0.2–0.5 and γ
from 10-6.3–10-6.7. If aspect is considered these levels shift, for the North location the
ideal conditions are Ω from 0.1–0.2 and γ from 10-6.3–10-7.2 and for the South location
Ω ranges from 0.2–0.4 and γ from 10-6.5–10-7.
Qualitatively, these results matched reasonably well with observed values for both
laboratory and field observations of near-surface facet formation. A strict, statistical
comparison was not possible due to the nature of the observed data. Hence, the
regions defined are not a definitive answer to the question of when near-surface facets
form. The regions are defined to offer a postulate that these regions are of importance
and require further research. Of particular interest are the two discrete regions mainly
observed in the South location results that indicated that near-surface facets form
generally under two scenarios: (1) with ρ and k following the basic trend of traditional
regression based relationships and (2) with low-density, high-thermal conductivity
snow, a postulate that is supported by the results presented in Chapter 4 of this
dissertation. That is, low-density snow subjected to a strong temperature gradient
includes significant heat-transfer due to vapor diffusion, thus the effective thermal
conductivity may be higher than expected.
243
CHAPTER 11
CONCLUSIONS
The review provided in Chapter 2 demonstrates the importance of studying the
metamorphic processes in snow, particularly surface hoar and near-surface facets.
This review set the stage for the main objective of the work presented in this disser-
tation: to further quantify the necessary conditions for the formation of both surface
hoar and near-surface facets.
To this end, weather stations and observations were established on a north- and
south-facing slope in southwest Montana. Throughout two winter seasons, 14 surface
hoar and 26 near-surface facet events were observed at one or both of the stations.
Beginning with the surface hoar events, see Chapter 3, it was determined that three
environmental parameters were statistically significant in leading to their formation:
long-wave radiation, snow temperature, and relative humidity. Based on percentiles
of recorded events, the ideal conditions to cause surface hoar formation were nightly
mean values of long-wave radiation of 220 W/m2, snow temperature of -16 ◦C, and
relative humidity of 65%.
In the case of near-surface facets, 26 events were observed during the two winter
seasons that data was collected, see Chapter 4. However, in this case the crystals
almost exclusively formed at the South weather station and were due to radiation-
recrystallization. The prevalence of near-surface facets formed due to radiation in-
dicated that the conditions in southwest Montana are sufficient for the development
of these crystals. A comparison of the mean daily values of all days with the values
from days with near-surface facet events suggested three parameters were statistically
significant: short- and long-wave radiation and relative humidity. Again, based on
percentiles of the observed events, the optimum conditions for facet formation, using
244
mean daily values at the South-facing weather station, were short-wave radiation of
620 W/m2, long-wave radiation of 220 W/m2, and relative humidity of 49%.
Using a thermal model (Chapter 5) and various numerical analysis techniques
(Chapters 6 and 7), an analytical investigation was conducted to meet the main
objective for both surface hoar formation (Chapter 8) and near-surface formation
due to radiation recrystallization (Chapters 9 and 10). In general, the surface hoar
analysis (Chapter 8) indicated that long-wave radiation and air temperature were
the most influential inputs affecting positive mass-fluxes at the snow surface; these
terms alone attributed to over 50% of the variance observed in positive values of
mass-flux. Also, Figures 8.9 and 8.12 were presented as possible tools for assessing
if the conditions are conducive to surface hoar formation. For example, at north-
facing aspects the following optimum conditions were presented: long-wave radiation
from 200–250 W/m2 and Π from 102.5–10−1.7 (see Figure 8.12), where Π is defined
in Equation (8.2). However, with respect to surface hoar size, mass-flux alone is
inadequate for determining the size of the crystals.
The results obtained for near-surface facets indicated that three parameters—
snow density, thermal conductivity, and incoming long-wave radiation—were the
most influential on the presence of a temperature gradient conducive to radiation-
recrystallization. In similar fashion as for surface hoar, Figures 10.6–10.8 were pre-
sented as tools for assessing the likelihood of facet formation. In the most general
case, irrespective of aspect, the optimum region for facet formation is Ω = SWLW (1−α)
from 0.2–0.5 and γ from 10-6.3–10-6.7, where Ω is defined by Equation (10.4) and γ
is the thermal diffusivity of the snow. If aspect is considered these levels shift; for
north-facing aspects the ideal conditions are Ω from 0.1–0.2 and γ from 10-6.3–10-7.2
and for south-facing aspects Ω from 0.2–0.4 and γ from 10-6.5–10-7.
245
Perhaps the most interesting result obtained throughout this dissertation was the
region of low-density, high-thermal conductivity snow conducive to the development of
near-surface facets (see Figure 10.8). Traditional relationships of density and thermal
conductivity, i.e., those defined by Sturm et al. (1997) indicate that this scenario
is unlikely. However, the analysis presented throughout this dissertation made no
assumptions regarding the mode of thermal conductivity. Thus, if heat transport due
to water vapor diffusion is significant, this density and thermal conductivity may be
reasonable. Considering that the majority of the observed near-surface facet events
described in Chapter 4 occurred with recently fallen snow, this scenario is likely a
real phenomenon.
If only one conclusion should be drawn from the work presented, it is the im-
portance of incoming long-wave radiation. Throughout the entire dissertation long-
wave radiation appeared as a dominant factor for both surface hoar and near-surface
facets. As presented here, long-wave is a crucial parameter and should be the focus
of additional research.
Finally, both sensitivity analysis and Monte Carlo simulations were presented
here as a tool for examining the conditions important to snow morphologies. And,
the work presented only scratches the surface of the potential use of these methods to
improve the current understanding of the most influential terms. Additional research
should be conducted exploring a variety of scenarios including layered snowpack, melt-
layers, and diurnal fluctuations to name a few. Furthermore, the analysis should be
applied to more detailed models such as SNOWPACK that include micro-structure
parameters. The potential applications are only limited by computation time and
imagination.
246
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258
APPENDICES
259
APPENDIX A
YELLOWSTONE CLUB WEATHER STATIONS
260
A.1 Introduction
Two weather stations were installed on Pioneer Mountain near Big Sky, Montana
at the Yellowstone Club. These stations were set up to gather data regarding the envi-
ronmental conditions that lead to the formation of near-surface faceted snow crystals
on two different aspects, north and south. The stations were originally established
during the winter season of 2005/2006. The instrumentation was changed since this
time and the information presented here focuses on the current configuration. The
purpose of this document is to provide details regarding the configuration and set up
of the Yellowstone Club weather stations.
A.2 Location
Two complete weather stations were established on Pioneer Mountain near Big
Sky, Montana. These sites are referred to as North and South in this document.
Both the North and South stations are on a slope of approximately 30◦. Addition-
ally, a third station shall be referred to: American Spirit (Aspirit). This station is
maintained by the Yellowstone Club Ski Patrol and is located near the top of the
American Spirit chair lift. Figure A.1 shows the location of each weather station on
Pioneer Mountain and Table A.1 details the location of each site.
Table A.1: Detailed location information on each of the three weather stations situ-
ated on Pioneer Mountain.
Station Latitude Longitude Elevation Aspect
Aspirit 45◦14′23.0′′N 111◦26′34.5′′W 2690 m n/a
South 45◦13′47.7′′N 111◦26′33.0′′W 2740 m 187◦
North 45◦14′52.3′′N 111◦27′21.8′′W 2530 m 0◦
261
S outh Weather Station
Pioneer M ountain
Lone Peak
Nor th Weather Station
(a)
Nor th Weather Station
Pioneer M ountain
Amer ican Spir it
S outh Weather Station
(b)
Figure A.1: Google Earth images of Pioneer Mountain showing the locations of (a)
the South and (b) the North and American Spirit weather stations.
262
A.3 Data Acquisition
The North and South stations use CR10(x) dataloggers that were programmed
with Campbell Scientific (CSI) PC208w software1. The connection between the com-
puter is made via a serial connection with a SC32B (CSI) between the CR10(x) and
computer. Data was acquired in the field using a PDA equipped with Campbell Scien-
tifics PConnect software. The PDA connection requires a PDA to SC I/0 connection,
which was included with the PConnect software. The dataloggers were powered with
CSI PS12 power supplies and utilized CSI AM416 multiplexers for additional mea-
surements. For instructions on using the PC208w or PConnect software, please refer
to the respective user manuals.
Weather data was recorded every 2 minutes during the 2005/2006 season and 3
minutes thereafter; these readings were averaged every 30 minutes. The weather data
was written to output array 100 on the dataloggers every thirty minutes. The data is
downloaded and saved as a *.dat file and acquired via a PDA. This file is a ASCII
comma delimited file. Table A.2 summarizes the data that is output for both the
North and South stations.
1www.campbellsci.com/pc208w.
263
Table A.2: List of output data from North and South weather stations, where LW =
longwave, SW = shortwave, and TC = thermocouple.
05/06 Season 05/06 & 07/08 Seasons 08/09 Season
Description Units Description Units Description Units
1 Array ID Array ID Array ID
2 Year yyyy Year yyyy Year yyyy
3 Day dd Day dd Day dd
4 Hour/minute hhmm Hour/minute hhmm Hour/minute hhmm
5 Battery V Battery V Battery V
6 Wind speed m/s Wind speed m/s Wind speed m/s
7 Wind direction deg Wind direction deg Wind direction deg
8 Surface #1 ◦C Surface Temp. ◦C Surface Temp. ◦C
9 Surface #2 ◦C Incoming SW W/m2 Incoming SW W/m2
10 Incoming SW W/m2 Reflected SW W/m2 Reflected SW W/m2
11 Reflected SW W/m2 LW W/m2 LW W/m2
12 LW W/m2 Depth cm Depth cm
13 Depth cm Air (NovaLynx) ◦C Air (CS215) ◦C
14 Air (NovaLynx) ◦C Air (CS215) ◦C Humidity %
15 Air (CS215) ◦C Humidity % LW voltage mV
16 Humidity % LW voltage mV LW resistance Ω
17 LW voltage mV LW resistance Ω TC (0 cm) ◦C
18 LW resistance Ω TC (0 cm) ◦C TC (2 cm) ◦C
19 TC (0 cm) ◦C TC (2 cm) ◦C TC (4 cm) ◦C
20 TC (1 cm) ◦C TC (4 cm) ◦C TC (6 cm) ◦C
21 TC (2 cm) ◦C TC (6 cm) ◦C TC (8 cm) ◦C
22 TC (3 cm) ◦C TC (8 cm) ◦C TC (10 cm) ◦C
23 TC (4 cm) ◦C TC (10 cm) ◦C TC (12 cm) ◦C
24 TC (5 cm) ◦C TC (12 cm) ◦C TC (14 cm) ◦C
25 TC (6 cm) ◦C TC (14 cm) ◦C TC (16 cm) ◦C
26 TC (7 cm) ◦C TC (16 cm) ◦C TC (18 cm) ◦C
27 TC (8 cm) ◦C TC (18 cm) ◦C TC (20 cm) ◦C
28 TC (9 cm) ◦C TC (20 cm) ◦C TC (22 cm) ◦C
29 TC (10 cm) ◦C TC (22 cm) ◦C TC (24 cm) ◦C
30 TC (11 cm) ◦C TC (24 cm) ◦C TC (26 cm) ◦C
31 TC (12 cm) ◦C TC (26 cm) ◦C TC (28 cm) ◦C
32 TC (13 cm) ◦C TC (28 cm) ◦C TC (30 cm) ◦C
33 TC (14 cm) ◦C TC (30 cm) ◦C TC (32 cm) ◦C
34 TC (15 cm) ◦C TC (32 cm) ◦C TC (34 cm) ◦C
35 TC (16 cm) ◦C TC (34 cm) ◦C TC (36 cm) ◦C
36 TC (17 cm) ◦C TC (36 cm) ◦C TC (38 cm) ◦C
37 TC (18 cm) ◦C TC (38 cm) ◦C TC (40 cm) ◦C
38 TC (19 cm) ◦C TC (40 cm) ◦C TC (ground) ◦C
39 TC (20 cm) ◦C TC (ground) ◦C
Continued on next page. . .
264
. . . continued from previous page
05/06 Season 05/06 & 07/08 Seasons 08/09 Season
Description Units Description Units Description Units
40 TC (21 cm) ◦C
41 TC (22 cm) ◦C
42 TC (23 cm) ◦C
43 TC (24 cm) ◦C
44 TC (25 cm) ◦C
45 TC (26 cm) ◦C
46 TC (27 cm) ◦C
47 TC (28 cm) ◦C
48 TC (29 cm) ◦C
49 TC (30 cm) ◦C
50 TC (ground) ◦C
A.4 Instrumentation
The weather station equipment was mounted on a cross arm and tower placed
on the slope before snow was present. Table A.3 provides the make and model of
each instrument implemented at each site for the different setups utilized during all
winter seasons. The North and South stations include incoming short-wave radiation,
reflected short-wave radiation, incoming long-wave radiation, air temperature, relative
humidity, snow surface temperature, wind speed, wind direction, ground temperature,
and a stack of type T thermocouples for measuring snow temperatures at depth.
265
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266
A.5 Programing and Wiring
A complete wiring diagram for the current (2008/2009) setup is included in Figure
A.2 and Table A.5 summarizes the wiring in tabular format. Generally, each sensor is
wired as defined in the sensor documentation and/or the Campbell Scientific litera-
ture. The following sections (A.5.1 and A.5.2) detail programs utilized at the weather
stations. The sections step through the entire program including the program ini-
tialization, sensor measurements, and data output. The only difference between the
North and South station program are the calibration constants for a few sensors. For
quick reference, Table A.4 includes the calibration multipliers that differ between the
stations. Finally, the complete programs for each station are included in Section A.6.
Programming with the CR10(x) was completed using PC208w, the following are
a few important points to understand when reading the programs described herein.
ˆ The “;” character indicates a comment. The comments are omitted in the
following sections but included in the complete programs in Section A.6.
ˆ Each action in the CR10(x) programs is sequentially numbered and also in-
cludes an instruction code. For example, 03: Temp (107) (P11) is the 3rd
instruction and has a code of P11. The options corresponding to this command
are indented underneath this first line. The instruction code is the important
identifier, and when referred to in this document are enclosed in brackets (e.g.,
[P11]).
267
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268
Table A.4: Summary of calibration constants of weather station sensors. The values
inside the brackets give the serial number of the sensor and all calibration numbers
are given as W/m2/mV.
North South Aspirit
CGR3 156.495 (070108) 84.531 (070112) n/a
CMP3, incoming 70.47 (080194) 70.87 (080193) n/a
CMP3, reflected 65.87 (080191) 69.74 (080192) n/a
Aspirt, PSP n/a n/a 122.55 (32530F3)
Aspirt, PIR n/a n/a 256.89 (33586F3)
269
T
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270
A.5.1 North and South Station Program
Program Initialization: The first three commands in the program establish the
program storage location on the CR10(x), measure the battery voltage with [P10],
and turn off the data logger when the battery voltage drops belows 11 volts with
[P89].
*Table 1 Program
01: 180 Execution Interval (seconds)
1: Batt Voltage (P10)
1: 14 Loc [ Battery ]
2: If (X<=>F) (P89)
1: 14 X Loc [ Battery ]
2: 4 <
3: 11 F
4: 0 Go to end of Program Table
Reference Temperature (CR10TCR): The reference temperature captured with
[P11], as measured by the CR10TCR, is used to adjust for the temperature of the
AM416 multiplexer terminals, where the thermocouples are attached. Thus, the
thermistor should be placed on the multiplexer.
3: Temp (107) (P11)
1: 1 Reps
2: 6 SE Channel
3: 1 Excite all reps w/E1
4: 1 Loc [ RefTemp__ ]
5: 1.0 Mult
6: 0.0 Offset
Activate AM416 Multiplexer: The AM416 multiplexer is activated by turning
the attached port (C1) to high with [P86], i.e., “on”. As will be detailed in the
next section, the multiplexers operate by first being activated. When the port (C2)
connected to CLK on the multiplexer is pulsed the signal form the first pair of channels
is transfered through the COM connections. The next time C2 is pulsed the second
pair is transfered and so forth.
271
4: Do (P86)
1: 41 Set Port 1 High
Thermocouple Array: The thermocouple array in the snow contains 20 sensors,
thus the 10 pairs of thermocouples must be measured. First, a loop is established
with [P87]. On each execution of this loop the C2 is pulsed with [P86] causing the
multiplexer to cycle through the first 10 terminal pairs. [P90] indicates that the
subsequent command, [P14], should be executed twice. Finally, the loop is ended
with [P95].
The −− in front of step 6 of [P14] indicates that each time this command is
executed that the storage location should be incremented. In this case, this results
in TC 1, TC 2, etc.
5: Beginning of Loop (P87)
1: 0 Delay
2: 10 Loop Count
6: Do (P86)
1: 72 Pulse Port 2
7: Step Loop Index (P90)
1: 2 Step
8: Thermocouple Temp (DIFF) (P14)
1: 2 Reps
2: 1 2.5 mV Slow Range
3: 01 DIFF Channel
4: 1 Type T (Copper-Constantan)
5: 1 Ref Temp (Deg. C) Loc [ RefTemp__ ]
6: 20 -- Loc [ TC_1 ]
7: 1.0 Mult
8: 0.0 Offset
9: End (P95)
Ground and Snow Surface Temperature: After reading the 20 thermocouples
the multiplexer is triggered again by pulsing C2 with [P86], this causes the 11th
terminal pair to be measured, which is the thermocouple at the ground ([P14]) and
272
the snow surface temperature ([P2]). The surface temperature requires a multiplier
of 0.1 ◦C/m, which is consistent between stations.
10: Do (P86)
1: 72 Pulse Port 2
11: Thermocouple Temp (DIFF) (P14)
1: 1 Reps
2: 1 2.5 mV Slow Range
3: 1 DIFF Channel
4: 1 Type T (Copper-Constantan)
5: 1 Ref Temp (Deg. C) Loc [ RefTemp__ ]
6: 40 Loc [ TCgnd ]
7: 1.0 Mult
8: 0.0 Offset
12: Volt (Diff) (P2)
1: 1 Reps
2: 5 2500 mV Slow Range
3: 2 DIFF Channel
4: 9 Loc [ SurTemp_1 ]
5: 0.1 Mult
6: 0.0 Offset
Incoming and Reflected Short-wave: As done for the previous readings, the
multiplexer is triggered by pulsing C2 with [P86], this causes the 12th terminal pair
to be measured, which is the two short-wave radiation sensors both of which are
voltages read using [P2].
13: Do (P86)
1: 72 Pulse Port 2
14: Volt (Diff) (P2)
1: 1 Reps
2: 3 25 mV Slow Range
3: 1 DIFF Channel
4: 4 Loc [ ShortUP ]
5: 72.43 Mult
6: 0.0 Offset
15: Volt (Diff) (P2)
1: 1 Reps
2: 22 7.5 mV 60 Hz Rejection Range
3: 2 DIFF Channel
4: 5 Loc [ ShortDOWN ]
5: 200 Mult
6: 0.0 Offset
Deactivate AM416 Multiplexer: The AM416 multiplexer is turned off be setting
Port 1 (C1) to low with [P86].
273
16: Do (P86)
1: 51 Set Port 1 Low
Wind Speed and Direction: The MetOne anemometer first measures the wind
speed by reading the value from pulse input 1 using [P3]. Then instructions [P89],
[P30], and [P95] (not shown) set negative values to zero, which occur in still conditions
due to instrument noise. These instructions are included in the complete program in
Section A.6.
17: Pulse (P3)
1: 1 Reps
2: 1 Pulse Input Channel
3: 22 Switch Closure, Output Hz
4: 2 Loc [ WindSpd ]
5: 0.7990 Mult
6: 0.2811 Offset
The wind direction is measured with [P5] via the voltage across a resistor in
the anemometer, which requires a current. The current is supplied as an excitation
voltage from E2 and the voltage measured on SE5. The offsets and multipliers are
consistent between the two stations.
21: AC Half Bridge (P5)
1: 1 Reps
2: 25 2500 mV 60 Hz Rejection Range
3: 5 SE Channel
4: 2 Excite all reps w/Exchan 2
5: 2500 mV Excitation
6: 3 Loc [ WindDir ]
7: 360 Mult
8: 0.0 Offset
4: Do (P86)
1: 41 Set Port 1 High
Long-wave Radiation: The Kipp & Zonen long-wave sensor requires two mea-
surements, a voltage and a resistance. The voltage is measured with [P2] and is used
to adjust for variations between the case and sensor temperatures. The resistance
measurement requires an excitation to acquire the voltage across the resistor using
274
[P5]. This value is then converted to a resistance with instruction [P59]. The multi-
plier for this instruction should be the value of the reference resistor wired from SE12
to E3.
These two values are converted to long-wave radiation (W/m2) using subroutines
#1 and #2 that are called with instruction [P86]. These subroutines use Equations
(A.1)–(A.3), where the incoming long-wave radiation Ld↓ is computed from the voltage
reading Uemf , the case resistance Rc, and the constants α, β, γ, and S. The constant
S is the sensor sensitivity included in Table A.4. The constants, for both the Eppley
and Kipp & Zonen sensors, are defined as α = 1.0295× 10−3, β = 2.391× 10−4, and
γ = 1.568× 10−7. The complete subroutines are provided in Section A.6.
Lnet =
Uemf
S
(A.1)
Tc =
1
α + [β · (ln(Rc) + γ(ln(Rc))3]
(A.2)
Ld↓ = Lnet + 5.67× 10
−8 · T 4b (A.3)
25: Volt (Diff) (P2)
1: 1 Reps
2: 1 2.5 mV Slow Range
3: 5 DIFF Channel
4: 100 Loc [ Uemf ]
5: 1.0 Mult
6: 0.0 Offset
26: AC Half Bridge (P5)
1: 1 Reps
2: 15 2500 mV Fast Range
3: 12 SE Channel
4: 3 Excite all reps w/Exchan 3
5: 2500 mV Excitation
6: 101 Loc [ Case_Res ]
7: 1.0 Mult
8: 0.0 Offset
27: BR Transform Rf[X/(1-X)] (P59)
1: 1 Reps
2: 101 Loc [ Case_Res ]
3: 1000 Multiplier (Rf)
28: Do (P86)
1: 1 Call Subroutine 1
275
29: Do (P86)
1: 2 Call Subroutine 2
Snow Depth: The Nova Lynx ultrasonic snow depth sensor operates in various
modes. The method presented in the Nova Lynx user manual proved to be unreliable
in the field. Thus, the method presented here is recommended. First, the sensor
is turned on using communication port 6 via [P22]. Then the program waits two
seconds, [P22], for the sensor to perform the measurement, which is accomplished
with the excitation with delay command, but notice that the excitation voltage is
set to zero. Next, the voltage returned from the sensor is collected via [P2] and the
sensor is powered off with [P86].
In order for the Nova Lynx sensor to operate correctly both dip switch #1 and
#3 must be in the “on” position, refer to the sensor user manual for setting these
switches. The voltage range should be from 0–5 V, which results in the use of the
multiplier of -0.25 cm/mV. The resulting depth is not temperature adjusted and
should be compensated for temperature using the multiplier (CF ) computed using
Equation (A.4) and the measured temperature (T ) from the CS215 sensor. This
sensor provides a more accurate reading of temperature than the Nova Lynx sensor
itself. This portion of the code is included in the complete program in Section A.6.
CF =
[
T + 273.15
273.15
] 1
2
(A.4)
The offset value should be set to the distance from the sensor to bare ground. A value
of 200 cm was used for both stations; this value was an assumed value because only
the occurrence of new snow was desired.
30: Do (P86)
1: 46 Set Port 6 High
31: Excitation with Delay (P22)
276
1: 2 Ex Channel
2: 200 Delay W/Ex (units = 0.01 sec)
3: 0000 Delay After Ex (units = 0.01 sec)
4: 0000 mV Excitation
32: Volt (Diff) (P2)
1: 1 Reps
2: 5 2500 mV Slow Range
3: 4 DIFF Channel
4: 116 Loc [ rawdepth ]
5: -0.25 Mult
6: 200 Offset
33: Do (P86)
1: 56 Set Port 6 Low
Temperature and Humidity: The CS215 sensor has a specific instruction, [P105],
designed for reading the sensor. The temperature is returned to the location specified
(6) and the humidity in the following location (7).
40: SDI-12 Recorder (P105)
1: 00 SDI-12 Address
2: 00 SDI-12 Command
3: 5 Port
4: 6 Loc [ TempCS215 ]
5: 1.0 Mult
6: 0.0 Offset
Data Storage: To store the data the output flags must be set to high, which is
accomplished with instruction [P92], in this case the data is written every 30 minutes.
Before writing the data, the storage location is set to 100 with [P80], which is an
arbitrary value. The storage location allows you to write various sets of data to a
single file. For example, it is common to write 30 minute data as well as the 24 hour
averages in different storage arrays.
First, the time stamp is output using [P77] as three values: year, day, and
hour/minute. Next, the 30 minute averages of all recorded data are output using
[P71], with the exception of the battery voltage in which the minimum is reported
with [P74]. Only two commands are shown in the code here, for the complete output
see Section A.6.
277
41: If time is (P92)
1: 0000 Minutes (Seconds --) into a
2: 30 Interval (same units as above)
3: 10 Set Output Flag High
42: Set Active Storage Area (P80)
1: 1 Final Storage Area 1
2: 100 Array ID
43: Real Time (P77)
1: 1220 Year,Day,Hour/Minute (midnight = 2400)
44: Minimum (P74)
1: 1 Reps
2: 00 Time Option
3: 14 Loc [ Battery ]
46: Average (P71)
1: 1 Reps
2: 9 Loc [ Surface ]
Complete Main Program: The main program does not require any ending state-
ment. However, the two commands shown here must be present. The first may be
used for a second program and the second indicates the beginning of the subroutines,
which must be present regardless of the presence of subroutines.
*Table 2 Program
02: 0.0000 Execution Interval (seconds)
*Table 3 Subroutines
Subroutines: The subroutines require a beginning statement, [P85], and an
ending statement of [P95]. The operations desired from the subroutine should be in
between these two statements. The subroutines have complete access to read and
write input values. For a complete example of a subroutine see the programs in
Section A.6.
1: Beginning of Subroutine (P85)
1: 1 Subroutine 1
13: End (P95)
278
A.5.2 American Spirit Weather Station
The Aspirit station and program are maintained by the Yellowstone Club Ski Pa-
trol. The Eppley PIR long-wave sensor at this station requires the same instructions
as the the North and South stations, as detailed in Section A.5.1.9. For the North
and South stations Kipp & Zonen sensors were purchased because of their ability to
adjust for solar contamination of the sensor, which is a problem with the Eppley PIR
sensors (Albrecht and Cox, 1977). To avoid this contamination with the Eppley PIR
long-wave sensor, an additional resistance measurement and computation is required.
Currently this additional adjustment is not included at the Aspirit station. A compar-
ison with the nearby Yellow Mule (YLWM8) RAWS2 weather station indicates that
solar contamination has a minimal effect. As shown in Figure A.3, the short-wave
solar irradiance is similar between the RAWS and Aspirit weather stations.
Nonetheless, the details regarding the implementation of this adjustment are pre-
sented here for future application by other researchers. Contrary to the CSI appli-
cation note recommended setup, in certain situations the dome thermistor correction
should be included in the computation of incoming long-wave radiation. Details
regarding this adjustment are given by Albrecht and Cox (1977), which explains that
under intense solar radiation the case and dome temperatures can differ by 10◦C re-
sulting in errors between 300 and 400 W/m2, such was the case at the South location
that motivated the usage of the Kipp & Zonen sensors.
Equation (A.5) is used to adjust for the dome temperature,
Ld↓ = Lnet + σT
4
b − kσ(T
4
d − T
4
b ), (A.5)
where σ = 5.67×10−8. This is an extension of Equation (A.3). Thus, the Eppley PIR
requires three measurements: Uemf , Rd, and Rc. The net radiation Lnet is calculated
2Remote Automated Weather Station (www.fs.fec.us/raws
279
03/16 12:00 03/17 12:00 03/18 12:00 03/19 12:00 03/20
0
100
200
300
400
500
600
700
800
900
Time
Ir
ra
di
an
ce
 (W
/m
2 )
 
 
Incoming Shortwave (Amer. Spirit)
Solar Radiation (Yellow mule)
Figure A.3: Comparison of incoming short-wave radiation at the Yellow Mule RAWS
and Aspirit weather stations.
using Equation (A.1) and the temperatures Td and Tc are computed using Equation
(A.2) using the appropriate resistance value. As with the Kipp & Zonen the sensitivity
S is provided by the manufacturer. Finally, the k is yet another constant that is not
provided; upon contacting a CSI representative it was recommended that a value
of k = 3.5 be used. This estimation can be eliminated by performing one of many
calibration procedures. Additional information on calibration may be found in Reda
et al. (2003) and Stoffel et al. (2006).
280
A.6 Complete Weather Station Programs
A.6.1 North Weather Station Program
1 ;{CR10X}
2 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
3 ; A PROGRAM BY:
4 ;
5 ; ANDREW E. SLAUGHTER
6 ; 205 Cobleigh Hall , MSU−Bozeman
7 ; P.O. Box 173900
8 ; Bozeman , MT 59717−3900
9 ; (406) 994−2293
10 ;
11 ;***********************************************************************************
12 ;
13 ; BACKGROUND:
14 ; The f o l l ow ing program u t i l i z e s the Campbell S c i e n t i f i c , Inc . ( SCI ) CR10(x )
15 ; data logge r and two CSI AM418 mul t ip l exe r to acqu i r e ba s i c weather data that
16 ; i n c l ude s snow surface temperature , snow depth , humidity , a i r temperature ,
17 ; wind speed , wind d i r e c t i on , longwave rad ia t i on , and shortwave r ad i a t i on .
18 ; Addi t iona l ly , the s t a t i on has an array o f thermocouples that measures the
19 ; snowpack temperature at var i ous depths below the surface .
20 ;
21 ; The ob j e c t i v e o f the s i t e i s to c o l l e c t f i e l d data regard ing the growth
22 ; o f near−surface f a c e t ed and surface hoar c r y s t a l s for the use in v e r i f y i n g both
23 ; lab and an a l y t i c a l models o f the near−surface pro c e s s e s .
24 ;
25 ; Last Updated : December 2008
26 ;
27 ;************************************************************************************
28 ;
29 ; WIRING SCHEME:
30 ;
31 ; −> CR10TCR Thermistor
32 ; Wht − AG
33 ; Blk − E1
34 ; Red − SE6
35 ;
36 ; −> AM418 MULTIPLEXIERS
37 ; Wiring (CR10 − AM416) :
38 ; C1 − REM
39 ; C2 − CLK
40 ; H1 − ComH1
41 ; L1 − ComL1
42 ; H2 − ComH2
43 ; L2 − ComL2
44 ;
45 ; −> THERMOCOUPLES ( wired to AM416)
46 ; 1H1, 1 L1 − TC 1
47 ; 1H2,1 L2 − TC 2
48 ; . . .
49 ; 11H1,11L1 − TC 21 (TC to Grnd)
50 ;
51 ; −> KIPP & ZONEN CMP3 ( shortwave , wired to AM416#1)
52 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
53 ; Up Mult . = 14.19 uV/W/mˆ2 = 70.47 W/mˆ2/mV (SN:080194)
54 ; Dn Mult . = 15.18 uV/W/mˆ2 = 65.87 W/mˆ2/mV (SN:080191)
55 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
56 ; Incoming (up) Re f l e c t ed (down)
57 ; Red − 12 :H1 Grn − 12 : L2
58 ; Blk − 12 : L1 Blu − 12 :H2
59 ;
60 ; −> KIPP & ZONEN CGR3 ( longwave ) ;
61 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
62 ; Mult . = 6 .39 uV/W/mˆ2 = 156.495 W/mˆ2/mV (SN:070108)
63 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
64 ; Red − L5
65 ; Blu − H5
66 ; Ylw − SE12
67 ; Grn − G
68 ; 1kOhm Res i s t o r from SE12 − E3
69 ;
70 ; −> METONE ANEMOMETER
71 ; Yel − SE5
72 ; Wht − AG
73 ; Grn − E2
74 ; Blk − G
75 ; Brn − G
76 ; Red − P1
77 ;
78 ; −> EVEREST INTERSCIENCE SNOW SURFACE TEMPERATURE
281
79 ; Blu − 11 :H2
80 ; Wht − 11 : L2
81 ; Red − 12V
82 ; Blk − G
83 ;
84 ; −> CAMPBELL SCIENTIFIC HUMIDITY AND TEMP
85 ; Blk ,Wht, Clr − G
86 ; Red − 12V
87 ; Grn − C5
88 ;
89 ; −> NOVALYNX ULTRASONIC SNOW DEPTH
90 ; Red − 12V
91 ; Blk − G
92 ; Clr − G
93 ; Grn − C6
94 ; Wht − 4H
95 ; Brn − 4L
96 ;
97 ;
98 ;********************************************************************
99 ;**** Begin Program *************************************************
100
101 ;EXECUTION INTERVAL IN SECONDS
102 *Table 1 Program
103 01 : 180 Execution I n t e r v a l ( seconds )
104
105
106 ;*********************************************************************
107 ;**** Battery Voltage ************************************************
108
109 1 : Batt Voltage (P10)
110 1 : 14 Loc [ Battery ]
111
112
113 ;*********************************************************************
114 ;**** Stop i f Battery < 11V ******************************************
115
116 2 : I f (X<=>F) (P89)
117 1 : 14 X Loc [ Battery ]
118 2 : 4 <
119 3 : 11 F
120 4 : 0 Go to end o f Program Table
121
122
123 ;*********************************************************************
124 ;**** Reference Temperature ******************************************
125 3 : Temp (107) (P11)
126 1 : 1 Reps
127 2 : 6 SE Channel
128 3 : 1 Exc i te a l l reps w/E1
129 4 : 1 Loc [ RefTemp ]
130 5 : 1 .0 Mult
131 6 : 0 .0 O f f s e t
132
133
134 ;*********************************************************************
135 ;**** Thermocouples , SW Radiation , & Sur face Temp ********************
136
137 ;TRIGGER MULTIPLEXER
138 4 : Do (P86)
139 1 : 41 Set Port 1 High
140
141 ;MEASURE THERCOUPLES 1 THRU 20
142 5 : Beginning o f Loop (P87)
143 1 : 0 Delay
144 2 : 10 Loop Count
145
146 6 : Do (P86)
147 1 : 72 Pulse Port 2
148
149 ; INDICATE TWO READINGS PER SET
150 7 : Step Loop Index (P90)
151 1 : 2 Step
152
153 ;READ 20 THERMOCOUPLES
154 8 : Thermocouple Temp (DIFF) (P14)
155 1 : 2 Reps
156 2 : 1 2 .5 mV Slow Range
157 3 : 01 DIFF Channel
158 4 : 1 Type T (Copper−Constantan )
159 5 : 1 Ref Temp (Deg . C) Loc [ RefTemp ]
160 6 : 20 −− Loc [ TC 1 ]
161 7 : 1 .0 Mult
162 8 : 0 .0 O f f s e t
163
164 ;END THE LOOP FOR READING THERMOCOULPES
165 9 : End (P95)
282
166
167 ;READ THE GROUND THERMOCOUPLE AND THE SURFACE TEMPERATURE
168 10 : Do (P86)
169 1 : 72 Pulse Port 2
170
171 11 : Thermocouple Temp (DIFF) (P14)
172 1 : 1 Reps
173 2 : 1 2 .5 mV Slow Range
174 3 : 1 DIFF Channel
175 4 : 1 Type T (Copper−Constantan )
176 5 : 1 Ref Temp (Deg . C) Loc [ RefTemp ]
177 6 : 40 Loc [ TCgnd ]
178 7 : 1 .0 Mult
179 8 : 0 .0 O f f s e t
180
181 12 : Volt ( D i f f ) (P2)
182 1 : 1 Reps
183 2 : 5 2500 mV Slow Range
184 3 : 2 DIFF Channel
185 4 : 9 Loc [ Sur face ]
186 5 : 0 .1 Mult
187 6 : 0 .0 O f f s e t
188
189 ;READ THE SHORTWAVE SENSORS (UP & DOWN)
190 13 : Do (P86)
191 1 : 72 Pulse Port 2
192
193 ; Incoming (up)
194 14 : Volt ( D i f f ) (P2)
195 1 : 1 Reps
196 2 : 3 25 mV Slow Range
197 3 : 1 DIFF Channel
198 4 : 4 Loc [ ShortUP ]
199 5 : 70 .47 Mult
200 6 : 0 .0 O f f s e t
201
202 ; Re f l e c t ed (Down)
203 15 : Volt ( D i f f ) (P2)
204 1 : 1 Reps
205 2 : 3 25 mV Slow Range
206 3 : 1 DIFF Channel
207 4 : 5 Loc [ ShortDOWN ]
208 5 : 65 .87 Mult
209 6 : 0 .0 O f f s e t
210
211 ;TURN OFF MULTIPLEXER
212 16 : Do (P86)
213 1 : 51 Set Port 1 Low
214
215
216 ;**************************************************************************
217 ;**** Wind Speed and Di r e c t i on ********************************************
218 ;−−−−−−−−−−−−−−−−−−−−−−−−−
219 ; WindSpd = m/s2
220 ; WindDir = 0−360 (N=0=360)
221 ;−−−−−−−−−−−−−−−−−−−−−−−−−−
222
223 ;MEASURE WIND SPEED
224 17 : Pulse (P3)
225 1 : 1 Reps
226 2 : 1 Pulse Input Channel
227 3 : 22 Switch Closure , Output Hz
228 4 : 2 Loc [ WindSpd ]
229 5 : 0 .7990 Mult
230 6 : 0 .2811 Of f s e t
231
232 ; IF WIND SPEED IS NEGATIVE SET TO ZERO
233 18 : I f (X<=>F) (P89)
234 1 : 2 X Loc [ WindSpd ]
235 2 : 1 =
236 3 : 0 .2811 F
237 4 : 30 Then Do
238
239 19 : Z=F (P30)
240 1 : 0 F
241 2 : 0 Exponent o f 10
242 3 : 2 Z Loc [ WindSpd ]
243
244 20 : End (P95)
245
246 21 : AC Half Bridge (P5)
247 1 : 1 Reps
248 2 : 25 2500 mV 60 Hz Re jec t i on Range
249 3 : 5 SE Channel
250 4 : 2 Exc i te a l l reps w/Exchan 2
251 5 : 2500 mV Exc i ta t i on
252 6 : 3 Loc [ WindDir ]
283
253 7 : 360 Mult
254 8 : 0 .0 O f f s e t
255
256
257 ;*************************************************************************
258 ;**** Longwave Radiat ion *************************************************
259 ;
260 ;MEASURE VOLTAGE FOR THERMOPILE
261 22 : Volt ( D i f f ) (P2)
262 1 : 1 Reps
263 2 : 1 2 .5 mV Slow Range
264 3 : 5 DIFF Channel
265 4 : 100 Loc [ Uemf ]
266 5 : 1 Mult
267 6 : 0 .0 O f f s e t
268
269 ;MEASURE THERMISTOR
270 23 : AC Half Bridge (P5)
271 1 : 1 Reps
272 2 : 15 2500 mV Fast Range
273 3 : 12 SE Channel
274 4 : 3 Exc i te a l l reps w/Exchan 3
275 5 : 2500 mV Exc i ta t i on
276 6 : 101 Loc [ Case Res ]
277 7 : 1 .0 Mult
278 8 : 0 .0 O f f s e t
279
280 ;CONVERT THERMISTOR MEASURE TO RESISTANCE
281 24 : BR Transform Rf [X/(1−X) ] (P59)
282 1 : 1 Reps
283 2 : 101 Loc [ Case Res ]
284 3 : 1000 Mu l t i p l i e r (Rf )
285
286 ;CONVERT RESISTANCE TO TEMPERATURE
287 25 : Do (P86)
288 1 : 1 Cal l Subroutine 1
289
290 ;CORRECT FOR CASE TEMPERATURE −> OUTPUT LONGWAVE RADIATION
291 26 : Do (P86)
292 1 : 2 Cal l Subroutine 2
293
294 ;************************************************************************
295 ;**** Snow Depth ********************************************************
296
297 27 : Do (P86)
298 1 : 46 Set Port 6 High
299
300 ; Wait 2 seconds for s ensor to warm−up and measure depth
301 28 : Exc i ta t i on with Delay (P22)
302 1 : 2 Ex Channel
303 2 : 200 Delay W/Ex ( un i t s = 0.01 sec )
304 3 : 0000 Delay After Ex ( un i t s = 0.01 sec )
305 4 : 0000 mV Exc i ta t i on
306
307 ; Depth given in mV and sca l ed to cm, o f f s e t = mounting he ight
308 29 : Volt ( D i f f ) (P2)
309 1 : 1 Reps
310 2 : 5 2500 mV Slow Range
311 3 : 4 DIFF Channel
312 4 : 116 Loc [ rawdepth ]
313 5 : −0.25 Mult
314 6 : 200 Of f s e t
315
316 30 : Do (P86)
317 1 : 56 Set Port 6 Low
318
319 ; Compute the temperature co r r e c t ed depth
320 31 : Z=X+F (P34)
321 1 : 6 X Loc [ TempCS215 ]
322 2 : 273 .15 F
323 3 : 117 Z Loc [ d1 ]
324
325 32 : Z=F (P30)
326 1 : 273 .15 F
327 2 : 00 Exponent o f 10
328 3 : 118 Z Loc [ k e l v i n ]
329
330 33 : Z=X/Y (P38)
331 1 : 117 X Loc [ d1 ]
332 2 : 118 Y Loc [ k e l v i n ]
333 3 : 119 Z Loc [ d2 ]
334
335 34 : Z=F (P30)
336 1 : 0 .5 F
337 2 : 00 Exponent o f 10
338 3 : 120 Z Loc [ exp ]
339
284
340 35 : Z=XˆY (P47)
341 1 : 119 X Loc [ d2 ]
342 2 : 120 Y Loc [ exp ]
343 3 : 121 Z Loc [ CF ]
344
345 36 : Z=X*Y (P36)
346 1 : 121 X Loc [ CF ]
347 2 : 116 Y Loc [ rawdepth ]
348 3 : 12 Z Loc [ Depth ]
349
350
351 ;*************************************************************************
352 ;**** Humidity & Air Temperature *****************************************
353
354 37 : SDI−12 Recorder (P105 )
355 1 : 00 SDI−12 Address
356 2 : 00 SDI−12 Command
357 3 : 5 Port
358 4 : 6 Loc [ TempCS215 ]
359 5 : 1 .0 Mult
360 6 : 0 .0 O f f s e t
361
362 ;*************************************************************************
363 ;**** Data Storage A l l o ca t i on ********************************************
364
365 ;SET TIME AND STORAGE
366 38 : I f time i s (P92)
367 1 : 0000 Minutes ( Seconds −−) in to a
368 2 : 30 I n t e r v a l ( same un i t s as above )
369 3 : 10 Set Output Flag High
370
371 39 : Set Active Storage Area (P80)
372 1 : 1 Fina l Storage Area 1
373 2 : 100 Array ID
374
375 40 : Real Time (P77)
376 1 : 1220 Year ,Day , Hour/Minute ( midnight = 2400)
377
378 ;OUTPUT MINIMUM BATTERY VOLTAGE
379 41 : Minimum (P74)
380 1 : 1 Reps
381 2 : 00 Time Option
382 3 : 14 Loc [ Battery ]
383
384 ;READ WIND SPEED AND DIRECTION
385 42 : Average (P71)
386 1 : 1 Reps
387 2 : 2 Loc [ WindSpd ]
388
389 43 : Average (P71)
390 1 : 1 Reps
391 2 : 3 Loc [ WindDir ]
392
393 ;SURFACE TEMPERATURE AVERAGES
394 44 : Average (P71)
395 1 : 1 Reps
396 2 : 9 Loc [ Sur face ]
397
398 ;AVERAGE LONG/SHORTWAVE RADATION
399 45 : Average (P71)
400 1 : 1 Reps
401 2 : 4 Loc [ ShortUP ]
402 46 : Average (P71)
403 1 : 1 Reps
404 2 : 5 Loc [ ShortDOWN ]
405
406 47 : Average (P71)
407 1 : 1 Reps
408 2 : 11 Loc [ Longwave ]
409
410 ;SNOW DEPTH AVERAGE
411 48 : Average (P71)
412 1 : 1 Reps
413 2 : 12 Loc [ Depth ]
414
415 49 : Average (P71)
416 1 : 1 Reps
417 2 : 116 Loc [ rawdepth ]
418
419 50 : Average (P71)
420 1 : 1 Reps
421 2 : 121 Loc [ CF ]
422
423 51 : Average (P71)
424 1 : 1 Reps
425 2 : 8 Loc [ DpthTEMP ]
426
285
427 ;ATMOSPHERE TEMP/HUMID AVERAGES
428 52 : Average (P71)
429 1 : 1 Reps
430 2 : 6 Loc [ TempCS215 ]
431
432 53 : Average (P71)
433 1 : 1 Reps
434 2 : 7 Loc [ HumdCS215 ]
435
436 ;AVERAGE RAW DATA FOR LONGWAVE
437 54 : Average (P71)
438 1 : 1 Reps
439 2 : 122 Loc [ ]
440
441 55 : Average (P71)
442 1 : 1 Reps
443 2 : 101 Loc [ Case Res ]
444
445 ;THERMOCOUPLE AVERAGES
446 56 : Average (P71)
447 1 : 1 Reps
448 2 : 20 Loc [ TC 1 ]
449 57 : Average (P71)
450 1 : 1 Reps
451 2 : 21 Loc [ TC 2 ]
452 58 : Average (P71)
453 1 : 1 Reps
454 2 : 22 Loc [ TC 3 ]
455 59 : Average (P71)
456 1 : 1 Reps
457 2 : 23 Loc [ TC 4 ]
458 60 : Average (P71)
459 1 : 1 Reps
460 2 : 24 Loc [ TC 5 ]
461 61 : Average (P71)
462 1 : 1 Reps
463 2 : 25 Loc [ TC 6 ]
464 62 : Average (P71)
465 1 : 1 Reps
466 2 : 26 Loc [ TC 7 ]
467 63 : Average (P71)
468 1 : 1 Reps
469 2 : 27 Loc [ TC 8 ]
470 64 : Average (P71)
471 1 : 1 Reps
472 2 : 28 Loc [ TC 9 ]
473 65 : Average (P71)
474 1 : 1 Reps
475 2 : 29 Loc [ TC 10 ]
476 66 : Average (P71)
477 1 : 1 Reps
478 2 : 30 Loc [ TC 11 ]
479 67 : Average (P71)
480 1 : 1 Reps
481 2 : 31 Loc [ TC 12 ]
482 68 : Average (P71)
483 1 : 1 Reps
484 2 : 32 Loc [ TC 13 ]
485 69 : Average (P71)
486 1 : 1 Reps
487 2 : 33 Loc [ TC 14 ]
488 70 : Average (P71)
489 1 : 1 Reps
490 2 : 34 Loc [ TC 15 ]
491 71 : Average (P71)
492 1 : 1 Reps
493 2 : 35 Loc [ TC 16 ]
494 72 : Average (P71)
495 1 : 1 Reps
496 2 : 36 Loc [ TC 17 ]
497 73 : Average (P71)
498 1 : 1 Reps
499 2 : 37 Loc [ TC 18 ]
500 74 : Average (P71)
501 1 : 1 Reps
502 2 : 38 Loc [ TC 19 ]
503 75 : Average (P71)
504 1 : 1 Reps
505 2 : 39 Loc [ TC 20 ]
506 76 : Average (P71)
507 1 : 1 Reps
508 2 : 40 Loc [ TCgnd ]
509
510 *Table 2 Program
511 02 : 0 .0000 Execution I n t e r v a l ( seconds )
512
513 *Table 3 Subrout ines
286
514
515
516 ;**** SUBROUTINE #1 ******************************************************************
517
518 1 : Beginning o f Subroutine (P85)
519 1 : 1 Subroutine 1
520
521 ;CONVERT RESISTANCE TO TEMPERATURE
522 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
523 ;T = 1/(A+B*Ln(R) + C*(Ln(R) ) ˆ3)
524 ;T = Temp in Deg . Kelvin
525 ;A = 0.0010295
526 ;B = 0.0002391
527 ;C = 0.0000001568
528 ;R = Measured r e s i s t a n c e o f the rmi s to r
529 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
530
531 ; Constant A
532 2 : Z=F (P30)
533 1 : 1 .0295 F
534 2 : −3 Exponent o f 10
535 3 : 102 Z Loc [ ConstA ]
536
537 ; Constant B
538 3 : Z=F (P30)
539 1 : 2 .391 F
540 2 : −4 Exponent o f 10
541 3 : 103 Z Loc [ ConstB ]
542
543 ; Constant C
544 4 : Z=F (P30)
545 1 : 1 .568 F
546 2 : −7 Exponent o f 10
547 3 : 104 Z Loc [ ConstC ]
548
549 ; Natural Log o f Res i s tance
550 5 : Z=LN(X) (P40)
551 1 : 101 X Loc [ Case Res ]
552 2 : 105 Z Loc [ Ln Res ]
553
554 ; B*Ln(R)
555 6 : Z=X*Y (P36)
556 1 : 103 X Loc [ ConstB ]
557 2 : 105 Y Loc [ Ln Res ]
558 3 : 107 Z Loc [ bLn res ]
559
560 ; Squre and Cube Natural Log
561 7 : Z=X*Y (P36)
562 1 : 105 X Loc [ Ln Res ]
563 2 : 105 Y Loc [ Ln Res ]
564 3 : 108 Z Loc [ Ln res2 ]
565
566 8 : Z=X*Y (P36)
567 1 : 105 X Loc [ Ln Res ]
568 2 : 108 Y Loc [ Ln res2 ]
569 3 : 109 Z Loc [ Ln res3 ]
570
571 ; C*(Ln(R) ) ˆ3
572 9 : Z=X*Y (P36)
573 1 : 104 X Loc [ ConstC ]
574 2 : 109 Y Loc [ Ln res3 ]
575 3 : 110 Z Loc [ CLn res3 ]
576
577 ; Add A and B Terms
578 10 : Z=X+Y (P33)
579 1 : 102 X Loc [ ConstA ]
580 2 : 107 Y Loc [ bLn res ]
581 3 : 111 Z Loc [ case temp ]
582
583 ; Add C Term to A/B Term
584 11 : Z=X+Y (P33)
585 1 : 111 X Loc [ case temp ]
586 2 : 110 Y Loc [ CLn res3 ]
587 3 : 111 Z Loc [ case temp ]
588
589 ; Take Rec iporca l o f case temp
590 12 : Z=1/X (P42)
591 1 : 111 X Loc [ case temp ]
592 2 : 111 Z Loc [ case temp ]
593
594 13 : End (P95)
595
596
597 ;**** SUBROUTINE #2 ***********************************************
598
599 14 : Beginning o f Subroutine (P85)
600 1 : 2 Subroutine 2
287
601
602 ;CORRECT PIR CASE TEMPERATURE
603 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
604 ; CorrectOutput = A + (C*Tˆ4)
605 ; A = Thermopile Output
606 ; C = Stefan−Boltzman Cnst = 5.6697 e−8 Wm−2K−4
607 ; T = Case Temperature in Kelvin
608 ; S = F = 6.39 uV/W/mˆ2 = 156.495 mV/W/mˆ2 (SN:070108)
609 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
610
611 ;CONVERT THERMOPILE TO Wm−2
612 15 : Z=X*F (P37)
613 1 : 100 X Loc [ Uemf ]
614 2 : 156.495 F
615 3 : 112 Z Loc [ PIR Aterm ]
616
617 ; Load 4 for r a i s i n g to f o r th
618 16 : Z=F (P30)
619 1 : 4 F
620 2 : 0 Exponent o f 10
621 3 : 113 Z Loc [ Power4 ]
622
623 ; Raise to 4 th Power
624 17 : Z=XˆY (P47)
625 1 : 111 X Loc [ case temp ]
626 2 : 113 Y Loc [ Power4 ]
627 3 : 111 Z Loc [ case temp ]
628
629 ; Load Boltzman
630 18 : Z=F (P30)
631 1 : 5 .669 F
632 2 : −8 Exponent o f 10
633 3 : 114 Z Loc [ PIR Bterm ]
634
635 ; Mutl ip ly Boltzman by Tˆ4
636 19 : Z=X*Y (P36)
637 1 : 114 X Loc [ PIR Bterm ]
638 2 : 111 Y Loc [ case temp ]
639 3 : 114 Z Loc [ PIR Bterm ]
640
641 ;Add A to Bterm −> OUTPUTS THE LONGWAVE RADIATION
642 20 : Z=X+Y (P33)
643 1 : 112 X Loc [ PIR Aterm ]
644 2 : 114 Y Loc [ PIR Bterm ]
645 3 : 11 Z Loc [ Longwave ]
646
647 21 : End (P95)
648
649 End Program
650
651 −Input Locat ions−
652 1 RefTemp 1 2 1
653 2 WindSpd 1 2 2
654 3 WindDir 1 1 1
655 4 ShortUP 5 1 1
656 5 ShortDOWN 1 1 2
657 6 TempCS215 9 2 2
658 7 HumdCS215 9 1 1
659 8 DpthTEMP 17 1 1
660 9 Sur face 1 1 1
661 10 1 0 0
662 11 Longwave 9 1 4
663 12 Depth 9 1 4
664 13 9 0 3
665 14 Battery 9 2 4
666 15 13 0 3
667 16 9 0 3
668 17 9 0 4
669 18 25 0 4
670 19 9 0 1
671 20 TC 1 13 1 2
672 21 TC 2 25 1 2
673 22 TC 3 9 1 1
674 23 TC 4 9 1 1
675 24 TC 5 9 1 1
676 25 TC 6 9 1 1
677 26 TC 7 9 1 1
678 27 TC 8 9 1 1
679 28 TC 9 9 1 1
680 29 TC 10 9 1 1
681 30 TC 11 17 1 1
682 31 TC 12 1 1 0
683 32 TC 13 1 1 0
684 33 TC 14 1 1 0
685 34 TC 15 1 1 0
686 35 TC 16 1 1 0
687 36 TC 17 1 1 0
288
688 37 TC 18 1 1 0
689 38 TC 19 1 1 0
690 39 TC 20 1 1 0
691 40 TCgnd 1 1 1
692 41 0 0 0
693 42 0 0 0
694 43 0 0 0
695 44 0 0 0
696 45 0 0 0
697 46 0 0 0
698 47 0 0 0
699 48 1 0 0
700 49 1 0 0
701 50 1 0 0
702 51 1 0 0
703 52 1 0 0
704 53 1 0 0
705 54 1 0 0
706 55 1 0 0
707 56 1 0 0
708 57 1 0 0
709 58 1 0 0
710 59 1 0 0
711 60 1 0 0
712 61 1 0 0
713 62 1 0 0
714 63 1 0 0
715 64 1 0 0
716 65 1 0 0
717 66 1 0 0
718 67 1 0 0
719 68 1 0 0
720 69 1 0 0
721 70 1 0 0
722 71 1 0 0
723 72 1 0 0
724 73 1 0 0
725 74 1 0 0
726 75 1 0 0
727 76 1 0 0
728 77 1 0 0
729 78 1 0 0
730 79 1 0 0
731 80 1 0 0
732 81 1 0 0
733 82 1 0 0
734 83 0 0 0
735 84 0 0 0
736 85 0 0 0
737 86 0 0 0
738 87 0 0 0
739 88 0 0 0
740 89 0 0 0
741 90 0 0 0
742 91 0 0 0
743 92 0 0 0
744 93 0 0 0
745 94 0 0 0
746 95 0 0 0
747 96 0 0 0
748 97 0 0 0
749 98 0 0 0
750 99 0 0 0
751 100 Uemf 1 1 1
752 101 Case Res 1 3 2
753 102 ConstA 1 1 1
754 103 ConstB 1 1 1
755 104 ConstC 1 1 1
756 105 Ln Res 1 4 1
757 106 1 0 0
758 107 bLn res 1 1 1
759 108 Ln res2 1 1 1
760 109 Ln res3 1 1 1
761 110 CLn res3 1 1 1
762 111 case temp 1 4 4
763 112 PIR Aterm 1 1 1
764 113 Power4 1 1 1
765 114 PIR Bterm 1 2 2
766 115 SurTemp 2 1 0 0
767 116 rawdepth 1 2 1
768 117 d1 1 1 1
769 118 ke l v i n 1 1 1
770 119 d2 1 1 1
771 120 exp 1 1 1
772 121 CF 1 2 1
773 122 1 1 0
774 −Program Secur i ty−
289
775 0000
776 0000
777 0000
778 −Mode 4−
779 −Fina l Storage Area 2−
780 0
781 −CR10X ID−
782 0
783 −CR10X Power Up−
784 3
A.6.2 South Weather Station Program
1 ;{CR10}
2 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
3 ; A PROGRAM BY:
4 ;
5 ; ANDREW E. SLAUGHTER
6 ; 205 Cobleigh Hall , MSU−Bozeman
7 ; P.O. Box 173900
8 ; Bozeman , MT 59717−3900
9 ; (406) 994−2293
10 ;
11 ;***********************************************************************************
12 ;
13 ; BACKGROUND:
14 ; The f o l l ow ing program u t i l i z e s the Campbell S c i e n t i f i c , Inc . ( SCI ) CR10(x )
15 ; data logge r and two CSI AM418 mul t ip l exe r to acqu i r e ba s i c weather data that
16 ; i n c l ude s snow surface temperature , snow depth , humidity , a i r temperature ,
17 ; wind speed , wind d i r e c t i on , longwave rad ia t i on , and shortwave r ad i a t i on .
18 ; Addi t iona l ly , the s t a t i on has an array o f thermocouples that measures the
19 ; snowpack temperature at var i ous depths below the surface .
20 ;
21 ; The ob j e c t i v e o f the s i t e i s to c o l l e c t f i e l d data regard ing the growth
22 ; o f near−surface f a c e t ed and surface hoar c r y s t a l s for the use in v e r i f y i n g both
23 ; lab and an a l y t i c a l models o f the near−surface pro c e s s e s .
24 ;
25 ; Last Updated : December 2008
26 ;
27 ;************************************************************************************
28 ;
29 ; WIRING SCHEME:
30 ;
31 ; −> CR10TCR Thermistor
32 ; Wht − AG
33 ; Blk − E1
34 ; Red − SE6
35 ;
36 ; −> AM418 MULTIPLEXIERS
37 ; Wiring (CR10 − AM416) :
38 ; C1 − REM
39 ; C2 − CLK
40 ; H1 − ComH1
41 ; L1 − ComL1
42 ; H2 − ComH2
43 ; L2 − ComL2
44 ;
45 ; −> THERMOCOUPLES ( wired to AM416)
46 ; 1H1, 1 L1 − TC 1
47 ; 1H2,1 L2 − TC 2
48 ; . . .
49 ; 11H1,11L1 − TC 21 (TC to Grnd)
50 ;
51 ; −> KIPP & ZONEN CMP3 ( shortwave , wired to AM416#1)
52 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
53 ; Up Mult . = 14.11 uV/W/mˆ2 = 70.87 W/mˆ2/mV (SN:080193)
54 ; Dn Mult . = 14.34 uV/W/mˆ2 = 69.74 W/mˆ2/mV (SN:080192)
55 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
56 ; Incoming (up) Re f l e c t ed (down)
57 ; Red − 12 :H1 Grn − 12 : L2
58 ; Blk − 12 : L1 Blu − 12 :H2
59 ;
60 ; −> KIPP & ZONEN CGR3 ( longwave ) ;
61 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
62 ; Mult . = 11.83 uV/W/mˆ2 = 84.531W/mˆ2/mV (SN:070108)
63 ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
64 ; Red − L5
65 ; Blu − H5
66 ; Ylw − SE12
290
67 ; Grn − G
68 ; 1kOhm Res i s t o r from SE12 − E3
69 ;
70 ; −> METONE ANEMOMETER
71 ; Yel − SE5
72 ; Wht − AG
73 ; Grn − E2
74 ; Blk − G
75 ; Brn − G
76 ; Red − P1
77 ;
78 ; −> EVEREST INTERSCIENCE SNOW SURFACE TEMPERATURE
79 ; Blu − 11 :H2
80 ; Wht − 11 : L2
81 ; Red − 12V
82 ; Blk − G
83 ;
84 ; −> CAMPBELL SCIENTIFIC HUMIDITY AND TEMP
85 ; Blk ,Wht, Clr − G
86 ; Red − 12V
87 ; Grn − C5
88 ;
89 ; −> NOVALYNX ULTRASONIC SNOW DEPTH
90 ; Red − 12V
91 ; Blk − G
92 ; Clr − G
93 ; Grn − C6
94 ; Wht − 4H
95 ; Brn − 4L
96 ;
97 ;
98 ;********************************************************************
99 ;**** Begin Program *************************************************
100
101 ;EXECUTION INTERVAL IN SECONDS
102 *Table 1 Program
103 01 : 180 Execution I n t e r v a l ( seconds )
104
105
106 ;*********************************************************************
107 ;**** Battery Voltage ************************************************
108
109 1 : Batt Voltage (P10)
110 1 : 14 Loc [ Battery ]
111
112
113 ;*********************************************************************
114 ;**** Stop i f Battery < 11V ******************************************
115
116 2 : I f (X<=>F) (P89)
117 1 : 14 X Loc [ Battery ]
118 2 : 4 <
119 3 : 11 F
120 4 : 0 Go to end o f Program Table
121
122
123 ;*********************************************************************
124 ;**** Reference Temperature ******************************************
125 3 : Temp (107) (P11)
126 1 : 1 Reps
127 2 : 6 SE Channel
128 3 : 1 Exc i te a l l reps w/E1
129 4 : 1 Loc [ RefTemp ]
130 5 : 1 .0 Mult
131 6 : 0 .0 O f f s e t
132
133
134 ;*********************************************************************
135 ;**** Thermocouples , SW Radiation , & Sur face Temp ********************
136
137 ;TRIGGER MULTIPLEXER
138 4 : Do (P86)
139 1 : 41 Set Port 1 High
140
141 ;MEASURE THERCOUPLES 1 THRU 20
142 5 : Beginning o f Loop (P87)
143 1 : 0 Delay
144 2 : 10 Loop Count
145
146 6 : Do (P86)
147 1 : 72 Pulse Port 2
148
149 ; INDICATE TWO READINGS PER SET
150 7 : Step Loop Index (P90)
151 1 : 2 Step
152
153 ;READ 20 THERMOCOUPLES
291
154 8 : Thermocouple Temp (DIFF) (P14)
155 1 : 2 Reps
156 2 : 1 2 .5 mV Slow Range
157 3 : 01 DIFF Channel
158 4 : 1 Type T (Copper−Constantan )
159 5 : 1 Ref Temp (Deg . C) Loc [ RefTemp ]
160 6 : 20 −− Loc [ TC 1 ]
161 7 : 1 .0 Mult
162 8 : 0 .0 O f f s e t
163
164 ;END THE LOOP FOR READING THERMOCOULPES
165 9 : End (P95)
166
167 ;READ THE GROUND THERMOCOUPLE AND THE SURFACE TEMPERATURE
168 10 : Do (P86)
169 1 : 72 Pulse Port 2
170
171 11 : Thermocouple Temp (DIFF) (P14)
172 1 : 1 Reps
173 2 : 1 2 .5 mV Slow Range
174 3 : 1 DIFF Channel
175 4 : 1 Type T (Copper−Constantan )
176 5 : 1 Ref Temp (Deg . C) Loc [ RefTemp ]
177 6 : 40 Loc [ TCgnd ]
178 7 : 1 .0 Mult
179 8 : 0 .0 O f f s e t
180
181 12 : Volt ( D i f f ) (P2)
182 1 : 1 Reps
183 2 : 5 2500 mV Slow Range
184 3 : 2 DIFF Channel
185 4 : 9 Loc [ Sur face ]
186 5 : 0 .1 Mult
187 6 : 0 .0 O f f s e t
188
189 ;READ THE SHORTWAVE SENSORS (UP & DOWN)
190 13 : Do (P86)
191 1 : 72 Pulse Port 2
192
193 ; Incoming (up)
194 14 : Volt ( D i f f ) (P2)
195 1 : 1 Reps
196 2 : 3 25 mV Slow Range
197 3 : 1 DIFF Channel
198 4 : 4 Loc [ ShortUP ]
199 5 : 70 .87 Mult
200 6 : 0 .0 O f f s e t
201
202 ; Re f l e c t ed (Down)
203 15 : Volt ( D i f f ) (P2)
204 1 : 1 Reps
205 2 : 3 25 mV Slow Range
206 3 : 1 DIFF Channel
207 4 : 5 Loc [ ShortDOWN ]
208 5 : 69 .74 Mult
209 6 : 0 .0 O f f s e t
210
211 ;TURN OFF MULTIPLEXER
212 16 : Do (P86)
213 1 : 51 Set Port 1 Low
214
215
216 ;**************************************************************************
217 ;**** Wind Speed and Di r e c t i on ********************************************
218 ;−−−−−−−−−−−−−−−−−−−−−−−−−
219 ; WindSpd = m/s2
220 ; WindDir = 0−360 (N=0=360)
221 ;−−−−−−−−−−−−−−−−−−−−−−−−−−
222
223 ;MEASURE WIND SPEED
224 17 : Pulse (P3)
225 1 : 1 Reps
226 2 : 1 Pulse Input Channel
227 3 : 22 Switch Closure , Output Hz
228 4 : 2 Loc [ WindSpd ]
229 5 : 0 .7990 Mult
230 6 : 0 .2811 Of f s e t
231
232 ; IF WIND SPEED IS NEGATIVE SET TO ZERO
233 18 : I f (X<=>F) (P89)
234 1 : 2 X Loc [ WindSpd ]
235 2 : 1 =
236 3 : 0 .2811 F
237 4 : 30 Then Do
238
239 19 : Z=F (P30)
240 1 : 0 F
292
241 2 : 0 Exponent o f 10
242 3 : 2 Z Loc [ WindSpd ]
243
244 20 : End (P95)
245
246 ; MEASURE WIND DIRECTION
247 21 : AC Half Bridge (P5)
248 1 : 1 Reps
249 2 : 25 2500 mV 60 Hz Re jec t i on Range
250 3 : 5 SE Channel
251 4 : 2 Exc i te a l l reps w/Exchan 2
252 5 : 2500 mV Exc i ta t i on
253 6 : 3 Loc [ WindDir ]
254 7 : 360 Mult
255 8 : 0 .0 O f f s e t
256
257
258 ;*************************************************************************
259 ;**** Longwave Radiat ion *************************************************
260 ;
261 ;MEASURE VOLTAGE FOR THERMOPILE
262 22 : Volt ( D i f f ) (P2)
263 1 : 1 Reps
264 2 : 1 2 .5 mV Slow Range
265 3 : 5 DIFF Channel
266 4 : 100 Loc [ Uemf ]
267 5 : 1 Mult
268 6 : 0 .0 O f f s e t
269
270 ;MEASURE THERMISTOR
271 23 : AC Half Bridge (P5)
272 1 : 1 Reps
273 2 : 15 2500 mV Fast Range
274 3 : 12 SE Channel
275 4 : 3 Exc i te a l l reps w/Exchan 3
276 5 : 2500 mV Exc i ta t i on
277 6 : 101 Loc [ Case Res ]
278 7 : 1 .0 Mult
279 8 : 0 .0 O f f s e t
280
281 ;CONVERT THERMISTOR MEASURE TO RESISTANCE
282 24 : BR Transform Rf [X/(1−X) ] (P59)
283 1 : 1 Reps
284 2 : 101 Loc [ Case Res ]
285 3 : 1000 Mu l t i p l i e r (Rf )
286
287 ;CONVERT RESISTANCE TO TEMPERATURE
288 25 : Do (P86)
289 1 : 1 Cal l Subroutine 1
290
291 ;CORRECT FOR CASE TEMPERATURE −> OUTPUT LONGWAVE RADIATION
292 26 : Do (P86)
293 1 : 2 Cal l Subroutine 2
294
295 ;************************************************************************
296 ;**** Snow Depth ********************************************************
297
298 27 : Do (P86)
299 1 : 46 Set Port 6 High
300
301 ; Wait 2 seconds for s ensor to warm−up and measure depth
302 28 : Exc i ta t i on with Delay (P22)
303 1 : 2 Ex Channel
304 2 : 200 Delay W/Ex ( un i t s = 0.01 sec )
305 3 : 0000 Delay After Ex ( un i t s = 0.01 sec )
306 4 : 0000 mV Exc i ta t i on
307
308 ; Depth given in mV and sca l ed to cm, o f f s e t = mounting he ight
309 29 : Volt ( D i f f ) (P2)
310 1 : 1 Reps
311 2 : 5 2500 mV Slow Range
312 3 : 4 DIFF Channel
313 4 : 116 Loc [ rawdepth ]
314 5 : −0.25 Mult
315 6 : 200 Of f s e t
316
317 30 : Do (P86)
318 1 : 56 Set Port 6 Low
319
320 ; Compute the temperature co r r e c t ed depth
321 31 : Z=X+F (P34)
322 1 : 6 X Loc [ TempCS215 ]
323 2 : 273 .15 F
324 3 : 117 Z Loc [ d1 ]
325
326 32 : Z=F (P30)
327 1 : 273 .15 F
293
328 2 : 00 Exponent o f 10
329 3 : 118 Z Loc [ k e l v i n ]
330
331 33 : Z=X/Y (P38)
332 1 : 117 X Loc [ d1 ]
333 2 : 118 Y Loc [ k e l v i n ]
334 3 : 119 Z Loc [ d2 ]
335
336 34 : Z=F (P30)
337 1 : 0 .5 F
338 2 : 00 Exponent o f 10
339 3 : 120 Z Loc [ exp ]
340
341 35 : Z=XˆY (P47)
342 1 : 119 X Loc [ d2 ]
343 2 : 120 Y Loc [ exp ]
344 3 : 121 Z Loc [ CF ]
345
346 36 : Z=X*Y (P36)
347 1 : 121 X Loc [ CF ]
348 2 : 116 Y Loc [ rawdepth ]
349 3 : 12 Z Loc [ Depth ]
350
351
352 ;*************************************************************************
353 ;**** Humidity & Air Temperature *****************************************
354
355 37 : SDI−12 Recorder (P105 )
356 1 : 00 SDI−12 Address
357 2 : 00 SDI−12 Command
358 3 : 5 Port
359 4 : 6 Loc [ TempCS215 ]
360 5 : 1 .0 Mult
361 6 : 0 .0 O f f s e t
362
363 ;*************************************************************************
364 ;**** Data Storage A l l o ca t i on ********************************************
365
366 ;SET TIME AND STORAGE
367 38 : I f time i s (P92)
368 1 : 0000 Minutes ( Seconds −−) in to a
369 2 : 30 I n t e r v a l ( same un i t s as above )
370 3 : 10 Set Output Flag High
371
372 39 : Set Active Storage Area (P80)
373 1 : 1 Fina l Storage Area 1
374 2 : 100 Array ID
375
376 40 : Real Time (P77)
377 1 : 1220 Year ,Day , Hour/Minute ( midnight = 2400)
378
379 ;OUTPUT MINIMUM BATTERY VOLTAGE
380 41 : Minimum (P74)
381 1 : 1 Reps
382 2 : 00 Time Option
383 3 : 14 Loc [ Battery ]
384
385 ;READ WIND SPEED AND DIRECTION
386 42 : Average (P71)
387 1 : 1 Reps
388 2 : 2 Loc [ WindSpd ]
389
390 43 : Average (P71)
391 1 : 1 Reps
392 2 : 3 Loc [ WindDir ]
393
394 ;SURFACE TEMPERATURE AVERAGES
395 44 : Average (P71)
396 1 : 1 Reps
397 2 : 9 Loc [ Sur face ]
398
399 ;AVERAGE LONG/SHORTWAVE RADATION
400 45 : Average (P71)
401 1 : 1 Reps
402 2 : 4 Loc [ ShortUP ]
403 46 : Average (P71)
404 1 : 1 Reps
405 2 : 5 Loc [ ShortDOWN ]
406
407 47 : Average (P71)
408 1 : 1 Reps
409 2 : 11 Loc [ Longwave ]
410
411 ;SNOW DEPTH AVERAGE
412 48 : Average (P71)
413 1 : 1 Reps
414 2 : 12 Loc [ Depth ]
294
415
416 49 : Average (P71)
417 1 : 1 Reps
418 2 : 116 Loc [ rawdepth ]
419
420 50 : Average (P71)
421 1 : 1 Reps
422 2 : 121 Loc [ CF ]
423
424 51 : Average (P71)
425 1 : 1 Reps
426 2 : 8 Loc [ DpthTEMP ]
427
428 ;ATMOSPHERE TEMP/HUMID AVERAGES
429 52 : Average (P71)
430 1 : 1 Reps
431 2 : 6 Loc [ TempCS215 ]
432
433 53 : Average (P71)
434 1 : 1 Reps
435 2 : 7 Loc [ HumdCS215 ]
436
437 ;AVERAGE RAW DATA FOR LONGWAVE
438 54 : Average (P71)
439 1 : 1 Reps
440 2 : 122 Loc [ ]
441
442 55 : Average (P71)
443 1 : 1 Reps
444 2 : 101 Loc [ Case Res ]
445
446 ;THERMOCOUPLE AVERAGES
447 56 : Average (P71)
448 1 : 1 Reps
449 2 : 20 Loc [ TC 1 ]
450 57 : Average (P71)
451 1 : 1 Reps
452 2 : 21 Loc [ TC 2 ]
453 58 : Average (P71)
454 1 : 1 Reps
455 2 : 22 Loc [ TC 3 ]
456 59 : Average (P71)
457 1 : 1 Reps
458 2 : 23 Loc [ TC 4 ]
459 60 : Average (P71)
460 1 : 1 Reps
461 2 : 24 Loc [ TC 5 ]
462 61 : Average (P71)
463 1 : 1 Reps
464 2 : 25 Loc [ TC 6 ]
465 62 : Average (P71)
466 1 : 1 Reps
467 2 : 26 Loc [ TC 7 ]
468 63 : Average (P71)
469 1 : 1 Reps
470 2 : 27 Loc [ TC 8 ]
471 64 : Average (P71)
472 1 : 1 Reps
473 2 : 28 Loc [ TC 9 ]
474 65 : Average (P71)
475 1 : 1 Reps
476 2 : 29 Loc [ TC 10 ]
477 66 : Average (P71)
478 1 : 1 Reps
479 2 : 30 Loc [ TC 11 ]
480 67 : Average (P71)
481 1 : 1 Reps
482 2 : 31 Loc [ TC 12 ]
483 68 : Average (P71)
484 1 : 1 Reps
485 2 : 32 Loc [ TC 13 ]
486 69 : Average (P71)
487 1 : 1 Reps
488 2 : 33 Loc [ TC 14 ]
489 70 : Average (P71)
490 1 : 1 Reps
491 2 : 34 Loc [ TC 15 ]
492 71 : Average (P71)
493 1 : 1 Reps
494 2 : 35 Loc [ TC 16 ]
495 72 : Average (P71)
496 1 : 1 Reps
497 2 : 36 Loc [ TC 17 ]
498 73 : Average (P71)
499 1 : 1 Reps
500 2 : 37 Loc [ TC 18 ]
501 74 : Average (P71)
295
502 1 : 1 Reps
503 2 : 38 Loc [ TC 19 ]
504 75 : Average (P71)
505 1 : 1 Reps
506 2 : 39 Loc [ TC 20 ]
507 76 : Average (P71)
508 1 : 1 Reps
509 2 : 40 Loc [ TCgnd ]
510
511 *Table 2 Program
512 02 : 0 .0000 Execution I n t e r v a l ( seconds )
513
514 *Table 3 Subrout ines
515
516
517 ;**** SUBROUTINE #1 ******************************************************************
518
519 1 : Beginning o f Subroutine (P85)
520 1 : 1 Subroutine 1
521
522 ;CONVERT RESISTANCE TO TEMPERATURE
523 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
524 ;T = 1/(A+B*Ln(R) + C*(Ln(R) ) ˆ3)
525 ;T = Temp in Deg . Kelvin
526 ;A = 0.0010295
527 ;B = 0.0002391
528 ;C = 0.0000001568
529 ;R = Measured r e s i s t a n c e o f the rmi s to r
530 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
531
532 ; Constant A
533 2 : Z=F (P30)
534 1 : 1 .0295 F
535 2 : −3 Exponent o f 10
536 3 : 102 Z Loc [ ConstA ]
537
538 ; Constant B
539 3 : Z=F (P30)
540 1 : 2 .391 F
541 2 : −4 Exponent o f 10
542 3 : 103 Z Loc [ ConstB ]
543
544 ; Constant C
545 4 : Z=F (P30)
546 1 : 1 .568 F
547 2 : −7 Exponent o f 10
548 3 : 104 Z Loc [ ConstC ]
549
550 ; Natural Log o f Res i s tance
551 5 : Z=LN(X) (P40)
552 1 : 101 X Loc [ Case Res ]
553 2 : 105 Z Loc [ Ln Res ]
554
555 ; B*Ln(R)
556 6 : Z=X*Y (P36)
557 1 : 103 X Loc [ ConstB ]
558 2 : 105 Y Loc [ Ln Res ]
559 3 : 107 Z Loc [ bLn res ]
560
561 ; Squre and Cube Natural Log
562 7 : Z=X*Y (P36)
563 1 : 105 X Loc [ Ln Res ]
564 2 : 105 Y Loc [ Ln Res ]
565 3 : 108 Z Loc [ Ln res2 ]
566
567 8 : Z=X*Y (P36)
568 1 : 105 X Loc [ Ln Res ]
569 2 : 108 Y Loc [ Ln res2 ]
570 3 : 109 Z Loc [ Ln res3 ]
571
572 ; C*(Ln(R) ) ˆ3
573 9 : Z=X*Y (P36)
574 1 : 104 X Loc [ ConstC ]
575 2 : 109 Y Loc [ Ln res3 ]
576 3 : 110 Z Loc [ CLn res3 ]
577
578 ; Add A and B Terms
579 10 : Z=X+Y (P33)
580 1 : 102 X Loc [ ConstA ]
581 2 : 107 Y Loc [ bLn res ]
582 3 : 111 Z Loc [ case temp ]
583
584 ; Add C Term to A/B Term
585 11 : Z=X+Y (P33)
586 1 : 111 X Loc [ case temp ]
587 2 : 110 Y Loc [ CLn res3 ]
588 3 : 111 Z Loc [ case temp ]
296
589
590 ; Take Rec iporca l o f case temp
591 12 : Z=1/X (P42)
592 1 : 111 X Loc [ case temp ]
593 2 : 111 Z Loc [ case temp ]
594
595 13 : End (P95)
596
597
598 ;**** SUBROUTINE #2 ***********************************************
599
600 14 : Beginning o f Subroutine (P85)
601 1 : 2 Subroutine 2
602
603 ;CORRECT PIR CASE TEMPERATURE
604 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
605 ; CorrectOutput = A + (C*Tˆ4)
606 ; A = Thermopile Output
607 ; C = Stefan−Boltzman Cnst = 5.6697 e−8 Wm−2K−4
608 ; T = Case Temperature in Kelvin
609 ; S = F = 11.83uV/W/mˆ2 = 84.531 W/mˆ2/mV (SN:070112)
610 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
611
612 ;CONVERT THERMOPILE TO Wm−2
613 15 : Z=X*F (P37)
614 1 : 100 X Loc [ Uemf ]
615 2 : 84 .531 F
616 3 : 112 Z Loc [ PIR Aterm ]
617
618 ; Load 4 for r a i s i n g to f o r th
619 16 : Z=F (P30)
620 1 : 4 F
621 2 : 0 Exponent o f 10
622 3 : 113 Z Loc [ Power4 ]
623
624 ; Raise to 4 th Power
625 17 : Z=XˆY (P47)
626 1 : 111 X Loc [ case temp ]
627 2 : 113 Y Loc [ Power4 ]
628 3 : 111 Z Loc [ case temp ]
629
630 ; Load Boltzman
631 18 : Z=F (P30)
632 1 : 5 .669 F
633 2 : −8 Exponent o f 10
634 3 : 114 Z Loc [ PIR Bterm ]
635
636 ; Mutl ip ly Boltzman by Tˆ4
637 19 : Z=X*Y (P36)
638 1 : 114 X Loc [ PIR Bterm ]
639 2 : 111 Y Loc [ case temp ]
640 3 : 114 Z Loc [ PIR Bterm ]
641
642 ;Add A to Bterm −> OUTPUTS THE LONGWAVE RADIATION
643 20 : Z=X+Y (P33)
644 1 : 112 X Loc [ PIR Aterm ]
645 2 : 114 Y Loc [ PIR Bterm ]
646 3 : 11 Z Loc [ Longwave ]
647
648 21 : End (P95)
649
650 End Program
651
652 −Input Locat ions−
653 1 RefTemp 1 2 1
654 2 WindSpd 1 2 2
655 3 WindDir 1 1 1
656 4 ShortUP 5 1 1
657 5 ShortDOWN 1 1 2
658 6 TempCS215 9 2 2
659 7 HumdCS215 9 1 1
660 8 DpthTEMP 17 1 1
661 9 Sur face 1 1 1
662 10 1 0 0
663 11 Longwave 9 1 4
664 12 Depth 9 1 4
665 13 9 0 3
666 14 Battery 9 2 4
667 15 13 0 3
668 16 9 0 3
669 17 9 0 4
670 18 25 0 4
671 19 9 0 1
672 20 TC 1 13 1 2
673 21 TC 2 25 1 2
674 22 TC 3 9 1 1
675 23 TC 4 9 1 1
297
676 24 TC 5 9 1 1
677 25 TC 6 9 1 1
678 26 TC 7 9 1 1
679 27 TC 8 9 1 1
680 28 TC 9 9 1 1
681 29 TC 10 9 1 1
682 30 TC 11 17 1 1
683 31 TC 12 1 1 0
684 32 TC 13 1 1 0
685 33 TC 14 1 1 0
686 34 TC 15 1 1 0
687 35 TC 16 1 1 0
688 36 TC 17 1 1 0
689 37 TC 18 1 1 0
690 38 TC 19 1 1 0
691 39 TC 20 1 1 0
692 40 TCgnd 1 1 1
693 41 0 0 0
694 42 0 0 0
695 43 0 0 0
696 44 0 0 0
697 45 0 0 0
698 46 0 0 0
699 47 0 0 0
700 48 1 0 0
701 49 1 0 0
702 50 1 0 0
703 51 1 0 0
704 52 1 0 0
705 53 1 0 0
706 54 1 0 0
707 55 1 0 0
708 56 1 0 0
709 57 1 0 0
710 58 1 0 0
711 59 1 0 0
712 60 1 0 0
713 61 1 0 0
714 62 1 0 0
715 63 1 0 0
716 64 1 0 0
717 65 1 0 0
718 66 1 0 0
719 67 1 0 0
720 68 1 0 0
721 69 1 0 0
722 70 1 0 0
723 71 1 0 0
724 72 1 0 0
725 73 1 0 0
726 74 1 0 0
727 75 1 0 0
728 76 1 0 0
729 77 1 0 0
730 78 1 0 0
731 79 1 0 0
732 80 1 0 0
733 81 1 0 0
734 82 1 0 0
735 83 0 0 0
736 84 0 0 0
737 85 0 0 0
738 86 0 0 0
739 87 0 0 0
740 88 0 0 0
741 89 0 0 0
742 90 0 0 0
743 91 0 0 0
744 92 0 0 0
745 93 0 0 0
746 94 0 0 0
747 95 0 0 0
748 96 0 0 0
749 97 0 0 0
750 98 0 0 0
751 99 0 0 0
752 100 Uemf 1 1 1
753 101 Case Res 1 3 2
754 102 ConstA 1 1 1
755 103 ConstB 1 1 1
756 104 ConstC 1 1 1
757 105 Ln Res 1 4 1
758 106 1 0 0
759 107 bLn res 1 1 1
760 108 Ln res2 1 1 1
761 109 Ln res3 1 1 1
762 110 CLn res3 1 1 1
298
763 111 case temp 1 4 4
764 112 PIR Aterm 1 1 1
765 113 Power4 1 1 1
766 114 PIR Bterm 1 2 2
767 115 SurTemp 2 1 0 0
768 116 rawdepth 1 2 1
769 117 d1 1 1 1
770 118 ke l v i n 1 1 1
771 119 d2 1 1 1
772 120 exp 1 1 1
773 121 CF 1 2 1
774 122 1 1 0
775 −Program Secur i ty−
776 0000
777 0000
778 0000
779 −Mode 4−
780 −Fina l Storage Area 2−
781 0
A.6.3 American Spirit Weather Station Program
1 ;{CR10X}
2 ; Program : Yel lowstone Club
3 ; American Sp i r i t L i f t − Tower 18
4 ; Elevat ion : 8840 ’
5 ;1−17−08
6 ; Phone Number : 993−2679
7 ; 45 . 2397 N 111.4429 W
8
9 ; Instrument Wiring
10
11 ; CS 500 Temp/RH Probe
12 ; Red 12V
13 ; Black 1H (SE chan 1 Temp)
14 ; Brown 1L (SE chan 2 RH)
15 ; Green G
16 ; Clear G
17
18 ; R. M. Young 05103 Wind Monitor−Tower 18
19 ; 1 − Clear G
20 ; 2 − Black G
21 ; 3 − Brown AG
22 ; 4 − Red 2H
23 ; 5 − Green E1
24 ; 6 − White P1
25
26 ; Epply PSP Shortwave So la r
27 ; Red H3
28 ; Black L3
29
30 ; Epply PIR Longwave So la r
31 ; Purple 4H
32 ; Grey 4L
33 ; Orange SE12
34 ; Black G
35 ; Yellow G
36 ; SE12 − SE12 wire a 1kOhm r e s i s t o r
37
38
39 ;FSL Tables :
40
41 ;60 Output Table 60 .00 Min
42 ; 1 60 L
43 ; 2 Year RTM L
44 ;3 Day RTM L
45 ;4 Hour Minute RTM L
46 ;5 Wind Spd S WVT L
47 ; 6 Wind Dir D1 WVT L
48 ; 7 Wind Spd MAX L
49 ; 8 Temp F AVG L
50 ; 9 Rel Humid L
51 ;10 Shortwave AVG L
52 ;11 Longwave AVG L
53 ;12 Battery MIN L
54
55 ;240 Output Table 1440.00 Min
56 ; 1 240 L
57 ; 2 Year RTM L
299
58 ; 3 Day RTM L
59 ;4 Hour Minute RTM L
60 ;5 Wind Spd S WVT L
61 ; 6 Wind Dir D1 WVT L
62 ; 7 Wind Spd MAX L
63 ; 8 Wind Spd Hr Min MAX L
64 ; 9 Temp F MIN L
65 ;10 Temp F Hr Min MIN L
66 ;11 Temp F MAX L
67 ;12 Temp F Hr Min MAX L
68 ;13 Temp F AVG L
69 ;14 Rel Humid AVG L
70 ;15 Shortwave AVG L
71 ;16 Longwave AVG L
72 ;17 Shortwave MIN L
73 ;18 Shortwave Hr Min MIN L
74 ;19 Shortwave MAX L
75 ;20 Shortwave Hr Min MAX L
76 ;21 Battery MIN L
77
78 ; Estimated Total F ina l Storage Locat ions used per day 309
79
80 *Table 1 Program
81 01 : 5 Execution I n t e r v a l ( seconds )
82
83 ; Measure wind speed −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
84
85 1 : Pulse (P3)
86 1 : 1 Reps
87 2 : 1 Pulse Channel 1
88 3 : 21 Low Level AC, Output Hz
89 4 : 1 Loc [ Wind Spd ]
90 5 : .2192 Mult
91 6 : 0 O f f s e t
92
93 ; Measure wind d i r e c t i o n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
94
95 2 : Excite−Delay (SE) (P4)
96 1 : 1 Reps
97 2 : 5 2500 mV Slow Range
98 3 : 3 SE Channel
99 4 : 1 Exc i te a l l reps w/Exchan 1
100 5 : 2 Delay ( un i t s 0 .01 sec )
101 6 : 2500 mV Exc i ta t i on
102 7 : 2 Loc [ Wind Dir ]
103 8 : .142 Mult
104 9 : 0 O f f s e t
105
106 ; Measure Datalogger i n t e r n a l temp in degree F−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
107
108 3 : I n t e rna l Temperature (P17)
109 1 : 6 Loc [ Ref Temp ]
110
111 4 : Z=X*F (P37)
112 1 : 6 X Loc [ Ref Temp ]
113 2 : 1 .8 F
114 3 : 6 Z Loc [ Ref Temp ]
115
116 5 : Z=X+F (P34)
117 1 : 6 X Loc [ Ref Temp ]
118 2 : 32 F
119 3 : 6 Z Loc [ Ref Temp ]
120
121 ; Measure a i r temp in degree F and r e l a t i v e humidity in % −−−−−−−−−−−−−−−−−−−−−−−−−
122
123 6 : Volt (SE) (P1)
124 1 : 1 Reps
125 2 : 25 2500 mV 60 Hz Re jec t i on Range
126 3 : 1 SE Channel
127 4 : 3 Loc [ Temp F ]
128 5 : 0 .18 Mult
129 6 : −40.0 O f f s e t
130
131 7 : Volt (SE) (P1)
132 1 : 1 Reps
133 2 : 25 2500 mV 60 Hz Re jec t i on Range
134 3 : 2 SE Channel
135 4 : 4 Loc [ Rel Humid ]
136 5 : . 1 Mult
137 6 : 0 O f f s e t
138
139 ; In s t . 8 − 11 l im i t r e l humidity to a max o f 100%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
140
141 8 : I f (X<=>F) (P89)
142 1 : 4 X Loc [ Rel Humid ]
143 2 : 3 >=
144 3 : 100 F
300
145 4 : 30 Then Do
146
147 9 : Z=F (P30)
148 1 : 100 F
149 2 : 0 Exponent o f 10
150 3 : 4 Z Loc [ Rel Humid ]
151
152 10 : End (P95)
153
154 11 : Batt Voltage (P10)
155 1 : 5 Loc [ Battery ]
156
157 ; Measure incoming shortwave s o l a r with Epply −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
158
159 12 : Volt ( D i f f ) (P2)
160 1 : 1 Reps
161 2 : 3 25 mV Slow Range
162 3 : 3 DIFF Channel
163 4 : 7 Loc [ Shortwave ]
164 5 : 122 .55 Mult
165 6 : 0 .0 O f f s e t
166
167 ; Measure incoming longwave with Epply −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
168
169 ;MEASURE VOLTAGE FOR THERMOPILE
170
171 13 : Volt ( D i f f ) (P2)
172 1 : 1 Reps
173 2 : 1 2 .5 mV Slow Range
174 3 : 4 DIFF Channel
175 4 : 19 Loc [ PIR mV ]
176 5 : 1 .0 Mult
177 6 : 0 .0 O f f s e t
178
179 ;MEASURE THERMISTOR
180
181 14 : AC Half Bridge (P5)
182 1 : 1 Reps
183 2 : 15 2500 mV Fast Range
184 3 : 12 SE Channel
185 4 : 3 Exc i te a l l reps w/Exchan 3
186 5 : 2500 mV Exc i ta t i on
187 6 : 12 Loc [ Case Res ]
188 7 : 1 .0 Mult
189 8 : 0 .0 O f f s e t
190
191 ;CONVERT THERMISTOR MEASURE TO RESISTANCE
192
193 15 : BR Transform Rf [X/(1−X) ] (P59)
194 1 : 1 Reps
195 2 : 12 Loc [ Case Res ]
196 3 : 1000 Mu l t i p l i e r (Rf )
197
198 ;CONVERT RESISTANCE TO TEMPERATURE
199
200 16 : Do (P86)
201 1 : 1 Cal l Subroutine 1
202
203 ;CORRECT FOR CASE TEMPERATURE −> OUTPUT LONGWAVE RADIATION
204
205 17 : Do (P86)
206 1 : 2 Cal l Subroutine 2
207
208 ; Output 1 hour data−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
209
210 18 : I f time i s (P92)
211 1 : 0 Minutes ( Seconds −−) in to a
212 2 : 60 I n t e r v a l ( same un i t s as above )
213 3 : 10 Set Output Flag High
214
215 19 : Set Active Storage Area (P80) ˆ6084
216 1 : 1 Fina l Storage Area 1
217 2 : 60 Array ID or Loc [ ]
218
219 20 : Real Time (P77) ˆ17297
220 1 : 1220 Year ,Day , Hour/Minute ( midnight = 2400)
221
222 21 : Wind Vector (P69 ) ˆ25081
223 1 : 1 Reps
224 2 : 0 Samples per Sub−I n t e r v a l
225 3 : 1 S , 1 Polar
226 4 : 1 Wind Speed/East Loc [ Wind Spd ]
227 5 : 2 Wind Di r e c t i on /North Loc [ Wind Dir ]
228
229 22 : Maximize (P73) ˆ30846
230 1 : 1 Reps
231 2 : 0 Value Only
301
232 3 : 1 Loc [ Wind Spd ]
233
234 23 : Average (P71) ˆ2969
235 1 : 1 Reps
236 2 : 3 Loc [ Temp F ]
237
238 24 : Sample (P70 ) ˆ2494
239 1 : 1 Reps
240 2 : 4 Loc [ Rel Humid ]
241
242 ; This i n s t e l im ina t e s −#’s - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
243
244 25:   If  ( X <= > F )  ( P89 )
245  1:  7         X  Loc  [  S h o r t w a v e  ]
246  2:  4         <
247  3:  0         F
248  4:  30        T h e n  Do
249
250      26:   Z = F  x  10^ n  ( P30 )
251       1:  0.0       F
252       2:  00        n ,  E x p o n e n t  of  10
253       3:  7         Z  Loc  [  S h o r t w a v e  ]
254
255 27:   End  ( P95 )
256
257 28:   A v e r a g e  ( P71 ) ^ 2 9 6 9
258  1:  1         R e p s
259  2:  7         Loc  [  S h o r t w a v e  ]
260
261 29:   A v e r a g e  ( P71 ) ^ 1 8 7 3 9
262  1:  1         R e p s
263  2:  8         Loc  [  L o n g w a v e   ]
264
265 30:   M i n i m u m  ( P74 ) ^ 2 9 8 1 3
266  1:  1         R e p s
267  2:  00        T i m e  O p t i o n
268  3:  5         Loc  [  B a t t e r y    ]
269
270 ; O u t p u t  24 hr  data - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
271
272 31:   If  t i m e  is  ( P92 )
273  1:  0         M i n u t e s  ( S e c o n d s  - -)  i n t o  a
274  2:  1 4 4 0      I n t e r v a l  ( s a m e  u n i t s  as  a b o v e )
275  3:  10        Set  O u t p u t  F l a g  H i g h
276
277 32:   Set  A c t i v e  S t o r a g e  A r e a  ( P80 ) ^ 3 2 7 1 5
278  1:  1         F i n a l  S t o r a g e  A r e a  1
279  2:  0 2 4 0      A r r a y  ID
280
281 33:   R e a l  T i m e  ( P77 ) ^ 2 4 6 1 3
282  1:  1 2 2 0      Year , Day , H o u r / M i n u t e  ( p r e v  day  at  m i d n i g h t ,  2 4 0 0  at  m i d n i g h t )
283
284 34:   W i n d  V e c t o r  ( P69 ) ^ 7 0 0 5
285  1:  1         R e p s
286  2:  0         S a m p l e s  per  Sub - I n t e r v a l
287  3:  1         S ,  1  P o l a r
288  4:  1         W i n d  S p e e d / E a s t  Loc  [  W i n d _ S p d   ]
289  5:  2         W i n d  D i r e c t i o n / N o r t h  Loc  [  W i n d _ D i r   ]
290
291 35:   M a x i m u m  ( P73 ) ^ 1 5 4 1 7
292  1:  1         R e p s
293  2:  10        V a l u e  w i t h  Hr - Min
294  3:  1         Loc  [  W i n d _ S p d   ]
295
296 36:   M i n i m i z e  ( P74 ) ^ 2 0 2 8
297  1:  1         R e p s
298  2:  10        V a l u e  w i t h  Hr - Min
299  3:  3         Loc  [  T e m p _ F     ]
300
301 37:   M a x i m u m  ( P73 ) ^ 2 4 6 6 3
302  1:  1         R e p s
303  2:  10        V a l u e  w i t h  Hr - Min
304  3:  3         Loc  [  T e m p _ F     ]
305
306 38:   A v e r a g e  ( P71 ) ^ 3 4 1 2
307  1:  1         R e p s
308  2:  3         Loc  [  T e m p _ F     ]
309
310 39:   A v e r a g e  ( P71 ) ^ 2 4 1 8 3
311  1:  1         R e p s
312  2:  4         Loc  [  R e l _ H u m i d  ]
313
314 ; T h i s  i n s t  e l i m i n a t e s  -# ’ s−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
315
316 40 : I f (X<=>F) (P89)
317 1 : 7 X Loc [ Shortwave ]
318 2 : 4 <
302
319 3 : 0 F
320 4 : 30 Then Do
321
322 41 : Z=F x 10ˆn (P30)
323 1 : 0 .0 F
324 2 : 00 n , Exponent o f 10
325 3 : 7 Z Loc [ Shortwave ]
326
327 42 : End (P95)
328
329 43 : Average (P71) ˆ2969
330 1 : 1 Reps
331 2 : 7 Loc [ Shortwave ]
332
333 44 : Do (P86)
334 1 : 29 Set Intermed . Proc . Disab le Flag Low ( Flag 9)
335
336 45 : Do (P86)
337 1 : 21 Set Flag 1 Low
338
339 46 : Average (P71) ˆ24559
340 1 : 1 Reps
341 2 : 8 Loc [ Longwave ]
342
343 47 : Minimum (P74) ˆ16675
344 1 : 1 Reps
345 2 : 10 Value with Hr−Min
346 3 : 7 Loc [ Shortwave ]
347
348 48 : Maximum (P73) ˆ8487
349 1 : 1 Reps
350 2 : 10 Value with Hr−Min
351 3 : 7 Loc [ Shortwave ]
352
353 49 : Minimum (P74) ˆ15889
354 1 : 1 Reps
355 2 : 00 Time Option
356 3 : 5 Loc [ Battery ]
357
358 *Table 2 Program
359 02 : 0 Execution I n t e r v a l ( seconds )
360
361 *Table 3 Subrout ines
362
363 ;**** SUBROUTINE #1 ******************************************************************
364
365 1 : Beginning o f Subroutine (P85)
366 1 : 1 Subroutine 1
367
368 ;CONVERT RESISTANCE TO TEMPERATURE
369 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
370 ;T = 1/(A+B*Ln(R) + C*(Ln(R) ) ˆ3)
371 ;T = Temp in Deg . Kelvin
372 ;A = 0.0010295
373 ;B = 0.0002391
374 ;C = 0.0000001568
375 ;R = Measured r e s i s t a n c e o f the rmi s to r
376 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
377
378 ; Constant A
379 2 : Z=F (P30)
380 1 : 1 .0295 F
381 2 : −3 Exponent o f 10
382 3 : 9 Z Loc [ ConstA ]
383
384 ; Constant B
385 3 : Z=F (P30)
386 1 : 2 .391 F
387 2 : −4 Exponent o f 10
388 3 : 10 Z Loc [ ConstB ]
389
390 ; Constant C
391 4 : Z=F (P30)
392 1 : 1 .568 F
393 2 : −7 Exponent o f 10
394 3 : 11 Z Loc [ ConstC ]
395
396 ; Natural Log o f Res i s tance
397 5 : Z=LN(X) (P40)
398 1 : 12 X Loc [ Case Res ]
399 2 : 13 Z Loc [ Ln Res ]
400
401 ; B*Ln(R)
402 6 : Z=X*Y (P36)
403 1 : 10 X Loc [ ConstB ]
404 2 : 13 Y Loc [ Ln Res ]
405 3 : 14 Z Loc [ bLn res ]
303
406
407 ; Squre and Cube Natural Log
408 7 : Z=X*Y (P36)
409 1 : 13 X Loc [ Ln Res ]
410 2 : 13 Y Loc [ Ln Res ]
411 3 : 15 Z Loc [ Ln res2 ]
412
413 8 : Z=X*Y (P36)
414 1 : 13 X Loc [ Ln Res ]
415 2 : 15 Y Loc [ Ln res2 ]
416 3 : 16 Z Loc [ Ln res3 ]
417
418 ; C*(Ln(R) ) ˆ3
419 9 : Z=X*Y (P36)
420 1 : 11 X Loc [ ConstC ]
421 2 : 16 Y Loc [ Ln res3 ]
422 3 : 17 Z Loc [ CLn res3 ]
423
424 ; Add A and B Terms
425 10 : Z=X+Y (P33)
426 1 : 9 X Loc [ ConstA ]
427 2 : 14 Y Loc [ bLn res ]
428 3 : 18 Z Loc [ case temp ]
429
430 ; Add C Term to A/B Term
431 11 : Z=X+Y (P33)
432 1 : 18 X Loc [ case temp ]
433 2 : 17 Y Loc [ CLn res3 ]
434 3 : 18 Z Loc [ case temp ]
435
436 ; Take Rec iporca l o f case temp
437 12 : Z=1/X (P42)
438 1 : 18 X Loc [ case temp ]
439 2 : 18 Z Loc [ case temp ]
440
441 13 : End (P95)
442
443 ;**** SUBROUTINE #2 ***********************************************
444
445 14 : Beginning o f Subroutine (P85)
446 1 : 2 Subroutine 2
447
448 ;CORRECT PIR CASE TEMPERATURE
449 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
450 ; CorrectOutput = A + (C*Tˆ4)
451 ; A = Thermipi le Output
452 ; C = Stefan−Boltzman Cnst = 5.6697 e−8 Wm−2K−4
453 ; T = Case Temperature in Kelvin
454 ;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
455
456 ;CONVERT THERMOPILE TO Wm−2
457 15 : Z=X*F (P37)
458 1 : 19 X Loc [ PIR mV ]
459 2 : 256.89 F
460 3 : 20 Z Loc [ PIR Aterm ]
461
462 ; Load 4 for r a i s i n g to f o r th
463 16 : Z=F (P30)
464 1 : 4 F
465 2 : 0 Exponent o f 10
466 3 : 21 Z Loc [ Power4 ]
467
468 ; Raise to 4 th Power
469 17 : Z=XˆY (P47)
470 1 : 18 X Loc [ case temp ]
471 2 : 21 Y Loc [ Power4 ]
472 3 : 18 Z Loc [ case temp ]
473
474 ; Load Boltzman
475 18 : Z=F (P30)
476 1 : 5 .669 F
477 2 : −8 Exponent o f 10
478 3 : 22 Z Loc [ PIR Bterm ]
479
480 ; Mutl ip ly Boltzman by Tˆ4
481 19 : Z=X*Y (P36)
482 1 : 22 X Loc [ PIR Bterm ]
483 2 : 18 Y Loc [ case temp ]
484 3 : 22 Z Loc [ PIR Bterm ]
485
486 ;Add A to Bterm −> OUTPUTS THE LONGWAVE RADIATION
487 20 : Z=X+Y (P33)
488 1 : 20 X Loc [ PIR Aterm ]
489 2 : 22 Y Loc [ PIR Bterm ]
490 3 : 8 Z Loc [ Longwave ]
491
492 21 : End (P95)
304
493
494 End Program
495
496 −Input Locat ions−
497 1 Wind Spd 1 4 1
498 2 Wind Dir 1 2 1
499 3 Temp F 1 4 1
500 4 Rel Humid 1 3 2
501 5 Battery 1 2 1
502 6 Ref Temp 1 2 3
503 7 Shortwave 1 6 3
504 8 Longwave 1 2 1
505 9 ConstA 1 1 1
506 10 ConstB 1 1 1
507 11 ConstC 1 1 1
508 12 Case Res 1 2 2
509 13 Ln Res 1 4 1
510 14 bLn res 1 1 1
511 15 Ln res2 1 1 1
512 16 Ln res3 1 1 1
513 17 CLn res3 1 1 1
514 18 case temp 1 4 4
515 19 PIR mV 1 1 1
516 20 PIR Aterm 1 1 1
517 21 Power4 1 1 1
518 22 PIR Bterm 1 2 2
519 23 0 0 0
520 24 0 0 0
521 25 0 0 0
522 26 0 0 0
523 27 0 0 0
524 28 0 0 0
525 −Program Secur i ty−
526 0000
527 0000
528 0000
529 −Mode 4−
530 −Fina l Storage Area 2−
531 0
532 −CR10X ID−
533 0
534 −CR10X Power Up−
535 3
536 −CR10X Compile Sett ing−
537 3
538 −CR10X RS−232 Sett ing−
539 −1
540 −DLD F i l e Labels−
541 0
542 −Fina l Storage Labels−
543 0 ,60 ,6084
544 1 ,Year RTM,17297
545 1 ,Day RTM
546 1 ,Hour Minute RTM
547 2 ,Wind Spd S WVT˜1 ,25081
548 2 ,Wind Dir D1 WVT˜2
549 3 ,Wind Spd MAX˜1 ,30846
550 4 , Shortwave AVG˜7 ,2969
551 5 ,Rel Humid ˜4 ,2494
552 6 , Battery MIN ˜5 ,29813
553 7 ,240 ,32715
554 8 ,Year RTM,24613
555 8 ,Day RTM
556 8 ,Hour Minute RTM
557 9 ,Wind Spd S WVT˜1 ,7005
558 9 ,Wind Dir D1 WVT˜2
559 10 ,Wind Spd MAX˜1 ,15417
560 10 ,Wind Spd Hr Min MAX˜1
561 11 ,Temp F MIN˜3 ,2028
562 11 ,Temp F Hr Min MIN˜3
563 12 ,Temp F MAX˜3 ,24663
564 12 ,Temp F Hr Min MAX˜3
565 13 ,Temp F AVG˜3 ,3412
566 14 ,Rel Humid AVG˜4 ,24183
567 15 , Battery MIN ˜5 ,15889
568 16 , Shortwave MIN˜7 ,16675
569 16 , Shortwave Hr Min MIN˜7
570 17 ,Shortwave MAX˜7 ,8487
571 17 , Shortwave Hr Min MAX˜7
572 18 ,Longwave AVG˜8 ,18739
573 19 ,Longwave AVG˜8 ,24559
305
APPENDIX B
YCWEATHER USER MANUAL
306
B.1 Installation
B.1.1 System Requirements
YCweather is a Windows based program that was compiled using MATLAB 2008b
(The Mathworks, Inc.) and requires MATLAB Component Runtime 7.9. YCweather
was designed to automatically update the software as well as the weather data files.
Thus, it is recommended that when using YCweather that the computer be connected
to the Internet. However, the Internet is not a requirement to run YCweather and for
this case the automatic data download option should be turned off, see the Section
B.7 for details.
B.1.2 Installing YCweather
YCweather is available for download from the website of the Subzero Science and
Engineering Research Facility1 at Montana State University. To install the software
the following steps must be followed.
1. Download the MATLAB Common Runtime (MCRinstaller.exe) software, this
is the background program necessary to run YCweather: www.coe.montana.
edu/ce/subzero/snow/MCRInstaller.exe.
2. Download the YCweather installer software package, YCinstaller.exe: www.coe.
montana.edu/ce/subzero/snow/YCinstaller.exe.
3. Execute the MCRinstaller.exe file, using the default settings for this program
is recommended.
1www.coe.montana.edu/ce/subzero/snow
307
4. Execute the YCinstaller.exe and follow the instructions, it installs similar to
most Windows programs.
5. When the installation process is complete, YCweather may be run by using the
YCweather.exe file located in the created program directory.
B.1.3 Updates
YCweather is a software package that is under development, as such updates will
be available periodically. When an update is available YCweather will provide the
user with a prompt, giving the user the option to update YCweather. It is strongly
recommend that if a new version is available that it be installed. The installation of
the update will occur automatically. Note, if the computer running YCweather does
not have Internet access the automatic update warnings will not be received. In this
case, the user should periodically check the download page for a newer version of the
software.
In order to install an updated version, simply download the file and install it as
explained in the installation instructions above. When installing allow the new version
to overwrite the old files, no data will be lost during this process. MCRinstaller.exe
only needs to be installed with the initial installation.
B.2 Program Control Window
Upon executing YCweather.exe, the window that appears is the Program Control
window, see Figure B.1. This window acts as the central controls for all operations
performed by YCweather. This section focuses on the main purpose of YCweather:
creating graphs of weather data. The Program Control window contains four basic
parts:
308
1. the menus, which are drop-down items at the top of the window (e.g., File menu
and Plot menu);
2. the toolbar, which contains the buttons just below the menus that act as short-
cuts to common menu items;
3. the Date/Time panel, which contains the options for selecting the date range
of interest; and
4. the Station panel, which lists the weather stations in the database.
Figure B.1: YCweather Program Control window.
B.3 Tutorial: Plotting Weather Data
To quickly create a simple plot of weather data:
309
1. Select a folder from the Folder/Season drop-down option on the Date/Time
panel.
2. Select a weather station from the buttons in the Station panel, for example
Ridge and Bridger from Bridger Bowl.
3. Choose a start and end date from the drop-down menus, be sure to select a range
that lies within the available data, which is given in the parenthesis adjacent to
the station radio buttons.
4. Select the Open Data List option. This is available by selecting Open Data List
option from the Data menu, pressing the Open Data List button on the toolbar,
or by pressing CTRL + V. This will open an additional window, as shown in
Figure B.2. Note, this window may take several seconds to open especially if
multiple stations are selected and/or if the stations contain a lot of data. The
reason being that when this window is open the program is recalling all of the
data for each station and storing it in a temporary location. This allows the
plots to be generated quickly.
5. In this new window (named Data List) select a weather parameter, such as Air
Temperature. Notice, that when you press a button that all variables without
the panel labeled Temperature disappear. This will prevent plotting of variables
with different units on the same axis.
6. Finally, plot the data. This can be done by pressing the Plot Weather Data
toolbar button on either Program Control or Data List window, by selecting
Weather Data from the Plot menu on either window, and by CTRL + W.
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Figure B.2: Example of the Data List window.
B.4 Creating Graphs
One of the main purposes of YCweather is to produce graphs, these graphs are
meant to be customizable and easily exportable. This section details the creation and
manipulation available in YCweather created graphs. Graphs are generated using the
Data List window as demonstrated in the previous section.
B.4.1 Dual-axis
When creating a graph, the Data List window (Figure B.2) displays two tabs.
Data selected via the Primary and Secondary tabs graph along the left-side and right-
side vertical axis, respectively. For example, Figure B.3 was created by selecting Air
Temperature under the Primary tab and Incoming Short-wave under the Secondary
Tab. The tick marks along the axes are setup to coincide, this sometimes results in
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illogical tick mark labels. This problem may be corrected by editing the limits and
step size, which is detailed in the following section.
03/13 12:00 03/14 12:00 03/15 12:00
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Time
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F)
 
 
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Irr
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Air Temperature (South) Incoming Shortwave (North)
Figure B.3: Example graph showing dual-axis capabilities.
B.4.2 Editing Axis Limits and Step-size
In many cases, especially when creating graphs for dissemination, it is desirable
to change the tick marks and limits on the graph. YCweather provides this capability
via two options: Limit Boxes and Step-size Boxes. These options are available on
the figure toolbar by the pairs of green arrows or by selecting the options from the
corresponding axis menu (e.g., the X-axis menu).
ˆ Limit Boxes: This option creates two text box items near the extremes of
the corresponding axis. Simply change the limits to the desired value and press
Enter. If the box is empty the axis limits are automatically determined based
on the data.
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ˆ Step-size Boxes: This toggle places a text box item near the lower axis limit.
This value dictates the step size between tick marks; leaving the value empty
results in automatic tick placement.
B.4.3 Exploring Data
The YCweather graphs allow the user to explore the data in various ways.
1. Limit/Step-size Boxes allow for custom control of axis limits and tick marks,
see Section B.4.2 for more details.
2. Zooming: This option is toggled by selecting the magnifying glass icon on the
figure toolbar.
3. Data Cursor allows the user to view the actual numbers associated with the
graph by selecting a portion of the plotted line. This option is available on the
figure toolbar.
4. Zoom Slider operates similar to the zooming feature but restricts the zoom to
the associated axis and has a slider bar that controls the zooming from 100% to
0.1% of the data range. The slider feature is available in the menus associated
with each axis (e.g., X-Axis menu).
5. Line highlighting is activated by left-clicking the mouse button on the desired
line, this will make the line large and display the actual data points that make
up the line. The highlighting is removed by left-clicking the line a second time.
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B.4.4 Context Menus
The lines, labels, and legends on YCweather graphs each have menus associated
that allow the user to manipulate the data. The menus for these items are accessed
by right-clicking on the object.
ˆ Line Context Menu: By right clicking on any line the user has control over
the appearance of the line for items such as the line thickness, style, color, or
markers. Additionally, a line may be deleted.
ˆ Label Context Menu: Each text item, such as the axis labels or annotations
(see Section B.4.5), allows for the user to edit the text, font, and location or
delete the item.
ˆ Legend Context Menu: The legend is also editable in its appearance includ-
ing options for editing the color of the box or the width of the bounding box.
Also, when two legends are present. as in Figure B.3. they may be combined
into a single legend by selecting the refresh option in this menu. Then simply
delete the unwanted legend box.
B.4.5 Figure Menus
The axes menus available from graphs created with YCweather include the Op-
tions menu as well as a menu for each axis. The Options menu provides generic
functionality that applies to the entire figure whereas the axis specific menus only
apply to that axis.
File menu (default MATLAB menu):
ˆ New: This option creates an empty figure, which is unaccessible with
YCweather.
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ˆ Open: Allows the user to open figure files that were saved with the *.fig ex-
tension.
ˆ Close: Closes the current figure.
ˆ Save: Saves the figure by overwriting the current file if the figure has previously
been saved, otherwise it envokes the Save as option.
ˆ Save as: Allows the user to save the figure in a vareity of formats, including
MATLAB *.fig format.
ˆ Export Setup: Opens MATLAB’s export user interface (see Section B.4.6).
ˆ Print Preview: Opens MATLAB’s print setup user interface (see Section
B.4.6).
ˆ Print: Sends the figure to the printer.
Options menu:
ˆ Add/Edit Labels: Allows user to add and/or edit the axes labels, figure name,
and figure title.
ˆ Interperter: Allows user to change the typesetting format, TEX and LATEX are
usefull when equations and units are being displayed.
ˆ Edit Font: Allows the user to change the font, style, and size for all text
objects in the figure.
ˆ Axes Color: Controls the background color of the figure.
ˆ Add Annotation: Enables user to insert items such as text boxes and arrows.
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ˆ Resize Figure: Allows for editing the size of the figure, which is useful for
exporting.
ˆ Tight Fit: Moves the axis labels to the outer extent to minimize whitespace
around the edges, this option is irreversible.
ˆ Export figure: Allows the user to export the figure as an image file (see Section
B.4.6).
X-,Y-, and Y2-Axis Menus:
ˆ Ticks/Labels: Allows for strict definition of the tick marks and labels used on
the associated axis.
ˆ Step Size Box: Toggle for the step size controls (see Section B.4.2).
ˆ Limit Boxes: Toggle for the axis limit controls (see Section B.4.2).
ˆ Zoom Slider: Toggles the presence of the zoom slider (see Section B.4.3).
ˆ Grid: Toggles the major grid lines.
ˆ Minor Grid: Toggles the minor grid lines.
ˆ Minor Ticks: Toggles the axis tick marks.
ˆ Reversed: Toggles the orientation of the tick marks and labels along the axis.
ˆ Add/Edit Legend: Allows the user to add or edit the legend entries.
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B.4.6 Exporting Figures
YCweather allows the user to output the graphs in a variety of formats. For those
familiar with MATLAB, it is possible save the figure as a *.fig file. The exporting/sav-
ing is accomplished in two ways. First, to simply create an image exactly as the figure
appears, select Export Figure from the Options Menu or press the associated Toolbar
button (see Section B.4.5). This option will output the figure exactly as it appears,
so it is imperative to setup the figure precisely as needed. The size of the exported
figure can be specified by editing the dimensions via the Resize Figure option in the
Options Menu.
The second option for saving/exporting figures is accomplished using the File
Menu (Section B.4.5), this menu is the default MATLAB figure menu; thus, for
users unfamiliar with MATLAB these options may be difficult to use. This menu
provides two options: one for printing the image that uses the Print Preview and
Print menu items and an Export Setup option for saving the figure as an image.
Both, the Print Preview and Export Setup open user interfaces with a variety of
options, details for using these items may be found in the online MATLAB help file:
http://www.mathworks.com/access/helpdesk/help/techdoc/index.html (in the
contents select “Graphics”; “Printing and Exporting”; “How to Print or Export”).
B.5 Workspaces
YCweather has the ability to save and load workspaces. A workspace is simply a
conglomeration windows including the Program Control, Data List, and any graphs.
For example, Figure B.4 is a workspace that includes graphs for air temperature and
short-wave irradiance.
To create a workspace consider the following example:
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Figure B.4: Example of a YCweather workspace.
1. Begin by creating a graph of some kind, see Section B.3 for instruction on
creating a graph.
2. To create a second plot the Clear Figures preference must be turned off, see
Section B.7 for details.
3. Arrange the windows as desired.
4. Save the workspace by selecting Save Workspace from the File menu in the
Program Control window. The default location is the \saved directory where
YCweather was installed. However, the workspace files (*.mat extension) may
be saved in any location.
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5. The workspace is now saved.
To load a workspace, simply select the Load Workspace option from the File
menu and locate the desired workspace file in the dialog box that appears. Once the
workspace has been selected YCweather will provide a prompt, as in Figure B.5, that
asks to use the current or stored time.
Figure B.5: Prompt that appears by default when opening a workspace.
The “current” option, by default, recalls the workspace using the most recent 48
hrs of data that exists. The stored time option uses the exact times set when the
workspace was created. These options exists so that the user can specify historical
(“stored”) workspaces of specific events or create workspaces (“current”) of commonly
used plots. The YCweather preferences (Section B.7) allow the number of hours to be
changed as well as the prompt appearance to be changed. For example, if a workspace
is created that is solely intended for a specific event, then the prompting preference
should be changed to “Stored” so that when this workspace is recalled it simply opens.
B.6 MesoWest Data
YCweather is capable of interfacing with weather data archived with MesoW-
est (mesowest.utah.edu/index.html). First, a text file names mesowest.txt must
be present in the season folder within the YCweather database, see Section B.11.2
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for details regarding the folder structure. This file contains two columns of comma
separated data: the first column contains a list of station identifiers as shown on
MesoWest. For example, YLWM8 is the identifier for the Yellow Mule station in
Southwest Montana. The second column contains a corresponding group name as-
sociated with the station identifier in the same row. For example, the Yellow Mule
station mentioned is a part of the RAWS network, thus an appropriate name may be
“RAWS Stations” and perhaps another group would include the National Weather
Service stations (e.g., “NWS Stations”).
The data available for download from MesoWest is limited to 30 days, as such if
changes to range on the Program Control is altered the MesoWest data may require
updating. When YCweather opens and MesoWest data is desired, it downloads the
data in the date range, by default this is the last 48 hours of the data. The MesoWest
data does not update automatically, but can easily be updated via the Toolbar button
or using the Data menu (see Section B.8).
B.7 Preferences
YCweather offers a variety of customizable options for controlling how the program
operates. These options are available in the Preferences, which may be opened via
the Program Control File menu or with the Toolbar button. Figure B.6 shows the
Preferences window. This window is divided into three parts, each of which has a
number of options as discussed below.
In order for the changes in preferences to take place, the Apply button must be
pressed. The changed setting will only apply to the current YCweather workspace
and will return to the default settings when YCweather is reopened. The selected
preferences may be defined as the default by selecting the Set Default option in
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Figure B.6: YCweather preferences window.
the Preferences window File menu. Additionally, if the workspace is saved (Section
B.5) the current setting are applied to that workspace file and will remain when this
workspace is opened in subsequent executions of this workspace file.
B.7.1 Stored Data Locations
As indicated in the Preferences window (Figure B.6) editing the “database” and
“saved” directory is not recommended unless a thorough understanding of file struc-
ture of YCweather is possessed. These details are included in Section B.11, which
discuss how YCweather operates. The “database” directory is where all the weather
data, images, and log files are stored. And the “saved” directory is the default location
for all YCweather related files created by the user.
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B.7.2 Weather Plotting Settings
The left-hand column in this section controls how the lines of weather graphs
will appear upon creation, allowing for the adjustment of the line style, line markers,
and line width for both the primary (left-side) and secondary (right-side) axes. The
right-hand column includes various options, which are summarized in the following
list.
ˆ Clear Current Figures: If this value is set to “Yes” then each time a graph
is created all others are deleted.
ˆ Type of Units: Specifies the units to display when graphs are created. Note,
if this is changed a new Data List window must be created because the unit
conversion occurs during the creation of this window.
ˆ Width/Height Units: Sets the units of the graph size upon creation, the
numeric value for this setting is provided in the following item.
ˆ Figure Width/Height: The width and height of a graph created based on
the units specified above.
B.7.3 Misc. Settings
The settings in Misc. Settings panel, as the name suggests, control various aspects
of YCweather. The first three items in the right-hand column toggle the appearance
of the corresponding side panels when YCweather opens. These panels are detailed
in Section B.9.
The left-hand column allows the user to determine the type of daily log that
YCweather should utilize, see Section B.8.2 for details. The Auto Weather Data Up-
date options toggles the automatic download of the latest available weather data, as
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detailed in Section B.8. The next two options in this column control the graphing start
and end times when graphs are created via workspaces, which is discussed in Section
B.5. Finally, the “Allow Mesowest” setting toggles the capability for YCweather to
communicate with the MesoWest database, which requires Internet access. Additional
details regarding the MesoWest feature may be found in Section B.6.
B.8 Data Menu
The Data menu in the Program Control window serves two functions: to update
the weather data and to access images and daily logs. This section briefly explains
data updating.
YCweather is designed to automatically collect the most recent weather data files;
however, this is only available from the Montana State University network. If off-
campus access is required please contact the MSU Department of Civil Engineering.
This automatic update may be disabled in the program Preferences (Section B.7) and
accessed manually via the “Check for new weather data” option in the Data menu.
This process includes updating the weather files as well as any daily log files that
exist. It is also possible to download images from a specific day and time via the
daily log viewer (see Section B.8.2).
B.8.1 Image Viewer
YCweather contains a basic image viewer for accessing images stored in the
YCweather database. Adding images to the database is explained in Section B.9.2
and the database file structure is explained in Section B.11. To access images, follow
these steps:
1. Select the station(s) of interest,
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2. Select the start date desired (the end date is not utilized), and
3. Select Open Images from the Data menu or Toolbar button on the Program
Control window.
If images for the selected station(s) exist a window will open displaying the images.
One window will appear for each station selected. Figure B.7 provides an example of
the image window. In the case where no images exist for selected date, but exist for
other dates at this station, an image window will open with the Select Date pop-up
menu (see Figure B.7) set to the first date available.
Figure B.7: Example of the image viewer for YCweather.
The YCweather image viewer offers the user the following functionality:
ˆ Toggles for cycling through images on the current date (right-hand buttons and
pop-up menu).
ˆ Toggles for changing the date being viewed (left-hand buttons and pop-up
menu).
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ˆ Zooming via the mouse cursor.
ˆ The ability to export the figure to another location via the Save image as. . . op-
tion in the File menu of the image viewer (this copies the image and does not
affect the original).
ˆ Capability of renaming an image in the database (Rename in the Options menu).
ˆ The ability of using the default Windows-based program for viewing images,
which is available as the Open with Windows item in the Options menu.
B.8.2 Daily Logs
One of YCweather’s main features is the daily logs, which are text notes associated
with each station and date. These logs are stored in the YCweather database (see
Section B.11) and added via the panel discussed in Section B.9.2. Two types of daily
logs are available, as shown in Figure B.8. The type of log displayed is controlled in
the preferences (Section B.7).
1. Yellowstone Club: A form specifically designed for usage with a research
project at the Yellowstone Club.
2. General: A simple form for typing notes.
To open the daily logs: (1) select the station(s) of interest, (2) select the start
date desired (the end date is not utilized), and (3) chose Open Daily Log(s) from the
Data menu or Toolbar on the Program Control window.
If daily logs exist then a window, similar to Figure B.8, will appear. The toggles
and pop-up menu on the right allow the user to cycle through all the logs for the
station. The daily log may be edited and the changes saved using the Save daily log
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option in the File menu. Additionally, the log may be opened in a traditional text
editor (Open log with Windows in the Open menu); however, this is not recommended
for the casual user. Editing the the log in this fashion may render the file unreadable
by YCweather.
It is possible to download images associated with the current station and date of
the daily log, this is accessible by selecting Download images from the Open menu.
Finally, the image viewer for the current station may be opened using the daily log
window by selecting Open images from the Open menu.
(a) Yellowstone Club (b) General
Figure B.8: Examples of the daily log options available in YCweather.
B.9 Panels Menu
Three additional panels for manipulating data within YCweather are available:
Thermocouple Plotter, Add Logs and Images, and Search. These features are avail-
able via the Panels menu on the Program Control window. These panels may be
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triggered automatically when YCweather opens via the program Preferences (Section
B.7). Figure B.9 shows the Program Control window with all the panels. Each of
these panels servers a specific function, as defined below, that a typical user will likely
not require.
Figure B.9: Program Control with all side panels showing.
B.9.1 Thermocouple Plotter
A weather station may contain thermocouple data that extends into the snowpack,
such as the North and South stations at the Yellowstone Club. In this case it is
desirable to graph temperature profiles of the snow pack at various intervals; the
Thermocouple Plotter panel serves this purpose. To understand this feature consider
the following tutorial, referring to Figure B.10.
1. Open the Yellowstone Club South weather station Data List, as in Figure B.10.
In the Data List window a list of thermocouple will be present in the Temper-
ature panel, this indicates that this station has thermocouple data.
2. Open the Thermocouple Plotter panel using the Panels menu on the Program
Control window.
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3. Select a start and end time on the Program Control for just a few hours as in
Figure B.10.
4. Change the Plot Interval pop-up menu to 30 minutes on the Thermocouple
Plotter panel.
5. Change the Exposed pop-up menu to a value of four.
6. Press the Plot TC Data button.
Figure B.10: Example workspace showing a graph of thermocouple data.
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Following the above steps should produce a graph similar to that displayed in
Figure B.10. This graph displays the theromocouple profiles at 30 minute intervals
over the specified time. The horizontal black line is meant to represent the snow
surface. For the North and South Yellowstone Club weather stations the number of
thermocouples exposed is recorded in the daily logs.
B.9.2 Add Logs and Images
This panel allows the user to add image files and daily logs to a specific station for
a specific date. Each weather station (e.g. South at the Yellowstone Club or Ridge at
Bridger Bowl) may have images and a daily log associated with the station for each
day. The Add Logs and Images panel allows this data to be assigned. For example,
to add a daily log for February 13 at the South Yellowstone Club station:
1. select the South station from the Yellowstone Club panel in the Program Control
window,
2. select the appropriate day from the Add Logs and Images panel, and
3. press the “Add Daily Log” button.
Performing these steps opens a window for adding and editing the daily log. Enter
the desired information and then select the Save daily log option from the File menu.
If a log already exists for the selected station and date, a warning will appear. If
it is desired to overwrite the log then continue, otherwise the log should be edited.
Editing daily logs is discussed in B.8.2.
Similarly, images can be added to the YCweather database. In this case the
program will prompt the user to select the desired images to include, these images
will be added to the database and accessible via the image viewer. No changes to the
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images occur, YCweather simply builds a reference to the image file(s). For specifics
on the YCweather file organization within the database see Section B.11.
B.9.3 Search
The Search panel provides the user a tool for searching all the daily log (Section
B.8.2) files for a specific folder/season. Type the desired keyword(s) in the window
with multiple words separated by a comma. If any matches for any of the keywords
exist the station and date will appear in the Results list. Selecting the desired result
opens the associated daily log.
B.10 Output Menu
B.10.1 Output data to file(s)
The weather data from YCweather may be exported as a comma separated text
(*.txt), Microsoft Excel 97-2003 (*.xls), or Microsoft Excel 2007 (*.xlsx) file via the
Output menu on the Program Control window. Selecting this option causes two
options to appear:
ˆ All data: This option exports all the available weather variables, as listed in
the Data list (Figure B.2), from the selected stations.
ˆ Selected data: This option only exports the data selected in the Data list
(Figure B.2).
After selecting one of these options YCweather will open a prompt asking: “Would
you like to crop the data between the selected date/times or write the entire data
set?” By selecting Crop only the data between the times selected on the Program
Control window are exported. Selecting Entire exports all available data.
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Next, YCweather will prompt for selecting the location and name of the output
file, this is where the file type may be specified. If either Excel file formats are selected
YCweather will create a single file with each selected station as Worksheets within
this file. Outputting as a text file (*.txt) results in a file for each selected station being
created, which will be named as <name> station.txt. The <name> is the filename
entered by the user and the station is the YCweather designation for the station.
B.10.2 Output to RadTherm
YCweather is capable of producing a text file for the use with RadTherm/RT
(ThermoAnalytics, Inc.), an example file is shown in Figure B.11b. This feature is
accessed from the Output menu on the Program Control window. This opens the
RadTherm Export window, as shown in Figure B.11a.
(a) (b)
Figure B.11: (a) RadTherm/RT file exporter and (b) an example output file.
When using the exporter, begin by selecting the desired station from the right-
column of pop-up menus. When a station is selected the corresponding Weather
Variable pop-up menu is changed to include weather variables with the necessary
units. The names that appear in both menus correspond to the tags assigned to the
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*.yc file for the stations, as discussed in Section B.11. The start and end date/time
values may be changed using the Program Control window and then by pressing the
Update Time button. Finally, the file is created by pressing Build File, a prompt will
open for selecting the location to save the file. The filename is dictated by the date.
The RadTherm/RT exporter also allows for the configuration of the pop-up menus
to be saved. This is available from the File menu on the exporter via the Save and
Open Settings options. These settings files utilize a *.rdt extension.
B.11 Application Details
The following section details of the operation of YCweather. This section is meant
to aid researchers at the Subzero Science and Engineering Research Facility maintain
and update the software. YCweather was written with MATLAB version 7.7.0.471
(R2008b). If you are using a newer version of MATLAB for editting YCweather
the MATLAB run time component must be updated on all machines attempting to
execute YCweather (see Section B.11.5 for additional details).
B.11.1 Basic YCweather Operations
The executable version of YCweather relies on four primary files that must
be located in the same location: YCweather.exe, YCmain.exe, default.mat, and
version.txt. YCweather.exe is a wrapper program that keeps the main program
YCmain.exe current based on the installed version (version.txt) and the available
version on the YCweather website (Section B.11.5). YCmain.exe may be executed
without YCweather.exe, but will never update in this case. The source code for
YCweather.exe and YCmain.exe are MATLAB m-files YCweather.m and MAIN.m,
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respectively. YCweather.m will not operate correctly via the m-file; MAIN.m may be
run via MATLAB if desired.
When YCmain.exe begins it opens the default.mat workspace file, which must
be located in the same directory. If this file does not exist or it is the first execution
of YCweather this file is created. The workspace file includes the location of the
database directory that contains all the weather data, daily logs, and image files.
This directory may be located anywhere on the machine as long as the workspace file
points to the correct location (see Sections B.7). However, when the default.mat
workspace file is created the location is initially set as the “database” folder in the
same directory as the YCweather executables.
When YCmain.exe (MAIN.m) begins operation, after opening the workspace file
(default.mat), it attempts to download the latest weather data. As mentioned in
Section B.8 this option may be turned off. The installation package, Section B.11.5,
includes the latest data from the current season. Thus, an Internet connection is
not required to run YCweather initially, but only to keep the program and data
current. Lastly, YCmain.m initializes by applying the default.mat workspace file
(callback readWS.m), prior to this the only two parameters in the workspace file
where utilized: the database location and the auto update trigger.
At this point, the YCweather is ready for manipulation by the user and the pro-
gram has opened all available data into the internal data structures.
B.11.2 Database Directory
The database directory must be organized in a specific fashion for YCweather to
operate correctly, most of this organization is handled automatically. The directory
tree for the YCweather database is shown in Figure B.12. The first level of folders in
the database directory are for each season of data, these exact folder names show up
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in the Program Control window in the Season/Folder pop-up menu. The initialization
of the pop-up menu occurs when a workspace file is opened, the source code being
callback readWS.m.
When the user selects the season via the pop-up menu YCweather accesses this
folder, inside of which the *.yc format files (see Section B.11.3) for all weather stations
are stored. These format files contain, among other things, an abbreviated station
name. This name is used in the internal data structure of YCweather as well as for
creating the next level of folders.
As described in Section B.8, YCweather acts as an archiving application for daily
logs and images. The daily logs, images, or image reference files are contained in
folders that exist within the station folders. The folders are named according the
aforementioned abbreviated station name. Within each of these folders two additional
folders exist: DailyLogs and Images. These folders, as the names suggest, store the
archived daily logs and image files. The station folders and sub-folders are created
automatically by YCweather when the user adds a daily log or image (see Section
B.9.2).
The DailyLog folder contains text files that store the daily log information, each
log must me named as mm-dd-yy.txt. The Images folder contains folders named as
mm-dd-yy. Within each folder the images are stored, the names are irrelevant, but the
files should be stored only in recognized formats, see MATLAB’s help on “imread”.
This directory may also contain a images.txt file which contains a list of image files
elsewhere on the computer that have been associated with the station and data by
the user, see Section B.9.2.
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database
07-08
08-09
YCnor th
YCsouth
YCsouth.yc
YCnor th.yc
I mages
DailyLogs
02-14-09
02-15-09
02-14-09.tx t
02-15-09.tx t
images.tx t
pic ture1.jpg
Figure B.12: Example of file structure of the database directory used by YCweather.
B.11.3 Weather Station Format Files (*.yc)
YCweather basis it’s entire operation on format files, which are simply text files
with a *.yc extension. An example, format file is include in Figure B.13. These
files communicate to YCweather the necessary information regarding the weather
data files. The weather data files may be any comma delimited text file completely
composed of columnar numeric entries.
Format files are composed of three parts, with the parts being separated by #
sign. The first portion consists of six lines that detail various parts of the data file.
Part two details each column of data present in the data file. Finally, part three
contains custom functions utilized for making calculations.
Part One: Data File Details: The following list is a line-by-line description of
the six components required in the first portion of the *.yc format files.
335
Figure B.13: Example format file utilzed by YCweather.
1. Station ID: The station id must be a single text string that uniquely identifies
the weather station associated with this data file. This ID must conform to
MATLAB’s variable naming convention, see MATLAB’s help file on “Naming
Variables”.
2. Station name: This string identifies the weather station and will appear next
to the toggle button within the Station Panel in the Program Control window.
3. Station location: This value is a text string that identifies the location of
the weather station, this name will appear in the Station Panel in the Program
Control Window.
4. Path to data file: The path to the data file must be any valid complete
or relative path and filename that references the data file associated with this
336
format file. In the example file, Figure B.13, the file alpine.dat must then
exist in the same directory as the *.yc format file.
5. Array ID: This value is useful for weather files that are composed of multiple
data arrays such as created via Campbell Scientific dataloggers. In many cases
data files of this type contain a identifier at the beginning of a row identify
the type of data. For example, a row beginning with 60 may represent hourly
data and those starting with 24 may indicate daily data. Thus, if the Array
ID is 60 in the *.yc format file then only the data marked with this ID would
be included for this station in YCweather. Another *.yc file would need to be
established to gather the data from the other array. Figure B.14 includes the
Array ID feature. In Array ID is present in the file, none should be entered in
this location.
6. Thermocouple ID: This identifier indicates if a thermocouple array within
the snowpack exists. If this data does not exist then 0 (zero) should be entered.
For stations with snowpack temperature arrays this ID should correspond the
the variable ID’s defined in part two of the *.yc file, as discussed below and
shown in Figure B.14. The variable ID also must contain a numeric portion
that indicates the location of each thermocouple in the snowpack. For example,
the thermocouples shown in Figure B.14 are space at 2 cm intervals.
Part Two: Weather Variables: Part two details the weather variables, there
should be one row for each column that exists in the corresponding data file. Each
row in this section has four comma separated values, as detailed below.
1. Variable Name: The first value is a single string of text that uniquely de-
fines the variable from others in the format file. This name must conform to
337
Figure B.14: Example format file utilized by YCweather that includes the thermo-
couple ID for plotting temperature profile data (this is not a complete file).
MATLAB’s variable naming convention, see MATLAB’s help file on “Naming
Variables”.
2. Inclusion Trigger: This value indicates to YCweather if the corresponding
column of data should be listed as a selectable option in the program. Entries
may be 0 or 1, where 0 excludes the data.
3. Units: The third entry communicates the units of the data; this value must
conform to the units specified in the units.txt file as detailed in Section B.11.4.
338
The units can be in either English or metric units, but must be included in the
aforementioned file.
4. Legend Label: The last value is a string describing the weather data that is
inserted into the legends of YCweather graphs.
Part Three: Custom Functions: This section allows for calculation to be done
on the weather variables listed above in Part Two. One function that is a critical com-
ponent of YCweather will be used here as an example, the calcwx time.m function.
YCweather requires that a variable named Time be present and contain
the time stamps for the weather data in MATLAB’s serial format. The
calcwx time.m function performs this operation. For information on this format see
MATLAB’s help on “Types of Date Formats”.
Taking a step back, the function read dat.m is responsible for reading the for-
mat *.yc files, this function outputs the weather data into a structure that is used
by YCweather; read dat.m also implements the custom functions listed in the *.yc
format files.
When the custom function are called from read dat.m they are implemented as
follows within MATLAB, using calcwx time.m as an example.
MATLAB
>> Time = calcwx_time(d,’year’,’month’,’day’,’hrmin’,’GNFAC’);
Comparing this functional operation to the row of inputs in the format file in
Figure B.13 shows that the first entry in the format *.yc file is the output variable
(Time), the next value is the function name (calcwx time), and the remaining items
are string input into the custom function. The input variable d is the data struc-
ture used for storing the weather data and is automatically inputed into the custom
339
function in read dat.m. This data structure contains all the data present in the
weather data file, as listed in Part Two. Hence, the function calcwx time.m uses
the input strings (’year’,’month’, etc.) to compute the new Time variable with the
appropriate time format required by YCweather. So, each custom function essentially
creates another weather variable for use by YCweather.
The custom functions were setup to allow YCweather to be expandable by the
user to perform calculations on the weather data. To best understand the custom
functions, it is best to examine the source code, specifically section five of read dat.m
and any of the existing custom functions: calcwx time.m, calcwx flux.m, and
calcwx labLW.m.
B.11.4 Variable Units
As mentioned in Section B.11.3, the format files require that the units for each
weather variable be defined. The units prescribed in the format file must be present in
the units.txt file, which is read by the getunit.m function. The units.txt defines
the units via text abbreviations (e.g., ’kPa’ for pressure) both in Metric and English,
the conversion factor between the units, and the appropriate axis labels for use in
YCweather generated graphs. The function getunit.m is utilized for extracting the
various unit related information in various portions of YCweather. Both the function
getunit.m and text file units.txt were designed to allow for additional units to be
added, which should only require adding a row to the text file.
The units.txt file should be composed of rows containing the following comma
separated information: 1) the Metric abbreviation, 2) the English abbreviation, 3)
text describing the unit, 4) the Metric abbreviation written in LATEX math format, 5)
the English abbreviation written in LATEX math format, 6) the Metric unit written as
TEX, 7) the English unit written as TEX, 7), and finally 8) the conversion multiplier
340
from English to Metric. Figure B.15 contains a portion of the units.txt file, refer
to the file itself for additional examples as well as additional information regarding
the format. Note, the # is the comment character within the file.
Figure B.15: Example entries for prescribing units within the units.txt file, which
is utilized by the getunit.m function.
B.11.5 Compiling and Implementing YCweather
The information in this section details the process for building a YCweather exe-
cutable file from the source code via MATLAB’s compiler. If any changes are made
to the source code of YCweather the following information will make the updates
available to all users running YCweather.
The function YCbuild.m acts to automate the process of compiling YCweather
into executable form as well as post the updated to the web folder. The function
requires two outside programs, WinSCP2 and InstallJammer3.
After changing the source code, implement the the following code from the MAT-
LAB command-line: >>YCbuild(’build’,0.5), where the second input is the new
version number. When this command executes the version is updated, YCweather.exe
2WinSCP: winscp.net
3InstallJammer: www.installjammer.com
341
and YCmean.exe are complied, YCmain.zip is packaged, the latest weather data from
the current season is prepared, and the installer is compiled (YCinstaller.exe). All
of these files are placed in the release directory, which are exactly the files need on
the YCweather website. Before compiling a new version, the version number in the
MAIN.m functions should be updated.
This process relies on three files. First, the two project files: YCweather.prj and
YCmain.prj. These files were created with MATLAB’s deploytool and dictate how
YCweather.exe and YCmain.exe are compilied. If any additional m-files are added to
YCweather then these files will needed to be added to the list of files in the YCmain.prj
file. The third file, is the InstallJammer installation file, YCinstaller.mpi, which is
stored in the YCinstallerFiles directory.
Once YCweather is complied it must also be uploaded to the website so that the
changes will be made available to all users of the program. This is done by executing
the following: >>YCbuild(’web’). This removes the old files from the web and adds
the new via WinSCP; access to the appropriate account on the MSU Department of
Civil Engineering server is required.
B.11.6 Website and Online Weather Database
The YCweather website, www.coe.montana.edu/ce/subzero/snow, is hosted by
the MSU College of Engineering and contains the necessary files for initially download-
ing YCweather and the files needed for automatically updating YCweather. These
files are automatically generated and created during the compilation and posting
process described in Section B.11.5.
YCweather also relies on an ftp accessible database also hosted by the College
of Engineering. This directory contains the weather files database directories that
are automatically accessed by YCweather for keeping the weather data up to date.
342
The MATLAB program GNAFC.m located on the server is executed hourly to keep the
weather data current via CRONTAB.
343
APPENDIX C
THERMAL MODEL SOFTWARE USER MANUAL
344
C.1 Introduction
A heat-equation based thermal model was presented in Chapter 5, which was
used as the basis of the sensitivity analysis and Monte Carlo simulations presented
in Chapters 7–10. This appendix details the implementation of the thermal model
software developed that works in conjunction with the sensitivity analysis software
detailed in Appendix D. Additionally, details are provided on other software devel-
oped for implementing this model, including a complete stand-alone graphical user
interface.
This appendix is divided into three main sections. Section C.2 explains the basic
operation of the thermal software. Section C.3 presents an interface that links the
thermal model with Microsoft Excel, allowing inputs to be easily modified. Figure
C.1 contains a flow chart demonstrating how the various functions detailed in the first
two sections (C.2 and C.3) interact. Finally, in Section C.4, a complete graphical user
interface is briefly presented that operates as a stand-alone Windows application.
A few notational conventions are utilized throughout this user manual:
ˆ Monospaced typeface indicates a MATLAB m-file, function, or variable (e.g.,
sobol.m).
ˆ MATLAB code is provided in figure windows (e.g., Figure D.2 when referenced
many times).
ˆ MATLAB code is also presented in-line with the text as:
>> 2+2
ans = 4
>>
345
 
ru n m o d el . m  
x ls_ in put . m  x ls_ p rep . m  therm al . m  
r ad _ calc . m  
c o n fin t. m  
alb edo .m a t  
T em p la te . xls x  
Figure C.1: Flow chart demonstrating how the various functions discussed in Section
C.2 and C.3 interact.
C.2 Basic Application
The basic operation of the thermal model is performed via the MATLAB
command-line using two functions: xls prep.m and thermal.m (the source code is
included in Section C.5). These two functions were utilized by saMODEL2.m as detailed
in Appendix D. First, xls prep.m is implemented, which requires three input arrays
that contain the snow properties, atmospheric conditions, and model constants. The
syntax for xls prep.m is as follows:
f unc t i on [ s , a ] = x l s p r e p ( snow , atm , cons tant s )
% XLS PREP b u i l d s a r rays f o r input ing in to thermal model
%
% SYNTAX:
% [ snow , atm ] = prep input ( snow , atm , cons tant s ) ;
The snow variable may be arranged in two ways: as a uniform or a varying snow-
pack. If the snowpack is assumed uniform then snow is a 1-D array with six values (in
order): depth (cm), density (kg/m3), thermal conductivity (W/(m·K)), specific heat
capacity (J/(gm·K)), snow temperature (◦C), and extinction coefficient (1/m). If the
snowpack varies then the array may be composed of any number of rows of the same
346
parameters that dictates the different layers. The following MATLAB code provides
example definitions of the snow variable:
>> snow = [ 5 0 , 130 , 0 . 06 , 2030 , −10, 70 ]
snow =
50 130 0 .06 2030 −10 70
>> snow = [ 0 , 130 , 0 . 06 , 2030 , −10, 70 ; 50 , 180 , 0 . 1 , 2030 , −5, 9 0 ; . . .
100 , 180 , 0 . 1 , 2030 , −5, 90 ]
snow =
0 130 0 .06 2030 −10 70
50 180 0 .1 2030 −5 90
100 180 0 .1 2030 −5 90
>>
The first example defines a 50 cm thick snow pack with constant properties. The
second example defines a 100 cm deep snowpack that increases in density, thermal
conductivity, temperature, and extinction coefficient from 0 to 50 cm. Then from
50 to 100 cm the conditions remain constant. The xls prep.m function performs
linear interpolation between the rows according to the layer thickness. An additional
seventh column is optional that specifies the extinction coefficient for the near-infrared
wavebands, in this case the extinction coefficient previously mentioned is used for the
visible waveband.
In similar fashion, the atmospheric conditions are defined in the atm variable,
which includes nine parameters (in order): time (hours), incoming long-wave radi-
ation (W/m2), incoming short-wave radiation (W/m2), albedo, wind speed (m/s),
air temperature (◦C), relative humidity (%), the lower boundary condition (◦C), and
air pressure (kPa). Two additional columns may also be defined that specifiy the
incoming short-wave radiation and albedo for the near-infrared wavebands. Again,
the short-wave radiation and albedo previously defined are then used as the values
for the visible spectrum.
The model constants are defined in the constants variable, which must include
the following (in order): latent heat of sublimation (kJ/kg), the latent heat trans-
fer coefficient, the sensible heat transfer coefficient, the ratio of molecular weights
347
of dry-air and water-vapor, the gas constant for water-vapor (kJ/(kg·K)), reference
temperature (◦C), reference vapor-pressure (kPa), the emissivity of snow, the layer
thickness (cm), and time step (s).
Once the three input variable arrays are defined the thermal model may be exe-
cuted, for example:
>> snow = [ 5 0 , 130 , 0 . 06 , 2030 , −10, 7 0 ] ;
>> atm = [0 ,240 ,0 , 0 .82 ,1 .7 , −10 , . 2 , −10 ,101 ; 10 ,240 ,500 ,0 .82 ,1 .7 , −10 , . 2 , −10 ,101 ] ;
>> contants = [ 2 8 3 3 , 0 . 0 0 2 3 , 0 . 0 0 2 3 , 0 . 6 2 2 , 0 . 4 6 2 , −5 , 0 . 4 0 2 , 0 . 9 5 , 1 , 6 0 , 1 ] ;
>> [ S ,A] = x l s p r e p ( snow , atm , cons tant s ) ;
>> [T,Q] = thermal (S , A, cons tant s ) ;
The thermal.m function implements the finite-difference solution presented in
Chapter 5. This function outputs an array containing snow temperatures (T) as
a function of model evaluation time (columns) and depth (rows). The various heat-
fluxes—long-wave, sensible, latent, short-wave—are output in the Q variable in similar
fashion.
C.3 Spreadsheet Application
C.3.1 General Application
To make the thermal model more powerful, two additional functions were
developed—xls input.m and runmodel.m—that provide an interface between MAT-
LAB and Microsoft Excel. This allows the various input matrices previously ex-
plained to be easily developed. First, the required structure of the Excel file must
be established. The Excel spreadsheet must be composed of three worksheets named
“SnowProperties”, “AtmosphericSettings”, and “Constants”. Each worksheet must
be formatted in a specific fashion, as shown in Figures C.2 and Figures C.3.1 Section
1A template may be downloaded at: www.coe.montana.edu/ce/subzero/snow/thermalmodel/
template.xlsx.
348
C.3.2 details some additional features available when using xls input.m, particularly
for the “Constants” worksheet.
(a) Snow Properties
(b) Atmospheric Settings
Figure C.2: Example of the (a) “SnowProperties” and (b)“AtmosphericSettings”
worksheets for Excel file read by xls input.m.
Once the Excel file is setup as desired, the function xls input.m is used to process
the data contained in the spreadsheet. As was the case for the basic operation, the
349
near-infrared columns are optional. For example, for the template.xlsx file available
for download, the following code implements the thermal model:
>> f i l ename = ’ template . x lsx ’ ;
>> [ s , a , c ] = x l s i n p u t ( f i l ename ) ;
>> [ S ,A] = x l s p r e p ( s , a , c ) ;
>> [T,Q] = thermal (S , A, c ) ;
The runmodel.m function performs the above actions, groups the results into a
data structure, and adds the ability to compute confidence level intervals for the
snow temperatures. The confidence intervals are explained further in the following
section. The code shown in Figure C.4 implements the thermal model via runmodel.m
and displays the data structure produced. The data structure and details regarding
various optional inputs are explained in the help associated with the runmodel.m.
Figure C.3: Example of the “Constants” worksheets for Excel file read by
xls input.m.
350
The data structure was designed to be implemented via the graphical user interface
(Section C.4), as such the data structure may contain many model runs, as shown in
Figure C.4.
1 >> f i l ename = ’template.xlsx’ ;
2 >> data = runmodel ( f i l ename )
3 data =
4 x l s : ’template.xlsx’
5 b o o t s e t t i n g s : [ ]
6 name : ’’
7 desc : ’’
8 time : ’22-Mar -2010 09:58:06 ’
9 T: [82 x601 double ]
10 Q: [81 x601x5 double ]
11 snw : [ 81 x7 double ]
12 atm : [601 x11 double ]
13 const : [ 1 x10 double ]
14 Tboot : [ ]
15 Qboot : [ ]
16 Sboot : [ ]
17 Aboot : [ ]
18 Cboot : [ ]
19 >> data (2 ) = runmodel ( f i l ename ) ; % mul t ip l e runs may be s to r ed
20 >>
Figure C.4: MATLAB implementation of runmodel.m and the resulting data struc-
ture.
C.3.2 Additional Features
The usage of the function xls input.m offers additional functionality for inputs,
including the usage of tabulated snow micro-structure data, input multipliers, and
confidence interval calculations.
Snow Micro-Structure: The snow albedo and extinction coefficient may be input
into the Excel document using keys: dXX, classX, or type.2 The dXX key allows
the snow grain diameter to be used to compute albedo and extinction coefficient
according to Armstrong and Brun (2008, Eq. 2.25, p. 56), were the XX is a number
2The italicized keys are used to reference the inputs, the actual text as would be entered in the
Excel worksheets is provide in quotes.
351
representing the size of the grain in millimeters (e.g., “d5”). Figure C.2b includes
the implementation of this option. The classX key uses the tabulated values from
Armstrong and Brun (2008, Tab. 2.6), where X is a value one to six (e.g., “class2”).
The type may be one of three strings: “fine”, “medium”, or “coarse”, this option is
only available for the computation of albedo. The usage of these keys results in the
computation of the albedo from the information provided in Baldridge et al. (2009).
The albedo and extinction coefficient calculations are preformed in the albedo and
extinction sub-functions of xls input.m.
In all cases, when the optional near-infrared columns are not utilized it is assumed
that albedo and extinction coefficients are defined for the “all-wave” spectrum that
includes both the visible and near-infrared spectrum. Using ASTM G-173 (2003)
the appropriated values are computed based on this spectrum via rad calc.m (see
Section C.5.5).
Both xls input.m and rad calc.m require the albedo.mat file that contains the
data structure shown in Figure C.5. Each component of this structure contains a
two-column numeric array, the the first column of which provides the wavelength in
nanometers. The second column of x.atsm contains solar irradiance as defined by
ASTM G-173 (2003). For the other items (e.g., x.fine), the second column contains
albedo values as defined in Baldridge et al. (2009).3
Finally, the snow density or the thermal conductivity may be automatically com-
puted by using “auto” in either column, but not both. The desired density or thermal
3This file may downloaded at www.coe.montana.edu/ce/subzero/snow/thermalmodel/
albedo.mat.
352
1 >> x = load ( ’albedo.mat’ )
2 x =
3 astm : [2002 x2 double ]
4 f i n e : [ 179 x2 double ]
5 medium : [179 x2 double ]
6 coa r s e : [ 179 x2 double ]
7 >>
Figure C.5: Required data structure of albedo.mat.
conductivity calculations are preformed using the relationships presented by Sturm
et al. (1997).
Input Multipliers: To enable simple modification of entire columns of data,
multipliers are provided on the “Constants” worksheet, as shown in Figure C.3. The
corresponding column from the other worksheets are simply multiplied by the values
listed, allowing the user to quickly modify the various inputs.
Confidence Intervals: The runmodel.m function includes the ability to compute
confidence intervals via confint.m, which uses the percentile bootstrap method pre-
sented by Efron and Tibshirani (1993). First, the percent error is prescribed by the
values listed in the “Error” column on the “Constants” worksheet, as shown in Figure
C.3. These values allow the parameter to vary plus or minus this amount according
to a normal distribution, such that the nσ tails of this distribution are at these limits,
where nσ is the number of standard deviations. The graphical user interface described
in the following sections provides the means for utilizing this feature.
C.4 Graphical User Interface
A graphical user interface (GUI), as shown in Figure C.6, was develop to act
as front-end to the software explained in the previous sections. This interface was
deployed via MATLAB’s deploytool tool and wrapped into an installable Windows-
353
based program. The complete installer, TMsetup.exe, may be downloaded at: www.
coe.montana.edu/ce/subzero/snow/thermalmodel/TMsetup.exe.
The stand-alone application may prompt the user to download a newer version,
which is recommended. By agreeing to this prompt the website listing the associated
files will automatically open in a browser. The only file that needs to be downloaded
is model.exe, this file should replace the original that is located in the installation
directory.
The GUI serves two functions, first it controls the operation of the runmodel.m
function and manages the data structure produced by this function (see Section C.3).
This is done through the use of projects, which are nothing more than MATLAB mat-
files that store the data structure produced by runmodel.m. However, the extension
was changed to *.prj. The GUI also provides tools for the visualization of the input
and output variables.
C.4.1 Performing Model Runs
The following briefly describes the basic steps of performing thermal model eval-
uations:
1. Select New Project from the file menu.
2. A prompt will appear that gives options regarding the Excel spreadsheet file to
utilize. Selecting New copies the template.xlsx file previously discussed to a file
selected by the user. Selecting Existing allows the user to select a previously
created file. In both cases the Excel file will open when the necessary actions
are complete.
3. Modify and saved the Excel file created for the desired conditions, as detailed
in Section C.3.
354
Figure C.6: Graphical user interface for implementing the snow thermal model.
4. Return to the GUI application and type a name for the current run as well
as a description. Also, if confidence levels are desired (see Section C.3.2.3)
the Compute Confidence Levels option should be checked at this time. The
computation of the confidence levels can be extremely time consuming, so begin
with a small number of re-samplings.
5. Press the Evaluate Model button on the GUI, this starts the model evaluations
which may take several minutes depending on the computer and model inputs. If
confidence levels are being computed a window will appear showing the progress
of the calculations.
355
6. When the run is complete it appears in the Project Run(s) menu on the left-side
of the GUI.
7. Additional runs may be computed by selecting the Create New Run button and
the Excel file may be changed by selecting the “...” button at the right-end
of the Base File text. This same button will also open the associated Excel
file when a model run is activated. It is not necessary to create a new Excel
file, but any changes made to the Excel file for additional model evaluations
must be saved, these changes will not be stored and cannot not be recalled (this
functionality may be available in future versions). Run names may be edited
or runs may be deleted by right-clicking on the run in the Project Run(s) list.
8. After all the desired runs are completed the project should be saved by selecting
Save Project from the File menu.
C.4.2 Graphing Results
It is possible to create graphs of both the model inputs and outputs, this is done
using the lower pane of the GUI shown in Figure C.6. First, a model run must be
selected in the Project Run(s) panel. The graphs created share all of the functionallity
of the graphs presented in Appendix B (Section B.4).
Model Inputs: The model inputs are graphed by selecting the Plot Input(s)
radio button, this will cause the Snow and Atmospheric parameter lists to become
activated. To create a graph simply select the desired item and press the Plot Results
button. A graph will appear for each item selected.
Model Outputs: Two different graph styles of model outputs are available:
profiles and coutours. Figure C.7 provides examples of the snowpack temperature
356
graphed with each of the different methods. When ploting profiles the interval, in
hours, must also be set (e.g., 2 results in profiles being plotted every 2 hours). It is
possible to graph a single profile directly from a contour plot, this is done be right-
clicking on the contour where the profile is desired and the selecting either a vertical
or horizontal profile.
−15 −10 −5 0
0
10
20
30
40
Temperature ( ◦C)
Depth(
cm
)
00:00
02:00
04:00
06:00
08:00
10:00
(a) Temperature Profiles
0 5 10
0
5
10
15
20
25
30
35
40
Time ( hr)
Depth(
cm
)
Tem
pera
ture
(
◦ C
)
−16
−14
−12
−10
−8
−6
−4
−2
(b) Temperature Contours
Figure C.7: Example graphs of snowpack temperatures demonstrated the two graph-
ing options available: (a) profiles and (b) contours.
In similar fashion, if confidence intervals were computed it is possible to graph
these intervals using the Output C.I. Profiles or Contours radio buttons. Figure C.8
provides examples of temperature data plots with confidence level intervals. Both
the profiles and the contours require the confidence level to be specified by a scalar
value (in percent) entered into the C.I. Level location. When profiles are plotted the
time(s) at which the profiles are desired must be specified in the Times (hr) location
(e.g., 2 or 2, 4). The confidence level contour graphs show the absolute value of the
largest deviation from the mean value.
357
−15 −10 −5 0
0
5
10
15
20
Temperature ( ◦ C)
Depth(
cm
)
04:00 5% C.I .
04:00
04:00 95% C.I .
(a) C.I. Profiles
0 1 2 3 4
0
5
10
15
20
Time ( hr)
Depth(
cm
)
Maxde
viatio
n(
◦ C
)
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) C.I. Contours
Figure C.8: Example graphs of snowpack temperature demonstrated the two graphing
options available for displaying confidence level intervals: (a) C.I. profiles and (b) C.I.
contours.
C.4.3 Closing Remarks
The information presented in this appendix explains the basic and advanced func-
tionality of the thermal model developed for the work presented throughout this
dissertation. The details presented as well as the entire software package was de-
veloped to make the model easily accessible, thus please contact the author if more
information is required.
358
C.5 Source Code
C.5.1 runmodel.m
1 func t i on data = runmodel ( vararg in )
2 % RUNMODEL program to exceucte thermal model us ing Excel input f i l e .
3 %
4 % SYNTAX:
5 % data = runmodel ;
6 % data = runmodel ( f i l ename ) ;
7 % data = runmodel ( f i l ename , name) ;
8 % data = runmodel ( f i l ename , name , desc ) ;
9 % data = runmodel ( . . . , [ B,N] ) ;
10 %
11 % DESCRIPTION:
12 % data = runmodel executes the thermal model v ia Excel input , prompting
13 % the user f o r a f i l ename .
14 % data = runmodel ( f i l ename ) executes the thermal model f o r supp l i ed name .
15 % data = runmodel ( f i l ename , name) same as above , but a l l ows the user to
16 % name the run ( e . g . name = ’Model Run #3 ’) ;
17 % data = runmodel ( f i l ename , name , desc ) same as above , but a l l ows user to
18 % a l s o add a d e s c r i p t i o n to the run ( e . g . desc = ’ This run mimics
19 % Feb−14−2008 at the South s t a t i on o f the YC’ ; )
20 % data = runmodel ( f i l ename , name , desc , [ B,N] ) runs the model and
21 % computes the bootst rap con f idence i n t e r v a l s , where B = number
22 % of resampl ings , N = number o f standard dev i a t i on s to assume f o r
23 % the t a i l s
24 %
25 % OUTPUT:
26 % The data s t ru c tu r e has the f o l l ow ing f i e ldnames
27 % x l s : Input Excel f i l ename
28 % boo t s e t t i n g s : Bootstrap s e t t i n g s
29 % name : Name o f cur rent run
30 % desc : Desc r ip t i on o f the cur rent run
31 % time : Star t time o f model execut ion
32 % T: Array o f snowpack temperatures
33 % Q: Array o f snowpack heat f l u x e s
34 % snw : Input array o f snow p r op e r t i e s
35 % atm : Input array o f atmospheric c ond i t i on s
36 % const : Array o f model constants
37 % Tboot : Bootstrap r e p l i c a t e s o f temperature
38 % Qboot : Bootstrap r e p l i c a t e s o f heat f l u x e s
39 % Sboot : Bootstrap r e p l i c a t e s o f snw inputs
40 % Aboot : Bootstrap r e p l i c a t e s o f atm inputs
41 % Cboot : Bootstrap r e p l i c a t e s o f const inputs
42 %
43
44 % 1 − GATHER OPTIONS
45 data = getopt i on s ( vararg in { :} ) ;
46
47 % 2 − EXECUTE MODEL
48 [ S ,A, data . const ] = x l s i npu t ( data . x l s ) ;
49 [ data . snw , data . atm ] = x l s p r ep (S ,A, data . const ) ;
50 [ data .T, data .Q] = thermal ( data . snw , data . atm , data . const ) ;
51
52 % 3 − RUN THE BOOSTRAP
53 i f ˜ isempty ( data . boo t s e t t i n g s ) ;
54 B = data . boo t s e t t i n g s ;
55 bootdata = con f i n t ( data . x l s ,B(1) ,B(2) ) ;
56 fn = f i e ldnames ( bootdata ) ;
57 f o r i = 1 : l ength ( fn ) ;
58 data . ( fn{ i }) = bootdata . ( fn{ i }) ;
59 end
60 end
61
62 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
63 func t i on [ data ,B] = ge topt i on s ( vararg in )
64 % GETOPTIONS determines / s e t s the input opt ions
65
66 % 1 − SET THE DEFAULTS
67 f i l ename = ’ ’ ;
68 name = ’ ’ ;
69 desc = ’ ’ ;
70 B = [ ] ;
71
72 % 2 − GATHER BOOTSTRAPPING DATA
73 idx = [ ] ;
74 f o r i = 1 : narg in ; idx ( i ) = isnumer ic ( vararg in { i }) ; end
75 ix = f ind ( idx , 1 , ’ f i r s t ’ ) ;
76 i f ˜ isempty ( ix ) ; B = vararg in { i x } ; end
77
78 % 3 − GATHER FILENAME, NAME, AND DESCRIPTION . .
79 i f isempty (B) ; rem = vararg in ; e l s e rem = vararg in ( 1 : nargin−1) ; end
359
80 i f l ength ( rem) >= 1; f i l ename = vararg in {1} ; end
81 i f l ength ( rem) >= 2; name = vararg in {2} ; end
82 i f l ength ( rem) == 3 ; desc = vararg in {3} ; end
83
84 % 4 − PROMPT FOR FILENAME
85 % 4.1 − Gather/ de f i n e the ” l a s t d i r ” p r e f e r en c e
86 i f i s p r e f ( ’ T h e r m a l M o d e l _ v 5 ’ , ’ l a s t d i r ’ ) ;
87 d e f d i r = ge tp r e f ( ’ T h e r m a l M o d e l _ v 5 ’ , ’ l a s t d i r ’ ) ;
88 e l s e
89 addpref ( ’ T h e r m a l M o d e l _ v 5 ’ , ’ l a s t d i r ’ , cd ) ;
90 d e f d i r = cd ;
91 end
92
93 % 4.2 − Prompt the user f o r a f i l ename
94 i f isempty ( f i l ename ) ;
95 F i l t e rSpe c = { ’ *. x l s x ’ , ’ E x c e l  W o r k b o o k  (*. x l s x ) ’ ; . . .
96 ’ *. xls ’ , ’ E x c e l  97 -2003  W o r k b o o k  (*. xls ) ’ ; . . .
97 ’ *.* ’ , ’ All  f i l e s  ( * . * ) ’ } ;
98 [ fn , pth ] = u i g e t f i l e ( F i l t e rSpec , ’ S e l e c t  f i l e ... ’ , d e f d i r ) ;
99 i f i snumer ic ( fn ) ; r e turn ; end
100 f i l ename = [ pth , fn ] ;
101 s e t p r e f ( ’ T h e r m a l M o d e l _ v 5 ’ , ’ l a s t d i r ’ , f i l e p a r t s ( f i l ename ) ) ;
102 end
103
104 % 5 − BUILD DATA STRUCTURE
105 % 5.1 − F i l e in format ion
106 data . x l s = f i l ename ;
107 data . boo t s e t t i n g s = B;
108 data . name = name ;
109 data . desc = desc ;
110 data . time = dat e s t r (now) ;
111
112 % 5.2 − Model eva luat i on
113 data .T = [ ] ; data .Q = [ ] ;
114 data . snw = [ ] ; data . atm = [ ] ; data . const = [ ] ;
115
116 % 5.3 − Bootstrap r e s u l t s
117 data . Tboot = [ ] ; data . Qboot = [ ] ;
118 data . Sboot = [ ] ; data . Aboot = [ ] ; data . Cboot = [ ] ;
C.5.2 xls input.m
1 func t i on [ S ,A,C,E] = x l s i npu t ( f i l ename )
2 % XLS INPUT bu i l d s input matr ics f o r usage with the thermal model ( v5 ) .
3 %
4 % SYNTAX:
5 % [ S ,A,C,E] = x l s i npu t ( f i l ename )
6 %
7
8 % 1 − CHECK FILE
9 i f narg in == 0 ; f i l ename = ’ i n p u t / W e t S n o w / b a s e . x l s x ’ ; end
10 i f ˜ e x i s t ( f i l ename , ’ f i l e ’ ) ;
11 e r r o r d l g ( ’ F i l e  d o e s  not  e x i s t ! ’ ) ; r e turn ;
12 end ;
13
14 % 2 − EXTRACT DATA FROM FILE
15 % 2.1 − Read f i l e s
16 [ S , snwTXT] = x l s r ead ( f i l ename , ’ S n o w P r o p e r t i e s ’ ) ;
17 [A,atmTXT] = x l s r ead ( f i l ename , ’ A t m o s p h e r i c S e t t i n g s ’ ) ;
18 [ const ] = x l s r ead ( f i l ename , ’ C o n s t a n t s ’ ) ;
19
20 % 2.2 − Seperate constants and mu l t i p l i e r s
21 C = const ( 1 : 1 0 ) ;
22 M = const ( 1 1 : l ength ( const ) ,1 ) ; M( isnan (M) ) = 0 ;
23 aM = M(1 : 1 0 ) ; % atmospher ic mu l t i p l i e r s
24 sM = M(11 : l ength (M) ) ; % snow mu l t i p l i e r s
25
26 % 2.3 − Seperate percent e r r o r va lues
27 Nc = s i z e ( const , 2 ) ;
28 i f Nc == 1 ;
29 E. atm = zero s ( l ength (aM) ,1) ; E . atm ( : , 1 ) = 0 . 0 5 ;
30 E. snow = ze ro s ( l ength (sM) ,1) ; E . snw ( : , 1 ) = 0 . 0 5 ;
31 E. const = ze ro s (10 ,1 ) ;
32 e l s e
33 E. atm = const ( 11 : l ength (aM)+10 ,2) /100 ;
34 E. snow = const ( l ength (aM)+11: l ength ( const ) ,2 ) /100 ;
35 E. const = const ( 1 : 1 0 , 2 ) /100 ;
36 end
37
360
38 % 3 − APPLY SPECIAL VALUES TO ALBEDO AND EXTICTION COLUMNS
39 A = albedo (A,atmTXT, S) ;
40 S = ex t i n c t i on (S , snwTXT) ;
41 S = dens i ty (S , snwTXT) ;
42
43 % 4 − APPLY MULTIPLERS
44 % 4.1 − Re−s i z e mu l t i p l i e r s a r rays to nece s sa ry s i z e
45 aM = [ 1 ;aM(1 : s i z e (A, 2 )−1) ] ; % 1 adds a column f o r time
46 sM = [ 1 ; sM( 1 : s i z e (S , 2 )−1) ] ; % 1 adds a column f o r the depth
47
48 % 4.2 − Apply mu l t i p l i e r s
49 f o r i = 1 : l ength (sM) ; S ( : , i ) = S ( : , i ) * sM( i ) ; end
50 f o r i = 1 : l ength (aM) ; A( : , i ) = A( : , i ) * aM( i ) ; end
51
52 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
53 func t i on A = albedo (A,atmTXT, S)
54 % ALBEDO app l i e s s p e c i a l input in to albedo column : dXX, classX , <type>
55 % Spec i a l va lues g iven in the albedo column (#4) assume that the shortwave
56 % column (3) i s an a l l−wave value , so i t i s d iv ided in to a VIS/NIR
57 % components as i s the albedo f o r a l l ” s p e c i a l ” ca s e s
58
59 % 1 − Determine ” s p e c i a l ” l o c a t i o n s
60 idx = f ind ( i snan (A( : , 4 ) ) ) ;
61
62 % 2 − Cycle through each s p e c i a l va lue and compute de s i r ed a lbedos
63 f o r i = 1 : l ength ( idx ) ;
64 va l = atmTXT{ idx ( i ) +3 ,4} ; % Current s p e c i a l case
65
66 % Opt ica l depth case : dXX
67 i f strcmpi ( ’ d ’ , va l (1 ) ) ; % Opt ica l depth caer
68 dopt = st r2doub l e ( va l ( 2 : l ength ( va l ) ) ) ;
69 i f i snan ( dopt ) ;
70 e r r o r ( ’ x l s _ i n p u t : a l b e d o ’ , ’ o p t i c a l  d e p t h  ill  d e f i n e . ’ ) ;
71 end
72 [A( idx ( i ) ,4 ) , b1 ,A( idx ( i ) ,11) ] = rad ca l c ( dopt , S (1 , 2 ) ) ;
73
74 % Class case : c lassX
75 e l s e i f l ength ( va l ) > 5 && strcmpi ( ’ c l a s s ’ , va l ( 1 : 5 ) ) ;
76 c l s = st r2doub l e ( va l ( 6 : l ength ( va l ) ) ) ;
77 i f i snan ( c l s ) ;
78 e r r o r ( ’ x l s _ i n p u t : a l b e d o ’ , ’ c l a s s  ill  d e f i n e . ’ ) ;
79 end
80 [A( idx ( i ) ,4 ) , b1 ,A( idx ( i ) ,11) ] = rad ca l c ( ’ c l a s s ’ , c l s ) ;
81
82 % Cuvre case : ’ f i n e ’ , ’ medium ’ , ’ coarse ’
83 e l s e i f sum( strcmpi ( val ,{ ’ f i n e ’ , ’ m e d i u m ’ , ’ c o a r s e ’}) ) == 1 ;
84 [A( idx ( i ) ,4 ) ,A( idx ( i ) ,11) ] = rad ca l c ( va l ) ;
85
86 % Record an e r r o r
87 e l s e
88 e r r o r ( ’ x l s _ i n p u t : a l b e d o ’ , ’ e r r o r  w i t h  a l b e d o  input ,  c o l u m  4! ’ ) ;
89 end
90
91 % Red i f ine a l l−wave shortwave to VIS/NIR components
92 [A( idx ( i ) ,3 ) ,A( idx ( i ) ,10) ] = rad ca l c (A( idx ( i ) ,3 ) ) ;
93 end
94
95 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
96 func t i on S = ex t i n c t i on (S , snwTXT)
97 % EXTINCTION app l i e s s p e c i a l input f o r e x t i n c t i on column : dXX or c lassX
98 % Spec i a l va lues g iven in the ex t e c t i on column (#6) overwr i t e VIS/NIR
99 % columns with the de s i r ed numeric value
100
101 % 1 − Determine ” s p e c i a l ” l o c a t i o n s
102 i f s i z e (S , 2 ) == 5 ; S ( : , 6 ) = NaN( s i z e (S , 1 ) ,1 ) ; end
103 idx = f ind ( i snan (S ( : , 6 ) ) ) ;
104
105 % 2 − Cycle through each s p e c i a l va lue and compute de s i r ed a lbedos
106 f o r i = 1 : l ength ( idx ) ;
107 va l = snwTXT{ idx ( i ) +3 ,6} ; % Current s p e c i a l case
108
109 % Opt ica l depth case : dXX
110 i f strcmpi ( ’ d ’ , va l (1 ) ) ; % Opt ica l depth caer
111 dopt = st r2doub l e ( va l ( 2 : l ength ( va l ) ) ) ;
112 i f i snan ( dopt ) ;
113 e r r o r ( ’ x l s _ i n p u t : e x t i n c t i o n ’ , ’ o p t i c a l  d e p t h  ill  d e f i n e . ’ ) ;
114 end
115 [ a1 , S( idx ( i ) ,6 ) , a2 , S( idx ( i ) ,7 ) ] = rad ca l c ( dopt , S (1 , 2 ) ) ;
116
117 % Class case : c lassX
118 e l s e i f l ength ( va l ) > 5 && strcmpi ( ’ c l a s s ’ , va l ( 1 : 5 ) ) ;
119 c l s = st r2doub l e ( va l ( 6 : l ength ( va l ) ) ) ;
120 i f i snan ( c l s ) ;
121 e r r o r ( ’ x l s _ i n p u t : e x t i n c t i o n ’ , ’ c l a s s  ill  d e f i n e . ’ ) ;
122 end
123 [ a1 , S( idx ( i ) ,6 ) , a2 , S( idx ( i ) ,7 ) ] = rad ca l c ( ’ c l a s s ’ , c l s ) ;
124
361
125 % Record an e r r o r
126 e l s e
127 e r r o r ( ’ x l s _ i n p u t : a l b e d o ’ , ’ e r r o r  w i t h  a l b e d o  input ,  c o l u m  4! ’ ) ;
128 end
129 end
130
131 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
132 func t i on S = dens i ty (S , snwTXT)
133 % DENSITY app l i e s s p e c i a l input f o r dens i ty and/or thermal conduct iv i ty
134 % columns , e i t h e r can be ’ auto ’ , j u s t not both . The auto va lues are
135 % rep laced i t the appopr iate value from Sturm , 1997 . The quadrat i c i s used
136 % fo r s o l v i ng f o r k and the exponent ia l when so l v i ng f o r dens i ty
137
138 f o r i = 1 : s i z e (S , 1 ) ;
139 rho = S( i , 2 ) /1000; k = S( i , 3 ) ;
140 i f i snumer ic ( rho ) && isnumer ic ( k ) ; % Both s p e c i f i e d
141 S( i , 2 ) = rho *1000; S( i , 3 ) = k ;
142 e l s e i f i snan ( rho ) && isnumer ic ( k ) ; % Compute dens i ty
143 S( i , 2 ) = ( log10 (k ) + 1 .652) / 2 .65 * 1000 ;
144 e l s e i f i snumer ic ( rho ) && isnan (k ) ; % Compute k
145 i f rho < 0 . 1 56 ;
146 S( i , 3 ) = 0.023 + 0.234 * rho ;
147 e l s e
148 S( i , 3 ) = 0.138 − 1 .01* rho + 3.233 * rho ˆ2 ;
149 end
150 e l s e
151 e r r o r ( ’ x l s _ i n p u t : d e n s i t y ’ , . . .
152 ’ e r r o r  w i t h  d e n s i t y / c o n d u c t i v i t y  input ,  c o l u m n  2  and / or  3! ’ ) ;
153 end
154 end
C.5.3 xls prep.m
1 func t i on [ s , a ] = x l s p r ep ( snow , atm , constants )
2 % XLS PREP bu i l d s ar rays f o r input ing in to thermal model
3 %
4 % SYNTAX:
5 % [ snow , atm ] = prep input ( snow , atm , constants ) ;
6 %
7 % INPUT:
8 % snow = matrix conta in ing snow data
9 % atm = matrix conta in ing atmospheric data
10 % constants = matrix conta in ing model constants
11 %
12 % EXAMPLE INPUT:
13 % snow = [50 ,130 ,0 .06 ,2030 , −10 ,70 ] ;
14 % atm = [6 ,240 ,500 ,0 .82 ,1 .7 , −10 , . 2 , −10 ,101 ] ;
15 % contants = [2833 , 0 . 0023 , 0 . 0023 , 0 . 622 , 0 . 462 , −5 , 0 . 402 , 0 . 95 , 1 , 60 , 1 ] ;
16 %
17
18 % 1 − F i l l in atmospheric data
19 atm ( : , 1 ) = atm ( : , 1 ) .* 3600 ; % Convert time to seconds
20 dt = constants (10) ; % Time step in seconds
21 a = f i l l a r r a y (atm , dt ) ;
22
23 % 2 − F i l l and snow p r op e r t i e s data
24 dz = constants (9 ) ;
25 s = f i l l a r r a y ( snow , dz ) ;
26
27 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
28 % SUBFUCTION: f i l l a r r a y
29 func t i on out = f i l l a r r a y ( in , i n t )
30 % FILL ARRAY bu i l d s an array from ” in ” us ing the i n t e r v a l in ” i n t ” based on
31 % the f i r s t column of data
32
33 % 1 − Build array f o r case when data i s only a s i n g l e row ( constant data )
34 l en = s i z e ( in , 1 ) ;
35 i f l en == 1 ;
36 in ( 2 , : ) = in ( 1 , : ) ;
37 in (1 , 1 ) = 0 ;
38 end
39
40 % 2 − Build new array with spac ing based on ” in t ”
41 % 2.1 − Build the f i r s t column of the new array ( e . g . time s t ep s )
42 n = s i z e ( in , 1 ) ;
43 x i = ( in (1 , 1 ) : i n t : in (n , 1 ) ) ’ ;
44
45 % 2.2 − I n t e r p o l a t e the remaining data based on the f i r s t column
46 x = in ( : , 1 ) ;
362
47 Y = in ( : , 2 : s i z e ( in , 2 ) ) ;
48 y i = in t e rp1 (x ,Y, xi , ’ l i n e a r ’ ) ;
49 out = [ xi , y i ] ;
C.5.4 thermal.m
1 func t i on [T,Q] = thermal ( snow , atm ,C)
2 % THERMAL executes 1−D heat equat ion based thermal model
3 %
4 % SYNTAX:
5 % [T,Q] = thermal ( snow , atm ,C)
6 %
7 % DESCRIPTION:
8 % [T,Q] = thermal ( snow , atm ,C) based on the in format ion provided in the
9 % numeric ar rays conta in ing snow p r op e r t i e s ( snow ) , atmospheric
10 % cond i t i on s (atm) , and model constants (C) a 1−D thermal ana l y s i s i s
11 % performed r e s u l t i n g in the snowpack temperatures (T) and a s s o c i a t ed
12 % heat f l u x e s (Q)
13 %
14
15 % 1 − PREPARE VARIABLES FOR CALCULATION
16 % 1.1 − Pre−de f i n e ar rays
17 i f s i z e (atm , 2 ) == 11 && s i z e ( snow , 2 ) == 7 ;
18 ndim = 2 ;
19 e l s e
20 ndim = 1 ;
21 end
22
23 nt = s i z e (atm , 1 ) ; % Number o f time s t ep s
24 ns = s i z e ( snow , 1 ) ; % Number o f snow elements
25
26 T = zero s ( ns , nt ) ; % Temperature array
27 q = ze ro s ( ns , nt , ndim) ; % Short−wave f l ux absorbed array
28 qs = ze ro s ( nt , 3 ) ; % Sur face f l ux array
29
30 A = zero s ( ns+1,ns+1) ; % A−matrix f o r temperature s o l u t i on
31 b = ze ro s ( ns+1 ,1) ; % b−vector f o r temperature s o l u t i on
32
33 % 1.2 − Estab l i sh user s p e c i f i e d constants
34 Ls = C(1) ;
35 Ke = C(2) ;
36 Kh = C(3) ;
37 MvMa = C(4) ;
38 Rv = C(5) ;
39 T0 = C(6) + 273 . 15 ;
40 e0 = C(7) ;
41 emis = C(8) ;
42 dz = C(9) /100 ;
43 dt = C(10) ;
44
45 % 1.3 − Def ine add i t i ona l constants needed
46 sb = 5.6696*10ˆ(−8) ; % Ste fan Bol tzmann constant (W/mˆ2/Kˆ4)
47 R = 0 . 2 87 ; % Gas constant f o r a i r ( kJ/kg/K)
48
49 % 1.4 − Compute the p r op e r t i e s o f a i r
50 Cp air = 1003; % Sp e c i f i c heat @−5C (J/kg/K)
51 r h o a i r = atm ( : , 9 ) . / (R*(atm ( : , 6 ) + 273 .15) ) ; % Density ( kg/mˆ2)
52
53 % 2 − INITILIZE ARRAYS FOR COMPUTATION
54 % 2.1 − I n i t i l i z e temperature array
55 T( : , 1 ) = snow ( : , 5 ) ; % I n i t i a l snow temperature
56 T( ns +1 , :) = atm ( : , 8 ) ; % Base
57
58 % 2.2 − General Matrix c o e f f i c i e n t s
59 Ca = snow ( : , 3 ) . / dz ˆ2 ; % a
60 Cb = ( snow ( : , 2 ) .* snow ( : , 4 ) ) . / dt ; % b
61 Cc = Cb + Ca ; % c
62 Cd = Cb − Ca ; % d
63
64 % 3 − BEGIN COMPUTING FOR EACH TIME STEP ( time step = index ” j ”)
65 f o r j = 2 : nt
66 % 3.1 − Estab l i sh a i r /snow su r f a c e temperatures
67 Ta = atm( j , 6 ) + 273 . 15 ;
68 Ts = T(1 , j−1) + 273 . 15 ;
69
70 % 3.2 − Compute longwave heat f l ux
71 qs ( j , 1 ) = atm( j , 2 ) − emis* sb*Tsˆ4 ;
72
73 % 3.3 − Compute the l a t en t heat f l ux
363
74 ea = e0*exp ( Ls/Rv *(1/T0 − 1/Ta) ) *atm( j , 7 ) /100 ;
75 es = e0*exp ( Ls/Rv *(1/T0 − 1/Ts) ) ;
76 qs ( j , 2 ) = 1000*MvMa* r h o a i r ( j ) *Ls*Ke*atm( j , 5 ) *( ea−es ) /atm( j , 9 ) ;
77
78 % 3.4 − Compute the s e n s i b l e heat f l ux
79 qs ( j , 3 ) = Kh* r h o a i r ( j ) *Cp air *atm( j , 5 ) *(Ta − Ts) ;
80
81 % 3.5 − Compute the absorbed shortwave and bu i ld s o l u t i on matrix
82 % f o r each l ay e r o f snow
83 % 3 . 5 . 1 − Compute shortwave absorbed in the top l ay e r
84 q (1 , j , 1 ) = atm( j , 3 ) *(1−atm( j , 4 ) )*(1−exp(−snow (1 ,6 ) *dz ) ) ;
85
86 % 3 . 5 . 2 − Compute shortave in NIR i f pre sent
87 i f ndim == 2 ;
88 q (1 , j , 2 ) = atm( j , 1 0 ) *(1−atm( j , 1 1 ) )*(1−exp(−snow (1 ,7 ) *dz ) ) ;
89 end
90
91 % 3 . 5 . 2 − Compute shortwave absorbed f o r lower l a y e r s and bu i ld
92 % so l u t i on matr i ces
93 f o r i = 2 : ns
94 % Short−wave r ad i a t i on absorbed
95 q ( i , j , 1 ) = q( i −1, j , 1 ) *exp(−snow ( i , 6 ) *dz ) ; % a l l−wave or VIS
96 i f ndim == 2 ;
97 q ( i , j , 2 ) = q( i −1, j , 2 ) *exp(−snow ( i , 7 ) *dz ) ; % NIR
98 end
99
100 % So lut i on matr i ces
101 A( i , i −1) = −Ca( i ) /2 ;
102 A( i , i ) = Cc( i ) ;
103 A( i , i +1) = −Ca( i ) /2 ;
104 b( i , 1 ) = Ca( i ) /2*T( i −1, j−1) + Cd( i ) *T( i , j−1) + . . .
105 Ca( i ) /2*T( i +1, j−1) + sum(q ( i , j , : ) ) /dz ;
106 end
107
108 % 3.6 − Compute the su r f a c e f l ux
109 s u r f l u x = sum( qs ( j , 1 : 3 ) ) ;
110
111 % 3.7 − I n s e r t matrix va lues f o r su r f a c e node ( i = 1)
112 A(1 ,1 ) = Cc(1) ;
113 A(1 ,2 ) = −Ca(1) ;
114 b (1) = Cd(1) *T(1 , j−1) + Ca(1) *T(2 , j−1) + 2* s u r f l u x /dz + . . .
115 sum(q (1 , j , : ) ) /dz ;
116
117 % 3.8 − I n s e r t matrix va lues f o r bottom boundary cond i t i on
118 A( ns+1,ns+1) = 1 ;
119 b( ns+1) = atm( j , 8 ) ;
120
121 % 3.9 − Calcu la te the new temperature p r o f i l e
122 Tnew = A\b ;
123 Tnew(Tnew>0) = 0 ;
124 T( : , j ) = Tnew ;
125 end ;
126
127 Q = ze ro s ( ns , nt , ndim+3) ;
128 Q( 1 , : , 1 : 3 ) = qs ;
129 Q( : , : , 4 : end ) = q ;
C.5.5 rad calc.m
1 func t i on varargout = rad ca l c ( vararg in )
2 % RAD CALC sp e c t r a l c a l c u l a t i o n s o f rad ia t i on , albedo , and e x t i c t i o n .
3 %
4 % SYNTAX:
5 % [ SWvis , SWnir , SWswir ] = rad ca l c ( SWall ) ;
6 % [ Avis , Anir , Aswir ] = rad ca l c ( curve ) ;
7 % [ Avis , Bvis , Anir , Bvis , Aswir , Bswir ] = rad ca l c ( dopt , rho ) ;
8 % [ Avis , Bvis , Anir , Bvis , Aswir , Bswir ] = rad ca l c ( ’ c l a s s ’ ,num) ;
9 %
10 % DESCRIPTION:
11 % [ SWvis , SWnir , SWswir ] = rad ca l c ( SWall ) computes s p e c t r a l components o f
12 % a l l−wave shortwave r ad i a t i on based on ASTM standard .
13 % [ Avis , Anir , Aswir ] = rad ca l c ( curve ) computes s p e c t r a l a lbedo components
14 % based on curves : ’ f i n e ’ , ’ medium ’ , ’ coarse ’
15 % [ Avis , Bvis , Anir , Bvis , Aswir , Bswir ] = rad ca l c ( dopt , rho ) computes
16 % sp e c t r a l components o f albedo and e x t i c t i o n based on Snow & Climate
17 % equat ions given on p . 5 6 .
18 % [ Avis , Bvis , Anir , Bvis , Aswir , Bswir ] = rad ca l c ( ’ c l a s s ’ ,num) computes
19 % sp e c t r a l components o f albedo and e x t i c t i o n based on Snow & Climate
20 % tab l e g iven on p . 57 , where num must be an i n t e g e r between 1 and 6 .
364
21 %
22
23 % 1 − Compute de s i r ed values , execute as order in SYNTAX/DESCRIPTION above
24 i f narg in == 1 && isnumer ic ( vararg in {1}) ;
25 output = shortwave ( vararg in {1}) ;
26 e l s e i f narg in == 1 && i s cha r ( vararg in {1}) ;
27 output = albedo curve ( vararg in {1}) ;
28 e l s e i f narg in == 2 && isnumer ic ( vararg in {1}) ;
29 output = albedo eqn ( vararg in { :} ) ;
30 e l s e i f narg in == 2 && i s cha r ( vararg in {1}) ;
31 output = a lbedo tab l e ( vararg in {2}) ;
32 end
33
34 % 2 − Produce output
35 varargout = num2cell ( output ) ;
36
37 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
38 func t i on out = a lbedo tab l e (N)
39 % ALBEDO TABLE computes albedo and ex c i c t i o n base on Snow&Climate (p . 5 7 )
40
41 % Error handl ing
42 i f N < 1 | | N > 6 ;
43 e r r o r ( ’ C l a s s  m u s t  be  an  i n t e r g e r  1  t h r o u g h  6! ’ ) ; out = NaN; return ;
44 end
45
46 % Build Table 2 .6 from Snow & Climate (2008) , p .57
47 C( : , 1 ) = [94 , 94 , 93 , 93 , 92 , 91 ] /100 ;
48 C(1 : 6 , 2 ) = 40 ;
49 C( : , 3 ) = [80 , 73 , 68 , 64 , 57 , 42 ] /100 ;
50 C( : , 4 ) = [110 , 136 , 190 , 110 , 112 , 127 ] ;
51 C( : , 5 ) = [59 , 49 , 42 , 37 , 30 , 18 ] /100 ;
52 C(1 : 6 , 6 ) = i n f ;
53
54 % Produce output
55 out = C(N, : ) ;
56
57 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
58 func t i on out = albedo eqn ( dopt , rho )
59 % ALBEDO EQN computes albedo and ex c i c t i o n base on Snow&Climate (p . 5 6 )
60
61 % Convert un i t s ( dopt mm−>m; rho kg/mˆ 3−>gm/cmˆ3)
62 dopt = dopt /1000; rho = rho /1000;
63
64 % VIS
65 out (1) = min ( 0 . 9 4 , 0 . 9 6 − 1 .58* sq r t ( dopt ) ) ;
66 out (2) = max(0 . 04 , 0 .0192* rho/ sq r t ( dopt ) ) *100 ;
67
68 % NIR
69 out (3) = 0.95 − 15 .4 * sq r t ( dopt ) ;
70 out (4) = max(1 , 0 .1098* rho/ sq r t ( dopt ) ) *100 ;
71
72 % SWIR
73 out (5) = 0.88 + 346.6* dopt − 32.31* sq r t ( dopt ) ;
74 out (6) = i n f ;
75
76 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
77 func t i on Aout = albedo curve ( curve )
78 % ALBEDOCURVE computes VIS ,NIR,& SWIR albedos based on input curve
79
80 % 1 − Load the de s i r ed curve
81 X = load ( ’ a l b e d o . mat ’ ) ;
82 A = X. ( curve ) ;
83
84 % 2 − Parse out the albedo f o r each wavelength group
85 L = [285 , 800 ; 800 ,1500 ; 1500 ,3500 ] ;
86 f o r i = 1 : s i z e (L , 1 ) ;
87 idx (1) = f i nd (A( : , 1 )>=L( i , 1 ) ,1 , ’ f i r s t ’ ) ;
88 idx (2) = f i nd (A( : , 1 )<=L( i , 2 ) ,1 , ’ l a s t ’ ) ;
89 Aout ( i ) = mean(A( idx (1) : idx (2) ,2) ) /100 ;
90 end
91
92 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
93 func t i on SWout = shortwave ( SWall )
94 % SHORTWAVE computes s p e c t r a l components o f a l l−wave based on ASTM standard
95
96 % 1 − Load the s o l a r spectrum de s i r ed
97 X = load ( ’ a l b e d o . mat ’ ) ;
98 S = X. astm ;
99
100 % 2 − Normalize s o l a r spectrum to inputed SW data
101 I = i n s o l a t i o n (S , [ 2 8 5 , 3 5 0 0 ] ) ;
102 S ( : , 2 ) = (S ( : , 2 ) / I ) *SWall ;
103
104 % 3 − Parse out wavelength groups
105 L = [285 , 800 ; 800 ,1500 ; 1500 ,3500 ] ;
106 f o r i = 1 : s i z e (L , 1 ) ; SWout( i ) = i n s o l a t i o n (S ,L( i , : ) ) ; end
107
365
108 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
109 func t i on I = i n s o l a t i o n (S ,L)
110 % POWER computers the i n s o l a t i o n between the a and b wavelenghts
111
112 % 1 − Locate i n d i c i e s o f wavelenghts
113 x = S ( : , 1 ) ; y = S ( : , 2 ) ;
114 i (1 ) = f ind (x>L(1) ,1 , ’ f i r s t ’ ) ;
115 i (2 ) = f ind (x<L(2) ,1 , ’ l a s t ’ ) ;
116 idx = i (1) : i (2 ) ;
117
118 % 2 − Compute the i n s o l a t i o n
119 I = sum(( y ( i (1 ) : i (2 )−1)+d i f f ( y ( idx ) ) ) .* d i f f ( x ( idx ) ) ) ;
C.5.6 confint.m
1 func t i on data = con f i n t ( f i l ename ,B, n)
2 % CONFINT computes the con f idence i n t e r v a l s f o r temp p r o f i l e s
3
4 % Read f i l e input
5 [ S ,A,C,E] = x l s i npu t ( f i l ename ) ;
6
7 % Compute the ac tua l temperature p r o f i l e
8 [ Sa ,Aa ] = x l s p r ep (S ,A,C) ;
9 T = thermal (Sa ,Aa ,C) ;
10
11 % Compute the standard dev i a t i on va lues
12 s = get s td (S ,E. snow , n , 1 ) ; % standard deva i t i on f o r snow p r op e r t i e s
13 a = get s td (A,E. atm , n , 1 ) ; % standard deva i t i on f o r atmospheric terms
14 c = get s td (C,E. const , n , 0 ) ; % standard dev i a t i on f o r constants
15
16 % Compute the Monte Carlo r e p l i c a t e s
17 data . Tboot = ze ro s ( [ s i z e (T) ,B ] ) ; % I n i t i l i z e s to rage array
18 h = waitbar (0 , ’ P l e a s e  w a i t ... ’ ) ;
19 f o r i = 1 :B;
20 r = rand (1) ;
21 S b = norminv ( r , S , abs ( s ) ) ; % Re−sample snow
22 S b ( : , 1 ) = S ( : , 1 ) ; % Snow depth does not change
23 S b ( i snan ( S b ) ) = S( i snan ( S b ) ) ;
24 A b = norminv ( r ,A, abs ( a ) ) ; % Re−sample atmosphere
25 A b ( : , 1 ) = A( : , 1 ) ; % Duration does not change
26 A b( isnan ( S b ) ) = A( isnan (A b) ) ;
27 C b = norminv ( r ,C, abs ( c ) ) ; % Constant resampl ing
28 C b (9) = C(9) ; % dz constant
29 C b (10) = C(10) ; % dt constant
30 C b ( isnan (C b ) ) = C( isnan (C b ) ) ;
31
32 [ SS ,AA] = x l s p r ep ( S b , A b , C b ) ; % Build input f o r eva luat i on
33 [ data . Tboot ( : , : , i ) , data . Qboot ( : , : , : , i ) ] = thermal (SS ,AA, C b ) ;
34 data . Sboot ( : , : , i ) = SS ;
35 data . Aboot ( : , : , i ) = AA;
36 data . Cboot ( : , : , i ) = C b ;
37 waitbar ( i /B, h) ;
38 end
39 c l o s e (h) ;
40
41 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
42 func t i on s = get s td (S ,E, n , o f f s e t )
43 % GETSTD returns the standard dev i a t i on o f the input items
44 i f o f f s e t == 1 ;
45 s ( : , 1 ) = S ( : , 1 ) ;
46 f o r i = 2 : s i z e (S , 2 ) ;
47 s ( : , i ) = S ( : , i ) .*E( i−o f f s e t ) /n ;
48 end
49 e l s e i f o f f s e t == 0 ;
50 f o r i = 1 : s i z e (S , 2 ) ;
51 s ( : , i ) = S ( : , i ) .*E( i ) /n ;
52 end
53 end
366
APPENDIX D
SENSITIVITY ANALYSIS SOFTWARE USER MANUAL
367
D.1 Introduction
The information presented here details the usage of sensitivity analysis software
developed to perform the analysis presented in Chapter 8 and 9. The software was
develop in MATLAB (The Mathworks, Inc.) based on the theory presented in Chap-
ter 6. The information presented here assumes the user is familiar with the basic
operation of the MATLAB language. As detailed in Chapter 6 the SOBOL method
is applicable to any function that utilizes discrete input and output. Therefore, the
general application of the software is presented followed by the specific application
to the thermal model used throughout this dissertation. Finally, the source code is
provided. This appendix follows the notational conventions outlined in Appendix C.
The information presented here only details the computation of the sensitivity
indices. However, in some respect the storage and visualization of this data is the
most difficult task. A variety of visualization tools were developed that yield the
various graphs and charts presented in Chapter 8 and 9. Including the instructions
for usage of these tools and/or providing the source code for these files was assumed
to be unnecessary. However, if this information is desired please contact the author.
D.2 General Application
D.2.1 Sensitivity Analysis Program
Inputs: The main function that executes the sensitivity analysis is the sobol.m
function (Section D.5.1). This function only has a single mandatory input, func,
which is a string or cell array such that it may be evaluated as defined in Section D.2.2.
The func variable may also be a data structure that contains the ~a output vectors
(see 6.4.2, Equation (6.32)). These vectors are computed with the sobolvec.mat
368
function. Section D.2.1.2 provides additional information on the function and Section
D.2.3 explains the and data structure. This feature allows the model evaluations to
be separated from the computation of sensitivity indices.
An additional five optional inputs may be specified, the syntax for which may be
retrieved by executing the help from the MATLAB command-line as:
>> help sobo l
SOBOL performs complete SOBOL s e n s i t i v i t y a n a l y s i s .
SYNTAX:
r = sobo l ( func ) ;
r = sobo l ( func , o u t f i l e ) ;
r = sobo l ( . . . ,K) ;
r = sobo l ( . . . , K,B) ;
r = sboo l ( . . . , K,B, ’ o f f ’ ) ;
[ r , o u t f i l e ] = sobo l ( . . . ) ;
A string may be specified in the variable outfile that contains the name of a
*.mat file where the output structure will be saved. The outfile is optional in all
cases and if an empty string is supplied the user will be prompted to select a file and
location to store the results. For this scenario, the outfile is the second output from
the sobol.m function. The default value for outfile is NaN, which does not produce
any output file, only the output structure r is returned to the command window.
The variable K specifies the number of Monte Carlo simulations to preform, i.e.,
the size for the input sample and re-sample matrices from which the desired function
evaluations will be computed (see Section 6.4.2). The default for K is 1,024. Similarly,
B specifies the number of bootstrap replicates to be performed when computing the
confidence level intervals (see Section 6.5). Specifying an empty numeric array skips
the computation of the confidence levels. The default value for B is 10,000. The final
optional input is a toggle that allows the user to turn ‘off’ the progress bar.
Program Flow: The primary sensitivity analysis function, sobol.m, relies on
several sub functions. A flow chart demonstrating the connection between the various
369
functions is provided in Figure D.1. A brief description of each function follows and
the complete source code is provided in Section D.5.
ˆ sobol.m: The main program of the sensitivity analysis software, which handles
the inputs; executes the model evaluations, sensitivity index computation, and
confidence interval calculations; and outputs the data structure and output file.
ˆ sobolvec.m: Performs the model evaluations according to the improved
SOBOL method detailed in Section 6.4.2.
ˆ sobolidx.m: Computes the total-effect as well as the first-, second-, and higher
order sensitivity indices via the methodology described in Section 6.4.2.
ˆ sobolci2.m: Calculates the confidence level intervals using the BCa bootstrap
methods detailed in Section 6.5.
ˆ sobol firstorder.m: Implements Equation (6.38) that computes the first-
order sensitivity index.
ˆ sobol secondorder.m: Implements Equation (6.39) that computes the second-
order sensitivity index.
ˆ sobol totaleffect.m: Implements Equation (6.40) that computes the total-
effect sensitivity index.
D.2.2 Input Function
To perform a sensitivity analysis a function that executes the desired calculation
must be written that is compatible with the SOBOL software package. This function
is passed to the sobol.m function via the func variable. This function should be
executable via the MATLAB command-line as either a string or cell array as:
370
 
so b o l . m  
so b o l v ec . m  so b o li d x . m  so b o lci2.m  
so b o l _first o rd er. m  
so b o l _se c o ndo rd er.
m  
so b o l _t o taleff ect . m  
Figure D.1: Flow chart of the main sensitivity analysis function sobol.m and associ-
ated sub functions.
>> [Y, x ] = f e v a l ( func , sk ) ; % Evaluat ion i f func i s o f c l a s s ‘ char ’
>> [Y, x ] = f e v a l ( func { :} , sk ) ; % Evaluat ion i f func i s o f c l a s s ‘ c e l l ’
The variable sk is defined and implemented during the execution of func in
sobolvec.m. When sk is an empty numeric, the evaluation of func must return
either a scalar that is equivalent the the number of input parameters (n; Section 6.2)
or a cell array of strings containing the variable names such that the length of this
cell array is equal to the number of input parameters. In the later case, the cell
array of strings is added to the output structure. The scalar or cell array should be
exported via the Y variable.
Using the number of input parameters determined by evaluating func with an
empty array for sk, in subsequent evaluations of func the variable sk is a numeric
array containing the Monte Carlo samplings of the inputs parameters such that the
size of sk is K × n (i.e., the W , W
′
, Ni, and N−i matrices defined in Section 6.4.2).
However, sobolvec.m develops these matrices as uniformly distributed values from
0 to 1. From this data func should convert these values to the desired distributions
(e.g., using MATLAB norminv function).
371
Figure D.2 is a simple example of an appropriate function for usage with sobol.m.
This function evaluates the simple equation presented in Fang et al. (2003). The g-
function utilized in Section 6.6 is also presented in Figure D.2 as another example
of the usage of sensitivity analysis software. Notice, the saG.m function contains a
second output not discussed in Chapter 6, but included here as an example.
1 f unc t i on [Y, x ] = saFANG( sk )
2 % SAFANG i s the func t i on used by Fang et a l . (2003) .
3
4 i f isempty ( sk ) Y = 5 ; x = [ ] ; r e turn ; end
5
6 % Input v a r i a b l e s
7 x ( : , 1 ) = expinv ( sk ( : , 1 ) , 0 . 5 ) ;
8 x ( : , 2 ) = wblinv ( sk ( : , 2 ) , 1 . 5 , 3 ) ;
9 x ( : , 3 ) = norminv ( sk ( : , 3 ) ,0 , s q r t ( 0 . 2 5 ) ) ;
10 x ( : , 4 ) = beta inv ( sk ( : , 4 ) , 1 . 5 , 2 . 5 ) ;
11 x ( : , 5 ) = gaminv ( sk ( : , 5 ) , 3 . 5 , 0 . 5 ) ;
12
13 % Analyze func t i on
14 Y( : , 1 ) = sum(x , 2 ) ;
Figure D.2: MATLAB code for saFANG.m function.
1 f unc t i on [ y , a ] = saG( sk )
2 % SAG execute s the g−f unc t i on from Chapter 6
3
4 % Return the number v a r i a b l e s or cont inue opera t i on
5 i f isempty ( sk ) ; y = 6 ; a = [ ] ; r e turn ; end
6
7 % Perform the c a l c u l a t i o n s
8 a = [ 0 , 0 . 5 , 3 , 9 , 9 9 , 9 9 ] ;
9 g = ze ro s ( s i z e ( sk ) ) ;
10 f o r i = 1 : s i z e ( sk , 2 ) ;
11 g ( : , i ) = ( abs (4* sk ( : , i ) − 2) + a ( i ) ) /(1 + a ( i ) ) ;
12 end
13
14 % Export the d e s i r e d output ( s )
15 y ( : , 1 ) = prod ( g , 2 ) ;
16 y ( : , 2 ) = sum( g , 2 ) ;
Figure D.3: MATLAB code for saG.m function.
When evaluated with the numeric arrays for sk, func should also provide two
outputs: Y and x. The Y variable should be a K × m array of the function output,
which are the ~a output vectors used by sobolidx.m (see Equation (6.32)). The x
372
variable is a K × n numeric array (i.e., the W , W
′
, Ni, and N−i matrices defined
in Section 6.4.2). The following section summarizes the storage of this data in the
output data structure.
D.2.3 Output Structure
The output data structure is composed of the all sensitivity analysis results for a
single execution of a function via sobol.m, this function contains n inputs factors, m
outputs factors, and is composed of K Monte Carlo replicates (see Chapter 6). The
following MATLAB code is an example of a data structure produced by sobol.m for
the saG.m function shown in Figure D.3, where n = 6, m = 2, and K = 500.
373
>> r = sobo l ( ’ saG ’ , 1000 , 500 )
r =
a 0 : [1000 x2 double ]
a K : [1000 x2 double ]
a i : [ 1000 x6x2 double ]
a n i : [1000 x6x2 double ]
S : [ 6 x6x2 double ]
ST : [ 6 x2 double ]
STci1 : [ 6 x2 double ]
STci2 : [ 6 x2 double ]
Sc i1 : [ 6 x6x2 double ]
Sc i2 : [ 6 x6x2 double ]
Sb ias : [ 6 x6x2 double ]
STbias : [ 6 x2 double ]
boots t rap : 500
The first four entries contain the ~a output vectors defined in Equation (6.32) on
page 150. The r.S and r.ST contain the sensitivity indices for each input. The
structure componets that included a “ci” are the confidence level intervals, where the
“1” is the lower or 5% confidence intervals and the “2” is the upper or 95% interval.
For example, r.Sci1 includes the 5% confidence level for r.S. The r.Sbias and
r.STbias are the bootstrap computed bias estimates (see Section 6.5), thus the bias
adjusted sensitivity indices are computed by r.S + r.Sbias and r.ST + r.STbias.
Finally, r.bootstrap contains the number of bootstrap replicates performed.
D.3 Thermal Model Application
This section briefly details the execution of sensitivity analysis software with the
thermal model presented in Chapter 5. The information presented only details a
single execution of the thermal model sensitivity analysis, which can be computation-
ally expensive. As such, additional tools to automate the sensitivity analysis were
developed. However, these programs are excluded from this discussion. If such an
application is desired please contact the author.
The main function is saMODEL2.m, which calls a sub-function, saMODEL2 output.m.
Thus, for implementing with sobol.m, the func input, in it’s simplest form is defined
374
as: func = ‘saMODEL2’. In addition to the information presented in this section, help
regarding the usage of saMODEL2.m may be obtained via the MATLAB command-line
by typing help saMODEL2.m. A portion of this help is displayed in Figure D.4, which
displays the various syntax options for saMODEL2.m. It is possible to execute the
function without any inputs, which uses the default values for the optional inputs as
shall be detailed next. For the saMODEL2.m function to operate correctly, the thermal
model software presented in Appendix C must be in a separate directory on the same
level as the directory containing the saMODEL2.m functions; the directory containing
the thermal model must be named “ThermalModel v5”.
1 f unc t i on [Y, x , user , varargout ] = saMODEL2( vararg in )
2 % SAMODEL2 performs a n a l y s i s f o r SOBOL and FAST s e n s i t i i v t y a n a l y s i s
3 %
4 % SYNTAX:
5 % [Y, x ] = saMODEL2( sk )
6 % [Y, x ] = saMODEL2( input , sk )
7 % [Y, x ] = saMODEL2( input , ’ PropertyName ’ , PropertyValue , . . . , sk )
8 % [Y, x , user ] = saMODEL2 ( . . . )
9 % L = saMODEL2 ( . . . , [ ] ) , where sk = [ ]
Figure D.4: Syntax for implementation of saMODEL2.m.
D.3.1 Input Files
The first optional input of saMODEL2.m (Figure D.4) is the input variable, which
is the name of a *.mat file containing a data structure of the various model inputs.
The file must be defined as a complete path or the path must be added to MATLAB
via the addpath function. The default file is control2.mat. A complete example of
the required elements of this structure are provided in help of saMODEL2.m. The code
in Figure D.5 is one portion of the complete file that is presented here to illustrate
the usage of these input files.
The input structure is composed of four sub structures: snow, atm, constant,
and dirt. Within these structures a list of variables is defined, as shown in Figure
375
1 >> w = load ( ’control2.mat’ )
2 w =
3 snow : [ 1 x1 s t r u c t ]
4 atm : [ 1 x1 s t r u c t ]
5 constant : [ 1 x1 s t r u c t ]
6 d i r t : [ 1 x1 s t r u c t ]
7
8 >> w. snow
9 ans =
10 depth : 50
11 dens i ty : {’unif’ [ 5 0 ] [ 5 0 0 ]}
12 conduc t i v i ty : {’unif’ [ 0 . 0 1 ] [ 0 . 7 ] }
13 s p e c i f i c : {’unif’ [ 1 7 9 5 ] [ 2 1 1 5 ]}
14 snowtemp : {’unif’ [−40] [ 0 ] }
15 kappa : {’unif’ [ 4 0 ] [ 2 0 0 ]}
16 kappaNIR : 0
Figure D.5: MATLAB code demonstrating the definition of the input *.mat files for
the saMODEL2.m function.
D.5. Each variable is either a scalar of cell array. Scalar values are evaluated as con-
stants and not considered for the sensitivity analysis. The cell arrays items determine
the variables to consider for the sensitivity analysis, which include the name of the
statistical distribution to be executed using MATLAB’s available inverse distributions
functions, e.g., unifinv and norminv. The “inv” portion of the function should not
be included.
After the distribution is defined the distribution parameters are defined, in the
case of unifinv this is simply the upper and lower limits. If the function is a nor-
mal distribution the inputs should be the mean and standard deviation, see help
norminv. It is also possible to add two additional inputs that limit the distribution
to these values. For example, w.snow.density = {’norm’, 150, 50, 50, 300}
would sample from a normal distribution with a mean of 150 and standard deviation
of 50, but the resulting sample would be limited to values between 50 and 300. Each
of the inputs in the data structure correspond—including the required units—with
the inputs defined in Appendix C, which details the usage of the thermal model itself.
376
For the data presented in Chapters 8 and 9, six input files were used that were defined
based on the distributions presented in Table 7.3.
The input labels may also be defined in a separate file: label.lbl. This file has
the same exact structure as shown in Figure D.5, but contains strings that provide
the name of the variable as shown in the following code.
>> L = load ( ’ l a b e l . l b l ’ , ’−mat ’ )
L =
snow : [ 1 x1 s t r u c t ]
atm : [ 1 x1 s t r u c t ]
constant : [ 1 x1 s t r u c t ]
d i r t : [ 1 x1 s t r u c t ]
p r o f i l e : [ 1 x1 s t r u c t ]
>> L . snow
ans =
depth : ’Snow depth , $z$ [ $cm$ ] ’
d ens i ty : ’Snow dens i ty , $\ rho$ [ $kg/mˆ3$ ] ’
c onduc t i v i ty : ’ Thermal conduct iv i ty , $k$ [$W/(mK) $ ] ’
s p e c i f i c : ’ S p e c i f i c heat , $C p$ [ $kJ /(kgK) $ ] ’
snowtemp : ’ I n i t i a l snow temp , $T s ˆ{ i n t }$ [ $ˆ{\ c i r c }$C ] ’
kappa : ’ Ext inc t i on c o e f f i c i e n t , $\kappa$ [$mˆ{−1}$ ] ’
kappaNIR : ’NIR e x t i n c t i o n c o e f f i c i e n t , $\kappaˆ{NIR}$ [$mˆ{−1}$ ] ’
These labels are used when producing the various graphs and plots. When execut-
ing saMODEL2.m, the function sobolvec.m includes two additional outputs—legend
and input—in the output data structure defined in Section D.2.3. These additional
outputs contain a cell array of legend entries gathered from the label.lbl file and
the input structure used for the analysis, respectively.
D.3.2 Evaluation Options
The saMODEL2.m is configurable such that the thermal model may be explored
beyond what is presented in Chapters 8 and 9. The options must be entered in pairs,
as shown in the following example execution:
>> L = saMODEL2( ’ c on t r o l 2 . mat ’ , ’ prog ’ , ’ on ’ , ’ d i r t ’ , ’ on ’ , [ ] ) ;
>> sk = rand (100 , l ength (L ) ) ;
>> [Y, x ] = saMODEL2( ’ c on t r o l 2 . mat ’ , ’ prog ’ , ’ on ’ , ’ type ’ , ’TG’ , ’ subtype ’ , ’ mean ’ , sk ) ;
The following list details the various options—including the default values—
available in saMODEL2.m. The help for the function (i.e., >> help saMODEL2.m) also
includes a brief description of these options.
377
ˆ ‘type’: Specifies the type of output to use for the sensitivity analysis, the avail-
able types include the snow temperature (‘T’), temperature gradient (‘TG’;
default), the snow temperature at the “knee” location (‘Tknee’, see Section
9.2), “knee” temperature gradient (‘KTG’), the “knee” depth (‘Kdepth’), the
“knee” duration (‘Kduration’), and mass-flux at the snow surface (‘MF’).
ˆ ‘subtype’: Specifies the calculation to perform on the output type, options in-
clude the mean, minimum, and maximum of the data (‘mean’ (default), ‘min’,
and ‘max’, respectively) in addition to the time at which the minimum and
maximum occur (‘min time’ and ‘max time’, respectively). Also, the output
type may be summed (‘total’) or the output may be provided as a function
of time using ‘all’.
ˆ ‘depth’: Defines the depth (in cm) at which the temperatures or temperature
gradients are considered when ‘T’ or ‘TG’ are specified. The default is 5 cm
and this property is input as a numeric scalar.
ˆ ‘inc’: defines the storage increment, in minutes, when the ‘all’ subtype is
specified. The default is 20 minutes and this property is input as a numeric
scalar.
ˆ ‘day’: A toggle that may be either ‘on’ (default) or ‘off’ that when set to
on modifies the short-wave to act as sine wave, with the mean of the sine-wave
to be equal to the inputed value.
ˆ ‘dirt’: A toggle that may be either ‘on’ or ‘off’ (default) that adds a layer
of dirt to the snowpack using the settings specified in the dirt structure of the
input file (see Figure D.5).
378
ˆ ‘profile’: Allows the user to turn on a snow profile feature by specifying
either ‘on’ or ‘off’ (default). Referring to Figure D.5, the top is assigned
temperature given by w.snow.snowtemp and the bottom by w.atm.bottom.
A linear profile between the top and bottom is constructed based on a tem-
perature within the snowpack defined by w.profile.temp at a depth of
w.profile.depth.
ˆ ‘prog’: A toggle that is either ‘on’ or ‘off’ (default) that controls the pres-
ence of a progress message.
D.4 Closing Remarks
The information presented in this appendix presents information for individuals
interested in implementing the SOBOL method of sensitivity analysis. The software
written to perform the work presented throughout this dissertation was designed to be
flexible, as it is the desire of the author that others would perform further analysis on
other functions as well as the thermal model presented in Chapter 5 and Appendix C.
The sensitivity analysis software is capable of evaluating any function with discrete
input and output, as explained in Section D.2. Additionally, a powerful function
was developed for exploration of the aforementioned thermal model, as detailed in
Section D.3. However, as mentioned, performing the analysis is only the first step,
management and visualization of the data is also required. Additional MATLAB
functions were develop for this purpose, but for reasons of brevity excluded, please
contact the author for further information.
379
D.5 Source Code
D.5.1 sobol.m
1 func t i on varargout = sobo l ( func , vararg in )
2 % SOBOL performs complete SOBOL s e n s i t i v i t y ana l y s i s .
3 %
4 % SYNTAX:
5 % r = sobo l ( func ) ;
6 % r = sobo l ( func , o u t f i l e ) ;
7 % r = sobo l ( . . . ,K) ;
8 % r = sobo l ( . . . ,K,B) ;
9 % r = sobo l ( . . . ,K,B, ’ o f f ’ ) ;
10 % [ r , o u t f i l e ] = sobo l ( . . . ) ;
11 %
12 % DESCRIPTION:
13 % r = sobo l ( func ) executes the func t i on de f ined in the s t r i n g func
14 % r = sobo l ( func , o u t f i l e ) executes the funt i on and saves the r e s u l t s in
15 % the * .mat f i l e s p e c i f i e d in o u t f i l e , an empty s t r i n g w i l l prompt
16 % the user f o r a f i l e and NaN ( de f au l t ) w i l l not produce an output
17 % f i l e
18 % r = sobo l ( . . . ,K) a l l ows the user to s p e c i f y the number o f Monte Carlo
19 % r e p l i c a t e s to u t l i z ed , the d e f au l t i s 1 ,024
20 % r = sobo l ( . . . ,K,B) a l l ows the user to a l s o s p e c i f i y the number o f
21 % bootst rap samples to use when computing the con f idence l e v e l s , the
22 % de f au l t i s 10 ,000
23 % r = sobo l ( . . . ,K,B, ’ o f f ’ ) t o g g l e s o f f the p rog r e s s bar
24 % [ r , o u t f i l e ] = sobo l ( . . . ) outputs the o u t f i l e name to the command window
25 %
26
27 % 1 − GATHER INPUT
28 % 1.1 − Determine m− f i l e /mat− f i l e source
29 spec = { ’ *. mat ’ , ’ MAT - f i l e  (*. mat ) ’ } ; % Also used in Sec . 3 .2
30 i f isempty ( func ) ;
31 l o c = [ cd , f i l e s e p , ’ vec ’ ] ;
32 [ fname , pth ] = u i g e t f i l e ( spec , ’ S e l e c t  MAT - f i l e ... ’ , l o c ) ;
33 i f i snumer ic ( fname ) ; varargout {1} = [ ] ; r e turn ; end
34 func = [ pth , fname ] ;
35 end
36
37 % 1.2 − Gather add i t i ona l input
38 [K,B, wtbar , o u t f i l e ] = inputopt ions ( vararg in { :} ) ;
39
40 % 2 − PERFORM SOBOL CALCULTIONS
41 % 2.1 − Account f o r d i r e c t s t ru c tu r e input
42 i f i s s t r u c t ( func ) ;
43 r = func ;
44 e l s e i f i s c ha r ( func ) | | ˜ i s c e l l ( func ) ;
45 [ p , f , e ] = f i l e p a r t s ( func ) ;
46 i f strcmpi ( e , ’ . m ’ ) ; r = load ( func ) ; end
47 end
48 i f ˜ e x i s t ( ’ r ’ , ’ var ’ ) ; r = sobo lvec ( func ,K, wtbar ) ; end
49
50 % 2.2 − Comput SOBOL ind i c e s
51 r = sobo l idx ( r ) ;
52
53 % 2.3 − Compute con f idence bounds
54 i f ˜ isempty (B) ; r = sobo l c i 2 ( r ,B, wtbar ) ; end
55
56 % 3 − OUTPUT DATA
57 % 3.1 − Output data s t ru c tu r e
58 r . bootst rap = B;
59 varargout {1} = r ;
60
61 % 3.2 − Output to f i l e , i f d e s i r ed
62 i f isempty ( o u t f i l e ) ;
63 [ fname , pth ] = u i p u t f i l e ( spec , ’ S a v e  as ... ’ , . . .
64 [ cd , f i l e s e p , ’ r e s u l t s ’ , f i l e s e p ] ) ;
65 i f i snumer ic ( fname ) ; re turn ; end
66 o u t f i l e = [ pth , fname ] ;
67 e l s e
68 i f i snan ( o u t f i l e ) ; varargout {2} = ’ ’ ; r e turn ; end
69 end
70
71 pth = f i l e p a r t s ( o u t f i l e ) ;
72 i f ˜ e x i s t ( pth , ’ dir ’ ) ; mkdir ( pth ) ; end
73 save ( o u t f i l e , ’ - mat ’ , ’ - s t r u c t ’ , ’ r ’ ) ;
74 varargout {2} = ou t f i l e ;
75
76 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
77 func t i on [K,B, wtbar , o u t f i l e ] = inputopt ions ( vararg in )
78 % INPUTOPTIONS gather s user input from command−l i n e
79
380
80 % 1 − SET THE DEFAULT VALUES, RETURN IF NO OPTIONS SPECIFIED
81 o u t f i l e = NaN; K = 1024; B = 10000; wtbar = ’ on ’ ;
82 i f narg in == 0 ; return ; end
83
84 % 2 − GATHER USER SUPPLIED OPTIONS
85 % 2.1 − Output f i l ename
86 i f i s c ha r ( vararg in {1}) ;
87 o u t f i l e = vararg in {1} ; N = 1 ;
88 e l s e N = 0 ;
89 end
90
91 % 2.2 − Addit iona l input opt ions
92 i f narg in >= N+1; K = vararg in {N+1}; end
93 i f narg in >= N+2; B = vararg in {N+2}; end
94 i f narg in == N+3; wtbar = vararg in {N+3}; end
D.5.2 sobolvec.m
1 func t i on output = sobo lvec ( func , vararg in )
2 %SOBOLVEC c r e a t e s the vec to r s needed f o r computing SOBOL ind i c e s
3 %
4 % SYNTAX:
5 % output = sobo lvec ( func ) ;
6 % output = sobo lvec ( func ,K) ;
7 % output = sobo lvec ( func ,K, ’ o f f ’ ) ;
8 %
9
10 % 1 − ASSIGN DEFAULTS AND/OR USER DEFINED OPTIONS
11 K = 1000; wtbar = ’ on ’ ;
12 N = length ( vararg in ) ;
13 i f N >= 1 && isnumer ic ( vararg in {1}) ; K = vararg in {1} ; end
14 i f N == 2 && i s cha r ( vararg in {2}) ; wtbar = vararg in {2} ; end
15
16 % 2 − BUILD/READ THE SOBOL VECTORS
17 % 2.1 − Case when ” func ” i s a f i l e conta in ing the vec to r s
18 i f i s c ha r ( func ) && ex i s t ( func , ’ f i l e ’ ) ;
19 [ pth , name , ext ] = f i l e p a r t s ( func ) ;
20 i f strcmpi ( ext , ’ . mat ’ ) ; output = load ( func ) ; re turn ; end
21 end
22
23 % 2.2 − Case when ” func ” must be eva luated
24 output = ana lyze func ( func ,K, wtbar ) ;
25
26 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
27 func t i on r = ana lyze func ( func ,K, wtbar )
28 % ANALYZE FUNC performs the func t i on eva lua t i on s
29
30 % 1 − DETERMINE NUMBER OF VARIABLES
31 addpath ( [ cd , f i l e s e p , ’ f u n c ’ ] ) ;
32 addpath ( [ cd , f i l e s e p , ’ i n p u t ’ ] ) ;
33 i f i s c ha r ( func ) ; func = { func } ; end
34 n func = narg in ( func {1}) ;
35 i f n func == −1 | | n func == 3 ;
36 [ n , x , input ] = f e v a l ( func { : } , [ ] ) ; r . input = input ;
37 e l s e
38 n = f e v a l ( func { : } , [ ] ) ;
39 end
40 i f i s c e l l (n) ; r . l egend = n ; n = length (n) ; end
41
42 % 2 − INTILIZE WAITBAR FOR FUNCTION EVALUATIONS
43 C = (n*2 + 2) ; c = 1 ;
44 hbar = updatebar (wtbar , 0 , ’ P e r f o r m i n g  f u n c t i o n  e v a l u a t i o n s ... ’ ) ;
45
46 % 3 − PERFORM MONTE CARLO SAMPLING AND RE−SAMPLING
47 M1 = rand (K, n) ; M2 = rand (K, n) ;
48
49 % 4 − NON−SUBSITITUTED FUNCTION EVALUATIONS
50 r . a 0 = f e v a l ( func { :} ,M2) ;
51 updatebar (wtbar , c/C, hbar ) ; c = c + 1 ;
52 r . a K = f e v a l ( func { :} ,M1) ;
53 updatebar (wtbar , c/C, hbar ) ; c = c + 1 ;
54
55 % 5 − PERFORM SOBOL FOR EACH i−th INPUT PARAMETER
56 r . a i = ze ro s (K, n , s i z e ( r . a 0 , 2 ) ) ; r . a n i = r . a i ;
57
58 f o r i = 1 : n
59 % 5.1 − Solve f o r M1−matrix , i−th s o l u t i o n s
60 Ni = M2; Ni ( : , i ) = M1( : , i ) ;
61 r . a i ( : , i , : ) = f e v a l ( func { :} , Ni ) ;
381
62 updatebar (wtbar , c/C, hbar ) ; c = c + 1 ;
63
64 % 5.2 − Solve f o r M2−matrix , −i−th s o l u t i o n s ( only in ’ improved ’ )
65 Nni = M1; Nni ( : , i ) = M2( : , i ) ;
66 r . a n i ( : , i , : ) = f e v a l ( func { :} , Nni ) ;
67 updatebar (wtbar , c/C, hbar ) ; c = c + 1 ;
68 end
69
70 c l o s e ( hbar ) ; drawnow ; % Closes waitbar
71
72 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
73 func t i on varargout = updatebar ( t r i g , progress , vararg in )
74 % UPDATEBAR operate s the waitbar a l l ow ing the user to turn i f o f f
75 varargout {1} = [ ] ;
76 i f strcmpi ( t r i g , ’ off ’ ) ; r e turn ; end ;
77
78 i f ˜ i s un i x && ˜ strcmpi ( t r i g , ’ s c r e e n ’ ) % Windows systems , show a graph i ca l waitbar
79 i f p rog r e s s == 0 ;
80 varargout {1} = waitbar (0 , vararg in {1}) ;
81 e l s e
82 waitbar ( progress , vararg in {1}) ;
83 end
84
85 e l s e i f i s un i x | | strcmpi ( t r i g , ’ s c r e e n ’ ) % Linux , p r in t p rog r e s s to the sc r een
86 i f p rog r e s s == 0 ;
87 t i c ;
88 d i sp ( ’ P e r f o r m i n g  m o d e l  e v a l u a t i o n s ,  p l e a s e  w a i t ... ’ ) ;
89 e l s e
90 e lp = toc ;
91 d i sp ( [ ’   ’ , num2str ( p rog r e s s *100) , ’ %  c o m p l e t e ;  ’ , . . .
92 num2str ( e lp /3600) , ’  h o u r s  e l a p s e d . ’ ] ) ;
93 end
94 end
D.5.3 sobolidx.m
1 func t i on r = sobo l idx ( r )
2 % SOBOLIDX computes the sobo l i n d i c e s f o r the input vec to r s .
3 %
4 % SYNTAX: r = sobo l idx ( r ) ;
5 %
6
7 % 1 − DEFINE THE VECTORS SIZES
8 [K, n ,m] = s i z e ( r . a i ) ; % m = number o f inputs ; n = number o f i n d i c e s
9
10 % 2 − COMPUTE INDICES FOR EACH INPUT (m)
11 f o r j = 1 :m; % loop number o f inputs
12 S = ze ro s (n , n) ; ST = ze ro s (n , 1 ) ; % c l e a r v a r i a b l e s from perv ious loop
13
14 % 2.1 − Seperate cur rent vec to r s
15 a 0 = r . a 0 ( : , j ) ; a K = r . a K ( : , j ) ;
16 a i = r . a i ( : , : , j ) ; a n i = r . a n i ( : , : , j ) ;
17
18 % 2.2 − Compute f i r s t and to ta l−e f f e c t i n d i c e s
19 f o r i = 1 : n ;
20 S( i , i ) = s o b o l f i r s t o r d e r ( a 0 , a K , a i ( : , i ) , a n i ( : , i ) ) ;
21 ST( i ) = s o b o l t o t a l e f f e c t ( a 0 , a K , a i ( : , i ) , a n i ( : , i ) ) ;
22 end
23
24 % 2.3 − Compute the second−order i n d i c e s
25 f o r i = 1 : n ;
26 f o r l = i +1:n
27 S( i , l ) = sobo l s e condorde r ( a i ( : , i ) , a n i ( : , i ) , a i ( : , l ) . . .
28 , a n i ( : , l ) ,S ( i , i ) ,S ( l , l ) ) ;
29 S( l , i ) = S( i , l ) ;
30 end
31 end
32
33 % 2.4 − Store data from current loop f o r output
34 r . S ( : , : , j ) = S ;
35 r .ST( : , j ) = ST;
36 end
382
D.5.4 sobolci2.m
1 func t i on r = sobo l c i 2 ( r , vararg in )
2 % SOBOLCI2 computes the bootst rap con f idence l e v e l i n t e r v a l s .
3 %
4 % SYNTAX:
5 % r = sobo l c i 2 ( r ) ;
6 % r = sobo l c i 2 ( r ,B) ;
7 % r = sobo l c i 2 ( r ,B, ’ o f f ’ ) ;
8 %
9
10 % 1 − PREPARE FOR ANALYSIS
11 % 1.1 − Gather the user input
12 B = 10000; wtbar = ’ on ’ ;
13 N = length ( vararg in ) ;
14 i f N >= 1 && isnumer ic ( vararg in {1}) ; B = vararg in {1} ; end
15 i f N == 2 && i s cha r ( vararg in {2}) ; wtbar = vararg in {2} ; end
16
17 % 1.2 − Determine s i z e o f input ar rays
18 [K, n ,m] = s i z e ( r . a i ) ;
19
20 % 2 − COMPUTE BOOTSTRAP REPLICATES
21 % 2.1 − I n i t i l i z e resample s to rage ar rays and rename o r i gna l va lues
22 S = ze ro s (n , n ,m,B) ; ST = ze ro s (n ,m,B) ;
23 zS = S ; zST = ST;
24 s = r . S ; s t = r .ST ; % o rg i n a l va lues
25
26 % 2.2 − Loop through number o f d e s i r e resampl ings
27 %hbar = updatebar (wtbar , 0 , ’ Computing bootst rap SOBOL ind i c e s . . . ’ ) ;
28 f o r i = 1 :B;
29 idx = randsample (K,K, t rue ) ;
30 [ S ( : , : , : , i ) ,ST ( : , : , i ) ] = sobo l r ep ( r , idx ) ;
31 zST ( : , : , i ) = ST ( : , : , i ) < s t ;
32 zS ( : , : , : , i ) = S ( : , : , : , i ) < s ;
33 %updatebar (wtbar , i /B, hbar ) ;
34 end
35 %c l o s e ( hbar ) ;
36
37 % 3 − COMPUTE BOOTSTRAP CONFIDENCE LEVEL INTERVALS
38 hbar = updatebar (wtbar , 0 , ’ C o m p u t i n g  c o n f i d e n c e  l e v e l  i n t e r v a l s ... ’ ) ;
39 f o r j = 1 :m % Loop through each output va r i ab l e
40
41 % 3.1 − I n i t i l i z e the v a r i a b l e s in use
42 Sj = so r t ( squeeze (S ( : , : , j , : ) ) ,3 ) ; % bootst rap r e p l i c a t e s o f S
43 STj = so r t ( squeeze (ST( : , j , : ) ) ,2 ) ; % bootst rap r e p l i c a t e s o f St
44 zSj = squeeze ( zS ( : , : , j , : ) ) ; % count o f va lues l e s s than o r i g i n a l S
45 zSTj = squeeze (zST ( : , j , : ) ) ; % coutn o f va lues l e s s than o r i g i n a l ST
46
47 % 3.2 − Compute the b ia s adjustment
48 S z0 = norminv (sum( zSj , 3 ) /B, 0 , 1 ) ;
49 ST z0 = norminv (sum( zSTj , 2 ) /B, 0 , 1 ) ;
50
51 % 3.3 − Compute the a c c e l e r a t i o n ( see Efron et a l . )
52 % 3 . 3 . 1 − Compute j a c kkn i f e s t a t i s t i c s
53 [ S jk , ST jk ] = j a c k c a l c ( r , j ) ;
54
55 % 3 . 3 . 2 − Compute the inner por t ion a c c e l e r a t i o n equat ion
56 f o r k = 1 :K
57 s c r S ( : , : , k ) = (mean( S jk , 3 ) − S jk ( : , : , k ) ) ;
58 scr ST ( : , k ) = (mean( ST jk , 2 ) − ST jk ( : , k ) ) ;
59 end
60
61 % 3 . 3 . 3 − Compute a c c e l e r a t i o n
62 aS = 1/6*sum( s c r S . ˆ3 , 3 ) . / ( sum( s c r S . ˆ2 , 3 ) . ˆ 1 . 5 ) ;
63 aST = 1/6*sum( scr ST .ˆ3 , 2 ) . / ( sum( scr ST .ˆ2 , 2 ) . ˆ 1 . 5 ) ;
64
65 % 3.4 − Compute the 95% upper and lower p e r c e n t i l e s
66 alpha1 = norminv ( 0 . 0 5 ) ; alpha2 = −alpha1 ;
67 f cn pc t = @( z0 , alpha , acc ) . . .
68 100*normcdf ( z0 +(z0+alpha ) ./(1− acc . * ( z0+alpha ) ) ) ;
69 S pct1 = f cn pc t ( S z0 , alpha1 , aS ) ;
70 S pct2 = f cn pc t ( S z0 , alpha2 , aS ) ;
71 ST pct1 = f cn pc t ( ST z0 , alpha1 , aST) ;
72 ST pct2 = f cn pc t ( ST z0 , alpha2 , aST) ;
73
74 % 3.5 − Gather the c o r r e c t value f o r the c a l cu l a t ed p e r c e n t i l e s
75 f o r i = 1 : n ;
76 r . STci1 ( i , j ) = p r c t i l e ( STj ( i , : ) , ST pct1 ( i ) ) ;
77 r . STci2 ( i , j ) = p r c t i l e ( STj ( i , : ) , ST pct2 ( i ) ) ;
78 f o r i i = 1 : n ;
79 r . Sc i1 ( i , i i , j ) = p r c t i l e ( Sj ( i , i i , : ) , S pct1 ( i , i i ) ) ;
80 r . Sc i2 ( i , i i , j ) = p r c t i l e ( Sj ( i , i i , : ) , S pct2 ( i , i i ) ) ;
81 end
82 end
83
84 % 3.6 − Compute b ia s from o r i g i n a l va lues
383
85 r . Sbias ( : , : , j ) = ( r . Sc i1 ( : , : , j ) + r . Sc i2 ( : , : , j ) ) /2 − s ( : , : , j ) ;
86 r . STbias ( : , j ) = ( r . STci1 ( : , j ) + r . STci2 ( : , j ) ) /2 − s t ( : , j ) ;
87
88 updatebar (wtbar , j /m, hbar ) ;
89 end
90 c l o s e ( hbar ) ;
91
92 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
93 func t i on [ S jk , ST jk ] = j a ck c a l c ( r , j )
94 % JACKCALC computes the j a c kkn i f e s t a t i s t i c
95
96 % 1 − I n i t i l i z e
97 [K, n ,m] = s i z e ( r . a i ) ;
98 S jk = ze ro s (n , n ,K) ; ST jk = ze ro s (n ,K) ;
99
100 % 2 − Loop through each parameter and r e c a l c u l a t e SOBOL ind i c e s
101 f o r k = 1 : l ength (K) ;
102 idx = true (K, 1 ) ; idx (k ) = f a l s e ;
103 r r . a 0 = r . a 0 ( idx , j ) ; r r . a K = r . a K( idx , j ) ;
104 r r . a i = r . a i ( idx , : , j ) ; r r . a n i = r . a n i ( idx , : , j ) ;
105 r r = sobo l idx ( r r ) ;
106 S jk ( : , : , k ) = r r . S ;
107 ST jk ( : , k ) = r r .ST ;
108 end
109
110 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
111 func t i on [ S ,ST ] = sobo l r ep ( r , idx )
112 % SOBOLREP executes sobo l idx func t i on
113 r r . a 0 = r . a 0 ( idx , : ) ; r r . a K = r . a K( idx , : ) ;
114 r r . a i = r . a i ( idx , : , : ) ; r r . a n i = r . a n i ( idx , : , : ) ;
115 r r = sobo l idx ( r r ) ;
116 S = r r . S ;
117 ST = rr .ST ;
118
119 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
120 func t i on varargout = updatebar ( t r i g , progress , vararg in )
121 % UPDATEBAR operate s the waitbar a l l ow ing the user to turn i f o f f
122 varargout {1} = [ ] ;
123 i f strcmpi ( t r i g , ’ off ’ ) ; r e turn ; end ;
124
125 i f ˜ i s un i x && ˜ strcmpi ( t r i g , ’ s c r e e n ’ ) % Windows systems , show a graph i ca l waitbar
126 i f p rog r e s s == 0 ;
127 varargout {1} = waitbar (0 , vararg in {1}) ;
128 e l s e
129 waitbar ( progress , vararg in {1}) ;
130 end
131
132 e l s e i f i s un i x | | strcmpi ( t r i g , ’ s c r e e n ’ ) % Linux , p r in t p rog r e s s to the sc r een
133 i f p rog r e s s == 0 ;
134 t i c ;
135 d i sp ( vararg in {1}) ;
136 e l s e
137 e lp = toc ;
138 d i sp ( [ ’   ’ , num2str ( p rog r e s s *100) , ’ %  c o m p l e t e ;  ’ , . . .
139 num2str ( e lp /3600) , ’  h o u r s  e l a p s e d . ’ ] ) ;
140 end
141 end
D.5.5 sobol firstorder.m
1 func t i on S = s o b o l f i r s t o r d e r ( a0 , aK, ai , ani )
2 % SOBOL FIRSTORDER computes the f i r s t order SOBOL s e n s i t i v i t y index .
3 K = length ( a0 ) ; % Number o f r e p l i c a t e s
4 E2 = 1/K * dot ( a0 , aK) ; % Square o f expected
5 V = 1/K * dot (aK,aK) − E2 ; % Total var iance
6 U(1) = 1/K * dot (aK, a i ) ; % F i r s t es t imate o f U
7 U(2) = 1/K * dot ( a0 , ani ) ; % Second est imate o f U
8 S = (mean(U) − E2) /V; % Fir s t−order index
384
D.5.6 sobol secondorder.m
1 func t i on S2 = sobo l s e condorde r ( ai , ani , aiL , aniL , Si , S l )
2 % SOBOL SECONDORDER computes the second−order SOBOL index .
3 K = length ( a i ) ; % Number o f r e p l i c a t e s
4 E2 = 1/K * dot ( ai , ani ) ; % Square o f expected value
5 V = 1/K * dot ( aniL , aniL ) − E2 ; % Total var iance
6 U1 = 1/K * dot ( ai , aniL ) ; % F i r s t es t imate o f Vi l
7 U2 = 1/K * dot ( aiL , ani ) ; % Second est imate o f Vi l
8 Sc = (mean ( [U1 ,U2 ] ) − E2) /V; % Closed second−order index
9 S2 = Sc−Si−Sl ; % Second−order index
D.5.7 sobol totaleffect.m
1 func t i on ST = s o b o l t o t a l e f f e c t ( a0 , aK, ai , ani )
2 % SOBOL TOTALEFFECT computes the SOBOL t o t a l e f f e c t index .
3 K = length ( a0 ) ; % Number o f r e p l i c a t e s
4 E2 = 1/K * dot ( a0 , aK) ; % Square o f expected
5 V = 1/K * dot ( a0 , a0 ) − E2 ; % Total var iance
6 U(1) = 1/K * dot ( a0 , a i ) ; % F i r s t e s t imate o f U
7 U(2) = 1/K * dot (aK, ani ) ; % Second est imate o f U
8 ST = 1 − (mean(U) − E2) /V; % Total−e f f e c t index
D.5.8 saMODEL2.m
1 func t i on [Y, x , user , varargout ] = saMODEL2( vararg in )
2 % SAMODEL2 performs ana l y s i s f o r SOBOL and FAST s e n s i t i i v t y ana l y s i s
3 %
4 % SYNTAX:
5 % [Y, x ] = saMODEL2( sk )
6 % [Y, x ] = saMODEL2( input , sk )
7 % [Y, x ] = saMODEL2( input , ’ PropertyName ’ , PropertyValue , . . . , sk )
8 % [Y, x , user ] = saMODEL2 ( . . . )
9 % L = saMODEL2 ( . . . , [ ] ) , where sk = [ ]
10 %
11 % DESCRIPTION:
12 % [Y, x ] = saMODEL2( sk ) runs thermal model us ing the ’ d e f au l t .mat ’ input
13 % f i l e ; a l l input f i l e s should be l o ca t ed in \ input d i r e c t o r y .
14 % [Y, x ] = saMODEL2( input , sk ) a l l ows user to s p e c i f y an input f i l e or
15 % st ruc ture , us ing input = ’ ’ w i l l prompt f o r a f i l e .
16 % [Y, x ] = saMODEL2( input , ’ PropertyName ’ , PropertyValue , . . . , sk ) a l l ows user
17 % to ad jus t the s e t t i n g s o f the model run , the a v a i l a b l e p r op e r t i e s
18 % are l i s t e d below .
19 % [Y, x , user ] = saMODEL2 ( . . . ) prov ides an add i t i ona l output that i s a data
20 % s t ru c tu r e that i n c l ude s the program opt ions s t r u c tu r e and the input
21 % data s t ru c tu r e .
22 % L = saMODEL ( . . . , sk ) , where sk = [ ] r e tu rns the v a r i a b l e s l a b e l s
23 % being explored given the de s i r ed input and s e t t i n g s . This value
24 % should be used to c r ea t e sk , n = length (L) .
25 %
26 % FIXED INPUTS:
27 % sk = a numeric array conta in ing rand numbers from 0 to 1 that i s
28 % [K x n ] in s i z e , where K i s the number o f model eva lua t i on s and n
29 % i s the number o f v a r i a b l e s being explored . n i s g iven by eva luat ing
30 % th i s funt i on as shown above .
31 % input = a * .mat f i l e conta in ing an input s t ru c tu r e or the s t ru c tu r e
32 % va r i ab l e i t s e l f , the s tuc tu r e must inc lude a l l snow , atm , and
33 % constant va lues shown below , the d i r t i s only need i f the ’ d i r t ’
34 % property i s ’ on ’ .
35 %
36 % X must be one o f two th ings . I f i t i s a s c a l a r value t h i s v a r i ab l e
37 % i s held constant at the s p e c i f i e d . Otherwise , X may be a c e l l
38 % array s t ru c tu r e as X = { ’FuncName ’ , Input1 , Input2 , . . . , Min ,Max} .
39 % ’FuncName ’ should be a s t r i n g d e f i n i n g the d i s t r i b u t i o n func t i on to
40 % analyze that matches an ava i l a b l e MATLAB inve r s e func t i on . For
41 % example , supply ’norm ’ would invoke the func t i on ’ norminv ’ . The
42 % inputs : Input1 , e tc . should be the exact number o f inputs r equ i r ed
43 % by the i nv e r s e funct ion , i . e . the d i s t r i b u t i o n parameters . The
44 % Min and Max va lues are opt iona l , i f the are inc luded the data i s
385
45 % r e s t r i c t e d to va lues between these two va lues .
46 %
47 % The f o l l ow ing i s an example input f i l e , the un i t s o f each va r i ab l e
48 % are con s i t en t with the thermal model input .
49 %
50 % >> input . snow
51 % ans =
52 % depth : 50
53 % dens i ty : { ’ uni f ’ [ 5 0 ] [ 5 00 ]}
54 % conduct iv i ty : { ’ uni f ’ [ 0 . 0 1 ] [ 0 . 7 ] }
55 % s p e c i f i c : 2030
56 % snowtemp : { ’ gev ’ [−0.39219] [ 5 . 7 9 5 1 ] [−16.339] [−20] [−5]}
57 % kappa : { ’ uni f ’ [ 4 0 ] [ 2 00 ]}
58 % kappaNIR : 0
59 %
60 % >> input . atm
61 % ans =
62 % time : 10
63 % longwave : { ’ gev ’ [−0.09476] [ 6 3 . 6 2 ] [ 2 8 7 . 9 7 ]}
64 % shortwave : { ’ gp ’ [−0.88865] [ 5 7 5 . 7 9 ] [ 3 9 . 0 9 4 ]}
65 % alpha : { ’ uni f ’ [ 0 . 4 ] [ 0 . 9 5 ] }
66 % wind : { ’ logn ’ [ 0 . 5 2 4 4 8 ] [ 0 . 3 3 004 ]}
67 % airtemp : { ’ gev ’ [−0.24391] [ 4 . 4 7 4 4 ] [−8.1885]}
68 % humidity : { ’ gev ’ [−0.65934] [ 1 5 . 9 1 8 ] [ 6 0 . 4 3 3 ]}
69 % bottom : 0
70 % pre s su r e : 85
71 % shortwaveNIR : 0
72 % alphaNIR : 0
73 %
74 % >> input . constant
75 % ans =
76 % Ls : 2833
77 % Ke : 0 .0023
78 % Kh: 0 .0023
79 % MvMa: 0 .622
80 % Rv : 0 .462
81 % T0 : −5
82 % e0 : 0 .402
83 % emis : 0 .9875
84 % dz : 0 .5
85 % dt : 60
86 %
87 % >> input . d i r t
88 % ans =
89 % depth : { ’ uni f ’ [ 1 ] [ 5 ] }
90 % kappa : { ’ uni f ’ [ 1 0 0 ] [ 1000 ]}
91 % kappaNIR : 0
92 %
93 % >> input . p r o f i l e
94 % ans =
95 % diurna l : { ’ uni f ’ [−20] [ 0 ] }
96 %
97 % OUTPUTS:
98 % Y = [K x p ] vec tor conta in ing de s i r ed output f o r each model evaulat ion ,
99 % with each output occupying a row of l ength p , where p i s the number
100 % output per eva luat i on . In the case o f output ( ’ type ’ ) , f o r ’TG’
101 % th i s p i s the number o f g rad i ent va lues s to r ed based on the model
102 % run durat ion and the s to rage increment ( ’ inc ’ ) .
103 % x = the sk input with a s s o c i a t ed va r i ab l e d i s t r i b u t i o n s app l i ed .
104 % user = a data s t ru c tu r e that i n c l ude s the program opt ions s t ru c tu r e and
105 % the input data s t ru c tu r e .
106 %
107 % AVAILABLE PROPERTIES ( Property and value pa i r s , s ee EXAMPLE fo r help )
108 % ’ type ’ s p e c i f i e s the type o f output to use f o r ana l y s i s ’TG’ i s the
109 % de f au l t value , the av a i l a b l e types indc lud : ’T’ , ’TG’ , ’Tknee ’ ,
110 % ’KTG’ , ’Kdepth ’ , ’ Kduration ’ , ’MF’
111 % ’ subtype ’ s p e c i f i e s the c a l c u l a t i o n to perform on the de s i r ed output ,
112 % opt ions inc lude : ’mean ’ , ’min ’ , ’ min time ’ , ’max ’ , ’max time ’ ,
113 % ’ to ta l ’ , or ’ a l l ’
114 % ’ depth ’ d e f i n e s the depth in cm f o r compute grad ients , t h i s va lues
115 % should be a sca l a r , the d e f au l t i s 5 cm.
116 % ’ inc ’ d e f i n e s the s to rage increment f o r ’TG’ and ’MF’ opt ions in
117 % minutes , the value should be s c a l a r and the de f au l t i s 20 min .
118 % ’day ’ mod i f i e s the short−wave ( in c lud ing NIR) to act as s i n e wave , with
119 % the mean o f the s i n e wave to be equal to the inputed short−wave
120 % value ( s ) . ’ day ’ i s an ’ on ’ / ’ o f f ’ togg le , the d e f au l t i s ’ on ’ .
121 % ’ d i r t ’ i s e i t h e r ’ on ’ or ’ o f f ’ ( d e f au l t ) that adds a l ay e r o f d i r t
122 % based on input . d i r t s e t t i n g s
123 % ’ p r o f i l e ’ a l l ows the user to turn on a snow p r o f i l e f ea ture , which
124 % i s an ’ on ’ / ’ o f f ’ togg le , the d e f au l t i s ’ o f f ’ . The top i s
125 % ass igned the value given by input . snow . snowtemp , the bottom by
126 % input . atm . bottom , and at a depth o f input . p r o f i l e . depth and
127 % temperature o f input . p r o f i l e . temp and a l i n e a r f i t in between .
128 % ’ prog ’ i s an ’ on ’ or ’ o f f ’ ( d e f au l t ) t ogg l e f o r the prog r e s s message
129 %
130 % EXAMPLE:
131 % >> L = saMODEL2( ’ con t ro l 2 .mat ’ , ’ prog ’ , ’ on ’ , ’ d i r t ’ , ’ on ’ , [ ] ) ;
386
132 % >> sk = rand (100 , l ength (L) ) ;
133 % >> [Y, x ] = saMODEL2( ’ con t ro l 2 .mat ’ , ’ prog ’ , ’ on ’ , ’ type ’ , ’TG’ , ’ subtype ’ ,
134 % ’mean ’ , sk ) ;
135 %
136 % NOTES FOR USER:
137 % (1) The \ func and \ func\ input d i r e c t o r i e s that conta in t h i s f i l e and
138 % the input f i l e s must be added to MATLAB’ s path ( see help addpath ) ,
139 % th i s i s automat i ca l l y done by sobo l .m.
140 % (2) This func t i on u t i l i z e d two m− f i l e s a s s o c i a t ed with ve r s i on 5 o f the
141 % thermal model , the path to the f i l e s thermal .m and x l s p r ep .m must
142 % a l s o be added ( see help addpath ) , the d e f au l t l o c a t i o n f o r these
143 % f i l e s i s a d i r e c t o r y that i s p a r a l l e l to the s e n s i t i v i t y d i r e c t o r y
144 % which conta ins sobo l .m named ThermalModel v5 , s ee Sect ion 1 .
145 % (3) I f input . snow . snowtemp = NaN, the snow temperature i s s e t to the
146 % a i r temperature .
147 % (4) I f input . atm . bottom = NaN, the bottom boundary cond i t i on s i s s e t to
148 % the snow temperature .
149 % (5) I f input . d i r t . kappaNIR = NAN, t h i s value i s s e t to input . d i r t . kappa
150 %
151 % PROGRAM OUTLINE:
152 % 1 − ADD DEFAULT PATHS TO THE THERMAL MODEL ( v5 )
153 % 2 − PREPARE INPUT FROM COMMAND LINE
154 % 3 − BUILD VARIABLE LIST AND RETURN LABELS ( sk = [ ] case )
155 % 4 − CONSTRUCT MODEL INPUT MATRICES
156 % 5 − PERFORM MODEL EVALUATIONS
157 % 6 − BUILD REMAINING OUTPUT
158 %
159 % SUBFUNCTIONS:
160 % GETUSEROPTIONS gather user input
161 % GETLABELS cons t ruc t s l i s t o f v a r i a b l e s with d i s t r i b u t i o n s
162 % BUILDINPUT cons tu r c t s input matr ices , one row per eva luat i on
163 % BUILDMATRIX eva lu t e s the d i s t r i b u t i o n func t i on s f o r sk va lues
164 % GETLIMTS ex t r a c t s min/max l im i t s supp l i ed with d i s t . f unc t i on input
165 % SPECIALINPUT app l i e s s p e c i a l c ond i t i on s f o r input data
166 %
167
168 % 1 − ADD DEFAULT PATHS TO THE THERMAL MODEL ( v5 )
169 l o c = cd ( ’ .. ’ ) ;
170 addpath ( [ cd , f i l e s e p , ’ T h e r m a l M o d e l _ v 5 ’ ] ) ;
171 cd ( l o c ) ;
172
173 % 2 − PREPARE INPUT FROM COMMAND LINE
174 [ in , sk , opt ] = ge tu s e r op t i on s ( vararg in { :} ) ;
175 user . input = in ;
176 user . opt ions = opt ;
177
178 % 3 − OPEN DATA STRUCTURE AND RETURN LABELS
179 % 3.1 − Load d i s t r i b u t i o n s t r u c t u r e s
180 addpath ( [ cd , f i l e s e p , ’ i n p u t ’ ] ) ;
181 i f ˜ i s s t r u c t ( in ) && ex i s t ( in , ’ f i l e ’ ) ; in = load ( in ) ; end
182
183 % 3.2 − Build l a b e l s ( stop operat ion in sk = [ ] case )
184 i f isempty ( sk ) ;
185 L = g e t l a b e l s ( in , opt ) ;
186 Y = L ; x = [ ] ; r e turn ;
187 end
188
189 % 4 − CONSTRUCT MODEL INPUT MATRICES
190 [ S ,A,C,D,P, x ] = bu i ld input ( in , opt , sk ) ;
191
192 % 5 − PERFORM MODEL EVALUATIONS
193 % 5.1 − Setup loop parameters and i n i t l i z e ar rays
194 K = s i z e ( sk , 1 ) ;
195 e = t i c ; k = K*0 . 0 1 ;
196
197 % 5.2 − I n i t i a l i z e ar rays
198 nn = in . atm . time /( opt . inc /60) + 1 ;
199 mm = 10/ in . constant . dz ;
200 ysz = 1 ;
201 i f strcmpi ( opt . subtype , ’ all ’ ) ; ysz = nn ; end
202 Y = zero s (K, ysz ) ;
203 Tout = ze ro s (K,mm, nn) ;
204 Qout = ze ro s (K, nn) ;
205 %Qout = ze ro s (K,mm, nn , 5 ) ;
206
207 % 5.3 − Loop through each model eva lua t i on
208 f o r i = 1 :K;
209 % 5 . 3 . 1 − Build f u l l matr i ces o f cur rent input parameters
210 c = C( i , : ) ; s = S( i , : ) ; a = A( i , : ) ;
211 i f ˜ isempty (P) ; p = P( i , : ) ; e l s e p = [ ] ; end
212 [ s , a ] = sp e c i a l i n pu t ( s , a , p , opt ) ;
213 [ s , a ] = x l s p r ep ( s , a , c ) ;
214
215 % 5 . 3 . 2 − Appl ies d i r t−l ayer , i f d e s i r ed
216 i f strcmpi ( opt . d i r t , ’ on ’ ) && ˜ isempty (D) ;
217 d = D( i , : ) ;
218 idx = f ind ( s ( : , 1 )>d (1) ,1 , ’ f i r s t ’ ) ;
387
219 S( idx , 6 ) = d (2) ; S( idx , 7 ) = d (3) ;
220 end
221
222 % 5 . 3 . 3 − Perform model c a l c u l a t i o n s and bu i ld de s i r ed output
223 [T,Q] = thermal ( s , a , c ) ;
224 Qs = squeeze (Q( 1 , : , 2 ) ) ; % Qs = l a t en t at su r f a c e
225 i f nargout == 5 ;
226 inc = ( opt . inc *60) /c (10) ; % Storage increment
227 idx = 1 : inc : s i z e (T, 2 ) ; % Ind i c e s o f s to rage increment
228 Tout ( i , : , : ) = T(1 :10/ c (9) , idx ) ; % Temps f o r montecarlo2
229 Qout ( i , : ) = squeeze (Qs ( : , idx ) ) ; % Flux usage , only l a t en t
230 e l s e
231 Y( i , : ) = saMODEL2 output (T,Qs , c , opt ) ;
232 end
233
234 % 5 . 3 . 4 − Print the progress , i f d e s i r ed
235 i f strcmpi ( opt . prog , ’ on ’ ) && round ( i /k ) == i /k ;
236 di sp ( [ num2str ( i /K*100) , ’ %  C o m p l e t e ,  ’ , . . .
237 num2str ( toc ( e ) /3600) , ’  hrs  e l a p s e d . ’ ] ) ;
238 end
239 end
240
241 % 6 − BUILD REMAINING OUTPUT
242 x = s i n g l e ( x ) ;
243 i f nargout == 5 ;
244 Y = [ ] ;
245 varargout {1} = s i n g l e (Tout ) ;
246 varargout {2} = s i n g l e (Qout ) ;
247 end
248
249 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
250 func t i on [ in , sk , opt ] = ge tu s e r op t i on s ( vararg in )
251 % GETUSEROPTIONS gather user input
252
253 % 1 − Extract mandotory ” sk” input and de f au l t f i l ename
254 n = narg in ; % Number o f input v a r i a b l e s
255 sk = vararg in {n} ;
256 in = ’ c o n t r o l 2 . mat ’ ;
257 q = {} ;
258
259 % 2 − Gather user s p e c i f i e d va lues
260 i f narg in >= 2; in = vararg in {1} ; end
261 i f narg in >= 3; q = vararg in ( 2 : nargin−1) ; end
262
263 % 3 − Set d e f au l t s
264 opt . type = ’ TG ’ ; opt . subtype = ’ m e a n ’ ;
265 opt . depth = 5 ;
266 opt . inc = 20 ;
267 opt . day = ’ on ’ ;
268 opt . p r o f i l e = ’ off ’ ;
269 opt . d i r t = ’ off ’ ;
270 opt . prog = ’ off ’ ;
271 opt . output = ’ ’ ;
272
273 % 4 − apply s e t t i n g s
274 n = length (q ) ; k = 1 ;
275 l i s t = f i e ldnames ( opt ) ;
276 whi le k < n
277 itm = q{k } ; va lue = q{k+1}; k = k + 2 ;
278 i f strmatch ( lower ( itm ) , l i s t , ’ e x a c t ’ ) ;
279 opt . ( itm ) = value ;
280 e l s e
281 mes = [ ’ The  option ,  ’ , itm , ’ ,  was  not  r e c o i g n i z e d . ’ ] ;
282 d i sp (mes ) ;
283 end
284 end
285
286 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
287 func t i on L = g e t l a b e l s (A, opt )
288 % GETLABELS cons t ruc t s l i s t o f v a r i a b l e s with d i s t r i b u t i o n s
289
290 % Pre−de f i n e output and load l a b e l s f i l e s
291 L = {} ; LB = load ( ’ l a b e l . lbl ’ , ’ - mat ’ ) ;
292 TF(1) = ˜ strcmpi ( opt . d i r t , ’ on ’ ) | | ˜ i s f i e l d (A, ’ d i r t ’ ) ;
293 TF(2) = ˜ strcmpi ( opt . p r o f i l e , ’ on ’ ) | | ˜ i s f i e l d (A, ’ p r o f i l e ’ ) ;
294
295 % Correct f i e ldnames f o r ex lu s i on o f d i r t
296 fnA = f i e ldnames (A) ;
297 i f TF(1) ; fnA ( strmatch ( ’ d i r t ’ , fnA ) ) = [ ] ; end
298 i f TF(2) ; fnA ( strmatch ( ’ p r o f i l e ’ , fnA ) ) = [ ] ; end
299
300 % Search input f o r c e l l s , buld ing
301 f o r i = 1 : l ength ( fnA ) ;
302 fnA2 = f i e ldnames (A. ( fnA{ i }) ) ;
303 f o r j = 1 : l ength ( fnA2 ) ;
304 itm = A. ( fnA{ i }) . ( fnA2{ j }) ;
305 i f i s c e l l ( itm ) ; L = [L ,LB . ( fnA{ i }) . ( fnA2{ j }) ] ; end
388
306 end
307 end
308
309 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
310 func t i on [ S ,A,C,D,P, x ] = bu i ld input ( IN , opt , sk )
311 % BUILDINPUT cons tu r c t s input matr ices , one row per eva luat i on
312
313 % 1 − Build data s t ru c tu r e conta in ing vec to r s va lues f o r each eva luato in
314 x = ze ro s ( s i z e ( sk ) ) ; % matrix o f input va lues
315 cnt = 1 ; % va r i ab l e counter
316
317 [X. snow , x , cnt ] = bui ldmatr ix ( IN . snow , sk , x , cnt ) ;
318 [X. atm , x , cnt ] = bui ldmatr ix ( IN . atm , sk , x , cnt ) ;
319 [X. constant , x , cnt ] = bui ldmatr ix ( IN . constant , sk , x , cnt ) ;
320 i f i s f i e l d ( IN , ’ d i r t ’ ) ;
321 [X. d i r t , x , cnt ] = bui ldmatr ix ( IN . d i r t , sk , x , cnt ) ;
322 end
323 i f i s f i e l d ( IN , ’ p r o f i l e ’ ) ;
324 [X. p r o f i l e , x ] = bui ldmatr ix ( IN . p r o f i l e , sk , x , cnt ) ;
325 end
326
327 % 2 − Build matr i ces ( ac tua l f i e ldnames used to ensure proper order )
328 s = X. snow ;
329 S = [ s . depth , s . dens i ty , s . conduct iv i ty , s . s p e c i f i c , s . snowtemp , . . .
330 s . kappa , s . kappaNIR ] ;
331
332 a = X. atm ;
333 A = [ a . time , a . longwave , a . shortwave , a . alpha , a . wind , a . airtemp , . . .
334 a . humidity , a . bottom , a . pressure , a . shortwaveNIR , a . alphaNIR ] ;
335
336 c = X. constant ;
337 C = [ c . Ls , c .Ke , c .Kh, c .MvMa, c .Rv , c .T0 , c . e0 , c . emis , c . dz , c . dt ] ;
338
339 i f strcmpi ( opt . d i r t , ’ on ’ ) && i s f i e l d (X, ’ d i r t ’ ) ;
340 d = X. d i r t ;
341 i f i snan (d . kappaNIR) ; d . kappaNIR = d . kappa ; end
342 D = [ d . depth , d . kappa , d . kappaNIR ] ;
343 e l s e
344 D = [ ] ;
345 end
346 i f strcmpi ( opt . p r o f i l e , ’ on ’ ) && i s f i e l d (X, ’ p r o f i l e ’ ) ;
347 p = X. p r o f i l e ;
348 P = [ p . temp , p . temp ] ;
349 e l s e
350 P = [ ] ;
351 end
352
353 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
354 func t i on [ IN ,X, cnt ] = bui ldmatr ix ( IN , sk ,X, cnt )
355 % BUILDMATRIX eva lu t e s the d i s t r i b u t i o n func t i on s f o r sk va lues
356
357 % 1 − Loop through each s t ru c tu r e item
358 K = s i z e ( sk , 1 ) ; % No . o f model eva lua t i on s
359 fn = f i e ldnames ( IN) ; % Fieldnames
360 f o r i = 1 : l ength ( fn ) ;
361 i f i snumer ic ( IN . ( fn{ i }) ) % Case when va r i ab l e i s constant
362 va l = IN . ( fn{ i }) ;
363 IN . ( fn{ i }) = ze ro s (K, 1 ) ;
364 IN . ( fn{ i }) ( : ) = va l ;
365
366 e l s e i f i s c e l l ( IN . ( fn{ i }) ) % Case when va r i ab l e i s a d i s t i b u t i o n
367 ev l = IN . ( fn{ i }) ; % Gathers d i s t r i b u t i o n i n f o
368 func = [ ev l {1} , ’ inv ’ ] ; % Extacts d i s t r i b u t i o n func t i on
369 input = ev l ( 2 : l ength ( ev l ) ) ; % Co l l e c t s d i s t . f unc t i on input
370 [ lim , n func ] = g e t l im i t s ( func , input ) ; % Gets l im i t s , i f e x i s t
371
372 x = f e v a l ( func , sk ( : , cnt ) , input {1 : n func −1}) ; % eva lu t e s fund
373 X( : , cnt ) = x ; % bu i l d s x−matrix f o r output
374 cnt = cnt + 1 ; % increments va r i ab l e counter
375
376 i f ˜ isempty ( l im ) ; % apply l im i t s i f the e x i s t
377 x (x<l im {1}) = lim {1} ;
378 x (x>l im {2}) = lim {2} ;
379 end
380 IN . ( fn{ i }) = x ;
381 end
382 end
383
384 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
385 func t i on [ lim , n func ] = g e t l im i t s ( func , input )
386 % GETLIMTS ex t r a c t s min/max l im i t s supp l i ed with d i s t . f unc t i on input
387
388 % Accounts f o r i nv e r s e func t i on with opt i ona l ”pcov” and ”alpha” inputs
389 switch func
390 case { ’ e v i n v ’ , ’ e x p i n v ’ , ’ g a m i n v ’ , ’ l o g n i n v ’ , ’ n o r m i n v ’ , ’ w b l i n v ’ } ;
391 n func = narg in ( func ) − 2 ;
392 otherwi se
389
393 n func = narg in ( func ) ;
394 end
395
396 % Extracts l im i t s
397 n input = length ( input ) ;
398 dn = n input − ( n func − 1) ;
399 i f dn == 2 ;
400 lim = input ( n input −1: n input ) ;
401 e l s e
402 lim = [ ] ;
403 end
404
405 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
406 func t i on [ s , a ] = sp e c i a l i n pu t (S ,A,P, opt )
407 % SPECIALINPUT app l i e s s p e c i a l c ond i t i on s f o r input data
408
409 % 1 − Apply the NaN cond i t i on s f o r bottom temperature and snow temp .
410 i f i snan (S ( : , 5 ) ) ; S ( : , 5 ) = A(1 ,6 ) ; end % Tsnow = Tair
411 i f i snan (A( : , 8 ) ) ; A( : , 8 ) = S ( : , 5 ) ; end % Tbottom = Tsnow
412 s = S ; a = A;
413
414 % 2 − Apply s ine−wave to shortwave input ( s )
415 i f strcmpi ( opt . day , ’ on ’ ) ;
416 dur = A(1) ; %Day−l i g h t durat ion ( hr )
417 t = (0 : 1 / 6 0 : dur ) ’ ; %1−min . i n t e r v a l s in hrs
418 a = ze ro s ( l ength ( t ) , l ength (A) ) ;
419 SW = A(3) * s i n (2* pi /dur* t − pi /2) + A(3) ;
420 NIR = A(10) * s i n (2* pi /dur* t − pi /2) + A(10) ;
421
422 a ( : , 1 ) = t ; a ( : , 3 ) = SW; a ( : , 1 0 ) = NIR ;
423 idx = [ 2 , 4 : 9 , 1 1 ] ;
424 f o r i = 1 : l ength ( idx ) ; a ( : , idx ( i ) ) = A( idx ( i ) ) ; end
425 end
426
427 % 3 − I n s e r t a temperature p r o f i l e
428 i f strcmpi ( opt . p r o f i l e , ’ on ’ ) && ˜ isempty (P) ;
429 s = [ ] ; % remove unmodif ied input
430 bot = A(1 ,8 ) ; % Bottom temp
431 diu = P(1) ; % Diurnal temp
432 s ( 1 , : ) = S ; s (1 , 1 ) = 0 ;
433 s ( 2 , : ) = S ; s (2 , 1 ) = P(2) ; s (2 , 5 ) = diu ;
434 s ( 3 , : ) = S ; s (3 , 1 ) = S (1) ; s (3 , 5 ) = bot ;
435 end
D.5.9 saMODEL2 output.m
1 func t i on y = saMODEL2 output (T,Q, c , opt , vararg in )
2 % SAMODEL2 OUTPUT produces the de s i r ed output
3 %
4 % SYNTAX:
5 % y = saMODEL2 output (T,Q, c , opt ) %saMODEL2.m
6 % y = saMODEL2 output (T,Q, c , opt , ’ raw ’ ) %saMODEL2sobol .m
7 %
8 % INPUTS:
9 % T,Q, c = Outputs from thermal model eva luat i on
10 % opt = data s t ru c tu r e conta in ing opt ions de f ined in saMODEL2.m
11 %
12
13 [ nk , nz , nt ] = s i z e (T) ;
14 i f nt > 1 ;
15 T = permute (T, [ 2 , 3 , 1 ] ) ; % Re−order so that the r e p l i c a t e s occupy the l a s t index
16 end
17 % 1 − GATHER DATA FOR EXPORT AND STORAGE (WHEN CALLED FROM SAMODEL2)
18 % Case when us ing s to red raw f i l e s , the data i s a l ready incremented
19 i f narg in == 5 && strcmpi ( ’ raw ’ , va rarg in {1}) ;
20 idx = 1 : s i z e (T, 2 ) ; % Does not increment the output
21 t s t ep = opt . inc *60 ; % Time step i s equal to the increment ( s )
22
23 % Case when running d i r e c t l y from SOBOL
24 e l s e
25 t s t ep = c (10) ; % Time step ( s )
26 inc = ( opt . inc *60) /c (10) ; % Storage increment
27 idx = 1 : inc : s i z e (T, 2 ) ; % Ind i c e s o f s to rage increment
28 end
29
30 % 2 − DEFINE THE DEPTH INDEX AND MEASURED DEPTH FOR GRAD. CALCULATIONS
31 z i = round ( opt . depth/c (9)+1) ; % Des ired depth index
32 dz = opt . depth /100 ; % Depth in meters
33
390
34 % 3 − COMPUTE THE DESIRED DATA TYPE
35 switch lower ( opt . type )
36 case ’ t ’ % snow temp .
37 y = squeeze (T( z i , : , : ) ) ;
38 case ’ tg ’ % temp . grad
39 y = squeeze ( ( d i f f (T( [ 1 , z i ] , : , : ) ) /dz ) ) ;
40 case ’ t k n e e ’ % temp . at knee
41 Tk = getknee (T) ; y = squeeze (Tk ( 2 , : , : ) ) ’ ;
42 case ’ ktg ’ % knee grad i ent
43 [Tk , d ] = getknee (T) ;
44 dz = (d−1)*c (9 ) /100 ;
45 y = squeeze ( d i f f (Tk , 1 , 1 ) ) ’ . / dz ;
46 %y( isnan (y ) ) = 0 ;
47 case ’ k d e p t h ’ % depth to knee
48 [Tk , d ] = getknee (T) ;
49 y = (d−1)*c (9 ) ;
50 case ’ k d u r a t i o n ’ ;
51 [Tk , d ] = getknee (T) ; d = d−1; d(d>0) = 1 ;
52 y = d* t s t ep /3600;
53 case ’ mf ’ % mass f l ux at su r f a c e
54 y = (Q/c (1) ) ;
55 end
56
57 % 4 − CORRECT FOR NaN VALUES FROM ‘ ‘KNEE’ ’ OUTPUTS
58 y ( i snan (y ) ) = 0 ;
59
60 % 5 − APPLY POS/NEG
61 i f i s c e l l ( opt . subtype ) ;
62 switch lower ( opt . subtype {2}) ;
63 case ’ pos ’ ; y (y<0) = 0 ;
64 case ’ neg ’ ; y (y>0) = 0 ;
65 end
66 t e s t = lower ( opt . subtype {1}) ;
67 e l s e
68 t e s t = lower ( opt . subtype ) ;
69 end
70
71 % 6 − COMPUTE THE SUBTYPE
72 % 6.1 − Re−o r i e n t s i n g l e column data
73 [ ry , cy ] = s i z e (y ) ;
74 i f cy == nk ; y = y ’ ; end
75
76 switch t e s t
77 case ’ m e a n ’ ; y = mean(y , 2 ) ;
78 case ’ min ’ ; y = min (y , [ ] , 2 ) ;
79 case ’ m i n _ t i m e ’ ; [ tmp , y ] = min (y , [ ] , 2 ) ; y = y*c (10) /3600;
80 case ’ max ’ ; y = max(y , [ ] , 2 ) ;
81 case ’ m a x _ t i m e ’ ; [ tmp , y ] = max(y , [ ] , 2 ) ; y = y*c (10) /3600;
82 case ’ t o t a l ’ ; y = sum(y , 2 ) ;
83 otherwi se % ALL
84 y = y ( : , idx , : ) ;
85 end
86
87 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
88 func t i on [ Tout , dpth idx , dur cnt ] = getknee (T)
89 % GETKNEE computes knee d i f f e r e n c e
90
91 dT = d i f f (T, 1 , 1 ) ;
92
93 % Correct f o r the ca s e s when the dT < 0 f o r the e n t i r e snowpack
94 A = sum(dT <= 0 ,1) == 0 ;
95 B = repmat (A, s i z e (dT, 1 ) ,1) ;
96 C = dT <= 0 + B;
97
98 [ rr , c ] = f i nd (C) ;
99 %[ rr , c ] = f i nd (dT <= 0) ;
100 [ c ,m, n ] = unique ( c , ’ f i r s t ’ ) ;
101 r ( 1 : l ength (m) ) = r r (m) ;
102
103 T = T(1 : end−1 , : , : ) ;
104 [ n1 , n2 , n3 ] = s i z e (T) ;
105
106 T2 = reshape (T , [ n1 , n2*n3 ] ) ;
107 ind = sub2ind ( s i z e (T2) , r , c ’ ) ;
108
109 Tout ( 1 , : , : ) = T( 1 , : , : ) ;
110 Tout ( 2 , : , : ) = reshape (T2( ind ) ,1 , n2 , n3 ) ;
111
112 dpth idx = reshape ( r , n2 , n3 ) ’ ;
113 dur cnt = squeeze (sum( ( d i f f (Tout , 1 , 1 ) > 0) ,2) ) ;
391
APPENDIX E
SENSITIVITY ANALYSIS RESULTS FOR SURFACE HOAR
392
E.1 Introduction
The tables presented in this appendix provide the complete sensitivity analysis re-
sults for Chapter 8 that explored surface hoar formation. The following tables include
the first-order, second-order, higher-order, and total-effect indices. The indices are
listed using the 90% confidence level intervals. The indices listed in Table 7.1 are used
in the tables presented here. The higher-order indices listed were computed from the
bias corrected first- and second-order and total-effect indices. The confidence levels
for this parameter were not computed, but would be of similar magnitude to the
confidence levels for the total-effect.
In this appendix only the night scenario results are listed, since the research focus
of Chapter 9 was on surface hoar. The three input “locations” were considered:
Control, North, and South. Chapter 7 includes the details on the development of the
input scenarios and locations. For each location four “classes” were considered: the
mass-flux, positive-only mass-flux, negative-only mass-flux, and snow temperature, all
at the snow surface. Note, the snow surface temperature results were not discussed
in Chapter 8, but included here since the Monte Carlo results presented in Chapter
8 utilized this parameter.
First, the sensitivity analysis was performed temporally at 20 minute intervals for
each of the classes mentioned, resulting in 30 sets of indices for each class. For these
temporal results, a table including only the total-effect indices as a function of time in
hours and the complete sensitivity results at mid-day are reported. Next, the mean,
maximum, and minimum were computed for each class. The caption for each table
is organized as location/class/type.
393
E.2 Mass Flux at Snow Surface
Table E.1: Control / Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 15.9–21.8 -1.5–4.8 -3.9–2.5 9.7–15.8 33.3–38.3 30.3–35.5 28.8–34.3 23.5–28.9
0.67 15.5–21.4 -2.5–4.0 -3.3–3.1 9.0–15.1 35.0–40.0 31.4–36.5 29.1–34.7 23.1–28.6
1.00 15.3–21.2 -2.6–3.9 -2.9–3.5 8.5–14.7 36.3–41.2 32.0–37.1 29.5–35.1 23.1–28.5
1.33 15.3–21.2 -2.5–3.9 -2.6–3.8 8.7–14.8 37.1–42.0 32.5–37.6 29.9–35.4 23.2–28.6
1.67 15.3–21.2 -2.5–3.9 -2.2–4.2 8.8–14.9 37.5–42.4 32.9–38.0 30.0–35.5 23.2–28.6
2.00 15.2–21.1 -2.5–3.8 -2.2–4.1 8.7–14.9 37.7–42.5 33.1–38.2 30.1–35.6 23.1–28.5
2.33 15.0–20.9 -2.6–3.7 -2.2–4.1 8.6–14.8 37.7–42.6 33.2–38.3 30.2–35.6 23.0–28.4
2.67 14.8–20.7 -2.8–3.6 -2.2–4.1 8.6–14.7 37.8–42.7 33.3–38.3 30.2–35.7 23.0–28.4
3.00 14.8–20.7 -2.7–3.6 -2.1–4.2 8.7–14.9 37.9–42.8 33.4–38.5 30.3–35.8 23.0–28.4
3.33 14.8–20.6 -2.7–3.6 -2.1–4.3 8.7–14.9 38.0–42.9 33.6–38.6 30.4–35.8 23.1–28.5
3.67 14.8–20.6 -2.7–3.6 -1.9–4.4 8.8–15.0 38.1–43.0 33.7–38.7 30.5–35.9 23.1–28.5
4.00 14.7–20.6 -2.7–3.6 -1.9–4.4 8.8–15.0 38.2–43.0 33.8–38.8 30.5–36.0 23.1–28.5
4.33 14.7–20.6 -2.7–3.6 -1.9–4.4 8.8–15.0 38.3–43.1 33.9–38.9 30.6–36.0 23.2–28.6
4.67 14.7–20.6 -2.7–3.7 -1.9–4.5 8.8–15.0 38.3–43.1 34.0–39.0 30.7–36.1 23.2–28.6
5.00 14.7–20.6 -2.6–3.7 -1.8–4.5 8.9–15.1 38.4–43.2 34.1–39.1 30.8–36.2 23.2–28.6
5.33 14.7–20.6 -2.6–3.7 -1.8–4.5 8.9–15.1 38.5–43.3 34.1–39.1 30.8–36.2 23.2–28.6
5.67 14.7–20.6 -2.6–3.7 -1.7–4.5 8.9–15.1 38.5–43.3 34.2–39.2 30.8–36.2 23.2–28.6
6.00 14.6–20.6 -2.5–3.8 -1.7–4.6 8.9–15.1 38.5–43.3 34.3–39.2 30.9–36.3 23.2–28.6
6.33 14.7–20.6 -2.4–3.9 -1.6–4.7 9.0–15.1 38.6–43.4 34.4–39.4 31.0–36.4 23.3–28.7
6.67 14.7–20.6 -2.4–3.9 -1.6–4.7 9.0–15.1 38.6–43.4 34.4–39.4 31.1–36.4 23.3–28.7
7.00 14.6–20.6 -2.4–3.8 -1.6–4.6 9.0–15.1 38.6–43.4 34.4–39.4 31.1–36.4 23.3–28.7
7.33 14.6–20.5 -2.4–3.8 -1.6–4.6 9.0–15.1 38.6–43.4 34.5–39.4 31.1–36.4 23.3–28.7
7.67 14.6–20.5 -2.5–3.9 -1.6–4.6 9.0–15.1 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
8.00 14.6–20.6 -2.5–3.9 -1.6–4.6 9.0–15.2 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
8.33 14.6–20.5 -2.5–3.9 -1.6–4.6 9.0–15.2 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
8.67 14.6–20.5 -2.5–3.9 -1.6–4.6 9.0–15.2 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
9.00 14.6–20.5 -2.5–3.8 -1.6–4.6 9.0–15.2 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
9.33 14.6–20.5 -2.5–3.8 -1.6–4.7 9.1–15.2 38.6–43.4 34.5–39.5 31.1–36.5 23.3–28.7
9.67 14.6–20.5 -2.4–3.9 -1.6–4.7 9.1–15.2 38.6–43.4 34.5–39.5 31.2–36.6 23.3–28.7
10.00 14.6–20.5 -2.5–3.9 -1.6–4.7 9.1–15.2 38.7–43.4 34.5–39.6 31.2–36.6 23.3–28.7
Table E.2: Control / Mass Flux / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 0.7–2.3 -0.7–2.1 -0.7–2.1 0.7–3.7 -1.5–2.3 0.1–3.1 -1.3–2.0 -0.6–2.5
2 -0.7–2.1 -0.0–0.2 -0.4–0.1 -0.4–0.1 -1.2–0.8 -1.2–0.1 -0.7–0.1 -0.4–0.4
3 -0.7–2.1 -0.4–0.1 -0.1–0.2 -0.4–0.3 -1.2–0.9 -1.1–0.2 -0.8–0.2 -0.5–0.4
4 0.7–3.7 -0.4–0.1 -0.4–0.3 -1.0–0.5 -1.4–2.0 -1.1–1.5 -1.1–1.9 -1.1–1.9
6 -1.5–2.3 -1.2–0.8 -1.2–0.9 -1.4–2.0 23.1–26.1 0.8–5.4 0.3–4.5 -2.0–1.9
9 0.1–3.1 -1.2–0.1 -1.1–0.2 -1.1–1.5 0.8–5.4 10.5–12.8 0.1–2.9 2.8–5.4
10 -1.3–2.0 -0.7–0.1 -0.8–0.2 -1.1–1.9 0.3–4.5 0.1–2.9 7.6–10.2 5.5–10.1
11 -0.6–2.5 -0.4–0.4 -0.5–0.4 -1.1–1.9 -2.0–1.9 2.8–5.4 5.5–10.1 9.6–11.7
Total 14.7–20.6 -2.6–3.7 -1.8–4.5 8.9–15.1 38.4–43.2 34.1–39.1 30.8–36.2 23.2–28.6
Higher 9.2 1.1 1.8 8.9 10.4 15.3 12.6 2.1
394
Table E.3: South / Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 38.2–45.9 1.7–11.2 -3.2–6.7 32.0–41.4 41.3–48.6 32.2–40.9 17.3–26.3 4.7–13.9
0.67 40.0–47.3 -1.7–8.1 -4.2–5.9 33.4–42.9 45.1–52.4 32.8–41.5 15.7–25.0 2.8–12.4
1.00 40.1–47.3 -3.2–6.6 -4.5–5.5 33.2–42.7 47.4–54.7 32.6–41.2 15.4–24.5 2.4–11.8
1.33 39.9–47.0 -4.2–5.7 -4.8–5.2 33.2–42.6 48.8–55.9 32.4–40.9 15.0–24.2 2.2–11.6
1.67 39.9–47.1 -3.8–6.0 -3.9–5.9 33.9–43.2 49.8–56.8 32.7–41.2 15.5–24.7 2.9–12.2
2.00 39.5–46.6 -3.9–5.8 -3.6–6.2 33.8–43.1 50.5–57.5 32.6–41.1 15.4–24.5 2.9–12.1
2.33 39.3–46.4 -4.1–5.6 -3.5–6.2 33.7–43.1 50.9–57.8 32.5–40.9 15.1–24.3 2.8–12.0
2.67 39.2–46.2 -4.1–5.5 -3.5–6.2 33.6–43.0 51.2–58.1 32.4–40.7 15.1–24.2 2.8–12.0
3.00 39.1–46.0 -4.1–5.4 -3.5–6.1 33.5–42.8 51.4–58.3 32.2–40.6 15.1–24.2 2.8–11.9
3.33 39.0–45.9 -4.2–5.3 -3.5–6.1 33.3–42.6 51.7–58.5 32.1–40.5 15.2–24.3 2.9–11.9
3.67 38.9–45.8 -4.3–5.2 -3.5–6.1 33.2–42.5 51.8–58.7 32.1–40.4 15.2–24.3 2.9–11.9
4.00 38.8–45.7 -4.4–5.1 -3.4–6.1 33.2–42.4 52.0–58.8 32.0–40.3 15.2–24.2 2.8–11.8
4.33 38.6–45.5 -4.6–5.0 -3.5–6.0 32.9–42.2 52.2–59.0 31.9–40.2 15.1–24.2 2.6–11.6
4.67 38.5–45.4 -4.6–5.0 -3.4–6.1 32.8–42.1 52.4–59.2 31.9–40.2 15.1–24.2 2.6–11.6
5.00 38.4–45.3 -4.5–5.0 -3.3–6.1 32.8–42.1 52.7–59.4 31.9–40.2 15.1–24.2 2.7–11.6
5.33 38.4–45.3 -4.5–5.0 -3.2–6.2 32.8–42.0 52.8–59.6 31.9–40.2 15.1–24.2 2.7–11.7
5.67 38.3–45.2 -4.5–5.0 -3.2–6.2 32.8–42.0 53.0–59.7 31.9–40.2 15.2–24.2 2.8–11.7
6.00 38.3–45.2 -4.4–5.1 -3.0–6.3 32.8–42.0 53.1–59.8 32.1–40.3 15.2–24.2 2.9–11.8
6.33 38.2–45.1 -4.5–5.0 -3.0–6.3 32.8–41.9 53.1–59.8 32.0–40.3 15.1–24.2 2.9–11.8
6.67 38.2–45.1 -4.5–5.0 -3.0–6.4 32.7–41.9 53.1–59.8 32.0–40.2 15.2–24.2 2.9–11.8
7.00 38.2–45.1 -4.4–5.1 -2.9–6.5 32.7–41.9 53.2–59.9 32.1–40.3 15.3–24.3 3.0–11.9
7.33 38.1–45.0 -4.4–5.1 -2.9–6.5 32.7–41.9 53.3–60.0 32.1–40.3 15.3–24.3 2.9–11.9
7.67 38.1–44.9 -4.4–5.1 -2.8–6.6 32.7–41.9 53.4–60.1 32.2–40.4 15.3–24.3 3.0–11.9
8.00 38.0–44.9 -4.4–5.1 -2.8–6.6 32.7–41.9 53.5–60.1 32.2–40.4 15.3–24.3 3.0–11.9
8.33 38.0–44.8 -4.4–5.1 -2.8–6.6 32.7–41.9 53.5–60.2 32.2–40.4 15.3–24.3 3.0–11.9
8.67 38.0–44.8 -4.3–5.2 -2.7–6.6 32.7–41.8 53.6–60.3 32.2–40.4 15.3–24.3 3.1–12.0
9.00 37.9–44.8 -4.3–5.2 -2.6–6.7 32.7–41.8 53.7–60.3 32.3–40.4 15.4–24.3 3.1–12.0
9.33 37.9–44.8 -4.3–5.2 -2.6–6.7 32.6–41.7 53.7–60.4 32.3–40.4 15.3–24.2 3.1–12.0
9.67 37.8–44.7 -4.3–5.1 -2.6–6.7 32.5–41.7 53.8–60.4 32.3–40.4 15.3–24.2 3.1–12.0
10.00 37.8–44.7 -4.3–5.1 -2.6–6.7 32.5–41.7 53.8–60.4 32.2–40.3 15.2–24.1 3.1–12.0
Table E.4: South / Mass Flux / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 0.4–3.6 -2.4–1.8 -2.3–1.9 4.0–9.4 -1.3–4.8 -1.0–4.5 -2.2–2.4 -2.1–2.2
2 -2.4–1.8 -0.2–0.1 -0.3–0.3 -0.3–0.4 -1.6–2.3 -0.5–0.6 -0.3–0.4 -0.3–0.4
3 -2.3–1.9 -0.3–0.3 -0.2–0.4 -0.9–0.2 -2.1–1.9 -1.1–0.3 -0.9–0.3 -0.8–0.4
4 4.0–9.4 -0.3–0.4 -0.9–0.2 -1.1–2.2 -4.7–1.9 -0.2–4.7 -2.6–2.4 -2.7–2.2
6 -1.3–4.8 -1.6–2.3 -2.1–1.9 -4.7–1.9 33.7–38.7 -2.2–5.2 -2.4–2.7 -1.7–1.8
9 -1.0–4.5 -0.5–0.6 -1.1–0.3 -0.2–4.7 -2.2–5.2 4.4–7.4 -2.3–1.6 -1.6–1.8
10 -2.2–2.4 -0.3–0.4 -0.9–0.3 -2.6–2.4 -2.4–2.7 -2.3–1.6 0.8–3.0 -1.8–1.9
11 -2.1–2.2 -0.3–0.4 -0.8–0.4 -2.7–2.2 -1.7–1.8 -1.6–1.8 -1.8–1.9 2.9–4.2
Total 38.4–45.3 -4.5–5.0 -3.3–6.1 32.8–42.1 52.7–59.4 31.9–40.2 15.1–24.2 2.7–11.6
Higher 30.0 0.1 2.9 30.0 17.5 25.2 18.1 3.7
395
Table E.5: North / Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 16.1–23.0 1.5–8.6 -4.1–3.1 16.2–23.1 57.2–62.2 10.0–17.2 15.0–21.7 2.0–8.9
0.67 20.4–27.0 0.1–7.1 -3.6–3.6 17.2–24.2 63.6–68.4 12.5–19.6 13.5–20.2 1.9–8.7
1.00 21.6–28.1 -0.9–6.0 -3.3–3.8 17.1–24.2 66.3–70.9 13.0–20.0 13.0–19.7 1.5–8.3
1.33 21.8–28.2 -1.9–5.1 -3.3–3.8 17.1–24.2 67.8–72.4 13.1–20.2 12.6–19.1 1.1–7.9
1.67 21.7–28.0 -2.9–4.2 -3.5–3.7 17.2–24.2 68.7–73.2 12.9–20.0 11.8–18.4 0.9–7.7
2.00 21.9–28.2 -3.1–3.9 -3.5–3.6 17.3–24.2 69.4–73.8 13.0–20.0 11.5–18.1 0.8–7.5
2.33 22.1–28.3 -3.5–3.6 -3.4–3.6 17.2–24.1 70.0–74.4 13.0–20.0 11.4–18.0 0.6–7.4
2.67 22.2–28.4 -3.4–3.6 -3.2–3.9 17.3–24.2 70.6–74.9 13.2–20.1 11.3–18.0 0.7–7.5
3.00 22.1–28.3 -3.5–3.5 -3.1–3.9 17.2–24.1 71.0–75.3 13.3–20.2 11.1–17.8 0.6–7.4
3.33 22.1–28.2 -3.7–3.3 -3.1–3.9 17.3–24.2 71.3–75.6 13.4–20.2 11.0–17.7 0.5–7.3
3.67 21.9–28.1 -3.8–3.2 -3.2–3.8 17.1–24.0 71.6–75.9 13.3–20.2 10.9–17.5 0.4–7.2
4.00 21.9–28.0 -3.8–3.1 -3.2–3.8 17.2–24.0 71.9–76.2 13.5–20.2 10.9–17.6 0.4–7.2
4.33 21.8–28.0 -3.9–3.0 -3.2–3.8 17.2–24.1 72.3–76.5 13.4–20.2 10.8–17.5 0.4–7.1
4.67 21.7–27.9 -4.1–2.9 -3.2–3.8 17.3–24.1 72.5–76.7 13.4–20.2 10.8–17.4 0.3–7.0
5.00 21.7–27.8 -4.1–2.8 -3.2–3.8 17.3–24.1 72.7–76.9 13.4–20.2 10.7–17.3 0.2–6.9
5.33 21.7–27.8 -4.1–2.9 -3.1–3.8 17.3–24.1 72.9–77.1 13.4–20.2 10.7–17.3 0.2–6.9
5.67 21.6–27.8 -4.1–2.8 -3.2–3.8 17.2–24.0 73.0–77.2 13.4–20.2 10.6–17.3 0.2–6.9
6.00 21.6–27.7 -4.2–2.7 -3.2–3.7 17.1–24.0 73.1–77.3 13.3–20.1 10.6–17.2 0.1–6.8
6.33 21.5–27.7 -4.2–2.7 -3.2–3.7 17.1–23.9 73.1–77.3 13.3–20.1 10.5–17.1 0.0–6.8
6.67 21.5–27.6 -4.3–2.7 -3.2–3.7 17.1–23.9 73.1–77.4 13.3–20.1 10.5–17.1 0.1–6.8
7.00 21.4–27.5 -4.3–2.7 -3.2–3.7 17.0–23.8 73.2–77.4 13.3–20.0 10.4–17.0 0.0–6.7
7.33 21.4–27.5 -4.2–2.7 -3.2–3.6 17.0–23.8 73.2–77.5 13.3–20.0 10.3–17.0 -0.0–6.7
7.67 21.3–27.5 -4.3–2.6 -3.3–3.6 16.9–23.7 73.3–77.5 13.2–20.0 10.3–16.9 -0.1–6.7
8.00 21.3–27.4 -4.3–2.6 -3.3–3.6 16.8–23.7 73.3–77.6 13.2–19.9 10.2–16.9 -0.1–6.6
8.33 21.3–27.4 -4.3–2.6 -3.3–3.5 16.8–23.6 73.4–77.6 13.1–19.9 10.2–16.8 -0.1–6.6
8.67 21.2–27.3 -4.4–2.6 -3.3–3.5 16.8–23.6 73.5–77.7 13.1–19.9 10.2–16.8 -0.1–6.6
9.00 21.2–27.3 -4.4–2.5 -3.4–3.5 16.8–23.6 73.5–77.7 13.1–19.8 10.1–16.8 -0.1–6.6
9.33 21.1–27.2 -4.4–2.5 -3.4–3.5 16.7–23.5 73.6–77.8 13.0–19.8 10.1–16.7 -0.1–6.6
9.67 21.1–27.2 -4.4–2.5 -3.4–3.4 16.7–23.5 73.6–77.8 13.0–19.8 10.1–16.7 -0.1–6.6
10.00 21.0–27.1 -4.4–2.5 -3.4–3.4 16.7–23.5 73.7–77.9 13.0–19.8 10.0–16.7 -0.1–6.5
Table E.6: North / Mass Flux / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 -0.8–1.8 -1.3–2.1 -0.9–2.5 4.6–8.8 -0.8–6.5 -0.8–3.2 -0.6–3.0 -0.7–2.7
2 -1.3–2.1 -0.2–0.2 -0.4–0.4 -0.5–0.3 -3.9–2.4 -0.4–0.4 -0.5–0.3 -0.4–0.4
3 -0.9–2.5 -0.4–0.4 -0.5–-0.0 0.1–1.1 -3.5–2.8 0.0–0.9 -0.2–0.8 0.0–0.9
4 4.6–8.8 -0.5–0.3 0.1–1.1 -0.8–0.8 -3.5–3.3 -1.5–1.6 -2.3–1.1 -2.6–0.7
6 -0.8–6.5 -3.9–2.4 -3.5–2.8 -3.5–3.3 54.4–59.6 -1.4–4.1 -2.0–2.0 -0.9–1.3
9 -0.8–3.2 -0.4–0.4 0.0–0.9 -1.5–1.6 -1.4–4.1 0.8–2.3 -0.9–1.6 -0.9–1.3
10 -0.6–3.0 -0.5–0.3 -0.2–0.8 -2.3–1.1 -2.0–2.0 -0.9–1.6 4.5–6.1 -0.6–2.2
11 -0.7–2.7 -0.4–0.4 0.0–0.9 -2.6–0.7 -0.9–1.3 -0.9–1.3 -0.6–2.2 3.1–3.9
Total 21.7–27.8 -4.1–2.8 -3.2–3.8 17.3–24.1 72.7–76.9 13.4–20.2 10.7–17.3 0.2–6.9
Higher 10.0 -0.2 -1.9 14.9 14.5 11.6 6.7 -1.7
396
Table E.7: Control / Mass Flux / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 0.8–2.5 -0.8–2.1 -0.8–2.1 0.8–3.7 -1.7–2.2 0.1–3.0 -1.3–1.9 -0.6–2.5
2 -0.8–2.1 0.0–0.2 -0.4–0.0 -0.3–0.1 -1.2–0.8 -1.1–0.1 -0.7–0.1 -0.4–0.4
3 -0.8–2.1 -0.4–0.0 -0.1–0.2 -0.4–0.3 -1.2–0.8 -1.0–0.2 -0.7–0.2 -0.5–0.4
4 0.8–3.7 -0.3–0.1 -0.4–0.3 -1.0–0.5 -1.6–1.9 -1.1–1.5 -1.1–1.9 -1.0–1.9
6 -1.7–2.2 -1.2–0.8 -1.2–0.8 -1.6–1.9 23.2–26.0 0.7–5.3 0.1–4.3 -2.0–1.8
9 0.1–3.0 -1.1–0.1 -1.0–0.2 -1.1–1.5 0.7–5.3 10.3–12.6 0.2–2.9 2.9–5.5
10 -1.3–1.9 -0.7–0.1 -0.7–0.2 -1.1–1.9 0.1–4.3 0.2–2.9 7.7–10.3 5.6–10.3
11 -0.6–2.5 -0.4–0.4 -0.5–0.4 -1.0–1.9 -2.0–1.8 2.9–5.5 5.6–10.3 9.7–11.8
Total 14.6–20.3 -2.8–3.5 -2.3–4.0 8.6–14.7 37.7–42.6 33.5–38.6 30.4–35.8 23.4–28.7
Higher 9.2 1.0 1.4 8.7 10.5 14.9 12.3 1.9
Table E.8: South / Mass Flux / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 0.4–3.7 -2.5–1.7 -2.4–1.8 4.1–9.4 -0.9–4.9 -1.2–4.5 -2.1–2.5 -2.1–2.2
2 -2.5–1.7 -0.1–0.2 -0.2–0.2 -0.4–0.2 -1.6–2.2 -0.7–0.4 -0.3–0.3 -0.3–0.2
3 -2.4–1.8 -0.2–0.2 -0.1–0.4 -0.9–0.2 -2.0–1.8 -1.1–0.3 -0.8–0.2 -0.8–0.3
4 4.1–9.4 -0.4–0.2 -0.9–0.2 -1.1–2.0 -4.5–1.7 -0.5–4.4 -2.7–2.3 -2.7–2.1
6 -0.9–4.9 -1.6–2.2 -2.0–1.8 -4.5–1.7 33.4–38.4 -2.3–5.2 -2.3–2.5 -1.6–1.8
9 -1.2–4.5 -0.7–0.4 -1.1–0.3 -0.5–4.4 -2.3–5.2 4.5–7.5 -2.2–1.5 -1.4–1.9
10 -2.1–2.5 -0.3–0.3 -0.8–0.2 -2.7–2.3 -2.3–2.5 -2.2–1.5 1.1–3.3 -1.6–2.1
11 -2.1–2.2 -0.3–0.2 -0.8–0.3 -2.7–2.1 -1.6–1.8 -1.4–1.9 -1.6–2.1 3.1–4.4
Total 37.9–45.1 -5.1–5.0 -4.1–5.8 32.3–41.8 51.6–58.4 31.7–40.1 14.5–23.8 2.4–11.9
Higher 29.5 0.2 2.4 30.2 16.7 25.4 17.3 3.4
Table E.9: North / Mass Flux / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 -0.7–1.9 -1.4–1.9 -1.1–2.2 4.5–8.6 -1.1–6.1 -0.9–2.9 -0.8–2.7 -0.9–2.4
2 -1.4–1.9 -0.1–0.2 -0.3–0.4 -0.3–0.3 -2.4–4.0 -0.4–0.2 -0.5–0.2 -0.3–0.3
3 -1.1–2.2 -0.3–0.4 -0.4–0.0 0.1–0.9 -3.9–2.4 -0.0–0.7 -0.3–0.6 -0.0–0.7
4 4.5–8.6 -0.3–0.3 0.1–0.9 -0.7–0.9 -3.8–2.9 -1.7–1.5 -2.4–0.9 -2.7–0.5
6 -1.1–6.1 -2.4–4.0 -3.9–2.4 -3.8–2.9 54.6–59.8 -1.5–3.9 -2.4–1.3 -0.9–1.3
9 -0.9–2.9 -0.4–0.2 -0.0–0.7 -1.7–1.5 -1.5–3.9 0.9–2.3 -1.0–1.2 -0.9–1.2
10 -0.8–2.7 -0.5–0.2 -0.3–0.6 -2.4–0.9 -2.4–1.3 -1.0–1.2 5.1–6.7 -0.5–2.2
11 -0.9–2.4 -0.3–0.3 -0.0–0.7 -2.7–0.5 -0.9–1.3 -0.9–1.2 -0.5–2.2 3.2–4.1
Total 20.7–27.1 -4.1–3.1 -3.7–3.5 16.3–23.3 71.7–75.8 12.6–19.4 10.4–17.3 0.3–7.3
Higher 10.7 -1.3 -1.1 15.1 13.5 11.8 7.4 -1.1
397
Table E.10: Control / Positive Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 -4.6–10.7 -10.5–5.5 -8.1–6.9 -4.6–11.2 10.1–23.7 11.6–26.1 63.8–71.2 47.7–56.9
0.67 -6.1–9.8 -6.6–8.3 -8.0–7.0 -6.3–9.7 11.5–25.2 11.9–26.4 64.1–71.7 49.7–58.8
1.00 -6.7–9.4 -6.9–8.1 -7.8–7.3 -6.9–9.3 11.9–25.6 11.9–26.6 64.4–72.0 50.6–59.7
1.33 -7.1–9.1 -7.2–8.0 -7.8–7.4 -7.3–9.0 11.8–25.6 11.9–26.7 64.7–72.3 51.2–60.3
1.67 -7.6–8.7 -7.5–7.8 -8.0–7.3 -7.6–8.7 11.6–25.5 11.7–26.7 64.8–72.4 51.5–60.6
2.00 -8.0–8.4 -7.7–7.6 -8.1–7.3 -7.9–8.5 11.5–25.5 11.5–26.6 64.9–72.5 51.7–60.9
2.33 -8.3–8.2 -7.9–7.5 -8.2–7.3 -8.2–8.3 11.4–25.4 11.4–26.7 65.0–72.6 51.9–61.0
2.67 -8.5–8.1 -8.0–7.4 -8.3–7.3 -8.3–8.2 11.3–25.4 11.4–26.7 65.1–72.6 51.9–61.1
3.00 -8.7–8.0 -8.1–7.4 -8.4–7.2 -8.4–8.2 11.2–25.3 11.4–26.7 65.1–72.7 52.1–61.2
3.33 -8.8–7.9 -8.2–7.3 -8.4–7.2 -8.6–8.1 11.2–25.3 11.4–26.7 65.1–72.7 52.1–61.3
3.67 -8.9–7.8 -8.2–7.3 -8.4–7.2 -8.6–8.1 11.1–25.3 11.4–26.7 65.1–72.7 52.2–61.4
4.00 -9.0–7.8 -8.3–7.3 -8.4–7.2 -8.7–8.0 11.0–25.2 11.4–26.7 65.1–72.7 52.2–61.4
4.33 -9.1–7.7 -8.3–7.3 -8.4–7.2 -8.8–8.0 11.0–25.2 11.4–26.7 65.1–72.7 52.3–61.5
4.67 -9.2–7.7 -8.4–7.3 -8.4–7.1 -8.8–8.0 10.9–25.2 11.3–26.7 65.2–72.8 52.3–61.5
5.00 -9.2–7.7 -8.4–7.3 -8.4–7.2 -8.9–8.0 10.9–25.1 11.3–26.8 65.2–72.8 52.4–61.6
5.33 -9.3–7.7 -8.5–7.2 -8.5–7.1 -8.9–8.0 10.9–25.1 11.3–26.8 65.2–72.8 52.4–61.6
5.67 -9.3–7.6 -8.5–7.2 -8.5–7.1 -8.9–8.0 10.8–25.0 11.3–26.7 65.2–72.8 52.5–61.6
6.00 -9.3–7.6 -8.5–7.2 -8.5–7.1 -8.9–8.0 10.8–25.0 11.3–26.8 65.2–72.8 52.5–61.7
6.33 -9.3–7.6 -8.5–7.1 -8.5–7.1 -8.9–8.0 10.7–25.0 11.3–26.7 65.2–72.8 52.5–61.7
6.67 -9.3–7.6 -8.5–7.1 -8.5–7.1 -8.9–8.0 10.7–25.0 11.3–26.8 65.2–72.8 52.5–61.7
7.00 -9.4–7.6 -8.5–7.2 -8.5–7.1 -8.9–8.0 10.7–24.9 11.3–26.8 65.2–72.8 52.5–61.7
7.33 -9.4–7.6 -8.5–7.1 -8.5–7.1 -8.9–7.9 10.6–24.9 11.2–26.8 65.2–72.9 52.6–61.7
7.67 -9.4–7.5 -8.6–7.1 -8.5–7.1 -9.0–7.9 10.6–24.9 11.3–26.8 65.2–72.9 52.6–61.7
8.00 -9.4–7.5 -8.6–7.1 -8.5–7.1 -9.0–7.9 10.5–24.8 11.3–26.8 65.2–72.9 52.6–61.8
8.33 -9.4–7.5 -8.6–7.1 -8.5–7.1 -9.0–7.9 10.5–24.8 11.3–26.8 65.2–72.9 52.6–61.8
8.67 -9.4–7.5 -8.6–7.1 -8.5–7.1 -9.0–7.9 10.4–24.8 11.3–26.8 65.2–72.9 52.6–61.8
9.00 -9.4–7.5 -8.6–7.1 -8.6–7.1 -9.0–7.9 10.4–24.8 11.3–26.8 65.2–72.9 52.6–61.8
9.33 -9.4–7.5 -8.6–7.1 -8.6–7.1 -9.0–7.9 10.4–24.8 11.3–26.8 65.2–72.9 52.7–61.8
9.67 -9.4–7.5 -8.6–7.1 -8.6–7.1 -9.0–7.9 10.4–24.7 11.3–26.8 65.2–72.9 52.6–61.8
10.00 -9.5–7.5 -8.6–7.1 -8.5–7.1 -9.1–7.9 10.3–24.7 11.3–26.8 65.3–72.9 52.7–61.8
Table E.11: Control / Mass Flux / Positive Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 -0.4–0.3 -0.4–0.9 -0.5–0.8 -0.2–1.2 -0.7–0.8 -0.5–1.0 -3.2–2.9 -0.7–2.6
2 -0.4–0.9 -0.1–0.1 -0.1–0.1 -0.1–0.2 -0.5–0.1 -0.2–0.2 -3.3–2.3 -0.9–1.9
3 -0.5–0.8 -0.1–0.1 -0.1–0.1 -0.3–0.1 -0.6–0.1 -0.5–0.1 -3.4–2.1 -0.9–2.0
4 -0.2–1.2 -0.1–0.2 -0.3–0.1 -0.7–0.0 -0.5–1.2 -0.2–1.2 -2.5–3.2 -0.6–2.4
6 -0.7–0.8 -0.5–0.1 -0.6–0.1 -0.5–1.2 4.2–6.3 -1.5–2.0 1.2–10.1 2.2–8.3
9 -0.5–1.0 -0.2–0.2 -0.5–0.1 -0.2–1.2 -1.5–2.0 1.0–2.7 0.5–8.0 1.2–5.4
10 -3.2–2.9 -3.3–2.3 -3.4–2.1 -2.5–3.2 1.2–10.1 0.5–8.0 16.8–22.3 24.0–34.9
11 -0.7–2.6 -0.9–1.9 -0.9–2.0 -0.6–2.4 2.2–8.3 1.2–5.4 24.0–34.9 8.5–12.0
Total -9.2–7.7 -8.4–7.3 -8.4–7.2 -8.9–8.0 10.9–25.1 11.3–26.8 65.2–72.8 52.4–61.6
Higher -2.7 -0.6 -0.3 -2.6 1.7 8.8 11.1 5.9
398
Table E.12: South / Positive Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 35.6–46.7 1.4–15.1 -5.5–9.2 32.6–44.1 36.6–47.7 24.3–36.2 42.8–53.4 11.0–23.3
0.67 41.6–52.0 -0.7–13.0 -3.5–10.4 30.9–42.6 43.9–54.2 28.2–39.6 38.5–49.3 10.4–22.7
1.00 43.4–53.3 -1.2–12.7 -2.7–11.4 30.5–42.3 47.0–57.0 29.8–41.3 36.4–47.6 10.4–22.9
1.33 44.2–53.8 -1.6–12.3 -2.5–11.7 30.4–42.2 48.8–58.5 30.5–41.9 35.3–46.6 10.4–22.7
1.67 44.5–54.2 -2.2–11.9 -2.3–11.9 29.9–41.8 49.8–59.5 30.7–42.0 34.6–45.9 10.4–22.7
2.00 44.8–54.4 -2.6–11.5 -2.1–12.0 29.8–41.7 50.4–60.0 30.7–42.0 33.6–45.2 10.3–22.6
2.33 45.1–54.6 -2.4–11.6 -1.6–12.3 29.9–41.9 50.9–60.5 31.0–42.3 33.3–44.9 10.4–22.8
2.67 45.4–54.8 -2.4–11.6 -1.4–12.5 30.1–42.0 51.4–61.0 31.3–42.5 32.9–44.6 10.3–22.6
3.00 45.4–54.9 -2.3–11.7 -1.3–12.5 30.2–42.2 51.8–61.4 31.4–42.6 32.8–44.4 10.3–22.6
3.33 45.5–55.0 -2.2–11.7 -1.2–12.5 30.4–42.4 52.2–61.7 31.5–42.8 32.7–44.3 10.2–22.6
3.67 45.4–55.1 -2.2–11.7 -1.1–12.6 30.5–42.5 52.4–61.9 31.7–42.9 32.5–44.1 10.3–22.5
4.00 45.5–55.1 -2.2–11.7 -1.2–12.5 30.5–42.5 52.6–62.1 31.7–42.9 32.3–43.9 10.3–22.5
4.33 45.6–55.1 -2.1–11.7 -1.1–12.5 30.5–42.5 52.9–62.3 31.8–43.0 32.2–43.7 10.4–22.6
4.67 45.5–55.1 -2.1–11.6 -1.2–12.5 30.4–42.4 53.0–62.4 31.8–43.0 32.0–43.5 10.4–22.5
5.00 45.5–55.0 -2.2–11.6 -1.3–12.4 30.3–42.3 53.1–62.5 31.8–43.0 31.8–43.4 10.3–22.5
5.33 45.4–55.0 -2.2–11.5 -1.3–12.4 30.3–42.3 53.2–62.6 31.8–43.0 31.6–43.2 10.2–22.4
5.67 45.4–55.0 -2.2–11.5 -1.2–12.4 30.3–42.3 53.4–62.7 31.8–43.0 31.5–43.2 10.3–22.4
6.00 45.4–55.0 -2.2–11.5 -1.2–12.4 30.3–42.3 53.5–62.8 31.9–43.1 31.5–43.1 10.3–22.4
6.33 45.4–54.9 -2.2–11.5 -1.2–12.3 30.3–42.3 53.6–62.9 31.9–43.1 31.4–43.0 10.2–22.4
6.67 45.4–54.9 -2.3–11.4 -1.3–12.3 30.2–42.2 53.6–62.9 31.9–43.1 31.3–42.9 10.1–22.4
7.00 45.3–54.9 -2.3–11.4 -1.2–12.3 30.2–42.2 53.7–63.0 31.9–43.1 31.2–42.8 10.1–22.4
7.33 45.3–54.9 -2.3–11.3 -1.2–12.4 30.2–42.1 53.8–63.1 31.9–43.1 31.1–42.8 10.0–22.3
7.67 45.3–54.9 -2.4–11.3 -1.2–12.4 30.2–42.1 53.8–63.1 31.9–43.1 31.0–42.7 10.0–22.3
8.00 45.3–54.8 -2.4–11.3 -1.1–12.4 30.1–42.0 53.9–63.2 31.9–43.2 30.9–42.6 10.0–22.3
8.33 45.3–54.8 -2.4–11.3 -1.1–12.4 30.1–42.0 53.9–63.2 32.0–43.2 30.9–42.6 10.0–22.3
8.67 45.3–54.8 -2.4–11.3 -1.1–12.4 30.1–42.0 54.0–63.3 32.0–43.2 30.8–42.6 10.0–22.3
9.00 45.3–54.8 -2.4–11.3 -1.1–12.5 30.0–42.0 54.0–63.3 32.0–43.2 30.8–42.5 10.0–22.3
9.33 45.3–54.8 -2.4–11.3 -1.1–12.5 30.0–42.0 54.1–63.4 32.0–43.2 30.7–42.5 10.0–22.3
9.67 45.3–54.8 -2.4–11.3 -1.0–12.5 30.0–42.0 54.1–63.4 32.0–43.2 30.7–42.4 10.0–22.3
10.00 45.2–54.8 -2.4–11.3 -1.0–12.5 30.0–42.0 54.1–63.4 32.0–43.2 30.6–42.4 9.9–22.3
Table E.13: South / Mass Flux / Positive Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 2.0–6.0 -0.3–3.4 -0.7–3.1 4.7–10.5 -0.4–6.9 5.3–11.3 -1.2–4.4 0.0–4.0
2 -0.3–3.4 -0.1–0.1 -0.3–0.2 -0.2–0.4 -3.4–1.2 -0.3–0.3 -1.2–0.1 -0.4–0.2
3 -0.7–3.1 -0.3–0.2 -0.3–0.3 -0.5–0.6 -3.4–1.2 -0.6–0.4 -1.6–0.1 -0.7–0.5
4 4.7–10.5 -0.2–0.4 -0.5–0.6 -1.0–1.6 -4.2–2.4 0.3–5.4 -2.3–2.5 -1.6–2.6
6 -0.4–6.9 -3.4–1.2 -3.4–1.2 -4.2–2.4 19.4–24.5 -3.3–3.5 1.6–8.2 -0.1–4.3
9 5.3–11.3 -0.3–0.3 -0.6–0.4 0.3–5.4 -3.3–3.5 0.1–2.5 -2.0–1.7 -1.1–2.1
10 -1.2–4.4 -1.2–0.1 -1.6–0.1 -2.3–2.5 1.6–8.2 -2.0–1.7 3.8–7.2 -0.7–3.2
11 0.0–4.0 -0.4–0.2 -0.7–0.5 -1.6–2.6 -0.1–4.3 -1.1–2.1 -0.7–3.2 1.0–2.4
Total 45.5–55.0 -2.2–11.6 -1.3–12.4 30.3–42.3 53.1–62.5 31.8–43.0 31.8–43.4 10.3–22.5
Higher 20.8 4.8 6.4 25.8 28.5 24.7 25.6 8.5
399
Table E.14: North / Positive Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 9.9–19.8 -2.8–8.1 -5.4–5.3 20.2–29.3 34.2–42.0 2.9–13.3 44.5–51.7 7.2–16.6
0.67 15.7–25.4 -4.7–6.3 -5.1–5.7 18.0–27.6 41.6–49.1 4.9–15.5 40.1–47.8 6.6–16.2
1.00 18.9–28.5 -5.0–5.9 -4.1–6.6 17.2–26.9 45.0–52.5 6.4–16.9 38.5–46.3 6.8–16.4
1.33 20.2–29.6 -5.7–5.3 -7.3–3.8 16.5–26.3 47.0–54.3 6.8–17.4 37.0–44.9 6.3–16.0
1.67 20.8–30.2 -6.1–4.9 -7.2–3.9 16.0–25.7 48.4–55.5 7.1–17.7 35.8–43.8 6.0–15.7
2.00 21.3–30.7 -6.4–4.7 -6.8–4.3 15.7–25.4 49.5–56.5 7.6–18.1 35.1–43.2 6.0–15.7
2.33 21.8–31.1 -6.4–4.7 -6.4–4.6 15.6–25.3 50.4–57.4 7.9–18.4 34.7–42.8 6.1–15.7
2.67 21.9–31.3 -6.4–4.6 -6.2–4.7 15.4–25.1 51.1–58.0 8.1–18.6 34.4–42.5 6.2–15.8
3.00 22.0–31.3 -6.5–4.5 -6.2–4.8 15.2–24.9 51.7–58.4 8.2–18.7 34.1–42.2 6.2–15.8
3.33 22.0–31.3 -6.7–4.3 -6.3–4.7 15.0–24.7 52.1–58.8 8.1–18.7 33.8–41.9 6.1–15.7
3.67 22.0–31.3 -6.8–4.2 -6.4–4.6 14.8–24.5 52.4–59.1 8.1–18.6 33.5–41.6 6.1–15.7
4.00 22.0–31.4 -6.9–4.1 -6.5–4.5 14.6–24.4 52.7–59.4 8.1–18.6 33.3–41.4 6.1–15.7
4.33 22.0–31.4 -7.0–4.1 -6.5–4.6 14.5–24.3 53.0–59.7 8.1–18.6 33.0–41.2 6.1–15.7
4.67 22.0–31.4 -7.0–4.0 -6.4–4.6 14.4–24.3 53.2–59.9 8.2–18.7 32.9–41.0 6.1–15.7
5.00 22.1–31.4 -7.0–4.0 -6.4–4.7 14.4–24.3 53.5–60.1 8.2–18.7 32.7–40.9 6.0–15.6
5.33 22.1–31.4 -7.0–4.0 -6.4–4.7 14.4–24.3 53.6–60.3 8.2–18.7 32.6–40.7 6.0–15.6
5.67 22.1–31.4 -7.1–3.9 -6.5–4.7 14.3–24.2 53.8–60.4 8.1–18.6 32.4–40.5 5.9–15.6
6.00 22.0–31.4 -7.2–3.8 -6.5–4.6 14.2–24.1 53.9–60.5 8.0–18.5 32.2–40.4 5.8–15.5
6.33 22.0–31.3 -7.3–3.8 -6.6–4.5 14.1–24.1 54.0–60.6 7.9–18.5 32.0–40.2 5.7–15.4
6.67 21.9–31.3 -4.1–6.5 -6.6–4.5 14.0–24.0 54.2–60.7 7.9–18.4 31.8–40.1 5.6–15.3
7.00 21.9–31.3 -4.1–6.4 -6.6–4.5 13.9–23.9 54.3–60.8 7.8–18.4 31.7–39.9 5.6–15.2
7.33 21.9–31.2 -4.2–6.3 -6.6–4.5 13.9–23.8 54.4–60.9 7.8–18.3 31.6–39.8 5.5–15.2
7.67 21.9–31.2 -4.3–6.3 -6.6–4.4 13.8–23.7 54.5–61.0 7.7–18.3 31.5–39.7 5.5–15.2
8.00 21.8–31.2 -4.3–6.2 -6.7–4.4 13.7–23.7 54.6–61.1 7.7–18.2 31.4–39.7 5.4–15.1
8.33 21.8–31.1 -4.4–6.2 -6.7–4.4 13.6–23.6 54.7–61.2 7.6–18.2 31.3–39.6 5.4–15.1
8.67 21.7–31.1 -4.4–6.1 -6.7–4.4 13.5–23.5 54.8–61.3 7.6–18.1 31.2–39.5 5.3–15.1
9.00 21.7–31.1 -4.5–6.1 -6.8–4.4 13.5–23.5 54.9–61.4 7.5–18.1 31.1–39.4 5.3–15.0
9.33 21.7–31.0 -4.5–6.1 -6.7–4.3 13.4–23.4 54.9–61.4 7.5–18.1 31.1–39.4 5.3–15.0
9.67 21.7–31.0 -4.5–6.1 -6.7–4.3 13.4–23.4 55.0–61.5 7.5–18.0 31.0–39.3 5.3–15.0
10.00 21.6–31.0 -4.5–6.0 -6.8–4.3 13.4–23.3 55.1–61.6 7.5–18.0 30.9–39.2 5.3–15.0
Table E.15: North / Mass Flux / Positive Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 2.1–5.0 -1.7–1.8 -1.6–1.9 1.8–6.0 -0.9–5.6 0.8–4.7 -1.7–2.7 -1.8–1.9
2 -1.7–1.8 -0.1–0.2 -0.4–0.1 -0.4–0.2 -2.9–1.8 -0.4–0.2 -1.8–0.6 -0.4–0.2
3 -1.6–1.9 -0.4–0.1 -0.0–0.4 -0.8–0.1 -3.0–1.8 -0.7–0.1 -2.2–0.3 -0.7–0.2
4 1.8–6.0 -0.4–0.2 -0.8–0.1 -1.1–0.7 -3.3–2.1 -2.0–1.2 -2.0–1.8 -1.3–2.0
6 -0.9–5.6 -2.9–1.8 -3.0–1.8 -3.3–2.1 27.9–32.3 -1.7–2.4 4.3–11.6 2.0–5.3
9 0.8–4.7 -0.4–0.2 -0.7–0.1 -2.0–1.2 -1.7–2.4 0.2–1.6 -2.4–1.1 -1.6–1.0
10 -1.7–2.7 -1.8–0.6 -2.2–0.3 -2.0–1.8 4.3–11.6 -2.4–1.1 15.2–18.3 0.7–3.5
11 -1.8–1.9 -0.4–0.2 -0.7–0.2 -1.3–2.0 2.0–5.3 -1.6–1.0 0.7–3.5 3.8–4.7
Total 22.1–31.4 -7.0–4.0 -6.4–4.7 14.4–24.3 53.5–60.1 8.2–18.7 32.7–40.9 6.0–15.6
Higher 13.5 -0.1 1.4 16.8 14.2 11.1 11.7 1.1
400
Table E.16: Control / Mass Flux / Positive Mean
@
@i
j
1 2 3 4 6 9 10 11
1 -0.4–0.4 -0.5–0.9 -0.5–0.8 -0.3–1.2 -0.7–0.8 -0.5–1.0 -3.0–2.8 -0.7–2.6
2 -0.5–0.9 -0.0–0.1 -0.2–0.1 -0.1–0.1 -0.6–0.1 -0.3–0.2 -3.3–2.3 -1.0–1.9
3 -0.5–0.8 -0.2–0.1 -0.1–0.1 -0.3–0.1 -0.6–0.1 -0.5–0.1 -3.4–2.2 -1.0–2.0
4 -0.3–1.2 -0.1–0.1 -0.3–0.1 -0.7–0.1 -0.5–1.2 -0.3–1.2 -2.5–3.2 -0.7–2.5
6 -0.7–0.8 -0.6–0.1 -0.6–0.1 -0.5–1.2 4.2–6.2 -1.5–2.0 1.6–10.1 2.2–8.3
9 -0.5–1.0 -0.3–0.2 -0.5–0.1 -0.3–1.2 -1.5–2.0 1.1–2.7 0.7–7.9 1.2–5.4
10 -3.0–2.8 -3.3–2.3 -3.4–2.2 -2.5–3.2 1.6–10.1 0.7–7.9 17.0–22.5 23.6–34.6
11 -0.7–2.6 -1.0–1.9 -1.0–2.0 -0.7–2.5 2.2–8.3 1.2–5.4 23.6–34.6 8.8–12.2
Total -8.6–7.7 -7.8–7.1 -8.0–6.9 -8.5–7.9 10.7–24.7 11.5–26.5 65.1–72.7 52.1–61.2
Higher -2.3 -0.1 -0.0 -2.4 1.2 8.7 10.8 5.8
Table E.17: South / Mass Flux / Positive Mean
@
@i
j
1 2 3 4 6 9 10 11
1 1.6–5.7 -0.4–3.4 -0.7–3.1 4.6–10.6 -0.2–7.0 5.6–11.3 -1.3–4.5 -0.1–3.9
2 -0.4–3.4 -0.2–0.1 -0.3–0.4 -0.4–0.3 -3.3–1.3 -0.5–0.3 -1.3–0.3 -0.4–0.4
3 -0.7–3.1 -0.3–0.4 -0.3–0.3 -0.5–0.6 -3.3–1.3 -0.5–0.5 -1.6–0.1 -0.6–0.5
4 4.6–10.6 -0.4–0.3 -0.5–0.6 -1.2–1.4 -3.9–2.8 0.5–5.6 -2.3–2.6 -1.5–2.8
6 -0.2–7.0 -3.3–1.3 -3.3–1.3 -3.9–2.8 18.8–24.0 -2.9–3.5 2.0–8.5 0.2–4.6
9 5.6–11.3 -0.5–0.3 -0.5–0.5 0.5–5.6 -2.9–3.5 -0.0–2.3 -2.1–1.8 -1.1–2.1
10 -1.3–4.5 -1.3–0.3 -1.6–0.1 -2.3–2.6 2.0–8.5 -2.1–1.8 4.4–8.0 -0.4–3.6
11 -0.1–3.9 -0.4–0.4 -0.6–0.5 -1.5–2.8 0.2–4.6 -1.1–2.1 -0.4–3.6 1.1–2.4
Total 44.6–54.3 -3.0–11.3 -2.4–11.9 29.6–41.9 51.7–61.5 31.0–42.3 31.8–43.8 9.7–22.5
Higher 20.1 4.2 5.3 24.6 26.4 23.5 24.3 7.3
Table E.18: North / Mass Flux / Positive Mean
@
@i
j
1 2 3 4 6 9 10 11
1 1.8–4.7 -1.7–1.8 -1.5–2.0 1.9–6.0 -0.8–5.5 0.6–4.5 -1.6–2.9 -1.7–1.9
2 -1.7–1.8 -0.0–0.2 -0.5–0.1 -0.4–0.2 -2.9–1.8 -0.4–0.2 -1.8–0.8 -0.5–0.2
3 -1.5–2.0 -0.5–0.1 -0.1–0.3 -0.7–0.2 -3.0–1.8 -0.6–0.1 -2.1–0.5 -0.6–0.3
4 1.9–6.0 -0.4–0.2 -0.7–0.2 -1.0–0.8 -3.3–2.2 -1.2–2.1 -1.9–2.1 -1.2–2.0
6 -0.8–5.5 -2.9–1.8 -3.0–1.8 -3.3–2.2 27.1–31.5 -1.7–2.4 4.7–11.9 1.9–5.2
9 0.6–4.5 -0.4–0.2 -0.6–0.1 -1.2–2.1 -1.7–2.4 0.3–1.7 -2.4–1.3 -1.5–1.0
10 -1.6–2.9 -1.8–0.8 -2.1–0.5 -1.9–2.1 4.7–11.9 -2.4–1.3 16.1–19.3 0.9–3.8
11 -1.7–1.9 -0.5–0.2 -0.6–0.3 -1.2–2.0 1.9–5.2 -1.5–1.0 0.9–3.8 3.9–4.9
Total 20.7–30.1 -7.1–3.7 -7.0–3.9 13.9–23.8 52.1–58.8 7.3–17.5 33.5–41.5 6.0–15.5
Higher 12.3 -0.3 0.3 14.9 13.2 9.2 10.3 0.6
401
Table E.19: Control / Negative Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 20.9–28.8 -2.0–6.8 -4.9–4.1 11.3–19.7 39.8–46.2 40.3–46.6 17.8–25.7 13.0–21.0
0.67 20.1–27.7 -3.1–5.6 -4.2–4.6 10.0–18.2 40.5–46.8 40.6–46.8 19.2–26.9 13.3–21.0
1.00 19.7–27.3 -3.1–5.5 -3.6–5.0 9.3–17.4 41.5–47.6 40.9–47.0 20.1–27.6 13.5–21.2
1.33 19.6–27.1 -3.0–5.5 -3.1–5.5 9.5–17.5 42.3–48.3 41.4–47.4 20.7–28.1 14.0–21.5
1.67 19.6–27.0 -2.9–5.5 -2.5–5.9 9.7–17.7 42.6–48.6 41.7–47.7 20.9–28.3 14.1–21.5
2.00 19.4–26.8 -3.0–5.4 -2.4–5.9 9.7–17.6 42.7–48.7 41.8–47.8 21.2–28.5 14.1–21.5
2.33 19.1–26.5 -3.1–5.2 -2.5–5.8 9.6–17.4 42.7–48.6 41.9–47.8 21.4–28.6 14.2–21.5
2.67 18.8–26.3 -3.3–5.1 -2.5–5.8 9.5–17.3 42.8–48.6 41.8–47.8 21.4–28.6 14.2–21.5
3.00 18.7–26.3 -3.2–5.1 -2.3–6.0 9.6–17.4 42.9–48.7 42.0–47.9 21.6–28.8 14.4–21.6
3.33 18.7–26.2 -3.2–5.1 -2.2–6.0 9.6–17.4 43.0–48.8 42.2–48.0 21.7–28.9 14.4–21.7
3.67 18.7–26.2 -3.1–5.2 -2.0–6.2 9.6–17.5 43.1–48.9 42.3–48.1 21.9–29.1 14.6–21.8
4.00 18.7–26.1 -3.1–5.1 -2.0–6.2 9.6–17.5 43.1–48.9 42.4–48.2 22.0–29.1 14.6–21.8
4.33 18.7–26.1 -3.1–5.2 -2.0–6.2 9.6–17.5 43.2–49.0 42.4–48.3 22.1–29.3 14.7–21.9
4.67 18.6–26.1 -3.0–5.2 -2.0–6.2 9.6–17.5 43.2–49.0 42.5–48.3 22.2–29.3 14.7–21.9
5.00 18.6–26.1 -3.0–5.3 -1.9–6.3 9.7–17.6 43.3–49.1 42.6–48.4 22.3–29.5 14.7–21.9
5.33 18.6–26.0 -3.0–5.3 -1.9–6.3 9.8–17.6 43.3–49.1 42.6–48.4 22.4–29.5 14.7–21.9
5.67 18.5–26.0 -3.0–5.3 -1.9–6.3 9.8–17.6 43.4–49.2 42.6–48.4 22.4–29.6 14.8–21.9
6.00 18.5–26.0 -2.9–5.3 -1.8–6.4 9.7–17.5 43.4–49.2 42.7–48.5 22.5–29.6 14.8–22.0
6.33 18.5–26.0 -2.8–5.5 -1.7–6.5 9.8–17.6 43.5–49.3 42.8–48.6 22.7–29.8 14.9–22.1
6.67 18.5–26.0 -2.8–5.5 -1.7–6.5 9.8–17.6 43.5–49.3 42.9–48.7 22.7–29.8 15.0–22.1
7.00 18.5–25.9 -2.8–5.5 -1.8–6.4 9.8–17.5 43.5–49.3 42.9–48.7 22.7–29.8 14.9–22.1
7.33 18.4–25.9 -2.8–5.4 -1.8–6.4 9.7–17.5 43.5–49.2 42.9–48.7 22.8–29.9 15.0–22.1
7.67 18.4–25.9 -2.7–5.5 -1.8–6.4 9.7–17.5 43.5–49.2 42.9–48.7 22.8–29.9 15.0–22.1
8.00 18.4–25.9 -2.7–5.5 -1.8–6.4 9.8–17.6 43.5–49.2 42.9–48.7 22.9–29.9 15.0–22.2
8.33 18.4–25.8 -2.7–5.5 -1.8–6.4 9.8–17.6 43.5–49.2 43.0–48.7 22.9–30.0 15.0–22.2
8.67 18.4–25.8 -2.7–5.5 -1.8–6.4 9.8–17.6 43.5–49.2 43.0–48.7 22.9–30.0 15.0–22.2
9.00 18.4–25.8 -2.7–5.4 -1.8–6.4 9.8–17.6 43.5–49.2 43.0–48.7 22.9–30.0 15.0–22.2
9.33 18.4–25.8 -2.7–5.4 -1.7–6.4 9.8–17.6 43.5–49.2 43.0–48.7 22.9–30.0 15.0–22.2
9.67 18.4–25.8 -2.7–5.5 -1.7–6.5 9.8–17.6 43.5–49.2 43.0–48.7 23.0–30.0 15.0–22.2
10.00 18.4–25.8 -2.7–5.5 -1.7–6.5 9.8–17.6 43.5–49.2 43.0–48.7 23.0–30.0 15.0–22.2
Table E.20: Control / Mass Flux / Negaitve Mid-Day
@
@i
j
1 2 3 4 6 9 10 11
1 1.0–3.0 -0.5–2.8 -0.5–2.8 1.0–4.4 -1.7–3.1 -0.3–3.7 -0.5–3.0 -0.5–3.0
2 -0.5–2.8 -0.1–0.2 -0.4–0.2 -0.4–0.2 -1.4–1.3 -2.2–0.4 -0.6–0.2 -0.5–0.3
3 -0.5–2.8 -0.4–0.2 -0.2–0.2 -0.4–0.5 -1.4–1.4 -2.1–0.5 -0.7–0.3 -0.6–0.4
4 1.0–4.4 -0.4–0.2 -0.4–0.5 -0.9–0.7 -2.3–2.0 -1.6–1.9 -1.8–1.6 -1.8–1.5
6 -1.7–3.1 -1.4–1.3 -1.4–1.4 -2.3–2.0 23.4–26.9 -1.0–5.7 0.6–5.1 -2.0–1.9
9 -0.3–3.7 -2.2–0.4 -2.1–0.5 -1.6–1.9 -1.0–5.7 18.8–21.9 -1.5–1.7 1.2–4.1
10 -0.5–3.0 -0.6–0.2 -0.7–0.3 -1.8–1.6 0.6–5.1 -1.5–1.7 2.4–4.7 0.2–4.5
11 -0.5–3.0 -0.5–0.3 -0.6–0.4 -1.8–1.5 -2.0–1.9 1.2–4.1 0.2–4.5 6.2–8.1
Total 18.6–26.1 -3.0–5.3 -1.9–6.3 9.7–17.6 43.3–49.1 42.6–48.4 22.3–29.5 14.7–21.9
Higher 10.4 1.4 2.3 11.3 15.4 19.9 16.2 5.3
402
Table E.21: South / Negative Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 39.0–48.8 -0.0–12.9 -5.4–8.1 24.4–36.3 37.4–46.6 36.6–47.8 9.9–22.8 1.9–14.6
0.67 40.2–49.7 -3.9–9.3 -6.8–6.9 24.5–36.5 38.9–48.0 36.5–47.3 8.9–21.6 -0.3–12.8
1.00 40.4–49.6 -5.8–7.6 -7.3–6.3 23.7–35.7 40.2–49.1 35.8–46.4 8.6–21.1 -1.1–11.9
1.33 40.2–49.3 -6.9–6.4 -7.4–5.8 23.3–35.3 41.1–49.8 35.3–45.8 8.5–20.8 -1.2–11.7
1.67 40.5–49.4 -6.3–6.8 -6.5–6.7 24.1–35.9 42.1–50.7 35.7–46.2 9.2–21.5 -0.1–12.5
2.00 40.0–48.9 -6.4–6.7 -6.1–7.1 24.1–35.7 42.7–51.2 35.6–46.0 9.1–21.4 -0.1–12.4
2.33 39.7–48.6 -6.7–6.4 -6.2–7.0 23.9–35.5 42.8–51.4 35.3–45.6 8.9–21.3 -0.1–12.3
2.67 39.5–48.4 -6.6–6.4 -6.1–7.0 23.7–35.3 43.1–51.6 35.1–45.4 8.9–21.2 -0.1–12.2
3.00 39.5–48.3 -6.7–6.2 -6.0–6.9 23.6–35.2 43.2–51.7 34.9–45.2 9.0–21.2 -0.1–12.2
3.33 39.3–48.2 -6.8–6.1 -5.9–7.0 23.3–34.9 43.3–51.8 34.9–45.1 9.1–21.3 0.0–12.2
3.67 39.2–48.0 -7.0–6.0 -5.9–6.9 23.1–34.8 43.5–51.9 34.8–45.0 9.1–21.2 0.1–12.2
4.00 39.1–47.9 -7.1–5.9 -5.8–6.9 23.0–34.6 43.5–51.9 34.7–44.8 9.1–21.3 -0.0–12.1
4.33 39.0–47.7 -7.3–5.7 -6.0–6.8 22.7–34.4 43.7–52.1 34.6–44.7 9.1–21.2 -0.3–11.8
4.67 38.9–47.7 -7.2–5.7 -5.9–6.8 22.7–34.3 43.9–52.3 34.5–44.7 9.2–21.2 -0.3–11.8
5.00 38.8–47.6 -7.2–5.8 -5.8–6.8 22.6–34.2 44.2–52.6 34.5–44.7 9.2–21.3 -0.2–11.9
5.33 38.8–47.5 -7.2–5.7 -5.7–6.9 22.6–34.2 44.4–52.7 34.5–44.7 9.3–21.3 -0.2–11.9
5.67 38.7–47.5 -7.2–5.7 -5.6–6.9 22.6–34.2 44.6–52.8 34.5–44.7 9.4–21.3 -0.1–12.0
6.00 38.8–47.5 -7.0–5.8 -5.5–7.0 22.7–34.2 44.8–53.0 34.7–44.7 9.5–21.4 0.1–12.1
6.33 38.7–47.4 -7.1–5.7 -5.5–7.0 22.6–34.1 44.8–53.0 34.7–44.7 9.5–21.4 0.0–12.1
6.67 38.7–47.4 -7.1–5.7 -5.3–7.1 22.6–34.1 44.8–52.9 34.7–44.7 9.6–21.4 0.0–12.1
7.00 38.7–47.4 -6.9–5.8 -5.3–7.2 22.6–34.1 44.8–53.0 34.7–44.8 9.7–21.5 0.2–12.3
7.33 38.6–47.3 -6.9–5.9 -5.2–7.2 22.6–34.1 44.9–53.1 34.8–44.8 9.8–21.5 0.2–12.3
7.67 38.6–47.3 -6.9–5.9 -5.1–7.3 22.7–34.1 45.0–53.1 34.9–44.8 10.0–21.6 0.2–12.3
8.00 38.5–47.2 -6.9–5.9 -5.0–7.3 22.7–34.1 45.1–53.2 34.9–44.8 10.0–21.6 0.2–12.3
8.33 38.4–47.2 -6.9–5.9 -5.0–7.4 22.6–34.0 45.2–53.3 34.9–44.8 10.0–21.6 0.2–12.3
8.67 38.4–47.1 -6.8–6.0 -4.9–7.4 22.6–34.0 45.3–53.3 35.0–44.9 10.1–21.7 0.3–12.4
9.00 38.5–47.2 -6.8–6.0 -4.8–7.5 22.6–33.9 45.4–53.4 35.0–44.9 10.1–21.7 0.3–12.4
9.33 38.4–47.1 -6.8–6.0 -4.8–7.5 22.5–33.9 45.4–53.4 35.0–44.9 10.0–21.6 0.3–12.4
9.67 38.4–47.0 -6.8–6.0 -4.8–7.5 22.5–33.9 45.4–53.5 35.0–44.8 9.9–21.5 0.3–12.4
10.00 38.3–47.0 -6.8–5.9 -4.8–7.5 22.4–33.8 45.5–53.5 34.9–44.8 9.9–21.5 0.3–12.4
Table E.22: South / Mass Flux / Negaitve Mid-Day
@
@i
j
1 2 3 4 6 9 10 11
1 4.5–8.0 -2.6–1.7 -2.5–2.0 1.6–6.6 -2.8–4.1 1.9–8.3 -2.4–2.3 -2.4–2.1
2 -2.6–1.7 -0.1–0.2 -0.3–0.4 -0.3–0.4 -2.9–1.7 -1.6–0.8 -0.4–0.3 -0.4–0.3
3 -2.5–2.0 -0.3–0.4 -0.3–0.4 -1.0–0.4 -2.1–2.5 -1.5–1.1 -0.8–0.5 -0.9–0.6
4 1.6–6.6 -0.3–0.4 -1.0–0.4 -1.0–1.8 -4.0–2.5 -1.3–3.7 -2.6–2.2 -2.6–2.1
6 -2.8–4.1 -2.9–1.7 -2.1–2.5 -4.0–2.5 29.8–35.3 -4.0–5.2 -2.3–3.0 -1.9–1.9
9 1.9–8.3 -1.6–0.8 -1.5–1.1 -1.3–3.7 -4.0–5.2 9.4–12.8 -2.9–1.0 -2.1–1.4
10 -2.4–2.3 -0.4–0.3 -0.8–0.5 -2.6–2.2 -2.3–3.0 -2.9–1.0 -0.7–1.4 -1.9–1.7
11 -2.4–2.1 -0.4–0.3 -0.9–0.6 -2.6–2.1 -1.9–1.9 -2.1–1.4 -1.9–1.7 2.3–3.5
Total 38.8–47.6 -7.2–5.8 -5.8–6.8 22.6–34.2 44.2–52.6 34.5–44.7 9.2–21.3 -0.2–11.9
Higher 28.0 0.7 1.3 24.2 15.4 23.4 16.0 4.0
403
Table E.23: North / Negative Mass Flux with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 16.9–27.8 0.4–12.1 -6.9–5.2 11.5–22.3 64.6–71.0 12.4–23.6 4.5–16.2 -2.2–9.4
0.67 21.5–31.5 -1.0–9.9 -5.8–5.5 11.2–21.8 67.9–73.8 15.2–25.7 4.4–15.6 -1.6–9.2
1.00 22.4–32.2 -2.6–8.5 -5.7–5.6 10.6–21.2 68.7–74.4 15.4–25.8 4.7–15.6 -2.1–8.7
1.33 22.5–32.2 -3.8–7.3 -5.6–5.6 10.2–20.8 69.2–74.8 15.5–25.8 4.5–15.4 -2.5–8.4
1.67 22.2–31.8 -5.1–6.0 -5.8–5.4 10.1–20.6 69.4–75.1 15.0–25.2 3.8–14.6 -2.8–8.1
2.00 22.4–31.9 -5.5–5.6 -6.0–5.2 10.0–20.4 69.6–75.2 14.8–25.0 3.6–14.5 -2.8–7.9
2.33 22.5–31.9 -5.8–5.1 -6.0–5.1 9.7–20.1 70.0–75.5 14.6–24.8 3.6–14.4 -3.0–7.7
2.67 22.7–32.0 -5.7–5.2 -5.5–5.4 9.8–20.1 70.4–75.9 14.8–25.0 3.8–14.6 -2.8–7.8
3.00 22.5–31.9 -5.9–4.9 -5.4–5.5 9.7–19.9 70.6–76.1 14.8–24.9 3.7–14.4 -3.0–7.7
3.33 22.5–31.8 -6.1–4.7 -5.4–5.4 9.7–19.9 70.8–76.3 14.9–25.0 3.7–14.3 -3.0–7.6
3.67 22.3–31.6 -6.3–4.5 -5.6–5.3 9.4–19.7 71.0–76.5 14.7–24.8 3.6–14.2 -3.2–7.3
4.00 22.4–31.7 -6.3–4.5 -5.5–5.3 9.5–19.7 71.3–76.8 14.9–25.0 3.7–14.3 -3.2–7.3
4.33 22.3–31.6 -6.4–4.3 -5.5–5.3 9.5–19.7 71.6–77.0 14.8–24.9 3.7–14.3 -3.4–7.2
4.67 22.2–31.5 -6.7–4.1 -5.5–5.2 9.4–19.6 71.7–77.2 14.8–24.9 3.5–14.2 -3.5–7.0
5.00 22.2–31.4 -6.7–4.1 -5.5–5.2 9.4–19.6 71.9–77.3 14.8–24.8 3.5–14.1 -3.6–6.9
5.33 22.2–31.4 -6.7–4.1 -5.4–5.2 9.5–19.6 72.1–77.5 14.9–24.9 3.6–14.2 -3.6–6.9
5.67 22.1–31.3 -6.8–4.1 -5.5–5.2 9.4–19.6 72.2–77.5 14.8–24.8 3.6–14.1 -3.6–6.9
6.00 22.0–31.3 -6.9–4.0 -5.6–5.1 9.3–19.5 72.2–77.5 14.7–24.7 3.5–14.0 -3.7–6.8
6.33 22.0–31.2 -6.9–3.9 -5.6–5.1 9.3–19.4 72.2–77.6 14.6–24.7 3.5–14.0 -3.7–6.8
6.67 21.9–31.2 -6.9–3.9 -5.5–5.1 9.2–19.4 72.2–77.5 14.7–24.7 3.5–14.0 -3.7–6.8
7.00 21.9–31.1 -6.9–3.9 -5.5–5.1 9.2–19.3 72.2–77.6 14.7–24.7 3.5–14.0 -3.7–6.7
7.33 21.9–31.1 -6.9–3.9 -5.6–5.0 9.1–19.3 72.1–77.5 14.6–24.7 3.5–13.9 -3.7–6.7
7.67 21.8–31.0 -6.9–3.9 -5.7–4.9 9.1–19.2 72.1–77.5 14.6–24.6 3.4–13.9 -3.8–6.6
8.00 21.8–30.9 -6.9–3.9 -5.8–4.9 9.0–19.1 72.2–77.6 14.6–24.6 3.4–13.8 -3.8–6.6
8.33 21.7–30.9 -6.9–3.8 -5.8–4.9 9.0–19.1 72.2–77.6 14.5–24.5 3.4–13.8 -3.8–6.6
8.67 21.7–30.9 -7.0–3.8 -5.8–4.8 8.9–19.1 72.3–77.6 14.5–24.5 3.3–13.8 -3.8–6.6
9.00 21.6–30.8 -7.0–3.8 -5.8–4.8 8.9–19.0 72.3–77.7 14.5–24.4 3.3–13.8 -3.8–6.6
9.33 21.6–30.8 -7.0–3.7 -5.9–4.7 8.8–19.0 72.3–77.7 14.4–24.4 3.3–13.7 -3.8–6.5
9.67 21.5–30.7 -7.0–3.7 -5.9–4.7 8.8–18.9 72.4–77.8 14.4–24.4 3.3–13.7 -3.8–6.5
10.00 21.5–30.7 -7.0–3.7 -5.9–4.7 8.8–18.9 72.4–77.8 14.4–24.4 3.3–13.7 -3.8–6.5
Table E.24: North / Mass Flux / Negaitve Mid-Day
@
@i
j
1 2 3 4 6 9 10 11
1 3.4–6.3 -1.6–2.1 -1.2–2.6 2.4–6.7 -3.9–6.9 0.5–4.8 -0.8–3.2 -1.1–2.6
2 -1.6–2.1 -0.3–0.2 -0.6–0.4 -0.5–0.5 -5.9–3.7 -0.6–0.6 -0.6–0.4 -0.6–0.4
3 -1.2–2.6 -0.6–0.4 -0.6–-0.1 0.3–1.4 -5.5–4.0 -0.1–1.1 0.1–1.4 0.2–1.2
4 2.4–6.7 -0.5–0.5 0.3–1.4 -0.9–0.8 -4.3–5.7 -1.4–1.8 -2.0–1.5 -2.5–0.8
6 -3.9–6.9 -5.9–3.7 -5.5–4.0 -4.3–5.7 52.0–59.3 -3.5–4.5 -2.9–2.2 -1.7–1.3
9 0.5–4.8 -0.6–0.6 -0.1–1.1 -1.4–1.8 -3.5–4.5 2.9–4.7 -1.0–1.7 -0.6–1.9
10 -0.8–3.2 -0.6–0.4 0.1–1.4 -2.0–1.5 -2.9–2.2 -1.0–1.7 -0.2–1.4 -0.9–1.8
11 -1.1–2.6 -0.6–0.4 0.2–1.2 -2.5–0.8 -1.7–1.3 -0.6–1.9 -0.9–1.8 1.6–2.3
Total 22.2–31.4 -6.7–4.1 -5.5–5.2 9.4–19.6 71.9–77.3 14.8–24.8 3.5–14.1 -3.6–6.9
Higher 10.3 -0.1 -2.5 9.4 18.6 11.1 6.2 -1.7
404
Table E.25: Control / Mass Flux / Negative Mean
@
@i
j
1 2 3 4 6 9 10 11
1 1.2–3.2 -0.6–2.7 -0.6–2.8 1.1–4.4 -1.8–3.0 -0.3–3.8 -0.4–3.1 -0.4–3.0
2 -0.6–2.7 -0.0–0.2 -0.4–0.1 -0.3–0.2 -1.4–1.3 -2.2–0.5 -0.5–0.1 -0.4–0.3
3 -0.6–2.8 -0.4–0.1 -0.2–0.2 -0.4–0.4 -1.4–1.3 -2.1–0.6 -0.6–0.2 -0.5–0.3
4 1.1–4.4 -0.3–0.2 -0.4–0.4 -0.9–0.7 -2.2–2.0 -1.6–2.0 -1.7–1.6 -1.7–1.5
6 -1.8–3.0 -1.4–1.3 -1.4–1.3 -2.2–2.0 23.5–26.9 -0.8–5.7 0.5–4.8 -2.0–1.9
9 -0.3–3.8 -2.2–0.5 -2.1–0.6 -1.6–2.0 -0.8–5.7 18.8–21.9 -1.4–1.7 1.3–4.1
10 -0.4–3.1 -0.5–0.1 -0.6–0.2 -1.7–1.6 0.5–4.8 -1.4–1.7 2.3–4.7 0.1–4.4
11 -0.4–3.0 -0.4–0.3 -0.5–0.3 -1.7–1.5 -2.0–1.9 1.3–4.1 0.1–4.4 6.2–8.1
Total 18.4–25.7 -3.4–4.9 -2.6–5.6 9.3–17.2 42.6–48.5 42.1–47.9 21.7–28.8 14.5–21.8
Higher 10.0 1.0 1.6 10.7 14.9 19.0 15.7 5.0
Table E.26: South / Mass Flux / Negative Mean
@
@i
j
1 2 3 4 6 9 10 11
1 4.8–8.1 -2.7–1.7 -2.5–1.9 1.6–6.6 -2.7–4.2 1.7–8.1 -2.3–2.3 -2.3–2.1
2 -2.7–1.7 -0.1–0.2 -0.2–0.3 -0.4–0.3 -2.9–1.7 -1.1–1.3 -0.2–0.4 -0.3–0.3
3 -2.5–1.9 -0.2–0.3 -0.2–0.4 -0.8–0.3 -2.0–2.7 -1.5–1.0 -0.7–0.4 -0.7–0.4
4 1.6–6.6 -0.4–0.3 -0.8–0.3 -1.1–1.7 -3.9–2.5 -1.5–3.5 -2.6–2.3 -2.7–2.0
6 -2.7–4.2 -2.9–1.7 -2.0–2.7 -3.9–2.5 29.7–35.0 -4.2–5.1 -2.4–2.8 -2.0–1.7
9 1.7–8.1 -1.1–1.3 -1.5–1.0 -1.5–3.5 -4.2–5.1 9.6–13.1 -2.8–1.0 -1.9–1.5
10 -2.3–2.3 -0.2–0.4 -0.7–0.4 -2.6–2.3 -2.4–2.8 -2.8–1.0 -0.7–1.5 -1.7–1.8
11 -2.3–2.1 -0.3–0.3 -0.7–0.4 -2.7–2.0 -2.0–1.7 -1.9–1.5 -1.7–1.8 2.5–3.7
Total 38.7–47.4 -6.9–5.8 -6.0–6.5 22.9–34.2 43.4–51.9 34.7–44.7 8.8–20.7 0.1–12.2
Higher 27.7 0.2 0.9 24.7 15.0 23.2 15.2 3.9
Table E.27: North / Mass Flux / Negative Mean
@
@i
j
1 2 3 4 6 9 10 11
1 3.5–6.3 -1.8–1.8 -1.4–2.3 2.1–6.3 -4.4–6.5 0.2–4.3 -1.0–2.8 -1.3–2.4
2 -1.8–1.8 -0.2–0.1 -0.4–0.4 -0.3–0.5 -3.6–6.1 -0.5–0.4 -0.5–0.3 -0.4–0.3
3 -1.4–2.3 -0.4–0.4 -0.5–-0.0 0.2–1.1 -6.2–3.5 -0.2–0.8 0.0–1.0 0.1–1.0
4 2.1–6.3 -0.3–0.5 0.2–1.1 -0.8–0.9 -4.8–5.3 -1.6–1.6 -2.2–1.1 -2.7–0.6
6 -4.4–6.5 -3.6–6.1 -6.2–3.5 -4.8–5.3 52.6–60.0 -3.7–4.2 -2.0–2.8 -1.5–1.3
9 0.2–4.3 -0.5–0.4 -0.2–0.8 -1.6–1.6 -3.7–4.2 3.2–4.9 -1.1–1.3 -0.5–1.7
10 -1.0–2.8 -0.5–0.3 0.0–1.0 -2.2–1.1 -2.0–2.8 -1.1–1.3 0.2–1.6 -0.8–1.7
11 -1.3–2.4 -0.4–0.3 0.1–1.0 -2.7–0.6 -1.5–1.3 -0.5–1.7 -0.8–1.7 1.7–2.3
Total 21.5–30.6 -6.3–4.5 -6.0–4.9 9.3–19.2 71.4–76.8 14.5–24.3 3.2–13.8 -3.2–7.4
Higher 11.7 -2.0 -1.4 10.6 16.1 11.8 5.9 -0.7
405
Table E.28: Control / Mass Flux / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 -0.6–0.5 -0.9–0.9 -0.9–0.8 -0.5–1.5 -1.0–0.9 -0.6–1.2 -3.2–2.8 -0.2–3.2
2 -0.9–0.9 -0.0–0.4 -0.5–0.3 -0.3–0.4 -1.0–0.2 -0.5–0.4 -3.3–2.3 -0.9–1.8
3 -0.9–0.8 -0.5–0.3 -0.1–0.0 -0.1–0.3 -0.5–0.2 -0.3–0.2 -3.4–2.1 -0.5–2.0
4 -0.5–1.5 -0.3–0.4 -0.1–0.3 -0.6–0.4 -0.4–1.4 -0.4–1.3 -2.5–3.3 0.0–3.3
6 -1.0–0.9 -1.0–0.2 -0.5–0.2 -0.4–1.4 4.5–6.2 -0.9–2.0 1.6–10.0 2.8–8.4
9 -0.6–1.2 -0.5–0.4 -0.3–0.2 -0.4–1.3 -0.9–2.0 1.5–3.0 0.3–7.3 1.7–5.3
10 -3.2–2.8 -3.3–2.3 -3.4–2.1 -2.5–3.3 1.6–10.0 0.3–7.3 17.9–23.0 21.2–30.9
11 -0.2–3.2 -0.9–1.8 -0.5–2.0 0.0–3.3 2.8–8.4 1.7–5.3 21.2–30.9 8.7–11.8
Total -4.5–10.6 -10.4–5.2 -7.8–6.9 -4.2–11.0 11.6–24.8 11.9–26.0 62.7–70.0 47.4–56.6
Higher 1.1 -2.3 -0.3 -0.3 0.9 8.1 11.2 2.1
Table E.29: South / Mass Flux / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 0.3–4.1 -0.7–3.3 -0.9–3.0 4.8–10.7 0.5–6.4 3.4–8.8 -1.3–5.1 -0.3–3.6
2 -0.7–3.3 -0.2–0.3 -0.6–0.4 -0.7–0.6 -2.7–0.8 -0.8–0.5 -2.0–0.5 -0.7–0.3
3 -0.9–3.0 -0.6–0.4 -0.3–0.3 -0.5–0.7 -2.4–1.2 -0.7–0.4 -2.4–0.1 -0.7–0.5
4 4.8–10.7 -0.7–0.6 -0.5–0.7 -1.5–1.5 -3.4–2.7 -1.0–4.2 -2.6–3.3 -1.8–2.7
6 0.5–6.4 -2.7–0.8 -2.4–1.2 -3.4–2.7 16.8–21.0 -2.2–3.4 -0.2–6.9 0.4–4.4
9 3.4–8.8 -0.8–0.5 -0.7–0.4 -1.0–4.2 -2.2–3.4 0.8–3.1 -1.9–2.7 -1.8–1.5
10 -1.3–5.1 -2.0–0.5 -2.4–0.1 -2.6–3.3 -0.2–6.9 -1.9–2.7 9.4–13.7 1.0–5.1
11 -0.3–3.6 -0.7–0.3 -0.7–0.5 -1.8–2.7 0.4–4.4 -1.8–1.5 1.0–5.1 1.8–3.1
Total 39.9–49.2 -0.3–12.1 -1.3–11.2 31.5–41.9 44.4–53.7 27.1–37.8 38.1–48.0 11.7–22.7
Higher 19.1 6.8 5.7 26.9 22.3 22.2 24.4 7.6
Table E.30: North / Mass Flux / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 1.4–4.0 -2.0–1.5 -1.6–1.9 1.9–5.9 0.3–5.7 0.4–4.1 -2.8–1.9 -1.8–1.8
2 -2.0–1.5 -0.1–0.4 -0.7–0.2 -0.6–0.4 -2.2–1.5 -0.7–0.3 -2.0–1.3 -0.6–0.4
3 -1.6–1.9 -0.7–0.2 -0.1–0.3 -0.6–0.4 -2.0–1.6 -0.5–0.3 -2.3–0.8 -0.5–0.4
4 1.9–5.9 -0.6–0.4 -0.6–0.4 0.6–2.6 -3.1–2.0 -1.9–1.5 -2.7–1.9 -1.5–1.9
6 0.3–5.7 -2.2–1.5 -2.0–1.6 -3.1–2.0 23.4–27.0 -1.1–2.5 2.4–9.7 1.5–4.6
9 0.4–4.1 -0.7–0.3 -0.5–0.3 -1.9–1.5 -1.1–2.5 1.2–2.5 -2.7–1.2 -1.6–0.9
10 -2.8–1.9 -2.0–1.3 -2.3–0.8 -2.7–1.9 2.4–9.7 -2.7–1.2 20.2–23.7 1.8–4.8
11 -1.8–1.8 -0.6–0.4 -0.5–0.4 -1.5–1.9 1.5–4.6 -1.6–0.9 1.8–4.8 4.7–5.8
Total 18.5–27.4 -5.3–4.9 -6.1–4.2 16.0–25.1 43.2–50.1 6.9–16.8 37.1–44.3 6.8–16.0
Higher 11.6 1.2 0.2 16.2 9.7 8.7 12.1 0.1
406
Table E.31: Control / Mass Flux / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 0.6–2.6 -0.2–3.2 -0.3–3.2 1.4–4.9 -1.6–3.3 -0.3–3.8 -0.3–3.3 -0.2–3.2
2 -0.2–3.2 -0.0–0.2 -0.4–0.1 -0.4–0.1 -1.4–1.2 -2.2–0.5 -0.6–0.0 -0.5–0.2
3 -0.3–3.2 -0.4–0.1 -0.2–0.3 -0.5–0.3 -1.3–1.5 -1.9–0.9 -0.5–0.4 -0.5–0.4
4 1.4–4.9 -0.4–0.1 -0.5–0.3 -1.1–0.6 -2.3–2.1 -1.4–2.3 -1.5–1.9 -1.6–1.7
6 -1.6–3.3 -1.4–1.2 -1.3–1.5 -2.3–2.1 22.6–26.0 -0.7–5.8 0.6–5.0 -2.3–1.6
9 -0.3–3.8 -2.2–0.5 -1.9–0.9 -1.4–2.3 -0.7–5.8 19.7–22.9 -1.3–1.9 1.4–4.2
10 -0.3–3.3 -0.6–0.0 -0.5–0.4 -1.5–1.9 0.6–5.0 -1.3–1.9 2.7–5.0 0.3–4.6
11 -0.2–3.2 -0.5–0.2 -0.5–0.4 -1.6–1.7 -2.3–1.6 1.4–4.2 0.3–4.6 6.5–8.4
Total 18.8–26.1 -2.7–5.5 -1.9–6.3 9.6–17.5 41.9–47.8 43.5–49.3 22.7–29.9 15.0–22.3
Higher 9.1 1.5 1.5 10.3 14.8 18.6 15.6 4.9
Table E.32: South / Mass Flux / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 4.0–7.5 -2.5–2.0 -2.5–2.2 1.8–7.0 -2.5–4.3 2.2–8.5 -2.2–2.7 -2.3–2.4
2 -2.5–2.0 -0.1–0.2 -0.4–0.3 -0.5–0.3 -1.8–2.7 -1.1–1.4 -0.3–0.4 -0.4–0.4
3 -2.5–2.2 -0.4–0.3 -0.3–0.4 -1.0–0.4 -2.3–2.4 -1.5–1.2 -0.9–0.5 -0.9–0.6
4 1.8–7.0 -0.5–0.3 -1.0–0.4 -1.1–1.7 -4.3–2.2 -1.4–3.6 -2.5–2.3 -2.6–2.2
6 -2.5–4.3 -1.8–2.7 -2.3–2.4 -4.3–2.2 28.9–34.3 -3.8–5.4 -2.1–3.3 -1.7–2.1
9 2.2–8.5 -1.1–1.4 -1.5–1.2 -1.4–3.6 -3.8–5.4 10.3–13.9 -2.7–1.3 -1.8–1.8
10 -2.2–2.7 -0.3–0.4 -0.9–0.5 -2.5–2.3 -2.1–3.3 -2.7–1.3 -0.2–1.8 -1.7–1.8
11 -2.3–2.4 -0.4–0.4 -0.9–0.6 -2.6–2.2 -1.7–2.1 -1.8–1.8 -1.7–1.8 2.7–3.9
Total 38.8–47.6 -6.7–6.0 -5.4–7.2 21.8–33.3 42.2–50.4 35.8–45.9 9.0–20.9 0.0–12.2
Higher 26.9 -0.7 1.7 23.6 12.8 22.1 14.1 2.9
Table E.33: North / Mass Flux / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 2.8–5.8 -1.5–2.5 -1.0–3.0 2.9–7.3 -3.8–6.4 0.7–5.0 -0.6–3.5 -1.0–2.9
2 -1.5–2.5 -0.3–0.1 -0.1–0.7 0.1–0.9 -5.7–3.3 -0.5–0.5 -0.1–0.7 -0.1–0.7
3 -1.0–3.0 -0.1–0.7 -0.6–0.0 0.1–1.3 -5.6–3.3 -0.3–1.0 -0.0–1.3 0.0–1.1
4 2.9–7.3 0.1–0.9 0.1–1.3 -0.8–1.1 -4.8–4.5 -1.4–1.9 -1.7–1.9 -2.5–1.1
6 -3.8–6.4 -5.7–3.3 -5.6–3.3 -4.8–4.5 50.2–57.1 -3.3–4.5 -3.2–2.0 -1.7–1.2
9 0.7–5.0 -0.5–0.5 -0.3–1.0 -1.4–1.9 -3.3–4.5 3.7–5.5 -1.2–1.5 -0.6–1.8
10 -0.6–3.5 -0.1–0.7 -0.0–1.3 -1.7–1.9 -3.2–2.0 -1.2–1.5 0.7–2.4 -0.8–2.0
11 -1.0–2.9 -0.1–0.7 0.0–1.1 -2.5–1.1 -1.7–1.2 -0.6–1.8 -0.8–2.0 1.9–2.6
Total 22.0–31.0 -6.8–3.9 -6.2–4.5 9.6–19.6 68.6–74.1 14.9–24.6 4.0–14.5 -3.7–6.7
Higher 9.1 -2.0 -3.0 8.6 19.1 10.3 5.1 -2.8
407
E.3 Snow Surface Temperature
Table E.34: Control / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 9.3–18.4 -4.7–4.8 -4.8–4.7 22.4–30.7 59.0–64.0 9.3–18.9 19.5–28.0 -4.0–5.4
0.67 10.1–19.0 -4.0–5.3 -5.5–4.0 17.4–25.9 63.9–68.7 9.9–19.2 21.0–29.3 -4.7–4.7
1.00 10.5–19.3 -4.6–4.7 -5.3–4.0 15.1–23.7 66.4–71.1 10.5–19.7 22.1–30.2 -4.6–4.6
1.33 11.1–19.7 -4.4–4.8 -4.8–4.4 14.6–23.3 67.9–72.5 11.3–20.5 23.0–31.0 -4.0–5.1
1.67 11.4–19.9 -4.4–4.8 -4.5–4.6 14.1–22.9 68.7–73.3 11.7–20.9 23.3–31.2 -3.9–5.2
2.00 11.4–19.8 -4.6–4.6 -4.7–4.5 13.7–22.4 69.2–73.7 11.9–21.1 23.4–31.3 -4.1–5.0
2.33 11.3–19.8 -4.7–4.5 -4.7–4.5 13.2–22.0 69.5–74.0 12.0–21.1 23.5–31.4 -4.2–4.9
2.67 11.4–19.8 -4.8–4.4 -4.7–4.4 13.0–21.7 69.9–74.4 12.2–21.3 23.6–31.4 -4.2–4.9
3.00 11.5–19.9 -4.8–4.4 -4.6–4.5 12.9–21.6 70.2–74.7 12.3–21.4 23.7–31.5 -4.1–5.0
3.33 11.5–20.0 -4.8–4.3 -4.6–4.5 12.7–21.4 70.4–74.9 12.5–21.6 23.7–31.5 -4.1–5.0
3.67 11.6–20.0 -4.8–4.3 -4.5–4.6 12.5–21.3 70.7–75.1 12.7–21.7 23.7–31.5 -4.1–5.0
4.00 11.6–20.0 -4.8–4.2 -4.5–4.6 12.4–21.2 70.8–75.2 12.8–21.8 23.7–31.5 -4.1–5.0
4.33 11.6–20.0 -4.8–4.3 -4.4–4.7 12.3–21.0 71.0–75.4 12.9–21.9 23.7–31.5 -4.0–5.0
4.67 11.7–20.1 -4.8–4.3 -4.3–4.8 12.2–21.0 71.1–75.5 13.1–22.0 23.8–31.6 -4.0–5.0
5.00 11.9–20.3 -4.6–4.5 -4.1–4.9 12.3–21.0 71.3–75.7 13.3–22.2 24.0–31.7 -3.8–5.2
5.33 11.8–20.3 -4.6–4.5 -4.1–5.0 12.2–21.0 71.5–75.8 13.4–22.3 24.0–31.8 -3.8–5.2
5.67 11.9–20.3 -4.6–4.4 -4.1–5.0 12.1–20.9 71.5–75.9 13.4–22.3 24.0–31.8 -3.8–5.2
6.00 11.9–20.3 -4.6–4.4 -4.0–5.0 12.0–20.8 71.6–76.0 13.6–22.4 24.1–31.8 -3.8–5.2
6.33 11.9–20.3 -4.6–4.4 -4.0–5.0 12.0–20.7 71.8–76.1 13.7–22.5 24.1–31.8 -3.7–5.2
6.67 11.9–20.3 -4.6–4.4 -4.0–5.0 11.9–20.6 71.9–76.1 13.8–22.6 24.1–31.8 -3.8–5.2
7.00 11.9–20.3 -4.6–4.4 -4.1–5.0 11.8–20.6 71.9–76.2 13.8–22.6 24.1–31.8 -3.8–5.2
7.33 11.9–20.3 -4.7–4.3 -4.1–4.9 11.8–20.5 71.9–76.2 13.8–22.6 24.1–31.8 -3.8–5.2
7.67 11.9–20.3 -4.7–4.3 -4.1–4.9 11.7–20.4 72.0–76.3 13.8–22.6 24.0–31.8 -3.8–5.1
8.00 11.9–20.3 -4.7–4.3 -4.1–4.9 11.6–20.4 72.1–76.3 13.8–22.6 24.0–31.8 -3.8–5.1
8.33 11.9–20.3 -4.7–4.3 -4.1–4.9 11.6–20.4 72.1–76.4 13.9–22.6 24.0–31.8 -3.8–5.1
8.67 11.9–20.3 -4.7–4.3 -4.1–4.9 11.6–20.3 72.1–76.4 13.9–22.6 24.0–31.7 -3.8–5.1
9.00 11.9–20.3 -4.7–4.2 -4.0–4.9 11.6–20.3 72.2–76.4 13.9–22.6 24.0–31.7 -3.7–5.2
9.33 12.0–20.3 -4.8–4.2 -4.0–5.0 11.6–20.3 72.2–76.5 13.9–22.6 24.0–31.7 -3.7–5.2
9.67 12.0–20.3 -4.7–4.3 -4.0–5.0 11.6–20.3 72.3–76.5 14.0–22.7 24.0–31.8 -3.7–5.2
10.00 12.0–20.3 -4.7–4.3 -4.0–5.0 11.6–20.3 72.3–76.6 14.0–22.7 24.0–31.7 -3.7–5.2
Table E.35: Control / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 -1.1–1.2 -1.0–2.9 -1.0–2.9 0.6–4.9 -3.8–3.8 -0.7–3.7 -0.5–3.6 -1.0–2.8
2 -1.0–2.9 -0.1–0.3 -0.5–0.3 -0.4–0.4 -4.2–3.1 -0.5–0.4 -0.9–0.6 -0.5–0.3
3 -1.0–2.9 -0.5–0.3 -0.1–0.3 -0.6–0.3 -4.1–3.3 -0.6–0.3 -0.8–0.8 -0.6–0.3
4 0.6–4.9 -0.4–0.4 -0.6–0.3 0.5–2.4 -3.0–4.9 -1.5–2.4 -1.1–2.4 -1.5–1.9
6 -3.8–3.8 -4.2–3.1 -4.1–3.3 -3.0–4.9 47.1–52.6 1.5–7.1 -1.1–5.9 -0.3–1.7
9 -0.7–3.7 -0.5–0.4 -0.6–0.3 -1.5–2.4 1.5–7.1 -1.7–0.8 1.5–5.8 -1.3–2.8
10 -0.5–3.6 -0.9–0.6 -0.8–0.8 -1.1–2.4 -1.1–5.9 1.5–5.8 10.1–12.6 -0.3–2.9
11 -1.0–2.8 -0.5–0.3 -0.6–0.3 -1.5–1.9 -0.3–1.7 -1.3–2.8 -0.3–2.9 -0.1–0.4
Total 11.9–20.3 -4.6–4.5 -4.1–4.9 12.3–21.0 71.3–75.7 13.3–22.2 24.0–31.7 -3.8–5.2
Higher 7.5 -0.3 0.3 10.3 16.2 7.7 7.0 -3.0
408
Table E.36: South / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 23.1–34.9 -2.3–11.6 -8.6–5.8 38.2–48.5 44.3–53.4 10.0–23.2 12.7–25.3 -8.1–6.4
0.67 25.7–36.9 -7.2–6.9 -6.4–7.2 32.7–43.5 50.6–59.1 10.0–22.8 14.4–26.7 -5.6–8.0
1.00 26.9–37.7 -5.2–7.9 -7.5–5.8 29.9–40.8 54.2–62.2 9.6–22.1 15.7–27.4 -6.6–6.7
1.33 27.4–38.1 -6.6–6.5 -8.0–5.1 28.2–39.2 56.2–64.0 9.5–21.9 16.3–27.8 -7.0–6.1
1.67 27.9–38.3 -6.6–6.3 -7.5–5.4 27.5–38.4 57.5–65.1 9.6–22.0 17.3–28.5 -6.8–6.2
2.00 27.8–38.1 -7.1–5.7 -7.6–5.2 26.4–37.3 58.1–65.6 9.3–21.6 17.1–28.3 -7.1–5.7
2.33 27.9–38.0 -7.6–5.2 -7.8–4.9 25.7–36.6 58.5–65.9 9.2–21.4 17.1–28.2 -7.1–5.6
2.67 27.8–37.9 -7.7–5.0 -7.9–4.7 25.1–36.0 59.0–66.3 9.2–21.3 17.2–28.2 -7.1–5.6
3.00 27.7–37.8 -7.8–4.9 -8.2–4.4 24.7–35.6 59.3–66.6 9.2–21.3 17.4–28.4 -7.1–5.5
3.33 27.8–37.8 -7.9–4.7 -8.2–4.4 24.3–35.3 59.7–66.9 9.4–21.3 17.7–28.5 -7.0–5.5
3.67 28.0–38.0 -7.7–4.8 -8.0–4.5 24.4–35.3 60.2–67.4 9.7–21.6 18.1–28.9 -6.7–5.8
4.00 28.2–38.1 -7.6–4.9 -7.6–4.8 24.4–35.3 60.6–67.8 10.1–21.8 18.5–29.2 -6.4–6.1
4.33 28.3–38.1 -7.7–4.7 -7.6–4.7 24.1–35.0 60.9–68.0 10.0–21.8 18.7–29.3 -6.4–5.9
4.67 28.4–38.1 -7.6–4.7 -7.5–4.8 24.0–34.9 61.2–68.2 10.1–21.9 18.9–29.4 -6.4–5.9
5.00 28.4–38.2 -7.6–4.7 -7.4–4.9 23.9–34.8 61.4–68.4 10.2–22.0 19.0–29.4 -6.3–6.0
5.33 28.5–38.1 -7.6–4.6 -7.3–4.9 23.8–34.6 61.6–68.5 10.3–22.0 19.0–29.5 -6.2–6.0
5.67 28.5–38.1 -7.7–4.5 -7.3–4.9 23.6–34.5 61.7–68.6 10.3–22.0 19.1–29.5 -6.2–6.0
6.00 28.5–38.1 -7.7–4.4 -7.2–4.9 23.6–34.4 61.7–68.7 10.3–22.0 19.2–29.6 -6.1–6.0
6.33 28.5–38.0 -7.7–4.4 -7.2–4.9 23.4–34.3 61.8–68.7 10.3–21.9 19.2–29.5 -6.1–6.0
6.67 28.4–38.0 -7.8–4.4 -7.2–4.9 23.3–34.1 61.8–68.8 10.3–21.9 19.3–29.6 -6.1–6.0
7.00 28.4–37.9 -7.7–4.4 -7.1–4.9 23.3–34.1 61.9–68.8 10.4–21.9 19.4–29.6 -6.0–6.0
7.33 28.4–37.9 -7.7–4.4 -7.0–4.9 23.3–34.0 62.0–68.9 10.4–21.9 19.5–29.7 -6.0–6.0
7.67 28.3–37.9 -7.7–4.4 -7.0–5.0 23.3–34.0 62.1–69.0 10.4–21.9 19.6–29.7 -6.0–6.0
8.00 28.3–37.8 -7.7–4.4 -6.9–5.0 23.2–33.9 62.2–69.0 10.4–21.9 19.6–29.7 -5.9–6.1
8.33 28.3–37.8 -7.6–4.4 -6.9–5.0 23.1–33.9 62.2–69.1 10.4–21.9 19.6–29.7 -5.9–6.1
8.67 28.3–37.8 -7.6–4.4 -6.8–5.1 23.1–33.8 62.2–69.1 10.5–22.0 19.6–29.8 -5.8–6.1
9.00 28.3–37.8 -7.5–4.5 -6.7–5.2 23.1–33.7 62.3–69.1 10.6–22.0 19.6–29.8 -5.8–6.1
9.33 28.3–37.8 -7.5–4.4 -6.7–5.2 23.0–33.6 62.3–69.1 10.5–22.0 19.6–29.7 -5.8–6.1
9.67 28.3–37.7 -7.6–4.4 -6.7–5.1 22.9–33.6 62.3–69.1 10.5–21.9 19.6–29.7 -5.8–6.1
10.00 28.2–37.7 -7.6–4.4 -6.7–5.1 22.9–33.5 62.3–69.1 10.5–21.9 19.6–29.6 -5.8–6.0
Table E.37: South / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 -3.0–1.8 -3.5–4.7 -3.2–4.9 3.2–11.6 0.4–10.2 -3.8–4.5 -2.5–5.3 -3.2–5.0
2 -3.5–4.7 -0.3–0.4 -0.7–0.7 -1.0–0.6 -3.7–3.9 -0.7–0.6 -0.9–1.0 -0.7–0.7
3 -3.2–4.9 -0.7–0.7 -0.4–0.6 -1.3–0.8 -3.9–3.7 -1.3–0.6 -1.4–1.0 -1.2–0.8
4 3.2–11.6 -1.0–0.6 -1.3–0.8 1.9–6.1 -5.4–4.8 -3.8–3.7 -2.8–4.4 -2.9–4.7
6 0.4–10.2 -3.7–3.9 -3.9–3.7 -5.4–4.8 35.9–42.1 -4.0–3.5 -5.4–2.6 -2.7–2.2
9 -3.8–4.5 -0.7–0.6 -1.3–0.6 -3.8–3.7 -4.0–3.5 -1.9–1.7 -2.9–3.4 -3.2–3.5
10 -2.5–5.3 -0.9–1.0 -1.4–1.0 -2.8–4.4 -5.4–2.6 -2.9–3.4 8.9–12.4 -3.4–1.9
11 -3.2–5.0 -0.7–0.7 -1.2–0.8 -2.9–4.7 -2.7–2.2 -3.2–3.5 -3.4–1.9 -0.6–0.4
Total 28.4–38.2 -7.6–4.7 -7.4–4.9 23.9–34.8 61.4–68.4 10.2–22.0 19.0–29.4 -6.3–6.0
Higher 17.0 -2.1 -1.1 17.0 22.8 16.2 13.3 -0.8
409
Table E.38: North / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8
0.33 11.3–23.5 -0.0–12.9 -6.7–6.8 28.3–38.5 56.2–63.2 1.9–14.8 6.9–19.3 -7.5–6.2
0.67 17.8–29.0 -1.6–10.8 -6.7–6.2 24.6–35.3 63.2–69.5 4.8–17.0 10.0–21.7 -7.0–6.0
1.00 19.3–30.1 -3.8–8.6 -7.0–5.6 22.3–33.1 65.6–71.7 4.9–16.9 11.3–22.6 -7.8–4.9
1.33 19.8–30.5 -5.0–7.2 -7.0–5.4 21.0–31.8 67.1–73.1 4.9–16.8 11.7–22.9 -8.0–4.5
1.67 20.1–30.6 -6.1–5.9 -7.1–5.1 20.2–30.9 67.9–73.8 4.7–16.6 11.6–22.6 -4.7–6.8
2.00 20.5–30.9 -6.8–5.1 -7.3–4.7 19.4–30.1 68.3–74.2 4.5–16.3 11.7–22.6 -5.0–6.4
2.33 20.9–31.2 -7.3–4.6 -7.4–4.6 18.8–29.5 69.0–74.9 4.7–16.3 12.0–22.8 -5.2–6.2
2.67 21.3–31.6 -7.5–4.5 -7.1–4.8 18.7–29.3 69.7–75.5 5.0–16.6 12.3–23.1 -5.0–6.3
3.00 21.5–31.7 -7.8–4.2 -7.1–4.8 18.4–29.0 70.1–75.9 5.1–16.7 12.4–23.1 -5.1–6.2
3.33 21.6–31.8 -4.5–6.8 -7.2–4.7 18.2–28.8 70.4–76.2 5.0–16.6 12.4–23.1 -5.2–6.2
3.67 21.5–31.8 -4.8–6.6 -7.3–4.6 17.7–28.3 70.6–76.4 4.8–16.4 12.3–23.0 -5.4–6.0
4.00 21.6–31.8 -5.0–6.3 -7.5–4.5 17.4–28.1 70.9–76.7 4.8–16.3 12.4–23.1 -5.5–5.8
4.33 21.5–31.8 -5.1–6.2 -7.5–4.4 17.2–27.9 71.1–76.9 4.8–16.3 12.5–23.2 -5.6–5.7
4.67 21.5–31.8 -5.3–6.0 -7.5–4.4 16.9–27.7 71.4–77.1 4.8–16.2 12.5–23.1 -5.7–5.6
5.00 21.5–31.7 -5.3–5.9 -7.5–4.4 16.7–27.5 71.6–77.3 4.8–16.2 12.6–23.1 -5.7–5.5
5.33 21.6–31.8 -5.3–5.9 -7.4–4.4 16.7–27.4 71.8–77.5 4.8–16.2 12.6–23.2 -5.7–5.5
5.67 21.6–31.7 -5.4–5.9 -7.4–4.4 16.6–27.2 71.9–77.6 4.7–16.2 12.6–23.1 -5.7–5.5
6.00 21.6–31.7 -5.4–5.8 -7.5–4.3 16.4–27.1 72.0–77.7 4.7–16.1 12.6–23.1 -5.8–5.4
6.33 21.5–31.6 -5.5–5.7 -7.5–4.3 16.2–26.9 72.1–77.7 4.7–16.0 12.6–23.1 -5.8–5.4
6.67 21.6–31.6 -5.5–5.6 -7.5–4.3 16.1–26.8 72.1–77.7 4.7–16.0 12.7–23.1 -5.8–5.3
7.00 21.6–31.6 -5.5–5.6 -7.5–4.2 16.1–26.7 72.2–77.8 4.7–16.0 12.6–23.0 -5.8–5.3
7.33 21.6–31.6 -5.6–5.6 -7.6–4.2 16.0–26.6 72.2–77.8 4.7–16.0 12.6–23.0 -5.9–5.2
7.67 21.6–31.6 -5.6–5.5 -7.6–4.1 16.0–26.5 72.3–77.9 4.7–16.0 12.6–23.0 -6.0–5.1
8.00 21.6–31.6 -5.7–5.4 -4.2–6.8 15.9–26.4 72.4–77.9 4.7–15.9 12.6–23.0 -6.0–5.1
8.33 21.6–31.6 -5.8–5.4 -4.3–6.7 15.8–26.4 72.4–78.0 4.6–15.9 12.5–22.9 -6.1–5.0
8.67 21.6–31.5 -5.9–5.3 -4.3–6.7 15.7–26.3 72.5–78.0 4.6–15.9 12.5–22.9 -6.2–4.9
9.00 21.5–31.5 -5.9–5.2 -4.4–6.6 15.6–26.2 72.5–78.1 4.6–15.8 12.5–22.9 -6.2–4.9
9.33 21.5–31.5 -6.0–5.2 -4.4–6.6 15.5–26.1 72.6–78.1 4.6–15.8 12.4–22.8 -6.3–4.9
9.67 21.5–31.5 -6.0–5.1 -4.5–6.5 15.4–26.0 72.6–78.1 4.5–15.8 12.4–22.8 -6.3–4.8
10.00 21.5–31.5 -6.0–5.0 -4.5–6.5 15.4–26.0 72.6–78.2 4.6–15.8 12.4–22.8 -6.3–4.8
Table E.39: North / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 6 9 10 11
1 -2.3–2.6 -5.0–3.6 -4.5–4.1 1.2–9.7 -1.5–10.2 -5.1–3.7 -3.3–5.1 -4.6–3.9
2 -5.0–3.6 -0.4–0.3 -0.6–0.8 -0.9–0.8 -5.0–5.0 -0.5–1.0 -0.9–0.8 -0.6–0.8
3 -4.5–4.1 -0.6–0.8 -0.6–0.6 -1.5–0.9 -5.0–4.9 -1.2–1.1 -0.9–1.6 -1.3–1.0
4 1.2–9.7 -0.9–0.8 -1.5–0.9 1.7–5.8 -4.7–6.9 -3.9–3.8 -4.1–3.3 -4.5–3.2
6 -1.5–10.2 -5.0–5.0 -5.0–4.9 -4.7–6.9 48.3–55.8 -3.6–2.9 -4.0–3.8 -1.4–1.7
9 -5.1–3.7 -0.5–1.0 -1.2–1.1 -3.9–3.8 -3.6–2.9 -1.5–1.4 -2.7–2.8 -2.7–3.0
10 -3.3–5.1 -0.9–0.8 -0.9–1.6 -4.1–3.3 -4.0–3.8 -2.7–2.8 6.5–9.8 -2.4–2.7
11 -4.6–3.9 -0.6–0.8 -1.3–1.0 -4.5–3.2 -1.4–1.7 -2.7–3.0 -2.4–2.7 -0.2–0.3
Total 21.5–31.7 -5.3–5.9 -7.5–4.4 16.7–27.5 71.6–77.3 4.8–16.2 12.6–23.1 -5.7–5.5
Higher 17.7 0.7 -1.3 13.2 17.2 11.2 8.8 0.4
410
Table E.40: Control / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 -1.2–1.2 -0.8–3.2 -1.0–3.0 0.7–4.9 -3.6–4.1 -0.6–3.9 -0.5–3.8 -1.0–3.0
2 -0.8–3.2 -0.2–0.3 -0.5–0.4 -0.6–0.4 -3.9–3.3 -0.5–0.5 -0.9–0.7 -0.5–0.3
3 -1.0–3.0 -0.5–0.4 -0.1–0.3 -0.5–0.4 -3.8–3.5 -0.5–0.4 -0.7–0.8 -0.5–0.3
4 0.7–4.9 -0.6–0.4 -0.5–0.4 2.2–4.2 -2.7–5.4 -1.4–2.4 -1.1–2.6 -1.5–2.0
6 -3.6–4.1 -3.9–3.3 -3.8–3.5 -2.7–5.4 46.3–51.9 1.2–6.9 -0.9–6.0 -0.2–1.8
9 -0.6–3.9 -0.5–0.5 -0.5–0.4 -1.4–2.4 1.2–6.9 -1.8–0.8 1.5–6.0 -1.4–3.0
10 -0.5–3.8 -0.9–0.7 -0.7–0.8 -1.1–2.6 -0.9–6.0 1.5–6.0 9.9–12.5 -0.2–3.1
11 -1.0–3.0 -0.5–0.3 -0.5–0.3 -1.5–2.0 -0.2–1.8 -1.4–3.0 -0.2–3.1 -0.1–0.4
Total 10.6–19.5 -5.0–4.3 -4.7–4.5 13.1–22.1 69.5–74.0 11.9–21.1 22.9–30.9 -4.2–4.9
Higher 5.5 -0.9 -0.8 8.9 14.1 6.2 5.6 -3.8
Table E.41: South / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 -2.8–2.1 -4.1–4.6 -3.8–4.8 2.2–11.2 0.0–10.1 -4.4–4.4 -3.0–5.1 -3.8–4.9
2 -4.1–4.6 -0.4–0.3 -0.9–0.5 -1.0–0.9 -3.6–3.9 -0.8–0.5 -1.0–0.9 -0.9–0.5
3 -3.8–4.8 -0.9–0.5 -0.3–0.6 -1.6–0.6 -3.9–3.8 -1.2–0.5 -1.5–0.8 -1.1–0.6
4 2.2–11.2 -1.0–0.9 -1.6–0.6 4.3–8.8 -5.8–4.9 -4.4–3.5 -3.3–4.1 -3.3–4.3
6 0.0–10.1 -3.6–3.9 -3.9–3.8 -5.8–4.9 35.4–42.0 -4.4–3.1 -6.0–2.0 -3.0–2.0
9 -4.4–4.4 -0.8–0.5 -1.2–0.5 -4.4–3.5 -4.4–3.1 -1.9–1.7 -3.0–3.4 -3.2–3.5
10 -3.0–5.1 -1.0–0.9 -1.5–0.8 -3.3–4.1 -6.0–2.0 -3.0–3.4 8.9–12.4 -2.1–3.3
11 -3.8–4.9 -0.9–0.5 -1.1–0.6 -3.3–4.3 -3.0–2.0 -3.2–3.5 -2.1–3.3 -0.5–0.5
Total 26.5–36.8 -7.9–4.8 -7.9–4.8 25.2–36.0 58.9–66.2 8.7–20.9 17.1–28.2 -6.6–5.9
Higher 17.9 -1.3 -0.5 17.8 22.3 16.2 12.2 -1.2
Table E.42: North / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 6 9 10 11
1 -2.4–2.5 -4.9–3.7 -4.5–4.1 0.7–9.4 -2.1–9.9 -5.2–3.5 -3.3–4.9 -4.7–3.9
2 -4.9–3.7 -0.4–0.3 -0.6–0.9 -1.1–0.7 -5.4–5.0 -0.5–1.0 -1.0–0.7 -0.6–0.9
3 -4.5–4.1 -0.6–0.9 -0.4–0.6 -1.1–1.2 -5.4–4.8 -1.2–0.9 -1.0–1.3 -1.2–0.8
4 0.7–9.4 -1.1–0.7 -1.1–1.2 3.8–8.1 -5.9–6.7 -3.9–3.7 -4.3–3.1 -4.4–3.1
6 -2.1–9.9 -5.4–5.0 -5.4–4.8 -5.9–6.7 48.5–56.1 -3.8–2.6 -4.8–3.2 -1.6–1.6
9 -5.2–3.5 -0.5–1.0 -1.2–0.9 -3.9–3.7 -3.8–2.6 -1.6–1.4 -2.7–2.8 -2.6–3.2
10 -3.3–4.9 -1.0–0.7 -1.0–1.3 -4.3–3.1 -4.8–3.2 -2.7–2.8 6.7–9.9 -2.5–2.6
11 -4.7–3.9 -0.6–0.9 -1.2–0.8 -4.4–3.1 -1.6–1.6 -2.6–3.2 -2.5–2.6 -0.2–0.2
Total 19.8–30.3 -5.1–6.4 -8.3–4.3 17.9–28.6 69.5–75.4 3.6–15.5 11.3–22.3 -5.5–5.9
Higher 17.4 1.4 -1.5 13.4 17.8 10.8 9.0 1.0
411
Table E.43: Control / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 -1.1–0.3 -0.2–2.4 -0.2–2.4 0.3–3.3 -1.4–3.9 -0.1–2.6 0.0–2.7 -0.3–2.3
2 -0.2–2.4 -0.1–0.1 -0.2–0.3 -0.8–1.0 -2.4–2.5 -0.3–0.2 -0.6–0.3 -0.2–0.3
3 -0.2–2.4 -0.2–0.3 -0.1–0.2 -0.9–0.9 -2.3–2.6 -0.5–0.3 -0.5–0.5 -0.5–0.2
4 0.3–3.3 -0.8–1.0 -0.9–0.9 12.7–15.2 7.8–14.9 -1.3–1.1 1.0–3.7 -0.6–1.3
6 -1.4–3.9 -2.4–2.5 -2.3–2.6 7.8–14.9 36.1–40.3 1.7–5.8 -0.2–4.5 -0.4–1.9
9 -0.1–2.6 -0.3–0.2 -0.5–0.3 -1.3–1.1 1.7–5.8 -1.1–0.8 1.6–4.9 -1.0–2.4
10 0.0–2.7 -0.6–0.3 -0.5–0.5 1.0–3.7 -0.2–4.5 1.6–4.9 8.0–10.2 -0.8–2.2
11 -0.3–2.3 -0.2–0.3 -0.5–0.2 -0.6–1.3 -0.4–1.9 -1.0–2.4 -0.8–2.2 -0.0–0.4
Total 3.5–11.1 -3.5–4.3 -3.3–4.4 32.6–38.7 64.7–68.9 8.2–15.8 20.7–27.4 -5.1–2.9
Higher -1.2 -0.8 -0.7 5.8 9.2 3.4 5.2 -4.8
Table E.44: South / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 -1.7–2.0 -2.9–3.6 -3.0–3.5 -4.3–4.1 -1.3–5.5 -3.1–3.6 -3.8–2.3 -2.9–3.7
2 -2.9–3.6 -0.2–0.3 -0.5–0.4 -3.7–2.0 -1.7–2.2 -0.5–0.5 -0.8–0.3 -0.5–0.5
3 -3.0–3.5 -0.5–0.4 -0.3–0.5 -3.9–1.8 -1.8–2.2 -1.0–0.6 -1.1–0.5 -1.0–0.5
4 -4.3–4.1 -3.7–2.0 -3.9–1.8 30.6–35.8 0.6–9.5 -2.1–2.8 -3.0–2.1 -1.9–2.2
6 -1.3–5.5 -1.7–2.2 -1.8–2.2 0.6–9.5 24.0–28.6 -2.4–3.3 -4.6–1.4 -4.2–0.6
9 -3.1–3.6 -0.5–0.5 -1.0–0.6 -2.1–2.8 -2.4–3.3 -1.2–1.5 -2.3–2.7 -2.5–2.7
10 -3.8–2.3 -0.8–0.3 -1.1–0.5 -3.0–2.1 -4.6–1.4 -2.3–2.7 5.7–8.4 -2.9–1.8
11 -2.9–3.7 -0.5–0.5 -1.0–0.5 -1.9–2.2 -4.2–0.6 -2.5–2.7 -2.9–1.8 -0.3–0.3
Total 12.7–21.8 -6.0–4.6 -5.6–4.9 50.2–56.8 45.0–51.7 5.9–15.6 13.1–22.1 -5.3–5.2
Higher 14.6 -0.2 0.9 17.2 17.3 9.5 14.3 1.9
Table E.45: North / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 6 9 10 11
1 -0.4–2.6 -3.0–2.4 -2.8–2.6 -5.1–3.0 -1.7–5.0 -2.8–2.7 -3.4–2.0 -2.9–2.4
2 -3.0–2.4 -0.1–0.3 -0.6–0.2 -3.5–2.6 -2.2–2.6 -0.6–0.3 -0.6–0.2 -0.6–0.2
3 -2.8–2.6 -0.6–0.2 -0.4–0.3 -3.6–2.6 -2.2–2.6 -0.7–0.9 -0.7–0.8 -0.7–0.9
4 -5.1–3.0 -3.5–2.6 -3.6–2.6 34.3–39.5 0.8–10.9 -2.3–1.5 -1.8–2.4 -1.8–1.5
6 -1.7–5.0 -2.2–2.6 -2.2–2.6 0.8–10.9 31.6–36.6 -2.7–1.4 -2.9–1.8 -2.8–0.8
9 -2.8–2.7 -0.6–0.3 -0.7–0.9 -2.3–1.5 -2.7–1.4 -0.8–1.1 -2.1–1.6 -2.2–1.6
10 -3.4–2.0 -0.6–0.2 -0.7–0.8 -1.8–2.4 -2.9–1.8 -2.1–1.6 1.6–3.8 -1.9–2.0
11 -2.9–2.4 -0.6–0.2 -0.7–0.9 -1.8–1.5 -2.8–0.8 -2.2–1.6 -1.9–2.0 -0.2–0.2
Total 8.5–17.5 -5.0–4.9 -4.2–5.6 49.6–55.7 49.3–55.3 1.4–10.8 6.5–15.7 -4.9–5.0
Higher 12.8 1.1 1.1 12.1 12.5 7.6 9.8 1.8
412
Table E.46: Control / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 -1.5–0.9 -1.2–3.2 -1.4–3.1 -6.0–8.8 -1.4–2.9 -1.1–3.5 -1.2–3.3 -1.4–3.1
2 -1.2–3.2 -0.2–0.3 -0.5–0.4 -7.2–7.7 -0.6–0.6 -0.5–0.5 -0.5–0.5 -0.5–0.4
3 -1.4–3.1 -0.5–0.4 -0.2–0.3 -7.4–7.6 -0.5–0.6 -0.5–0.5 -0.5–0.5 -0.5–0.4
4 -6.0–8.8 -7.2–7.7 -7.4–7.6 68.9–77.9 -1.3–8.6 -3.3–3.8 -2.1–5.2 -0.5–1.5
6 -1.4–2.9 -0.6–0.6 -0.5–0.6 -1.3–8.6 4.3–7.6 -0.3–5.5 -1.2–4.7 -2.3–3.5
9 -1.1–3.5 -0.5–0.5 -0.5–0.5 -3.3–3.8 -0.3–5.5 -1.7–0.7 -1.2–3.5 -1.4–3.2
10 -1.2–3.3 -0.5–0.5 -0.5–0.5 -2.1–5.2 -1.2–4.7 -1.2–3.5 -0.6–1.7 -1.1–3.2
11 -1.4–3.1 -0.5–0.4 -0.5–0.4 -0.5–1.5 -2.3–3.5 -1.4–3.2 -1.1–3.2 -0.3–0.2
Total 2.3–13.2 -6.9–4.6 -6.6–4.9 85.0–89.0 13.8–23.6 1.3–12.6 2.8–13.7 -6.4–4.9
Higher 1.0 -2.4 -1.8 6.0 3.3 1.4 1.2 -4.5
Table E.47: South / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 -2.5–2.0 -2.1–6.3 -1.6–6.6 -9.4–7.2 -0.9–7.4 -1.1–7.1 -1.4–6.7 -1.7–6.7
2 -2.1–6.3 -0.6–0.2 -0.3–1.2 -13.5–1.9 -1.2–1.9 -0.2–1.3 -0.3–1.2 -0.3–1.2
3 -1.6–6.6 -0.3–1.2 -0.4–0.6 -14.1–1.5 -1.3–1.8 -1.2–0.8 -1.1–0.9 -1.2–0.8
4 -9.4–7.2 -13.5–1.9 -14.1–1.5 51.3–62.5 -4.8–11.2 -4.4–5.4 -5.8–4.8 -2.8–2.9
6 -0.9–7.4 -1.2–1.9 -1.3–1.8 -4.8–11.2 9.6–15.0 -5.6–3.2 -3.4–5.4 -3.8–4.8
9 -1.1–7.1 -0.2–1.3 -1.2–0.8 -4.4–5.4 -5.6–3.2 -1.5–2.1 -4.2–2.7 -4.5–2.7
10 -1.4–6.7 -0.3–1.2 -1.1–0.9 -5.8–4.8 -3.4–5.4 -4.2–2.7 1.0–4.7 -4.2–2.7
11 -1.7–6.7 -0.3–1.2 -1.2–0.8 -2.8–2.9 -3.8–4.8 -4.5–2.7 -4.2–2.7 -0.7–0.4
Total 10.9–25.8 -9.8–7.3 -10.2–7.0 73.3–80.2 20.6–34.2 -0.3–16.2 4.1–19.8 -9.4–7.7
Higher 3.6 0.3 1.9 29.8 7.7 6.7 7.1 -2.3
Table E.48: North / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 6 9 10 11
1 -2.6–2.6 -3.4–6.3 -3.0–6.6 -10.3–6.1 -2.1–7.6 -2.9–6.7 -2.7–6.5 -3.0–6.5
2 -3.4–6.3 -0.5–0.3 -0.5–1.1 -14.7–-0.4 -1.6–2.8 -0.5–1.2 -0.8–1.0 -0.6–1.1
3 -3.0–6.6 -0.5–1.1 -0.7–0.4 -15.0–-0.6 -2.4–2.2 -0.9–1.4 -1.0–1.4 -0.9–1.3
4 -10.3–6.1 -14.7–-0.4 -15.0–-0.6 44.9–56.5 -1.9–16.6 -4.6–5.4 -7.4–4.3 -2.4–3.9
6 -2.1–7.6 -1.6–2.8 -2.4–2.2 -1.9–16.6 12.7–19.2 -2.1–7.3 -2.8–6.8 -2.6–6.3
9 -2.9–6.7 -0.5–1.2 -0.9–1.4 -4.6–5.4 -2.1–7.3 -1.9–1.7 -3.3–3.5 -3.4–3.7
10 -2.7–6.5 -0.8–1.0 -1.0–1.4 -7.4–4.3 -2.8–6.8 -3.3–3.5 2.5–6.6 -4.7–2.8
11 -3.0–6.5 -0.6–1.1 -0.9–1.3 -2.4–3.9 -2.6–6.3 -3.4–3.7 -4.7–2.8 -0.1–0.5
Total 8.2–25.1 -9.9–8.8 -9.8–8.6 65.4–74.1 30.2–43.3 -2.1–16.3 3.8–20.8 -11.1–7.8
Higher 7.2 4.0 4.7 29.6 3.8 1.4 5.9 -5.8
413
APPENDIX F
SENSITIVITY ANALYSIS RESULTS FOR NEAR-SURFACE FACETS
414
F.1 Introduction
The tables presented in this appendix provide the complete sensitivity analysis
results for Chapter 9 that explored near-surface facet formation. The following ta-
bles include the first-order, second-order, higher-order, and total-effect indices. The
indices are listed using the 90% confidence level intervals. Each table uses the indices
listed in Table 7.1 that correspond to the specific parameters.
In this appendix only the day-light scenario results are listed, since the research
focus of Chapter 9 is on radiation-recrystallization. Refer to Section 9.2 for details
regarding the nomenclature used for labeling the tables.
415
F.2 Snow Surface Temperature
Table F.1: Control / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 9.1–17.9 -5.0–4.6 -5.9–3.6 22.0–29.8 -6.3–3.2 60.3–65.4 -6.3–3.2 -6.3–3.2 7.4–16.1 17.2–25.1 -5.2–4.3
0.67 9.6–18.3 -4.2–5.0 -6.5–2.9 17.0–25.3 -6.8–2.7 65.9–70.7 -6.8–2.7 -6.8–2.7 9.3–17.9 18.5–26.2 -5.6–3.8
1.00 10.0–18.6 -4.7–4.5 -6.3–3.2 14.8–23.2 -6.5–2.9 68.4–73.1 -6.4–3.0 -6.5–2.9 10.7–19.1 19.7–27.3 -5.4–4.0
1.33 9.3–17.9 -5.8–3.6 -6.9–2.6 12.5–21.0 -7.4–2.2 68.9–73.5 -6.7–2.8 -7.0–2.5 10.1–18.7 19.8–27.5 -6.4–3.1
1.67 8.8–17.5 -6.4–3.1 -7.3–2.3 11.1–19.8 -7.7–1.9 68.6–73.2 -6.1–3.4 -6.5–3.1 9.9–18.6 20.3–28.0 -6.8–2.7
2.00 8.0–16.9 -7.3–2.4 -8.0–1.8 9.6–18.6 -8.3–1.6 67.4–72.2 -5.1–4.5 -5.9–3.9 9.2–18.1 20.4–28.2 -7.6–2.2
2.33 7.2–16.4 -7.9–2.0 -8.6–1.5 8.4–17.5 -8.8–1.3 65.8–70.9 -6.0–4.0 -4.4–5.4 8.7–17.7 20.5–28.4 -8.3–1.7
2.67 6.5–16.0 -8.4–1.7 -9.2–1.2 7.3–16.7 -9.3–1.1 63.9–69.3 -3.6–6.5 -5.6–4.8 8.4–17.5 20.5–28.4 -9.0–1.3
3.00 5.8–15.5 -9.0–1.3 -9.9–0.8 6.3–15.9 -9.7–1.1 62.1–67.8 -1.4–8.8 -4.1–6.5 7.8–17.1 20.2–28.4 -9.7–1.0
3.33 5.0–15.1 -9.8–1.0 -10.6–0.4 5.3–15.2 -10.3–0.8 60.5–66.5 0.6–10.9 -2.7–8.0 7.0–16.7 19.9–28.4 -10.4–0.6
3.67 4.5–14.8 -10.3–0.7 -11.1–0.2 4.5–14.6 -10.7–0.8 59.3–65.5 2.5–12.9 -1.6–9.3 6.6–16.6 19.8–28.5 -11.0–0.4
4.00 4.1–14.7 -10.5–0.8 -11.4–0.3 3.9–14.2 -11.0–0.9 58.2–64.7 4.0–14.4 -0.6–10.4 6.3–16.6 19.6–28.5 -11.4–0.2
4.33 3.8–14.6 -10.8–0.7 -11.7–0.2 3.4–14.0 -11.3–0.8 57.4–64.1 5.0–15.5 0.1–11.3 6.1–16.6 19.5–28.6 -11.9–0.1
4.67 3.8–14.6 -10.9–0.8 -11.9–0.2 3.1–13.8 -11.4–0.9 56.9–63.8 5.6–16.2 0.5–11.9 6.2–16.7 19.6–28.8 -12.0–0.1
5.00 3.6–14.5 -11.2–0.7 -12.2–0.1 2.7–13.6 -11.7–0.8 56.6–63.5 5.7–16.5 0.4–12.0 6.1–16.8 19.5–28.8 -12.2–0.1
5.33 3.6–14.6 -11.4–0.6 -12.4–-0.0 2.6–13.6 -11.9–0.7 56.5–63.4 5.5–16.3 0.2–11.9 6.1–16.9 19.5–28.9 -12.4–0.0
5.67 3.7–14.7 -11.3–0.7 -12.4–-0.1 2.8–13.8 -11.9–0.7 56.5–63.5 5.0–16.0 -0.1–11.6 6.3–17.0 19.7–29.1 -12.3–0.0
6.00 4.0–15.0 -11.2–0.7 -12.4–-0.1 3.1–14.0 -11.9–0.6 56.7–63.7 4.2–15.2 -0.7–11.1 6.5–17.2 20.0–29.3 -12.3–0.1
6.33 4.4–15.4 -11.1–0.8 -12.2–-0.0 3.6–14.3 -11.9–0.6 57.2–64.0 3.2–14.2 -1.4–10.3 6.8–17.4 20.3–29.6 -12.1–0.1
6.67 4.9–15.8 -10.8–0.8 -12.0–-0.1 4.3–14.8 -11.7–0.5 57.9–64.6 1.8–12.9 -2.3–9.4 7.2–17.7 20.8–29.9 -11.7–0.3
7.00 5.6–16.3 -10.5–1.0 -11.7–0.1 5.0–15.4 -11.5–0.4 58.8–65.4 0.3–11.3 -3.3–8.3 7.8–18.0 21.3–30.2 -11.3–0.5
7.33 6.4–16.8 -10.0–1.2 -11.2–0.4 5.9–16.1 -11.2–0.5 60.0–66.3 -1.3–9.6 -4.4–7.1 8.4–18.4 21.9–30.6 -10.8–0.8
7.67 7.3–17.5 -9.6–1.4 -10.5–0.7 7.0–16.9 -10.8–0.6 61.5–67.5 -3.0–7.8 -5.5–5.8 9.1–18.8 22.5–31.0 -10.1–1.2
8.00 8.3–18.2 -9.1–1.6 -9.8–1.1 8.1–17.7 -10.3–0.9 63.2–68.9 -4.8–5.9 -6.6–4.5 9.8–19.3 23.1–31.5 -9.4–1.6
8.33 9.3–19.0 -8.5–1.9 -9.0–1.7 9.3–18.6 -9.5–1.2 65.0–70.5 -6.3–4.3 -4.3–6.2 10.4–19.8 23.7–31.9 -8.5–2.2
8.67 10.4–19.7 -7.8–2.3 -8.0–2.4 10.1–19.3 -8.7–1.8 67.0–72.2 -4.4–5.6 -4.9–5.4 11.2–20.2 24.3–32.2 -7.7–2.7
9.00 11.3–20.4 -7.1–2.9 -7.1–3.1 10.6–19.7 -7.9–2.4 68.7–73.7 -5.1–4.9 -5.2–4.9 11.9–20.8 24.7–32.4 -6.8–3.4
9.33 11.9–20.8 -6.4–3.4 -6.4–3.7 11.0–20.0 -7.1–2.9 70.1–75.0 -5.3–4.6 -5.2–4.7 12.4–21.2 24.9–32.5 -6.1–3.9
9.67 12.2–21.0 -5.8–3.9 -5.8–4.2 11.1–20.0 -6.6–3.4 71.2–75.9 -5.3–4.5 -5.1–4.7 12.8–21.5 24.8–32.4 -5.6–4.3
10.00 12.3–20.9 -5.6–4.1 -5.4–4.5 10.9–19.8 -6.3–3.5 71.8–76.5 -5.3–4.5 -5.1–4.7 13.0–21.7 24.6–32.2 -5.3–4.5
Table F.2: Control / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -1.0–0.6 -1.5–1.5 -1.5–1.5 -0.4–2.4 -1.6–1.4 -1.8–3.7 -1.7–1.4 -1.7–1.3 -1.4–1.9 -1.9–1.4 -1.6–1.4
2 -1.5–1.5 -0.1–0.3 -0.5–0.4 -0.4–0.6 -0.4–0.4 -1.4–3.7 -0.5–0.7 -0.6–0.5 -0.4–0.6 -0.3–1.3 -0.5–0.4
3 -1.5–1.5 -0.5–0.4 -0.1–0.2 -0.3–0.4 -0.4–0.3 -1.6–3.5 -0.2–0.7 -0.5–0.3 -0.3–0.4 -0.5–0.9 -0.4–0.3
4 -0.4–2.4 -0.4–0.6 -0.3–0.4 0.5–1.9 -1.7–1.0 -2.2–3.5 -1.5–1.2 -1.6–1.2 -1.9–1.0 -1.7–1.2 -1.7–1.0
5 -1.6–1.4 -0.4–0.4 -0.4–0.3 -1.7–1.0 -0.1–0.2 -1.5–3.5 -0.1–0.9 -0.1–0.7 -0.1–0.7 -0.2–1.3 -0.1–0.6
6 -1.8–3.7 -1.4–3.7 -1.6–3.5 -2.2–3.5 -1.5–3.5 24.8–29.2 3.7–7.8 2.3–6.2 -0.9–3.9 2.7–8.6 -0.9–1.3
7 -1.7–1.4 -0.5–0.7 -0.2–0.7 -1.5–1.2 -0.1–0.9 3.7–7.8 4.5–6.0 -0.8–2.1 -1.0–1.8 0.5–3.6 -0.6–1.6
8 -1.7–1.3 -0.6–0.5 -0.5–0.3 -1.6–1.2 -0.1–0.7 2.3–6.2 -0.8–2.1 3.7–5.1 -0.1–2.5 0.4–3.8 -0.3–1.8
9 -1.4–1.9 -0.4–0.6 -0.3–0.4 -1.9–1.0 -0.1–0.7 -0.9–3.9 -1.0–1.8 -0.1–2.5 0.3–2.1 1.9–5.2 -1.5–1.4
10 -1.9–1.4 -0.3–1.3 -0.5–0.9 -1.7–1.2 -0.2–1.3 2.7–8.6 0.5–3.6 0.4–3.8 1.9–5.2 7.2–9.3 -0.9–1.5
11 -1.6–1.4 -0.5–0.4 -0.4–0.3 -1.7–1.0 -0.1–0.6 -0.9–1.3 -0.6–1.6 -0.3–1.8 -1.5–1.4 -0.9–1.5 -0.1–0.2
Total 3.6–14.5 -11.2–0.7 -12.2–0.1 2.7–13.6 -11.7–0.8 56.6–63.5 5.7–16.5 0.4–12.0 6.1–16.8 19.5–28.8 -12.2–0.1
Higher 7.9 -7.2 -7.3 6.9 -7.7 10.9 -3.9 -6.9 3.2 1.4 -7.6
416
Table F.3: South / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 16.4–25.7 -0.7–9.8 -5.8–5.0 19.0–28.2 -6.1–4.8 59.8–65.1 -6.0–4.8 -6.1–4.8 2.5–12.9 11.3–20.7 -5.2–5.6
0.67 17.8–26.8 -3.1–7.1 -5.7–4.6 15.3–24.8 -6.3–4.1 65.3–70.2 -6.2–4.1 -6.4–4.0 3.8–13.5 13.5–22.4 -5.5–4.9
1.00 16.2–25.0 -5.8–4.5 -4.5–5.6 12.4–21.9 -4.9–5.0 66.7–71.4 -4.3–5.7 -4.7–5.3 2.2–11.9 14.0–22.6 -3.9–6.1
1.33 13.8–22.7 -4.6–5.4 -5.7–4.2 10.0–19.6 -6.3–3.7 66.2–71.0 -4.0–5.9 -4.8–5.0 1.5–10.9 14.1–22.5 -5.3–4.8
1.67 11.4–20.2 -5.8–4.0 -6.8–3.1 7.5–17.2 -7.2–2.8 64.0–69.0 -4.8–5.1 -6.3–3.8 0.2–9.7 13.6–22.0 -6.6–3.6
2.00 9.3–18.1 -6.8–3.0 -7.6–2.2 5.2–15.0 -7.6–2.3 60.6–65.9 -0.9–8.6 -3.4–6.5 -0.5–8.9 12.7–21.1 -7.3–2.7
2.33 7.3–16.5 -7.5–2.4 -8.4–1.6 3.4–13.4 -8.1–1.9 56.9–62.6 3.5–12.7 0.0–9.7 -1.2–8.3 11.8–20.3 -8.0–2.1
2.67 5.9–15.3 -8.2–2.1 -9.0–1.2 1.7–12.0 -8.5–1.7 53.6–59.7 7.6–16.6 3.1–12.7 -1.8–7.9 10.9–19.7 -8.7–1.6
3.00 4.8–14.4 -8.8–1.7 -9.7–1.0 0.3–11.0 -9.0–1.5 51.0–57.5 11.1–20.1 5.5–15.0 -2.5–7.6 10.1–19.1 -9.4–1.3
3.33 3.7–13.7 -9.4–1.5 -10.3–0.6 -0.8–10.2 -9.5–1.4 49.0–55.9 13.9–22.9 7.1–16.8 -3.0–7.4 9.4–18.7 -10.0–0.8
3.67 2.9–13.3 -9.8–1.3 -10.8–0.6 -1.5–9.7 -9.7–1.5 47.7–54.8 16.1–25.1 8.6–18.4 -3.3–7.3 8.9–18.5 -10.5–0.7
4.00 2.4–13.0 -10.1–1.3 -11.2–0.4 -2.2–9.3 -10.0–1.4 46.8–54.1 17.6–26.8 9.5–19.5 -3.7–7.2 8.3–18.2 -10.9–0.5
4.33 2.1–12.9 -10.4–1.3 -11.7–0.2 -2.8–9.1 -10.2–1.5 46.2–53.7 18.6–27.9 10.1–20.3 -4.1–7.1 7.9–18.1 -11.1–0.5
4.67 1.8–12.8 -10.7–1.2 -12.2–-0.1 -3.3–8.8 -10.6–1.3 45.7–53.4 19.1–28.4 10.2–20.6 -4.6–7.0 7.7–18.1 -11.5–0.4
5.00 1.7–12.9 -10.9–1.1 -12.5–-0.3 -3.5–8.7 -10.8–1.2 45.6–53.3 19.2–28.7 10.1–20.7 -4.9–6.9 7.6–18.1 -11.7–0.4
5.33 1.8–13.0 -10.9–1.1 -12.6–-0.2 -3.6–8.7 -10.9–1.2 45.6–53.3 19.1–28.6 9.9–20.5 -4.8–7.0 7.7–18.2 -11.8–0.4
5.67 2.1–13.3 -10.8–1.2 -12.5–-0.2 -3.4–8.8 -10.8–1.2 45.7–53.5 18.6–28.1 9.5–20.2 -4.7–7.2 7.8–18.3 -11.8–0.4
6.00 2.5–13.6 -10.7–1.2 -12.5–-0.2 -3.2–9.0 -10.8–1.3 46.2–53.9 17.6–27.2 8.8–19.5 -4.5–7.3 8.0–18.5 -11.7–0.4
6.33 3.2–14.1 -10.4–1.4 -12.2–-0.1 -2.7–9.3 -10.6–1.3 46.8–54.5 16.3–25.9 8.0–18.5 -4.1–7.5 8.4–18.7 -11.4–0.6
6.67 4.0–14.8 -10.1–1.6 -11.8–0.2 -2.2–9.8 -10.4–1.4 47.8–55.3 14.7–24.3 6.9–17.4 -3.6–7.7 9.0–19.1 -11.0–0.8
7.00 5.1–15.6 -9.7–1.9 -11.2–0.5 -1.4–10.3 -10.0–1.6 49.2–56.4 12.8–22.4 5.8–16.2 -2.9–8.1 9.7–19.6 -10.4–1.1
7.33 6.4–16.6 -9.2–2.2 -10.5–1.0 -0.4–11.0 -9.6–1.8 50.9–57.9 10.6–20.1 4.4–14.7 -2.3–8.5 10.7–20.2 -9.8–1.6
7.67 8.0–17.9 -8.6–2.5 -9.6–1.6 0.7–11.9 -9.1–2.0 53.1–59.9 8.0–17.5 2.8–13.1 -1.5–8.9 11.8–21.0 -9.1–2.0
8.00 9.7–19.3 -8.0–2.8 -8.6–2.3 1.9–12.8 -8.5–2.4 55.9–62.3 5.2–14.8 0.9–11.3 -0.8–9.4 12.9–22.0 -8.3–2.6
8.33 11.6–21.0 -7.3–3.3 -7.5–3.1 3.1–13.9 -7.6–3.0 59.1–65.1 2.3–12.1 -0.9–9.5 -0.3–9.9 14.3–23.1 -7.4–3.3
8.67 13.8–22.9 -6.2–4.2 -6.1–4.4 4.3–14.8 -6.5–3.9 62.6–68.2 -0.2–9.8 -2.4–8.0 0.7–10.8 15.7–24.4 -6.2–4.3
9.00 15.8–24.7 -4.8–5.4 -4.5–5.9 5.3–15.7 -5.1–5.2 65.8–71.1 -1.7–8.3 -3.1–7.3 1.9–11.9 17.3–25.9 -4.8–5.6
9.33 17.3–26.2 -6.4–4.2 -6.1–4.6 6.0–16.4 -6.7–3.9 68.3–73.5 -2.5–7.7 -3.3–7.1 3.0–13.0 18.5–27.2 -6.6–4.3
9.67 18.3–27.3 -5.2–5.3 -4.8–5.8 6.4–16.9 -5.6–4.9 69.9–75.0 -2.5–7.8 -3.1–7.4 3.8–13.8 19.4–28.1 -5.5–5.4
10.00 18.7–27.6 -4.6–5.9 -4.1–6.4 6.5–17.0 -5.0–5.5 70.9–75.9 -2.4–7.9 -2.9–7.6 4.2–14.2 19.9–28.6 -4.9–5.9
Table F.4: South / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.2–1.7 -1.7–0.8 -2.0–0.5 -0.9–1.5 -1.9–0.6 -2.9–1.7 -1.6–1.6 -1.6–1.3 -1.4–1.1 -1.6–1.0 -1.9–0.6
2 -1.7–0.8 -0.1–0.5 -0.5–0.6 -0.5–0.7 -0.4–0.7 -1.8–2.4 -0.5–1.5 -0.9–0.7 -0.8–0.5 -0.5–0.9 -0.5–0.6
3 -2.0–0.5 -0.5–0.6 -0.2–0.1 -0.2–0.3 -0.2–0.3 -1.8–2.3 -0.1–1.4 -0.4–0.6 -0.1–0.4 -0.0–0.8 -0.1–0.3
4 -0.9–1.5 -0.5–0.7 -0.2–0.3 0.1–1.4 -1.4–1.0 -2.1–2.7 -1.8–1.2 -1.1–1.4 -1.4–1.1 -1.3–1.1 -1.5–1.0
5 -1.9–0.6 -0.4–0.7 -0.2–0.3 -1.4–1.0 0.1–0.5 -1.6–2.6 -0.2–1.8 -0.4–1.0 -0.2–0.8 -0.2–1.0 -0.3–0.6
6 -2.9–1.7 -1.8–2.4 -1.8–2.3 -2.1–2.7 -1.6–2.6 21.2–25.1 6.0–10.8 4.1–8.5 -0.5–2.8 1.9–6.1 -1.0–1.2
7 -1.6–1.6 -0.5–1.5 -0.1–1.4 -1.8–1.2 -0.2–1.8 6.0–10.8 10.4–12.5 -0.8–3.0 -1.0–1.6 1.2–4.3 -0.9–1.3
8 -1.6–1.3 -0.9–0.7 -0.4–0.6 -1.1–1.4 -0.4–1.0 4.1–8.5 -0.8–3.0 7.9–9.8 -0.1–2.5 0.9–4.0 -0.4–1.9
9 -1.4–1.1 -0.8–0.5 -0.1–0.4 -1.4–1.1 -0.2–0.8 -0.5–2.8 -1.0–1.6 -0.1–2.5 0.5–1.6 0.4–2.6 -0.6–1.4
10 -1.6–1.0 -0.5–0.9 -0.0–0.8 -1.3–1.1 -0.2–1.0 1.9–6.1 1.2–4.3 0.9–4.0 0.4–2.6 4.7–6.1 -0.6–1.3
11 -1.9–0.6 -0.5–0.6 -0.1–0.3 -1.5–1.0 -0.3–0.6 -1.0–1.2 -0.9–1.3 -0.4–1.9 -0.6–1.4 -0.6–1.3 -0.0–0.3
Total 1.7–12.9 -10.9–1.1 -12.5–-0.3 -3.5–8.7 -10.8–1.2 45.6–53.3 19.2–28.7 10.1–20.7 -4.9–6.9 7.6–18.1 -11.7–0.4
Higher 9.6 -5.6 -7.3 2.0 -6.8 5.5 -1.9 -5.5 -4.6 -4.2 -7.1
417
Table F.5: North / Temp. at 0cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 17.9–30.0 4.8–17.9 -4.4–9.8 39.4–48.7 -4.8–9.2 45.6–54.1 -4.8–9.2 -4.8–9.2 4.0–17.3 9.3–21.7 -3.6–10.2
0.67 21.6–32.9 -0.4–12.7 -6.2–7.6 31.0–41.6 -6.7–7.1 53.2–60.7 -6.7–7.1 -6.7–7.1 3.4–16.5 10.7–22.6 -5.3–8.4
1.00 22.9–34.0 -2.8–10.2 -6.0–7.6 28.7–39.4 -7.6–6.1 56.6–63.8 -7.4–6.2 -7.5–6.2 2.8–15.9 11.7–23.4 -5.5–8.0
1.33 22.1–32.9 -4.9–8.0 -7.1–6.3 25.9–36.8 -8.3–5.0 57.7–64.8 -8.0–5.4 -7.8–5.5 1.8–14.7 11.9–23.1 -6.4–6.8
1.67 21.0–31.6 -6.2–6.5 -7.5–5.5 23.5–34.4 -4.9–7.2 58.4–65.3 -7.9–5.0 -7.7–5.2 1.1–13.6 11.9–22.9 -7.1–5.8
2.00 18.9–29.3 -7.3–5.1 -8.2–4.4 20.8–31.7 -5.9–6.2 58.3–65.0 -7.8–4.7 -7.6–4.9 0.2–12.3 11.3–22.2 -8.1–4.5
2.33 17.2–27.4 -7.6–4.4 -4.7–6.8 18.8–29.5 -5.9–5.7 57.7–64.2 -6.3–5.7 -5.9–6.0 -0.0–11.6 11.5–22.0 -4.7–7.0
2.67 15.5–25.5 -4.6–6.5 -5.2–5.9 16.8–27.1 -6.2–5.0 56.6–62.9 -4.6–6.8 -3.8–7.4 -0.8–10.5 11.2–21.5 -5.2–6.2
3.00 14.0–23.7 -4.9–5.8 -5.5–5.2 14.8–24.9 -6.2–4.6 55.0–61.1 -2.7–8.2 -1.4–9.3 -1.4–9.5 11.1–20.9 -5.3–5.6
3.33 12.8–22.3 -5.1–5.4 -5.5–4.9 13.1–23.0 -6.0–4.4 53.2–59.4 -0.5–9.8 1.3–11.5 -1.7–8.8 10.9–20.4 -5.3–5.2
3.67 11.7–21.1 -5.3–4.8 -5.5–4.7 11.5–21.4 -5.8–4.3 51.3–57.5 1.3–11.1 3.9–13.6 -2.0–8.3 10.8–20.1 -5.4–4.8
4.00 10.9–20.2 -5.5–4.5 -5.6–4.4 10.3–20.1 -5.8–4.1 49.7–55.9 2.6–12.2 6.0–15.4 -2.3–7.8 10.7–19.8 -5.5–4.5
4.33 10.2–19.4 -5.8–4.1 -5.8–4.1 9.3–19.1 -6.0–3.8 48.3–54.5 3.6–13.0 7.5–16.7 -2.6–7.4 10.4–19.4 -5.8–4.0
4.67 9.7–18.9 -6.0–3.9 -6.0–3.8 8.8–18.5 -6.4–3.5 47.2–53.5 4.2–13.5 8.6–17.6 -2.9–7.0 10.1–19.1 -6.2–3.7
5.00 9.4–18.5 -6.2–3.7 -6.2–3.6 8.4–18.1 -6.6–3.3 46.6–52.9 4.5–13.7 9.3–18.2 -3.2–6.7 10.0–19.0 -6.6–3.3
5.33 9.4–18.5 -6.2–3.7 -6.2–3.5 8.3–18.1 -6.7–3.3 46.4–52.8 4.6–13.8 9.6–18.5 -3.1–6.8 10.0–19.0 -6.6–3.3
5.67 9.7–18.8 -6.2–3.7 -6.2–3.7 8.6–18.3 -6.7–3.3 46.7–53.0 4.5–13.7 9.5–18.4 -2.9–7.0 10.2–19.2 -6.5–3.5
6.00 10.2–19.3 -6.3–3.7 -6.2–3.8 9.0–18.8 -6.7–3.3 47.2–53.6 3.9–13.3 8.8–17.8 -2.8–7.1 10.5–19.6 -6.5–3.6
6.33 10.8–20.0 -6.3–3.7 -6.2–3.8 9.6–19.4 -6.8–3.3 48.2–54.6 3.0–12.6 7.6–16.7 -2.8–7.3 10.9–20.0 -6.5–3.7
6.67 11.8–21.0 -6.4–3.8 -6.1–4.0 10.4–20.3 -6.9–3.4 49.6–55.9 1.9–11.6 5.9–15.4 -2.6–7.7 11.3–20.5 -6.4–4.0
7.00 13.0–22.4 -6.4–3.9 -6.1–4.2 11.6–21.6 -7.0–3.5 51.3–57.7 0.5–10.5 3.9–13.7 -2.3–8.1 11.8–21.1 -6.2–4.2
7.33 14.7–24.1 -6.5–4.1 -6.0–4.6 13.1–23.1 -7.0–3.7 53.3–59.6 -1.1–9.3 1.6–11.8 -2.0–8.6 12.4–21.8 -6.1–4.6
7.67 16.6–26.0 -6.5–4.4 -5.8–5.0 14.7–24.9 -7.1–3.9 55.4–61.8 -2.9–7.9 -0.8–9.9 -1.7–9.1 13.0–22.6 -5.9–5.0
8.00 18.7–28.2 -6.5–4.7 -5.6–5.5 16.6–26.8 -7.0–4.2 57.6–63.9 -4.7–6.6 -2.9–8.2 -1.3–9.8 13.7–23.4 -5.6–5.5
8.33 20.8–30.4 -6.5–5.0 -5.4–6.0 18.4–28.7 -6.9–4.6 59.5–65.9 -6.2–5.5 -4.9–6.7 -0.8–10.4 14.2–24.2 -5.3–6.2
8.67 22.8–32.5 -6.4–5.3 -5.2–6.5 20.0–30.5 -6.8–4.9 60.9–67.5 -7.5–4.6 -6.4–5.6 -0.4–11.1 14.7–24.9 -5.1–6.7
9.00 24.5–34.2 -6.4–5.6 -4.9–7.1 21.2–31.9 -6.7–5.3 62.0–68.6 -4.7–7.1 -7.6–4.7 -0.0–11.7 15.1–25.4 -4.8–7.1
9.33 25.7–35.5 -6.4–5.9 -8.1–4.3 22.1–32.9 -6.6–5.6 62.7–69.4 -5.3–6.8 -4.7–7.4 0.3–12.2 15.3–25.7 -4.7–7.4
9.67 26.5–36.4 -6.3–6.1 -7.8–4.6 22.6–33.5 -6.5–5.9 63.2–69.9 -5.7–6.6 -5.0–7.2 0.7–12.7 15.4–26.0 -8.3–4.5
10.00 26.9–36.9 -6.2–6.3 -7.7–4.9 22.8–33.8 -6.4–6.0 63.5–70.3 -5.8–6.6 -5.1–7.2 1.1–13.0 15.4–26.1 -8.1–4.6
Table F.6: North / Temp. at 0cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–2.3 -2.5–1.7 -2.5–1.8 -0.8–3.5 -2.6–1.7 -0.8–5.2 -2.6–1.5 -2.5–1.5 -2.6–1.7 -2.1–2.1 -2.9–1.4
2 -2.5–1.7 -0.3–0.3 -0.4–0.8 -0.2–1.1 -0.3–0.9 0.1–5.1 -0.3–1.1 -0.4–1.1 -0.4–0.9 -0.4–1.2 -0.4–0.9
3 -2.5–1.8 -0.4–0.8 -0.3–0.4 -0.5–0.8 -0.6–0.6 -0.4–4.6 -0.6–0.6 -0.8–0.8 -0.6–0.7 -0.8–0.7 -0.7–0.6
4 -0.8–3.5 -0.2–1.1 -0.5–0.8 1.4–3.4 -1.8–1.9 -0.6–5.3 -1.5–2.0 -1.8–1.6 -2.0–1.8 -1.1–2.6 -1.6–2.1
5 -2.6–1.7 -0.3–0.9 -0.6–0.6 -1.8–1.9 0.1–0.5 -0.2–4.8 -0.4–0.6 -0.6–0.8 -0.3–0.5 -0.5–0.9 -0.4–0.4
6 -0.8–5.2 0.1–5.1 -0.4–4.6 -0.6–5.3 -0.2–4.8 32.5–36.9 -0.4–3.1 -0.4–3.9 -0.8–2.9 -0.9–4.2 -0.7–1.8
7 -2.6–1.5 -0.3–1.1 -0.6–0.6 -1.5–2.0 -0.4–0.6 -0.4–3.1 8.1–9.8 -2.1–1.3 -1.4–1.1 -1.6–1.4 -1.4–0.9
8 -2.5–1.5 -0.4–1.1 -0.8–0.8 -1.8–1.6 -0.6–0.8 -0.4–3.9 -2.1–1.3 11.5–13.4 -1.3–1.4 -2.0–1.3 -1.3–1.2
9 -2.6–1.7 -0.4–0.9 -0.6–0.7 -2.0–1.8 -0.3–0.5 -0.8–2.9 -1.4–1.1 -1.3–1.4 -0.7–0.8 -1.1–1.8 -1.1–1.8
10 -2.1–2.1 -0.4–1.2 -0.8–0.7 -1.1–2.6 -0.5–0.9 -0.9–4.2 -1.6–1.4 -2.0–1.3 -1.1–1.8 9.5–11.5 -1.6–1.2
11 -2.9–1.4 -0.4–0.9 -0.7–0.6 -1.6–2.1 -0.4–0.4 -0.7–1.8 -1.4–0.9 -1.3–1.2 -1.1–1.8 -1.6–1.2 -0.3–0.3
Total 9.4–18.5 -6.2–3.7 -6.2–3.6 8.4–18.1 -6.6–3.3 46.6–52.9 4.5–13.7 9.3–18.2 -3.2–6.7 10.0–19.0 -6.6–3.3
Higher 12.9 -6.0 -3.4 5.6 -4.6 -2.8 -0.5 0.4 0.2 1.3 -1.8
418
Table F.7: Control / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.8–1.1 -2.2–1.3 -2.2–1.2 0.1–3.4 -2.3–1.2 -1.6–5.6 -1.4–1.9 -1.4–1.9 -2.3–1.5 -1.7–1.9 -2.3–1.2
2 -2.2–1.3 -0.1–0.3 -0.4–0.4 -0.2–0.7 -0.3–0.4 -1.6–5.5 -0.4–0.4 -0.3–0.4 -0.3–0.5 -0.2–1.5 -0.3–0.4
3 -2.2–1.2 -0.4–0.4 -0.2–0.2 -0.3–0.5 -0.4–0.3 -1.7–5.4 -0.3–0.4 -0.4–0.3 -0.3–0.4 -0.4–1.2 -0.4–0.3
4 0.1–3.4 -0.2–0.7 -0.3–0.5 2.9–4.6 -1.7–1.2 -2.0–5.8 -1.7–1.3 -1.7–1.3 -1.9–1.4 -1.4–2.0 -1.7–1.2
5 -2.3–1.2 -0.3–0.4 -0.4–0.3 -1.7–1.2 -0.1–0.1 -1.8–5.2 -0.1–0.4 -0.1–0.4 -0.1–0.4 -0.3–1.3 -0.1–0.3
6 -1.6–5.6 -1.6–5.5 -1.7–5.4 -2.0–5.8 -1.8–5.2 36.9–42.3 0.5–3.6 -0.2–2.7 0.4–5.9 2.4–9.5 -1.1–1.0
7 -1.4–1.9 -0.4–0.4 -0.3–0.4 -1.7–1.3 -0.1–0.4 0.5–3.6 1.5–2.3 -0.5–0.9 -0.7–0.8 -0.2–2.1 -0.5–0.8
8 -1.4–1.9 -0.3–0.4 -0.4–0.3 -1.7–1.3 -0.1–0.4 -0.2–2.7 -0.5–0.9 1.1–1.8 0.0–1.5 0.1–2.5 -0.1–1.1
9 -2.3–1.5 -0.3–0.5 -0.3–0.4 -1.9–1.4 -0.1–0.4 0.4–5.9 -0.7–0.8 0.0–1.5 -0.4–1.5 2.1–5.9 -1.7–1.7
10 -1.7–1.9 -0.2–1.5 -0.4–1.2 -1.4–2.0 -0.3–1.3 2.4–9.5 -0.2–2.1 0.1–2.5 2.1–5.9 9.6–11.9 -0.8–1.8
11 -2.3–1.2 -0.3–0.4 -0.4–0.3 -1.7–1.2 -0.1–0.3 -1.1–1.0 -0.5–0.8 -0.1–1.1 -1.7–1.7 -0.8–1.8 -0.1–0.3
Total 5.9–16.4 -9.9–1.4 -10.6–0.8 8.6–18.6 -10.7–0.9 62.2–68.1 -7.7–4.0 -6.0–5.3 6.3–16.7 19.6–28.7 -10.1–1.3
Higher 8.9 -7.1 -6.7 6.7 -6.8 3.7 -7.4 -6.0 3.3 -1.2 -5.0
Table F.8: South / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.7–2.8 -1.8–1.4 -2.0–1.3 -1.4–1.8 -2.0–1.3 -4.2–3.2 -2.1–1.3 -1.8–1.5 -2.1–1.4 -1.6–1.9 -2.1–1.2
2 -1.8–1.4 -0.1–0.5 -0.8–0.3 -0.9–0.3 -0.7–0.4 -4.1–3.1 -0.7–0.6 -0.8–0.4 -0.7–0.5 -1.0–0.6 -0.8–0.4
3 -2.0–1.3 -0.8–0.3 -0.2–0.2 -0.4–0.5 -0.4–0.4 -3.9–3.2 -0.4–0.5 -0.5–0.4 -0.3–0.5 -0.4–0.9 -0.3–0.4
4 -1.4–1.8 -0.9–0.3 -0.4–0.5 2.3–3.9 -1.8–1.1 -3.9–4.0 -1.7–1.4 -1.7–1.3 -1.6–1.4 -1.5–1.7 -1.9–1.1
5 -2.0–1.3 -0.7–0.4 -0.4–0.4 -1.8–1.1 -0.0–0.4 -4.0–3.1 -0.6–0.4 -0.4–0.5 -0.5–0.3 -0.7–0.7 -0.4–0.4
6 -4.2–3.2 -4.1–3.1 -3.9–3.2 -3.9–4.0 -4.0–3.1 37.9–43.5 0.3–4.7 -0.3–3.6 -0.9–3.6 0.8–6.8 -1.2–1.0
7 -2.1–1.3 -0.7–0.6 -0.4–0.5 -1.7–1.4 -0.6–0.4 0.3–4.7 4.6–5.8 -1.0–1.1 -1.0–0.9 -0.2–2.2 -0.8–0.9
8 -1.8–1.5 -0.8–0.4 -0.5–0.4 -1.7–1.3 -0.4–0.5 -0.3–3.6 -1.0–1.1 3.6–4.7 -0.5–1.4 -0.2–2.1 -0.5–1.2
9 -2.1–1.4 -0.7–0.5 -0.3–0.5 -1.6–1.4 -0.5–0.3 -0.9–3.6 -1.0–0.9 -0.5–1.4 0.5–1.9 -0.4–2.4 -1.7–0.9
10 -1.6–1.9 -1.0–0.6 -0.4–0.9 -1.5–1.7 -0.7–0.7 0.8–6.8 -0.2–2.2 -0.2–2.1 -0.4–2.4 8.3–10.2 -1.1–1.4
11 -2.1–1.2 -0.8–0.4 -0.3–0.4 -1.9–1.1 -0.4–0.4 -1.2–1.0 -0.8–0.9 -0.5–1.2 -1.7–0.9 -1.1–1.4 0.1–0.6
Total 6.7–17.7 -9.7–2.4 -10.3–1.9 2.5–14.6 -9.9–2.2 56.0–62.7 -0.6–11.0 -4.5–7.5 -4.0–7.9 9.7–20.4 -9.7–2.5
Higher 12.9 -1.7 -3.7 6.6 -2.5 11.3 -2.8 -5.5 -1.1 -1.4 -2.9
Table F.9: North / Temp. at 0cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.4–3.3 -2.4–3.6 -2.4–3.6 0.3–6.5 -2.5–3.5 -1.3–7.6 -2.7–3.2 -2.6–3.1 -2.5–3.6 -3.3–2.2 -3.0–3.1
2 -2.4–3.6 -0.4–0.3 -0.8–0.6 -0.7–1.0 -0.8–0.6 -1.5–6.1 -0.8–0.6 -0.8–0.6 -0.7–0.7 -1.0–0.9 -0.8–0.6
3 -2.4–3.6 -0.8–0.6 -0.2–0.5 -0.8–0.9 -0.9–0.5 -1.8–5.8 -0.9–0.5 -0.9–0.5 -0.9–0.6 -0.9–1.0 -0.9–0.5
4 0.3–6.5 -0.7–1.0 -0.8–0.9 4.4–7.4 -1.4–3.5 -1.5–7.6 -1.2–3.4 -1.5–3.1 -1.7–3.3 -0.9–4.0 -1.2–3.7
5 -2.5–3.5 -0.8–0.6 -0.9–0.5 -1.4–3.5 -0.1–0.2 -1.6–5.9 -0.4–0.2 -0.4–0.2 -0.2–0.3 -0.9–0.6 -0.2–0.3
6 -1.3–7.6 -1.5–6.1 -1.8–5.8 -1.5–7.6 -1.6–5.9 39.4–45.5 -1.3–2.2 -1.3–2.6 -1.9–3.0 -2.9–3.8 -1.0–1.9
7 -2.7–3.2 -0.8–0.6 -0.9–0.5 -1.2–3.4 -0.4–0.2 -1.3–2.2 3.1–4.0 -1.1–0.9 -1.2–0.6 -1.2–1.2 -1.2–0.5
8 -2.6–3.1 -0.8–0.6 -0.9–0.5 -1.5–3.1 -0.4–0.2 -1.3–2.6 -1.1–0.9 4.0–5.1 -1.1–0.8 -1.3–1.3 -1.0–0.8
9 -2.5–3.6 -0.7–0.7 -0.9–0.6 -1.7–3.3 -0.2–0.3 -1.9–3.0 -1.2–0.6 -1.1–0.8 -0.9–1.1 -2.4–1.5 -1.8–2.1
10 -3.3–2.2 -1.0–0.9 -0.9–1.0 -0.9–4.0 -0.9–0.6 -2.9–3.8 -1.2–1.2 -1.3–1.3 -2.4–1.5 10.7–13.4 -1.5–2.1
11 -3.0–3.1 -0.8–0.6 -0.9–0.5 -1.2–3.7 -0.2–0.3 -1.0–1.9 -1.2–0.5 -1.0–0.8 -1.8–2.1 -1.5–2.1 -0.3–0.4
Total 14.6–25.5 -5.9–6.1 -5.9–6.1 17.3–28.2 -6.6–5.5 54.7–61.7 -5.4–7.0 -3.6–8.4 -2.7–9.5 11.1–21.9 -5.5–6.4
Higher 9.8 -2.4 -1.6 3.7 -3.8 0.6 -3.4 -3.1 2.2 3.3 -1.0
419
Table F.10: Control / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.7–0.7 -1.5–1.1 -1.6–1.0 -0.5–2.0 -1.7–1.0 -0.6–3.9 -1.1–1.6 -1.1–1.7 -1.3–1.6 -1.5–1.2 -1.7–0.9
2 -1.5–1.1 -0.2–0.3 -0.3–0.7 -0.3–0.9 -0.2–0.8 -0.1–3.8 -0.3–0.9 -0.3–0.8 -0.2–0.8 -0.2–1.2 -0.3–0.7
3 -1.6–1.0 -0.3–0.7 -0.1–0.1 -0.3–0.4 -0.3–0.2 -0.4–3.3 -0.3–0.3 -0.3–0.3 -0.2–0.2 -0.5–0.5 -0.2–0.3
4 -0.5–2.0 -0.3–0.9 -0.3–0.4 4.2–5.9 -1.2–0.8 3.2–8.4 -0.8–1.5 -1.4–1.0 -1.1–1.3 0.0–2.6 -1.3–0.7
5 -1.7–1.0 -0.2–0.8 -0.3–0.2 -1.2–0.8 -0.2–0.2 -0.4–3.4 -0.1–0.9 -0.1–0.8 -0.2–0.6 -0.3–0.9 -0.2–0.6
6 -0.6–3.9 -0.1–3.8 -0.4–3.3 3.2–8.4 -0.4–3.4 22.3–25.8 2.6–6.2 2.1–5.6 -1.0–2.8 2.1–6.7 -1.5–0.9
7 -1.1–1.6 -0.3–0.9 -0.3–0.3 -0.8–1.5 -0.1–0.9 2.6–6.2 3.7–5.2 -0.1–2.8 -0.7–2.0 0.3–3.2 -0.3–2.0
8 -1.1–1.7 -0.3–0.8 -0.3–0.3 -1.4–1.0 -0.1–0.8 2.1–5.6 -0.1–2.8 3.1–4.6 -0.1–2.6 0.2–3.1 0.0–2.3
9 -1.3–1.6 -0.2–0.8 -0.2–0.2 -1.1–1.3 -0.2–0.6 -1.0–2.8 -0.7–2.0 -0.1–2.6 0.1–1.9 1.9–5.1 -1.1–1.9
10 -1.5–1.2 -0.2–1.2 -0.5–0.5 0.0–2.6 -0.3–0.9 2.1–6.7 0.3–3.2 0.2–3.1 1.9–5.1 6.5–8.4 -1.2–1.5
11 -1.7–0.9 -0.3–0.7 -0.2–0.3 -1.3–0.7 -0.2–0.6 -1.5–0.9 -0.3–2.0 0.0–2.3 -1.1–1.9 -1.2–1.5 -0.1–0.2
Total 2.2–11.8 -6.4–3.5 -7.4–2.6 20.5–29.2 -6.4–3.4 55.9–61.9 9.8–18.7 6.1–15.4 8.9–18.1 21.2–29.3 -6.8–3.1
Higher 5.4 -5.5 -3.8 11.8 -4.2 9.3 -0.5 -3.1 5.0 4.4 -4.0
Table F.11: South / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.0–1.3 -1.9–0.6 -1.9–0.5 -1.1–1.2 -1.9–0.5 -2.6–1.6 -2.1–0.8 -1.7–1.0 -1.6–0.8 -1.8–0.6 -1.9–0.5
2 -1.9–0.6 -0.2–0.4 -0.6–0.6 -0.7–0.6 -0.5–0.7 -1.4–2.2 -0.6–1.2 -0.7–0.9 -0.6–0.7 -0.5–0.9 -0.6–0.6
3 -1.9–0.5 -0.6–0.6 -0.1–0.1 -0.2–0.2 -0.2–0.3 -1.7–1.8 -0.6–0.7 -0.6–0.4 -0.3–0.2 -0.4–0.3 -0.1–0.3
4 -1.1–1.2 -0.7–0.6 -0.2–0.2 0.9–2.1 -1.5–0.5 -1.8–2.4 -1.2–1.2 -1.2–1.0 -1.4–0.6 -1.0–1.0 -1.6–0.4
5 -1.9–0.5 -0.5–0.7 -0.2–0.3 -1.5–0.5 0.1–0.6 -1.4–2.2 -0.6–1.2 -0.5–0.9 -0.2–0.7 -0.4–0.7 -0.4–0.5
6 -2.6–1.6 -1.4–2.2 -1.7–1.8 -1.8–2.4 -1.4–2.2 22.8–26.3 4.5–9.0 3.7–7.8 -0.7–2.2 0.4–4.2 -1.1–1.1
7 -2.1–0.8 -0.6–1.2 -0.6–0.7 -1.2–1.2 -0.6–1.2 4.5–9.0 10.9–13.0 -0.6–3.1 -1.0–1.7 0.4–3.3 -0.9–1.4
8 -1.7–1.0 -0.7–0.9 -0.6–0.4 -1.2–1.0 -0.5–0.9 3.7–7.8 -0.6–3.1 8.6–10.5 -0.5–2.2 0.0–3.0 -0.7–1.8
9 -1.6–0.8 -0.6–0.7 -0.3–0.2 -1.4–0.6 -0.2–0.7 -0.7–2.2 -1.0–1.7 -0.5–2.2 0.8–2.0 -0.3–1.9 -1.1–1.1
10 -1.8–0.6 -0.5–0.9 -0.4–0.3 -1.0–1.0 -0.4–0.7 0.4–4.2 0.4–3.3 0.0–3.0 -0.3–1.9 4.9–6.4 -0.6–1.4
11 -1.9–0.5 -0.6–0.6 -0.1–0.3 -1.6–0.4 -0.4–0.5 -1.1–1.1 -0.9–1.4 -0.7–1.8 -1.1–1.1 -0.6–1.4 -0.0–0.3
Total 0.9–10.6 -7.0–2.6 -8.7–1.2 0.6–10.7 -6.5–3.3 47.9–54.4 21.6–29.6 14.4–22.9 -2.5–7.1 8.7–17.8 -8.1–1.7
Higher 10.4 -2.6 -3.0 5.4 -2.2 10.4 3.2 -0.6 -1.3 1.1 -3.3
Table F.12: North / Temp. at 0cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -1.0–1.1 -2.2–1.5 -2.2–1.4 -0.4–3.4 -2.3–1.4 0.3–5.4 -1.4–2.1 -1.4–2.1 -2.4–1.4 -1.9–1.6 -1.4–2.3
2 -2.2–1.5 -0.4–0.2 -0.1–1.0 -0.1–1.2 -0.1–1.1 0.2–4.4 -0.1–1.2 -0.1–1.4 -0.2–1.0 -0.2–1.1 -0.1–1.0
3 -2.2–1.4 -0.1–1.0 -0.3–0.3 -0.6–0.8 -0.6–0.6 -0.4–3.9 -0.6–0.6 -0.8–0.6 -0.6–0.6 -0.6–0.7 -0.6–0.6
4 -0.4–3.4 -0.1–1.2 -0.6–0.8 6.2–8.3 -0.1–2.9 2.5–7.9 0.4–3.4 0.3–3.4 -0.2–2.8 0.9–3.8 0.0–3.0
5 -2.3–1.4 -0.1–1.1 -0.6–0.6 -0.1–2.9 0.1–0.5 -0.2–3.9 -0.5–0.5 -0.5–0.8 -0.5–0.4 -0.7–0.5 -0.5–0.3
6 0.3–5.4 0.2–4.4 -0.4–3.9 2.5–7.9 -0.2–3.9 31.4–35.2 -0.6–2.8 -0.2–3.8 -0.6–2.5 -1.2–3.1 -0.6–1.8
7 -1.4–2.1 -0.1–1.2 -0.6–0.6 0.4–3.4 -0.5–0.5 -0.6–2.8 7.5–9.3 -0.9–2.7 -1.0–1.8 -1.8–1.2 -1.0–1.8
8 -1.4–2.1 -0.1–1.4 -0.8–0.6 0.3–3.4 -0.5–0.8 -0.2–3.8 -0.9–2.7 10.8–12.8 -1.4–1.6 -1.8–1.5 -1.3–1.6
9 -2.4–1.4 -0.2–1.0 -0.6–0.6 -0.2–2.8 -0.5–0.4 -0.6–2.5 -1.0–1.8 -1.4–1.6 -0.5–0.8 -0.8–1.7 -1.1–1.4
10 -1.9–1.6 -0.2–1.1 -0.6–0.7 0.9–3.8 -0.7–0.5 -1.2–3.1 -1.8–1.2 -1.8–1.5 -0.8–1.7 8.7–10.5 -1.1–1.6
11 -1.4–2.3 -0.1–1.0 -0.6–0.6 0.0–3.0 -0.5–0.3 -0.6–1.8 -1.0–1.8 -1.3–1.6 -1.1–1.4 -1.1–1.6 -0.3–0.2
Total 5.7–14.0 -5.7–3.1 -3.9–4.6 18.0–25.6 -3.8–4.6 46.2–51.6 7.8–15.7 12.2–19.7 -2.0–6.7 10.1–18.0 -4.0–4.5
Higher 6.1 -7.3 -1.4 -3.0 -3.0 -3.8 -2.0 -1.4 -1.0 0.7 -3.4
420
Table F.13: Control / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -1.0–1.2 -1.7–2.5 -1.8–2.5 -8.2–7.9 -1.8–2.5 -1.3–2.9 -1.7–2.6 -1.8–2.5 -1.9–2.5 -1.5–2.7 -1.6–2.6
2 -1.7–2.5 -0.4–0.3 -0.5–0.8 -9.0–7.1 -0.5–0.8 -0.8–0.6 -0.5–0.8 -0.5–0.8 -0.5–0.8 -0.8–0.5 -0.5–0.8
3 -1.8–2.5 -0.5–0.8 -0.4–0.2 -8.8–7.3 -0.4–0.7 -0.3–0.9 -0.4–0.7 -0.4–0.7 -0.4–0.8 -0.4–0.8 -0.4–0.7
4 -8.2–7.9 -9.0–7.1 -8.8–7.3 73.0–82.8 -1.3–0.5 -2.4–8.1 -1.2–0.8 -0.9–1.3 -3.9–3.5 -1.1–7.1 -1.3–0.8
5 -1.8–2.5 -0.5–0.8 -0.4–0.7 -1.3–0.5 -0.3–0.3 -0.7–0.5 -0.5–0.5 -0.5–0.5 -0.5–0.7 -0.5–0.6 -0.5–0.5
6 -1.3–2.9 -0.8–0.6 -0.3–0.9 -2.4–8.1 -0.7–0.5 3.9–7.1 -3.2–2.6 -3.3–2.5 -2.4–3.5 -3.6–2.2 -3.3–2.5
7 -1.7–2.6 -0.5–0.8 -0.4–0.7 -1.2–0.8 -0.5–0.5 -3.2–2.6 -0.2–0.4 -0.8–0.5 -0.7–0.5 -0.7–0.6 -0.8–0.5
8 -1.8–2.5 -0.5–0.8 -0.4–0.7 -0.9–1.3 -0.5–0.5 -3.3–2.5 -0.8–0.5 -0.4–0.2 -0.4–0.7 -0.4–0.8 -0.5–0.7
9 -1.9–2.5 -0.5–0.8 -0.4–0.8 -3.9–3.5 -0.5–0.7 -2.4–3.5 -0.7–0.5 -0.4–0.7 -1.0–1.1 -1.6–2.5 -2.0–2.2
10 -1.5–2.7 -0.8–0.5 -0.4–0.8 -1.1–7.1 -0.5–0.6 -3.6–2.2 -0.7–0.6 -0.4–0.8 -1.6–2.5 -0.5–1.6 -1.8–2.3
11 -1.6–2.6 -0.5–0.8 -0.4–0.7 -1.3–0.8 -0.5–0.5 -3.3–2.5 -0.8–0.5 -0.5–0.7 -2.0–2.2 -1.8–2.3 -0.4–0.2
Total 3.1–13.6 -5.8–5.4 -6.0–5.3 87.1–90.5 -6.4–4.9 11.1–21.0 -6.3–5.1 -6.1–5.2 0.3–10.9 1.6–12.1 -6.1–5.3
Higher 4.2 -0.2 -1.4 7.7 -1.1 7.9 -0.5 -1.1 3.7 2.4 -0.7
Table F.14: South / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.2–3.7 -2.8–3.8 -2.7–3.9 -24.1–0.4 -2.7–3.8 -3.0–3.4 -2.7–3.9 -2.6–3.9 -4.1–2.6 -2.5–4.0 -2.7–3.8
2 -2.8–3.8 -0.7–0.4 -1.2–0.8 -23.3–0.4 -1.3–0.8 -0.8–1.4 -1.3–0.8 -1.2–0.8 -1.2–0.8 -1.2–0.9 -1.2–0.8
3 -2.7–3.9 -1.2–0.8 -0.5–0.7 -23.5–0.4 -1.3–1.0 -1.2–1.1 -1.3–1.0 -1.3–1.0 -1.3–1.1 -1.3–1.0 -1.3–1.0
4 -24.1–0.4 -23.3–0.4 -23.5–0.4 79.7–94.7 -1.6–2.0 -4.9–7.8 -1.5–2.7 -2.1–2.6 -3.0–4.7 -3.0–6.5 -2.1–0.9
5 -2.7–3.8 -1.3–0.8 -1.3–1.0 -1.6–2.0 -0.6–0.5 -0.9–1.3 -0.9–1.3 -1.0–1.2 -0.9–1.3 -1.0–1.2 -1.0–1.2
6 -3.0–3.4 -0.8–1.4 -1.2–1.1 -4.9–7.8 -0.9–1.3 1.3–4.8 -5.9–1.2 -5.7–1.3 -5.7–1.3 -5.3–1.7 -5.8–1.2
7 -2.7–3.9 -1.3–0.8 -1.3–1.0 -1.5–2.7 -0.9–1.3 -5.9–1.2 -0.6–0.5 -1.1–1.2 -1.0–1.2 -1.1–1.1 -1.1–1.1
8 -2.6–3.9 -1.2–0.8 -1.3–1.0 -2.1–2.6 -1.0–1.2 -5.7–1.3 -1.1–1.2 -0.6–0.6 -1.1–1.2 -1.1–1.1 -1.1–1.1
9 -4.1–2.6 -1.2–0.8 -1.3–1.1 -3.0–4.7 -0.9–1.3 -5.7–1.3 -1.0–1.2 -1.1–1.2 -0.3–1.9 -3.6–0.8 -3.6–0.8
10 -2.5–4.0 -1.2–0.9 -1.3–1.0 -3.0–6.5 -1.0–1.2 -5.3–1.7 -1.1–1.1 -1.1–1.1 -3.6–0.8 -1.0–1.4 -3.2–1.7
11 -2.7–3.8 -1.2–0.8 -1.3–1.0 -2.1–0.9 -1.0–1.2 -5.8–1.2 -1.1–1.1 -1.1–1.1 -3.6–0.8 -3.2–1.7 -0.6–0.4
Total 5.0–19.2 -5.4–9.8 -5.7–9.6 88.5–92.1 -5.7–9.7 3.3–17.6 -5.8–9.6 -5.9–9.5 -3.5–11.6 -0.8–13.8 -5.7–9.7
Higher 18.3 14.4 13.9 33.4 0.8 16.1 3.0 3.3 8.2 8.0 6.8
Table F.15: North / Temp. at 0cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.6–4.1 -2.4–4.0 -2.5–3.9 -17.2–7.9 -2.5–3.9 -3.4–2.9 -2.5–3.9 -2.5–3.9 -4.0–2.4 -3.8–2.6 -2.7–3.8
2 -2.4–4.0 -0.3–0.3 -0.7–0.6 -17.3–7.0 -0.7–0.7 -0.5–0.9 -0.7–0.7 -0.7–0.7 -0.6–0.7 -0.6–0.7 -0.7–0.7
3 -2.5–3.9 -0.7–0.6 -0.3–0.6 -17.5–7.0 -1.2–0.7 -1.1–0.9 -1.2–0.7 -1.2–0.7 -1.1–0.8 -1.2–0.7 -1.2–0.7
4 -17.2–7.9 -17.3–7.0 -17.5–7.0 74.4–88.8 -1.0–0.6 -2.6–9.0 -0.8–1.0 -1.3–0.9 -4.2–3.3 -3.6–5.3 -0.8–1.8
5 -2.5–3.9 -0.7–0.7 -1.2–0.7 -1.0–0.6 -0.0–0.3 -0.5–0.3 -0.7–0.0 -0.7–0.0 -0.7–0.1 -0.7–0.1 -0.7–0.0
6 -3.4–2.9 -0.5–0.9 -1.1–0.9 -2.6–9.0 -0.5–0.3 0.9–4.8 -3.6–3.6 -3.7–3.5 -3.7–3.6 -3.2–4.1 -3.7–3.6
7 -2.5–3.9 -0.7–0.7 -1.2–0.7 -0.8–1.0 -0.7–0.0 -3.6–3.6 -0.0–0.4 -0.8–0.1 -0.8–0.1 -0.8–0.1 -0.8–0.1
8 -2.5–3.9 -0.7–0.7 -1.2–0.7 -1.3–0.9 -0.7–0.0 -3.7–3.5 -0.8–0.1 -0.2–0.3 -0.6–0.4 -0.6–0.4 -0.7–0.4
9 -4.0–2.4 -0.6–0.7 -1.1–0.8 -4.2–3.3 -0.7–0.1 -3.7–3.6 -0.8–0.1 -0.6–0.4 -0.9–1.2 -2.2–2.0 -2.4–1.8
10 -3.8–2.6 -0.6–0.7 -1.2–0.7 -3.6–5.3 -0.7–0.1 -3.2–4.1 -0.8–0.1 -0.6–0.4 -2.2–2.0 -0.8–2.0 -3.2–2.1
11 -2.7–3.8 -0.7–0.7 -1.2–0.7 -0.8–1.8 -0.7–0.0 -3.7–3.6 -0.8–0.1 -0.7–0.4 -2.4–1.8 -3.2–2.1 -0.6–0.2
Total 7.7–21.7 -5.2–10.1 -5.1–10.2 89.7–93.7 -5.7–9.7 7.5–21.6 -5.6–9.7 -5.4–9.9 -2.3–12.7 1.0–15.6 -4.8–10.5
Higher 14.5 6.6 8.5 21.5 3.3 8.6 3.1 3.0 7.6 8.5 4.0
421
F.3 Snow Temperature at 2 cm
Table F.16: Control / Temp. at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 4.0–12.7 0.5–9.5 -3.6–5.6 61.8–65.8 -3.6–5.6 22.2–29.4 -3.6–5.6 -3.6–5.6 -0.2–8.8 4.0–12.5 -3.4–5.7
0.67 1.5–10.2 -0.4–8.5 -5.0–4.1 39.0–44.7 -5.0–4.1 39.6–45.4 -5.0–4.1 -5.0–4.1 0.8–9.6 8.0–16.1 -4.8–4.3
1.00 -0.2–8.5 -1.5–7.2 -5.8–3.2 26.7–33.4 -5.8–3.2 49.1–54.1 -5.6–3.4 -5.7–3.3 1.3–10.0 10.3–18.1 -5.5–3.5
1.33 -1.1–7.4 -2.6–6.0 -3.6–4.7 19.3–26.4 -3.6–4.7 54.2–58.7 -5.3–3.4 -5.6–3.3 1.6–10.0 11.7–19.3 -3.3–5.0
1.67 -1.6–6.8 -3.5–5.0 -4.1–4.0 14.4–21.7 -4.0–4.1 56.5–60.8 -4.2–4.3 -4.5–4.1 1.5–9.7 12.7–20.0 -3.8–4.3
2.00 -2.2–6.1 -4.2–4.2 -4.6–3.4 10.6–18.3 -4.5–3.5 56.9–61.2 -2.1–6.2 -2.5–5.8 0.9–9.2 13.1–20.3 -4.3–3.7
2.33 -2.9–5.6 -5.1–3.6 -5.2–3.0 7.7–15.6 -4.9–3.2 56.1–60.5 1.1–9.1 0.3–8.4 0.3–8.6 13.2–20.5 -4.8–3.2
2.67 -3.5–5.1 -5.5–3.2 -5.6–2.5 5.3–13.4 -5.3–2.9 54.7–59.3 4.9–12.9 3.7–11.7 -0.2–8.3 13.2–20.5 -5.3–2.8
3.00 -4.3–4.5 -6.0–2.9 -6.2–2.3 3.3–11.7 -5.7–2.7 53.3–58.1 8.9–16.7 6.9–14.9 -0.8–7.9 13.0–20.6 -5.9–2.5
3.33 -5.1–4.0 -3.6–4.9 -6.7–2.1 1.6–10.3 -6.2–2.5 52.2–57.2 12.6–20.4 9.8–17.7 -1.3–7.6 12.9–20.7 -6.4–2.4
3.67 -5.9–3.4 -3.9–4.9 -7.1–1.9 0.2–9.2 -6.6–2.4 51.4–56.6 15.7–23.5 12.1–20.0 -1.6–7.5 12.9–20.9 -6.8–2.1
4.00 -3.8–5.2 -4.2–4.9 -7.5–1.8 -1.2–8.2 -7.0–2.3 50.7–56.2 18.2–26.1 14.0–21.9 -2.0–7.4 12.9–21.0 -7.2–2.0
4.33 -4.5–4.8 -4.4–4.9 -7.9–1.7 -2.3–7.4 -7.4–2.1 50.4–56.1 20.1–28.1 15.3–23.4 -2.3–7.4 12.9–21.2 -7.6–1.9
4.67 -5.1–4.3 -4.4–5.0 -8.1–1.6 -3.2–6.7 -7.7–2.0 50.3–56.1 21.6–29.7 16.4–24.5 -2.6–7.4 12.9–21.5 -7.9–1.8
5.00 -5.7–4.0 -4.4–5.1 -8.4–1.5 -4.0–6.1 -7.9–2.0 50.4–56.3 22.6–30.7 17.1–25.3 -2.6–7.5 13.0–21.7 -8.2–1.7
5.33 -6.2–3.6 -4.4–5.3 -8.6–1.4 -4.7–5.6 -8.1–1.8 50.5–56.5 23.1–31.3 17.5–25.8 -2.7–7.5 13.1–22.0 -8.3–1.6
5.67 -6.6–3.3 -4.3–5.4 -8.7–1.4 -5.3–5.1 -8.3–1.7 50.8–56.8 23.2–31.5 17.6–25.9 -2.7–7.6 13.3–22.2 -8.4–1.6
6.00 -6.9–3.0 -4.1–5.6 -8.8–1.3 -5.8–4.7 -8.3–1.7 51.2–57.1 22.8–31.2 17.4–25.7 -2.6–7.7 13.6–22.4 -8.5–1.5
6.33 -7.2–2.7 -4.0–5.7 -8.8–1.2 -6.2–4.2 -8.4–1.6 51.6–57.5 22.1–30.4 16.7–25.1 -2.6–7.7 13.8–22.7 -8.5–1.5
6.67 -7.4–2.4 -3.8–5.8 -8.8–1.1 -6.6–3.8 -8.4–1.6 52.2–58.0 20.8–29.1 15.7–24.1 -2.4–7.8 14.2–22.9 -8.5–1.4
7.00 -7.5–2.2 -6.8–3.4 -8.8–1.1 -6.8–3.4 -8.4–1.4 53.0–58.6 18.9–27.3 14.2–22.7 -2.2–7.8 14.5–23.1 -8.5–1.3
7.33 -7.6–2.1 -6.6–3.4 -8.7–1.0 -4.0–5.5 -8.3–1.3 54.0–59.5 16.4–24.8 12.2–20.7 -2.0–7.9 14.8–23.2 -8.4–1.2
7.67 -7.5–2.0 -6.4–3.5 -8.6–0.9 -4.2–5.2 -8.2–1.2 55.4–60.6 13.3–21.7 9.6–18.1 -1.6–8.0 15.3–23.5 -8.3–1.2
8.00 -7.3–2.0 -6.1–3.5 -8.3–1.0 -4.2–4.9 -8.0–1.3 57.3–62.2 9.5–18.1 6.4–15.0 -1.2–8.1 15.8–23.8 -8.0–1.3
8.33 -6.9–2.2 -5.7–3.6 -8.0–1.1 -4.1–4.9 -7.7–1.4 59.6–64.3 5.5–14.2 3.0–11.6 -0.8–8.3 16.4–24.2 -7.6–1.5
8.67 -6.3–2.5 -5.5–3.7 -7.5–1.4 -3.8–5.0 -7.3–1.6 62.5–66.8 1.7–10.4 -0.4–8.4 -0.3–8.6 17.0–24.6 -7.1–1.8
9.00 -5.7–3.0 -5.4–3.6 -6.9–1.8 -3.5–5.1 -6.8–1.9 65.5–69.4 -1.6–7.2 -3.1–5.7 0.3–9.0 17.5–25.0 -6.5–2.1
9.33 -5.1–3.4 -5.5–3.4 -6.4–2.1 -6.0–3.0 -6.3–2.2 68.2–71.9 -3.9–4.9 -5.0–3.7 0.9–9.5 17.8–25.2 -6.0–2.5
9.67 -4.6–3.7 -5.7–3.0 -6.0–2.3 -5.7–3.1 -6.0–2.4 70.5–73.9 -5.5–3.3 -3.6–4.7 1.5–10.0 18.0–25.2 -5.6–2.8
10.00 -4.4–3.9 -3.5–4.7 -5.8–2.5 -5.6–3.2 -5.7–2.5 72.0–75.3 -3.6–4.6 -4.3–3.9 2.1–10.4 18.0–25.2 -5.3–2.9
Table F.17: Control / Temp. at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.1–0.5 -0.4–0.8 -0.3–0.8 0.0–1.1 -0.3–0.8 -1.3–2.7 -0.0–1.6 0.0–1.6 -0.2–1.0 0.0–1.4 -0.3–0.8
2 -0.4–0.8 0.0–0.7 -0.5–0.7 -0.6–0.6 -0.5–0.7 0.1–4.2 -0.2–1.6 -0.3–1.3 -0.4–0.9 0.1–1.4 -0.5–0.7
3 -0.3–0.8 -0.5–0.7 -0.0–0.0 -0.1–0.1 -0.0–0.1 -1.2–2.6 -0.1–0.9 -0.2–0.7 -0.0–0.1 -0.1–0.5 -0.0–0.1
4 0.0–1.1 -0.6–0.6 -0.1–0.1 2.1–2.8 -0.6–0.5 -1.7–2.8 -0.2–1.7 -0.5–1.2 -0.6–0.6 -0.5–1.2 -0.6–0.5
5 -0.3–0.8 -0.5–0.7 -0.0–0.1 -0.6–0.5 -0.0–0.2 -1.4–2.4 -0.3–0.9 -0.4–0.6 -0.3–0.3 -0.2–0.6 -0.2–0.3
6 -1.3–2.7 0.1–4.2 -1.2–2.6 -1.7–2.8 -1.4–2.4 22.7–26.1 7.9–12.3 6.1–10.0 -1.0–1.4 2.0–5.6 -1.3–0.3
7 -0.0–1.6 -0.2–1.6 -0.1–0.9 -0.2–1.7 -0.3–0.9 7.9–12.3 9.0–11.1 0.3–4.2 -0.6–2.2 2.0–5.2 -0.4–2.2
8 0.0–1.6 -0.3–1.3 -0.2–0.7 -0.5–1.2 -0.4–0.6 6.1–10.0 0.3–4.2 7.8–9.7 -0.4–2.3 1.1–4.1 -0.6–1.9
9 -0.2–1.0 -0.4–0.9 -0.0–0.1 -0.6–0.6 -0.3–0.3 -1.0–1.4 -0.6–2.2 -0.4–2.3 -0.5–0.5 0.9–3.0 -1.1–0.9
10 0.0–1.4 0.1–1.4 -0.1–0.5 -0.5–1.2 -0.2–0.6 2.0–5.6 2.0–5.2 1.1–4.1 0.9–3.0 5.7–7.1 -0.2–1.5
11 -0.3–0.8 -0.5–0.7 -0.0–0.1 -0.6–0.5 -0.2–0.3 -1.3–0.3 -0.4–2.2 -0.6–1.9 -1.1–0.9 -0.2–1.5 -0.0–0.2
Total -5.7–4.0 -4.4–5.1 -8.4–1.5 -4.0–6.1 -7.9–2.0 50.4–56.3 22.6–30.7 17.1–25.3 -2.6–7.5 13.0–21.7 -8.2–1.7
Higher -5.9 -4.9 -5.5 -3.9 -4.6 2.5 -3.9 -4.1 -2.1 -4.0 -5.2
422
Table F.18: South / Temp. at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 6.5–18.1 1.8–13.9 -3.5–9.1 58.3–63.9 -3.5–9.1 22.8–32.4 -3.5–9.1 -3.5–9.1 -2.7–9.8 1.5–13.5 -3.4–9.2
0.67 4.6–15.3 1.2–12.2 -4.6–7.0 36.6–44.1 -4.6–6.9 39.4–46.5 -4.6–7.0 -4.6–7.0 -3.3–8.2 4.3–14.9 -4.4–7.1
1.00 3.4–13.5 -0.2–10.2 -5.3–5.5 25.2–33.3 -5.3–5.5 47.7–53.7 -4.6–6.2 -4.8–6.0 -3.7–7.0 5.9–15.6 -5.1–5.7
1.33 3.3–12.7 -1.4–8.4 -5.8–4.4 18.0–26.2 -5.7–4.5 50.9–56.1 -2.8–7.0 -3.4–6.6 -4.1–6.0 6.7–15.7 -5.6–4.6
1.67 3.2–12.0 -2.3–6.9 -6.2–3.3 12.8–21.0 -5.9–3.5 50.1–55.1 1.1–9.9 0.1–9.1 -4.4–5.0 6.7–15.2 -6.0–3.5
2.00 2.4–10.8 -3.2–5.6 -3.9–4.6 8.9–16.9 -3.5–5.0 47.0–52.1 6.5–14.4 4.8–12.9 -4.9–4.0 6.3–14.3 -3.6–4.8
2.33 1.2–9.3 -3.8–4.6 -4.5–3.7 5.8–13.8 -3.9–4.2 43.6–48.9 12.7–19.8 10.1–17.5 -5.4–3.2 5.7–13.4 -4.2–3.9
2.67 0.0–8.1 -4.3–4.0 -4.9–3.2 3.5–11.5 -4.3–3.8 40.8–46.2 18.3–25.0 14.6–21.5 -3.3–4.8 5.0–12.8 -4.7–3.4
3.00 -1.1–7.2 -4.5–3.8 -5.3–2.9 1.8–10.0 -4.6–3.6 38.9–44.5 23.2–29.7 18.2–24.9 -3.6–4.5 4.8–12.6 -5.1–3.1
3.33 -2.1–6.3 -5.0–3.6 -5.7–2.7 0.5–8.9 -5.0–3.3 37.6–43.6 27.1–33.4 20.7–27.4 -3.9–4.4 4.6–12.6 -5.5–2.9
3.67 -3.2–5.5 -5.3–3.6 -6.0–2.5 -0.7–7.9 -5.3–3.2 37.0–43.2 30.0–36.4 22.6–29.4 -4.1–4.4 4.4–12.7 -5.8–2.7
4.00 -4.1–4.9 -5.4–3.6 -6.3–2.5 -1.8–7.2 -5.6–3.1 36.8–43.2 32.2–38.6 24.0–30.9 -4.2–4.5 4.2–12.8 -6.1–2.7
4.33 -5.0–4.4 -5.6–3.8 -6.6–2.5 -2.6–6.6 -5.9–3.1 36.9–43.5 33.7–40.3 25.0–32.1 -4.4–4.5 4.2–13.0 -6.4–2.7
4.67 -5.7–3.9 -5.5–4.0 -6.8–2.5 -3.4–6.1 -6.1–3.1 37.2–44.0 34.9–41.6 25.7–32.9 -4.5–4.6 4.2–13.3 -6.6–2.6
5.00 -6.4–3.5 -5.5–4.3 -7.0–2.5 -4.1–5.7 -6.3–3.0 37.6–44.4 35.7–42.4 26.3–33.5 -4.6–4.7 4.2–13.5 -6.8–2.7
5.33 -3.9–5.4 -5.4–4.5 -7.1–2.5 -4.6–5.4 -6.5–3.0 38.0–44.9 36.1–42.9 26.6–33.9 -4.6–4.9 4.3–13.7 -6.9–2.7
5.67 -4.4–5.1 -5.2–4.8 -7.2–2.4 -5.1–5.1 -6.6–3.0 38.5–45.5 36.2–43.0 26.6–34.0 -4.6–4.9 4.4–13.9 -7.0–2.7
6.00 -4.8–4.7 -5.0–5.1 -7.3–2.4 -5.5–4.7 -6.7–2.9 39.1–46.0 35.9–42.8 26.4–33.9 -4.6–4.9 4.6–14.1 -7.0–2.7
6.33 -5.2–4.3 -4.7–5.2 -7.4–2.3 -5.8–4.4 -6.7–2.9 39.7–46.6 35.2–42.2 26.0–33.4 -4.6–4.9 4.8–14.3 -7.0–2.6
6.67 -5.5–3.9 -4.4–5.4 -7.4–2.3 -6.1–4.0 -6.7–2.9 40.4–47.2 34.2–41.1 25.2–32.7 -4.6–4.9 5.0–14.4 -7.0–2.6
7.00 -5.8–3.5 -4.2–5.6 -7.3–2.1 -6.3–3.7 -6.7–2.7 41.3–47.8 32.6–39.5 24.0–31.5 -4.5–4.8 5.3–14.5 -7.0–2.4
7.33 -6.0–3.2 -3.8–5.7 -7.3–2.0 -6.5–3.4 -6.7–2.6 42.4–48.7 30.3–37.3 22.3–29.8 -4.4–4.7 5.6–14.6 -7.0–2.3
7.67 -6.2–2.9 -3.6–5.8 -7.3–1.9 -3.6–5.4 -6.7–2.4 43.8–49.8 27.1–34.2 19.9–27.4 -4.4–4.5 5.9–14.6 -6.9–2.2
8.00 -6.2–2.7 -3.3–5.8 -7.2–1.7 -3.8–5.1 -6.6–2.3 45.8–51.6 22.9–30.1 16.5–24.2 -4.4–4.4 6.3–14.8 -6.9–2.0
8.33 -6.2–2.6 -3.1–5.9 -7.2–1.6 -3.8–4.9 -6.6–2.2 48.9–54.2 17.4–24.9 12.1–20.1 -4.4–4.3 6.9–15.2 -6.8–2.0
8.67 -5.9–2.9 -2.8–6.2 -7.0–1.8 -3.7–5.0 -6.5–2.3 53.2–58.1 11.3–19.2 7.1–15.4 -4.2–4.5 7.6–15.9 -6.6–2.2
9.00 -5.3–3.4 -2.6–6.4 -6.7–2.3 -3.5–5.2 -6.1–2.7 58.6–63.0 5.2–13.6 2.2–10.8 -3.8–4.9 8.6–16.9 -6.2–2.6
9.33 -4.7–4.2 -2.9–6.3 -6.2–2.7 -6.2–3.3 -5.8–3.2 64.0–67.9 0.2–9.1 -1.8–7.3 -3.4–5.3 9.8–18.0 -5.8–3.2
9.67 -4.1–4.7 -3.6–5.7 -5.8–3.1 -5.8–3.6 -5.5–3.5 68.7–72.2 -3.2–6.1 -4.6–4.9 -6.1–3.4 10.7–18.9 -5.4–3.6
10.00 -3.9–5.0 -4.5–5.0 -5.6–3.5 -5.7–3.9 -5.3–3.7 71.8–75.0 -5.0–4.4 -5.8–3.7 -5.8–3.8 11.3–19.5 -5.1–3.9
Table F.19: South / Temp. at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.4–1.0 -0.3–0.9 -0.3–0.9 0.1–1.2 -0.2–1.0 -0.8–2.1 0.2–2.8 0.2–2.3 -0.2–1.0 -0.0–1.3 -0.3–0.9
2 -0.3–0.9 0.2–0.9 -0.6–0.7 -0.7–0.6 -0.6–0.7 0.1–3.0 -0.1–2.6 -0.2–1.9 -0.6–0.7 -0.2–1.1 -0.6–0.6
3 -0.3–0.9 -0.6–0.7 -0.0–0.0 -0.1–0.1 -0.0–0.1 -0.9–1.5 -0.3–1.8 -0.2–1.3 -0.0–0.1 -0.0–0.3 -0.0–0.1
4 0.1–1.2 -0.7–0.6 -0.1–0.1 1.7–2.3 -0.4–0.6 -1.8–1.3 0.1–2.9 -0.1–2.0 -0.4–0.6 -0.4–0.9 -0.5–0.5
5 -0.2–1.0 -0.6–0.7 -0.0–0.1 -0.4–0.6 -0.0–0.3 -1.1–1.4 -0.3–1.9 -0.4–1.3 -0.4–0.3 -0.2–0.5 -0.4–0.2
6 -0.8–2.1 0.1–3.0 -0.9–1.5 -1.8–1.3 -1.1–1.4 16.3–19.0 9.0–13.9 6.3–10.5 -0.7–0.9 0.0–2.6 -0.9–0.6
7 0.2–2.8 -0.1–2.6 -0.3–1.8 0.1–2.9 -0.3–1.9 9.0–13.9 16.8–19.6 2.3–7.3 -0.5–2.2 2.7–5.8 -0.2–2.3
8 0.2–2.3 -0.2–1.9 -0.2–1.3 -0.1–2.0 -0.4–1.3 6.3–10.5 2.3–7.3 13.1–15.5 -0.2–2.5 1.2–4.3 -0.3–2.2
9 -0.2–1.0 -0.6–0.7 -0.0–0.1 -0.4–0.6 -0.4–0.3 -0.7–0.9 -0.5–2.2 -0.2–2.5 -0.2–0.3 -0.1–1.0 -0.4–0.7
10 -0.0–1.3 -0.2–1.1 -0.0–0.3 -0.4–0.9 -0.2–0.5 0.0–2.6 2.7–5.8 1.2–4.3 -0.1–1.0 3.2–4.1 -0.1–1.1
11 -0.3–0.9 -0.6–0.6 -0.0–0.1 -0.5–0.5 -0.4–0.2 -0.9–0.6 -0.2–2.3 -0.3–2.2 -0.4–0.7 -0.1–1.1 0.0–0.2
Total -6.4–3.5 -5.5–4.3 -7.0–2.5 -4.1–5.7 -6.3–3.0 37.6–44.4 35.7–42.4 26.3–33.5 -4.6–4.7 4.2–13.5 -6.8–2.7
Higher -8.8 -5.6 -4.4 -4.5 -3.7 -0.1 -7.4 -6.4 -3.2 -5.8 -5.0
423
Table F.20: North / Temp. at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 2.2–15.6 -0.8–13.0 -2.6–11.3 80.3–83.6 -2.6–11.3 10.7–22.9 -2.5–11.3 -2.6–11.3 -2.1–11.7 0.3–13.9 -2.5–11.3
0.67 1.1–14.9 -1.6–12.5 -4.2–10.2 61.0–67.0 -4.1–10.3 24.0–34.6 -4.1–10.3 -4.1–10.2 -3.1–11.1 2.6–16.1 -4.0–10.3
1.00 0.3–13.9 -2.5–11.5 -4.8–9.3 47.3–55.0 -4.8–9.3 33.8–43.0 -4.8–9.3 -4.8–9.4 -3.5–10.4 4.7–17.6 -4.7–9.4
1.33 -0.5–12.9 -3.3–10.4 -5.5–8.4 37.3–46.1 -5.5–8.4 40.4–48.6 -5.1–8.7 -5.0–8.7 -4.2–9.6 6.2–18.6 -5.3–8.6
1.67 -0.6–12.2 -3.6–9.5 -5.7–7.6 29.9–39.1 -5.7–7.6 44.6–51.9 -4.6–8.6 -4.6–8.6 -4.4–8.8 7.3–19.1 -5.6–7.8
2.00 -0.5–11.7 -3.9–8.6 -5.9–6.7 23.9–33.4 -5.9–6.8 46.5–53.3 -3.4–8.9 -3.2–9.2 -4.6–8.0 7.9–19.0 -5.7–6.9
2.33 -0.2–11.2 -3.6–8.0 -6.0–5.9 19.2–28.6 -5.9–6.0 46.6–52.9 -1.5–10.0 -0.7–10.6 -4.7–7.1 8.2–18.6 -5.8–6.1
2.67 -0.0–10.6 -3.2–7.7 -6.0–5.1 15.3–24.5 -5.7–5.4 45.2–51.3 1.1–11.5 2.6–12.8 -4.8–6.3 8.2–17.9 -5.8–5.3
3.00 -0.1–9.9 -2.6–7.5 -5.9–4.4 12.2–21.1 -5.6–4.7 43.0–49.1 3.9–13.4 6.2–15.5 -4.8–5.5 8.0–17.1 -5.8–4.6
3.33 -0.3–9.1 -2.2–7.3 -6.0–3.8 9.6–18.2 -5.6–4.1 40.7–46.7 6.5–15.2 9.8–18.2 -5.1–4.8 7.7–16.3 -5.8–4.0
3.67 -0.8–8.3 -1.8–7.2 -6.2–3.2 7.4–15.8 -5.8–3.5 38.6–44.4 8.7–16.9 13.0–20.8 -5.2–4.1 7.2–15.5 -6.0–3.4
4.00 -1.3–7.4 -1.6–7.0 -3.6–4.8 5.6–13.8 -3.2–5.2 36.6–42.5 10.5–18.2 15.9–23.2 -5.5–3.5 6.8–14.8 -3.4–5.0
4.33 -1.9–6.6 -1.5–6.9 -3.9–4.3 4.1–12.2 -3.5–4.7 35.1–40.9 11.9–19.3 18.2–25.2 -5.7–3.0 6.4–14.3 -3.7–4.5
4.67 -2.5–5.9 -1.4–6.9 -4.2–3.9 2.9–11.0 -3.8–4.3 34.0–39.8 13.0–20.1 20.1–26.8 -3.3–4.8 6.2–13.9 -4.0–4.1
5.00 -3.0–5.2 -1.1–7.0 -4.3–3.7 1.9–9.9 -3.9–4.0 33.3–39.1 13.6–20.7 21.5–28.0 -3.5–4.5 6.0–13.6 -4.2–3.8
5.33 -3.6–4.6 -0.9–7.2 -4.5–3.4 1.1–9.2 -4.1–3.8 33.0–38.8 14.0–21.0 22.4–28.8 -3.6–4.3 5.9–13.5 -4.3–3.6
5.67 -4.1–4.1 -0.6–7.4 -4.6–3.3 0.5–8.5 -4.2–3.7 33.1–38.9 14.1–21.1 22.6–29.0 -3.7–4.2 5.9–13.5 -4.4–3.5
6.00 -4.5–3.7 -0.3–7.7 -4.6–3.3 -0.0–8.1 -4.2–3.6 33.5–39.3 13.9–20.9 22.4–28.8 -3.7–4.1 6.1–13.6 -4.4–3.4
6.33 -5.0–3.4 -0.0–8.0 -4.7–3.3 -0.4–7.7 -4.3–3.6 34.3–40.1 13.5–20.5 21.6–28.1 -3.7–4.2 6.3–13.8 -4.5–3.5
6.67 -5.3–3.1 0.3–8.4 -4.7–3.3 -0.8–7.5 -4.3–3.7 35.6–41.3 12.7–19.9 20.3–27.0 -3.7–4.3 6.6–14.2 -4.5–3.6
7.00 -5.7–2.9 0.7–8.9 -4.7–3.5 -1.0–7.4 -4.3–3.9 37.3–42.9 11.6–18.9 18.4–25.3 -3.7–4.4 7.0–14.7 -4.5–3.7
7.33 -3.2–5.0 1.0–9.4 -4.6–3.7 -1.2–7.4 -4.2–4.0 39.5–45.0 10.0–17.6 16.1–23.3 -3.6–4.7 7.5–15.3 -4.4–3.9
7.67 -3.4–5.1 1.3–9.8 -4.6–3.9 -1.3–7.5 -4.2–4.3 42.2–47.7 7.9–16.0 13.2–20.9 -3.5–5.1 8.1–16.1 -4.4–4.2
8.00 -3.4–5.4 1.3–10.2 -4.5–4.4 -1.4–7.8 -4.1–4.8 45.5–50.9 5.6–14.2 10.1–18.2 -3.4–5.4 8.8–17.0 -4.3–4.6
8.33 -6.4–3.5 1.1–10.4 -4.4–4.9 -1.5–8.1 -4.1–5.2 49.1–54.4 3.2–12.2 6.6–15.5 -6.3–3.6 9.5–18.0 -4.2–5.1
8.67 -6.4–3.9 0.6–10.3 -4.4–5.3 -1.5–8.5 -4.0–5.6 53.0–58.0 0.8–10.4 3.5–13.0 -6.3–4.1 10.3–19.1 -4.1–5.6
9.00 -6.3–4.4 -0.2–10.0 -4.3–5.7 -1.5–8.8 -4.0–6.1 56.7–61.6 -1.5–8.8 0.7–10.8 -6.3–4.6 11.1–20.2 -4.0–6.1
9.33 -6.3–4.9 -1.3–9.5 -4.3–6.2 -1.6–9.2 -7.5–3.8 60.1–64.7 -3.4–7.5 -1.6–9.1 -6.1–5.0 11.7–21.1 -7.4–3.9
9.67 -6.2–5.3 -2.3–8.9 -4.2–6.6 -1.7–9.4 -7.5–4.1 62.9–67.4 -4.9–6.6 -3.4–7.9 -6.1–5.4 12.2–21.9 -7.4–4.2
10.00 -6.2–5.6 -3.3–8.2 -7.8–4.1 -1.8–9.5 -7.6–4.2 65.0–69.3 -5.7–6.0 -4.6–7.1 -5.9–5.8 12.7–22.4 -7.5–4.4
Table F.21: North / Temp. at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.5–2.4 -0.8–1.0 -0.9–0.8 -0.9–0.7 -1.0–0.8 -1.2–2.7 -1.2–0.8 -1.4–1.2 -0.9–0.8 -0.9–1.0 -0.9–0.8
2 -0.8–1.0 1.8–2.8 -0.9–1.1 -1.2–0.8 -0.7–1.3 -0.0–3.8 -0.9–1.3 -0.5–2.3 -0.8–1.1 -0.5–1.5 -0.8–1.1
3 -0.9–0.8 -0.9–1.1 -0.0–0.1 -0.4–0.0 -0.1–0.1 -0.7–2.6 -0.4–0.5 -0.5–1.2 -0.1–0.1 -0.2–0.4 -0.1–0.1
4 -0.9–0.7 -1.2–0.8 -0.4–0.0 6.2–7.2 -0.9–0.6 -1.6–2.8 -1.1–0.9 -1.2–1.5 -0.8–0.7 -0.9–1.0 -0.8–0.6
5 -1.0–0.8 -0.7–1.3 -0.1–0.1 -0.9–0.6 -0.2–0.2 -0.5–2.8 -0.4–0.8 -0.5–1.4 -0.4–0.4 -0.3–0.7 -0.4–0.4
6 -1.2–2.7 -0.0–3.8 -0.7–2.6 -1.6–2.8 -0.5–2.8 29.2–32.2 -0.1–3.0 0.2–4.6 -1.1–0.6 -0.6–2.5 -0.8–0.8
7 -1.2–0.8 -0.9–1.3 -0.4–0.5 -1.1–0.9 -0.4–0.8 -0.1–3.0 12.8–14.8 -1.6–2.9 -1.7–1.1 -1.9–1.1 -1.8–1.0
8 -1.4–1.2 -0.5–2.3 -0.5–1.2 -1.2–1.5 -0.5–1.4 0.2–4.6 -1.6–2.9 19.2–21.6 -1.6–1.2 -1.5–1.7 -1.6–1.2
9 -0.9–0.8 -0.8–1.1 -0.1–0.1 -0.8–0.7 -0.4–0.4 -1.1–0.6 -1.7–1.1 -1.6–1.2 -0.2–0.3 -0.2–1.0 -0.6–0.4
10 -0.9–1.0 -0.5–1.5 -0.2–0.4 -0.9–1.0 -0.3–0.7 -0.6–2.5 -1.9–1.1 -1.5–1.7 -0.2–1.0 8.6–9.8 -0.9–0.6
11 -0.9–0.8 -0.8–1.1 -0.1–0.1 -0.8–0.6 -0.4–0.4 -0.8–0.8 -1.8–1.0 -1.6–1.2 -0.6–0.4 -0.9–0.6 0.1–0.2
Total -3.0–5.2 -1.1–7.0 -4.3–3.7 1.9–9.9 -3.9–4.0 33.3–39.1 13.6–20.7 21.5–28.0 -3.5–4.5 6.0–13.6 -4.2–3.8
Higher -1.1 -3.5 -1.8 -0.6 -2.2 -4.4 2.2 -0.2 0.8 -1.2 0.4
424
Table F.22: Control / Temp. at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.2–0.9 -0.5–0.8 -0.5–0.7 -0.7–0.7 -0.6–0.7 -1.8–5.0 -0.6–0.7 -0.7–0.6 -0.5–0.9 -0.6–1.0 -0.5–0.7
2 -0.5–0.8 -0.2–0.4 -0.5–0.8 -0.8–0.6 -0.5–0.8 -0.4–6.4 -0.7–0.7 -0.8–0.6 -0.8–0.5 -0.3–1.3 -0.5–0.8
3 -0.5–0.7 -0.5–0.8 -0.0–0.0 -0.7–0.1 -0.1–0.1 -1.5–5.0 -0.1–0.3 -0.1–0.2 -0.1–0.1 -0.3–0.7 -0.1–0.1
4 -0.7–0.7 -0.8–0.6 -0.7–0.1 9.7–11.1 -0.9–0.7 -2.4–6.1 -1.0–1.1 -1.1–0.9 -0.8–1.1 -0.8–1.9 -0.9–0.7
5 -0.6–0.7 -0.5–0.8 -0.1–0.1 -0.9–0.7 -0.0–0.2 -1.6–5.0 -0.2–0.3 -0.3–0.2 -0.2–0.3 -0.3–0.8 -0.2–0.2
6 -1.8–5.0 -0.4–6.4 -1.5–5.0 -2.4–6.1 -1.6–5.0 38.5–43.1 2.2–5.9 1.4–4.8 -0.5–2.5 1.8–6.6 -1.4–0.1
7 -0.6–0.7 -0.7–0.7 -0.1–0.3 -1.0–1.1 -0.2–0.3 2.2–5.9 4.1–5.2 -0.3–1.8 -0.6–1.2 0.3–2.6 -0.4–1.2
8 -0.7–0.6 -0.8–0.6 -0.1–0.2 -1.1–0.9 -0.3–0.2 1.4–4.8 -0.3–1.8 3.6–4.7 -0.5–1.2 -0.1–2.1 -0.6–1.0
9 -0.5–0.9 -0.8–0.5 -0.1–0.1 -0.8–1.1 -0.2–0.3 -0.5–2.5 -0.6–1.2 -0.5–1.2 -0.8–0.5 1.7–4.3 -1.4–1.2
10 -0.6–1.0 -0.3–1.3 -0.3–0.7 -0.8–1.9 -0.3–0.8 1.8–6.6 0.3–2.6 -0.1–2.1 1.7–4.3 9.5–11.2 -0.4–1.6
11 -0.5–0.7 -0.5–0.8 -0.1–0.1 -0.9–0.7 -0.2–0.2 -1.4–0.1 -0.4–1.2 -0.6–1.0 -1.4–1.2 -0.4–1.6 0.0–0.3
Total -5.1–4.6 -4.9–4.8 -7.3–2.6 4.5–13.9 -7.2–2.7 54.5–59.7 2.7–12.2 0.2–9.9 -2.3–7.7 13.0–21.7 -7.0–2.8
Higher -3.0 -3.9 -4.5 -3.1 -4.5 -5.3 -4.3 -4.4 -1.9 -4.9 -2.8
Table F.23: South / Temp. at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.1–1.8 -0.9–0.6 -0.9–0.5 -0.8–0.7 -0.9–0.5 -2.0–4.5 -0.5–1.4 -0.6–1.1 -0.9–0.6 -0.7–1.0 -0.9–0.6
2 -0.9–0.6 -0.2–0.6 -0.9–0.6 -0.9–0.8 -0.8–0.7 -0.4–5.8 -0.6–1.4 -0.7–1.1 -0.9–0.6 -0.3–1.3 -0.9–0.6
3 -0.9–0.5 -0.9–0.6 -0.0–0.0 -0.7–0.1 -0.1–0.1 -1.7–4.3 -0.2–0.9 -0.1–0.7 -0.1–0.1 -0.1–0.6 -0.1–0.1
4 -0.8–0.7 -0.9–0.8 -0.7–0.1 9.5–11.0 -1.0–0.5 -3.4–4.5 -0.9–1.9 -1.0–1.4 -0.9–0.7 -0.8–1.5 -0.9–0.6
5 -0.9–0.5 -0.8–0.7 -0.1–0.1 -1.0–0.5 -0.0–0.3 -1.7–4.3 -0.3–1.0 -0.3–0.7 -0.4–0.2 -0.2–0.7 -0.4–0.2
6 -2.0–4.5 -0.4–5.8 -1.7–4.3 -3.4–4.5 -1.7–4.3 34.8–39.4 3.3–8.3 2.5–6.8 -1.0–0.9 0.3–4.3 -1.1–0.5
7 -0.5–1.4 -0.6–1.4 -0.2–0.9 -0.9–1.9 -0.3–1.0 3.3–8.3 9.7–11.5 0.2–3.5 -0.3–1.7 1.0–3.5 -0.2–1.7
8 -0.6–1.1 -0.7–1.1 -0.1–0.7 -1.0–1.4 -0.3–0.7 2.5–6.8 0.2–3.5 7.8–9.4 -0.5–1.6 0.1–2.5 -0.5–1.4
9 -0.9–0.6 -0.9–0.6 -0.1–0.1 -0.9–0.7 -0.4–0.2 -1.0–0.9 -0.3–1.7 -0.5–1.6 -0.4–0.3 0.3–1.7 -0.5–0.9
10 -0.7–1.0 -0.3–1.3 -0.1–0.6 -0.8–1.5 -0.2–0.7 0.3–4.3 1.0–3.5 0.1–2.5 0.3–1.7 7.2–8.5 -0.3–1.1
11 -0.9–0.6 -0.9–0.6 -0.1–0.1 -0.9–0.6 -0.4–0.2 -1.1–0.5 -0.2–1.7 -0.5–1.4 -0.5–0.9 -0.3–1.1 0.1–0.3
Total -7.1–4.1 -4.2–6.3 -7.0–3.8 3.5–13.8 -6.7–4.1 45.8–52.4 11.1–20.8 6.9–16.8 -4.9–5.8 4.8–14.9 -6.7–4.1
Higher -4.4 -2.3 -3.1 -2.3 -2.9 -7.6 -8.1 -6.8 -1.5 -6.7 -2.4
Table F.24: North / Temp. at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.7–1.8 -1.4–0.7 -1.5–0.6 -2.0–0.6 -1.5–0.6 -2.5–4.9 -1.3–0.8 -1.4–0.8 -1.5–0.6 -1.4–0.9 -1.5–0.6
2 -1.4–0.7 0.8–1.9 -1.0–1.1 -2.0–0.8 -0.9–1.2 -1.3–6.0 -1.0–1.1 -0.8–1.6 -1.0–1.1 -0.8–1.4 -1.0–1.1
3 -1.5–0.6 -1.0–1.1 -0.1–0.1 -1.9–-0.1 -0.1–0.1 -2.0–5.0 -0.2–0.3 -0.4–0.4 -0.1–0.1 -0.6–0.5 -0.1–0.1
4 -2.0–0.6 -2.0–0.8 -1.9–-0.1 16.0–18.4 -1.2–1.0 -3.6–6.3 -1.1–1.6 -2.0–1.2 -1.1–1.2 -1.8–1.7 -1.1–1.1
5 -1.5–0.6 -0.9–1.2 -0.1–0.1 -1.2–1.0 -0.2–0.2 -1.9–5.1 -0.5–0.5 -0.5–0.6 -0.4–0.4 -0.6–0.7 -0.4–0.4
6 -2.5–4.9 -1.3–6.0 -2.0–5.0 -3.6–6.3 -1.9–5.1 40.3–45.5 -1.8–1.9 -1.5–2.9 -0.8–1.3 -2.4–2.8 -0.9–0.8
7 -1.3–0.8 -1.0–1.1 -0.2–0.3 -1.1–1.6 -0.5–0.5 -1.8–1.9 6.2–7.8 -1.8–1.3 -1.6–0.8 -1.5–1.4 -1.6–0.7
8 -1.4–0.8 -0.8–1.6 -0.4–0.4 -2.0–1.2 -0.5–0.6 -1.5–2.9 -1.8–1.3 8.7–10.6 -1.4–1.2 -1.6–1.5 -1.4–1.1
9 -1.5–0.6 -1.0–1.1 -0.1–0.1 -1.1–1.2 -0.4–0.4 -0.8–1.3 -1.6–0.8 -1.4–1.2 -0.3–0.5 -0.6–1.2 -0.9–0.6
10 -1.4–0.9 -0.8–1.4 -0.6–0.5 -1.8–1.7 -0.6–0.7 -2.4–2.8 -1.5–1.4 -1.6–1.5 -0.6–1.2 11.1–12.9 -1.0–0.9
11 -1.5–0.6 -1.0–1.1 -0.1–0.1 -1.1–1.1 -0.4–0.4 -0.9–0.8 -1.6–0.7 -1.4–1.1 -0.9–0.6 -1.0–0.9 0.1–0.3
Total -4.6–6.8 -3.3–8.0 -7.3–4.4 12.4–22.0 -7.0–4.6 43.0–49.6 2.4–13.0 5.7–16.1 -6.2–5.3 7.5–17.7 -7.0–4.6
Higher 2.2 -1.5 -1.5 1.3 -2.5 -5.7 1.7 1.3 -0.1 0.2 -0.2
425
Table F.25: Control / Temp. at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.1–0.5 -0.3–0.6 -0.3–0.5 -0.5–0.6 -0.3–0.6 -0.6–2.8 0.0–1.4 -0.0–1.3 -0.3–0.6 0.0–1.1 -0.3–0.5
2 -0.3–0.6 0.1–0.9 -0.6–0.7 -0.9–0.6 -0.6–0.7 0.8–4.4 -0.3–1.6 -0.6–1.3 -0.5–0.9 -0.2–1.2 -0.6–0.7
3 -0.3–0.5 -0.6–0.7 -0.0–0.0 -0.5–0.1 -0.1–0.0 -0.4–2.8 -0.0–0.9 -0.2–0.6 -0.0–0.1 -0.1–0.4 -0.0–0.0
4 -0.5–0.6 -0.9–0.6 -0.5–0.1 4.8–6.2 -0.5–0.4 2.9–7.5 1.3–3.5 0.5–2.5 -0.7–0.5 0.8–2.5 -0.5–0.4
5 -0.3–0.6 -0.6–0.7 -0.1–0.0 -0.5–0.4 -0.1–0.2 -0.5–2.7 -0.2–1.0 -0.4–0.6 -0.4–0.2 -0.2–0.6 -0.3–0.2
6 -0.6–2.8 0.8–4.4 -0.4–2.8 2.9–7.5 -0.5–2.7 18.0–21.1 6.5–10.6 4.8–8.5 -1.4–1.0 1.2–4.5 -1.2–0.5
7 0.0–1.4 -0.3–1.6 -0.0–0.9 1.3–3.5 -0.2–1.0 6.5–10.6 7.6–9.7 0.0–4.1 -0.8–2.1 1.4–4.6 -0.5–2.1
8 -0.0–1.3 -0.6–1.3 -0.2–0.6 0.5–2.5 -0.4–0.6 4.8–8.5 0.0–4.1 6.9–8.9 -0.8–2.1 0.3–3.5 -0.8–1.8
9 -0.3–0.6 -0.5–0.9 -0.0–0.1 -0.7–0.5 -0.4–0.2 -1.4–1.0 -0.8–2.1 -0.8–2.1 -0.2–0.8 0.6–2.7 -0.8–1.2
10 0.0–1.1 -0.2–1.2 -0.1–0.4 0.8–2.5 -0.2–0.6 1.2–4.5 1.4–4.6 0.3–3.5 0.6–2.7 4.4–5.8 -0.4–1.3
11 -0.3–0.5 -0.6–0.7 -0.0–0.0 -0.5–0.4 -0.3–0.2 -1.2–0.5 -0.5–2.1 -0.8–1.8 -0.8–1.2 -0.4–1.3 -0.1–0.1
Total -5.6–4.3 -5.9–4.3 -7.6–2.5 13.6–22.9 -7.1–2.9 47.7–54.2 23.6–32.0 17.7–26.3 -1.4–8.6 13.2–22.0 -7.3–2.7
Higher -4.7 -5.6 -4.6 2.6 -4.0 2.7 -0.6 -0.5 0.1 -0.5 -4.0
Table F.26: South / Temp. at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.2–0.7 -0.1–0.8 -0.1–0.8 0.1–1.0 -0.1–0.9 -0.5–2.2 0.4–2.8 0.2–2.2 -0.1–0.9 0.1–1.2 -0.1–0.8
2 -0.1–0.8 0.4–1.1 -0.6–0.7 -0.5–0.8 -0.5–0.8 0.4–3.3 0.1–2.8 -0.1–2.1 -0.5–0.8 -0.2–1.2 -0.6–0.7
3 -0.1–0.8 -0.6–0.7 -0.0–0.0 -0.1–0.1 -0.0–0.1 -0.7–1.6 -0.2–1.9 -0.3–1.2 -0.0–0.1 0.0–0.3 -0.0–0.1
4 0.1–1.0 -0.5–0.8 -0.1–0.1 1.8–2.4 -0.5–0.4 -1.2–1.7 0.3–2.9 -0.2–1.9 -0.5–0.4 -0.4–0.8 -0.5–0.3
5 -0.1–0.9 -0.5–0.8 -0.0–0.1 -0.5–0.4 0.0–0.3 -0.9–1.5 -0.2–2.0 -0.4–1.3 -0.4–0.2 -0.2–0.5 -0.2–0.3
6 -0.5–2.2 0.4–3.3 -0.7–1.6 -1.2–1.7 -0.9–1.5 15.6–18.3 9.1–14.1 6.0–10.0 -0.8–0.9 -0.2–2.3 -0.9–0.5
7 0.4–2.8 0.1–2.8 -0.2–1.9 0.3–2.9 -0.2–2.0 9.1–14.1 16.0–18.8 2.3–7.4 -0.5–2.2 2.7–5.9 -0.2–2.3
8 0.2–2.2 -0.1–2.1 -0.3–1.2 -0.2–1.9 -0.4–1.3 6.0–10.0 2.3–7.4 12.6–15.0 -0.3–2.4 0.9–4.1 -0.4–2.2
9 -0.1–0.9 -0.5–0.8 -0.0–0.1 -0.5–0.4 -0.4–0.2 -0.8–0.9 -0.5–2.2 -0.3–2.4 -0.2–0.4 -0.1–0.9 -0.4–0.6
10 0.1–1.2 -0.2–1.2 0.0–0.3 -0.4–0.8 -0.2–0.5 -0.2–2.3 2.7–5.9 0.9–4.1 -0.1–0.9 3.1–4.1 -0.3–1.0
11 -0.1–0.8 -0.6–0.7 -0.0–0.1 -0.5–0.3 -0.2–0.3 -0.9–0.5 -0.2–2.3 -0.4–2.2 -0.4–0.6 -0.3–1.0 -0.0–0.1
Total -5.0–4.3 -4.6–4.9 -7.5–2.1 -2.8–7.1 -6.8–2.6 38.0–44.7 36.5–43.3 26.9–34.2 -4.8–4.5 4.9–13.8 -7.2–2.3
Higher -7.5 -6.3 -5.2 -3.4 -4.4 0.3 -6.4 -4.3 -3.2 -4.4 -5.1
Table F.27: North / Temp. at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.1–1.8 -0.8–0.7 -0.8–0.6 -0.9–0.6 -0.8–0.6 -0.9–2.4 -0.9–0.8 -1.4–0.8 -0.8–0.6 -0.8–0.7 -0.8–0.6
2 -0.8–0.7 2.6–3.7 -1.2–0.9 -1.2–0.9 -1.0–1.1 0.0–3.6 -1.0–1.3 -0.7–2.2 -1.2–0.9 -0.9–1.1 -1.1–0.9
3 -0.8–0.6 -1.2–0.9 -0.0–0.1 -0.8–-0.1 -0.1–0.1 -0.4–2.4 -0.3–0.5 -0.6–1.0 -0.1–0.1 -0.3–0.3 -0.1–0.1
4 -0.9–0.6 -1.2–0.9 -0.8–-0.1 9.0–10.4 -0.6–0.6 0.5–4.5 0.1–2.1 -0.0–2.6 -0.6–0.7 -0.0–1.7 -0.5–0.7
5 -0.8–0.6 -1.0–1.1 -0.1–0.1 -0.6–0.6 -0.1–0.3 -0.3–2.6 -0.4–0.8 -0.7–1.2 -0.5–0.3 -0.4–0.5 -0.5–0.3
6 -0.9–2.4 0.0–3.6 -0.4–2.4 0.5–4.5 -0.3–2.6 25.7–28.5 0.3–3.4 -0.0–4.0 -0.9–0.9 -0.7–2.2 -0.9–0.8
7 -0.9–0.8 -1.0–1.3 -0.3–0.5 0.1–2.1 -0.4–0.8 0.3–3.4 12.0–14.1 -1.2–3.2 -1.4–1.5 -1.8–1.2 -1.4–1.6
8 -1.4–0.8 -0.7–2.2 -0.6–1.0 -0.0–2.6 -0.7–1.2 -0.0–4.0 -1.2–3.2 18.3–20.7 -1.8–1.3 -2.0–1.3 -1.7–1.3
9 -0.8–0.6 -1.2–0.9 -0.1–0.1 -0.6–0.7 -0.5–0.3 -0.9–0.9 -1.4–1.5 -1.8–1.3 -0.3–0.2 -0.3–0.8 -0.4–0.7
10 -0.8–0.7 -0.9–1.1 -0.3–0.3 -0.0–1.7 -0.4–0.5 -0.7–2.2 -1.8–1.2 -2.0–1.3 -0.3–0.8 7.5–8.7 -1.0–0.8
11 -0.8–0.6 -1.1–0.9 -0.1–0.1 -0.5–0.7 -0.5–0.3 -0.9–0.8 -1.4–1.6 -1.7–1.3 -0.4–0.7 -1.0–0.8 0.0–0.2
Total -2.9–4.9 1.3–8.8 -3.4–4.2 10.5–17.9 -3.0–4.6 32.6–38.3 15.5–22.1 22.7–28.9 -4.7–3.3 6.3–13.4 -3.3–4.4
Higher -0.2 -0.3 -0.1 -0.7 -0.7 -3.3 1.6 1.9 -0.5 0.5 0.8
426
Table F.28: Control / Temp. at 2cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.0–0.5 -0.9–0.2 -0.9–0.2 -14.5–3.0 -0.9–0.2 -0.8–0.3 -0.9–0.2 -0.9–0.2 -0.8–0.2 -0.8–0.2 -0.9–0.2
2 -0.9–0.2 -0.2–0.2 -0.4–0.5 -14.1–3.4 -0.4–0.5 -0.2–0.6 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.3–0.5 -0.4–0.5
3 -0.9–0.2 -0.4–0.5 -0.0–0.0 -14.3–3.2 -0.1–0.1 -0.2–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -14.5–3.0 -14.1–3.4 -14.3–3.2 86.5–96.2 -0.3–0.2 -1.6–4.6 -0.7–0.7 -0.5–0.7 -1.5–1.1 -1.3–2.5 -0.3–0.2
5 -0.9–0.2 -0.4–0.5 -0.1–0.1 -0.3–0.2 -0.0–0.0 -0.1–0.1 -0.1–0.0 -0.1–0.0 -0.1–0.0 -0.1–0.0 -0.1–0.0
6 -0.8–0.3 -0.2–0.6 -0.2–0.1 -1.6–4.6 -0.1–0.1 2.8–5.3 -3.6–1.1 -3.7–1.0 -3.2–1.4 -3.0–1.7 -3.7–1.0
7 -0.9–0.2 -0.4–0.5 -0.1–0.1 -0.7–0.7 -0.1–0.0 -3.6–1.1 -0.2–0.2 -0.4–0.4 -0.4–0.4 -0.4–0.4 -0.4–0.4
8 -0.9–0.2 -0.4–0.5 -0.1–0.1 -0.5–0.7 -0.1–0.0 -3.7–1.0 -0.4–0.4 -0.2–0.2 -0.4–0.4 -0.4–0.4 -0.4–0.4
9 -0.8–0.2 -0.4–0.5 -0.1–0.1 -1.5–1.1 -0.1–0.0 -3.2–1.4 -0.4–0.4 -0.4–0.4 -0.5–0.4 -0.7–1.2 -0.9–1.1
10 -0.8–0.2 -0.3–0.5 -0.1–0.1 -1.3–2.5 -0.1–0.0 -3.0–1.7 -0.4–0.4 -0.4–0.4 -0.7–1.2 -0.5–0.9 -0.9–1.9
11 -0.9–0.2 -0.4–0.5 -0.1–0.1 -0.3–0.2 -0.1–0.0 -3.7–1.0 -0.4–0.4 -0.4–0.4 -0.9–1.1 -0.9–1.9 -0.1–0.0
Total -2.4–7.4 -2.5–7.3 -2.8–7.0 92.1–94.6 -2.8–7.0 5.9–15.2 -2.6–7.2 -2.6–7.2 -1.4–8.4 1.0–10.5 -2.8–7.1
Higher 10.9 7.4 8.1 16.8 2.5 10.6 3.8 3.8 4.5 5.0 3.4
Table F.29: South / Temp. at 2cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.3 -0.6–0.4 -0.6–0.4 -27.9–2.6 -0.6–0.4 -0.5–0.4 -0.6–0.4 -0.6–0.4 -0.6–0.4 -0.5–0.4 -0.6–0.4
2 -0.6–0.4 -0.2–0.2 -0.3–0.4 -27.8–2.6 -0.3–0.4 -0.3–0.4 -0.3–0.4 -0.3–0.4 -0.3–0.4 -0.3–0.4 -0.3–0.4
3 -0.6–0.4 -0.3–0.4 -0.0–0.0 -27.9–2.7 -0.1–0.1 -0.0–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -27.9–2.6 -27.8–2.6 -27.9–2.7 95.6–112.9 -0.1–0.2 -1.3–3.4 -0.7–0.4 -0.5–0.4 -0.5–0.6 -1.1–2.0 -0.3–0.2
5 -0.6–0.4 -0.3–0.4 -0.1–0.1 -0.1–0.2 -0.0–0.0 -0.0–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.5–0.4 -0.3–0.4 -0.0–0.1 -1.3–3.4 -0.0–0.1 -0.3–1.5 -1.6–1.9 -1.6–1.9 -1.6–1.9 -1.4–2.1 -1.6–1.9
7 -0.6–0.4 -0.3–0.4 -0.1–0.1 -0.7–0.4 -0.1–0.1 -1.6–1.9 -0.1–0.2 -0.4–0.3 -0.4–0.3 -0.4–0.3 -0.4–0.3
8 -0.6–0.4 -0.3–0.4 -0.1–0.1 -0.5–0.4 -0.1–0.1 -1.6–1.9 -0.4–0.3 -0.1–0.2 -0.3–0.3 -0.3–0.3 -0.3–0.3
9 -0.6–0.4 -0.3–0.4 -0.1–0.1 -0.5–0.6 -0.1–0.1 -1.6–1.9 -0.4–0.3 -0.3–0.3 -0.2–0.2 -0.4–0.3 -0.4–0.3
10 -0.5–0.4 -0.3–0.4 -0.1–0.1 -1.1–2.0 -0.1–0.1 -1.4–2.1 -0.4–0.3 -0.3–0.3 -0.4–0.3 -0.8–0.2 -0.2–1.9
11 -0.6–0.4 -0.3–0.4 -0.1–0.1 -0.3–0.2 -0.1–0.1 -1.6–1.9 -0.4–0.3 -0.3–0.3 -0.4–0.3 -0.2–1.9 -0.1–0.1
Total -1.2–12.6 -1.3–12.5 -1.4–12.4 97.7–99.3 -1.4–12.4 1.2–14.7 -1.3–12.5 -1.3–12.5 -1.2–12.5 -0.1–13.5 -1.4–12.4
Higher 18.9 18.2 18.2 30.7 5.5 5.2 5.8 5.6 5.6 5.5 4.7
Table F.30: North / Temp. at 2cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.5 -0.9–0.5 -0.9–0.5 -26.9–2.4 -0.9–0.5 -0.9–0.5 -0.9–0.5 -0.9–0.5 -0.9–0.5 -0.9–0.5 -0.9–0.5
2 -0.9–0.5 -0.3–0.2 -0.4–0.7 -26.4–2.7 -0.4–0.7 -0.3–0.8 -0.4–0.7 -0.4–0.7 -0.4–0.7 -0.4–0.7 -0.4–0.7
3 -0.9–0.5 -0.4–0.7 -0.1–0.1 -26.7–2.4 -0.1–0.2 -0.2–0.1 -0.1–0.2 -0.2–0.1 -0.1–0.2 -0.2–0.1 -0.1–0.2
4 -26.9–2.4 -26.4–2.7 -26.7–2.4 90.6–107.4 -0.2–0.2 -1.4–5.5 -0.6–0.5 -0.7–0.7 -0.6–0.8 -1.9–3.0 -0.4–0.3
5 -0.9–0.5 -0.4–0.7 -0.1–0.2 -0.2–0.2 -0.0–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.0
6 -0.9–0.5 -0.3–0.8 -0.2–0.1 -1.4–5.5 -0.1–0.1 1.3–4.0 -3.7–1.5 -3.7–1.5 -3.7–1.5 -3.3–1.9 -3.8–1.5
7 -0.9–0.5 -0.4–0.7 -0.1–0.2 -0.6–0.5 -0.1–0.1 -3.7–1.5 -0.2–0.2 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5
8 -0.9–0.5 -0.4–0.7 -0.2–0.1 -0.7–0.7 -0.1–0.1 -3.7–1.5 -0.4–0.5 -0.2–0.2 -0.4–0.5 -0.4–0.5 -0.4–0.5
9 -0.9–0.5 -0.4–0.7 -0.1–0.2 -0.6–0.8 -0.1–0.1 -3.7–1.5 -0.4–0.5 -0.4–0.5 -0.3–0.3 -0.4–0.6 -0.5–0.6
10 -0.9–0.5 -0.4–0.7 -0.2–0.1 -1.9–3.0 -0.1–0.1 -3.3–1.9 -0.4–0.5 -0.4–0.5 -0.4–0.6 -0.8–0.8 -0.8–2.4
11 -0.9–0.5 -0.4–0.7 -0.1–0.2 -0.4–0.3 -0.1–0.0 -3.8–1.5 -0.4–0.5 -0.4–0.5 -0.5–0.6 -0.8–2.4 -0.1–0.1
Total -1.5–12.6 -1.7–12.6 -1.9–12.3 94.7–97.3 -1.9–12.3 3.2–16.9 -1.8–12.4 -1.8–12.4 -1.5–12.5 0.8–14.8 -1.8–12.4
Higher 19.6 16.3 17.4 30.7 5.5 10.5 6.4 6.4 6.4 7.1 5.7
427
F.4 Snow Temperature at 5 cm
Table F.31: Control / Temp. at 5cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 3.6–11.9 1.2–9.7 -0.9–7.6 91.5–92.7 -1.0–7.6 3.5–11.7 -1.0–7.6 -1.0–7.6 -0.5–8.1 0.1–8.7 -1.0–7.6
0.67 5.3–13.6 3.2–11.7 -1.6–7.2 79.7–82.1 -1.6–7.2 10.5–18.4 -1.6–7.2 -1.7–7.2 -0.1–8.6 1.6–10.2 -1.6–7.2
1.00 5.3–13.7 4.3–12.9 -2.3–6.6 69.2–72.6 -2.3–6.6 16.9–24.4 -2.3–6.6 -2.3–6.6 0.1–8.9 3.0–11.5 -2.2–6.7
1.33 4.9–13.4 4.8–13.4 -3.1–6.0 60.6–64.7 -3.1–5.9 21.9–29.0 -2.9–6.1 -3.0–6.0 0.1–9.0 4.0–12.6 -3.0–6.1
1.67 4.8–13.2 5.0–13.6 -3.6–5.4 53.6–58.2 -3.7–5.4 25.4–32.3 -3.1–5.9 -3.3–5.7 0.1–8.9 4.9–13.3 -3.6–5.4
2.00 4.8–13.1 5.1–13.6 -4.3–4.7 47.7–52.9 -4.1–4.9 27.5–34.3 -2.8–6.1 -3.2–5.9 -0.1–8.5 5.4–13.7 -4.2–4.8
2.33 4.9–13.2 4.9–13.3 -4.9–4.1 42.8–48.4 -4.4–4.6 28.5–35.2 -1.9–6.9 -2.4–6.4 -0.5–8.1 5.6–13.9 -4.8–4.2
2.67 5.0–13.2 4.7–13.0 -5.4–3.5 38.7–44.6 -4.5–4.4 28.7–35.3 -0.5–8.2 -1.3–7.5 -1.0–7.6 5.7–13.9 -5.3–3.6
3.00 4.9–13.0 4.4–12.8 -3.4–5.2 35.2–41.3 -4.5–4.4 28.4–35.1 1.4–9.9 0.2–8.8 -1.4–7.1 5.6–13.7 -3.2–5.4
3.33 4.6–12.7 4.0–12.4 -3.9–4.7 32.1–38.5 -4.4–4.5 27.9–34.7 3.5–11.8 1.9–10.3 -2.0–6.7 5.4–13.6 -3.8–4.9
3.67 4.2–12.4 3.6–12.1 -4.4–4.4 29.5–36.1 -4.2–4.7 27.6–34.4 5.5–13.7 3.4–11.8 -2.4–6.4 5.3–13.6 -4.2–4.6
4.00 3.6–11.9 3.3–11.8 -4.8–4.1 27.2–34.1 -4.1–4.9 27.4–34.3 7.4–15.6 4.8–13.2 -2.7–6.1 5.3–13.6 -4.7–4.2
4.33 3.0–11.4 2.8–11.5 -5.2–3.9 25.2–32.3 -3.9–5.1 27.3–34.3 9.0–17.2 6.2–14.4 -3.1–5.9 5.3–13.6 -5.1–4.0
4.67 2.3–10.9 2.3–11.0 -5.6–3.6 23.4–30.7 -3.8–5.2 27.3–34.4 10.5–18.7 7.3–15.6 -3.3–5.8 5.3–13.7 -5.5–3.7
5.00 1.5–10.3 1.7–10.6 -5.9–3.3 21.7–29.3 -3.8–5.3 27.6–34.8 11.7–19.9 8.2–16.5 -3.5–5.7 5.4–13.9 -5.8–3.4
5.33 0.7–9.6 1.1–10.2 -6.2–3.1 20.2–28.0 -3.9–5.3 28.1–35.4 12.8–21.0 9.0–17.4 -3.7–5.6 5.6–14.2 -6.1–3.3
5.67 -0.1–8.9 0.6–9.7 -6.5–2.9 18.8–26.7 -4.1–5.2 28.8–36.1 13.5–21.8 9.7–18.0 -3.7–5.7 5.8–14.5 -6.4–3.0
6.00 -1.0–8.2 -0.0–9.2 -6.8–2.8 17.4–25.5 -4.4–5.0 29.7–37.0 14.1–22.4 10.1–18.5 -3.8–5.8 6.2–14.9 -6.6–2.9
6.33 -1.9–7.5 -0.6–8.8 -7.0–2.6 16.1–24.3 -4.7–4.8 30.8–38.1 14.5–22.7 10.4–18.9 -3.8–5.8 6.6–15.3 -6.8–2.7
6.67 -2.8–6.7 -1.1–8.3 -7.2–2.5 14.8–23.2 -5.1–4.5 32.2–39.4 14.5–22.8 10.5–19.0 -3.7–5.9 7.0–15.8 -7.0–2.7
7.00 -3.6–6.0 -1.6–7.9 -7.3–2.4 13.6–22.0 -5.5–4.2 33.7–40.8 14.2–22.5 10.2–18.7 -3.6–6.0 7.5–16.3 -7.2–2.6
7.33 -4.4–5.2 -2.1–7.4 -7.5–2.3 12.4–20.9 -6.0–3.8 35.5–42.5 13.4–21.8 9.7–18.2 -3.6–6.2 8.1–16.9 -7.4–2.4
7.67 -5.2–4.6 -2.6–7.1 -7.6–2.2 11.2–19.9 -3.6–5.9 37.6–44.3 12.1–20.5 8.5–17.2 -3.4–6.3 8.7–17.5 -7.5–2.4
8.00 -5.8–4.0 -2.8–6.8 -7.7–2.2 10.3–18.9 -4.0–5.5 39.9–46.5 10.1–18.7 6.8–15.6 -3.1–6.5 9.4–18.2 -7.5–2.4
8.33 -6.1–3.7 -3.1–6.6 -7.6–2.2 9.3–18.0 -4.4–5.1 42.8–49.1 7.6–16.4 4.7–13.6 -2.9–6.8 10.2–18.9 -7.4–2.4
8.67 -6.4–3.5 -3.0–6.6 -7.5–2.4 8.5–17.3 -4.7–4.8 46.0–52.0 4.7–13.8 2.2–11.4 -2.5–7.2 11.0–19.7 -7.2–2.6
9.00 -6.3–3.5 -2.9–6.7 -7.2–2.6 7.7–16.6 -4.9–4.7 49.5–55.1 1.8–11.2 -0.3–9.1 -2.1–7.6 11.9–20.5 -7.0–2.9
9.33 -6.2–3.6 -2.9–6.6 -7.0–2.8 7.0–15.9 -5.1–4.5 53.0–58.3 -0.6–8.8 -2.3–7.2 -1.6–8.0 12.7–21.1 -6.6–3.1
9.67 -6.1–3.8 -3.0–6.5 -6.7–3.1 6.4–15.3 -5.2–4.4 56.1–61.1 -2.7–6.9 -4.1–5.6 -1.0–8.5 13.3–21.7 -6.3–3.4
10.00 -5.8–3.9 -3.2–6.4 -6.4–3.3 5.9–14.8 -5.2–4.4 58.7–63.4 -4.0–5.5 -5.3–4.5 -0.4–9.0 13.8–22.1 -6.0–3.6
Table F.32: Control / Temp. at 5cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 4.0–5.2 -0.8–1.3 -1.2–0.7 -0.4–2.3 -0.8–1.1 -1.6–1.9 -0.8–1.5 -0.9–1.4 -1.1–0.9 -1.0–1.2 -1.1–0.8
2 -0.8–1.3 2.7–4.1 -1.4–0.8 -1.1–1.8 -0.8–1.4 -1.1–2.5 -1.4–1.1 -1.4–1.1 -1.5–0.9 -1.0–1.5 -1.5–0.8
3 -1.2–0.7 -1.4–0.8 -0.0–0.1 -1.4–0.4 -0.1–0.1 -0.5–1.7 -0.0–0.6 -0.1–0.5 -0.1–0.1 -0.1–0.3 -0.1–0.1
4 -0.4–2.3 -1.1–1.8 -1.4–0.4 18.0–20.3 -0.8–1.0 -2.3–2.7 -0.8–2.2 -0.7–2.1 -0.9–1.1 -1.0–1.7 -1.0–0.6
5 -0.8–1.1 -0.8–1.4 -0.1–0.1 -0.8–1.0 1.6–2.4 -1.2–1.6 -1.0–0.8 -0.7–1.1 -0.9–0.6 -0.6–0.9 -0.9–0.5
6 -1.6–1.9 -1.1–2.5 -0.5–1.7 -2.3–2.7 -1.2–1.6 16.7–19.4 1.8–5.0 1.1–4.1 -2.1–0.6 -0.9–2.0 -2.8–-0.3
7 -0.8–1.5 -1.4–1.1 -0.0–0.6 -0.8–2.2 -1.0–0.8 1.8–5.0 6.8–8.5 -0.1–3.1 -0.6–1.9 0.7–3.3 -0.6–1.9
8 -0.9–1.4 -1.4–1.1 -0.1–0.5 -0.7–2.1 -0.7–1.1 1.1–4.1 -0.1–3.1 6.1–7.7 -1.0–1.5 -0.3–2.2 -1.1–1.2
9 -1.1–0.9 -1.5–0.9 -0.1–0.1 -0.9–1.1 -0.9–0.6 -2.1–0.6 -0.6–1.9 -1.0–1.5 -0.5–0.5 0.1–2.0 -0.7–1.2
10 -1.0–1.2 -1.0–1.5 -0.1–0.3 -1.0–1.7 -0.6–0.9 -0.9–2.0 0.7–3.3 -0.3–2.2 0.1–2.0 3.6–4.9 -0.4–1.6
11 -1.1–0.8 -1.5–0.8 -0.1–0.1 -1.0–0.6 -0.9–0.5 -2.8–-0.3 -0.6–1.9 -1.1–1.2 -0.7–1.2 -0.4–1.6 -0.0–0.2
Total 1.5–10.3 1.7–10.6 -5.9–3.3 21.7–29.3 -3.8–5.3 27.6–34.8 11.7–19.9 8.2–16.5 -3.5–5.7 5.4–13.9 -5.8–3.4
Higher -0.5 2.2 -1.6 3.5 -1.9 6.9 -1.1 -1.1 0.1 -0.8 -0.3
428
Table F.33: South / Temp. at 5cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 5.9–17.8 2.9–15.2 -0.5–12.2 88.8–90.7 -0.5–12.2 4.4–16.5 -0.5–12.2 -0.5–12.2 -0.4–12.2 0.3–12.9 -0.5–12.2
0.67 8.2–19.8 5.5–17.4 -1.3–11.2 75.0–78.6 -1.4–11.2 11.0–22.2 -1.3–11.2 -1.4–11.1 -1.1–11.5 0.9–13.1 -1.3–11.2
1.00 8.8–19.9 7.0–18.4 -2.4–10.0 63.5–68.4 -2.4–10.0 16.8–27.0 -2.3–10.1 -2.4–10.0 -1.7–10.5 1.4–13.3 -2.3–10.0
1.33 9.5–20.1 7.8–18.8 -3.1–8.9 54.0–59.9 -3.1–8.9 20.7–30.1 -2.4–9.4 -2.7–9.2 -2.2–9.5 1.8–13.2 -3.0–8.9
1.67 10.5–20.5 8.2–18.7 -3.6–7.8 46.0–52.5 -3.4–8.0 22.7–31.5 -1.6–9.6 -2.3–9.1 -2.8–8.6 2.0–12.9 -3.6–7.9
2.00 11.4–20.8 8.5–18.4 -4.1–6.8 39.2–46.1 -3.5–7.4 22.8–31.3 0.1–10.6 -1.0–9.8 -3.3–7.6 1.9–12.3 -4.1–6.8
2.33 11.8–20.8 8.5–17.9 -4.7–5.7 33.6–40.7 -3.4–6.9 21.9–30.1 2.5–12.4 1.1–11.1 -3.9–6.5 1.5–11.5 -4.6–5.8
2.67 11.6–20.3 8.2–17.4 -5.3–4.7 29.0–36.3 -3.1–6.8 20.5–28.5 5.4–14.6 3.4–12.7 -4.4–5.6 1.0–10.6 -5.3–4.8
3.00 11.2–19.6 8.0–16.9 -5.8–4.0 25.3–32.8 -2.6–7.1 19.1–27.1 8.2–16.9 5.6–14.5 -4.9–4.8 0.5–9.9 -5.8–4.1
3.33 10.6–18.9 7.9–16.5 -6.3–3.4 22.6–30.1 -1.9–7.4 18.0–25.9 10.8–19.0 7.6–16.1 -5.3–4.3 0.2–9.4 -6.3–3.5
3.67 9.8–18.1 7.5–16.2 -3.8–5.5 20.3–28.0 -1.2–7.9 17.1–25.0 12.8–20.9 9.2–17.6 -5.6–3.9 -0.1–9.0 -3.7–5.5
4.00 8.9–17.3 7.1–15.8 -4.1–5.1 18.5–26.4 -0.7–8.4 16.5–24.6 14.6–22.6 10.6–18.8 -5.8–3.6 -0.3–8.8 -4.0–5.2
4.33 8.1–16.5 6.7–15.4 -4.5–4.9 17.0–25.0 -0.3–8.8 16.2–24.3 16.1–24.1 11.7–19.9 -3.4–5.9 -0.5–8.7 -4.4–4.9
4.67 7.3–15.7 6.1–15.0 -4.8–4.6 15.7–23.8 0.0–9.1 16.2–24.4 17.4–25.3 12.7–20.9 -3.6–5.7 -0.7–8.5 -4.7–4.7
5.00 6.4–15.0 5.6–14.5 -5.0–4.5 14.5–22.8 0.2–9.4 16.4–24.6 18.6–26.4 13.5–21.7 -3.8–5.6 -0.7–8.5 -4.9–4.5
5.33 5.5–14.2 5.1–14.0 -5.3–4.3 13.5–21.9 0.3–9.5 16.8–25.0 19.5–27.4 14.3–22.4 -4.0–5.5 -0.7–8.6 -5.2–4.4
5.67 4.5–13.4 4.4–13.5 -5.5–4.1 12.6–21.1 0.3–9.4 17.4–25.7 20.4–28.3 15.0–23.1 -4.2–5.3 -0.7–8.7 -5.4–4.3
6.00 3.6–12.5 3.8–12.9 -5.7–4.0 11.7–20.3 0.1–9.3 18.3–26.5 21.2–29.1 15.6–23.7 -4.3–5.3 -0.6–8.9 -5.6–4.2
6.33 2.6–11.6 3.2–12.3 -5.8–3.9 10.9–19.6 -0.2–9.1 19.3–27.6 21.9–29.7 16.1–24.2 -4.3–5.3 -0.3–9.2 -5.7–4.0
6.67 1.5–10.6 2.5–11.7 -6.0–3.7 10.1–18.8 -0.5–8.7 20.6–28.8 22.4–30.2 16.5–24.6 -4.3–5.4 -0.0–9.5 -5.9–3.8
7.00 0.4–9.6 1.9–11.1 -6.1–3.7 9.3–18.1 -1.0–8.3 22.1–30.1 22.6–30.4 16.7–24.8 -4.4–5.3 0.4–9.8 -6.0–3.8
7.33 -0.7–8.6 1.2–10.5 -6.3–3.6 8.6–17.5 -1.6–7.8 23.9–31.8 22.5–30.2 16.7–24.7 -4.5–5.3 0.8–10.2 -6.1–3.7
7.67 -1.8–7.6 0.5–9.9 -6.5–3.4 7.9–16.8 -2.3–7.2 25.9–33.6 21.8–29.6 16.2–24.2 -4.5–5.3 1.2–10.6 -6.3–3.6
8.00 -2.9–6.7 -0.1–9.4 -6.7–3.2 7.2–16.2 -3.1–6.5 28.4–35.9 20.4–28.3 15.0–23.1 -4.6–5.2 1.7–11.0 -6.4–3.4
8.33 -3.8–5.9 -0.7–8.9 -6.8–3.1 6.6–15.7 -3.9–5.9 31.5–38.8 18.0–26.1 12.9–21.3 -4.7–5.2 2.3–11.6 -6.5–3.3
8.67 -4.6–5.4 -1.0–8.8 -6.8–3.2 6.1–15.3 -4.5–5.4 35.4–42.5 14.6–23.0 9.9–18.8 -4.6–5.3 2.9–12.4 -6.6–3.4
9.00 -5.0–5.1 -1.1–8.9 -6.8–3.5 5.7–15.1 -5.2–5.0 40.2–47.0 10.4–19.3 6.5–15.8 -4.6–5.5 3.8–13.3 -6.5–3.7
9.33 -5.3–5.2 -1.1–9.1 -6.7–3.7 5.4–15.1 -6.0–4.5 45.5–51.9 6.1–15.6 3.0–12.8 -4.5–5.8 4.7–14.5 -6.4–4.0
9.67 -5.5–5.3 -1.3–9.1 -6.6–4.0 5.1–15.0 -6.5–4.3 50.6–56.6 2.3–12.3 -0.2–10.1 -4.2–6.3 5.7–15.5 -6.3–4.3
10.00 -5.5–5.4 -1.6–9.0 -6.4–4.3 4.7–14.9 -7.0–4.0 55.0–60.5 -0.7–9.8 -2.5–8.2 -4.0–6.5 6.5–16.4 -6.0–4.6
Table F.34: South / Temp. at 5cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 7.8–9.2 -0.3–2.1 -1.0–1.0 -0.6–2.2 -1.2–0.9 -1.3–1.8 -0.6–2.5 -0.5–2.3 -0.9–1.1 -0.9–1.3 -1.0–1.0
2 -0.3–2.1 5.3–7.0 -1.2–1.2 -1.2–1.6 -0.6–1.8 -1.2–2.0 -1.0–2.2 -1.7–1.2 -1.2–1.2 -1.5–1.1 -1.2–1.2
3 -1.0–1.0 -1.2–1.2 -0.0–0.1 -1.0–0.4 -0.2–0.1 -0.4–0.9 -0.2–1.1 -0.0–1.0 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -0.6–2.2 -1.2–1.6 -1.0–0.4 13.5–15.3 -0.7–1.0 -1.3–2.1 -0.5–2.6 -0.2–2.6 -0.7–0.8 -0.8–1.1 -0.8–0.7
5 -1.2–0.9 -0.6–1.8 -0.2–0.1 -0.7–1.0 3.1–4.3 -1.3–1.2 -0.9–1.9 -0.9–1.7 -0.9–1.2 -1.2–0.9 -0.9–1.1
6 -1.3–1.8 -1.2–2.0 -0.4–0.9 -1.3–2.1 -1.3–1.2 10.7–12.9 2.4–5.5 1.8–4.6 -1.9–0.6 -1.0–1.5 -2.0–0.4
7 -0.6–2.5 -1.0–2.2 -0.2–1.1 -0.5–2.6 -0.9–1.9 2.4–5.5 12.5–14.7 0.9–4.7 -0.4–2.3 1.0–3.8 -0.3–2.4
8 -0.5–2.3 -1.7–1.2 -0.0–1.0 -0.2–2.6 -0.9–1.7 1.8–4.6 0.9–4.7 10.0–12.0 -0.8–1.8 -0.2–2.5 -0.9–1.7
9 -0.9–1.1 -1.2–1.2 -0.1–0.1 -0.7–0.8 -0.9–1.2 -1.9–0.6 -0.4–2.3 -0.8–1.8 -0.3–0.2 -0.2–0.7 -0.4–0.6
10 -0.9–1.3 -1.5–1.1 -0.1–0.2 -0.8–1.1 -1.2–0.9 -1.0–1.5 1.0–3.8 -0.2–2.5 -0.2–0.7 1.7–2.5 -0.1–1.3
11 -1.0–1.0 -1.2–1.2 -0.1–0.1 -0.8–0.7 -0.9–1.1 -2.0–0.4 -0.3–2.4 -0.9–1.7 -0.4–0.6 -0.1–1.3 -0.0–0.1
Total 6.4–15.0 5.6–14.5 -5.0–4.5 14.5–22.8 0.2–9.4 16.4–24.6 18.6–26.4 13.5–21.7 -3.8–5.6 -0.7–8.5 -4.9–4.5
Higher -1.8 1.6 -1.2 0.6 -0.5 1.5 -5.8 -4.1 -0.5 -2.8 -1.6
429
Table F.35: North / Temp. at 5cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 1.7–13.8 0.6–12.7 -0.8–11.6 96.0–97.0 -0.8–11.7 1.5–13.6 -0.8–11.7 -0.8–11.7 -0.7–11.7 -0.4–11.9 -0.7–11.7
0.67 3.1–15.5 1.4–14.1 -1.2–11.7 89.4–91.5 -1.2–11.7 5.3–17.5 -1.2–11.7 -1.2–11.7 -1.1–11.8 -0.0–12.8 -1.2–11.7
1.00 3.4–16.2 1.8–14.8 -1.9–11.4 82.4–85.4 -2.0–11.4 9.5–21.5 -1.9–11.4 -2.0–11.4 -1.6–11.7 0.6–13.6 -1.9–11.4
1.33 3.5–16.3 2.0–15.2 -2.5–11.0 75.4–79.4 -2.5–10.9 13.6–25.1 -2.5–11.0 -2.5–11.0 -2.0–11.3 1.2–14.2 -2.5–11.0
1.67 3.7–16.4 2.2–15.3 -2.9–10.5 68.7–73.5 -3.0–10.5 17.1–28.2 -2.7–10.7 -2.8–10.6 -2.5–10.9 1.8–14.7 -3.0–10.5
2.00 4.0–16.5 2.3–15.2 -3.4–9.9 62.3–67.9 -3.4–9.9 20.3–30.8 -2.7–10.5 -3.0–10.4 -2.9–10.4 2.4–15.0 -3.4–10.0
2.33 4.6–16.7 2.4–15.0 -3.7–9.4 56.3–62.4 -3.6–9.5 22.5–32.5 -2.4–10.6 -2.6–10.6 -3.1–9.9 3.0–15.3 -3.6–9.4
2.67 5.2–16.9 2.4–14.5 -3.9–8.7 50.5–57.1 -3.6–9.0 24.0–33.5 -1.4–11.0 -1.4–11.1 -3.3–9.3 3.3–15.2 -3.9–8.8
3.00 5.9–17.0 2.3–14.0 -4.2–8.0 45.2–52.2 -3.7–8.5 24.6–33.8 -0.2–11.7 0.2–12.0 -3.5–8.7 3.5–14.9 -4.1–8.1
3.33 6.4–17.1 2.3–13.5 -4.4–7.4 40.5–47.7 -3.6–8.1 24.7–33.6 1.3–12.5 2.1–13.3 -3.8–8.0 3.7–14.6 -4.3–7.5
3.67 6.6–16.9 2.0–12.9 -4.6–6.7 36.3–43.6 -3.5–7.8 24.5–33.1 2.9–13.5 4.1–14.7 -4.0–7.3 3.8–14.2 -4.5–6.8
4.00 6.7–16.5 1.8–12.2 -4.9–6.1 32.6–40.1 -3.5–7.4 24.2–32.5 4.4–14.4 6.3–16.2 -4.3–6.6 3.7–13.8 -4.8–6.2
4.33 6.4–15.9 1.4–11.6 -5.2–5.5 29.3–37.0 -3.5–7.1 23.8–31.9 5.7–15.3 8.3–17.7 -4.5–6.0 3.5–13.4 -5.1–5.5
4.67 5.8–15.2 1.1–11.0 -5.4–4.9 26.5–34.3 -3.4–6.8 23.5–31.4 6.8–16.1 10.1–19.2 -4.8–5.5 3.4–13.0 -5.4–5.0
5.00 5.2–14.3 0.7–10.4 -5.7–4.5 24.2–32.0 -3.4–6.6 23.3–31.0 7.8–16.9 11.8–20.5 -5.1–5.0 3.3–12.7 -5.7–4.5
5.33 4.3–13.4 0.3–9.9 -6.0–4.1 22.2–30.0 -3.5–6.4 23.3–30.9 8.6–17.4 13.1–21.6 -5.3–4.6 3.2–12.5 -5.9–4.1
5.67 3.5–12.6 0.0–9.5 -6.2–3.8 20.5–28.4 -3.5–6.2 23.6–31.1 9.2–17.9 14.3–22.5 -5.5–4.4 3.2–12.4 -6.1–3.8
6.00 2.7–11.7 -0.2–9.2 -6.4–3.5 19.1–27.1 -3.6–6.1 24.1–31.5 9.6–18.2 15.0–23.1 -5.7–4.2 3.3–12.5 -6.3–3.6
6.33 1.8–10.9 -0.5–9.0 -3.6–5.8 17.9–25.9 -3.7–5.9 24.9–32.2 9.7–18.3 15.4–23.4 -5.8–4.0 3.5–12.6 -6.5–3.5
6.67 0.8–10.0 -0.7–8.9 -3.7–5.7 16.9–25.0 -3.9–5.8 26.0–33.3 9.7–18.3 15.4–23.4 -6.0–3.9 3.8–12.8 -3.6–5.9
7.00 -0.1–9.2 -0.8–8.9 -3.8–5.7 16.2–24.4 -4.1–5.7 27.4–34.6 9.3–18.1 14.9–23.1 -6.1–3.9 4.1–13.1 -3.7–5.8
7.33 -1.0–8.6 -0.8–9.0 -3.9–5.7 15.5–23.9 -4.3–5.6 29.2–36.3 8.7–17.6 14.0–22.3 -6.2–4.0 4.4–13.6 -3.8–5.9
7.67 -1.9–8.0 -0.7–9.3 -3.9–5.8 15.0–23.5 -4.6–5.6 31.2–38.3 7.7–16.9 12.5–21.2 -6.3–4.1 4.9–14.2 -3.8–6.0
8.00 -2.7–7.5 -0.6–9.6 -4.0–6.0 14.6–23.4 -4.9–5.6 33.6–40.6 6.3–15.8 10.7–19.7 -6.4–4.2 5.3–14.8 -3.8–6.2
8.33 -3.4–7.2 -0.5–10.0 -4.0–6.2 14.2–23.4 -5.3–5.6 36.2–43.2 4.5–14.5 8.4–18.0 -6.5–4.5 5.8–15.6 -7.2–3.8
8.67 -3.9–7.1 -0.5–10.4 -7.6–3.8 14.0–23.4 -5.6–5.6 39.1–46.0 2.7–13.2 6.1–16.3 -6.6–4.7 6.4–16.3 -7.3–4.0
9.00 -4.3–7.1 -0.4–10.7 -7.7–4.1 13.9–23.5 -6.0–5.7 42.0–48.8 0.9–11.9 3.9–14.5 -6.6–5.0 6.9–17.2 -7.4–4.3
9.33 -4.6–7.3 -0.5–11.0 -7.8–4.3 13.7–23.6 -6.3–5.7 44.9–51.6 -0.8–10.7 1.9–13.0 -6.6–5.4 7.4–18.0 -7.6–4.6
9.67 -4.7–7.4 -0.6–11.1 -7.8–4.6 13.4–23.7 -6.5–5.8 47.5–54.0 -2.2–9.7 0.1–11.7 -6.6–5.7 8.0–18.7 -7.5–4.9
10.00 -4.7–7.6 -0.9–11.1 -7.8–4.8 13.1–23.6 -6.7–5.9 49.8–56.2 -3.3–8.9 -1.3–10.7 -6.6–6.1 8.5–19.4 -7.5–5.1
Table F.36: North / Temp. at 5cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 6.1–8.0 -1.6–1.7 -2.2–0.9 -2.9–1.6 -2.4–0.8 -2.5–1.9 -1.7–1.6 -1.9–1.7 -2.2–0.9 -2.2–1.1 -2.2–0.9
2 -1.6–1.7 0.2–1.9 -1.8–1.2 -1.8–2.1 -1.2–1.7 -0.5–3.4 -1.5–1.4 -1.7–1.3 -1.8–1.2 -1.5–1.5 -1.8–1.2
3 -2.2–0.9 -1.8–1.2 -0.1–0.1 -2.9–0.1 -0.2–0.2 -0.8–1.9 -0.3–0.4 -0.5–0.7 -0.1–0.2 -0.2–0.4 -0.1–0.2
4 -2.9–1.6 -1.8–2.1 -2.9–0.1 23.3–26.5 -1.2–1.0 -2.6–3.2 -1.9–1.3 -2.2–1.7 -1.0–1.2 -1.5–1.7 -1.0–1.1
5 -2.4–0.8 -1.2–1.7 -0.2–0.2 -1.2–1.0 0.5–1.4 -1.2–1.9 -1.2–0.8 -1.2–1.0 -0.8–1.0 -1.1–0.7 -0.8–1.0
6 -2.5–1.9 -0.5–3.4 -0.8–1.9 -2.6–3.2 -1.2–1.9 22.1–25.2 -1.4–1.6 -1.3–2.4 -1.3–1.2 -1.6–1.4 -1.4–1.1
7 -1.7–1.6 -1.5–1.4 -0.3–0.4 -1.9–1.3 -1.2–0.8 -1.4–1.6 8.3–10.3 -1.8–2.2 -1.5–1.6 -1.2–1.8 -1.5–1.5
8 -1.9–1.7 -1.7–1.3 -0.5–0.7 -2.2–1.7 -1.2–1.0 -1.3–2.4 -1.8–2.2 12.4–14.7 -2.6–0.6 -2.5–0.8 -2.6–0.6
9 -2.2–0.9 -1.8–1.2 -0.1–0.2 -1.0–1.2 -0.8–1.0 -1.3–1.2 -1.5–1.6 -2.6–0.6 -0.3–0.2 -0.5–0.7 -0.5–0.7
10 -2.2–1.1 -1.5–1.5 -0.2–0.4 -1.5–1.7 -1.1–0.7 -1.6–1.4 -1.2–1.8 -2.5–0.8 -0.5–0.7 6.1–7.4 -1.2–0.7
11 -2.2–0.9 -1.8–1.2 -0.1–0.2 -1.0–1.1 -0.8–1.0 -1.4–1.1 -1.5–1.5 -2.6–0.6 -0.5–0.7 -1.2–0.7 0.0–0.2
Total 5.2–14.3 0.7–10.4 -5.7–4.5 24.2–32.0 -3.4–6.6 23.3–31.0 7.8–16.9 11.8–20.5 -5.1–5.0 3.3–12.7 -5.7–4.5
Higher 7.1 3.6 0.8 5.1 1.3 0.8 2.9 5.1 1.5 2.5 1.5
430
Table F.37: Control / Temp. at 5cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 2.5–3.7 -1.0–1.1 -1.1–0.9 -2.5–2.2 -1.3–0.8 -1.5–2.6 -0.9–1.3 -1.0–1.1 -1.1–1.0 -0.8–1.5 -1.1–0.9
2 -1.0–1.1 1.2–2.7 -1.5–1.2 -2.7–2.1 -1.2–1.4 -0.4–3.9 -1.2–1.5 -1.5–1.2 -1.6–1.1 -1.6–1.1 -1.5–1.1
3 -1.1–0.9 -1.5–1.2 -0.1–0.0 -3.8–0.4 -0.1–0.1 -0.7–2.3 -0.0–0.3 -0.1–0.2 -0.1–0.1 -0.1–0.4 -0.1–0.1
4 -2.5–2.2 -2.7–2.1 -3.8–0.4 33.7–37.7 -1.1–0.8 -2.8–5.0 -0.9–2.2 -1.2–1.8 -1.4–1.0 -1.0–2.7 -1.0–0.7
5 -1.3–0.8 -1.2–1.4 -0.1–0.1 -1.1–0.8 0.5–1.1 -1.0–2.3 -0.8–0.4 -0.5–0.7 -0.6–0.5 -0.6–0.5 -0.6–0.5
6 -1.5–2.6 -0.4–3.9 -0.7–2.3 -2.8–5.0 -1.0–2.3 21.7–24.9 -0.7–2.4 -1.1–1.9 -1.8–1.3 -1.0–2.4 -2.8–0.0
7 -0.9–1.3 -1.2–1.5 -0.0–0.3 -0.9–2.2 -0.8–0.4 -0.7–2.4 3.0–4.1 -0.3–1.8 -0.5–1.3 0.1–2.0 -0.5–1.3
8 -1.0–1.1 -1.5–1.2 -0.1–0.2 -1.2–1.8 -0.5–0.7 -1.1–1.9 -0.3–1.8 2.9–3.9 -0.9–0.9 -0.6–1.3 -1.0–0.7
9 -1.1–1.0 -1.6–1.1 -0.1–0.1 -1.4–1.0 -0.6–0.5 -1.8–1.3 -0.5–1.3 -0.9–0.9 -0.6–0.5 0.3–2.5 -0.9–1.5
10 -0.8–1.5 -1.6–1.1 -0.1–0.4 -1.0–2.7 -0.6–0.5 -1.0–2.4 0.1–2.0 -0.6–1.3 0.3–2.5 4.9–6.4 -0.8–1.5
11 -1.1–0.9 -1.5–1.1 -0.1–0.1 -1.0–0.7 -0.6–0.5 -2.8–0.0 -0.5–1.3 -1.0–0.7 -0.9–1.5 -0.8–1.5 -0.0–0.2
Total -1.2–8.6 -0.6–9.3 -4.6–4.9 37.4–43.9 -6.0–4.2 28.1–35.7 1.4–10.9 -0.4–9.2 -3.1–6.8 5.3–14.5 -4.5–5.0
Higher 0.1 1.6 1.0 4.6 -1.9 3.4 -1.7 -0.6 0.6 -0.5 1.0
Table F.38: South / Temp. at 5cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 6.1–7.6 -1.4–1.1 -1.1–1.2 -3.4–2.1 -1.3–1.0 -1.7–2.5 -1.4–1.4 -1.4–1.2 -1.0–1.3 -1.6–0.9 -1.1–1.2
2 -1.4–1.1 3.0–4.8 -1.2–1.9 -3.3–2.0 -1.4–1.6 -0.5–3.9 -1.4–1.8 -2.0–1.2 -1.2–1.9 -1.6–1.6 -1.2–1.8
3 -1.1–1.2 -1.2–1.9 -0.0–0.1 -4.0–0.3 -0.1–0.2 -0.8–1.8 -0.2–0.6 -0.1–0.5 -0.1–0.1 -0.1–0.3 -0.1–0.1
4 -3.4–2.1 -3.3–2.0 -4.0–0.3 29.0–33.3 -1.4–0.7 -3.4–3.4 -1.3–2.8 -1.1–2.6 -0.8–1.0 -1.0–2.0 -1.0–0.6
5 -1.3–1.0 -1.4–1.6 -0.1–0.2 -1.4–0.7 1.3–2.2 -1.1–2.0 -0.9–1.2 -1.0–1.0 -0.7–1.1 -1.0–0.8 -0.7–1.1
6 -1.7–2.5 -0.5–3.9 -0.8–1.8 -3.4–3.4 -1.1–2.0 17.8–21.0 0.2–3.7 -0.2–3.3 -1.9–1.2 -1.8–1.5 -2.2–0.9
7 -1.4–1.4 -1.4–1.8 -0.2–0.6 -1.3–2.8 -0.9–1.2 0.2–3.7 6.9–8.6 0.1–3.1 -0.3–2.1 0.4–2.9 -0.2–2.1
8 -1.4–1.2 -2.0–1.2 -0.1–0.5 -1.1–2.6 -1.0–1.0 -0.2–3.3 0.1–3.1 5.9–7.4 -1.1–1.2 -0.7–1.6 -1.1–1.2
9 -1.0–1.3 -1.2–1.9 -0.1–0.1 -0.8–1.0 -0.7–1.1 -1.9–1.2 -0.3–2.1 -1.1–1.2 -0.4–0.3 -0.2–1.0 -0.6–0.7
10 -1.6–0.9 -1.6–1.6 -0.1–0.3 -1.0–2.0 -1.0–0.8 -1.8–1.5 0.4–2.9 -0.7–1.6 -0.2–1.0 3.2–4.4 -0.7–1.3
11 -1.1–1.2 -1.2–1.8 -0.1–0.1 -1.0–0.6 -0.7–1.1 -2.2–0.9 -0.2–2.1 -1.1–1.2 -0.6–0.7 -0.7–1.3 0.0–0.2
Total 2.6–13.6 2.0–13.2 -7.8–4.3 30.6–38.9 -3.9–7.7 20.9–30.4 6.9–17.4 4.0–14.9 -6.6–5.4 0.1–11.3 -7.7–4.4
Higher 2.1 1.9 -1.3 5.1 -0.4 0.8 -4.0 -1.5 -2.3 -1.0 -2.7
Table F.39: North / Temp. at 5cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 2.7–4.4 -2.1–1.1 -2.4–0.7 -7.4–1.0 -2.5–0.6 -2.6–2.7 -2.2–1.0 -2.4–0.8 -2.5–0.7 -2.3–1.0 -2.4–0.7
2 -2.1–1.1 -0.9–1.0 -2.0–1.6 -6.1–1.8 -1.6–2.0 -0.6–4.4 -1.9–1.7 -1.9–1.6 -2.0–1.6 -1.5–2.0 -2.0–1.6
3 -2.4–0.7 -2.0–1.6 -0.1–0.1 -7.6–-0.1 -0.2–0.2 -1.2–2.5 -0.2–0.3 -0.3–0.3 -0.1–0.2 -0.4–0.3 -0.1–0.2
4 -7.4–1.0 -6.1–1.8 -7.6–-0.1 40.0–46.3 -1.6–0.8 -4.1–5.9 -2.4–1.4 -2.6–1.9 -1.0–1.3 -2.3–2.4 -1.0–1.2
5 -2.5–0.6 -1.6–2.0 -0.2–0.2 -1.6–0.8 0.0–0.8 -1.4–2.6 -1.0–0.6 -1.0–0.7 -0.6–1.0 -1.0–0.6 -0.6–0.9
6 -2.6–2.7 -0.6–4.4 -1.2–2.5 -4.1–5.9 -1.4–2.6 23.9–28.0 -2.1–1.3 -1.9–1.8 -1.3–1.7 -2.3–1.5 -1.4–1.5
7 -2.2–1.0 -1.9–1.7 -0.2–0.3 -2.4–1.4 -1.0–0.6 -2.1–1.3 4.2–5.6 -1.7–1.2 -1.3–1.2 -1.2–1.2 -1.3–1.1
8 -2.4–0.8 -1.9–1.6 -0.3–0.3 -2.6–1.9 -1.0–0.7 -1.9–1.8 -1.7–1.2 6.2–7.9 -2.0–0.6 -2.0–0.8 -2.0–0.6
9 -2.5–0.7 -2.0–1.6 -0.1–0.2 -1.0–1.3 -0.6–1.0 -1.3–1.7 -1.3–1.2 -2.0–0.6 -0.4–0.3 -0.7–0.8 -0.6–0.9
10 -2.3–1.0 -1.5–2.0 -0.4–0.3 -2.3–2.4 -1.0–0.6 -2.3–1.5 -1.2–1.2 -2.0–0.8 -0.7–0.8 6.4–7.9 -1.3–1.0
11 -2.4–0.7 -2.0–1.6 -0.1–0.2 -1.0–1.2 -0.6–0.9 -1.4–1.5 -1.3–1.1 -2.0–0.6 -0.6–0.9 -1.3–1.0 0.0–0.2
Total 0.5–12.7 -0.8–11.6 -5.5–7.3 41.9–49.5 -4.3–8.3 24.9–34.3 1.1–13.2 3.1–14.9 -4.9–7.8 3.3–15.1 -5.5–7.4
Higher 12.2 6.6 5.1 11.8 2.3 0.2 4.4 5.7 2.6 3.8 2.4
431
Table F.40: Control / Temp. at 5cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 2.7–3.6 -0.6–1.0 -0.9–0.7 -1.7–1.4 -1.0–0.6 -0.9–2.3 -0.5–1.6 -0.6–1.4 -0.9–0.7 -0.6–1.2 -0.9–0.7
2 -0.6–1.0 0.9–2.0 -1.3–0.8 -1.9–1.2 -1.2–0.8 -0.5–2.8 -1.2–1.1 -1.2–1.0 -1.3–0.8 -0.8–1.3 -1.3–0.7
3 -0.9–0.7 -1.3–0.8 -0.0–0.1 -2.0–0.5 -0.1–0.1 -0.2–2.0 -0.0–0.7 -0.2–0.5 -0.1–0.1 -0.1–0.3 -0.1–0.1
4 -1.7–1.4 -1.9–1.2 -2.0–0.5 19.7–22.4 -0.5–0.7 1.5–6.8 1.1–4.2 0.5–3.3 -1.0–0.6 0.4–2.9 -0.4–0.6
5 -1.0–0.6 -1.2–0.8 -0.1–0.1 -0.5–0.7 1.5–2.3 -0.9–1.9 -1.0–1.0 -0.7–1.1 -1.0–0.6 -0.6–1.0 -0.9–0.6
6 -0.9–2.3 -0.5–2.8 -0.2–2.0 1.5–6.8 -0.9–1.9 13.9–16.5 2.5–5.8 1.5–4.6 -1.8–0.7 -0.8–2.1 -2.2–0.0
7 -0.5–1.6 -1.2–1.1 -0.0–0.7 1.1–4.2 -1.0–1.0 2.5–5.8 6.3–8.1 0.1–3.6 -0.5–2.2 0.9–3.6 -0.3–2.2
8 -0.6–1.4 -1.2–1.0 -0.2–0.5 0.5–3.3 -0.7–1.1 1.5–4.6 0.1–3.6 6.0–7.7 -1.2–1.4 -0.6–2.1 -1.2–1.2
9 -0.9–0.7 -1.3–0.8 -0.1–0.1 -1.0–0.6 -1.0–0.6 -1.8–0.7 -0.5–2.2 -1.2–1.4 -0.2–0.8 -0.1–1.8 -1.0–0.9
10 -0.6–1.2 -0.8–1.3 -0.1–0.3 0.4–2.9 -0.6–1.0 -0.8–2.1 0.9–3.6 -0.6–2.1 -0.1–1.8 3.0–4.2 -0.3–1.6
11 -0.9–0.7 -1.3–0.7 -0.1–0.1 -0.4–0.6 -0.9–0.6 -2.2–0.0 -0.3–2.2 -1.2–1.2 -1.0–0.9 -0.3–1.6 -0.0–0.2
Total -2.6–7.1 -2.8–7.0 -6.5–3.4 30.3–37.9 -4.9–5.1 29.3–36.8 14.2–22.8 9.9–18.7 -3.5–6.5 5.7–15.0 -6.3–3.5
Higher -2.5 0.6 -1.9 3.9 -2.0 4.3 -2.2 -0.9 0.6 -0.9 -1.3
Table F.41: South / Temp. at 5cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 4.9–6.0 -0.4–1.4 -0.8–0.9 -0.5–1.7 -0.9–0.9 -1.0–2.1 -0.3–2.8 -0.3–2.4 -0.8–1.0 -0.6–1.4 -0.8–0.9
2 -0.4–1.4 1.5–2.8 -1.1–1.2 -1.5–0.9 -0.5–1.7 -0.7–2.3 -0.8–2.3 -1.4–1.4 -1.1–1.1 -1.4–1.0 -1.1–1.1
3 -0.8–0.9 -1.1–1.2 -0.0–0.1 -0.8–0.3 -0.1–0.1 -0.4–1.2 -0.2–1.5 -0.1–1.2 -0.1–0.1 -0.0–0.2 -0.1–0.1
4 -0.5–1.7 -1.5–0.9 -0.8–0.3 11.5–13.2 -0.7–0.8 -1.8–1.7 -0.0–3.5 -0.1–2.9 -0.7–0.7 -0.6–1.3 -0.7–0.6
5 -0.9–0.9 -0.5–1.7 -0.1–0.1 -0.7–0.8 3.1–4.2 -1.3–1.4 -0.8–2.2 -1.0–1.7 -1.0–0.9 -1.2–0.8 -1.0–0.9
6 -1.0–2.1 -0.7–2.3 -0.4–1.2 -1.8–1.7 -1.3–1.4 11.6–14.0 4.3–8.1 2.8–6.1 -1.8–0.6 -1.6–1.0 -1.9–0.4
7 -0.3–2.8 -0.8–2.3 -0.2–1.5 -0.0–3.5 -0.8–2.2 4.3–8.1 13.9–16.4 1.5–6.0 -0.0–2.7 1.8–4.8 0.0–2.7
8 -0.3–2.4 -1.4–1.4 -0.1–1.2 -0.1–2.9 -1.0–1.7 2.8–6.1 1.5–6.0 11.2–13.4 -0.7–2.1 0.1–3.0 -0.8–1.9
9 -0.8–1.0 -1.1–1.1 -0.1–0.1 -0.7–0.7 -1.0–0.9 -1.8–0.6 -0.0–2.7 -0.7–2.1 -0.2–0.3 -0.3–0.8 -0.5–0.6
10 -0.6–1.4 -1.4–1.0 -0.0–0.2 -0.6–1.3 -1.2–0.8 -1.6–1.0 1.8–4.8 0.1–3.0 -0.3–0.8 1.9–2.8 -0.0–1.5
11 -0.8–0.9 -1.1–1.1 -0.1–0.1 -0.7–0.6 -1.0–0.9 -1.9–0.4 0.0–2.7 -0.8–1.9 -0.5–0.6 -0.0–1.5 -0.0–0.1
Total -0.1–9.5 -1.4–8.3 -6.5–3.4 11.2–20.2 -2.2–7.5 21.5–29.5 24.0–31.7 17.3–25.4 -4.8–4.9 -0.1–9.5 -6.3–3.5
Higher -5.3 -1.0 -3.1 -0.2 -2.3 1.9 -8.4 -5.4 -1.9 -3.5 -3.4
Table F.42: North / Temp. at 5cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 3.9–5.2 -1.3–1.1 -1.6–0.7 -3.0–0.8 -1.8–0.6 -1.8–2.1 -1.2–1.5 -1.6–1.4 -1.7–0.6 -1.7–0.7 -1.6–0.7
2 -1.3–1.1 -0.7–0.7 -1.0–1.7 -2.8–0.7 -1.0–1.6 -0.4–3.0 -1.4–1.2 -1.5–1.2 -0.9–1.7 -1.4–1.1 -1.0–1.7
3 -1.6–0.7 -1.0–1.7 -0.0–0.1 -3.0–-0.2 -0.1–0.1 -0.6–2.0 -0.3–0.5 -0.6–0.8 -0.2–0.1 -0.2–0.3 -0.2–0.1
4 -3.0–0.8 -2.8–0.7 -3.0–-0.2 23.4–26.5 -1.0–0.7 -0.5–5.0 -0.6–2.5 -0.9–3.0 -0.8–0.8 -0.4–2.2 -0.7–0.9
5 -1.8–0.6 -1.0–1.6 -0.1–0.1 -1.0–0.7 0.6–1.5 -0.8–2.2 -1.1–1.0 -1.2–1.2 -0.8–1.0 -1.1–0.7 -0.8–1.0
6 -1.8–2.1 -0.4–3.0 -0.6–2.0 -0.5–5.0 -0.8–2.2 21.0–23.8 -0.5–2.6 -0.8–3.2 -1.3–1.0 -1.1–1.8 -1.3–0.9
7 -1.2–1.5 -1.4–1.2 -0.3–0.5 -0.6–2.5 -1.1–1.0 -0.5–2.6 9.2–11.3 -1.2–3.1 -1.2–1.9 -1.9–1.3 -1.2–1.9
8 -1.6–1.4 -1.5–1.2 -0.6–0.8 -0.9–3.0 -1.2–1.2 -0.8–3.2 -1.2–3.1 14.1–16.5 -2.4–0.8 -2.5–0.8 -2.4–0.8
9 -1.7–0.6 -0.9–1.7 -0.2–0.1 -0.8–0.8 -0.8–1.0 -1.3–1.0 -1.2–1.9 -2.4–0.8 -0.3–0.2 -0.5–0.7 -0.5–0.7
10 -1.7–0.7 -1.4–1.1 -0.2–0.3 -0.4–2.2 -1.1–0.7 -1.1–1.8 -1.9–1.3 -2.5–0.8 -0.5–0.7 5.9–7.1 -0.9–0.9
11 -1.6–0.7 -1.0–1.7 -0.2–0.1 -0.7–0.9 -0.8–1.0 -1.3–0.9 -1.2–1.9 -2.4–0.8 -0.5–0.7 -0.9–0.9 -0.0–0.2
Total 1.3–10.4 -1.2–8.0 -5.3–4.3 26.8–34.0 -2.9–6.5 25.3–32.5 11.2–19.4 16.0–23.7 -4.6–5.0 3.7–12.7 -5.2–4.4
Higher 4.9 2.3 0.3 4.0 0.5 -0.8 1.6 4.0 0.7 2.3 -0.0
432
Table F.43: Control / Temp. at 5cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.4 -0.7–0.4 -0.7–0.4 -15.1–3.1 -0.7–0.4 -0.6–0.5 -0.7–0.4 -0.7–0.4 -0.7–0.4 -0.6–0.5 -0.7–0.4
2 -0.7–0.4 -0.4–0.2 -0.6–0.5 -14.8–3.3 -0.6–0.5 -0.5–0.7 -0.6–0.5 -0.6–0.5 -0.6–0.5 -0.6–0.6 -0.6–0.5
3 -0.7–0.4 -0.6–0.5 -0.0–0.0 -15.0–3.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -15.1–3.1 -14.8–3.3 -15.0–3.1 91.1–101.0 -0.1–0.1 -1.9–2.7 -0.8–0.5 -0.5–0.7 -0.8–1.0 -0.9–1.9 -0.2–0.1
5 -0.7–0.4 -0.6–0.5 -0.1–0.1 -0.1–0.1 -0.0–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.6–0.5 -0.5–0.7 -0.1–0.1 -1.9–2.7 -0.1–0.1 1.3–3.1 -2.7–0.9 -2.8–0.8 -2.6–1.0 -2.4–1.1 -2.8–0.8
7 -0.7–0.4 -0.6–0.5 -0.1–0.1 -0.8–0.5 -0.1–0.1 -2.7–0.9 -0.3–0.2 -0.4–0.6 -0.4–0.6 -0.4–0.6 -0.4–0.6
8 -0.7–0.4 -0.6–0.5 -0.1–0.1 -0.5–0.7 -0.1–0.1 -2.8–0.8 -0.4–0.6 -0.3–0.2 -0.5–0.4 -0.5–0.4 -0.5–0.3
9 -0.7–0.4 -0.6–0.5 -0.1–0.1 -0.8–1.0 -0.1–0.1 -2.6–1.0 -0.4–0.6 -0.5–0.4 -0.4–0.3 -0.5–0.9 -0.6–0.8
10 -0.6–0.5 -0.6–0.6 -0.1–0.1 -0.9–1.9 -0.1–0.1 -2.4–1.1 -0.4–0.6 -0.5–0.4 -0.5–0.9 -0.6–0.5 -0.7–1.5
11 -0.7–0.4 -0.6–0.5 -0.1–0.1 -0.2–0.1 -0.1–0.1 -2.8–0.8 -0.4–0.6 -0.5–0.3 -0.6–0.8 -0.7–1.5 -0.1–0.0
Total -2.0–7.5 -2.1–7.5 -2.5–7.1 95.1–97.0 -2.5–7.1 2.5–11.7 -2.2–7.4 -2.2–7.4 -1.6–7.9 -0.3–9.1 -2.5–7.1
Higher 9.6 8.8 8.4 16.9 2.5 8.8 3.5 3.8 3.7 4.1 3.0
Table F.44: South / Temp. at 5cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.2 -0.4–0.5 -0.4–0.5 -27.8–2.7 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5
2 -0.4–0.5 -0.2–0.2 -0.5–0.4 -27.8–2.8 -0.5–0.4 -0.5–0.4 -0.5–0.4 -0.5–0.4 -0.5–0.4 -0.5–0.4 -0.5–0.4
3 -0.4–0.5 -0.5–0.4 -0.0–0.0 -27.9–2.7 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1
4 -27.8–2.7 -27.8–2.8 -27.9–2.7 96.9–114.2 -0.1–0.2 -1.0–2.4 -0.7–0.5 -0.5–0.5 -0.3–0.5 -0.8–1.3 -0.3–0.2
5 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.1–0.2 -0.0–0.0 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.4–0.5 -0.5–0.4 -0.0–0.1 -1.0–2.4 -0.1–0.1 -0.5–0.8 -1.0–1.6 -1.0–1.6 -1.0–1.6 -1.6–1.0 -1.0–1.6
7 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.7–0.5 -0.1–0.1 -1.0–1.6 -0.2–0.1 -0.3–0.5 -0.3–0.5 -0.3–0.5 -0.3–0.5
8 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.5–0.5 -0.1–0.1 -1.0–1.6 -0.3–0.5 -0.1–0.2 -0.4–0.3 -0.4–0.3 -0.4–0.3
9 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.3–0.5 -0.1–0.1 -1.0–1.6 -0.3–0.5 -0.4–0.3 -0.2–0.1 -0.2–0.3 -0.2–0.3
10 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.8–1.3 -0.1–0.1 -1.6–1.0 -0.3–0.5 -0.4–0.3 -0.2–0.3 -0.6–0.2 -0.2–1.3
11 -0.4–0.5 -0.5–0.4 -0.0–0.1 -0.3–0.2 -0.1–0.1 -1.0–1.6 -0.3–0.5 -0.4–0.3 -0.2–0.3 -0.2–1.3 -0.0–0.1
Total -1.1–12.5 -1.2–12.4 -1.4–12.2 98.6–99.7 -1.4–12.2 0.1–13.6 -1.4–12.3 -1.3–12.4 -1.2–12.4 -0.7–12.9 -1.4–12.2
Higher 18.0 18.3 18.0 30.3 5.3 5.0 4.9 5.2 5.0 5.7 4.5
Table F.45: North / Temp. at 5cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.3–0.4 -0.7–0.8 -0.7–0.7 -26.4–3.0 -0.7–0.7 -0.6–0.8 -0.7–0.8 -0.7–0.8 -0.7–0.8 -0.7–0.8 -0.7–0.8
2 -0.7–0.8 -0.5–0.3 -0.5–1.0 -26.0–3.2 -0.5–1.0 -0.4–1.1 -0.5–1.0 -0.5–1.1 -0.5–1.0 -0.5–1.1 -0.5–1.0
3 -0.7–0.7 -0.5–1.0 -0.1–0.0 -26.4–2.9 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -26.4–3.0 -26.0–3.2 -26.4–2.9 93.4–109.9 -0.2–0.2 -1.1–3.9 -0.5–0.6 -0.8–0.6 -0.5–0.6 -1.2–2.3 -0.3–0.2
5 -0.7–0.7 -0.5–1.0 -0.1–0.1 -0.2–0.2 -0.1–0.1 -0.1–0.2 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.6–0.8 -0.4–1.1 -0.1–0.1 -1.1–3.9 -0.1–0.2 0.5–2.5 -2.8–1.2 -2.8–1.2 -2.8–1.2 -2.6–1.4 -2.8–1.2
7 -0.7–0.8 -0.5–1.0 -0.1–0.1 -0.5–0.6 -0.1–0.1 -2.8–1.2 -0.3–0.2 -0.5–0.3 -0.5–0.3 -0.5–0.3 -0.5–0.3
8 -0.7–0.8 -0.5–1.1 -0.1–0.1 -0.8–0.6 -0.1–0.1 -2.8–1.2 -0.5–0.3 -0.3–0.2 -0.4–0.6 -0.4–0.6 -0.4–0.6
9 -0.7–0.8 -0.5–1.0 -0.1–0.1 -0.5–0.6 -0.1–0.1 -2.8–1.2 -0.5–0.3 -0.4–0.6 -0.2–0.2 -0.4–0.3 -0.4–0.3
10 -0.7–0.8 -0.5–1.1 -0.1–0.1 -1.2–2.3 -0.1–0.1 -2.6–1.4 -0.5–0.3 -0.4–0.6 -0.4–0.3 -0.5–0.7 -0.6–2.0
11 -0.7–0.8 -0.5–1.0 -0.1–0.1 -0.3–0.2 -0.1–0.1 -2.8–1.2 -0.5–0.3 -0.4–0.6 -0.4–0.3 -0.6–2.0 -0.1–0.1
Total -1.3–12.2 -1.3–12.2 -1.9–11.8 96.5–98.5 -1.9–11.8 1.2–14.5 -1.7–11.9 -1.6–12.0 -1.7–11.9 -0.1–13.3 -1.8–11.8
Higher 16.7 14.6 16.3 28.8 4.6 8.2 6.0 5.5 5.5 5.4 4.9
433
F.5 Snow Temperature at 8 cm
Table F.46: Control / Temp. at 8cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 0.9–9.4 -0.2–8.3 -1.0–7.7 97.6–98.1 -1.0–7.6 0.2–8.6 -1.0–7.6 -1.0–7.6 -0.9–7.7 -0.7–7.8 -0.9–7.6
0.67 3.4–11.9 1.4–10.0 -1.0–7.9 92.6–93.7 -1.0–7.8 2.8–11.4 -1.0–7.8 -1.0–7.8 -0.7–8.2 -0.2–8.6 -1.0–7.8
1.00 4.8–13.3 2.8–11.5 -1.1–7.8 87.1–88.7 -1.2–7.8 5.8–14.3 -1.2–7.8 -1.2–7.7 -0.4–8.6 0.5–9.4 -1.2–7.8
1.33 5.5–14.0 4.0–12.7 -1.4–7.6 81.6–83.8 -1.5–7.5 8.6–16.9 -1.4–7.6 -1.4–7.5 -0.1–8.8 1.1–10.0 -1.4–7.6
1.67 6.0–14.5 5.0–13.7 -1.7–7.3 76.5–79.2 -1.8–7.3 11.0–19.1 -1.6–7.4 -1.7–7.3 -0.1–8.8 1.7–10.5 -1.7–7.3
2.00 6.3–14.8 6.0–14.5 -2.1–7.0 71.8–74.9 -2.1–7.0 12.8–20.7 -1.7–7.4 -1.8–7.2 -0.2–8.7 2.2–10.9 -2.1–7.0
2.33 6.7–15.2 6.8–15.3 -2.4–6.7 67.5–71.0 -2.4–6.7 14.0–21.8 -1.5–7.5 -1.8–7.2 -0.4–8.6 2.5–11.2 -2.4–6.7
2.67 7.2–15.5 7.5–16.0 -2.7–6.3 63.6–67.4 -2.6–6.5 14.6–22.4 -1.2–7.8 -1.5–7.5 -0.6–8.3 2.6–11.3 -2.7–6.3
3.00 7.5–15.9 8.2–16.6 -3.1–5.9 60.0–64.2 -2.8–6.3 14.9–22.7 -0.5–8.3 -1.0–7.8 -0.9–8.0 2.6–11.2 -3.1–5.9
3.33 7.7–16.0 8.9–17.1 -3.5–5.5 56.9–61.4 -2.9–6.1 15.0–22.7 0.2–9.0 -0.5–8.3 -1.2–7.7 2.5–11.1 -3.4–5.6
3.67 7.9–16.1 9.4–17.6 -3.8–5.2 54.2–58.9 -2.9–6.0 14.8–22.6 1.0–9.7 0.0–8.8 -1.5–7.4 2.5–11.0 -3.8–5.2
4.00 7.9–16.0 9.9–18.0 -4.3–4.8 51.8–56.6 -3.0–6.0 14.6–22.5 1.7–10.4 0.6–9.3 -1.9–7.1 2.3–10.9 -4.2–4.8
4.33 7.7–15.9 10.2–18.4 -4.6–4.5 49.6–54.7 -3.0–6.0 14.5–22.4 2.6–11.2 1.1–9.8 -2.2–6.8 2.3–10.8 -4.6–4.5
4.67 7.4–15.7 10.4–18.7 -5.0–4.2 47.6–52.9 -2.9–6.1 14.5–22.3 3.3–11.9 1.6–10.2 -2.4–6.6 2.2–10.7 -4.9–4.2
5.00 7.0–15.4 10.5–18.8 -5.3–3.9 45.9–51.4 -2.9–6.1 14.4–22.4 4.0–12.6 2.1–10.7 -2.7–6.4 2.1–10.7 -5.3–3.9
5.33 6.6–15.0 10.6–18.8 -5.7–3.6 44.3–50.0 -3.0–6.1 14.5–22.5 4.6–13.2 2.4–11.1 -2.9–6.2 2.0–10.7 -5.7–3.6
5.67 6.1–14.6 10.4–18.8 -3.4–5.7 42.8–48.7 -3.0–6.1 14.8–22.8 5.1–13.7 2.7–11.5 -3.1–6.1 2.0–10.7 -3.4–5.8
6.00 5.5–14.1 10.2–18.6 -3.8–5.5 41.4–47.4 -3.1–6.1 15.3–23.3 5.6–14.2 3.1–11.8 -3.3–6.0 2.0–10.8 -3.7–5.5
6.33 4.8–13.5 9.9–18.4 -4.1–5.2 40.1–46.3 -3.3–6.0 15.8–23.9 5.9–14.6 3.4–12.2 -3.5–5.8 2.1–11.0 -4.0–5.3
6.67 3.9–12.8 9.5–18.1 -4.4–5.0 38.9–45.1 -3.5–5.9 16.6–24.6 6.2–14.8 3.6–12.4 -3.7–5.8 2.3–11.2 -4.4–5.0
7.00 3.1–12.1 9.0–17.6 -4.7–4.7 37.6–44.0 -3.8–5.7 17.5–25.6 6.3–15.0 3.7–12.6 -3.8–5.7 2.5–11.5 -4.6–4.8
7.33 2.2–11.3 8.4–17.1 -5.0–4.6 36.4–42.9 -4.2–5.4 18.7–26.8 6.3–15.0 3.7–12.6 -3.9–5.7 2.9–11.8 -4.9–4.7
7.67 1.4–10.5 7.6–16.5 -5.2–4.4 35.2–41.9 -4.6–5.1 20.3–28.2 5.9–14.8 3.4–12.4 -3.9–5.8 3.3–12.3 -5.1–4.5
8.00 0.4–9.8 6.8–15.8 -5.5–4.2 34.0–40.8 -5.1–4.7 22.1–29.9 5.4–14.2 2.8–12.0 -4.0–5.8 3.8–12.8 -5.4–4.3
8.33 -0.4–9.0 6.1–15.1 -5.6–4.1 32.9–39.8 -5.5–4.4 24.3–32.0 4.4–13.4 2.0–11.3 -3.9–6.0 4.4–13.4 -5.5–4.2
8.67 -1.1–8.4 5.3–14.4 -5.8–4.0 31.7–38.8 -5.9–4.1 26.9–34.4 3.1–12.3 0.9–10.3 -3.7–6.1 5.0–14.1 -5.6–4.1
9.00 -1.7–7.9 4.5–13.7 -5.9–4.0 30.6–37.7 -6.2–3.8 29.8–37.1 1.6–11.0 -0.4–9.1 -3.5–6.4 5.8–14.8 -5.7–4.1
9.33 -2.1–7.6 3.8–13.2 -5.9–3.9 29.4–36.7 -3.6–6.1 32.9–40.0 0.1–9.6 -1.8–7.9 -3.2–6.7 6.6–15.6 -5.7–4.1
9.67 -2.4–7.4 3.3–12.6 -5.9–4.0 28.1–35.5 -3.9–5.8 36.0–42.9 -1.3–8.3 -3.0–6.8 -2.9–7.0 7.3–16.3 -5.7–4.2
10.00 -2.6–7.1 2.7–12.1 -5.8–4.1 26.9–34.4 -4.1–5.6 38.9–45.6 -2.5–7.2 -4.0–5.9 -2.4–7.4 8.0–16.9 -5.6–4.3
Table F.47: Control / Temp. at 8cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 6.7–8.2 -1.1–1.6 -1.1–1.0 -2.3–3.8 -1.4–0.8 -1.1–1.8 -1.4–1.0 -1.0–1.4 -1.0–1.2 -0.9–1.5 -1.1–1.0
2 -1.1–1.6 7.7–9.6 -1.8–0.8 -3.7–2.7 -1.7–0.9 -1.5–2.0 -1.3–1.6 -1.7–1.1 -1.9–0.8 -1.4–1.4 -1.8–0.8
3 -1.1–1.0 -1.8–0.8 -0.1–0.1 -4.3–0.8 -0.1–0.1 -0.3–0.8 -0.0–0.3 -0.0–0.2 -0.1–0.2 -0.0–0.2 -0.1–0.2
4 -2.3–3.8 -3.7–2.7 -4.3–0.8 37.9–42.2 -1.1–0.5 -2.3–2.6 -0.8–2.1 -1.0–1.7 -1.1–0.8 -0.7–2.0 -0.8–0.5
5 -1.4–0.8 -1.7–0.9 -0.1–0.1 -1.1–0.5 1.2–1.9 -0.8–1.0 -0.9–0.5 -1.0–0.4 -1.1–0.2 -1.0–0.4 -1.1–0.2
6 -1.1–1.8 -1.5–2.0 -0.3–0.8 -2.3–2.6 -0.8–1.0 10.6–12.7 -1.6–1.4 -1.8–1.0 -2.5–0.4 -1.5–1.3 -3.0–-0.1
7 -1.4–1.0 -1.3–1.6 -0.0–0.3 -0.8–2.1 -0.9–0.5 -1.6–1.4 2.7–4.0 -0.4–2.0 -0.7–1.6 -0.1–2.2 -0.6–1.6
8 -1.0–1.4 -1.7–1.1 -0.0–0.2 -1.0–1.7 -1.0–0.4 -1.8–1.0 -0.4–2.0 2.7–4.0 -0.9–1.3 -1.2–1.0 -1.0–1.2
9 -1.0–1.2 -1.9–0.8 -0.1–0.2 -1.1–0.8 -1.1–0.2 -2.5–0.4 -0.7–1.6 -0.9–1.3 -0.4–0.5 -0.4–1.3 -0.8–0.9
10 -0.9–1.5 -1.4–1.4 -0.0–0.2 -0.7–2.0 -1.0–0.4 -1.5–1.3 -0.1–2.2 -1.2–1.0 -0.4–1.3 2.0–3.1 -0.8–1.3
11 -1.1–1.0 -1.8–0.8 -0.1–0.2 -0.8–0.5 -1.1–0.2 -3.0–-0.1 -0.6–1.6 -1.0–1.2 -0.8–0.9 -0.8–1.3 -0.1–0.1
Total 7.0–15.4 10.5–18.8 -5.3–3.9 45.9–51.4 -2.9–6.1 14.4–22.4 4.0–12.6 2.1–10.7 -2.7–6.4 2.1–10.7 -5.3–3.9
Higher 2.3 8.1 0.8 8.7 2.6 8.9 1.6 2.2 2.7 1.5 1.0
434
Table F.48: South / Temp. at 8cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 2.0–14.9 0.8–13.6 -0.5–12.6 96.6–97.3 -0.5–12.6 0.8–13.7 -0.5–12.6 -0.5–12.6 -0.4–12.6 -0.4–12.7 -0.5–12.6
0.67 5.7–18.0 3.0–15.7 -0.5–12.5 89.7–91.4 -0.5–12.5 3.7–16.2 -0.5–12.5 -0.5–12.5 -0.4–12.6 0.1–13.0 -0.5–12.5
1.00 7.9–19.8 5.0–17.3 -0.8–12.2 82.5–85.2 -0.8–12.1 6.5–18.6 -0.7–12.2 -0.8–12.1 -0.6–12.3 0.5–13.2 -0.7–12.2
1.33 9.5–20.9 7.0–18.8 -1.2–11.6 75.5–79.0 -1.3–11.5 9.0–20.6 -1.1–11.7 -1.1–11.6 -0.9–11.8 0.5–13.1 -1.2–11.5
1.67 10.8–21.9 8.6–20.0 -1.6–10.9 68.7–73.0 -1.6–10.9 10.6–21.8 -1.1–11.3 -1.3–11.1 -1.3–11.2 0.9–13.0 -1.6–10.9
2.00 12.3–22.8 10.1–21.0 -1.9–10.2 62.3–67.2 -1.9–10.2 11.5–22.3 -0.6–11.3 -1.1–10.9 -1.6–10.5 0.9–12.7 -2.0–10.2
2.33 13.4–23.5 11.5–21.9 -2.4–9.4 56.3–61.8 -2.1–9.7 11.8–22.1 0.1–11.6 -0.6–11.0 -2.0–9.8 0.8–12.2 -2.4–9.4
2.67 14.0–23.9 12.7–22.7 -2.9–8.6 51.1–57.0 -2.3–9.2 11.3–21.5 1.0–12.1 0.1–11.3 -2.4–9.0 0.4–11.5 -2.9–8.6
3.00 14.4–24.0 13.6–23.3 -3.3–7.9 46.5–52.9 -2.3–8.8 10.7–20.7 2.1–12.7 0.9–11.7 -2.9–8.3 0.0–10.9 -3.3–7.8
3.33 14.6–23.9 14.6–24.0 -3.7–7.3 42.8–49.5 -2.2–8.7 10.0–19.8 3.2–13.6 1.6–12.2 -3.2–7.7 -0.3–10.3 -3.7–7.2
3.67 14.6–23.7 15.4–24.6 -4.1–6.7 39.8–46.6 -2.0–8.7 9.3–19.0 4.3–14.3 2.4–12.7 -3.5–7.2 -0.6–9.9 -4.1–6.7
4.00 14.3–23.3 16.2–25.1 -4.4–6.3 37.2–44.2 -1.7–8.8 8.7–18.4 5.2–15.1 3.1–13.1 -3.8–6.8 -0.8–9.5 -4.4–6.3
4.33 14.0–22.9 16.8–25.5 -4.7–5.9 35.1–42.3 -1.3–9.1 8.3–17.9 6.1–15.7 3.6–13.5 -4.2–6.4 -1.1–9.1 -4.8–5.9
4.67 13.5–22.5 17.1–25.8 -5.1–5.5 33.3–40.6 -1.0–9.3 7.9–17.5 6.8–16.3 4.0–13.9 -4.5–6.0 -1.3–8.8 -5.1–5.5
5.00 12.9–21.9 17.3–26.0 -5.5–5.1 31.6–39.1 -0.8–9.4 7.7–17.2 7.4–16.8 4.4–14.2 -4.8–5.7 -1.6–8.5 -5.5–5.1
5.33 12.3–21.3 17.4–26.1 -5.8–4.8 30.2–37.9 -0.5–9.6 7.5–17.1 7.9–17.2 4.8–14.5 -5.1–5.4 -1.9–8.4 -5.8–4.8
5.67 11.6–20.6 17.3–26.0 -6.1–4.5 29.0–36.8 -0.4–9.7 7.5–17.2 8.3–17.7 5.2–14.8 -5.4–5.2 -2.0–8.2 -6.2–4.6
6.00 10.7–19.9 17.1–25.9 -6.5–4.3 28.0–35.9 -0.4–9.8 7.7–17.4 8.7–18.2 5.4–15.2 -5.6–5.0 -2.1–8.1 -6.4–4.3
6.33 9.8–19.2 16.8–25.6 -6.7–4.0 27.1–35.1 -0.4–9.8 8.1–17.8 9.2–18.6 5.8–15.5 -5.9–4.8 -2.2–8.1 -6.7–4.1
6.67 9.0–18.4 16.4–25.3 -3.9–6.4 26.3–34.4 -0.5–9.7 8.6–18.3 9.6–19.1 6.1–15.9 -6.1–4.6 -2.2–8.2 -3.8–6.5
7.00 8.0–17.5 15.8–24.8 -4.2–6.2 25.6–33.8 -0.8–9.6 9.4–19.1 10.0–19.5 6.4–16.2 -6.3–4.5 -2.2–8.2 -4.1–6.3
7.33 6.8–16.6 15.0–24.1 -4.4–6.0 24.9–33.2 -1.0–9.3 10.4–20.0 10.3–19.8 6.7–16.5 -6.5–4.3 -2.2–8.4 -4.3–6.1
7.67 5.6–15.6 14.1–23.4 -4.7–5.9 24.3–32.7 -1.5–9.0 11.6–21.2 10.5–20.0 6.9–16.7 -6.8–4.2 -2.1–8.6 -4.6–6.0
8.00 4.4–14.5 13.0–22.5 -5.0–5.7 23.8–32.2 -1.9–8.7 13.1–22.7 10.3–19.8 6.8–16.7 -7.0–4.1 -1.9–8.8 -4.9–5.8
8.33 3.2–13.5 11.9–21.5 -5.3–5.6 23.3–31.9 -2.5–8.2 15.1–24.6 9.8–19.4 6.3–16.3 -4.1–6.7 -1.6–9.1 -5.1–5.8
8.67 1.9–12.5 10.6–20.6 -5.6–5.5 23.0–31.8 -3.2–7.8 17.8–27.1 8.7–18.5 5.4–15.6 -4.2–6.7 -1.2–9.6 -5.4–5.7
9.00 0.8–11.7 9.4–19.6 -5.8–5.5 22.9–31.7 -3.9–7.4 21.0–30.2 7.0–17.2 3.9–14.4 -4.4–6.8 -0.8–10.2 -5.6–5.7
9.33 -0.3–11.0 8.3–18.8 -5.9–5.5 22.7–31.8 -4.6–6.9 24.9–33.8 4.9–15.5 2.1–13.0 -4.5–6.9 -0.1–11.0 -5.8–5.7
9.67 -1.0–10.6 7.3–18.0 -6.1–5.6 22.5–31.8 -5.1–6.6 29.0–37.6 2.6–13.6 0.3–11.6 -4.4–7.1 0.6–11.8 -5.8–5.9
10.00 -1.5–10.2 6.2–17.2 -6.0–5.8 22.2–31.6 -5.6–6.4 33.0–41.3 0.7–12.0 -1.3–10.2 -7.8–4.4 1.4–12.7 -5.8–6.0
Table F.49: South / Temp. at 8cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 12.8–15.0 -2.4–1.9 -1.0–1.1 -2.9–3.2 -1.6–0.8 -1.9–1.1 -1.3–1.6 -1.5–1.2 -1.0–1.3 -1.5–0.9 -1.0–1.1
2 -2.4–1.9 14.4–17.0 -1.4–1.3 -2.8–3.6 -1.2–1.5 -1.5–2.1 -1.3–2.0 -2.0–1.2 -1.4–1.3 -1.8–1.1 -1.4–1.3
3 -1.0–1.1 -1.4–1.3 -0.0–0.1 -3.4–0.7 -0.2–0.1 -0.2–0.5 -0.1–0.4 -0.1–0.4 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -2.9–3.2 -2.8–3.6 -3.4–0.7 27.6–31.4 -1.2–0.6 -1.4–2.0 -0.9–2.1 -0.8–1.8 -0.6–0.7 -0.7–1.2 -0.7–0.5
5 -1.6–0.8 -1.2–1.5 -0.2–0.1 -1.2–0.6 2.5–3.6 -1.3–0.7 -1.0–1.3 -1.2–1.0 -1.5–0.5 -1.4–0.6 -1.5–0.4
6 -1.9–1.1 -1.5–2.1 -0.2–0.5 -1.4–2.0 -1.3–0.7 6.7–8.6 -0.6–2.1 -0.9–1.8 -2.0–0.8 -1.6–1.2 -2.1–0.7
7 -1.3–1.6 -1.3–2.0 -0.1–0.4 -0.9–2.1 -1.0–1.3 -0.6–2.1 5.4–7.2 -0.2–2.9 -0.6–2.3 -0.0–2.8 -0.5–2.3
8 -1.5–1.2 -2.0–1.2 -0.1–0.4 -0.8–1.8 -1.2–1.0 -0.9–1.8 -0.2–2.9 4.8–6.4 -1.0–1.6 -1.5–1.1 -1.1–1.6
9 -1.0–1.3 -1.4–1.3 -0.1–0.1 -0.6–0.7 -1.5–0.5 -2.0–0.8 -0.6–2.3 -1.0–1.6 -0.3–0.2 -0.4–0.5 -0.5–0.4
10 -1.5–0.9 -1.8–1.1 -0.1–0.2 -0.7–1.2 -1.4–0.6 -1.6–1.2 -0.0–2.8 -1.5–1.1 -0.4–0.5 0.8–1.6 -0.4–1.2
11 -1.0–1.1 -1.4–1.3 -0.1–0.1 -0.7–0.5 -1.5–0.4 -2.1–0.7 -0.5–2.3 -1.1–1.6 -0.5–0.4 -0.4–1.2 -0.0–0.1
Total 12.9–21.9 17.3–26.0 -5.5–5.1 31.6–39.1 -0.8–9.4 7.7–17.2 7.4–16.8 4.4–14.2 -4.8–5.7 -1.6–8.5 -5.5–5.1
Higher 4.5 6.0 0.7 5.5 3.5 5.1 -0.8 1.6 0.3 1.4 -0.3
435
Table F.50: North / Temp. at 8cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 0.2–12.4 -0.4–11.8 -0.8–11.5 98.8–99.2 -0.7–11.5 -0.1–12.0 -0.7–11.5 -0.7–11.5 -0.8–11.5 -0.7–11.5 -0.7–11.5
0.67 1.6–14.0 0.5–13.0 -0.8–11.8 96.2–97.2 -0.7–11.8 1.3–13.6 -0.8–11.8 -0.7–11.8 -0.7–11.8 -0.5–12.1 -0.8–11.8
1.00 2.7–15.3 1.4–14.0 -1.0–11.9 93.0–94.5 -1.0–11.9 3.1–15.5 -1.0–11.9 -1.0–11.9 -0.9–12.0 -0.4–12.5 -1.0–11.9
1.33 3.4–16.1 2.0–14.8 -1.1–11.9 89.4–91.5 -1.2–11.9 4.9–17.4 -1.1–12.0 -1.2–11.9 -1.1–12.0 -0.0–12.9 -1.2–11.9
1.67 4.0–16.6 2.5–15.4 -1.5–11.7 85.6–88.2 -1.5–11.7 6.8–19.2 -1.4–11.8 -1.5–11.8 -1.3–11.9 0.1–13.3 -1.5–11.8
2.00 4.6–17.1 3.1–16.0 -1.8–11.5 81.6–84.7 -1.9–11.5 8.6–20.8 -1.7–11.7 -1.8–11.6 -1.6–11.7 0.5–13.6 -1.8–11.5
2.33 5.2–17.6 3.7–16.4 -2.2–11.2 77.3–81.0 -2.2–11.2 10.2–22.2 -1.7–11.6 -1.8–11.5 -1.9–11.4 0.8–13.8 -2.1–11.2
2.67 5.8–18.1 4.2–16.8 -2.3–10.9 73.0–77.1 -2.3–10.9 11.6–23.2 -1.6–11.6 -1.6–11.5 -2.0–11.1 1.0–13.8 -2.3–10.9
3.00 6.6–18.6 4.8–17.1 -2.6–10.5 68.6–73.2 -2.5–10.5 12.6–24.0 -1.2–11.7 -1.2–11.7 -2.3–10.7 1.3–13.8 -2.5–10.5
3.33 7.4–19.1 5.2–17.2 -2.8–10.0 64.3–69.3 -2.6–10.2 13.3–24.4 -0.7–11.9 -0.5–12.0 -2.5–10.3 1.4–13.7 -2.8–10.0
3.67 8.1–19.4 5.7–17.4 -3.0–9.6 60.2–65.6 -2.7–9.8 13.8–24.6 0.1–12.3 0.4–12.6 -2.7–9.8 1.5–13.6 -2.9–9.6
4.00 8.5–19.6 6.1–17.5 -3.1–9.1 56.3–62.1 -2.6–9.5 14.2–24.7 0.9–12.7 1.5–13.2 -2.8–9.4 1.6–13.4 -3.1–9.1
4.33 8.9–19.6 6.3–17.5 -3.3–8.7 52.8–58.8 -2.7–9.2 14.4–24.6 1.6–13.1 2.6–13.9 -3.0–9.0 1.7–13.2 -3.3–8.7
4.67 9.0–19.5 6.4–17.4 -3.5–8.2 49.6–55.9 -2.8–8.9 14.5–24.5 2.3–13.5 3.6–14.6 -3.2–8.5 1.7–12.9 -3.5–8.2
5.00 8.9–19.2 6.5–17.2 -3.9–7.8 46.8–53.3 -2.9–8.6 14.7–24.4 3.0–13.9 4.6–15.3 -3.5–8.1 1.6–12.6 -3.9–7.8
5.33 8.4–18.7 6.4–17.0 -4.1–7.4 44.3–51.0 -3.0–8.4 14.8–24.4 3.6–14.3 5.5–15.9 -3.7–7.7 1.6–12.4 -4.1–7.4
5.67 7.9–18.2 6.3–16.7 -4.4–7.0 42.2–49.0 -3.0–8.2 15.0–24.5 4.1–14.6 6.3–16.5 -4.0–7.3 1.6–12.3 -4.3–7.0
6.00 7.3–17.5 6.0–16.4 -4.7–6.6 40.3–47.4 -3.1–7.9 15.4–24.7 4.5–14.9 7.0–17.0 -4.3–7.0 1.6–12.2 -4.6–6.7
6.33 6.6–16.8 5.8–16.1 -4.8–6.4 38.8–45.9 -3.3–7.8 15.9–25.2 4.8–15.1 7.4–17.5 -4.5–6.7 1.6–12.2 -4.8–6.4
6.67 5.7–16.0 5.4–15.8 -5.1–6.1 37.5–44.8 -3.4–7.6 16.6–25.8 5.0–15.2 7.8–17.7 -4.7–6.5 1.7–12.3 -5.1–6.2
7.00 4.8–15.2 5.1–15.5 -5.4–5.9 36.4–43.8 -3.6–7.5 17.6–26.7 4.9–15.3 7.9–17.9 -4.8–6.4 1.8–12.4 -5.3–6.0
7.33 3.9–14.4 4.7–15.2 -5.5–5.8 35.7–43.1 -3.8–7.3 18.8–27.8 4.8–15.2 7.6–17.8 -5.0–6.3 2.0–12.6 -5.5–5.9
7.67 2.9–13.6 4.3–14.9 -5.8–5.7 35.0–42.6 -4.0–7.3 20.2–29.2 4.4–15.0 7.2–17.5 -5.2–6.2 2.2–12.9 -5.7–5.8
8.00 1.9–12.9 3.9–14.6 -5.9–5.7 34.6–42.3 -4.2–7.2 21.8–30.8 3.8–14.5 6.5–16.9 -5.4–6.2 2.5–13.3 -5.9–5.7
8.33 1.0–12.2 3.5–14.5 -6.1–5.7 34.3–42.1 -4.5–7.2 23.7–32.6 3.1–14.0 5.5–16.2 -5.5–6.2 2.9–13.8 -6.0–5.8
8.67 0.2–11.7 3.0–14.3 -6.3–5.7 34.1–42.1 -4.8–7.1 25.6–34.6 2.2–13.3 4.3–15.3 -5.6–6.3 3.2–14.3 -6.1–5.8
9.00 -0.5–11.3 2.7–14.2 -6.5–5.8 33.8–42.0 -5.0–7.1 27.8–36.7 1.1–12.5 3.0–14.4 -5.7–6.5 3.7–14.9 -6.3–5.9
9.33 -1.0–11.1 2.2–14.1 -6.5–5.9 33.7–42.0 -5.3–7.2 30.0–38.8 0.0–11.8 1.7–13.4 -5.8–6.6 4.1–15.4 -6.3–6.1
9.67 -1.5–10.9 1.9–14.1 -6.7–6.1 33.4–41.9 -5.6–7.2 32.1–40.8 -1.1–11.1 0.4–12.6 -5.9–6.8 4.5–16.0 -6.5–6.2
10.00 -1.8–10.7 1.5–14.0 -6.7–6.2 33.0–41.7 -5.7–7.2 34.1–42.7 -2.1–10.4 -0.7–11.8 -5.9–7.0 4.9–16.5 -6.6–6.3
Table F.51: North / Temp. at 8cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 8.0–10.3 -2.5–1.1 -2.7–0.8 -7.8–2.0 -3.0–0.6 -1.9–2.4 -2.3–1.4 -2.4–1.3 -2.8–0.7 -2.6–1.0 -2.7–0.7
2 -2.5–1.1 3.8–5.9 -2.2–1.5 -6.2–3.1 -2.1–1.5 -0.7–3.5 -1.9–1.8 -2.2–1.5 -2.3–1.4 -1.9–1.8 -2.2–1.5
3 -2.7–0.8 -2.2–1.5 -0.1–0.1 -7.7–0.4 -0.1–0.2 -0.6–1.0 -0.2–0.3 -0.2–0.4 -0.1–0.3 -0.1–0.3 -0.1–0.2
4 -7.8–2.0 -6.2–3.1 -7.7–0.4 42.4–48.7 -1.4–0.6 -3.2–3.2 -1.9–1.4 -1.5–2.4 -0.7–1.2 -1.5–1.9 -0.7–1.0
5 -3.0–0.6 -2.1–1.5 -0.1–0.2 -1.4–0.6 0.4–1.1 -0.8–1.3 -0.9–0.7 -0.9–0.7 -1.0–0.5 -0.9–0.6 -1.0–0.5
6 -1.9–2.4 -0.7–3.5 -0.6–1.0 -3.2–3.2 -0.8–1.3 14.0–16.7 -2.0–1.3 -2.0–1.3 -1.4–1.8 -2.2–1.2 -1.5–1.7
7 -2.3–1.4 -1.9–1.8 -0.2–0.3 -1.9–1.4 -0.9–0.7 -2.0–1.3 3.6–5.2 -1.6–1.6 -1.8–1.1 -1.7–1.1 -1.8–1.1
8 -2.4–1.3 -2.2–1.5 -0.2–0.4 -1.5–2.4 -0.9–0.7 -2.0–1.3 -1.6–1.6 6.1–8.0 -2.9–0.3 -2.8–0.4 -2.9–0.3
9 -2.8–0.7 -2.3–1.4 -0.1–0.3 -0.7–1.2 -1.0–0.5 -1.4–1.8 -1.8–1.1 -2.9–0.3 -0.3–0.3 -0.7–0.4 -0.6–0.5
10 -2.6–1.0 -1.9–1.8 -0.1–0.3 -1.5–1.9 -0.9–0.6 -2.2–1.2 -1.7–1.1 -2.8–0.4 -0.7–0.4 3.6–4.9 -0.8–1.4
11 -2.7–0.7 -2.2–1.5 -0.1–0.2 -0.7–1.0 -1.0–0.5 -1.5–1.7 -1.8–1.1 -2.9–0.3 -0.6–0.5 -0.8–1.4 -0.0–0.2
Total 8.9–19.2 6.5–17.2 -3.9–7.8 46.8–53.3 -2.9–8.6 14.7–24.4 3.0–13.9 4.6–15.3 -3.5–8.1 1.6–12.6 -3.9–7.8
Higher 14.2 9.7 6.3 12.2 4.6 2.8 6.3 7.6 5.4 5.5 4.6
436
Table F.52: Control / Temp. at 8cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 3.8–5.0 -1.3–1.0 -1.2–0.9 -6.1–3.1 -1.4–0.7 -1.4–1.4 -1.1–1.2 -1.2–1.0 -1.2–1.0 -1.0–1.3 -1.2–0.9
2 -1.3–1.0 4.0–5.6 -1.5–1.1 -5.7–3.5 -1.5–1.2 -0.8–2.4 -1.3–1.4 -1.6–1.1 -1.7–1.0 -1.2–1.6 -1.6–1.1
3 -1.2–0.9 -1.5–1.1 -0.1–0.0 -7.5–0.9 -0.1–0.1 -0.3–0.9 -0.0–0.2 -0.1–0.2 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -6.1–3.1 -5.7–3.5 -7.5–0.9 54.2–60.3 -1.3–0.3 -2.7–3.8 -0.8–2.3 -1.2–1.7 -1.1–1.1 -0.8–2.7 -0.6–0.4
5 -1.4–0.7 -1.5–1.2 -0.1–0.1 -1.3–0.3 0.5–0.9 -0.8–0.7 -0.6–0.5 -0.6–0.4 -0.6–0.3 -0.5–0.4 -0.6–0.3
6 -1.4–1.4 -0.8–2.4 -0.3–0.9 -2.7–3.8 -0.8–0.7 11.0–13.3 -1.4–1.7 -1.7–1.4 -2.4–0.9 -1.5–1.6 -2.8–0.4
7 -1.1–1.2 -1.3–1.4 -0.0–0.2 -0.8–2.3 -0.6–0.5 -1.4–1.7 1.2–2.1 -0.4–1.4 -0.5–1.2 -0.2–1.5 -0.5–1.3
8 -1.2–1.0 -1.6–1.1 -0.1–0.2 -1.2–1.7 -0.6–0.4 -1.7–1.4 -0.4–1.4 1.4–2.3 -0.7–1.0 -1.0–0.6 -0.7–0.9
9 -1.2–1.0 -1.7–1.0 -0.1–0.1 -1.1–1.1 -0.6–0.3 -2.4–0.9 -0.5–1.2 -0.7–1.0 -0.5–0.5 -0.4–1.4 -0.8–1.1
10 -1.0–1.3 -1.2–1.6 -0.1–0.2 -0.8–2.7 -0.5–0.4 -1.5–1.6 -0.2–1.5 -1.0–0.6 -0.4–1.4 2.2–3.4 -1.1–1.2
11 -1.2–0.9 -1.6–1.1 -0.1–0.1 -0.6–0.4 -0.6–0.3 -2.8–0.4 -0.5–1.3 -0.7–0.9 -0.8–1.1 -1.1–1.2 -0.1–0.1
Total 2.9–12.1 5.2–14.2 -4.9–4.8 60.8–65.1 -3.9–5.8 13.4–21.8 -0.2–9.2 -1.2–8.2 -2.5–7.0 1.6–10.9 -4.9–4.9
Higher 5.5 6.2 3.2 9.7 1.8 5.8 -0.2 1.3 2.5 1.0 1.2
Table F.53: South / Temp. at 8cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 8.7–10.6 -1.5–1.8 -1.3–1.0 -8.0–2.8 -1.7–0.7 -1.7–1.6 -1.6–1.1 -1.1–1.5 -1.3–1.1 -1.1–1.4 -1.3–1.0
2 -1.5–1.8 8.7–11.0 -1.2–1.8 -7.7–3.1 -1.9–1.1 -0.8–3.1 -1.5–1.8 -2.0–1.2 -1.2–1.8 -1.6–1.5 -1.2–1.8
3 -1.3–1.0 -1.2–1.8 -0.0–0.1 -8.1–0.9 -0.1–0.1 -0.3–0.7 -0.1–0.3 -0.1–0.3 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -8.0–2.8 -7.7–3.1 -8.1–0.9 45.1–52.0 -1.8–0.3 -2.9–2.9 -1.2–2.7 -1.1–2.3 -0.5–0.9 -0.7–2.0 -0.7–0.4
5 -1.7–0.7 -1.9–1.1 -0.1–0.1 -1.8–0.3 1.1–2.0 -0.7–1.1 -0.7–1.1 -0.8–0.9 -1.0–0.6 -0.9–0.8 -1.0–0.6
6 -1.7–1.6 -0.8–3.1 -0.3–0.7 -2.9–2.9 -0.7–1.1 8.6–11.1 -1.5–1.9 -1.8–1.6 -2.1–1.4 -1.6–1.8 -2.2–1.3
7 -1.6–1.1 -1.5–1.8 -0.1–0.3 -1.2–2.7 -0.7–1.1 -1.5–1.9 3.1–4.5 -0.3–2.4 -0.4–2.0 -0.1–2.3 -0.4–2.0
8 -1.1–1.5 -2.0–1.2 -0.1–0.3 -1.1–2.3 -0.8–0.9 -1.8–1.6 -0.3–2.4 2.9–4.2 -0.9–1.3 -1.4–0.9 -0.9–1.3
9 -1.3–1.1 -1.2–1.8 -0.1–0.1 -0.5–0.9 -1.0–0.6 -2.1–1.4 -0.4–2.0 -0.9–1.3 -0.3–0.2 -0.4–0.6 -0.6–0.4
10 -1.1–1.4 -1.6–1.5 -0.1–0.2 -0.7–2.0 -0.9–0.8 -1.6–1.8 -0.1–2.3 -1.4–0.9 -0.4–0.6 1.3–2.3 -0.9–1.1
11 -1.3–1.0 -1.2–1.8 -0.1–0.1 -0.7–0.4 -1.0–0.6 -2.2–1.3 -0.4–2.0 -0.9–1.3 -0.6–0.4 -0.9–1.1 -0.0–0.1
Total 7.2–18.3 10.0–20.9 -5.4–7.1 49.5–56.0 -3.0–9.2 9.2–20.2 2.5–14.0 0.6–12.5 -4.8–7.6 -1.5–10.6 -5.4–7.1
Higher 6.5 6.3 3.8 11.4 3.2 4.0 -0.4 1.3 0.5 0.8 0.4
Table F.54: North / Temp. at 8cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 3.5–5.4 -2.4–0.9 -2.6–0.7 -13.3–2.1 -2.7–0.6 -1.4–2.6 -2.4–1.0 -2.5–0.8 -2.6–0.7 -2.4–0.9 -2.6–0.7
2 -2.4–0.9 0.9–2.9 -1.9–1.8 -11.9–3.0 -1.8–1.9 -0.3–3.8 -1.8–1.9 -1.9–1.7 -1.9–1.8 -1.5–2.1 -1.9–1.8
3 -2.6–0.7 -1.9–1.8 -0.1–0.1 -13.5–0.8 -0.2–0.1 -0.6–1.1 -0.2–0.2 -0.2–0.2 -0.2–0.1 -0.2–0.2 -0.2–0.1
4 -13.3–2.1 -11.9–3.0 -13.5–0.8 58.8–68.6 -1.6–0.2 -3.7–5.1 -2.2–1.5 -1.6–2.6 -0.7–1.0 -2.2–2.3 -0.6–0.8
5 -2.7–0.6 -1.8–1.9 -0.2–0.1 -1.6–0.2 0.1–0.7 -1.0–0.9 -0.7–0.6 -0.7–0.6 -0.7–0.5 -0.6–0.6 -0.7–0.5
6 -1.4–2.6 -0.3–3.8 -0.6–1.1 -3.7–5.1 -1.0–0.9 12.6–15.6 -2.0–1.4 -2.0–1.5 -1.3–2.3 -2.1–1.4 -1.3–2.2
7 -2.4–1.0 -1.8–1.9 -0.2–0.2 -2.2–1.5 -0.7–0.6 -2.0–1.4 1.9–3.1 -1.3–1.1 -0.8–1.4 -0.8–1.4 -0.8–1.4
8 -2.5–0.8 -1.9–1.7 -0.2–0.2 -1.6–2.6 -0.7–0.6 -2.0–1.5 -1.3–1.1 3.4–4.8 -2.1–0.2 -2.1–0.2 -2.1–0.2
9 -2.6–0.7 -1.9–1.8 -0.2–0.1 -0.7–1.0 -0.7–0.5 -1.3–2.3 -0.8–1.4 -2.1–0.2 -0.3–0.3 -0.7–0.5 -0.5–0.6
10 -2.4–0.9 -1.5–2.1 -0.2–0.2 -2.2–2.3 -0.6–0.6 -2.1–1.4 -0.8–1.4 -2.1–0.2 -0.7–0.5 3.2–4.5 -1.4–0.9
11 -2.6–0.7 -1.9–1.8 -0.2–0.1 -0.6–0.8 -0.7–0.5 -1.3–2.2 -0.8–1.4 -2.1–0.2 -0.5–0.6 -1.4–0.9 -0.0–0.2
Total 3.2–15.4 2.7–15.1 -3.8–9.3 63.3–68.4 -3.4–9.7 13.5–24.5 -0.1–12.5 0.5–13.1 -3.5–9.5 1.1–13.5 -3.8–9.3
Higher 16.8 10.3 9.9 18.0 4.8 1.7 4.3 6.3 4.2 5.3 4.1
437
Table F.55: Control / Temp. at 8cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 4.8–5.9 -0.8–1.1 -0.6–0.9 -2.6–3.3 -0.8–0.8 -1.1–1.5 -0.8–1.2 -1.0–1.0 -0.6–1.0 -0.9–1.0 -0.6–1.0
2 -0.8–1.1 5.6–7.3 -1.6–0.9 -2.7–3.4 -1.2–1.2 -1.2–2.2 -1.0–1.7 -1.5–1.2 -1.6–0.9 -1.2–1.4 -1.6–0.9
3 -0.6–0.9 -1.6–0.9 -0.0–0.1 -4.1–0.9 -0.1–0.1 -0.1–1.0 -0.0–0.3 -0.1–0.2 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -2.6–3.3 -2.7–3.4 -4.1–0.9 36.2–40.5 -0.6–0.8 0.3–5.7 0.6–3.8 -0.1–2.8 -1.0–0.8 0.2–3.0 -0.4–0.4
5 -0.8–0.8 -1.2–1.2 -0.1–0.1 -0.6–0.8 1.3–2.0 -0.9–0.9 -0.6–0.9 -0.8–0.7 -0.9–0.4 -0.7–0.6 -0.9–0.4
6 -1.1–1.5 -1.2–2.2 -0.1–1.0 0.3–5.7 -0.9–0.9 9.6–11.7 -0.2–2.5 -0.7–2.0 -2.2–0.6 -1.1–1.6 -2.5–0.1
7 -0.8–1.2 -1.0–1.7 -0.0–0.3 0.6–3.8 -0.6–0.9 -0.2–2.5 3.1–4.5 0.0–2.7 -0.3–2.1 0.4–2.8 -0.2–2.1
8 -1.0–1.0 -1.5–1.2 -0.1–0.2 -0.1–2.8 -0.8–0.7 -0.7–2.0 0.0–2.7 3.3–4.6 -1.4–0.9 -1.1–1.1 -0.8–1.4
9 -0.6–1.0 -1.6–0.9 -0.1–0.1 -1.0–0.8 -0.9–0.4 -2.2–0.6 -0.3–2.1 -1.4–0.9 -0.2–0.7 -0.6–1.2 -1.1–0.8
10 -0.9–1.0 -1.2–1.4 -0.1–0.2 0.2–3.0 -0.7–0.6 -1.1–1.6 0.4–2.8 -1.1–1.1 -0.6–1.2 1.8–2.9 -0.6–1.4
11 -0.6–1.0 -1.6–0.9 -0.1–0.1 -0.4–0.4 -0.9–0.4 -2.5–0.1 -0.2–2.1 -0.8–1.4 -1.1–0.8 -0.6–1.4 -0.1–0.1
Total 1.9–11.0 5.6–14.8 -4.5–4.8 47.5–53.2 -4.1–5.6 16.4–24.8 5.7–14.8 2.9–12.2 -3.8–6.0 1.9–11.1 -4.3–4.9
Higher -0.4 3.5 1.2 4.9 -0.6 5.7 -2.5 0.3 1.4 -0.1 0.4
Table F.56: South / Temp. at 8cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 8.8–10.3 -0.7–2.1 -1.1–0.8 -1.7–2.9 -0.9–1.1 -1.2–1.7 -0.7–2.2 -0.7–1.9 -1.0–1.0 -0.9–1.3 -1.1–0.8
2 -0.7–2.1 9.6–11.8 -1.2–1.5 -1.8–3.1 -1.2–1.4 -1.1–2.5 -0.9–2.5 -1.6–1.6 -1.2–1.6 -1.5–1.4 -1.2–1.5
3 -1.1–0.8 -1.2–1.5 -0.0–0.1 -2.6–0.7 -0.2–0.1 -0.2–0.7 -0.1–0.8 -0.0–0.6 -0.1–0.1 -0.1–0.2 -0.1–0.1
4 -1.7–2.9 -1.8–3.1 -2.6–0.7 23.4–26.7 -1.1–0.7 -2.0–1.8 -0.5–3.0 -0.5–2.7 -0.7–0.7 -0.5–1.5 -0.8–0.4
5 -0.9–1.1 -1.2–1.4 -0.2–0.1 -1.1–0.7 2.8–3.9 -0.8–1.4 -1.1–1.3 -1.3–1.0 -1.2–0.7 -1.0–0.9 -1.2–0.7
6 -1.2–1.7 -1.1–2.5 -0.2–0.7 -2.0–1.8 -0.8–1.4 8.2–10.3 1.1–4.2 0.3–3.3 -2.1–0.9 -1.5–1.4 -2.2–0.7
7 -0.7–2.2 -0.9–2.5 -0.1–0.8 -0.5–3.0 -1.1–1.3 1.1–4.2 7.8–9.8 0.9–4.3 -0.0–2.9 1.0–3.9 0.0–2.9
8 -0.7–1.9 -1.6–1.6 -0.0–0.6 -0.5–2.7 -1.3–1.0 0.3–3.3 0.9–4.3 6.6–8.4 -1.2–1.6 -0.7–2.0 -1.2–1.5
9 -1.0–1.0 -1.2–1.6 -0.1–0.1 -0.7–0.7 -1.2–0.7 -2.1–0.9 -0.0–2.9 -1.2–1.6 -0.3–0.3 -0.3–0.7 -0.5–0.5
10 -0.9–1.3 -1.5–1.4 -0.1–0.2 -0.5–1.5 -1.0–0.9 -1.5–1.4 1.0–3.9 -0.7–2.0 -0.3–0.7 1.0–1.9 -0.2–1.5
11 -1.1–0.8 -1.2–1.5 -0.1–0.1 -0.8–0.4 -1.2–0.7 -2.2–0.7 0.0–2.9 -1.2–1.5 -0.5–0.5 -0.2–1.5 -0.1–0.1
Total 5.6–15.6 10.1–19.9 -4.8–5.5 26.1–34.3 -2.3–8.4 11.2–20.8 11.5–21.0 7.2–17.1 -7.1–4.0 -2.2–8.3 -4.7–5.6
Higher -1.9 0.8 0.3 2.5 0.0 2.3 -6.4 -2.6 -2.7 -3.0 -0.8
Table F.57: North / Temp. at 8cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 5.6–7.3 -1.7–1.1 -2.0–0.6 -6.4–1.2 -2.2–0.5 -2.4–1.4 -1.6–1.3 -2.0–1.1 -2.1–0.5 -2.1–0.7 -2.1–0.6
2 -1.7–1.1 1.6–3.3 -1.2–1.9 -5.5–1.6 -1.7–1.4 -0.7–3.1 -1.7–1.4 -1.3–1.9 -1.2–1.9 -1.9–1.3 -1.2–1.9
3 -2.0–0.6 -1.2–1.9 -0.1–0.1 -6.3–0.2 -0.2–0.1 -0.5–1.3 -0.2–0.3 -0.4–0.4 -0.1–0.1 -0.2–0.2 -0.1–0.1
4 -6.4–1.2 -5.5–1.6 -6.3–0.2 38.6–43.8 -0.9–0.9 -1.5–4.9 -1.0–2.4 -1.8–2.3 -0.9–0.7 -0.8–2.3 -0.7–0.7
5 -2.2–0.5 -1.7–1.4 -0.2–0.1 -0.9–0.9 0.5–1.3 -1.1–1.3 -0.7–1.1 -0.8–1.1 -0.9–0.7 -0.7–0.8 -0.9–0.8
6 -2.4–1.4 -0.7–3.1 -0.5–1.3 -1.5–4.9 -1.1–1.3 15.5–18.1 -0.9–2.0 -1.2–2.1 -1.5–1.2 -1.3–1.6 -1.5–1.1
7 -1.6–1.3 -1.7–1.4 -0.2–0.3 -1.0–2.4 -0.7–1.1 -0.9–2.0 5.4–7.2 -1.2–2.3 -1.7–1.3 -1.6–1.3 -1.7–1.3
8 -2.0–1.1 -1.3–1.9 -0.4–0.4 -1.8–2.3 -0.8–1.1 -1.2–2.1 -1.2–2.3 8.8–10.9 -2.9–0.3 -2.9–0.3 -2.8–0.3
9 -2.1–0.5 -1.2–1.9 -0.1–0.1 -0.9–0.7 -0.9–0.7 -1.5–1.2 -1.7–1.3 -2.9–0.3 -0.3–0.3 -0.7–0.5 -0.5–0.6
10 -2.1–0.7 -1.9–1.3 -0.2–0.2 -0.8–2.3 -0.7–0.8 -1.3–1.6 -1.6–1.3 -2.9–0.3 -0.7–0.5 4.2–5.4 -1.0–0.9
11 -2.1–0.6 -1.2–1.9 -0.1–0.1 -0.7–0.7 -0.9–0.8 -1.5–1.1 -1.7–1.3 -2.8–0.3 -0.5–0.6 -1.0–0.9 -0.0–0.2
Total 4.1–14.2 2.3–12.6 -4.6–6.3 43.3–49.8 -3.0–7.7 18.1–26.9 5.8–15.7 8.1–17.9 -4.1–6.7 2.0–12.2 -4.6–6.3
Higher 10.6 5.3 3.8 9.8 2.1 2.0 3.3 5.8 3.7 4.0 2.9
438
Table F.58: Control / Temp. at 8cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.3 -0.6–0.5 -0.6–0.5 -15.4–2.8 -0.6–0.5 -0.5–0.5 -0.6–0.5 -0.5–0.5 -0.6–0.5 -0.5–0.5 -0.6–0.5
2 -0.6–0.5 -0.3–0.2 -0.6–0.4 -15.3–2.9 -0.6–0.4 -0.5–0.5 -0.6–0.4 -0.6–0.4 -0.6–0.4 -0.6–0.5 -0.6–0.4
3 -0.6–0.5 -0.6–0.4 -0.0–0.0 -15.3–2.9 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -15.4–2.8 -15.3–2.9 -15.3–2.9 93.8–103.7 -0.1–0.1 -1.7–1.8 -0.7–0.5 -0.3–0.7 -0.5–0.9 -0.7–1.4 -0.1–0.1
5 -0.6–0.5 -0.6–0.4 -0.1–0.1 -0.1–0.1 -0.0–0.0 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.5–0.5 -0.5–0.5 -0.1–0.1 -1.7–1.8 -0.1–0.1 0.6–2.0 -2.1–0.7 -2.2–0.6 -2.1–0.7 -2.0–0.8 -2.2–0.6
7 -0.6–0.5 -0.6–0.4 -0.1–0.1 -0.7–0.5 -0.1–0.1 -2.1–0.7 -0.2–0.2 -0.4–0.5 -0.4–0.5 -0.3–0.5 -0.4–0.5
8 -0.5–0.5 -0.6–0.4 -0.1–0.1 -0.3–0.7 -0.1–0.1 -2.2–0.6 -0.4–0.5 -0.2–0.2 -0.4–0.4 -0.4–0.4 -0.4–0.4
9 -0.6–0.5 -0.6–0.4 -0.1–0.1 -0.5–0.9 -0.1–0.1 -2.1–0.7 -0.4–0.5 -0.4–0.4 -0.4–0.2 -0.3–0.8 -0.3–0.7
10 -0.5–0.5 -0.6–0.5 -0.1–0.1 -0.7–1.4 -0.1–0.1 -2.0–0.8 -0.3–0.5 -0.4–0.4 -0.3–0.8 -0.3–0.6 -0.6–1.2
11 -0.6–0.5 -0.6–0.4 -0.1–0.1 -0.1–0.1 -0.1–0.1 -2.2–0.6 -0.4–0.5 -0.4–0.4 -0.3–0.7 -0.6–1.2 -0.1–0.0
Total -1.6–7.6 -1.7–7.5 -2.0–7.1 96.6–98.1 -2.1–7.1 1.0–10.0 -1.8–7.4 -1.8–7.4 -1.5–7.7 -0.6–8.5 -2.1–7.1
Higher 9.5 9.9 9.0 16.6 2.7 7.7 3.4 3.4 3.3 3.5 3.0
Table F.59: South / Temp. at 8cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.2 -0.4–0.5 -0.4–0.5 -27.9–2.3 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5 -0.4–0.5
2 -0.4–0.5 -0.2–0.2 -0.5–0.3 -27.9–2.4 -0.5–0.3 -0.4–0.3 -0.5–0.3 -0.5–0.3 -0.5–0.3 -0.5–0.3 -0.5–0.3
3 -0.4–0.5 -0.5–0.3 -0.0–0.0 -28.0–2.4 -0.0–0.0 -0.0–0.0 -0.0–0.0 -0.0–0.0 -0.0–0.0 -0.0–0.0 -0.0–0.0
4 -27.9–2.3 -27.9–2.4 -28.0–2.4 97.8–115.0 -0.2–0.1 -0.9–1.7 -0.6–0.4 -0.5–0.4 -0.2–0.4 -0.7–0.9 -0.2–0.1
5 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.2–0.1 -0.0–0.0 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1 -0.0–0.1
6 -0.4–0.5 -0.4–0.3 -0.0–0.0 -0.9–1.7 -0.0–0.1 -0.5–0.5 -1.3–0.8 -1.3–0.8 -1.3–0.7 -1.2–0.8 -1.3–0.8
7 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.6–0.4 -0.0–0.1 -1.3–0.8 -0.2–0.1 -0.3–0.4 -0.3–0.4 -0.3–0.4 -0.3–0.4
8 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.5–0.4 -0.0–0.1 -1.3–0.8 -0.3–0.4 -0.1–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.2
9 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.2–0.4 -0.0–0.1 -1.3–0.7 -0.3–0.4 -0.3–0.2 -0.2–0.1 -0.1–0.3 -0.1–0.3
10 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.7–0.9 -0.0–0.1 -1.2–0.8 -0.3–0.4 -0.3–0.2 -0.1–0.3 -0.4–0.2 -0.3–0.9
11 -0.4–0.5 -0.5–0.3 -0.0–0.0 -0.2–0.1 -0.0–0.1 -1.3–0.8 -0.3–0.4 -0.3–0.2 -0.1–0.3 -0.3–0.9 -0.0–0.1
Total -0.9–12.4 -1.0–12.3 -1.1–12.2 99.0–99.9 -1.1–12.2 -0.2–13.1 -1.0–12.3 -1.0–12.3 -1.0–12.3 -0.6–12.6 -1.1–12.2
Higher 18.2 18.9 18.4 31.0 5.5 7.3 5.7 6.0 5.7 5.8 5.4
Table F.60: North / Temp. at 8cm / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.3–0.4 -0.6–0.8 -0.6–0.8 -26.3–2.3 -0.6–0.8 -0.5–0.8 -0.6–0.8 -0.6–0.8 -0.6–0.8 -0.6–0.8 -0.6–0.8
2 -0.6–0.8 -0.4–0.3 -0.5–0.9 -26.1–2.4 -0.5–0.9 -0.4–1.0 -0.5–0.9 -0.5–0.9 -0.5–0.9 -0.5–1.0 -0.5–0.9
3 -0.6–0.8 -0.5–0.9 -0.1–0.0 -26.4–2.0 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
4 -26.3–2.3 -26.1–2.4 -26.4–2.0 95.4–111.6 -0.1–0.2 -1.1–2.8 -0.4–0.6 -0.6–0.7 -0.3–0.5 -0.9–1.8 -0.1–0.3
5 -0.6–0.8 -0.5–0.9 -0.1–0.1 -0.1–0.2 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1 -0.1–0.1
6 -0.5–0.8 -0.4–1.0 -0.1–0.1 -1.1–2.8 -0.1–0.1 0.1–1.7 -2.1–1.0 -2.1–1.0 -2.1–1.0 -2.0–1.1 -2.1–1.0
7 -0.6–0.8 -0.5–0.9 -0.1–0.1 -0.4–0.6 -0.1–0.1 -2.1–1.0 -0.2–0.2 -0.5–0.3 -0.5–0.3 -0.5–0.3 -0.5–0.3
8 -0.6–0.8 -0.5–0.9 -0.1–0.1 -0.6–0.7 -0.1–0.1 -2.1–1.0 -0.5–0.3 -0.3–0.2 -0.6–0.3 -0.6–0.4 -0.4–0.6
9 -0.6–0.8 -0.5–0.9 -0.1–0.1 -0.3–0.5 -0.1–0.1 -2.1–1.0 -0.5–0.3 -0.6–0.3 -0.1–0.2 -0.3–0.2 -0.4–0.2
10 -0.6–0.8 -0.5–1.0 -0.1–0.1 -0.9–1.8 -0.1–0.1 -2.0–1.1 -0.5–0.3 -0.6–0.4 -0.3–0.2 -0.5–0.5 -0.5–1.5
11 -0.6–0.8 -0.5–0.9 -0.1–0.1 -0.1–0.3 -0.1–0.1 -2.1–1.0 -0.5–0.3 -0.4–0.6 -0.4–0.2 -0.5–1.5 -0.1–0.1
Total -1.1–12.0 -1.0–12.1 -1.5–11.6 97.5–99.1 -1.5–11.6 0.5–13.4 -1.4–11.8 -1.3–11.8 -1.5–11.7 -0.3–12.7 -1.5–11.7
Higher 16.6 15.6 16.9 29.2 4.6 7.5 5.8 5.7 5.6 5.6 4.8
439
F.6 Temperature at “Knee”
Table F.61: Control / “Knee” Temp. with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 0.6–8.2 -0.7–6.9 -4.5–3.4 27.2–33.3 -4.6–3.2 59.4–63.3 -4.6–3.2 -4.7–3.2 6.3–13.7 13.8–20.5 -4.0–3.7
0.67 0.8–8.2 -2.5–5.1 -3.0–4.6 17.1–23.8 -3.1–4.5 67.4–70.8 -3.1–4.5 -3.1–4.5 7.8–15.0 15.3–21.8 -4.7–3.1
1.00 1.1–8.4 -3.3–4.4 -3.3–4.2 11.7–18.8 -3.4–4.0 70.6–73.8 -3.2–4.2 -3.4–4.1 8.2–15.4 16.0–22.5 -2.8–4.6
1.33 1.2–8.5 -3.8–3.8 -3.6–3.9 8.3–15.5 -3.8–3.6 71.5–74.7 -2.9–4.6 -3.1–4.4 8.0–15.1 16.4–22.9 -3.2–4.2
1.67 0.9–8.4 -4.3–3.4 -4.1–3.5 5.8–13.1 -4.3–3.2 70.9–74.2 -4.0–3.7 -4.5–3.2 7.1–14.4 16.6–23.0 -3.8–3.7
2.00 0.7–8.3 -4.6–3.2 -4.7–3.0 4.0–11.5 -4.9–2.8 69.4–72.7 -1.7–5.9 -2.6–5.1 6.1–13.4 16.5–23.1 -4.5–3.2
2.33 0.3–8.0 -5.0–3.0 -5.5–2.5 2.3–10.1 -5.5–2.4 67.1–70.7 1.5–9.1 0.1–7.8 4.9–12.5 16.3–23.0 -5.3–2.6
2.67 -0.2–7.8 -3.0–5.0 -6.0–2.2 0.9–9.1 -6.1–2.1 64.7–68.6 5.5–13.2 3.4–11.2 4.0–11.9 15.9–22.9 -5.9–2.4
3.00 -0.9–7.5 -3.4–5.0 -6.7–1.9 -0.3–8.2 -6.7–1.8 62.5–66.8 9.3–17.1 6.4–14.3 2.9–11.2 15.4–22.7 -6.6–2.0
3.33 -1.7–7.1 -4.0–4.8 -7.6–1.4 -1.6–7.4 -7.5–1.4 60.6–65.2 12.5–20.4 8.8–16.9 1.8–10.6 14.9–22.5 -7.4–1.6
3.67 -2.5–6.7 -4.3–4.9 -8.2–1.2 -2.7–6.8 -8.2–1.3 59.2–64.1 15.3–23.4 10.9–19.2 1.0–10.2 14.6–22.5 -8.1–1.3
4.00 -3.0–6.6 -4.5–5.0 -8.5–1.2 -3.4–6.4 -8.5–1.2 58.1–63.3 17.8–26.0 12.8–21.3 0.6–10.2 14.3–22.5 -8.4–1.3
4.33 -3.5–6.4 -4.6–5.2 -8.8–1.2 -4.1–6.1 -8.8–1.2 57.3–62.7 19.6–27.9 14.3–22.9 0.3–10.1 14.1–22.6 -8.7–1.3
4.67 -3.9–6.2 -4.7–5.3 -9.1–1.2 -4.6–5.9 -9.0–1.2 56.8–62.4 20.9–29.3 15.4–24.0 0.1–10.1 14.0–22.6 -8.9–1.3
5.00 -4.3–6.1 -4.8–5.4 -9.4–1.0 -5.1–5.6 -9.4–1.0 56.3–62.1 21.7–30.3 15.9–24.7 -0.0–10.2 13.9–22.7 -9.2–1.1
5.33 -4.5–5.9 -4.7–5.5 -9.5–1.0 -5.4–5.4 -9.5–1.0 56.1–62.0 22.3–30.9 16.3–25.1 -0.1–10.2 13.8–22.7 -9.4–1.1
5.67 -4.7–5.8 -4.6–5.6 -9.6–0.9 -5.6–5.2 -9.6–0.9 56.0–61.9 22.4–31.0 16.4–25.2 -0.2–10.2 13.8–22.7 -9.5–1.0
6.00 -5.0–5.6 -4.6–5.7 -9.7–0.8 -5.8–5.0 -9.7–0.9 56.0–61.9 22.1–30.7 16.1–25.0 -0.2–10.2 13.9–22.7 -9.6–0.9
6.33 -5.1–5.4 -4.5–5.8 -9.8–0.6 -5.9–4.8 -9.7–0.8 56.1–61.9 21.3–29.9 15.5–24.4 -0.1–10.2 13.9–22.7 -9.7–0.8
6.67 -5.0–5.4 -4.1–5.9 -9.7–0.7 -5.6–4.9 -9.4–0.9 56.3–62.1 20.2–28.8 14.8–23.6 0.3–10.4 14.1–22.8 -9.5–0.8
7.00 -4.8–5.4 -3.9–6.0 -9.5–0.7 -5.3–5.0 -9.3–0.9 56.8–62.3 18.7–27.3 13.7–22.4 0.6–10.6 14.3–22.8 -9.4–0.7
7.33 -4.4–5.5 -6.5–3.6 -9.2–0.8 -4.7–5.2 -8.9–1.0 57.4–62.8 16.7–25.2 12.0–20.7 1.3–10.9 14.5–22.9 -9.1–0.8
7.67 -3.7–5.9 -5.7–4.1 -8.6–1.1 -3.9–5.7 -8.3–1.3 58.2–63.5 14.5–22.9 10.1–18.7 1.9–11.3 15.0–23.1 -8.5–1.2
8.00 -3.1–6.2 -5.0–4.4 -8.3–1.1 -3.4–5.9 -8.0–1.4 59.5–64.5 11.2–19.6 7.5–16.1 2.6–11.6 15.4–23.3 -8.1–1.2
8.33 -2.1–6.9 -4.1–5.0 -7.5–1.5 -2.4–6.6 -7.4–1.6 61.6–66.3 7.9–16.2 4.5–13.1 3.3–12.2 16.3–23.8 -7.5–1.6
8.67 -0.7–7.8 -2.8–5.9 -6.6–2.1 -1.0–7.6 -6.7–2.0 64.0–68.3 4.6–12.9 1.9–10.3 4.5–13.0 17.1–24.4 -6.6–2.0
9.00 0.1–8.4 -2.0–6.5 -6.2–2.2 -0.3–8.1 -6.4–2.0 66.4–70.3 1.4–9.7 -0.8–7.6 5.6–13.8 17.6–24.6 -6.2–2.2
9.33 1.2–9.3 -1.6–6.7 -5.6–2.5 0.2–8.4 -5.8–2.4 68.6–72.3 -1.3–7.0 -2.9–5.5 6.7–14.6 18.1–25.1 -5.5–2.7
9.67 1.9–9.9 -1.0–7.0 -5.0–3.0 0.5–8.6 -5.3–2.7 70.5–74.0 -2.7–5.5 -3.8–4.3 7.4–15.2 18.6–25.3 -4.9–3.1
10.00 2.3–10.0 -0.9–7.0 -4.7–3.2 0.6–8.5 -5.2–2.7 72.0–75.5 -3.5–4.6 -4.6–3.5 8.2–15.8 18.7–25.3 -4.6–3.3
Table F.62: Control / “Knee” Temp. / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.4–1.3 -1.2–0.6 -1.1–0.6 -0.7–1.1 -1.1–0.6 -0.9–3.5 -1.1–1.0 -0.9–1.1 -1.2–0.7 -0.5–1.4 -1.1–0.6
2 -1.2–0.6 0.5–1.1 -0.3–0.7 -0.3–0.8 -0.3–0.7 -0.6–3.7 -0.2–1.6 -0.2–1.4 -0.3–0.9 -0.0–1.3 -0.3–0.7
3 -1.1–0.6 -0.3–0.7 -0.1–0.1 -0.3–0.2 -0.2–0.2 -1.7–2.3 -0.4–0.7 -0.3–0.5 -0.2–0.2 -0.4–0.3 -0.2–0.2
4 -0.7–1.1 -0.3–0.8 -0.3–0.2 0.9–1.7 -1.1–0.5 -2.0–2.4 -1.0–1.0 -1.2–0.7 -1.3–0.4 -1.1–0.7 -1.1–0.4
5 -1.1–0.6 -0.3–0.7 -0.2–0.2 -1.1–0.5 -0.1–0.1 -1.9–2.0 -0.2–0.7 -0.2–0.6 -0.2–0.2 -0.3–0.4 -0.2–0.1
6 -0.9–3.5 -0.6–3.7 -1.7–2.3 -2.0–2.4 -1.9–2.0 21.5–25.1 8.5–12.9 6.3–10.3 -0.7–2.3 1.7–5.3 -1.2–0.5
7 -1.1–1.0 -0.2–1.6 -0.4–0.7 -1.0–1.0 -0.2–0.7 8.5–12.9 6.9–9.0 -0.4–3.5 -1.3–1.7 0.8–4.1 -0.9–1.7
8 -0.9–1.1 -0.2–1.4 -0.3–0.5 -1.2–0.7 -0.2–0.6 6.3–10.3 -0.4–3.5 6.0–7.8 -1.1–1.7 0.4–3.5 -1.0–1.4
9 -1.2–0.7 -0.3–0.9 -0.2–0.2 -1.3–0.4 -0.2–0.2 -0.7–2.3 -1.3–1.7 -1.1–1.7 -0.7–0.5 0.2–2.6 -1.1–1.1
10 -0.5–1.4 -0.0–1.3 -0.4–0.3 -1.1–0.7 -0.3–0.4 1.7–5.3 0.8–4.1 0.4–3.5 0.2–2.6 5.2–6.7 -1.0–1.0
11 -1.1–0.6 -0.3–0.7 -0.2–0.2 -1.1–0.4 -0.2–0.1 -1.2–0.5 -0.9–1.7 -1.0–1.4 -1.1–1.1 -1.0–1.0 -0.0–0.2
Total -4.3–6.1 -4.8–5.4 -9.4–1.0 -5.1–5.6 -9.4–1.0 56.3–62.1 21.7–30.3 15.9–24.7 -0.0–10.2 13.9–22.7 -9.2–1.1
Higher -0.7 -4.9 -4.5 -0.3 -4.3 9.6 1.8 0.4 2.9 2.1 -4.0
440
Table F.63: South / “Knee” Temp. with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 1.1–9.8 1.6–10.2 -4.0–5.0 15.9–23.7 -4.1–4.9 64.8–68.5 -4.1–4.9 -4.1–4.9 -0.8–8.0 10.3–18.2 -3.7–5.3
0.67 2.0–10.1 -1.0–7.4 -4.5–4.1 9.1–17.0 -4.6–3.9 71.3–74.3 -4.5–4.0 -4.6–3.9 -0.5–7.8 12.1–19.6 -4.2–4.3
1.00 2.1–10.0 -2.4–5.8 -4.8–3.5 6.1–13.9 -4.9–3.5 72.5–75.4 -4.2–4.1 -4.5–3.9 -0.9–7.2 12.9–20.0 -4.5–3.8
1.33 1.9–9.6 -2.9–5.1 -5.2–2.9 4.0–11.7 -5.4–2.8 70.7–73.7 -2.4–5.6 -2.9–5.1 -1.3–6.5 12.9–19.9 -5.0–3.1
1.67 1.2–8.8 -3.1–4.7 -3.8–3.9 2.0–9.6 -3.6–4.0 66.3–69.6 1.1–8.7 -0.0–7.6 -2.5–5.2 12.0–18.9 -3.5–4.2
2.00 0.3–7.9 -3.3–4.4 -4.7–3.0 0.3–8.0 -4.4–3.2 61.0–64.8 6.5–13.7 4.3–11.6 -3.5–4.3 10.8–17.7 -4.3–3.4
2.33 -0.5–7.2 -3.6–4.2 -5.4–2.4 -1.0–6.9 -5.1–2.7 56.2–60.4 12.7–19.4 9.1–16.2 -4.2–3.6 9.8–16.8 -5.0–2.9
2.67 -1.6–6.5 -4.1–4.0 -6.0–2.1 -2.2–6.0 -5.8–2.3 52.2–56.9 18.2–24.8 13.0–20.2 -5.1–3.1 8.7–16.1 -5.7–2.4
3.00 -2.5–6.1 -4.6–3.9 -6.6–1.9 -3.2–5.4 -6.5–2.0 49.4–54.5 22.7–29.3 16.3–23.5 -3.2–5.1 8.1–15.8 -6.2–2.2
3.33 -3.4–5.6 -5.1–3.8 -7.2–1.7 -4.2–4.8 -7.0–1.8 47.5–53.0 26.1–32.9 18.8–26.0 -3.7–4.9 7.5–15.6 -6.9–1.9
3.67 -4.3–5.0 -5.6–3.7 -7.8–1.5 -5.0–4.3 -7.7–1.4 46.2–52.0 28.8–35.6 20.5–27.9 -4.1–4.8 6.8–15.2 -7.5–1.6
4.00 -5.1–4.6 -5.7–3.8 -8.2–1.4 -5.7–4.1 -8.2–1.3 45.4–51.5 30.8–37.9 21.9–29.5 -4.5–4.8 6.4–15.2 -7.9–1.5
4.33 -5.7–4.4 -6.1–3.8 -8.6–1.3 -6.3–3.8 -8.6–1.2 44.8–51.2 32.2–39.3 22.9–30.7 -4.9–4.7 6.1–15.2 -8.3–1.5
4.67 -6.3–4.2 -6.2–3.9 -8.9–1.3 -6.8–3.6 -8.9–1.1 44.5–51.0 33.1–40.5 23.6–31.6 -5.1–4.7 5.9–15.3 -8.6–1.5
5.00 -6.7–4.0 -6.2–4.2 -9.1–1.3 -4.0–6.1 -9.2–1.1 44.3–51.0 33.8–41.3 24.2–32.2 -5.2–4.8 5.8–15.4 -8.9–1.4
5.33 -7.0–3.9 -6.2–4.3 -9.3–1.3 -4.2–6.0 -9.3–1.1 44.2–51.0 34.2–41.7 24.4–32.6 -5.3–4.8 5.7–15.4 -9.0–1.4
5.67 -4.0–6.3 -6.1–4.4 -9.4–1.2 -4.3–5.9 -9.4–1.0 44.1–51.0 34.3–41.8 24.5–32.7 -5.4–4.8 5.7–15.4 -9.2–1.3
6.00 -4.4–5.9 -6.1–4.4 -9.6–1.0 -4.6–5.7 -9.6–0.9 44.1–50.9 34.0–41.5 24.1–32.4 -5.5–4.7 5.5–15.3 -9.3–1.2
6.33 -4.6–5.7 -6.0–4.5 -9.6–0.8 -4.5–5.7 -9.6–0.8 44.3–51.1 33.3–40.8 23.7–31.9 -5.5–4.6 5.6–15.3 -9.3–1.1
6.67 -4.7–5.4 -5.6–4.6 -9.5–0.8 -4.3–5.7 -9.4–0.9 44.6–51.3 32.3–39.7 23.1–31.1 -5.4–4.6 5.8–15.3 -9.3–0.9
7.00 -4.4–5.4 -5.0–5.0 -9.1–0.8 -3.7–6.0 -9.0–1.0 45.1–51.6 30.9–38.2 22.1–30.1 -5.0–4.8 6.2–15.4 -8.9–1.0
7.33 -4.2–5.3 -4.4–5.3 -8.8–0.9 -6.3–3.7 -8.6–1.1 45.8–52.0 29.0–36.3 20.7–28.6 -4.5–4.9 6.5–15.5 -8.6–1.1
7.67 -4.0–5.3 -4.0–5.5 -8.6–0.9 -5.8–3.9 -8.3–1.2 46.8–52.7 26.3–33.6 18.6–26.4 -4.4–4.8 6.8–15.5 -8.4–1.1
8.00 -6.3–3.3 -3.0–6.0 -8.1–1.0 -5.2–4.2 -7.7–1.4 48.3–54.0 22.7–30.0 16.0–23.8 -3.9–5.0 7.3–15.7 -7.9–1.3
8.33 -5.5–3.7 -2.7–6.2 -7.7–1.2 -4.4–4.7 -7.3–1.6 50.7–56.1 18.2–25.7 12.4–20.3 -3.4–5.3 7.8–16.0 -7.5–1.4
8.67 -3.8–5.2 -1.2–7.4 -6.6–2.1 -2.9–5.9 -6.3–2.3 54.4–59.3 13.2–20.9 8.9–16.9 -5.2–3.8 9.7–17.6 -6.6–2.1
9.00 -1.8–6.9 0.2–8.7 -6.2–2.4 -2.0–6.8 -5.7–2.8 59.2–63.6 8.4–16.3 4.4–12.7 -4.3–4.5 10.9–18.6 -6.0–2.6
9.33 -0.3–8.2 1.0–9.4 -5.6–2.9 -1.2–7.5 -5.3–3.2 63.8–67.8 3.4–11.6 0.7–9.2 -3.7–5.0 12.2–19.7 -5.6–2.9
9.67 1.9–10.2 2.4–10.6 -3.9–4.3 0.3–8.8 -3.8–4.5 68.4–72.0 1.0–9.3 -1.0–7.5 -1.8–6.8 14.1–21.4 -4.0–4.3
10.00 2.6–10.9 1.8–10.0 -3.1–5.0 0.4–9.0 -3.2–5.0 71.1–74.6 -0.8–7.5 -2.1–6.5 -1.1–7.4 15.1–22.3 -3.3–5.0
Table F.64: South / “Knee” Temp. / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.6–1.4 -1.0–0.4 -0.9–0.4 -1.2–0.4 -0.9–0.5 -1.0–2.2 -0.7–1.7 -0.8–1.3 -0.9–0.4 -0.8–0.6 -0.9–0.4
2 -1.0–0.4 0.7–1.4 -0.5–0.7 -0.4–0.7 -0.5–0.7 -0.4–2.8 0.1–2.7 -0.1–2.0 -0.6–0.7 -0.2–1.2 -0.5–0.7
3 -0.9–0.4 -0.5–0.7 -0.0–0.0 -0.1–0.0 -0.1–0.0 -1.2–1.5 -0.3–1.5 -0.2–1.1 -0.1–0.1 -0.0–0.4 -0.1–0.0
4 -1.2–0.4 -0.4–0.7 -0.1–0.0 0.6–1.3 -0.7–0.6 -1.6–1.7 -0.7–1.8 -0.5–1.4 -0.8–0.7 -0.7–0.8 -0.8–0.6
5 -0.9–0.5 -0.5–0.7 -0.1–0.0 -0.7–0.6 -0.1–0.0 -1.3–1.5 -0.1–1.7 -0.1–1.3 -0.0–0.3 0.0–0.5 -0.0–0.3
6 -1.0–2.2 -0.4–2.8 -1.2–1.5 -1.6–1.7 -1.3–1.5 16.4–19.4 9.8–14.9 7.5–11.6 -0.7–1.1 0.5–3.4 -1.0–0.5
7 -0.7–1.7 0.1–2.7 -0.3–1.5 -0.7–1.8 -0.1–1.7 9.8–14.9 13.3–15.9 1.7–6.5 -0.8–2.1 2.4–5.7 -0.7–1.9
8 -0.8–1.3 -0.1–2.0 -0.2–1.1 -0.5–1.4 -0.1–1.3 7.5–11.6 1.7–6.5 10.3–12.5 -0.7–2.1 1.2–4.3 -0.7–1.8
9 -0.9–0.4 -0.6–0.7 -0.1–0.1 -0.8–0.7 -0.0–0.3 -0.7–1.1 -0.8–2.1 -0.7–2.1 -0.3–0.4 0.1–1.4 -0.2–1.1
10 -0.8–0.6 -0.2–1.2 -0.0–0.4 -0.7–0.8 0.0–0.5 0.5–3.4 2.4–5.7 1.2–4.3 0.1–1.4 3.4–4.5 -0.9–0.6
11 -0.9–0.4 -0.5–0.7 -0.1–0.0 -0.8–0.6 -0.0–0.3 -1.0–0.5 -0.7–1.9 -0.7–1.8 -0.2–1.1 -0.9–0.6 0.0–0.3
Total -6.7–4.0 -6.2–4.2 -9.1–1.3 -4.0–6.1 -9.2–1.1 44.3–51.0 33.8–41.3 24.2–32.2 -5.2–4.8 5.8–15.4 -8.9–1.4
Higher -1.8 -6.3 -5.1 -0.6 -5.8 3.8 -2.6 -3.4 -2.9 -3.6 -4.9
441
Table F.65: North / “Knee” Temp. with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 1.6–13.7 1.8–13.7 -4.4–8.3 37.2–45.3 -4.4–8.2 44.1–51.3 -4.5–8.2 -4.4–8.2 -1.0–11.2 7.5–18.7 -4.1–8.5
0.67 2.8–14.1 -0.9–10.8 -5.8–6.3 23.0–32.4 -5.8–6.3 54.7–60.5 -6.0–6.1 -6.0–6.2 -1.3–10.4 9.5–20.1 -5.5–6.7
1.00 3.6–14.5 -2.3–9.1 -6.1–5.6 16.8–26.5 -6.5–5.3 59.3–64.5 -6.4–5.4 -6.5–5.3 -1.5–9.9 10.7–20.8 -5.9–5.8
1.33 4.0–14.4 -3.4–7.8 -6.3–5.1 13.0–22.7 -6.8–4.6 61.3–66.2 -6.6–4.8 -6.5–4.9 -1.6–9.3 11.2–20.9 -6.3–5.1
1.67 4.0–14.1 -3.8–6.9 -6.2–4.7 10.4–20.0 -6.8–4.1 61.6–66.2 -5.9–5.0 -5.6–5.2 -2.1–8.5 11.3–20.6 -6.5–4.5
2.00 3.6–13.3 -3.9–6.4 -6.3–4.1 8.3–17.7 -3.9–6.1 60.2–64.8 -4.6–5.7 -4.2–6.1 -2.6–7.6 11.3–20.3 -6.6–3.8
2.33 3.3–12.4 -3.2–6.4 -6.2–3.6 6.5–15.6 -3.8–5.7 57.6–62.3 -2.1–7.4 -1.4–8.1 -2.8–6.8 11.1–19.6 -6.4–3.4
2.67 2.9–11.5 -2.5–6.6 -3.6–5.5 5.1–13.8 -4.0–5.1 54.1–58.9 0.4–9.3 1.7–10.6 -3.3–5.9 10.6–18.7 -3.8–5.3
3.00 2.5–10.8 -1.9–6.8 -3.8–4.9 3.9–12.2 -4.2–4.5 50.7–55.5 3.1–11.4 5.2–13.3 -3.7–5.1 10.0–17.7 -3.9–4.8
3.33 1.9–9.9 -1.3–7.0 -4.0–4.4 2.8–11.0 -4.3–4.1 47.4–52.4 5.7–13.4 8.6–16.2 -4.0–4.4 9.3–16.8 -4.1–4.3
3.67 1.5–9.3 -0.8–7.2 -4.1–4.0 2.0–9.9 -4.3–3.8 44.5–49.5 8.0–15.3 11.7–18.9 -4.1–4.0 8.8–16.1 -4.2–3.9
4.00 1.1–8.7 -0.4–7.3 -4.2–3.7 1.2–9.0 -4.5–3.5 42.1–47.2 9.9–16.8 14.6–21.4 -4.3–3.7 8.4–15.5 -4.4–3.5
4.33 0.5–8.0 -0.3–7.3 -4.4–3.4 0.5–8.3 -4.8–3.1 40.2–45.3 11.2–18.0 16.8–23.4 -4.6–3.3 7.8–15.0 -4.6–3.2
4.67 0.0–7.6 -0.2–7.3 -4.6–3.1 0.1–7.8 -4.9–2.8 38.8–44.0 12.2–18.8 18.5–25.0 -4.7–3.1 7.5–14.6 -4.8–2.9
5.00 -0.3–7.2 -0.1–7.3 -4.7–2.9 -0.3–7.3 -5.1–2.6 37.9–43.1 12.8–19.4 19.8–26.1 -4.9–2.9 7.3–14.4 -4.9–2.7
5.33 -0.6–6.9 0.1–7.5 -4.8–2.8 -0.5–7.1 -5.1–2.5 37.4–42.7 13.2–19.8 20.6–26.8 -4.9–2.8 7.3–14.3 -5.0–2.6
5.67 -0.8–6.7 0.3–7.7 -4.8–2.8 -0.7–7.0 -5.2–2.4 37.3–42.5 13.4–19.9 20.9–27.0 -4.9–2.8 7.3–14.3 -5.0–2.6
6.00 -1.0–6.6 0.5–7.9 -4.7–2.8 -0.7–7.0 -5.1–2.5 37.6–42.8 13.3–19.8 20.6–26.8 -4.9–2.9 7.5–14.5 -5.0–2.6
6.33 -1.0–6.5 0.7–8.1 -4.8–2.9 -0.7–7.0 -5.1–2.5 38.2–43.4 12.8–19.4 19.8–26.1 -4.9–2.9 7.6–14.7 -5.0–2.6
6.67 -1.0–6.7 1.0–8.5 -4.7–3.0 -0.4–7.3 -5.1–2.6 39.2–44.4 12.1–18.8 18.5–25.0 -4.7–3.1 8.0–15.1 -5.0–2.7
7.00 -0.8–6.8 1.2–8.7 -4.7–3.1 -0.2–7.6 -5.0–2.8 40.7–45.8 11.0–17.8 16.7–23.4 -4.7–3.3 8.2–15.4 -5.0–2.8
7.33 -0.6–7.2 1.5–9.2 -4.5–3.5 0.5–8.3 -4.8–3.2 42.5–47.6 9.6–16.7 14.7–21.6 -4.5–3.6 8.8–16.1 -4.8–3.2
7.67 0.0–8.0 2.0–9.9 -4.1–4.0 1.3–9.3 -4.6–3.5 44.8–49.9 8.1–15.5 12.4–19.6 -4.0–4.2 9.6–17.0 -4.5–3.6
8.00 0.8–8.9 2.5–10.5 -3.8–4.5 2.4–10.5 -4.3–4.0 47.5–52.5 6.2–13.9 9.7–17.2 -3.7–4.8 10.4–17.8 -4.2–4.1
8.33 1.2–9.7 2.4–10.7 -4.0–4.6 3.2–11.6 -4.4–4.3 50.5–55.4 3.6–11.8 6.5–14.6 -3.7–5.1 10.7–18.5 -4.4–4.2
8.67 2.4–11.1 2.7–11.3 -3.7–5.1 4.4–12.9 -4.1–4.8 53.9–58.7 1.5–10.2 3.9–12.5 -3.2–5.8 11.8–19.7 -4.2–4.7
9.00 3.9–12.7 2.8–11.6 -6.1–3.4 5.7–14.4 -3.8–5.3 56.9–61.6 -0.2–8.9 1.8–10.7 -2.7–6.6 13.0–21.0 -3.8–5.2
9.33 5.0–14.0 2.5–11.6 -5.7–4.0 6.7–15.6 -3.6–5.7 59.7–64.3 -1.5–7.9 -0.0–9.2 -2.1–7.4 14.0–22.1 -6.1–3.5
9.67 5.9–15.0 1.9–11.3 -5.7–4.2 6.8–15.8 -6.4–3.5 62.2–66.7 -3.0–6.7 -1.8–7.9 -1.9–7.8 14.2–22.5 -6.2–3.8
10.00 6.5–15.6 1.3–10.8 -5.3–4.7 6.9–16.1 -6.2–3.9 63.9–68.4 -3.6–6.2 -2.7–7.1 -1.6–8.1 14.7–23.1 -5.8–4.1
Table F.66: North / “Knee” Temp. / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 2.0–3.2 -1.8–0.4 -1.8–0.3 -1.6–0.8 -1.8–0.3 -1.1–3.0 -1.7–0.6 -1.5–1.1 -2.0–0.3 -1.4–0.7 -1.8–0.3
2 -1.8–0.4 3.1–4.2 -0.8–0.9 -0.9–0.9 -0.8–0.9 -0.5–3.5 -0.9–1.3 -0.6–2.0 -0.8–0.9 -0.7–1.2 -0.8–0.9
3 -1.8–0.3 -0.8–0.9 -0.2–0.2 -0.4–0.5 -0.3–0.5 -1.0–2.3 -0.6–0.5 -0.8–0.8 -0.3–0.5 -0.5–0.4 -0.3–0.5
4 -1.6–0.8 -0.9–0.9 -0.4–0.5 2.9–3.9 -1.1–0.8 -1.7–2.4 -0.9–1.2 -1.1–1.3 -1.2–0.7 -1.2–0.8 -1.1–0.8
5 -1.8–0.3 -0.8–0.9 -0.3–0.5 -1.1–0.8 -0.1–0.1 -0.8–2.5 -0.3–0.6 -0.6–0.9 -0.2–0.3 -0.3–0.5 -0.2–0.3
6 -1.1–3.0 -0.5–3.5 -1.0–2.3 -1.7–2.4 -0.8–2.5 29.5–32.5 0.4–3.3 0.6–4.5 -0.7–1.1 -0.6–2.6 -0.9–0.8
7 -1.7–0.6 -0.9–1.3 -0.6–0.5 -0.9–1.2 -0.3–0.6 0.4–3.3 11.6–13.5 -2.2–1.9 -1.8–0.9 -1.1–1.8 -1.8–0.8
8 -1.5–1.1 -0.6–2.0 -0.8–0.8 -1.1–1.3 -0.6–0.9 0.6–4.5 -2.2–1.9 17.2–19.4 -1.6–1.1 -1.5–1.7 -1.6–1.1
9 -2.0–0.3 -0.8–0.9 -0.3–0.5 -1.2–0.7 -0.2–0.3 -0.7–1.1 -1.8–0.9 -1.6–1.1 -0.3–0.5 -0.6–1.0 -1.0–0.6
10 -1.4–0.7 -0.7–1.2 -0.5–0.4 -1.2–0.8 -0.3–0.5 -0.6–2.6 -1.1–1.8 -1.5–1.7 -0.6–1.0 8.9–10.2 -0.8–0.9
11 -1.8–0.3 -0.8–0.9 -0.3–0.5 -1.1–0.8 -0.2–0.3 -0.9–0.8 -1.8–0.8 -1.6–1.1 -1.0–0.6 -0.8–0.9 -0.1–0.2
Total -0.3–7.2 -0.1–7.3 -4.7–2.9 -0.3–7.3 -5.1–2.6 37.9–43.1 12.8–19.4 19.8–26.1 -4.9–2.9 7.3–14.4 -4.9–2.7
Higher 5.1 -2.2 -1.1 0.7 -1.8 -0.4 2.6 1.9 0.3 -0.1 0.6
442
Table F.67: Control / “Knee” Temp. / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.1–1.2 -1.2–0.8 -1.3–0.8 -1.3–0.9 -1.2–0.8 -1.4–5.3 -0.7–1.3 -1.2–0.8 -1.2–1.1 -0.6–1.5 -1.2–0.8
2 -1.2–0.8 1.0–1.5 -0.3–0.5 -0.6–0.5 -0.4–0.5 -1.1–5.6 -0.2–0.9 -0.3–0.8 -0.3–0.6 -0.2–1.2 -0.3–0.5
3 -1.3–0.8 -0.3–0.5 -0.1–0.1 -0.4–0.2 -0.2–0.2 -2.1–4.4 -0.4–0.3 -0.4–0.2 -0.3–0.2 -0.5–0.5 -0.2–0.2
4 -1.3–0.9 -0.6–0.5 -0.4–0.2 4.5–5.7 -1.1–0.8 -1.8–5.7 -0.8–1.3 -0.9–1.2 -1.3–0.9 -1.0–1.3 -1.1–0.8
5 -1.2–0.8 -0.4–0.5 -0.2–0.2 -1.1–0.8 -0.1–0.1 -2.3–4.1 -0.2–0.2 -0.3–0.1 -0.1–0.2 -0.5–0.5 -0.2–0.2
6 -1.4–5.3 -1.1–5.6 -2.1–4.4 -1.8–5.7 -2.3–4.1 38.5–43.2 2.9–6.5 1.9–5.2 0.7–4.2 1.8–6.5 -0.8–0.7
7 -0.7–1.3 -0.2–0.9 -0.4–0.3 -0.8–1.3 -0.2–0.2 2.9–6.5 3.1–4.2 -0.6–1.5 -0.9–0.9 -0.1–2.0 -0.6–1.0
8 -1.2–0.8 -0.3–0.8 -0.4–0.2 -0.9–1.2 -0.3–0.1 1.9–5.2 -0.6–1.5 2.8–3.8 -0.7–1.0 -0.4–1.7 -0.8–0.8
9 -1.2–1.1 -0.3–0.6 -0.3–0.2 -1.3–0.9 -0.1–0.2 0.7–4.2 -0.9–0.9 -0.7–1.0 -0.8–0.7 1.1–3.9 -1.2–1.6
10 -0.6–1.5 -0.2–1.2 -0.5–0.5 -1.0–1.3 -0.5–0.5 1.8–6.5 -0.1–2.0 -0.4–1.7 1.1–3.9 8.9–10.7 -0.9–1.2
11 -1.2–0.8 -0.3–0.5 -0.2–0.2 -1.1–0.8 -0.2–0.2 -0.8–0.7 -0.6–1.0 -0.8–0.8 -1.2–1.6 -0.9–1.2 0.0–0.3
Total -3.3–6.4 -4.6–4.8 -7.7–1.9 1.3–10.9 -7.9–1.7 61.8–66.5 2.3–11.7 -0.4–9.1 1.2–10.9 14.0–22.4 -7.5–2.1
Higher -0.4 -4.5 -3.6 -0.7 -3.8 0.3 -3.8 -3.8 0.8 -1.0 -3.2
Table F.68: South / “Knee” Temp. / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.7–1.8 -1.5–0.4 -1.5–0.5 -1.7–0.6 -1.6–0.5 -2.6–4.3 -1.2–1.0 -1.2–0.9 -1.6–0.5 -0.9–1.1 -1.5–0.5
2 -1.5–0.4 1.7–2.3 -0.4–0.6 -0.6–0.4 -0.4–0.6 -1.9–4.9 -0.3–1.4 -0.3–1.1 -0.4–0.6 -0.2–1.2 -0.4–0.6
3 -1.5–0.5 -0.4–0.6 -0.1–0.1 -0.4–0.1 -0.2–0.2 -2.8–3.6 -0.3–0.7 -0.3–0.4 -0.2–0.2 -0.3–0.6 -0.2–0.2
4 -1.7–0.6 -0.6–0.4 -0.4–0.1 4.0–5.1 -1.2–0.6 -3.0–4.3 -1.4–0.9 -1.3–0.8 -1.2–0.7 -0.9–1.2 -1.2–0.7
5 -1.6–0.5 -0.4–0.6 -0.2–0.2 -1.2–0.6 -0.1–0.1 -2.9–3.6 -0.3–0.6 -0.3–0.4 -0.2–0.2 -0.3–0.6 -0.2–0.1
6 -2.6–4.3 -1.9–4.9 -2.8–3.6 -3.0–4.3 -2.9–3.6 38.0–42.6 4.0–8.7 3.0–7.1 -0.6–1.6 1.1–5.4 -1.0–0.5
7 -1.2–1.0 -0.3–1.4 -0.3–0.7 -1.4–0.9 -0.3–0.6 4.0–8.7 7.6–9.2 -0.2–2.7 -0.6–1.5 0.8–3.3 -0.4–1.5
8 -1.2–0.9 -0.3–1.1 -0.3–0.4 -1.3–0.8 -0.3–0.4 3.0–7.1 -0.2–2.7 6.1–7.5 -0.7–1.2 0.1–2.4 -0.6–1.2
9 -1.6–0.5 -0.4–0.6 -0.2–0.2 -1.2–0.7 -0.2–0.2 -0.6–1.6 -0.6–1.5 -0.7–1.2 -0.3–0.5 0.1–1.9 -0.6–1.1
10 -0.9–1.1 -0.2–1.2 -0.3–0.6 -0.9–1.2 -0.3–0.6 1.1–5.4 0.8–3.3 0.1–2.4 0.1–1.9 8.0–9.4 -0.8–0.8
11 -1.5–0.5 -0.4–0.6 -0.2–0.2 -1.2–0.7 -0.2–0.1 -1.0–0.5 -0.4–1.5 -0.6–1.2 -0.6–1.1 -0.8–0.8 0.2–0.5
Total -6.2–4.6 -4.0–6.0 -8.3–2.1 -2.6–8.0 -8.4–1.9 54.3–59.8 9.3–18.6 4.5–14.3 -4.9–5.2 6.8–16.5 -7.9–2.3
Higher 0.6 -3.7 -3.4 -0.5 -3.1 -1.9 -5.6 -5.7 -1.7 -5.7 -3.2
Table F.69: North / “Knee” Temp. / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.4–3.1 -2.1–1.0 -2.1–1.1 -1.6–2.0 -2.2–1.0 -1.9–5.6 -2.1–0.9 -2.1–1.0 -2.3–0.9 -1.5–1.5 -2.2–1.0
2 -2.1–1.0 3.1–4.1 -1.0–0.8 -1.0–1.0 -1.0–0.8 -2.1–5.4 -1.1–0.9 -0.9–1.3 -1.0–0.8 -1.0–1.1 -1.0–0.8
3 -2.1–1.1 -1.0–0.8 -0.3–0.2 -0.7–0.5 -0.4–0.7 -2.2–4.6 -0.7–0.4 -0.5–0.7 -0.4–0.7 -0.8–0.6 -0.4–0.6
4 -1.6–2.0 -1.0–1.0 -0.7–0.5 7.1–8.8 -1.4–1.4 -2.8–5.5 -1.4–1.5 -1.7–1.3 -1.6–1.3 -1.2–2.1 -1.3–1.5
5 -2.2–1.0 -1.0–0.8 -0.4–0.7 -1.4–1.4 -0.1–0.1 -2.1–4.7 -0.3–0.4 -0.4–0.4 -0.3–0.3 -0.7–0.5 -0.3–0.3
6 -1.9–5.6 -2.1–5.4 -2.2–4.6 -2.8–5.5 -2.1–4.7 42.2–47.2 -1.2–2.1 -1.4–2.5 -1.2–1.2 -2.1–3.0 -1.0–0.7
7 -2.1–0.9 -1.1–0.9 -0.7–0.4 -1.4–1.5 -0.3–0.4 -1.2–2.1 5.5–6.9 -1.2–1.6 -1.5–0.7 -1.4–1.3 -1.5–0.7
8 -2.1–1.0 -0.9–1.3 -0.5–0.7 -1.7–1.3 -0.4–0.4 -1.4–2.5 -1.2–1.6 7.5–9.1 -1.2–1.2 -1.4–1.5 -1.2–1.2
9 -2.3–0.9 -1.0–0.8 -0.4–0.7 -1.6–1.3 -0.3–0.3 -1.2–1.2 -1.5–0.7 -1.2–1.2 -0.6–0.6 -1.4–1.1 -1.0–1.4
10 -1.5–1.5 -1.0–1.1 -0.8–0.6 -1.2–2.1 -0.7–0.5 -2.1–3.0 -1.4–1.3 -1.4–1.5 -1.4–1.1 11.6–13.5 -0.8–1.4
11 -2.2–1.0 -1.0–0.8 -0.4–0.6 -1.3–1.5 -0.3–0.3 -1.0–0.7 -1.5–0.7 -1.2–1.2 -1.0–1.4 -0.8–1.4 -0.2–0.3
Total -0.1–9.7 -2.0–8.1 -4.2–5.7 5.8–15.2 -4.7–5.2 49.7–55.2 1.4–11.0 4.2–13.6 -4.8–5.5 9.3–18.4 -4.3–5.6
Higher 4.6 -1.4 0.2 0.9 -0.5 -1.0 1.0 0.1 1.6 0.4 1.1
443
Table F.70: Control / “Knee” Temp. / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.1–0.9 -0.7–1.1 -0.6–1.1 -0.9–0.9 -0.6–1.1 -0.3–3.4 -0.9–1.0 -0.8–1.1 -1.0–0.9 -0.7–1.1 -0.6–1.0
2 -0.7–1.1 0.4–1.1 -0.5–0.7 -0.6–0.8 -0.5–0.7 0.4–4.2 -0.3–1.5 -0.5–1.3 -0.4–0.9 -0.3–1.2 -0.5–0.7
3 -0.6–1.1 -0.5–0.7 -0.1–0.1 -0.5–0.1 -0.2–0.1 -0.8–2.5 -0.3–0.6 -0.4–0.4 -0.2–0.1 -0.3–0.3 -0.2–0.1
4 -0.9–0.9 -0.6–0.8 -0.5–0.1 3.4–4.9 -0.7–0.7 2.9–7.5 0.4–2.7 -0.3–1.8 -1.0–0.7 0.2–2.2 -0.8–0.6
5 -0.6–1.1 -0.5–0.7 -0.2–0.1 -0.7–0.7 -0.1–0.1 -0.8–2.4 -0.2–0.8 -0.3–0.6 -0.2–0.2 -0.1–0.5 -0.2–0.2
6 -0.3–3.4 0.4–4.2 -0.8–2.5 2.9–7.5 -0.8–2.4 17.1–20.5 6.6–10.8 4.8–8.7 -1.5–1.3 1.4–4.9 -1.2–0.7
7 -0.9–1.0 -0.3–1.5 -0.3–0.6 0.4–2.7 -0.2–0.8 6.6–10.8 5.9–7.9 -0.2–3.7 -1.1–2.0 1.0–4.2 -0.6–2.2
8 -0.8–1.1 -0.5–1.3 -0.4–0.4 -0.3–1.8 -0.3–0.6 4.8–8.7 -0.2–3.7 5.5–7.5 -1.2–1.7 -0.0–3.3 -1.1–1.6
9 -1.0–0.9 -0.4–0.9 -0.2–0.1 -1.0–0.7 -0.2–0.2 -1.5–1.3 -1.1–2.0 -1.2–1.7 -0.6–0.6 0.2–2.5 -1.3–1.0
10 -0.7–1.1 -0.3–1.2 -0.3–0.3 0.2–2.2 -0.1–0.5 1.4–4.9 1.0–4.2 -0.0–3.3 0.2–2.5 4.1–5.6 -1.0–1.1
11 -0.6–1.0 -0.5–0.7 -0.2–0.1 -0.8–0.6 -0.2–0.2 -1.2–0.7 -0.6–2.2 -1.1–1.6 -1.3–1.0 -1.0–1.1 -0.0–0.2
Total -4.5–6.0 -5.4–5.1 -7.4–3.2 14.3–24.1 -7.5–3.2 53.7–59.9 23.2–31.9 17.4–26.2 1.1–11.2 15.4–24.4 -7.2–3.4
Higher -2.6 -5.6 -3.2 6.7 -3.8 8.9 3.6 3.2 4.2 4.2 -2.9
Table F.71: South / “Knee” Temp. / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 0.5–1.3 -1.1–0.3 -1.1–0.2 -1.3–0.2 -1.1–0.3 -1.1–1.9 -0.9–1.5 -1.0–1.0 -1.1–0.3 -0.6–0.8 -1.1–0.3
2 -1.1–0.3 0.8–1.5 -0.5–0.7 -0.4–0.9 -0.5–0.7 -0.1–3.0 0.2–2.8 -0.1–2.1 -0.5–0.7 -0.4–1.0 -0.5–0.7
3 -1.1–0.2 -0.5–0.7 -0.0–0.1 -0.2–-0.0 -0.1–0.0 -1.1–1.5 -0.4–1.4 -0.3–1.0 -0.2–0.0 -0.2–0.2 -0.1–0.0
4 -1.3–0.2 -0.4–0.9 -0.2–-0.0 1.0–1.8 -1.0–0.4 -1.0–2.1 -0.7–1.9 -0.6–1.4 -1.0–0.4 -0.9–0.6 -1.0–0.4
5 -1.1–0.3 -0.5–0.7 -0.1–0.0 -1.0–0.4 -0.1–0.1 -1.0–1.5 -0.3–1.6 -0.2–1.1 -0.2–0.2 -0.1–0.4 -0.2–0.2
6 -1.1–1.9 -0.1–3.0 -1.1–1.5 -1.0–2.1 -1.0–1.5 15.6–18.6 9.0–14.0 6.5–10.7 -1.1–0.9 -0.2–2.7 -1.2–0.5
7 -0.9–1.5 0.2–2.8 -0.4–1.4 -0.7–1.9 -0.3–1.6 9.0–14.0 13.3–16.0 1.6–6.5 -1.0–1.9 2.1–5.4 -0.8–1.9
8 -1.0–1.0 -0.1–2.1 -0.3–1.0 -0.6–1.4 -0.2–1.1 6.5–10.7 1.6–6.5 10.4–12.7 -1.0–1.9 0.6–3.9 -1.0–1.7
9 -1.1–0.3 -0.5–0.7 -0.2–0.0 -1.0–0.4 -0.2–0.2 -1.1–0.9 -1.0–1.9 -1.0–1.9 -0.3–0.4 -0.2–1.2 -0.4–0.9
10 -0.6–0.8 -0.4–1.0 -0.2–0.2 -0.9–0.6 -0.1–0.4 -0.2–2.7 2.1–5.4 0.6–3.9 -0.2–1.2 3.2–4.4 -0.6–1.0
11 -1.1–0.3 -0.5–0.7 -0.1–0.0 -1.0–0.4 -0.2–0.2 -1.2–0.5 -0.8–1.9 -1.0–1.7 -0.4–0.9 -0.6–1.0 -0.0–0.3
Total -6.4–3.8 -4.8–5.2 -8.5–1.6 -3.7–6.8 -8.5–1.5 43.6–50.2 35.1–42.4 25.9–33.5 -4.8–5.0 6.5–15.7 -8.1–1.9
Higher -0.4 -5.4 -3.9 0.1 -4.4 6.1 0.1 0.2 -0.8 -1.0 -3.5
Table F.72: North / “Knee” Temp. / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.5–2.6 -1.4–0.6 -1.4–0.6 -1.3–1.0 -1.4–0.6 -0.4–3.1 -1.1–1.0 -1.2–1.3 -1.5–0.6 -1.1–0.9 -1.4–0.6
2 -1.4–0.6 3.5–4.6 -1.2–0.7 -0.8–1.2 -1.2–0.7 -0.0–3.7 -1.0–1.2 -0.8–1.9 -1.2–0.7 -1.0–1.0 -1.2–0.7
3 -1.4–0.6 -1.2–0.7 -0.2–0.2 -0.6–0.3 -0.3–0.4 -0.4–2.5 -0.5–0.5 -0.7–0.8 -0.3–0.4 -0.5–0.3 -0.3–0.4
4 -1.3–1.0 -0.8–1.2 -0.6–0.3 6.2–7.6 -0.5–1.3 1.4–5.2 0.2–2.3 -0.1–2.5 -0.5–1.3 0.4–2.4 -0.4–1.4
5 -1.4–0.6 -1.2–0.7 -0.3–0.4 -0.5–1.3 -0.1–0.1 -0.3–2.6 -0.4–0.5 -0.7–0.8 -0.3–0.3 -0.4–0.3 -0.3–0.2
6 -0.4–3.1 -0.0–3.7 -0.4–2.5 1.4–5.2 -0.3–2.6 25.4–28.4 0.8–3.7 0.4–4.2 -0.6–1.5 -0.5–2.7 -0.6–1.3
7 -1.1–1.0 -1.0–1.2 -0.5–0.5 0.2–2.3 -0.4–0.5 0.8–3.7 11.0–12.9 -1.5–2.6 -1.4–1.4 -1.9–1.1 -1.4–1.3
8 -1.2–1.3 -0.8–1.9 -0.7–0.8 -0.1–2.5 -0.7–0.8 0.4–4.2 -1.5–2.6 16.6–18.8 -1.7–1.2 -1.9–1.3 -1.6–1.2
9 -1.5–0.6 -1.2–0.7 -0.3–0.4 -0.5–1.3 -0.3–0.3 -0.6–1.5 -1.4–1.4 -1.7–1.2 -0.4–0.4 -0.5–1.0 -0.9–0.7
10 -1.1–0.9 -1.0–1.0 -0.5–0.3 0.4–2.4 -0.4–0.3 -0.5–2.7 -1.9–1.1 -1.9–1.3 -0.5–1.0 7.3–8.7 -1.0–1.0
11 -1.4–0.6 -1.2–0.7 -0.3–0.4 -0.4–1.4 -0.3–0.2 -0.6–1.3 -1.4–1.3 -1.6–1.2 -0.9–0.7 -1.0–1.0 -0.1–0.2
Total -0.2–7.4 1.5–9.0 -3.6–4.1 10.3–17.3 -3.9–3.8 36.4–41.8 14.2–21.0 20.9–27.3 -4.0–3.8 7.4–14.5 -3.6–4.0
Higher 2.5 -0.1 -0.0 -1.4 -1.0 -2.9 1.9 2.6 -0.2 1.2 0.2
444
Table F.73: Control / “Knee” Temp. / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.6–0.5 -0.8–1.3 -0.8–1.3 -13.3–4.2 -0.8–1.3 -1.2–0.9 -0.9–1.3 -0.8–1.3 -1.3–0.8 -1.3–0.8 -0.8–1.3
2 -0.8–1.3 -0.3–0.4 -0.7–0.8 -13.2–4.2 -0.7–0.8 -0.6–0.9 -0.7–0.8 -0.7–0.8 -0.7–0.8 -0.7–0.8 -0.7–0.8
3 -0.8–1.3 -0.7–0.8 -0.1–0.1 -13.4–4.0 -0.2–0.3 -0.2–0.3 -0.2–0.3 -0.2–0.3 -0.2–0.3 -0.2–0.3 -0.2–0.3
4 -13.3–4.2 -13.2–4.2 -13.4–4.0 85.6–95.2 -0.5–0.1 -2.8–3.6 -0.8–0.9 -0.8–0.5 -2.0–1.4 -1.4–2.2 -0.4–0.2
5 -0.8–1.3 -0.7–0.8 -0.2–0.3 -0.5–0.1 -0.1–0.2 -0.4–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.3 -0.3–0.2 -0.3–0.2
6 -1.2–0.9 -0.6–0.9 -0.2–0.3 -2.8–3.6 -0.4–0.2 2.7–5.2 -3.8–0.9 -3.9–0.8 -3.1–1.6 -3.2–1.4 -3.9–0.8
7 -0.9–1.3 -0.7–0.8 -0.2–0.3 -0.8–0.9 -0.3–0.2 -3.8–0.9 -0.2–0.3 -0.5–0.5 -0.5–0.6 -0.5–0.5 -0.5–0.5
8 -0.8–1.3 -0.7–0.8 -0.2–0.3 -0.8–0.5 -0.3–0.2 -3.9–0.8 -0.5–0.5 -0.2–0.2 -0.4–0.5 -0.4–0.5 -0.4–0.5
9 -1.3–0.8 -0.7–0.8 -0.2–0.3 -2.0–1.4 -0.3–0.3 -3.1–1.6 -0.5–0.6 -0.4–0.5 -0.7–0.6 -0.9–1.6 -1.0–1.5
10 -1.3–0.8 -0.7–0.8 -0.2–0.3 -1.4–2.2 -0.3–0.2 -3.2–1.4 -0.5–0.5 -0.4–0.5 -0.9–1.6 -0.5–0.9 -1.3–1.4
11 -0.8–1.3 -0.7–0.8 -0.2–0.3 -0.4–0.2 -0.3–0.2 -3.9–0.8 -0.5–0.5 -0.4–0.5 -1.0–1.5 -1.3–1.4 -0.2–0.1
Total -1.8–8.0 -2.4–7.5 -3.2–6.7 92.1–94.6 -3.2–6.6 5.6–14.8 -2.9–6.9 -3.0–6.9 -0.6–9.0 -0.0–9.6 -3.2–6.7
Higher 7.0 6.4 5.9 16.5 2.0 12.0 3.1 3.3 4.8 4.8 2.9
Table F.74: South / “Knee” Temp. / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.2–0.6 -1.1–0.4 -1.1–0.4 -27.9–2.6 -1.1–0.4 -1.0–0.5 -1.1–0.4 -1.1–0.4 -1.1–0.4 -1.1–0.4 -1.1–0.4
2 -1.1–0.4 -0.2–0.3 -0.6–0.5 -27.8–2.7 -0.6–0.5 -0.6–0.5 -0.6–0.5 -0.6–0.5 -0.6–0.5 -0.6–0.5 -0.6–0.5
3 -1.1–0.4 -0.6–0.5 -0.1–0.1 -27.8–2.5 -0.3–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.2 -0.3–0.2
4 -27.9–2.6 -27.8–2.7 -27.8–2.5 94.6–111.9 -0.4–0.4 -3.3–2.1 -0.8–0.3 -0.6–0.5 -1.0–0.9 -1.7–1.7 -0.5–0.2
5 -1.1–0.4 -0.6–0.5 -0.3–0.2 -0.4–0.4 -0.2–0.1 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.3–0.3
6 -1.0–0.5 -0.6–0.5 -0.3–0.2 -3.3–2.1 -0.3–0.3 0.3–2.4 -2.9–1.2 -2.9–1.2 -2.9–1.2 -2.6–1.4 -2.9–1.2
7 -1.1–0.4 -0.6–0.5 -0.3–0.2 -0.8–0.3 -0.3–0.3 -2.9–1.2 -0.1–0.3 -0.6–0.1 -0.6–0.1 -0.6–0.1 -0.6–0.1
8 -1.1–0.4 -0.6–0.5 -0.3–0.2 -0.6–0.5 -0.3–0.3 -2.9–1.2 -0.6–0.1 -0.2–0.2 -0.4–0.3 -0.4–0.3 -0.4–0.3
9 -1.1–0.4 -0.6–0.5 -0.3–0.2 -1.0–0.9 -0.3–0.3 -2.9–1.2 -0.6–0.1 -0.4–0.3 -0.2–0.5 -1.0–0.3 -1.0–0.3
10 -1.1–0.4 -0.6–0.5 -0.3–0.2 -1.7–1.7 -0.3–0.3 -2.6–1.4 -0.6–0.1 -0.4–0.3 -1.0–0.3 -0.6–0.6 -0.8–1.6
11 -1.1–0.4 -0.6–0.5 -0.3–0.2 -0.5–0.2 -0.3–0.3 -2.9–1.2 -0.6–0.1 -0.4–0.3 -1.0–0.3 -0.8–1.6 -0.0–0.2
Total -1.9–12.3 -1.7–12.3 -1.9–12.2 97.2–99.0 -1.9–12.2 1.0–14.6 -2.0–12.1 -1.9–12.2 -1.8–12.3 -0.5–13.3 -2.0–12.1
Higher 20.8 18.8 18.3 33.9 5.5 11.3 7.4 6.8 7.3 7.6 6.6
Table F.75: North / “Knee” Temp. / Minimum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -0.3–0.8 -1.3–0.9 -1.3–0.9 -25.8–3.1 -1.3–0.9 -1.2–1.0 -1.3–0.9 -1.3–0.9 -1.3–0.9 -1.2–1.0 -1.4–0.9
2 -1.3–0.9 -0.3–0.5 -0.9–0.7 -25.5–3.2 -0.9–0.7 -0.8–0.7 -0.9–0.7 -0.9–0.7 -0.9–0.7 -0.9–0.7 -0.9–0.7
3 -1.3–0.9 -0.9–0.7 -0.1–0.1 -25.6–3.2 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.3–0.3 -0.2–0.3 -0.3–0.3
4 -25.8–3.1 -25.5–3.2 -25.6–3.2 87.6–104.2 -0.4–0.3 -2.7–5.2 -0.8–0.4 -1.0–0.3 -1.2–1.2 -2.6–2.8 -0.5–0.3
5 -1.3–0.9 -0.9–0.7 -0.3–0.3 -0.4–0.3 -0.2–0.1 -0.3–0.5 -0.3–0.5 -0.3–0.5 -0.3–0.5 -0.3–0.5 -0.3–0.5
6 -1.2–1.0 -0.8–0.7 -0.3–0.3 -2.7–5.2 -0.3–0.5 2.0–5.0 -4.5–1.4 -4.5–1.3 -4.4–1.5 -4.0–1.9 -4.5–1.4
7 -1.3–0.9 -0.9–0.7 -0.3–0.3 -0.8–0.4 -0.3–0.5 -4.5–1.4 -0.2–0.2 -0.4–0.4 -0.4–0.4 -0.4–0.4 -0.4–0.4
8 -1.3–0.9 -0.9–0.7 -0.3–0.3 -1.0–0.3 -0.3–0.5 -4.5–1.3 -0.4–0.4 -0.3–0.2 -0.3–0.6 -0.6–0.3 -0.3–0.6
9 -1.3–0.9 -0.9–0.7 -0.3–0.3 -1.2–1.2 -0.3–0.5 -4.4–1.5 -0.4–0.4 -0.3–0.6 -0.4–0.5 -0.9–0.9 -0.9–0.9
10 -1.2–1.0 -0.9–0.7 -0.2–0.3 -2.6–2.8 -0.3–0.5 -4.0–1.9 -0.4–0.4 -0.6–0.3 -0.9–0.9 -0.7–1.2 -1.4–2.3
11 -1.4–0.9 -0.9–0.7 -0.3–0.3 -0.5–0.3 -0.3–0.5 -4.5–1.4 -0.4–0.4 -0.3–0.6 -0.9–0.9 -1.4–2.3 -0.2–0.1
Total -2.1–12.4 -2.3–12.3 -2.7–11.9 93.6–96.6 -2.8–11.8 4.0–17.8 -2.7–11.9 -2.6–12.0 -2.2–12.3 0.6–14.8 -2.7–11.9
Higher 18.1 17.1 16.1 32.2 4.4 13.6 6.5 6.8 6.6 8.1 5.9
445
F.7 Temperature Gradient at 2 cm
Table F.76: Control / Temp. Gradient at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 38.4–44.0 11.6–17.9 -3.7–3.3 46.7–52.5 -4.4–2.5 38.9–44.6 -4.4–2.5 -4.4–2.4 14.6–20.7 17.2–23.3 -3.3–3.6
0.67 50.3–56.3 8.5–15.9 -4.9–3.1 50.6–57.7 -6.0–2.1 40.5–47.3 -6.0–2.1 -6.0–2.1 19.2–26.1 20.0–27.2 -4.6–3.3
1.00 57.6–63.7 6.3–14.7 -5.3–4.1 52.3–60.3 -6.7–2.7 42.4–49.9 -6.8–2.6 -6.7–2.6 22.4–30.1 22.9–30.7 -5.4–3.8
1.33 62.6–69.0 3.6–13.3 -6.3–4.6 52.5–61.3 -7.9–3.2 42.8–51.1 -8.0–2.9 -7.9–2.9 23.6–32.1 24.6–33.2 -6.8–3.8
1.67 65.6–72.3 1.5–12.3 -7.0–5.3 52.1–61.8 -8.2–4.3 43.3–52.4 -8.4–3.6 -8.0–4.0 24.8–34.1 26.0–35.4 -7.8–4.1
2.00 67.1–74.1 -0.7–11.4 -8.2–5.3 51.1–61.4 -8.8–5.0 43.5–53.3 -8.5–4.7 -8.3–4.8 25.3–35.3 26.9–37.0 -9.1–4.1
2.33 67.2–74.5 -1.8–11.2 -8.8–5.7 49.6–60.6 -8.3–6.4 43.9–54.3 -7.6–6.4 -7.6–6.4 25.8–36.4 27.9–38.6 -9.7–4.4
2.67 66.0–73.5 -1.8–11.7 -8.9–6.2 47.9–59.3 -7.4–7.9 44.8–55.6 -5.8–8.7 -6.2–8.4 26.5–37.3 29.3–40.2 -10.1–4.7
3.00 64.2–72.0 -2.0–12.2 -9.2–6.5 45.7–57.4 -6.9–9.1 45.6–56.9 -9.3–6.8 -10.4–6.2 26.8–37.9 29.9–41.4 -10.5–4.9
3.33 62.1–70.4 -2.4–12.4 -9.5–6.7 43.4–55.6 -6.9–9.7 46.4–58.0 -8.4–8.2 -10.2–6.9 26.6–38.2 30.5–42.4 -11.0–4.8
3.67 60.4–68.9 -2.6–12.9 -9.6–7.1 41.7–54.3 -6.5–10.6 47.2–59.1 -7.6–9.3 -10.1–7.6 27.1–38.9 31.2–43.4 -11.2–5.0
4.00 58.5–67.4 -2.2–13.5 -9.7–7.3 39.5–52.6 -11.5–6.7 47.8–59.9 -6.9–10.3 -9.7–8.1 27.2–39.2 31.3–43.9 -11.3–5.3
4.33 56.8–66.0 -1.7–14.2 -9.9–7.4 37.8–51.2 -11.7–6.9 48.4–60.6 -6.4–10.9 -9.4–8.6 27.0–39.3 31.6–44.4 -11.5–5.4
4.67 55.1–64.6 -0.8–15.0 -9.9–7.5 36.2–49.8 -11.7–7.1 48.9–61.3 -5.7–11.6 -9.0–9.0 26.9–39.4 31.9–45.0 -11.4–5.8
5.00 53.4–63.3 -0.6–15.3 -10.3–7.2 34.6–48.4 -12.0–6.9 49.3–61.7 -5.6–11.8 -9.1–9.0 26.4–39.1 32.0–45.1 -11.4–5.8
5.33 52.1–62.2 -0.2–15.6 -10.6–7.0 33.8–47.6 -7.0–11.2 49.7–62.0 -5.7–11.7 -9.1–9.0 26.0–38.8 32.1–45.2 -11.4–5.8
5.67 51.1–61.2 0.3–16.1 -10.5–6.9 33.3–47.1 -7.0–11.0 50.3–62.5 -5.8–11.5 -8.9–9.0 25.9–38.6 32.5–45.3 -11.2–5.9
6.00 50.4–60.5 0.8–16.3 -10.3–6.9 33.2–46.8 -7.2–10.5 50.8–62.8 -6.2–11.1 -8.9–8.8 25.8–38.3 32.6–45.3 -10.9–5.9
6.33 49.9–60.0 1.0–16.3 -10.1–6.8 33.4–46.7 -7.6–9.8 51.4–63.1 -6.7–10.4 -9.0–8.6 25.6–38.0 32.7–45.2 -10.7–5.9
6.67 49.6–59.6 1.3–16.3 -10.0–6.8 33.8–46.8 -8.1–9.0 52.0–63.4 -7.4–9.5 -9.0–8.3 25.5–37.7 32.9–45.1 -10.6–5.8
7.00 49.6–59.5 1.6–16.3 -9.6–6.7 34.5–47.2 -8.3–8.4 52.8–63.8 -8.0–8.8 -9.0–8.0 25.3–37.4 33.2–45.1 -10.2–5.8
7.33 50.0–59.8 2.0–16.3 -9.3–6.6 35.6–47.9 -8.5–7.8 53.6–64.3 -8.3–8.2 -8.9–7.7 25.2–37.2 33.6–45.1 -9.8–5.8
7.67 50.8–60.4 2.3–16.3 -8.9–6.6 37.1–49.0 -8.8–7.1 54.2–64.7 -8.7–7.5 -8.7–7.3 25.5–37.2 33.8–45.1 -9.3–5.9
8.00 52.1–61.4 2.5–16.1 -8.7–6.5 38.8–50.4 -9.0–6.4 54.9–65.1 -8.9–7.0 -8.8–6.9 26.0–37.5 34.2–45.2 -9.0–6.0
8.33 54.0–63.1 2.7–16.1 -8.0–6.9 41.0–52.4 -8.7–6.3 55.9–65.8 -8.9–6.7 -8.6–6.9 26.9–38.1 34.8–45.6 -8.3–6.3
8.67 56.6–65.3 3.3–16.2 -7.0–7.5 43.4–54.7 -7.9–6.7 57.1–66.7 -8.5–6.9 -8.0–7.3 28.3–39.3 35.7–46.3 -7.4–7.0
9.00 59.5–68.0 3.7–16.4 -5.8–8.5 46.1–57.4 -6.8–7.6 58.4–67.8 -7.6–7.6 -6.9–8.1 30.3–41.2 37.2–47.7 -6.0–8.1
9.33 62.8–71.0 3.8–16.5 -9.0–6.2 49.0–60.3 -10.1–5.1 59.7–69.1 -6.5–8.4 -6.0–8.8 32.8–43.5 39.0–49.5 -9.0–5.9
9.67 66.2–74.0 3.5–16.2 -7.5–7.4 51.8–63.3 -8.8–6.2 61.0–70.3 -5.7–9.1 -5.1–9.5 35.3–45.8 40.8–51.3 -7.5–7.1
10.00 69.3–76.8 2.8–15.6 -6.6–8.2 54.2–65.8 -7.7–6.9 61.9–71.3 -5.2–9.6 -4.6–9.9 37.4–47.8 42.2–52.8 -6.6–8.0
Table F.77: Control / Temp. Gradient at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -2.0–2.0 -3.0–1.5 -1.5–2.0 4.3–9.8 -2.1–1.9 -3.4–3.2 -1.5–2.4 -2.6–1.6 -3.0–2.5 -3.4–2.7 -1.9–1.6
2 -3.0–1.5 2.4–3.8 -1.1–0.7 -1.1–1.4 -1.3–0.7 2.4–6.3 -0.9–1.1 -1.0–0.9 -1.1–2.0 0.6–3.6 -1.1–0.7
3 -1.5–2.0 -1.1–0.7 -0.2–0.1 -0.4–0.4 -0.3–0.3 -0.6–1.1 -0.2–0.3 -0.3–0.3 -0.3–0.5 -0.5–0.5 -0.3–0.3
4 4.3–9.8 -1.1–1.4 -0.4–0.4 -1.1–0.7 -1.8–1.4 -2.0–2.2 -1.8–1.4 -2.1–1.2 -2.2–1.4 -2.5–1.6 -1.8–1.2
5 -2.1–1.9 -1.3–0.7 -0.3–0.3 -1.8–1.4 1.5–2.4 -1.4–1.4 -1.2–0.4 -1.2–0.3 -0.8–1.3 -1.2–1.0 -1.1–0.4
6 -3.4–3.2 2.4–6.3 -0.6–1.1 -2.0–2.2 -1.4–1.4 4.0–6.8 -2.0–1.1 -1.5–1.5 -1.9–2.7 -2.1–2.8 -1.7–1.1
7 -1.5–2.4 -0.9–1.1 -0.2–0.3 -1.8–1.4 -1.2–0.4 -2.0–1.1 -0.3–0.7 -0.9–1.0 -1.1–1.2 -1.2–1.0 -0.7–1.1
8 -2.6–1.6 -1.0–0.9 -0.3–0.3 -2.1–1.2 -1.2–0.3 -1.5–1.5 -0.9–1.0 -0.5–0.6 -1.2–1.1 -0.7–1.6 -0.8–1.1
9 -3.0–2.5 -1.1–2.0 -0.3–0.5 -2.2–1.4 -0.8–1.3 -1.9–2.7 -1.1–1.2 -1.2–1.1 1.1–3.1 -1.2–3.4 -1.8–1.1
10 -3.4–2.7 0.6–3.6 -0.5–0.5 -2.5–1.6 -1.2–1.0 -2.1–2.8 -1.2–1.0 -0.7–1.6 -1.2–3.4 1.5–3.5 -1.2–1.4
11 -1.9–1.6 -1.1–0.7 -0.3–0.3 -1.8–1.2 -1.1–0.4 -1.7–1.1 -0.7–1.1 -0.8–1.1 -1.8–1.1 -1.2–1.4 -0.0–0.2
Total 53.4–63.3 -0.6–15.3 -10.3–7.2 34.6–48.4 -12.0–6.9 49.3–61.7 -5.6–11.8 -9.1–9.0 26.4–39.1 32.0–45.1 -11.4–5.8
Higher 52.8 -1.4 -2.0 36.4 -3.0 45.5 3.2 0.8 29.3 32.9 -1.8
446
Table F.78: South / Temp. Gradient at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 46.5–51.5 17.2–23.1 -3.9–2.9 31.7–38.1 -2.6–4.0 34.1–39.8 -2.6–4.0 -2.6–4.0 8.4–14.8 10.8–17.2 -3.6–3.3
0.67 57.3–62.2 13.2–19.7 -4.2–3.0 34.5–41.8 -3.3–3.7 34.5–41.0 -3.3–3.6 -3.4–3.6 10.8–17.5 12.9–19.8 -4.3–3.0
1.00 63.4–68.3 9.6–16.7 -3.3–4.3 36.1–44.2 -4.4–3.3 35.0–42.0 -4.5–3.2 -4.6–3.0 11.0–18.1 13.8–21.3 -3.2–4.7
1.33 67.7–72.5 6.5–14.6 -4.3–4.0 37.3–46.1 -5.1–3.6 34.8–42.6 -5.0–3.5 -5.4–3.0 11.1–18.9 14.3–22.5 -4.3–4.5
1.67 69.8–74.8 4.7–13.6 -5.0–4.2 38.0–47.6 -4.6–4.9 34.7–43.3 -4.2–5.0 -4.7–4.4 11.2–19.5 14.9–23.6 -5.0–4.7
2.00 70.2–75.4 3.6–13.2 -5.0–4.9 37.8–48.1 -5.9–4.6 35.0–44.3 -5.0–5.4 -5.5–4.7 11.5–20.5 15.5–24.8 -4.9–5.5
2.33 68.9–74.4 3.0–13.5 -5.4–5.2 36.9–47.8 -4.7–6.5 35.5–45.5 -2.4–8.6 -3.3–7.6 11.6–21.3 16.0–26.0 -5.1–6.0
2.67 66.9–72.7 3.0–14.2 -5.6–5.9 35.6–47.1 -3.7–8.3 36.5–47.1 0.6–12.1 -1.4–10.2 11.6–22.2 16.1–26.9 -5.3–6.7
3.00 64.7–71.2 3.1–15.1 -5.5–6.6 34.4–46.5 -2.9–9.8 37.8–48.8 3.2–15.2 0.1–12.4 11.8–23.0 16.6–28.1 -5.2–7.6
3.33 62.5–69.5 2.9–15.7 -6.1–6.8 32.6–45.5 -2.6–10.9 38.6–50.1 4.8–17.5 0.6–13.6 11.6–23.6 16.8–28.9 -5.6–7.8
3.67 60.4–67.9 3.2–16.5 -6.7–7.0 31.1–44.6 -2.4–11.7 39.6–51.3 6.4–19.6 1.2–14.9 11.5–24.2 17.1–29.9 -6.0–8.3
4.00 58.4–66.3 3.9–17.4 -7.1–7.1 29.6–43.5 -2.6–12.2 40.4–52.4 7.7–21.3 1.7–15.9 11.1–24.6 17.4–30.6 -6.3–8.5
4.33 56.4–64.8 4.6–18.5 -7.5–7.2 28.2–42.5 -2.6–12.6 41.3–53.5 9.0–22.8 2.3–16.9 10.7–24.7 17.6–31.2 -6.3–8.9
4.67 54.5–63.2 4.7–19.0 -8.1–7.0 26.7–41.4 -3.2–12.4 41.9–54.3 9.4–23.6 2.3–17.3 10.2–24.6 17.9–31.7 -6.7–8.9
5.00 52.8–61.8 5.2–19.6 -8.4–7.0 25.5–40.4 -3.6–12.3 42.5–55.0 9.7–24.0 2.5–17.7 9.6–24.5 18.1–32.1 -6.7–9.2
5.33 51.2–60.4 5.6–19.9 -8.4–7.0 24.7–39.6 -3.9–12.1 43.1–55.6 9.6–23.9 2.3–17.7 9.5–24.4 18.3–32.3 -6.7–9.2
5.67 50.1–59.4 6.1–20.3 -8.2–7.1 24.2–39.1 -4.1–11.8 43.9–56.1 9.2–23.6 2.1–17.5 9.5–24.3 18.5–32.3 -6.6–9.2
6.00 49.2–58.5 6.3–20.4 -8.0–7.1 23.8–38.6 -4.4–11.4 44.7–56.7 8.5–22.8 1.8–17.0 9.5–24.0 18.5–32.3 -6.4–9.1
6.33 48.6–57.9 6.7–20.4 -7.6–7.1 23.9–38.3 -4.6–10.8 45.7–57.4 7.7–21.6 1.5–16.3 9.6–23.9 18.9–32.2 -6.1–9.1
6.67 48.6–57.5 7.1–20.4 -7.0–7.3 24.1–38.1 -4.9–10.2 46.7–58.1 6.6–20.3 1.3–15.7 10.0–23.7 19.2–32.2 -10.3–5.4
7.00 48.8–57.6 7.6–20.4 -6.1–7.7 24.8–38.3 -4.8–9.7 48.0–59.0 5.7–19.0 1.2–15.3 10.6–23.6 19.8–32.4 -9.4–5.6
7.33 49.5–57.9 8.0–20.4 -9.5–4.9 25.6–38.6 -4.7–9.2 49.3–59.8 4.7–17.7 1.4–14.8 11.1–23.5 20.6–32.6 -8.5–5.8
7.67 50.4–58.6 8.4–20.5 -8.3–5.5 26.6–39.2 -4.4–8.9 50.6–60.6 3.9–16.5 1.6–14.5 11.6–23.5 21.3–32.9 -7.7–6.1
8.00 52.0–59.8 9.1–20.7 -7.0–6.3 27.9–40.1 -4.0–8.8 52.0–61.5 3.5–15.8 2.1–14.5 12.4–23.7 22.2–33.3 -6.6–6.6
8.33 54.3–61.6 10.4–21.5 -5.0–7.7 29.3–41.3 -3.1–9.2 53.0–62.4 3.7–15.7 3.0–15.1 13.6–24.3 23.4–34.1 -5.1–7.5
8.67 57.5–64.4 12.2–22.9 -2.2–9.9 31.1–42.8 -1.1–10.7 54.2–63.4 5.1–16.8 5.2–16.7 15.7–26.2 25.5–35.6 -2.8–9.1
9.00 61.3–67.8 14.3–24.7 1.0–12.8 33.0–44.4 1.6–13.1 55.8–64.8 7.6–18.9 7.6–19.0 19.0–29.2 27.9–37.9 -0.1–11.5
9.33 65.5–71.7 16.2–26.4 4.6–16.0 34.8–46.4 4.6–16.0 58.0–66.7 10.7–21.7 10.6–21.7 22.8–32.7 31.1–40.9 2.8–14.1
9.67 69.9–75.7 17.0–27.2 8.0–19.1 37.0–48.6 7.7–18.9 60.3–68.9 13.5–24.4 13.2–24.2 26.4–36.2 34.2–43.9 5.6–16.7
10.00 73.8–79.2 16.8–27.2 10.1–21.2 39.1–50.7 9.8–20.9 62.3–70.8 15.4–26.2 14.9–25.8 29.2–39.0 36.4–46.2 7.5–18.6
Table F.79: South / Temp. Gradient at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 1.9–6.1 -0.2–4.3 -0.8–2.4 3.3–8.4 -1.2–2.9 -3.0–3.5 1.1–5.3 0.8–4.9 -0.1–4.5 -2.0–3.3 -0.8–2.6
2 -0.2–4.3 3.0–4.6 -1.0–0.9 -1.4–1.2 -0.5–1.7 2.2–5.9 -0.7–1.6 -0.9–1.2 -0.3–2.2 0.3–2.8 -1.1–0.7
3 -0.8–2.4 -1.0–0.9 -0.2–0.1 -0.0–0.7 -0.2–0.5 -0.6–1.0 -0.1–0.5 -0.2–0.5 -0.2–0.6 -0.1–0.6 -0.1–0.5
4 3.3–8.4 -1.4–1.2 -0.0–0.7 -1.2–0.3 -0.7–2.0 -1.8–2.2 -0.6–2.1 -1.0–1.7 -0.6–2.0 -0.6–2.7 -0.8–1.7
5 -1.2–2.9 -0.5–1.7 -0.2–0.5 -0.7–2.0 2.9–4.2 -1.9–1.3 -1.3–0.9 -1.6–0.5 -1.4–0.8 -1.2–1.2 -1.3–0.5
6 -3.0–3.5 2.2–5.9 -0.6–1.0 -1.8–2.2 -1.9–1.3 4.4–7.8 -2.1–1.7 -1.5–2.1 -0.3–3.9 0.3–5.0 -0.9–2.3
7 1.1–5.3 -0.7–1.6 -0.1–0.5 -0.6–2.1 -1.3–0.9 -2.1–1.7 -0.8–0.7 -0.2–2.3 -1.4–1.4 -0.9–1.8 -0.9–1.4
8 0.8–4.9 -0.9–1.2 -0.2–0.5 -1.0–1.7 -1.6–0.5 -1.5–2.1 -0.2–2.3 -0.7–0.6 -1.4–1.1 -1.3–1.2 -1.0–1.2
9 -0.1–4.5 -0.3–2.2 -0.2–0.6 -0.6–2.0 -1.4–0.8 -0.3–3.9 -1.4–1.4 -1.4–1.1 0.5–2.0 0.8–3.5 0.1–2.4
10 -2.0–3.3 0.3–2.8 -0.1–0.6 -0.6–2.7 -1.2–1.2 0.3–5.0 -0.9–1.8 -1.3–1.2 0.8–3.5 0.3–2.0 -0.3–2.0
11 -0.8–2.6 -1.1–0.7 -0.1–0.5 -0.8–1.7 -1.3–0.5 -0.9–2.3 -0.9–1.4 -1.0–1.2 0.1–2.4 -0.3–2.0 -0.2–0.2
Total 52.8–61.8 5.2–19.6 -8.4–7.0 25.5–40.4 -3.6–12.3 42.5–55.0 9.7–24.0 2.5–17.7 9.6–24.5 18.1–32.1 -6.7–9.2
Higher 33.6 -0.9 -3.0 23.2 0.2 33.1 10.9 5.9 7.0 14.4 -2.9
447
Table F.80: North / Temp. Gradient at 2cm with Time (Total-effect)
@
@t
i
1 2 3 4 5 6 7 8 9 10 11
0.33 44.6–50.9 22.2–28.6 -2.5–5.1 48.3–55.1 -3.7–3.7 31.5–37.9 -3.8–3.7 -3.8–3.7 9.6–16.8 9.4–16.7 -2.7–4.9
0.67 60.0–65.5 16.5–23.6 -3.3–4.8 50.2–58.0 -5.3–2.9 33.0–40.2 -5.2–2.9 -5.3–2.8 11.7–19.3 11.0–19.1 -3.7–4.4
1.00 68.3–73.4 12.1–19.7 -2.2–6.3 52.7–61.4 -5.5–3.2 34.8–42.6 -5.5–3.2 -5.5–3.1 13.7–22.0 12.8–21.4 -3.2–5.6
1.33 73.0–77.9 8.7–16.9 -2.8–6.4 54.4–63.6 -5.8–3.3 35.7–44.0 -5.8–3.3 -5.8–3.3 14.6–23.2 13.7–22.6 -3.5–5.7
1.67 76.4–81.1 6.9–15.4 -2.5–7.1 55.5–65.3 -5.5–4.1 36.6–45.2 -5.5–3.9 -5.4–4.1 15.5–24.3 14.9–24.0 -3.6–6.0
2.00 78.6–83.1 5.5–14.4 -2.7–7.3 56.0–66.0 -5.7–4.5 36.8–45.7 -5.8–4.2 -5.4–4.5 15.8–24.8 15.2–24.8 -4.1–6.0
2.33 79.7–84.1 5.1–14.4 -2.4–7.8 55.7–66.0 -4.7–5.8 36.9–46.2 -4.8–5.4 -4.3–5.8 16.5–25.6 16.0–25.9 -3.7–6.6
2.67 79.7–84.2 4.9–14.6 -3.0–7.5 55.0–65.3 -4.4–6.3 36.9–46.2 -4.3–6.0 -3.7–6.5 16.0–25.4 16.1–26.2 -4.3–6.4
3.00 78.8–83.5 5.2–15.1 -3.6–7.3 53.7–64.3 -4.0–7.0 36.7–46.1 -3.8–6.8 -3.3–7.4 15.3–24.9 15.9–26.2 -4.8–6.3
3.33 77.4–82.2 6.0–16.0 -3.8–7.3 52.1–62.9 -3.2–7.8 36.6–46.1 -3.2–7.7 -2.0–8.9 14.8–24.5 15.8–26.3 -5.0–6.3
3.67 75.4–80.5 6.3–16.7 -4.1–7.2 50.3–61.1 -3.0–8.3 36.0–45.7 -3.2–8.0 -1.0–10.1 14.2–24.0 15.4–26.0 -5.6–5.9
4.00 73.2–78.5 7.4–17.8 -4.4–7.0 48.1–59.0 -2.6–8.7 35.8–45.4 -3.1–8.3 -0.0–11.1 13.4–23.5 15.2–25.9 -5.9–5.7
4.33 71.1–76.6 8.3–18.8 -4.8–6.9 46.1–57.3 -2.5–8.9 35.5–45.3 -2.9–8.6 0.7–11.9 12.9–23.1 15.1–25.9 -6.3–5.5
4.67 69.1–74.9 9.2–19.8 -5.3–6.5 44.7–56.0 -2.9–8.7 35.2–45.1 -2.9–8.7 1.0–12.4 12.3–22.6 14.9–25.7 -7.0–5.1
5.00 67.5–73.4 10.2–20.7 -5.7–6.3 43.5–54.8 -3.3–8.5 35.2–45.1 -3.0–8.8 1.0–12.6 11.6–22.1 14.7–25.6 -7.5–4.7
5.33 66.3–72.3 11.0–21.5 -5.8–6.2 42.8–54.1 -3.4–8.4 35.4–45.4 -3.0–8.9 1.2–12.8 11.5–22.0 14.7–25.7 -7.5–4.7
5.67 65.4–71.5 11.7–22.1 -5.6–6.3 42.5–53.8 -3.6–8.2 35.8–45.8 -2.9–8.9 1.4–12.9 11.5–22.0 14.7–25.8 -7.4–4.8
6.00 64.8–70.9 12.2–22.5 -5.8–6.1 42.5–53.6 -4.0–7.8 36.0–46.0 -3.0–8.7 1.2–12.8 11.3–21.8 14.9–25.9 -7.4–4.8
6.33 64.6–70.7 12.5–22.7 -6.0–5.9 42.5–53.7 -4.6–7.1 36.3–46.2 -3.2–8.4 0.8–12.3 11.1–21.6 15.0–26.0 -7.4–4.7
6.67 64.9–70.9 12.6–22.7 -5.9–5.7 43.1–54.2 -5.2–6.5 36.6–46.4 -3.4–8.1 0.4–11.8 11.2–21.5 15.1–26.1 -7.3–4.7
7.00 65.6–71.5 12.6–22.6 -5.8–5.6 44.0–54.9 -5.7–5.9 37.0–46.8 -3.6–7.8 -0.1–11.2 11.4–21.6 15.5–26.3 -7.1–4.9
7.33 66.7–72.5 12.3–22.2 -5.6–5.7 45.2–56.1 -6.1–5.4 37.5–47.3 -3.8–7.6 -0.6–10.6 11.7–21.9 15.9–26.6 -6.8–5.0
7.67 68.3–73.8 11.7–21.7 -5.2–6.0 47.0–57.7 -6.5–4.9 38.2–47.9 -3.8–7.5 -0.9–10.2 12.3–22.3 16.5–27.1 -6.5–5.2
8.00 70.2–75.6 11.0–20.9 -4.9–6.3 49.2–59.7 -6.8–4.6 39.0–48.7 -3.7–7.5 -1.1–9.8 12.9–22.9 17.3–27.7 -6.1–5.5
8.33 72.5–77.7 10.0–19.9 -4.4–6.8 51.6–62.0 -6.9–4.5 40.1–49.8 -3.6–7.5 -1.2–9.6 13.7–23.6 18.3–28.6 -5.6–5.9
8.67 75.0–80.0 8.7–18.6 -3.7–7.3 54.0–64.4 -6.9–4.5 41.3–51.0 -3.5–7.5 -1.4–9.3 14.6–24.4 19.3–29.6 -5.1–6.4
9.00 77.8–82.6 7.0–17.1 -2.9–8.0 56.6–67.1 -6.7–4.5 42.6–52.4 -3.6–7.4 -1.7–9.0 15.7–25.4 20.4–30.7 -4.5–6.8
9.33 80.6–85.2 5.3–15.5 -2.2–8.7 59.0–69.7 -6.6–4.7 44.0–53.8 -3.7–7.3 -1.9–8.7 16.9–26.6 21.5–31.8 -3.9–7.3
9.67 83.3–87.7 3.7–14.1 -1.5–9.4 61.1–71.9 -6.2–5.0 45.2–55.2 -3.8–7.2 -2.1–8.6 18.1–27.8 22.6–32.9 -3.4–7.8
10.00 85.4–89.7 2.4–12.9 -0.9–10.0 62.9–73.7 -5.8–5.3 46.2–56.2 -3.9–7.1 -2.2–8.5 19.0–28.8 23.4–33.8 -2.9–8.3
Table F.81: North / Temp. Gradient at 2cm / Mid-day
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 4.5–10.6 -2.8–1.8 -1.7–2.3 10.1–17.7 -2.7–1.6 0.1–7.8 -2.6–1.7 -2.8–1.9 -3.2–2.4 -3.1–3.3 -2.6–1.6
2 -2.8–1.8 6.9–8.9 -1.5–0.7 -2.0–1.4 -0.4–1.8 1.0–4.4 -1.0–1.4 -1.0–1.5 -1.2–1.8 -0.9–1.7 -1.3–0.9
3 -1.7–2.3 -1.5–0.7 -0.4–0.5 -0.0–1.5 -0.2–1.1 -0.1–1.7 -0.3–0.9 -0.3–1.1 -0.1–1.3 -0.4–1.1 -0.4–0.9
4 10.1–17.7 -2.0–1.4 -0.0–1.5 -0.7–2.1 -0.4–3.0 -0.5–4.1 -0.2–3.4 -0.3–3.3 -0.3–3.6 -0.3–3.5 -0.1–3.4
5 -2.7–1.6 -0.4–1.8 -0.2–1.1 -0.4–3.0 2.5–3.6 -1.2–1.2 -0.7–1.1 -1.1–0.9 -1.3–0.8 -1.3–0.8 -1.0–0.8
6 0.1–7.8 1.0–4.4 -0.1–1.7 -0.5–4.1 -1.2–1.2 0.9–3.2 -0.5–2.7 -0.6–2.9 -1.2–2.5 -0.4–3.4 -0.4–2.6
7 -2.6–1.7 -1.0–1.4 -0.3–0.9 -0.2–3.4 -0.7–1.1 -0.5–2.7 -0.2–0.6 -0.4–1.4 -0.2–1.7 -0.3–1.6 -0.4–1.3
8 -2.8–1.9 -1.0–1.5 -0.3–1.1 -0.3–3.3 -1.1–0.9 -0.6–2.9 -0.4–1.4 0.3–1.3 -1.1–1.0 -1.3–0.8 -0.8–1.1
9 -3.2–2.4 -1.2–1.8 -0.1–1.3 -0.3–3.6 -1.3–0.8 -1.2–2.5 -0.2–1.7 -1.1–1.0 -0.2–1.2 -1.3–1.4 -1.4–1.1
10 -3.1–3.3 -0.9–1.7 -0.4–1.1 -0.3–3.5 -1.3–0.8 -0.4–3.4 -0.3–1.6 -1.3–0.8 -1.3–1.4 -0.4–1.2 -1.0–1.7
11 -2.6–1.6 -1.3–0.9 -0.4–0.9 -0.1–3.4 -1.0–0.8 -0.4–2.6 -0.4–1.3 -0.8–1.1 -1.4–1.1 -1.0–1.7 -0.4–0.3
Total 67.5–73.4 10.2–20.7 -5.7–6.3 43.5–54.8 -3.3–8.5 35.2–45.1 -3.0–8.8 1.0–12.6 11.6–22.1 14.7–25.6 -7.5–4.7
Higher 47.5 4.5 -3.6 23.2 -1.8 23.4 -2.6 2.9 13.3 15.2 -4.4
448
Table F.82: Control / Temp. Gradient at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 -1.3–3.0 -2.1–2.1 -2.1–1.4 7.6–13.4 -1.5–2.1 -1.4–5.4 -2.3–1.5 -2.3–1.3 -3.3–2.8 -3.6–2.6 -2.2–1.3
2 -2.1–2.1 0.2–1.3 -1.0–0.5 -0.5–1.8 -0.8–0.8 3.4–6.5 -0.9–0.7 -0.8–0.8 -0.8–1.7 1.0–3.4 -1.0–0.6
3 -2.1–1.4 -1.0–0.5 -0.2–0.2 -0.5–0.5 -0.4–0.3 -0.2–1.5 -0.3–0.3 -0.5–0.3 -0.1–0.6 -0.4–0.7 -0.3–0.4
4 7.6–13.4 -0.5–1.8 -0.5–0.5 -1.2–1.2 -1.5–1.7 -2.6–2.2 -1.4–1.7 -1.9–1.4 -1.5–2.4 -1.8–2.2 -1.5–1.6
5 -1.5–2.1 -0.8–0.8 -0.4–0.3 -1.5–1.7 0.6–1.1 -1.1–1.1 -0.8–0.3 -0.7–0.3 -0.6–0.7 -0.9–0.6 -0.8–0.3
6 -1.4–5.4 3.4–6.5 -0.2–1.5 -2.6–2.2 -1.1–1.1 5.1–8.0 -1.9–1.4 -2.1–1.2 -1.8–2.8 -2.3–2.7 -2.0–1.3
7 -2.3–1.5 -0.9–0.7 -0.3–0.3 -1.4–1.7 -0.8–0.3 -1.9–1.4 -0.1–0.5 -0.6–0.6 -0.6–0.9 -0.5–0.9 -0.6–0.6
8 -2.3–1.3 -0.8–0.8 -0.5–0.3 -1.9–1.4 -0.7–0.3 -2.1–1.2 -0.6–0.6 -0.4–0.3 -0.7–0.9 -0.2–1.4 -0.4–0.8
9 -3.3–2.8 -0.8–1.7 -0.1–0.6 -1.5–2.4 -0.6–0.7 -1.8–2.8 -0.6–0.9 -0.7–0.9 -0.0–1.8 -1.1–3.3 -1.0–1.8
10 -3.6–2.6 1.0–3.4 -0.4–0.7 -1.8–2.2 -0.9–0.6 -2.3–2.7 -0.5–0.9 -0.2–1.4 -1.1–3.3 1.2–3.2 -1.1–1.8
11 -2.2–1.3 -1.0–0.6 -0.3–0.4 -1.5–1.6 -0.8–0.3 -2.0–1.3 -0.6–0.6 -0.4–0.8 -1.0–1.8 -1.1–1.8 -0.1–0.2
Total 61.1–69.2 -3.5–10.8 -9.9–5.7 44.2–56.0 -9.6–6.5 51.4–61.9 -7.9–7.1 -8.4–7.0 26.3–37.6 31.7–42.7 -10.1–5.2
Higher 53.9 -4.7 -2.4 38.5 -1.9 43.1 -0.1 -0.2 27.8 30.6 -2.2
Table F.83: South / Temp. Gradient at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 7.6–12.4 -0.9–3.2 -1.3–1.9 4.9–10.6 -1.5–2.2 -3.5–3.9 -1.0–2.7 -0.8–2.8 0.6–5.3 -2.9–2.9 -1.6–1.8
2 -0.9–3.2 0.6–1.6 -1.1–0.3 -0.8–1.4 -0.6–0.9 2.2–5.1 -0.9–0.7 -1.1–0.5 -1.0–0.9 -0.2–1.7 -1.0–0.5
3 -1.3–1.9 -1.1–0.3 -0.2–0.2 -0.4–0.5 -0.5–0.3 -0.7–1.0 -0.5–0.3 -0.5–0.3 -0.5–0.4 -0.5–0.4 -0.4–0.4
4 4.9–10.6 -0.8–1.4 -0.4–0.5 -1.0–1.1 -0.3–2.7 -1.2–3.5 -0.0–2.9 -0.5–2.4 0.2–3.5 -0.1–4.1 -0.3–2.6
5 -1.5–2.2 -0.6–0.9 -0.5–0.3 -0.3–2.7 1.6–2.4 -1.7–0.8 -1.4–0.2 -1.5–0.1 -1.2–0.6 -1.5–0.3 -1.4–0.1
6 -3.5–3.9 2.2–5.1 -0.7–1.0 -1.2–3.5 -1.7–0.8 6.5–9.8 -1.7–2.0 -1.3–2.3 -0.8–3.4 -0.3–4.9 -1.3–2.2
7 -1.0–2.7 -0.9–0.7 -0.5–0.3 -0.0–2.9 -1.4–0.2 -1.7–2.0 -0.2–0.6 -0.8–0.9 -1.1–0.7 -1.1–0.8 -1.0–0.6
8 -0.8–2.8 -1.1–0.5 -0.5–0.3 -0.5–2.4 -1.5–0.1 -1.3–2.3 -0.8–0.9 -0.4–0.5 -1.1–0.7 -1.0–0.7 -0.8–0.7
9 0.6–5.3 -1.0–0.9 -0.5–0.4 0.2–3.5 -1.2–0.6 -0.8–3.4 -1.1–0.7 -1.1–0.7 0.3–1.8 0.0–2.9 -0.3–2.0
10 -2.9–2.9 -0.2–1.7 -0.5–0.4 -0.1–4.1 -1.5–0.3 -0.3–4.9 -1.1–0.8 -1.0–0.7 0.0–2.9 0.7–2.5 -0.4–2.1
11 -1.6–1.8 -1.0–0.5 -0.4–0.4 -0.3–2.6 -1.4–0.1 -1.3–2.2 -1.0–0.6 -0.8–0.7 -0.3–2.0 -0.4–2.1 -0.1–0.4
Total 63.7–70.1 1.0–13.2 -5.6–6.9 33.9–46.0 -6.2–7.0 46.3–56.5 -2.6–10.2 -4.5–8.5 12.1–23.4 20.0–30.9 -8.4–5.1
Higher 42.2 1.1 1.0 22.0 0.0 33.8 2.5 0.9 9.1 17.5 -4.0
Table F.84: North / Temp. Gradient at 2cm / Mean
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 4.9–11.7 -2.5–1.9 -1.9–1.9 15.2–23.7 -1.4–2.2 1.6–10.2 -2.2–1.4 -2.4–1.4 -3.2–2.7 -2.5–4.0 -2.7–1.3
2 -2.5–1.9 2.6–3.8 -1.2–0.5 -0.7–1.9 -1.0–0.7 0.9–3.4 -0.9–0.8 -1.0–0.8 -1.1–1.2 -1.0–0.9 -1.2–0.5
3 -1.9–1.9 -1.2–0.5 -0.4–0.3 -0.1–1.4 -0.3–0.9 -0.1–1.7 -0.3–0.9 -0.2–1.0 -0.1–1.2 -0.4–0.8 -0.3–0.9
4 15.2–23.7 -0.7–1.9 -0.1–1.4 0.2–3.1 -0.8–2.3 -0.6–4.1 -0.5–2.7 -0.5–2.6 -1.0–2.6 -0.8–2.9 -0.5–2.6
5 -1.4–2.2 -1.0–0.7 -0.3–0.9 -0.8–2.3 0.8–1.4 -1.0–0.5 -0.7–0.5 -0.7–0.5 -0.8–0.5 -1.1–0.2 -0.8–0.3
6 1.6–10.2 0.9–3.4 -0.1–1.7 -0.6–4.1 -1.0–0.5 1.4–3.8 -0.3–3.0 -0.2–3.2 -1.3–2.8 -0.3–3.7 -0.2–3.0
7 -2.2–1.4 -0.9–0.8 -0.3–0.9 -0.5–2.7 -0.7–0.5 -0.3–3.0 -0.2–0.4 -0.2–0.9 -0.1–1.1 -0.1–1.1 -0.2–0.8
8 -2.4–1.4 -1.0–0.8 -0.2–1.0 -0.5–2.6 -0.7–0.5 -0.2–3.2 -0.2–0.9 0.2–0.9 -0.7–0.7 -0.6–0.8 -0.6–0.7
9 -3.2–2.7 -1.1–1.2 -0.1–1.2 -1.0–2.6 -0.8–0.5 -1.3–2.8 -0.1–1.1 -0.7–0.7 -0.2–1.2 -1.2–1.3 -1.3–1.0
10 -2.5–4.0 -1.0–0.9 -0.4–0.8 -0.8–2.9 -1.1–0.2 -0.3–3.7 -0.1–1.1 -0.6–0.8 -1.2–1.3 -0.2–1.3 -1.3–1.2
11 -2.7–1.3 -1.2–0.5 -0.3–0.9 -0.5–2.6 -0.8–0.3 -0.2–3.0 -0.2–0.8 -0.6–0.7 -1.3–1.0 -1.3–1.2 -0.3–0.3
Total 77.1–81.8 4.3–14.7 -5.2–6.2 52.7–63.4 -5.8–5.6 39.2–48.6 -5.5–5.8 -3.7–7.5 13.4–23.2 16.3–26.7 -5.5–6.0
Higher 46.8 4.9 -2.7 28.2 -1.3 24.2 -3.9 -1.4 15.8 17.2 -1.4
449
Table F.85: Control / Temp. Gradient at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 2.6–6.1 -1.6–2.8 -1.7–1.8 4.1–8.9 -1.7–2.0 0.2–6.4 -1.5–2.2 -1.9–1.8 0.8–5.7 -1.1–3.9 -1.7–2.0
2 -1.6–2.8 4.3–6.0 -1.0–0.7 -0.6–2.0 -0.8–0.9 3.8–9.2 -0.9–0.8 -0.6–1.1 -0.8–2.0 1.5–4.7 -0.9–0.8
3 -1.7–1.8 -1.0–0.7 -0.3–0.2 -0.4–0.8 -0.4–0.5 -0.8–2.8 -0.3–0.5 -0.4–0.5 -0.1–1.1 -0.2–1.2 -0.3–0.5
4 4.1–8.9 -0.6–2.0 -0.4–0.8 -1.0–1.1 -1.2–2.2 -2.8–2.3 -2.0–1.4 -1.4–2.1 -1.9–1.8 -1.6–2.2 -1.2–2.2
5 -1.7–2.0 -0.8–0.9 -0.4–0.5 -1.2–2.2 0.0–0.6 -1.1–2.6 -0.6–0.5 -0.5–0.6 -0.6–0.7 -0.9–0.9 -0.6–0.5
6 0.2–6.4 3.8–9.2 -0.8–2.8 -2.8–2.3 -1.1–2.6 11.8–15.7 -1.3–1.5 -1.6–1.2 -1.1–3.6 -0.7–4.6 -1.7–1.1
7 -1.5–2.2 -0.9–0.8 -0.3–0.5 -2.0–1.4 -0.6–0.5 -1.3–1.5 0.1–0.9 -0.8–0.6 -0.8–0.9 -0.8–1.2 -0.7–0.6
8 -1.9–1.8 -0.6–1.1 -0.4–0.5 -1.4–2.1 -0.5–0.6 -1.6–1.2 -0.8–0.6 -0.1–0.5 -0.9–0.8 -0.2–1.7 -0.6–0.7
9 0.8–5.7 -0.8–2.0 -0.1–1.1 -1.9–1.8 -0.6–0.7 -1.1–3.6 -0.8–0.9 -0.9–0.8 1.4–3.4 0.7–4.5 -1.7–1.2
10 -1.1–3.9 1.5–4.7 -0.2–1.2 -1.6–2.2 -0.9–0.9 -0.7–4.6 -0.8–1.2 -0.2–1.7 0.7–4.5 3.5–5.6 -1.4–1.4
11 -1.7–2.0 -0.9–0.8 -0.3–0.5 -1.2–2.2 -0.6–0.5 -1.7–1.1 -0.7–0.6 -0.6–0.7 -1.7–1.2 -1.4–1.4 -0.2–0.3
Total 40.4–52.5 3.1–18.3 -11.5–6.3 21.6–33.9 -11.3–6.4 46.9–58.0 -9.2–8.3 -9.0–8.3 18.3–32.2 20.8–34.4 -11.1–6.5
Higher 26.4 -5.9 -5.0 19.3 -4.2 24.5 -1.3 -1.7 14.9 12.3 -2.5
Table F.86: South / Temp. Gradient at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 15.9–20.6 -1.9–2.9 -0.8–2.4 0.9–5.5 -0.8–2.5 0.6–7.3 -1.3–2.1 -1.1–2.1 2.4–6.6 -0.6–4.4 -1.0–2.1
2 -1.9–2.9 6.2–7.9 -1.1–0.8 -1.2–1.1 -0.9–1.1 0.1–4.5 -0.9–1.1 -1.3–0.7 -0.9–1.5 -0.1–2.4 -1.0–0.9
3 -0.8–2.4 -1.1–0.8 -0.4–0.3 -0.7–0.5 -0.6–0.7 -1.5–1.2 -0.5–0.8 -0.5–0.8 -0.4–0.8 -0.5–0.7 -0.5–0.7
4 0.9–5.5 -1.2–1.1 -0.7–0.5 -0.8–0.7 -1.7–0.9 -2.0–1.7 -1.5–1.1 -1.2–1.5 -1.6–1.2 -1.8–1.4 -1.2–1.6
5 -0.8–2.5 -0.9–1.1 -0.6–0.7 -1.7–0.9 0.4–1.2 -1.9–1.1 -0.9–0.6 -1.0–0.4 -0.8–0.7 -1.1–0.4 -1.0–0.4
6 0.6–7.3 0.1–4.5 -1.5–1.2 -2.0–1.7 -1.9–1.1 12.2–15.4 -1.0–1.8 -1.3–1.6 -1.2–2.4 -0.4–3.4 -1.3–1.4
7 -1.3–2.1 -0.9–1.1 -0.5–0.8 -1.5–1.1 -0.9–0.6 -1.0–1.8 0.5–1.4 -0.9–0.8 -0.7–1.0 -1.1–0.9 -1.1–0.6
8 -1.1–2.1 -1.3–0.7 -0.5–0.8 -1.2–1.5 -1.0–0.4 -1.3–1.6 -0.9–0.8 0.3–1.2 -0.9–0.7 -1.1–0.6 -1.0–0.6
9 2.4–6.6 -0.9–1.5 -0.4–0.8 -1.6–1.2 -0.8–0.7 -1.2–2.4 -0.7–1.0 -0.9–0.7 1.2–2.7 -0.7–1.8 -1.3–1.0
10 -0.6–4.4 -0.1–2.4 -0.5–0.7 -1.8–1.4 -1.1–0.4 -0.4–3.4 -1.1–0.9 -1.1–0.6 -0.7–1.8 2.4–4.0 -1.0–1.3
11 -1.0–2.1 -1.0–0.9 -0.5–0.7 -1.2–1.6 -1.0–0.4 -1.3–1.4 -1.1–0.6 -1.0–0.6 -1.3–1.0 -1.0–1.3 -0.1–0.3
Total 51.2–59.3 6.2–18.7 -5.7–7.7 8.4–21.8 -8.5–5.6 39.2–49.6 -8.1–6.1 -7.7–6.3 5.5–18.2 8.7–20.9 -6.5–7.1
Higher 19.8 1.6 -0.0 12.9 -1.4 22.4 -2.4 -1.2 4.2 7.2 0.1
Table F.87: North / Temp. Gradient at 2cm / Maximum
@
@i
j
1 2 3 4 5 6 7 8 9 10 11
1 21.4–27.6 -2.2–3.0 -1.7–1.9 8.6–15.1 -1.8–1.7 0.7–6.8 -1.3–2.3 -2.0–1.6 2.1–7.3 -0.8–4.1 -1.7–2.0
2 -2.2–3.0 11.5–14.1 -1.8–0.7 -2.0–1.8 -1.5–0.9 0.1–3.8 -1.2–1.3 -1.0–1.5 -2.0–1.3 -1.8–1.2 -1.7–0.7
3 -1.7–1.9 -1.8–0.7 -0.1–0.6 -0.6–0.7 -0.5–0.5 -0.8–0.5 -0.5–0.7 -0.6–0.6 -0.5–0.7 -0.7–0.6 -0.6–0.6
4 8.6–15.1 -2.0–1.8 -0.6–0.7 -0.8–1.4 -0.4–2.8 -0.3–3.3 -0.2–3.1 -0.0–3.1 -0.4–3.0 -0.2–3.1 -0.2–3.0
5 -1.8–1.7 -1.5–0.9 -0.5–0.5 -0.4–2.8 0.6–1.3 -1.1–0.4 -0.9–0.5 -0.9–0.4 -1.2–0.4 -1.2–0.3 -1.0–0.3
6 0.7–6.8 0.1–3.8 -0.8–0.5 -0.3–3.3 -1.1–0.4 1.4–3.5 -0.2–2.7 -0.2–2.6 -0.7–2.5 -0.1–3.1 -0.4–2.4
7


















468
APPENDIX G
YELLOWSTONE CLUB DAILY LOGS