A model for predicting ion concentrations in the Yellowstone River between Billings and Miles City,
Montana : a management tool
by Richard William Karp
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Industrial and Management Engineering
Montana State University
© Copyright by Richard William Karp (1976)
Abstract:
Regression equations were developed for the individual common ions versus electrical conductivity
(EC), and the individual common ions versus EC and flow (Q). These regression equations relate the
instantaneous concentration of common ions in Meq/L to the instantaneous EC and Q in μmhos/ cm @
25°c and cfs, respectively. The regression equations relating the common ion concentrations to EC
produced relatively good results. A cation-anion balance was used to determine the quality of the
simulated common ion concentrations.
Water quantity and quality models based upon the principle of conservation of matter and employing
the macro-to-micro philosophy were developed on an annual basis. A methodology for obtaining the
quality model from the quantity model was developed. The quantity model simulates the amounts of
flow associated with the various flow components; and the quality model simulates the EC associated
with the various flow components. The use of EC in the quality model provides the necessary
parameter for the use of the regression equations involving the common ion concentrations.
Both the quantity and quality models provide reasonable results that would be expected on an annual
basis. 
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In presenting this thesis in partial fulfillment of the 
requirements for an advanced degree at Montana State University,
I agree that the Library shall make it freely available for inspection 
I further agree that permission for extensive copying of this thesis 
for scholarly purposes may be granted by my major professor, or, in 
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any copying or publication of this thesis for financial gain shall 
not be allowed without my written permission.
Date
Signa
/
'A MODEL FOR PREDICTING ION CONCENTRATIONS IN THE 
■ YELLOWSTONE RIVER BETWEEN BILLINGS AND 
MILES CITY, MONTANA: A MANAGEMENT TOOL
"by
RICHARD WILLIAM KARP
A thesis submitted in partial fulfillment 
of the requirements for the degree
of
MASTER OF SCIENCE 
in
Industrial and Management Engineering
Approved:
lttee
Major Department
Graduate Dean
MONTANA STATE UNIVERSITY 
Bozeman, Montana
December, 1976
ill
ACKNOWLEDGMENTS
The author wishes to express his appreciation to the Montana 
State Department of Health and Environmental Sciences, Water Quality. 
Bureau, the United States Environmental Protection Agency, and the Montana 
University Joint Water Resources Research Center at Montana State Univer­
sity for the financial support in the development of this thesis.
To Dr. Donald W. Boyd, Chairman of my graduate committee, and to 
the other members of my committee, I extend my thanks for the help they 
have provided.
Finally, to my wife Kathy, for spending many lonely nights and 
tolerating my anxiety when progress faltered, I give my sincerest thanks.
TABLE OF CONTENTS
Page
V I T A ....................     ii
ACKNOWLEDGMENT ......................    ill
LIST OF TABLES............ ■...................... .............. vi
LIST OF F I G U R E S ........................  ix
ABSTRACT ......................................................... x
CHAPTER I - Introduction ............................  . . . . . .  I
Scope and Objectives ................    3
Physical Description of the Study Area . , , ............. 4
CHAPTER II - Ion Concentration Versus Electrical Conductivity
and Flow Relationships   7
Data Availability and Preparation.................. . . 10
Regression Analysis . . . . . . . . . . . . . . . . . . .  22
CHAPTER III - Large Scale Water Quantity and Quality Models . . .  ?2
Quantity Model ...........................................  73
Quality M o d e l ................   84
CHAPTER IV - Concluding Remarks ............  . . . . . . . . . .  101
Current Utility . ................    101
Recommendations for Future Research . . . . .  ............  102
REFERENCES ....................  . . . . . . . .  ................  103
APPENDICES........................................ .............. 108
APPENDIX I - Raw D a t a ........................   10Q
APPENDIX II - Chi-Square Goodness of Fit Test, Z-Test
Statistic..........................  l4l
VPage
APPENDIX III - Quantity Model; 1944 to 1973 ............  144
APPENDIX IV - Quality Model; 1944 to 1968, 1944 to 1973 • 154
vi
LIST OF TABLES
Table Page
1 Equivalent Weights and Conductance Factors for the
Common I o n s ..........................................   8
2 Data Availability for the Four Sites.......... .. . . .  0 12
3 Summary of Hypothesis Testing for the Various Sites . . .  20
4 Summary of Linear Regression Equations with Iog(ECM)
as the Independent Variable for the Yellowstone River 
at Billings . . . . . . . . . . .  ....................  23
5 Summary of Linear Regression Equations with Iog(ECM)
as the Independent Variable for the Bighorn River
at Bighorn . . . . . . . . . . . . . . . . . . . . . .  24
6 Summary of linear Regression Equations with Iog(ECM)
as the Independent Variable for the Tongue River
at Miles City ........................   25
7 Summary of Linear Regression Equations with Iog(ECM)
as the Independent Variable for the Yellowstone River
at Miles C i t y ..................   26
8 Simulated Common Ion Data in Mg/L for the Yellowstone
River at Billings ................................... . 29
9 Simulated Common Ion Data in Mg/L for the Bighorn
River at Bighorn . . . . . . . .  ....................  33
10 Simulated Common Ion Data in Mg/L for the Tongue
River at Miles City ............................. 37
11 Simulated Common Ion Data in Mg/L for the Yellowstone
River at Miles City ............................. .. 4l
12 Log(ECM) Versus Iog(Q) Linear Regression Equations for
the Sites ..............     46
13 Log(ion) Versus Iog(ECM) and Iog(Q) Linear Regression
Equations for the Yellowstone River at Billings . . . .  47
Table Page
14 Log(ion) Versus Iog(ECM) and Iog(Q) Linear Regression
Equations for the Bighorn River at Bighorn.......... .' ' 48
15 Log(ion) Versus Iog(ECM) and Iog(Q) Linear Regression
Equations for the Tongue River at Miles City . . . . . .  4^
16 Log(ion) Versus Iog(ECM) and Iog(Q) Linear Regression
Equations.for the Yellowstone River at Miles City . . .  $0
I? Simulated Common Ion Data for the Yellowstone River
at Billings in Mg/L ............  . . . . . . . . . . .  52
18 Simulated Common Ion Data for the Bighorn River
at Bighorn in M g / L ................................   5&
19 Simulated Common Ion Data for the Tongue River
at Miles City in Mg/L .................................  60
20 Simulated Common Ion Data for the Yellowstone River
at Miles City in M g / L ........ ........................ 6.4
21 Composite Data from the U. S, Geological Survey and
Instantaneous Data from the Montana State W, Q. B. for
the Tongue River at Miles City ......................... 67
22 Simulated Common Ion Data for Composite U. S. Geological
Survey and Instantaneous Montana State W. Q, B. Data
for the Tongue River at Miles City .................... 69
23 Summary of Equations for Quantity Model  ..........  85
24 Summary of Equations for Quality Model . . . . . . . . . .  94
25 ECM Values for the Essential Parameters of the Quality
Model, 1944 to 1968, Using L2 and L5 as Exogenous
(units = /mhos/cm @ 25°c) ....................... 96
26 ECM Values for the Essential Parameters of the Quality
Model, 1944 to 1968, Employing Perturbed Values of L5
(units - jumhos/cm @ 25° c ) ............ . 9?
vii
viii
Table Page
A Raw Data for the Yellowstone River at Billings H O
B Raw Data for the Bighorn River at Bighorn . . . . . . . .  H 4
G Raw Data for the Tongue River at Miles City . . . . . . .  H S
D Raw Data for the Yellowstone River at Miles City . . . . .  122
E Common Ion Data for the Yellowstone River at Billings
in Meq/L .......................................  124
F Common Ion Data for the Bighorn River at Bighorn
in Meq/L . . . . . . . . . . . . . . . . . . . . . . .  128
G Common Ion Data for the Tongue River at Miles City
in Meq/L . . . . . . . . . . . . . . . . . . . . . . .  132
H Common Ion Data for the Yellowstone River at Miles City
in M e q / L ............................................. 136
I Primary Flow Data 30 Years, 1944 - 1973 ........... 138
J Simulated Values of ECM for 25 Years 1944 - 1968 . . . . .  1-40
• LIST OF FIGURES
Table Page
1 Schematic Map Showing Middle Yellowstone. River Basin . . . .  5
2 Grouped ERR Data for the Yellowstone River at Billings . . .  i6
3 . Grouped ERR Data for the Bighorn River at B i g h o r n ........  17
4 Grouped ERR Data for the Tongue River at Miles City . . . .  18
5 Grouped ERR Data for the Yellowstone River at Miles City , . 19
6 Schematic of a Sub-basin for an Annual Flow Model . . . . .  74
7 Schematic of Quantity Model Used in Sub-basin 42KJA . . . .  77
8 Schematic of Surface Water Storage ..............   79
9 Schematic of Ground Water Storage ............... - 80
10 Schematic of Quality Model Used in Sub-basin 42KJA......... 88
ix
XABSTRACT
Regression equations were developed for the individual common ions 
versus electrical conductivity (EC), and the individual common ions versus 
EC and flow (Q). These regression equations relate the instantaneous con­
centration of common ions in Meq/L to the instantaneous EC and Q in pmhos/ 
cm @ 25°c and cfs, respectively. The regression equations relating the 
common ion concentrations to EC produced relatively good results. A cation- 
anion balance was used to determine the quality of the simulated common ion 
concentrations.
Water quantity and quality models based upon the principle of con­
servation of matter and employing the macro-to-micro philosophy were 
developed on an annual basis. A methodology for obtaining the quality 
model from the quantity model was developed. The quantity model simulates 
the amounts of flow associated with the various flow components; and the 
quality model simulates the EC associated with the various flow components. 
The use of EC in the quality model provides the necessary parameter for the 
use of the regression equations involving the common ion concentrations.
Both the quantity and quality models provide reasonable results that would 
be expected on an annual basis.
. >
CHAPTER I
Introduction ..
The passage of the Federal Water Pollution Control Act Amend­
ments of 1972, Public Law 92-500, exemplifies- a major concern for the 
protection of water quality in the nation's waters. Congress stated 
that the goals of the Act were to restore and maintain the chemical, 
physical, and biological integrity of these waters (Ref. l). Thus, 
the Act implied the need to describe natural variation in the chemical, 
physical, and biological behavior of the water ecosystem. These 
descriptions are needed in order that rational management decisions, 
can be made with respect to water quality.
In response to P. L. 92-500, many governmental agencies that 
are concerned with water resource management have been collecting water 
quality data on the nation's waterways. As water quality data increases 
in amount, it is necessary to analyze the data for relationships which 
can be used to determine what the natural variations of the various 
factors are in the ecosystem. These relationships can also provide 
predictions needed in water quality management. Further, such relation­
ships may identify those parameters for which more data is needed and 
those parameters for which data collection can be discontinued.
Recent concern for the water quality in the Yellowstone River 
Basin in Montana has increased due to the increase in coal and energy 
related developments in the middle portion of the basin. An intent of 
this study was to analyze the available data for relations that would
I2
aid in the water quality management of this area of the Yellowstone 
River Basin.
The inverse relationship between the concentration of dissolved 
constituents and flow STA has been well documented in the literature. 
This hyperbolic relationship has been pursued in a number of inves­
tigations on natural waters to determine regression equations. These 
regression equations are generally in terms of the logarithm of con­
stituent concentration versus the logarithm of the flow (Ref. 7-11)• 
Total dissolved solids (TBG) has received by far the most attention in 
its inverse relationship with Q (Ref. 8, 9)• To date, these regression 
equations have involved the concentration of dissolved constituents in 
mass per volume units. In this study, however, the concentration of 
the common ions used in regression equations are in terms of milli- 
equivalence per litre (Meq/b) which is a measure of the number of 
charged ion particles per volume. The regression equations are in terms 
of these concentrations for the common ions versus the measured electri­
cal conductivity (ECM) and Q, In a recent investigation by Leonard 
Frost, Jr. (Ref. 5)* linear regression analysis was performed on a 
number of dissolved constituents, including the common ions in mass per 
volume units of concentration, with ECM.
The majority of water quantity and quality models documented 
in the literature develop an extremely detailed model of a small incre­
mental element of a basin. Often times, because of the extremely 
complex nature of the interacting components of a real world hydrologic
• 3 ’
system, the model is either unsoIvable, or because of generalizations 
and approximations does not give realistic results. In this study, 
the macro-to-micro philosophy of modeling of large scale systems was 
employed. This approach provides a number of advantages. First of 
all, it provides a working model of the system from the very beginning. 
Even though the model at the most macro level may not produce usable 
results, it provides a stepping stone toward the micro level. The 
formulation of relationships at the macro level provide structure that 
can be built upon at the next level of modeling. Thus, a systematic 
development of working models is produced which allows for feedback of 
information. The next level of modeling, toward the micro, provides a 
check on the model at the previous level. Another advantage to this 
method of modeling is that in many instances usable results can be 
obtained from the first few levels (Ref. 2).
Scope and Objectives
The scope and objectives of this study were:
I. To develop a system of regression equations for predicting 
the level (concentration) of selected water quality parameters at 
specific points along the Yellowstone River and, its major tributaries 
between Billings and Miles City, Montana. Relationships were sought 
between the common ions - calcium (Ca), magnesium (Mg), sodium (Na), 
potassium (K), bicarbonate (HCO3), carbonate (CO3), sulfate (SO4), and 
chloride (Cl) - and the measured electrical conductivity (ECM), and flow
4(Q). The specific points on the Yellowstone River were the U. S . 
Geological Survey water quality and quantity stations at Billings and 
Miles City. The points on the major tributaries were the U. S . Geolog­
ical Survey water quality and quantity stations on the Bighorn River 
near its confluence with the Yellowstone River at Bighorn, Montana, and 
on the Tongue River near its confluence with the Yellowstone River at 
Miles City.
2. To initialize an approach to the large scale modeling 
required to describe incremental changes in water quality parameters as 
a result of various water use categories in the Yellowstone River Basin. 
The Montana State Department of Natural Resources and Conservation's ■ 
water planning model was calibrated on an annual basis for the Yellow-" 
stone River sub-basin 42KJA (Ref. 2, 3)• The water planning model is a 
flow, hydrologic, or "quantity" model which has the potential of provid­
ing the flow component quantities required in modeling water quality.
■ After the calibration of the water quantity model for sub-basin 42KJA 
on an annual.basis, this model was employed in developing a water quality 
model.
Physical Description of the Study Area
Figure I shows a sketch map of the middle portion of the Yellow­
stone River Basin. This portion of the Yellowstone River Basin is on 
the western edge of the Great Plains. The basin is generally an area of 
rolling hills with gentle to moderate relief. The major surface water
Figure I — Sketch map showing Middle Yellowstone River Basin
(Taken from Ref. 12)
6inflows to the study area are the Yellowstone and Bighorn Rivers. The 
confluence of the Yellowstone and Tongue Rivers mark the eastern bound­
ary of the study area. The Tongue River is not included as an inflow to 
sub-basin 42KJA.
Mountains to the west and southwest of the middle portion of 
the Yellowstone River Basin create a moisture shadow. Air masses 
moving toward the basin from these directions must rise to get over 
the mountains, and thus lose most of their moisture before reaching the 
basin. Average annual precipitation for the area ranges from 11 to l6 
inches per year (Ref. 4).
About 75 percent of the land in the middle portion of the Yel­
lowstone River Basin is used for pasture and range land in the production 
of livestock. Of the irrigated lands, about 70 percent is used in the 
production of hay, pasture and sugar beets. There is also a substantial 
amount of dryland farming for the production of grains. With the renewed 
interest in coal development, certain areas of this portion of the basin
may, in the future, be supporting vast strip mines.
*
Agricultural irrigation is the primary water use in the basin. 
With the increase in coal and energy development expected to occur, the 
amount of water needed for industrial use "will increase, but agricultural 
uses will probably still dominate because of existing water rights.
CHAPTER II
Ion Concentration Versus Electrical 
Conductivity and Flow Relationships
In this study, the dissolved constituents of interest are the 
common ions. The common ions represent the major cations and anions 
dissolved in natural waters. These ionic species are the positive 
(cations) and negative (anions) components of salts that have become 
dissolved in the water. These common ion constituents of natural waters 
are most likely derived from direct chemical solution of minerals in 
rocks and in soils in which the water comes into contact.
The common ion concentrations are generally expressed in milli­
grams per litre (Mg/l). This is a mass (weight) concentration. The 
mass of an ionic constituent divided by its equivalent combining mass 
(weight), termed equivalent weight, gives a measure of the number of 
ionic charges (positive or negative) of that specie in the water solution. 
The equivalent weights of the common ion, listed in Table I, can be 
found in any general chemistry textbook. The result of the division of 
the mass concentration of an ionic specie in Mg/L by its equivalent 
weight has the units milli-equivalence per litre (Meq/b). In all natural 
water solutions there exists electrical neutrality, i.e. the number of 
positive charges equals the number of negative charges. Thus, by 
summing the concentration in Meq/L, of all the positively charged ions 
(cations) and subtracting the sum of all the negatively charged ions 
(anions) the result should be zero. Of course, there are many other
8TABLE I
Equivalent Weights and Conductance 
Factors for the"Common Ions
Common Ion 
(abbr.)
Equivalent Weight 
(Mg/Meq)
Conductance Factor 
(jimhos/cm @ 25°c/Meq/L)
Ca 20.040 52.0
Mg 12.156 46.6
Na 22.990 48.9
K 39.102 72.0
HCO3 60.998 43.6
CO3 29.995 84.6
SO4 48.030 73.9
Cl 35.453 75.9
\
9constituents dissolved in natural waters besides the common ions. 
Generally, the total amount of these other constituents is so small 
compared to the total amount of the common ions that they add little to 
the sum of either cations or anions and are well within the experimental 
error involved in measuring the common ions. Thus, in determining the 
sum of cations and anions in natural waters only the common ions are 
considered. The difference between the sum of cations and the sum of 
anions is termed the cation-anion balance. The cation-anion balance is 
generally used to indicate the quality of a chemical analysis performed 
on a sample of natural water.
Since the common ions exist in natural waters as charged species, 
natural waters are electrolytic; i.e., they are capable of carrying an 
electric current. The conductance of a water solution is related to the 
amount and type of dissolved ionic constituents. For example, a solu­
tion with a high concentration of common ions has a higher conductance 
than does a solution with a low concentration of common ions. Also, 
under identical conditions, potassium ions are better conductors than 
calcium ions. This is an innate feature of the individual ions. Thus, 
each ionic specie in a solution of natural water makes a contribution to 
the overall conductance of that solution. The conductance for strong 
electrolytes as the common ions are classified, approaches a definite 
value at infinite dillution. That is, when no other ions are present 
and the concentration of a particular ion is decreased, the conductivity 
due to that ion approaches a limiting value. This limiting value is
10
generally termed the conductance factor of the particular ion. Values of 
the conductance factor for the common ions are given in Table I in the
I
units of micro-pmhos per centimeter at 25°c per Meq/L (/mhos/cm @ 25°c/ 
Meq/L). The common unit of conductance reported in the analysis of natu­
ral waters is /mhos/cm @ 25°c. Of course, solutions of natural waters are 
not infinitely dilute and thus, an electrical conductivity calculated from 
the amounts of the various common ions (ECC) is generally higher in value 
than the actual measured electrical conductivity (ECM), This is primar­
ily due to the perturbing effect the ions have on one another, which 
decreases their mobility, As a part of this study, an investigation was 
made to determine whether a statistical relationship existed between ECM 
and ECC, and the individual common ions. All relationships between common 
ions, electrical conductivity and flow rate employ Meq/L for common ion 
concentration, jumhos/cm @ 25°c for electrical conductivity, and cubic feet 
per second (cfs) for flow.
Data Availability and Preparation
For the common ion and ECM data available at the selected sites, 
there are two types, composite and instantaneous. Composite data are 
obtained by combining a number of samples together according to the 
value of the flow at the instant each sample was taken. Thus, a com­
posite gives a flow weighted average over a particular time interval. 
Composite data for the selected sites are available from U. S. Geological 
Survey annual water resources reports (Ref. 6). Instantaneous data are 
obtained from individual samples taken at any instant in time. The term
11
grab sample data is often used to imply instantaneous data. Instanta­
neous data is available from the U. S . Geological Survey annual reports 
and from the Montana State Water Quality Bureau. Since one of the 
objectives of the study was to determine relationships between common 
ion concentration, EGM1 and Q, the instantaneous data were preferred to 
the composite. Also, once the regression equations for the common ions 
were developed the composite data would possibly provide a check, much 
like the split record approach of validation in modeling. Data from 
the Water Quality Bureau was generally incomplete; i.e., missing certain 
values of common ion parameters or not having instantaneous flow values. 
The period of record for which data are available at the four sites is 
tabulated in Table 2. Only data predating Yellowtail Reservior (196?) 
were used for the site on the Bighorn River. The data is usually taken 
at monthly intervals. The earlier data are generally composite and 
the later data, instantaneous. ' Another reason for preferring the 
instantaneous data in the construction of the regression equations is 
that the most current data has the benefit of better technology.
As previously mentioned, the cation-anion balance is commonly 
used to determine the quality of a common ion analysis. This same idea 
was extended to determine the quality of the available instantaneous 
data. By using the cation-anion balance, a screening process was devel­
oped to exclude any sample data for which there was a high probability 
of error either in the chemical analysis or in the printing of the data 
for publication. A data point consists of the measured concentrations of
12
Data Availability for the Four Sites *
TABLE 2
Site Period of Collection Type of Data
Yellowstone River 10/50 - 9/58 Composite
at Billings
7/6] - 12/6? Composite
l/?0 - Current Instantaneous
Bighorn River 11/45 - 8/47 Composite
at Bighorn
3/49 - 12/50 Composite
1/51 - 12/69 Composite
l/70 - Current. Instantaneous
Tongue River 9/48 - 9/49 Composite
at Miles City
1/51 - 12/69 Composite
l/70 - Current Instantaneous
Yellowstone River 10/68 - Current Composite/
at Miles City Ins tantaneous
*For the site, Yellowstone River at Miles City, the flow is 
taken below the confluence with the Tongue River and the common ion 
and electrical conductivity data are taken above the confluence.
13
the various common ions in Meq/L, the measured electrical conductivity 
(ECM) in pmhos/cm @ 25°c and the measured flow ST A in cfs. The cation- 
anion balance which was termed the difference between the sum of the 
cations and the sum of the anions is not in itself a good gage. For 
example, suppose that the sum of the cations and the sum of the anions 
were 2.00 and 1.50 Meq/L, respectively. Then the cation-anion balance 
would be .50 Meq/L. Now suppose for another sample the sum of the 
cations and the sum of the anions were 5»00 and 4.50 Meq/L respectively. 
Again, the cation-anion balance would be .50 Meg/L, but clearly the - 
cation-anion balance in the former case is more seriously in -error than 
in the latter case. In order to use the cation-anion balance for com­
parison to determine quality, the balance was normalized by dividing by the 
average of the sum of cations and anions. This normalized cation-anion 
balance was termed the error of balance (ERR) and is given algebraically 
by:
u) ■ERR
(s  cations - a anions)
( cations - s anions)/2
Tables A through D in Appendix I are tabulations of the raw data for 
each of the four' sites. The concentration of the common ions for this 
data is in Mg/L. Tables E through H in Appendix I give the concentration 
of the common ions in Meq/L, the value of ERR, the ERR sample mean
(ERR), and the ERR sample standard deviation (SgTffl) for each data point 
at the four sites.
For a^given data point, the values of bicarbonate (HOC^) and
14
carbonate (CO3) together make up what is termed the alkalinity (ALK).
In natural waters, especially in this region, HGO3 is by far the major 
contributor to ALK. This is basically due to the fact that GO3 can only 
exist in natural waters when the pH is greater than 8.3. At a pH of 8.3» 
any CO3 present is chemically converted to HGO3. HGO3 also has another 
source besides the solution of minerals from rocks and soils, photo­
synthesis. Photosynthesis produces carbon dioxide (COg) which combines 
with water (H2O) to produce HCO3. CO3 seems to be significant when pre­
sent, but most often it is not present. ALK and HGO3 were used in this 
study rather than HGO3 and CO3. When good regression equations exist for 
ALK and HGO3, the concentration of CO3 can be determined from the dif­
ference in ALK and HCO3.
For the data points at each of the four sites, the ERR values 
were statistically tested for normal distribution with population mean 
SEMN0 A equal to zero. A chi-square goodness of fit was employed to test 
this hypothesis. Rejection of the hypothesis could imply the following 
conditions:
1. The sample is not representative of one having an underlying 
normal distribution.
2. The population mean (jUgpp) is not zero.
3. Both, A^ERR is not equal to zero and the underlying distri­
bution is not normal.
A rejection of the hypothesis because Ugpp is significantly different than 
zero could mean that the assumption of the common ions being the major
15
cations and anions in natural water solutions is not upheld. Neglecting 
a significant constituent in the determination of ERE would cause 
to "be different than zero. Figures 2 through 5 show plots on normal 
probability paper of the ERR data associated with each site. If the 
ERR data falls along a straight line this indicates the possibility of an 
underlying normal distribution. The chi-square goodness of fit test for 
the ERR data for the Yellowstone River at Billings is shown in Appendix 
II. The chi-square goodness of fit tests for the other three sites 
were performed similarly. For each site, except the Yellowstone River 
at Miles City, the. underlying distribution was determined to be normal 
with ^ ERR equal to zero. The level of significance S?PA for the chi- 
square goodness of fit tests was set at .05. Since the hypothesis of 
a normal distribution with RIONN equal to zero was rejected for the Yel­
lowstone River at Miles City, a hypothesis was set up to test whether or 
not CIO00 was significantly different than zero. The z test statistic 
was employed for testing this hypothesis. The actual hypothesis test is 
shown in Appendix II. The results of these hypothesis tests are summa­
rized in Table 3.*
Using the normal distribution, upper and lower limits were 
determined on the ERR data for a 95 percent probability of occurrence.
Pr(L < ERR < u) = .95
P r /  1 ~ ^ERR /  ERR -  PERR /  u ~ ' ^ERR 
\  tfERR tfERR tfERR
Thus, the lower limit, L, is given by:
Gr
ou
pe
d 
in
te
rv
al
s
20 30 40 50 60 ?0 80
Gummulative Probability
98 995 10 90 95
Figure 2. Grouped ERR Data for Yellowstone River at Billings
Gr
ou
pe
d 
Da
ta
 I
nt
er
va
ls 02 -
20 30 40 50 60 70 80
Gummulatlve Probability
98 9990 95
Figure 3. Group ERR Data for Bighorn at Bighorn
20 30 40 50 60 70 80
Cummulative Probability
95 98 99
Figure 4. Grouped ERR Data for Tongue River at Miles City
OO -
20 30 40 50 60 70 80 90 95 98 99
Gummulative Probability
Figure 5. Grouped ERR Data for the Yellowstone River at Miles City
TABLE 3
Summaxy of Hypothesis Testing for the Various Sites *
Critical Test*
Site Hypothesis Test Test Statistic Degrees of Freedom Statistic Limit
Yellowstone River Chi-square X2 = 5.28 5 = 11.0?
at Billings Goodness of Fit
Bighorn River Chi-square X2 = 2.90 3 X2 = 7.81
. at Bighorn Goodness of Fit
Tongue River Chi-square X2 = 3.67 3
C
OclI
at Miles City Goodness of Fit
Yellowstone River Chi-square X2 - 17.77 3
C
OclI
at Miles City Goodness of Fit
Yellowstone River, Z Test Z = 0.51 N < ± 1.96at Miles City for Means
* ,X= .05 level of significance for all tests
21
Y n prestighO00
and the upper limit, U, is given try
a n restltf00
The sample standard deviation (Sgpp) was used as an approximation to 
<5O 0 0 e These upper and lower limits, termed acceptance limits, were used 
to screen the data. For example, consider a data point whose ERR value 
falls outside the interval created by the acceptance limits. Such a 
data point has a probability of occurrence of only .05. Since errors can 
be made in the chemical analysis of a water sample or in the tabulation 
and publishing of the data, there is a relatively high probability that 
some of the data for this data point is incorrect. By rejecting data 
points whose ERR values lie outside the interval, only good quality data 
is used in the development of the regression equations. One of the char­
acteristics of least-square regression analysis is that minimum variance 
rather than minimum deviation is used. Thus, if erroneous data is used 
there is the possibility of the analysis showing little or no statistical 
correlation when in fact one does exist. Since the hypothesis of a 
normal distribution for ERR was rejected for the Yellowstone River at Miles 
City, no attempt was made to screen this data. This screening process 
resulted in the rejection of 3 data points from the 68 data points for 
the Yellowstone River at Billings, 3 data points from the 66 data points 
for the Bighorn River at Bighorn, and 2 data points from the 66 data 
points for the Tongue River at Miles City.
22
Regression Analysis
At each of the four sites, linear regression analyses were per­
formed on the acceptable data.' A value for the calculated electrical ■ 
conductivity (ECC) was determined for each data point at each site using 
the conductance factors’ in Table I.
.ECC = K i  Ki (iqn)i ' (2) . .
Where Ki is the conductance factor for the ith ion, and •
(ion)i is the concentration in Meg/L of the ith ion.
The ECC along with the common ions were regressed individually.with the . 
natural logarithm of the ECM as the independent variable. The results of 
the linear regression analyses for the four sites are shown in Tables 4 
through 7» The general form of the regression■equations is*
(ion)i = bl Iog(ECM)■ - at (3)
Where:
(ion) ^ is concentration of the 'ith 
ion in Meg/L or ECC
ai and bi are the regression .
■constants for the ith ppn
ECM- is in /mhos/cm @ 25°c ■
The regression equations have high values except for. those.of K and. . 
Cl. • The R^ values for these two ions are consistently low for each site. 
The t-values listed in Tables 4 through 7 refer to the significance of 
the coefficient bi on the independent variable. The..05 .significance 
level (<<) for the regression : . •. .
TABLE 4
Dependent t
Summary of Linear Regression Equations with Log (ECM) as the
Independent Variable for the Yellowstone River at Billings
Variable Regression Equation R2 Value
ECC ECC = 359.68' Log (ECM) - 1699.03 .959 38.60
Ca Ca = 1.322 • Log (ECM) - 6.004 .923 27.45
Mg Mg = 0.854?' Log (ECM) - 4.022 .939 31.02
Na Na = 0.9595* Log (ECM) - 4.595 .891 22.68
K K = .05645 • Log (ECM) - .2509 .561 8.97
ALK ALK = 1.543 • Log (ECM) - 6.716 - .963 40.41
HCO3 HCO3= 1.538' Log (ECM) - 6.686 .962 39.65
SO4 SO4 = 1.454 • Log (ECM) - 7.153 .908 24.8?
Cl Cl = .1409 ' Log (ECM) - .6565 .669 11.29
TABLE 5
Summary of Linear Regression Equations with Log (ECM) as the
Independent Variable for the Bighorn River at Bighorn
Dependent t
Variable ' Regression Equation 0 2 Value
ECC ECC = 1116 • Log (ECM) - 6479 .926 27.66
Ca Ca = 3.04? . Log (ECM) - 16.91 .822 16.75
Mg Mg = 2.420 • Log (ECM) - 14.17 .791 15.19
Na Na = 4.194 • Log (ECM) - 24.90
CO
18.85
K K = .07011 • Log (ECM) - .3750 .396 6.33
ALK ALK = 2.132 • Log (ECM) - 11.04 .736 13.03
HCO3 HCO3= 2.092 • Log (ECM) - 10.78 .707 12.13
SO4 SO4 = 7.018 • Log (ECM) - 41.66 .892 22.50
Cl Cl = .2846 • Log (ECM) - 1.635 .571 9.01
TABLE 6
Summary of Linear Regression Equations with Log (ECM) as the
Independent Variable for the Tongue River at Miles City
Dependent t
Variable . Regression Equation R2 Value
ECC ECC = 1023 * Log (ECM) - 5813 .940 31.27
Ca Ca = 2.278 • Log (ECM) - 12.15 .80? 16.09
Mg Mg = 3.756 • Log (ECM) - 21.64 .898 23.35
Na Na = 3.067 . Log (ECM) - 17.84 .811 16.30
K ' K = .09219 • Log (ECM) - .4877 ' .514 8.09
ALK ALK = 3.448 • Log (ECM) - 18.56 .830 17.37
HCO3 HCO3= 3.411 • Log (ECM) - 18.33 .822 16.94
SO4 SO4 = 5.595 * Log (ECM) - 32.65 .898 23.38
Cl Cl = .09498 • Log (ECM) - .5214 • 535 8.45
Dependent t
Variable 1 Regression Equation 02 Value
TABLE 7
Summary of Linear Regression Equations with Log (ECM) as the
Independent Variable for the Yellowstone River at Miles City
ECC ECC “ 625.6 • Log (ECM) - 3272 .960 26.86
Ca Ca = 1.834 • Log (ECM) - 9.130 .933 20.50
Mg Mg = 1.355 • Log (ECM) - 7.070 .907 17.14
Na Na = 2.20? • Log (ECM) - 11.83 .948 23.34
K K = .0620 • Log (ECM) - .3069 .837 12.41
ALK ALK = 1.660 • Log (ECM) - 7.816 .960 26.84
HCO3 HCO3= 1.637 • Log (ECM) - 7.688 .957 25.92
SO4 SO4 = 3.610 ' Log (ECM) - 19.56 • 927 19.57
Cl Cl = .1926 • Log (ECM) - 1.006 .840 12.56
2?
constant, "bi, is + I.96. If the t-value is greater than I.96 or less 
than -I.96, then the regression coefficient has statistical significance. 
Thus, the coefficients, hi, are all significant, even those for the 
regression equations of K and Cl. The low values for the K and Cl ■ 
regression equations may suggest that the variations in K and Cl are not 
sufficiently measurable with variations in ECM. The magnitude of the K 
and Cl concentrations are relatively small in comparison to those of the 
other common ions. K and Cl concentrations are reported generally to 
only two significant figures. The current analytical methods for deter­
mining K and Cl concentrations can yield more accurate results than 
have been published. The publishing of K and Cl concentrations to only 
two significant figures is probably due to historical publishing and the 
fact that more accurate results have little effect on the cation-anion 
balance. By replacing the regression equations for K and Cl with the 
regression equation for ECC and the cation-anion balance equation, a 
better set of regressions might be produced. The procedure would be to 
calculate all the common ion concentrations, except for K and Cl., from 
their respective regression equations and then solve the ECC regression 
equation and the cation-anion balance equation simultaneously for K and 
Cl. In the ECC regression equation, ECC would be replaced by its 
expression from equation (2). The undesirable aspect of this system of 
equations is that the cation-anion balance equation is lost as a check 
on the predicted values of the common ions. The possibility of using 
the ECC regression equation to show consistency is also lost.
28
The form of the regression equation for ECG may he quite realis­
tic, i.e. a physical regression equation as opposed to a purely 
statistical regression equation. For high ECM values the EGC values 
would tend to be quite a bit larger than the ECM values, whereas at 
low ECM values the ECC values would tend to be closer to the ECM values. 
Physically, this may be due to increased interference with ion mobility 
and the possible lack of full ionization of the salt molecules at high 
ECM values. The high values associated with the regression equations 
for the common ions may indicate that the relative proportions of the 
common ions in natural water solutions remain somewhat constant. For 
example, suppose that the fraction, (ion)^/ECC, equaled a constant:
(Ion)iZECC = Ci (4)
Then from the regression equation for ECC
ECC = B log (ECM) - A
upon substituting for ECC from equation (4) and rearranging terms, the 
following relation is obtained:
■ (Ion)i = (Ci1B) log (ECM) - (Ci-A)
This is the form of the regression equations for the common ions, equa­
tion (3) where (Ci-B) would be bi and (Ci-A) would be ai.
Using the regression equations in Tables 4 to 7* and the ECM data 
from which the regression equations were originally generated, values of 
the common ions were simulated. These simulated values for the common 
ions for the four sites are shown in Tables 8-through 11. The common ion 
concentrations have been converted back to Mg/L for comparison with
TABLE 8
Simulated Common ion Data in Mg/L for
the Yellowstone River at Billings
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO 3 CO 3 S04
(Mg/L) (Mg/L) (Mg/L)
Cl
(Mg/L) ERR
(jumho/cm 
ECM'
@ 25°c) 
ECM
37.9 13.2 26.1 3.4 152.7 .0 73.6 6.6 -.0046 391 393
37.7 13.1 25.9 3.4 151.7 .0 72.9 6.5 -.0048 38? 389
38.3 13.3 26.4 3.4 153.8 .0 74.5 6.6 -.0044 396 398
42.4 14.9 29.8 3.7 168.4 .0 85.4 7.4 -.0021 459 465
40.8 14.3 28.5 3.6 162.8 .0 81.2 7.1 -.0029 434 438
43.2 15.2 30.5 3.8 171.2 .0 8?.4 7.6 -.0018 473 479
38.7 13.5 26.8 3.4 155.5 .0 75.7 6.7 -.0041 402 405
32.1 10.9 21.3 2.9 131.9 .0 58.2 5.5 -.0090 316 315
6.1 .7 - #4 .7 39.7 .0 -10.4 .6 -.2120 123 118
18.0 5.3 9-5 1.7 81.9 .0 21.0 2.8 -.0332 190 185
31.1 10.5 20.5 2.8 128.6 .0 55-7 5-3 -.0098 306 304
34.5 11.8 23.3 3.1 140.4 .0 64.5 5.9 -.0070 345 345
37.5 13.0 25.7 3.3 151.0 .0 72.4 6.5 -.0049 384 386
38.3 13.3 26.5 3.4 154.1 .0 74.7 6.6 -.0043 397 399
37.5 13.0 25-7 3.3 151.0 .0 72.4 6.5 -.0049 384 386
34.1 11.7 22.9 3.1 139.1 .0 63.5 5.8 -.0073 340 340
Table 8 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3 CO 3 S04
(Mg/L) (Mg/L) (Mg/L)
Cl
(Mg/L) ERR
Oumho/cm 
ECM'
@ 250c) 
ECM
37.9 13.1 26.1 3.4 152.4 .0 73.5 6.6 -.0046 390 392
40.9 14.3 28.6 3.6 163.3 .0 81.5 7.1 -.0029 436 440
40.3 14.1 28.1 3.6 160.9 .0 79.8 7.0 -.0032 425 429
23.2 7.4 13.8 2.1 100.3 .0 34.7 3.8 —.0204 229 225
14.3 3.9 6.5 1.4 68.9 .0 11.3 2.1 -.0492 l66 l6l
17.1 5.0 8.8 1.6 78.8 .0 18.7 2.6 -.0363 184 179
27.3 9.0 17.3 2.5 115.0 .0 45.6 4.6 -.0141 266 263
32.3 11.0 21.5 2.9 132.8 .0 58.8 5.5 -.0087 319 318
33.0 11.2 22.0 3.0 135.1 .0 60.6 5.6 -.0082 327 326
36.5 12.6 24.9 3-3 147.5 .0 69.8 6.3 -.0055 371 372
38.3 13.3 26.4 3.4 153.8 .0 74.5 6.6 -.0044 396 398
43.5 15.4 30.8 3.8 172.4 .0 88.3 7.6 -.0016 478 485
40.3 14.1 28.1 3.6 160.9 .0 79.8 7.0 -.0032 425 429
39.8 13.9 27.7 3.5 159.3 .0 78.6 6.9 -.0035 419 422
40.9 14.3 28.6 3.6 163.0 .0 81.4 7.1 -.0029 435 439
46.8 16.6 33.5 41 184.0 .0 97.0 8.2 -.0002 539 549
22.2 7.0 13.0 2.1 96.9 .0 32.2 3.6 -.0223 221 217
16.8 4.9 8.5 1.6 77.8 .0 17.9 2.6 -.0374 182 177
Table 8 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3 CO3
(Mg/L) (Mg/L)
SO4
(Mg/L)
37.5 13.0 25.7 3.3 151.0 .0 72.4
37.2 12.9 25.5 3.3 150.0 .0 71.6
43.1 15.2 30.4 3.8 170.8 .0 87.2
49.2 17.6 35.5 4.3 192.7 .0 103.4
39.0 13.6 27.0 3.5 156.4 .0 76.4
41.2 14.4 28.8 3.6 164.1 .0 82.2
41.3 14.5 28.9 3.7 164.5 .0 82.5
44.3 15.7 31.4 3.9 175.1 .0 90.3
23.2 7.4 13.8 2.1 100.3 .0 34.7
18.5 5.6 10.0 1.8 84.0 .0 22.5
28.6 9.5 18.3 2.6 119.5 .0 48.9
36.1 12.5 24.6 3.2 146.2 .0 68.8
38.4 13.4 26.5 3.4 154.3 .0 74.9
38.5 13.4 26.6 3.4 154.8 .0 75-2
41.9 14.7 29.5 3.7 166.8 .0 84.2
19.9 6.1 11.1 1.9 88.8 .0 26.1
40.8 14.3 28.5 3.6 162.8 .0 81.2
40.7 14.3 28.4 3.6 162.4 .0 80.9
Cl (jimho/cm @ 25°c)
(Mg/L) ERR ECM' ECM
6.5 -.0049 384 386
6.4 -.0051 380 382
7.5 -.0018 471 477
8.7 .0007 588 602
6.8 -.0040 406 409
7.2 -.0028 439 444
7.2 -.0027 u m 446
7.8 -.0013 492 499
3.8 -.0204 229 225
2.9 -.0314 194 189
4.8 -.0125 279 276
6.2 -.0058 366 367
6.7 -.0043 398 400
6.7 -.0042 400 402
7.3 -.0024 452 457
3.2 -.0275 204 199
7.1 -.0029 434 438
7.1 -.0030 432 436
Table 8 (Cont.)
Ga
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
GO3 S04
(Mg/L) (Mg/L)
Cl
(Mg/L) ERR
(^unho/cm @ 25°c) 
EGM’ EGM
43.9 15.5 31.1 3-9 173.7 .0 89.3 7.7 -.0014 485 492
43.0 15.2 30.4 3.8 170.6 .0 87.0 7.5 -.0018 470 476
43.4 15.3 30.7 3.8 172.0 .0 83.0 7.6 -.0017 476 483
39.1 13.6 27.1 3.5 156.9 .0 76.8 6.8 -.0039 408 411
30.3 10.2 19.8 2.7 125.7 .0 53-6 5.1 -.0106 297 295
12.1 3.0 4.6 1.2 6l.0 .0 5.4 1.7 -.0651 153 148
12.8 3.3 5.2 1.3 63.5 .0 7.3 1.8 -.0593 157 152
26.2 8.6 16.3 2.4 110.9 .0 42.6 4.3 -.0156 255 252
36.6 12.6 25.0 3.3 147.8 .0 70.0 6.3 -.0055 372 373
40.9 14.3 28.6 3.6 I63.O .0 81.4 7.1 -.0029 435 439
38.8 13.5 26.9 3.4 155.7 .0 75.9 6.7 -.0041 403 406
40.2 14.1 28.0 3.6 160.7 .0 79.6 7.0 -.0033 424 428
39.8 13.9 27.7 3.5 159.1 .0 78.4 6.9 -.0035 418 421
39.5 13.8 27.4 3.5 158.2 .0 77.8 6.9 -.0037 4l4 417
19.9 6.1 11.1 1.9 88.8 .0 26.1 3.2 -.0275 204 199
32.2 10.9 21.4 2.9 132.5 .0 58.6 5-5 -.0088 318 317
Simulated Common ion Data in Mg/L for 
the Bighorn River at Bighorn
TABLE 9
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO 3 CO3 S04
(Mg/L) (Mg/L) (Mg/L)
Cl
(Mg/L) ERR
/umho/cm 
ECM'
@ 25°c) 
ECM
81.8 30.4 91.8 4.2 221.7 .0 321.4 11.5 .0021 982 982
76.9 28.1 84.1 4.0 211.4 .0 294.2 10.7 -.0001 906 906
82.2 30.6 92.4 4.2 222.5 .0 323.4 11.6 .0023 988 988
80.2 29.7 89.4 4.2 218.4 .0 312.7 11.3 .0015 957 957
79.8 29.4 88.6 4.1 217-5 .0 310.2 11.2 .0013 950 950
80.5 29.8 89.8 4.2 218.9 .0 314.1 11.3 .0016 961 961
81.1 30.1 90.7 4.2 220.1 .0 317.2 11.4 .0018 970 970
79.5 29.3 88.2 4.1 216.9 .0 308.8 11.2 .0011 946 946
75.8 27.5 82.4 4.0 209.1 .0 288.2 10.6 -.0007 890 890
72.3 25.8 76.8 3-8 201.7 .0 268.7 10.0 -.0026 841 840
64.5 22.1 64.6 3.4 185.6 .0 226.0 8.7 -.0077 741 740
49.7 14.9 41.1 2.8 154.5 .0 143.9 6.2 -.0237 581 580
77.2 28.2 84.5 4.0 212.0 .0 295.7 10.8 .0000 910 910
65.5 22.6 66.1 3.5 187.6 .0 231.4 8.9 -.0069 753 752
76.2 27.7 82.9 4.0 209.8 .0 290.1 10.6 -.0005 895 895
65.1 22.4 65.5 3.5 186.8 .0 229.2 8.8 -.0072 748 747
Table 9 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
CO3
(Mg/L)
S04
(Mg/L)
Cl
(Mg/L) ERR
(jumbo/cm 
ECM'
@ 25°c) 
ECM
72.0 25.7 76.3 3-8 201.1 .0 267.1 9.9 -.0027 837 836
71.0 25.2 74.8 3.7 199.1 .0 261.8 9.8 -.0033 824 823
76.6 27.9 83.7 4.0 210.8 .0 292.7 10.7 -.0002 902 902
77-9 28.5 85.7 4.0 213.5 .0 299.8 10.9 .0004 921 921
92.5 35.6 108.7 4.7 244.0 1.0 380.4 13.3 .0061 1173 1170
68.8 24.1 71.2 3.6 194.4 .0 249.3 9.4 -.0047 794 793
73.6 26.5 78.8 3.9 204.5 .0 275.9 10.2 -.0018 859 858
77.4 28.3 84.9 4.0 212.5 .0 297.2 10.8 .0002 914 914
71.0 25.2 74.8 3.7 199.1 .0 261.8 9.8 -.0033 824 823
75.0 27.1 81.1 3.9 207.4 .0 283.7 10.4 -.0011 879 878
69.8 24.6 72.8 3.7 196.5 .0 254.8 9.6 -.0041 807 806
73.6 26.5 78.9 3.9 204.6 .0 276.3 10.2 -.0018 860 859
66.3 23.0 67.4 3.5 189.3 .0 235.9 9.0 -.0064 763 762
78.2 28.7 86.2 4.1 214.2 .0 301.6 11.0 .0005 926 926
73.3 26.3 78.4 3.8 203.9 .0 274.3 10.1 -.0020 855 854
74.5 26.9 80.3 3.9 206.4 .0 281.0 10.3 -.0013 872 871
76.6 27.9 83.5 4.0 210.7 .0 292.4 10.7 -.0003 901 901
78.9 29.0 87.2 4.1 215.6 .0 305.2 11.1 .0008 936 936
Table 9 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3 CO3
(Mg/L) (Mg/L)
SOi4.
(Mg/L)
Cl
(Mg/L) ERR
(pmho/cm 
ECM'
@ 250c)
ECM
79.5 29.3 88.2 4.1 216.9 .0 308.8 11.2 .0011 946 946
?8.0 28.6 85.9 4.1 213.8 .0 300.5 10.9 .0004 923 923
64.5 22.1 64.6 3-4 185.6 .0 226.0 8.7 -.0077 741 740
65.1 22.4 65.5 3-5 186.8 .0 229.2 8.8 -.0072 748 747
81.9 30.5 91.9 4.2 221.8 .0 321.7 11.6 .0022 983 983
79.3 29.2 87.9 4.1 216.5 .0 307.7 11.1 .0011 943 943
80.0 29.6 89.I 4.1 218.0 .0 311.6 11.3 .0014 954 954
78.2 28.7 86.2 .4.1 214.2 .0 301.6 11.0 .0005 926 926
51.6 15.9 44.2 2.9 158.6 .0 154.8 6.6 -.0208 600 599
51.5 15.8 . 44.0 2.9 158.4 .0 154.2 6.5 -.0210 599 598
52.6 16.4 45.8 2.9 160.7 .0 160.3 6.7 -.0195 610 609
82.1 30.5 92.2 4.2 222.2 .0 322.8 11.6 .0022 986 986
82.6 30.8 93-0 4.3 223.2 .0 325.5 11.7 .0024 994 994
80.6 29.8 90.0 4.2 219.2 .0 314.8 11.4 .0016 963 963
83.5 31.3 94.6 4.3 225.3 1.0 330.9 11.8 .0028 1013 1010
93.0 35.8 109.6 4.7 245.1 1.0 383.3 13.4 .0062 1183 1180
85.9 32.4 98.3 4.4 230.2 1.0 344.0 12.2 .0038 1053 1050
78.7 28.9 86.9 4.1 215.1 .0 304.1 11.0 .0008 933 933
Table 9 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
oqdv
(Mg/L)
qdv
(Mg/L)
SOz4.
(Mg/L)
Cl
(Mg/L) ERR
(jumho/cm 
ECM'
@ 25°c) 
ECM
76.4 27.8 83.3 4.0 210.4 .0 291.6 10.7 -.0003 899 899
74.2 26.8 79-8 3.9 205.8 .0 279.4 10.3 -.0015 868 86?
66.2 22.9 67.3 3-5 189.1 .0 235.4 9.0 -.0064 762 76l
76.5 27.9 83.4 4.0 210.6 .0 292.0 10.7 -.0003 900 900
77.4 28.3 84.9 4.0 212.5 .0 297.2 10.8 .0002 914 914
74.1 26.7 79.7 3.9 205.6 .0 279.0 10.3 -.0015 867 866
75.9 27.6 82.6 4.0 209.4 .0 289.0 10.6 -.0006 892 892
78.8 28.9 87.0 4.1 215.3 .0 304.5 11.0 .0008 934 934
75.0 27.1 81.1 3.9 207.4 .0 283.7 10.4 -.0011 879 878
63.1 21.4 62.3 3.4 182.6 .0 218.2 8.5 -.0088 724 723
92.0 35.3 107.9 4.7 242.9 1.0 377.5 13.2 .0059 1163 Il60
89.8 34.3 104.5 4.6 238.5 1.0 365.7 12.9 .0052 1123 1120
55.4 17.7 50.1 3.0 l66.4 .0 175.5 7.2 -.0162 638 637
87.1 32.9 100.1 4.5 232.6 1.0 350.3 12.4 .0042 1073 1070
TABLE 10
Simulated Common ion Data in Mg/L for
the Tongue River in Miles City
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCOo COo SOk
(Mg/L) (Mg/L) (Mg/L)
Cl
(Mg/L) ERR
4Si @ 25°c) 
ECM
59.0 39.5 57.1 4.8 260.8 .0 212.6 3.8 -.0023 752 755
61.8 42.3 6l.4 5.0 273.3 1.0 228.8 4.0 -.0018 801 802
77.0 57.5 84.9 6.2 342.9 1.0 318.6 5.2 .0001 1117 1120
81.3 6i.8 91.5 6.6 362.4 1.0 343.8 5.5 .0005 1227 1230
71.3 51.8 76.1 5.8 316.8 1.0 284.9 4.7 -.0005 986 988
69.8 50.2 73.7 '5.7 309.7 1.0 275.8 4.6 -.0007 953 955
77.8 58.3 86.2 6.3 346.5 1.0 323.4 5.2 .0002 1137 1140
67.8 48.2 . 70.6 5.5 300.6 1.0 264.0 4.5 -.0009 912 914
50.4 30.9 43.6 4.1 221.4 .0 l6l.8 3.2 -.0044 623 625
25.9 6.3 5.9 2.2 109.5 .0 17-3 1.4 -.0282 364 365
50.6 31.1 44.1 4.2 222.4 .0 163.1 3.2 -.0043 626 628
54.3 34.7 49.7 4.4 239.0 .0 184.5 3.5 -.0033 677 680
70.9 51.4 75.5 5.8 315.1 1.0 282.7 4.7 -.0005 978 980
64.4 44.9 65.4 5.2 285.2 1.0 244.1 4.2 -.0014 848 849
65-9 46.3 67.7 5-4 292.0 1.0 2#.9 4.3 -.0012 875 877
73.2 53.7 79.0 5.9 325.4 1.0 296.1 4.9 -.0003 1028 1030
V)
-<?
Table 10 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCOo
(Mg/L)
COo SO4
(Mg/L) (Mg/L)
Cl
(Mg/L) ERR
(jumbo/cm 
ECM'
@ 25°c) 
ECM
77.4 57.9 85.5 6.3 344.7 1.0 321.0 5.2 .0001 1127 1130
74.1 54.6 80.4 6.0 329.4 1.0 301.3 4.9 -.0002 1048 1050
73.7 54.1 79.7 6.0 327.4 1.0 298.7 4.9 -.0002 1038 1040
79.0 59.5 88.0 6.4 352.0 1.0 330.3 5.3 .0003 1167 1170
66.3 46.8 68.4 5.4 294.1 1.0 255.6 4.4 -.0011 884 886
34.9 15.4 19.8 2.9 150.7 .0 70.5 2.0 -.0125 444 445
39.3 19.8 26.6 3.3 170.8 .0 96.4 2.4 -.0091 489 490
48.5 28.9 40.8 4.0 212.6 .0 150.4 3.0 -.0050 597 599
55.8 36.3 52.2 4.6 246.2 .0 193.8 3.6 -.0030 701 704
57.1 37.6 54.2 4.7 252.1 .0 201.3 3.7 -.0027 721 724
73.2 53.7 79.0 5.9 325.4 1.0 296.1 4.9 -.0003 1028 1030
75.4 55.9 82.4 6.1 335.3 1.0 308.8 5.0 —.0001 1077 1080
77.0 57.5 84.9 6.2 342.9 1.0 318.6 5.2 .0001 1117 1120
76.2 56.7 83.6 6.2 339.1 1.0 313.8 5.1 .0000 1097 1100
36.7 17.2 22.6 3.1 159.0 .0 81.2 2.2 -.0110 462 463
67.0 47.5 69.4 5.4 297.1 1.0 259.5 4.4 -.0010 897 899
76.2 56.7 83.6 6.2 339.1 1.0 313.8 5.1 .0000 1097 1100
43.3 23.8 32.8 3.6 189.1 .0 120.0 2.7 -.0070 533 535
ITable 10 (Cont.)
Ga
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO 3 
(Mg/L)
GO 3 SO4
(Mg/L) (Mg/L)
Cl
(Mg/L) ERR
(jumho/cm 
ECM'
@ 25°c) 
ECM
62.9 43.4 63.1 5.1 278.5 1.0 235.5 4.1 -.0016 821 822
67.2 47.6 69.7 5.5 297.8 1.0 260.4 4.4 -.0010 900 902
71.3 51.8 76.1 5.8 316.8 1.0 284.9 4.7 -.0005 986 988
70.3 50.8 74.5 5.7 312.1 1.0 278.8 4.7 -.0006 964 966
57.2 37.7 54.3 4.7 252.3 .0 201.7 3.7 -.0027 722 725
77.8 58.3 86.2 6.3 346.5 1.0 323.4 5.2 .0002 1137 1140
68.2 48.6 71.2 5.5 302.4 1.0 266.3 4.5 -.0009 920 922
37.1 17.6 23.2 3.1 160.8 .0 83.5 2.2 -.0106 466 46?
52.4 32.9 46.9 4.3 230.6 .0 173.6 3.3 -.0038 651 653
66.1 46.6 68.0 5.4 292.9 1.0 254.1 4.3 -.0012 879 881
59.3 39.7 57.5 4.8 261.9 .0 214.0 3.8 -.0023 756 759
79.0 59.5 88.0 6.4 352.0 1.0 330.3 5.3 .0003 H 67 1170
78.2 58.7 86.8 6.3 348.4 1.0 325.7 5.2 .0002 1147 1150
71.9 52.3 76.9 5.8 319.3 1.0 288.1 4.8 —.0004 998 1000
69.0 49.5 72.5 5.6 306.2 1.0 271.2 4.6 -.0008 937 939
69.9 50.4 73.9 5.7 310.3 1.0 276.6 4.6 -.0007 956 958
66,4 46.9 68.5 5.4 294.3 1.0 255.9 4.4 -.0011 885 887
40.5 21.0 28.5 3.4 176.2 .0 103.5 2.5 -.0084 502 503
Table 10 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
49.3 29.8 42.1 4.0
47.0 27.5 38.5 ' 3.9
56.8 37-3 53.7 4.6
71.7 52.2 76.7 5.8
68.3 48.7 71.4 5.5
62.4 42.9 62.3 5.1
64.9 45.4 66.2 5.3
70.1 50.6 74.3 5.7
53.4 33.8 48.4 4.4
50.6 31.0 44.0 4.1
72.8 53.2 78.3 5.9
70.4 50.9 74.7 5.7
59.3 39.7 57.5 4.8
67.1 47.5 69.5 5.5
HCO3 CO 3 SO4 Cl 
(Mg/L) (Mg/1) (Mg/L) (Mg/L)
216.4 .0 155.3 3.1
205.9 .0 141.7 2.9
250.6 .0 199.5 3-7
318.6 1.0 287.3 4.8
302.8 1.0 266.9 4.5
276.2 1.0 232.5 4.1
287.6 1.0 247.3 4.3
311.4 1.0 278.0 4.6
235.0 .0 179.3 3.4
222.1 .0 162.7 3.2
323.4 1.0 293.5 4.8
312.7 1.0 279.7 4.7
261.9 .0 214.0 3-8
297.3 1.0 259.8 4.4
ERR
(jumho/cm 
ECM'
@ 25°c) 
ECM
-.0047 608 610
-.0055 578 580
-.0028 716 719
-.0004 995 997
-.0009 922 924
-.0017 812 813
-.0013 857 859
-.0006 961 963
-.0036 664 667
-.0043 625 627
-.0003 1018 1020
-.0006 967 969
-.0023 756 759
-.0010 898 900
TABLE 11
Simulated Common ion Data In Mg/L for 
the Yellowstone River at Miles City
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCOo
(Mg/L)
COo
(Mg/L)
S04
(Mg/L)
Cl
(Mg/L) ERR
(jamho/cm 
ECM'
@ 25°c) 
ECM
56.8 21.5 59.0 3.8 182.5 1.0 191.7 8.9 .0021 681 681
60.9 23.4 64.7 4.1 193.7 1.0 211.1 9.6 .0019 762 762
64.0 24.7 68.9 4.3 202.0 1.0 225.5 10.2 .0017 827 828
65.2 25.3 70.7 4.4 205.5 1.0 231.5 10.4 .0016 856 857
62.7 24.2 67.2 4.2 198.6 1.0 219.6 10.0 .0018 800 800
50.0 18.5 49.6 3.4 164.0 1.0 159.6 7.6 .0026 566 566
33.9 11.2 27.4 2.3 120.2 .0 83.5 4.6 .0047 364 365
23.3 6.5 12.8 1.6 91.6 .0 33.8 2.7 .0084 274 274
35.8 12.1 30.1 2.4 125.5 .0 92.8 5.0 .0043 384 385
43.9 15.7 41.3 3.0 147.6 .0 131.0 6.5 .0031 478 480
55.6 21.0 57.4 3.7 179.4 1.0 186.2 8.7 .0021 660 660
63.8 24.6 68.7 4.3 201.6 1.0 224.7 10.2 .0017 824 824
56.3 21.3 58.3 3.8 181.0 1.0 189.1 8.8 .0021 671 671
Table 11 (Cent.)
Ga
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HGO3
(Mg/L)
CO3
(Mg/L)
S04
(Mg/L)
Cl
(Mg/L) ERR
(jumho/cm 
ECM'
@ 25°c) 
ECM
36.2 12.3 30.6 2.5 126.6 .0 94.6 5.1 .0042 388 389
57-9 22.0 60.5 3.9 185.4 1.0 196.7 9.1 .0020 701 701
55.7 21.0 57.5 3.7 179.5 1.0 136.5 8.7 .0021 661 66l
19.5 4.8 7.6 1.4 81.2 .0 15.8 2.0 .0115 247 247
43.6 15.6 40.7 2.9 146.5 .0 129.2 6.4 .0032 473 475
56.1 21.2 58.0 3.8 180.6 1.0 188.3 8.7 .0021 668 668
60.1 23.0 63.7 4.0 191.6 1.0 207.5 9.5 .0019 746 746
58.0 22.0 60.6 3.9 185.7 1.0 197.2 9.1 .0020 703 703
37.5 12.9 32.4 2.5 130.1 .0 100.7 5.3 .0040 402 403
47.9 17.5 46.7 3.2 158.2 .0 149.5 7.2 .0027 532 534
6l.O 23.4 64.9 4.1 194.0 1.0 211.6 9.7 .0018 764 764
66.3 25.8 72.2 4.4 208.5 1.0 236.7 10.7 .0016 882 883
40.7 14.3 36.7 2.8 138.7 .0 115.5 5.9 .0035 438 439
53.6 20.1 54.7 3.6 173.9 1.0 176.8 8.3 .0023 625 625
50.3 18.6 50.1 3.4 164.9 1.0 161.1 7.7 .0025 571 571
57.9 22.0 60.6 3.9 185.5 1.0 196.9 9.1 .0020 702 702
60.2 23.1 63.8 4.0 191.9 1.0 . 207.9 9.5 .0019 748 748
19.8 4.9 8.0 1.4 82.0 .0 17.2 2.0 .0112 249 249
59.0 22.5 62.1 4.0 188.6 1.0 202.3 9.3 .0019 724 724
43
Tables A to D in Appendix I. Included in Tables 8 to Ii are the values■ 
of the EEH determined from the simulated values for the common ions. Also 
included in these Tables is a predicated value," ECMf„■of EGM and the 
actual value of EGM. EGMe was obtained by determining a value for EGG, 
EGG*-, from the simulated results :df the common ion concentrations (equa- 
tion 2J and then by solving the regression, equation for EGG (equation 4) 
for ECM': EGG* = S iK1(Ion)1 (2)
EGG* = B log (ECfT) - A (4)
EGM' = EXP.( (l/B) • EGG* + . Pr / ~ ~ (5)
By substituting into equation (2) the regression equation expression for 
each of the common ions given by equation (3) and then substituting this 
into equation (5)s the following expression -is obtained: —
EGM' = ( (ECM)1ZB(SKiti) ) (EXp . P r / ~ (EXP (-l/B. (^K1Si) ) (6)
In order for EGM' to equal EGM in equation (6), the following must hold:
. SiKibi " B (7)
S 1Kiai = A  (8)
The closeness of the values of ECM1 and ECK in Tables 8 to 11 further 
supports the conjecture that the relative proportions of the common ion 
concentrations remain somewhat constant in natural waters„
The Yellowstone River at Billings is the only site of the four, 
above which there is no major regulation of flow. ' The Bighorn River at 
Bighorn is regulated quite heavily by Yellowtail Reservoir which is on ■ 
the Bighorn River mainstem approximately 100 river miles upstream. The •, 
Tongue River at Miles City is regulated to some extent by the Tongue River
Reservoir, "but not to the degree of the Bighorn River (Ref. 12). The 
site on the Yellowstone River at Miles City being above the confluence 
with the.Tongue River is only influenced by Yellowtail Reservoir. Tables 
8 to 11 may indicate that flow regulation reinforces or maintains this 
relative proportioning of the common ions. The simulated values in 
Tables 8 to 11, with the observed data in Tables A to D in Appendix I, 
compare favorably in most cases. The values of the ERR in Tables 8 to 11 
give an indication of the quality of the simulated common ion concentra­
tions. Of particular interest is the fact that in Tables 8 to 11, the 
largest absolute values of ERR occur consistently for the lowest EGM 
values. The relative proportions of the common ions break down or are 
altered at apparently low EGM conditions. Low EGM values are generally 
associated with relatively high flow conditions. To improve the regres­
sion equations at low ECM conditions, the addition of a flow dimension was 
considered.
As previously mentioned, as Q increases the concentration of 
dissolved constituents decreases. Regression equations involving the 
concentration of- dissolved constituents with Q are generally developed on 
the basis of the logarithm of the concentration in Mg/L versus the loga­
rithm of Q as the independent variable. The form of these equations is;
log(ion)i = log(ai) - bi Iog(Q) (9)
These regression equations are then solved for the ion concentration to 
give:
(Ion)i = ai(Q)"bi (10)
Where:
(ion)  ^is the concentration of the i^’n ion 
Q is the flow
'ai and hi are regression constants, . •
By altering the regression equations, the corresponding R2- values are not 
necessarily applicable, as 'is especially true when the regression equa­
tions are more statistical than physical in character, As previously 
mentioned, TDS has received the most interest in its inverse relationship 
with Q. EGM and TDS correlate very strongly as might'he expected since 
they essentially measure the same thing. Table 12 shows regression equa- . 
tions of the form:
log (EGM) = log (A) - B Iog(Q) ■ ' (ll)
for the four sites. The extremely low R2 values for the Bighorn-and. Ton­
gue Rivers are most likely indicative of their upstream regulation. Note• 
that the t~ value for the Bighorn River at Bighorn, indicates-,-that .the 
coefficient on the Iog(Q) term is hot significantly different than zero. 
This indicates no appreciable relationship between EGM and Q. This same 
situation of low R2 values was found for regressions involving the natural, 
logarithm of the individual common ions, in Meq/L concentrations, with -the 
natural logarithm of flow.
Tables 13'through Io show sets "of regression equations for each of 
the four sites. These regression equations involve the natural logarithms 
of EGM and Q as independent variables and the natural, logarithm of the 
individual common ion concentrations as dependent variables. The form
TABLE 12
Log (ECM) vs. Log Q Linear Regression Equations for the Four Sites
Site Regression Equation R2 t-Value
Yellowstone at 
Billings
log (ECM) = 9.501 - .4218 " log Q .871 -20,66 '
Bighorn at 
Bighorn
log (ECM) = 7.175 - .04815*log Q .023 - 1.202
Tongue at
Miles City
log (ECM) = 7.764 - .1763 • log Q .322 - 5.423
Yellowstone at 
Miles City
log (ECM) = 10.57 - .4444 • log Q .706 - 8.043
TABLE 13
Log (ion) vs. Log (ECM) & Log (Q) Regression Equations
for Yellowstone River at Billings
Dependent
Variable ■ Regression Equations 02 t-Value
log (ECC) log (ECC) = faO
iO%-4 (ECM)
log (Ca) log (Ca) .8l29*log (ECM)
log (Mg) log (Mg) .99^8'log (ECM)
log (Na) log (Na) I.308-log (ECM)
log (K) log (K) .5696‘log (ECM)
log (ALK) log (ALK) .8578*log (ECM)
log (HCO3) log (HCO3) .8529-log (ECM)
log (SO4) log (SO4) = 1.450*log (ECM)
log (Cl) log (Cl) = .5552-log (ECM)
.01531‘log (Q) - .03149 .992 31.29 
- 1.016
.05760-log (Q) - 3.760 .963 12.49 
- 1.958
.05535‘log (Q) - 5.430 .983 19.43
- 2.393
.03793’log (Q) - 8.042 .964 15.50
.9949
.1226* log (Q) - 4.859 .759 3.442 
- 1.639
.02842*log (Q) - 4.462 .978 20.21
1.482
.02709‘log (Q) - 4.423 .977 19.65
1.38
.02274*log (Q) - 8.098 .967 14.80 
. 514
.2867' log (Q) - 2.641 .798 2.663
- 3.043
TABLE 14
Log (Ion) vs. Log (ECM) & Log (Q) Regression Equation
for Bighorn River at Bighorn
Dependent
Variable Regression Equations R2 t-Value
log (ECC) log (ECC) = I.096-log (ECM) -
log (Ca) log (Ca) “ .8956*log (ECM) +
log (Mg) log (Mg) = 1.117-log (ECM) -
log Na) log (Na) = 1.2?4-log (ECM) -
log (K) Iog(K) = .7306-log (ECM) +
log (ALK) log (ALK) = .6712-log (ECM) +
log (HCO3) log (HCO3) “ .6612-log (ECM) +
log (SO4) log (SO4) - I;271-log (ECM) -
log (Cl) log (Cl) = 1.117'log (ECM) +
60T-IOOI—I
OO
(Q) - .4425 .947 32.45
.0936
.03341 • log (Q) - 5.037 .842 17.87
2.106
.005305-log (Q) - 6.739 .844 17.77 
- .267
.04434 • log (Q) - 7.033 .867 19.12 
- 2.103
.05444 • log (Q) - 7.718 .431 6.713
1.581
.03059 • log (Q) - 3.583 .772 14.23
2.05
.03386 • log (Q) - 3.545 .746 13.29
2.15
.01324- log (Q) - 6.747 .929 27.53 
- .906
.04076 • log (Q) - 9.153 .606 9.597
1.107
TABLE 15
Log (ion) vs. Log (ECM) & Log (Q) Regression Equations
for Tongue River at Miles City
Dependent
Variable Regression Equation 02 t-Value
log (ECC) log (ECC) = 1.124'log (ECM) -
log (Ca) log (Ca) .8215'log (ECM) +
log (Mg) log (Mg) = I.273-log (ECM) +
log (Na) log (Na) = I.207-log (ECM) -
log (K) log (K) - .8247-log (ECM) +
log (ALK) log (ALK) = .7586-log (ECM) -
log (HCO3) log a 8
V
) = .7511-log (ECM) -
log (SO4) log
OCO = 1.403-log (ECM) +
log (Cl) log (Cl) - .9806-log (ECM) -
.001299-log (Q) - .6289 .977 42.06
- .16
.007594-log (Q) - 4.433 .852 15.67
.466
.02187 • log (Q) - 7.458 .940 26.22
1.45
.06028- log (Q) - 6.807 .902 17-73 
- 2.85
.02084 • log (Q) - 7.729 .635 8.87
• 72
,04305 • log (Q) - 3.344 .896 16.97 
- 3.10
.04319• log (Q) - 3.297 .892 16.55 
- 3.06
.02590 • log (Q) - 8.049
COxnOx 31.87
I.89hOOrHv-fO0 (Q) - 8.787 •550 7.09 
- . 04
ITABLE 16
Log (ion) vs. Log (ECM) & Log (Q) Regression Equations 
for Yellowstone River at Miles City
Dependent
Variable Regression Equations 0 2 t-Value
log (ECC) log (ECC) I.105-log (ECM)
log (Ca) log (Ca) .9648-log (ECM)
log (Mg) log (Mg) I.012-log (ECM)
log (Na) log (Na) = I.252-log (ECM)
log (K) log (K) .8509-log (ECM)
log (ALK) log (ALK) .6012-log (ECM)
log (HCO3) log (HCO3) = .6089-log (ECM)
log (SO4) log (SO4) 1.468.log (ECM)
log (Cl) log (Cl) = I.003-log (ECM)
.003845-log (Q) - •5665 .994 36.57
0.23
.06474 • log (Q) - 5.860 • 954 14.10
1.79
.05641 • log (Q) - 5.539 .935 9.65 
- 1.02
.02602 • log (Q) - 7.016 .989 25.63 
- 1.01
.01299 * log (Q) - 7.779 .864 6.79
- .20
.07205 • log (Q) - 2.165 .970 10.72 
- 2.43
.0624 • log (Q) - 2.310 .966 12.48 
- 2.29
.05844 • log (Q) - 8.765
T—4
COON 21.20
1.60
.1048 • log (Q) - 6.977 .868 6.06 
- 1.20
51
of these regression equations is given by:
log(ion)i = ai Iog(EGM) ± bi Iog(Q) - log(ci) (12)
Where ai, bi and ci are regression constants. Comparing these sets of 
regression equations with those in Tables 4 to 11, it is apparent that 
the values are generally larger for the sets involving Q. Many of the 
t-values for the regression constants bi, which are the coefficients on the 
Iog(Q) term, indicate that bi is not significantly different than zero. 
These t-values are the lower values shown for each regression equation in 
Tables 13 to l6. These regression equations reduce to:
Iog(Ion)i = ai Iog(ECM) - log(ci) (13)
From which the following relation for (Ion)i is obtained:
(Ion)i = IyZci(ECM)a^ (14)
Since the regression equations in Tables 13 to l6 must be altered to 
obtain values for the common ion concentrations, the corresponding 
values are not directly applicable. Although, these R% values should 
give an indication of the fit of the altered regression equations, espe­
cially if the equations involve some underlying physical relation.
Using the regression equations in Tables 13 to I6 and the ECM 
and Q data from which the regression equations were generated, values 
for the common ion concentrations were simulated. These simulated values 
for the common ions for the four sites are shown in Tables 17 through 20. 
Included in these Tables are values of ERR calculated from the simulated 
common ion concentrations. The values in Tables. 17 to 20 are comparable 
to the values in Tables 8 to 11 and Tables A to D observed in Appendix
TABLE I?
Simulated Common Ion Data for the 
Yellowstone River at Billings in Mg/L
Ca Mg Na K HCO3 CO3 S04 Cl ERR
(/umho/<
ECM
2m @ 25°c)
, Q
(cfs)
36.3 12.5 25.5 3.1 151.3 .0 69.3 5.7 -.0142 393 6090
■ 36.6 12.6 24.9 3.2 148.9 .0 68.7 6.2 -.0054 389 4640
37.7 13.0 25.4 3-4 150.9 .0 71.3 6.7 -.0004 398 3700
44.2 15.7 30.5 3.9 169.9 .0 90.5 8.4 .0024 465 2180
41.3 14.5 28.6 3.6 162.8 .0 82.4 7.5 -.0011 438 2970
43.1 15.4 32.7 3.6 178.2 .0 92.7 6.8 -.0222 479 5000
38.3 13.3 26.0 3.4 153.1 .0 73.2 6.7 -.0015 405 3690
30.3 10.0 19.1 2.8 125.4 .0 50.2 5.0 -.0018 315 6230
12.2 3.4 5.7 1.2 57.1 .0 11.6 1.7 -.0456 118 41300
18.6 5.6 9.9 1.8 81.8 .0 22.7 2.8 -.0150 185 16500
29.2 9.6 18.3 2.7 122.0 .0 47.6 4.8 -.0037 304 7020
32.5 11.0 21.5 2.9 135.6 .0 57.2 5.2 -.0083 345 6510
36.2 12.5 24.7 3.2 148.2 .0 67.8 6.0 -.0067 386 4930
37.6 13.0 25.6 3.3 151.7 .0 71.4 6.4 -.0041 399 4180
36.6 12.6 24.5 3.2 147.5 .0 68.1 6.3 -.0017 386 4160
32.7 11.0 20.9 3.0 132.9 .0 56.4 5.6 .0018 340 4840
%
Table I? (Cont.)
Ca Mg Na
37.7 13.0 24.7
41.5 14.6 28.7
40.4 14.1 27.9
22.0 6.9 12.7
16.2 4.8 8.4
17.6 5.3 9.6
25.6 8.2 15.3
30.2 10.0 19.5
31.1 10.4 20.0
35.0 11.9 23.6
37.4 12.9 25.6
45.7 16.4 32.2
40.5 14.2 27.9
39.7 13.9 27.4
40.7 14.3 29.0
46.7 17.2 39.9
21.3 6.6 12.1
17.8 5.3 9.4
36.8 12.7 24.4
K HCO3 CO3
3.4 148.3 .0
3.7 163.4 .0
3.5 160.4 .0
2.1 96.1 .0
1.6 73.4 .0
1.7 80.6 .0
2.4 108.5 .0
2.7 137.0 .0
2.8 129.2 .0
3.1 143.9 .0
3.3 151.6 .0
4.0 176.0 .0
3.6 160.2 .0
3.5 158.3 .0
3.5 164.4 .0
3-6 203.1 .0
2.0 93.5 .0
1.7 79.0 .0
3.3 147.0 .0
SO4 Cl ERR
70.1 
82.9
79.7
30.3
18.4
21.4
38.3
50.7
52.7
64.1
71.0
96.2
79.8
77.8
82.1
111.6
28.7
21.2
68.3
ECM Q
(pmho/cm @ 25°c) (cfs)
6.9 .0057 392 3110
7.5 -.0012 440 2940
7.2 -.0027 429 3330
3.3 -.0140 225 13700
2.3 -.0274 161 24400
2.4 -.0337 179 27000
4.1 -.0047 263 8890
4.8 -.0080 318 7440
5.1 -.0048 326 6460
5.8 -.0069 372 5370
6.3 -.0056 398 4440
8.7 -.0005 485 2140
7.3 -.0013 429 3160
7.0 -.0027 422 3470
6.9 -.0093 439 3970
6.3 -.0469 549 8500
3.2 -.0166 217 15200
2.7 -.0183 177 18500
6.6 .0021 386 3650
Table I? (Cont.)
Ca Mg Na K HCO3 CO3
35.5 12.2 24.5 3.1 147.6 .0
44.2 15.8 32.0 . 3.8 175.1 .0
56.0 20.8 42.0 4.8 208.9 .0
39.0 13.6 26.1 3.5 153.6 .0
41.8 14.7 29.1 3-7 164.7 .0
41.5 14.6 29.4 3.6 166.1 .0
45.0 16.2 34.3 3.8 183-7 .0
21.3 6.7 12.9 1.9 97.6 .0
18.8 5.7 10.2 1.8 83.4 .0
26.6 8.6 16.3 2.4 113.1 .0
35.2 12.0 22.9 3.2 141.0 .0
37.8 13.1 25.6 3.3 151.7 .0
38.0 13.2 25.7 3.4 152.2 .0
42.6 15.1 30.3 3.7 169.0 .0
18.9 5.8 11.2 1.7 88.7 .0
40.3 14.2 29.0 3.4 164.7 .0
40.2 14.1 28.8 3.5 163.9 .0
45.7 16.4 33.1 3.9 179.2 .0
44.4 15.9 31.7 3-9 174.3 .0
SO4 Cl ERR ECM Q
(pnho/cm @ 25°c) (cfs)
66.5 5-7 -.0118 382 5970
93.2 7.8 -.0083 477 3030
133.0 11.2 -.0087 602 1330
74.5 7.2 .0034 409 3030
84.0 7.5 -.0022 444 2970
84.2 7.2 -.0075 446 3560
98.7 7.3 -.0208 499 4240
29.9 2.8 -.0353 225 24200
23.4 2.8 -.0181 189 17600
41.1 4.2 -.0073 276 9000
63.3 6.3 .0036 367 3890
71.8 6.6 -.0017 400 3830
72.4 6.7 -.0012 402 3720
87.5 7.5 -.0059 457 3140
24.8 2.4 -.0446 199 33400
81.6 6.6 -.0134 438 4610
81.1 6.6 -.0121 436 4450
97.7 8.2 -.0075 492 2680
93.1 8.0 -.0053 476 2730
Table I? (Cont.)
Ca Mg Na K HCO3 CO3
45.1 16.1 32.3 3.9 176.2 .0
38.7 13.5 26.5 . 3-4 155.2 .0
28.7 9.4 17.5 2.6 118.7 .0
14.8 4.3 7.6 1.4 69.2 .0
15.3 4.5 7.8 1.5 70.3 .0
24.7 7.9 14.5 2.3 104.7 .0
35.2 12.0 23.6 3.1 144.0 .0
40.7 14.3 29.0 3.5 164.4 .0
38.7 13.4 25.9 3.5 152.8 .0
39.8 13.9 28.1 3.4 161.1 .0
39.4 13.8 27.4 3.4 158.5 .0
39.4 13.7 26.9 3-5 156.6 .0
18.9 5.8 11.2 1.7 88.7 .0
30.0 10.0 19.4 2.7 127.0 .0
SO4 Cl ERR ECM
(}mho/cm @ 25°c)
95.3 8.2 -.0047 483
74.7 6.7 -.0031 411
45.6 4.8 .0006 295
16.1 2.0 -.0451 148
16.8 2.1 -.0346 152
36.0 4.0 -.0041 252
64.5 5.9 -.0051 373
82.1 6.9 -.0092 439
73.7 7.0 .0028 406
79.1 6.7 -.0088 428
77-3 6.8 -.0054 421
76.5 7.0 -.0006 417
24.8 2.4 -.0446 199
50.4 4.7 -.0112 317
, Q(cfs)
2550
3770
6440
38600
29800
9210
5040
3950
3150
4160
3850
3320
33400
8250
TABLE 18
Simulated Common ion Data for the 
Bighorn River at Bighorn in Mg/L
Ca Mg Na K HCO 3 CO 3 S04 Cl ERR
(jumho/<
ECM
:m @ 25°c)
Q
(cfs)
83.2 30.2 89-5 4.3 225.0 .0 319.4 11.8 -.0035 982 5970
77.4 27.6 80.8 4.0 213.1 .0 288.4 10.7 -.0056 906 5830
80.5 30.6 94.9 4.0 217.3 1.0 326.8 11.3 .0056 988 1900
79.9 29.4 88.7 4.1 217.2 .0 311.3 11.2 -.0001 957 3500
79.5 29.2 87.7 4.1 216.5 .0 308.2 11.1 -.0007 950 3650
8l.l 29.5 87.8 4.2 220.4 .0 311.5 11.4 -.0027 961 4960
81.9 29.8 88.6 4.2 222.2 .0 315.0 11.5 -.0029 970 5230
80.9 29.0 84.7 4.2 220.6 .0 304.0 11.4 -.0057 946 6950
76.9 27.0 78.0 4.0 212.8 .0 280.8 10.7 -.0084 890 7900
72.5 25.4 73.2 3.8 203.2 .0 261.7 9.9 -.0085 840 6270
63.6 22.1 63.7 3.4 183.7 .0 224.3 8.4 -.0093 740 3740
51.0 16.8 46.9 2.8 155-9 .0 164.7 6.4 -.0202 580 3470
75.9 27.9 83.8 3-9 208.8 .0 292.7 10.5 -.0001 910 2900
64.3 22.5 65.4 3.4 184.9 .0 229.3 8.5 -.0077 752 3310
72.7 27.5 85.1 3.7 200.8 1.0 289.7 10.0 .0062 895 1280
64.4 22.3 64.1 3.4 185.7 .0 226.6 8.6 -.0100 747 4300
Table 18 (Cont.)
Ga Mg Na K HCO3 CO3
71.2 25.3 74.0 3.7 200.0 .0
69.9 24.9 72.9 . 3.7 197.0 .0
75.6 27.6 82.5 3.9 208.3 .0
77.1 28.2 84.6 4.0 211.4 .0
98.0 36.7 110.9 4.9 254.2 .0
69.2 23.8 67.6 3.7 196.6 .0
72.6 26.1 76.9 3.8 202.6 .0
74.5 28.1 86.8 3.8 204.7 1.0
69.8 24.9 73.1 3.7 196.7 .0
75.1 26.7 77.9 3.9 208.3 .0
69.4 24.2 69.9 3.7 196.6 .0
72.4 26.1 77.4 3.8 202.0 .0
65.5 22.8 65.9 3-5 187.8 .0
77.9 28.4 84.5 4.0 213.5 .0
73.9 25.8 74.3 3.9 206.3 .0
75.2 26.4 76.2 4.0 209.1 .0
77.4 27.4 79.7 4.1 213.4 .0
77.7 28.8 87.1 4.0 212.3 .0
78.6 29.1 88.0 4.0 214.4 .0
SO4
261.5
256.8
289.0
296.7
398.0
242.8
270.7
297.0
256.9
277.3
248.9
271.5
232.6
298.0
266.8
273.6
285.8
303.6
307.4
Cl ERR
(Jimho/
ECM
cm @ 25°c)
, Q 
(cfs)
9.7 -.0058 836 4260
9.5 -.0053 823 3740
10.4 -.0012 902 3240
10.7 -.0008 921 3320
14.4 -.0013 1170 7220
9.3 -.0116 793 7260
9.9 -.0039 858 3760
10.3 .0055 914 1490
9.5 -.0048 823 3540
10.3 -.0061 878 5500
9.4 -.0087 806 5280
9.9 -.0030 859 3360
8.7 -.0087 762 4060
10.8 -.0021 926 4000
10.1 -.0089 854 7070
10.4 -.0083 871 7170
10.7 -.0068 901 6750
10.8 .0011 936 2760
11.0 .0009 946 2970
5^
Table 18 (Coni.)
Ca Mg Na K HCO3 CO3 S04 Cl ERR ECM
(jumho/cm @ 25°c)
, Q
(cfs)
77.0 28.3 85.1 4.0 211.2 .0 297.8 10.7 -.0002 923 3100
63.7 22.1 63.6 . 3.4 184.0 .0 224.2 8.4 -.0097 740 3930
65.3 22.3 62.9 3-5 188.3 .0 225.3 8.7 -.0131 747 6500
82.7 30.3 90.4 4.2 223.5 .0 320.7 11.7 -.0019 983 4820
80.5 28.9 84.7 4.2 219.6 .0 303.1 11.3 -.0052 943 6420
81.3 29.2 86.0 4.2 221.2 .0 307.6 11.4 -.0047 954 6340
80.1 28.2 81.5 4.2 219.6 .0 294.7 11.2 -.0083 926 9160
53.5 17.4 47.5 3.0 162.5 .0 170.3 6.8 -.0231 599 6300
53.1 17.4 47.9 3.0 161.2 .0 170.4 6.7 —.02l6 598 5100
53.0 17.8 50.2 2.9 160.3 .0 175.6 6.7 -.0165 609 3030
81.9 30.4 92.3 4.2 221.2 .0 323.5 11.5 .0010 986 3340
82.9 30.7 92.6 4.2 223.5 .0 326.2 11.7 .0000 994 3900
80.5 29.6 89.1 4.1 218.6 .0 313.5 11.3 -.0004 963 3720
84.9 31.2 93.4 4.3 227.9 .0 331.8 12.1 -.0016 1010 5040
94.6 37.3 118.6 4.6 245.0 1.0 409.2 13.8 .0087 1180 2040
86.2 32.7 100.6 4.3 229.4 1.0 351.1 12.3 .0037 1050 2880
79.3 28.5 84.1 4.1 217.0 .0 299.5 11.1 -.0044 933 5610
78.5 27.3 77.8 4.2 216.7 .0 283.1 10.9 -.0105 899 11100
72.8 26.4 78.6 3.8 202.6 .0 275.0 10.0 -.0020 867 3080
oo
Table 18 (Cont.)
Ca Mg Na K HCO3 CO3 S04 Cl ERR ECM
(fimho/cm @ 25°c)
, Q
(cfs)
63.9 22.9 67.8 3-4 183.3 .0 234.3 8.5 -.0034 76l 2060
?6.i 27.5 81.3 . 3.9 209.9 .0 287.2 10.5 -.0033 900 4230
77.6 27.9 82.3 4.0 213.3 .0 292.2 10.8 -.0042 914 5030
73.2 26.3 77.8 3.8 203.8 .0 273.9 10.0 -.0036 866 3750
76.3 27.1 79.3 4.0 210.9 .0 282.8 10.6 -.0060 892 5800
78.8 28.6 85.0 4.1 215.6 .0 300.8 11.0 -.0027 934 4500
77.2 26.6 75.0 4.1 214.4 .0 274.3 10.7 -.0122 878 12800
62.1 21.5 62.1 3.3 180.3 .0 218.0 8.2 -.0095 723 3420
91.8 36.7 H 8.3 4.5 238.6 1.0 402.7 13.3 .0122 1160 1310
91.4 35.1 109.3 4.5 239.4 1.0 381.2 13.2 .0051 1120 2860
55.0 18.7 53.5 3.0 164.4 .0 186.3 7.0 -.0131 637 2630
84.2 33.6 IO8.9 4.1 222.8 1.0 365.6 12.0 .0145 1070 840
TABLE 19
Simulated Common ion Data for the 
Tongue River at Miles City in Mg/L
Ca Mg Na K HCO3 CO 3 S04 Cl ERR
I
ECM
'jimho/cm @ 25°c)
, Q
(cfs)
57.7 36.9 52.4 4.6 251.7 .0 196.0 3-6 -.0008 755 444
60.7 40.0 55.7 4.9 261.2 1.0 214.4 3.8 .0004 802 539
79.2 59.9 88.8 6.3 351.3 1.0 333.3 5-2 .0017 1120 188
85.7 67.7 98.5 6.8 374.2 1.0 381.8 5.8 .0011 1230 222
71.8 51.7 73.7 5.7 311.6 1.0 283.9 4.6 .0010 988 340
69.9 49.7 70.2 5.6 302.0 1.0 271.6 4.5 .0010 955 389
80.5 61.6 89.3 6.4 351.9 1.0 344.0 5.3 .0012 1140 245
67.6 47.4 65.0 5.4 287.5 1.0 257.9 4.3 .0011 914 569
49.6 29.4 40.2 4.0 212.7 .0 152.7 3.0 -.0014 62$ 815
32.0 15.0 20.4 2.6 139.2 .0 72.7 1.7 -.0143 365 1310
49.2 28.6 44.4 3.9 228.3 .0 147.7 3.0 -.0097 628 173
52.7 32.0 47.6 4.2 237.7 .0 167.1 3.2 -.0049 680 270
70.8 50.2 77.0 5.6 321.8 1.0 274.3 4.6 .0011 980 140
63.4 42.? 6l.O 5.1 277.0 1.0 230.0 4.0 .0002 849 372
65.2 44.6 63.2 5.2 283.1 1.0 241.1 4.1 .0006 877 395
73-5 52.8 84.7 5.7 342.6 1.0 289.7 4.8 .0021 1030 78
&
Table 19 (Gont1)
Ca Mg Na
79.8 60.6 89.6
75.3 55.6 80.4
75.0 55.5 77.3
82.1 63.2 93.9
66.2 46.1 60.6
37.8 19.5 25.3
40.5 21.4 30.7
47.5 27.2 41.0
54.4 33.6 48.8
55.7 34.9 50.1
74.3 54.5 ' 77.5
77.3 57.9 82.0
79.0 59.4 90.9
78.1 58.6 86.7
39-1 20.6 26.0
67.1 47.2 60.8
78.8 60.0 81.0
43.9 24.5 31.9
6l.6 40.7 59.9
K HCO3 CO3
6.3 353.1 1.0
6.0 329.5 1.0
6.0 320.6 1.0
6.5 363.8 1.0
5.4 274.3 1.0
3.1 158.7 .0
3.2 180.2 .0
3.8 216.6 .0
‘4.3 241.0 .0
4.5 244.7 .0
5.9 321.5 1.0
6.2 333.0 1.0
6.2 357.2 1.0
6.2 345.8 1.0
3.2 161.0 .0
5.4 274.6 1.0
6.3 329.5 1.0
3.6 18].6 .0
4.9 274.2 1.0
SO4 Cl ERR ECM Q
(/imho/cm © 25°c) (cfs)
337.8 5-3 .0017 1130 195
307.3 4.9 .0012 1050 270
306.9 4.9 .0009 1040 430
353.9 5.5 .0018 1170 179
250.4 4.2 .0017 886 983
97.0 2.1 -.0029 445 1950
107.5 2.3 -.0110 490 554
139.7 2.8 -.0091 599 257
176.7 3.3 -.0028 704 361
184.4 3.4 -.0018 724 410
301.0 4.8 .0010 1030 341
321.7 5.1 .0009 1080 345
330.0 5.3 .0024 1120 128
325.4 5.2 .0016 1100 198
103.5 2.2 .0014 463 2800
257.1 4.2 .0019 899 1230
335.0 5.1 .0000 1100 604
125.1 2.5 .0001 535 1660
218.0 3.9 -.0008 822 269
Table 19, (Cont.)
Ca Mg Na K HCO3
8
S04 Cl ERR ECM
Oumho/cm @ 25°c)
, Q
(cfs)
66.5 45.8 66.9 5.3 293.8 1.0 248.4 4.2 .0004 902 272
71.6 51.2 75.5 5.7 317.1 1.0 280.9 4.6 .0011 988 227
70.1 49.5 74.7 5.5 315.5 1.0 270.2 4.5 .0009 966 172
56.2 35.8 47.3 4.5 235.0 .0 189.4 3.4 .0017 725 1070
81.4 63.4 82.4 6.6 332.2 1.0 356.1 5.3 -.0012 1140 930
68.4 48.5 63.4 5.5 281.9 1.0 265.2 4.3 .0015 922 1040
39.3 20.7 26.9 3.2 165.0 .0 103.6 2.2 -.0021 467 1840
5l.o 30.4 45.0 4.1 229.6 .0 158.3 3.1 -.0055 653 300
64.6 43.2 70.6 5.0 306.3 1.0 231.9 4.2 -.0015 881 69
57.9 37.2 52.7 4.6 252.6 .0 197.5 3.6 -.0007 759 447
81.9 62.7 96.0 6.5 369.4 1.0 350.6 5.5 .0026 1170 125
80.8 61.6 93.0 6.4 361.8 1.0 343.9 5.4 .0022 1150 150
72.4 52.2 75.9 5.8 317.9 1.0 286.9 4.7 .0011 1000 265
68.9 48.5 69.2 5.5 299.6 1.0 264.5 4.4 .0009 939 350
70.2 50.2 69.1 5.6 298.8 1.0 274.9 4.5 .0010 958 526
66.0 45.6 62.5 5.3 280.5 1.0 247.6 4.2 .0011 887 596
42.0 23.0 28.4 3.4 170.0 .0 116.8 2.4 .0044 503 3360
48.3 28.0 41.0 3.9 216.3 .0 144.5 2.9 -•OO65 610 363
46.2 26.1 39.5 3.7 211.9 .0 133.3 2.8 -.0107 580 243
Table 19 (Cont.)
Ca Mg Na K HCO3 CO3
55.1 34.2 51.3 4.4 249.4 .0
72.2 52.0 75.6 . 5.7 317.0 1.0
67.7 46.9 70.2 5.4 303.2 1.0
60.9 39.8 60.4 4.8 276.4 1.0
64.1 43.4 61.8 5.1 279.2 1.0
70.6 50.6 69.3 5.7 299.0 1.0
52.5 32.3 42.4 4.3 219.2 .0
49.6 29.3 41.2 4.0 216.5 .0
72.4 51.2 88.1 5.6 353.0 1.0
70.7 50.5 71.6 5.6 305.9 1.0
57.5 36.3 56.3 4.5 264.8 .0
65.6 44.1 73.7 5.1 315.1 1.0
S04 Cl ERR ECM Q
(jumho/cm @ 25°c) (cfs)
180.0 3.4 -.0040 719 235
285.7 4.7 .0011 997 267
254.9 4.3 .0004 924 200
212.5 3.8 —.0018 813 184
233.9 4.0 .0004 859 380
277.5 4.5 .0011 963 567
169.3 3.1 .0017 667 1270
152.0 3.0 -.0032 627 575
279.4 4.8 .0025 1020 33
276.9 4.5 .0010 969 373
192.0 3.6 -.0041 759 150
237.2 4.2 -.0012 900 52
TABLE 20
Simulated Common ion Data for the 
Yellowstone River at Miles City in Mg/L
Ca Mg Na K HCO 3 CO3 S04 Cl ERR ECM
(pmho/cm @ 25°c)
Q
(cfs)
55.1 21.3 57.7 3.8 184.4 1.0 182.1 9.0 .0013 681 7460
61.9 23.7 66.1 4.1 195.8 1.0 216.5 10.0 .0003 762 8530
66.8 25.9 73.5 4.4 207.0 1.0 243.4 10.9 -.0023 828 7880
69.1 26.7 76.7 4.6 211.1 1.0 256.2 11.3 -.0036 857 8000
65.0 24.8 70.3 4.3 201.5 1.0 232.7 10.4 -.0011 800 8680
48.2 17.0 44.9 3.2 157.8 .0 144.5 7.0 .0061 566 14800
33.7 10.3 25.3 2.2 113.5 .0 80.4 4.0 .0100 365 40200
26.1 7.5 17.5 1.7 93-1 .0 54.0 2.9 .0009 274 58400
33.9 11.3 27.5 2.3 122.5 .0 83.5 4.6 -.0022 385 19900
40.2 14.6 36.9 2.8 145.8 1.0 111.2 6.1 -.0022 480 10600
53.6 20.6 55.4 3.7 180.3 1.0 174.5 8.7 .0017 660 7860
65.9 25.9 73.3 4.4 208.1 1.0 239.8 11.0 -.0020 824 6900
55.0 20.7 56.3 3.7 180.6 1.0 180.2 8.7 .0024 671 9000
35.0 11.2 27.6 2.3 120.6 .0 86.6 4.5 .0059 389 28600
58.0 21.5 59.2 3.8 183.5 .0 194.1 8.9 .0027 701 10700
57.4 19.4 54.0 3.6 169.2 .0 185.7 7.8 .0070 661 22000
Table 20 (Cont.)
Ca Mg Na K HCO3 CO3 S04 Cl ERR ECM
(pmho/cm @ 25°c)
, Q
(cfs)
23.4 6.9 15.4 1.5 88.5 .0 45.8 2.7 -.0148 247 48400
39.8 14.5 36.4 . 2.7 145.0 1.0 109.4 6.1 -.0030 475 10400
53.8 21.0 56.4 3.7 183.2 1.0 176.1 8.9 .0008 668 6840
60.5 23.2 64.5 4.1 193.9 1.0 209.2 9.8 .0006 746 8100
59.3 21.2 59.0 3.8 180.4 .0 198.3 8.7 .0036 703 14400
36.5 11.5 28.8 2.4 122.3 .0 91.8 4.6 .0096 403 32000
44.5 16.3 42.2 3.0 155.8 1.0 129.9 6.8 .0012 534 10300
61.2 24.0 66.7 4.1 198.9 1.0 214.4 10.2 -.0002 764 6800
71.4 27.5 79.5 4.7 214.1 1.0 268.9 11.5 -.0051 883 8600
39.6 12.6 32.1 2.5 129.1 .0 103.9 5.0 .0119 439 31100
50.3 19.6 51.9 3-5 176.3 1.0 159.4 8.4 -.0000 625 6610
47.6 17.4 45.8 3.2 161.9 1.0 143.6 7.3 .0030 571 10700
56.8 21.9 59.9 3.8 187.5 1.0 190.7 9.3 .0013 702 7640
6l.4 23.0 64.4 4.0 191.9 1.0 212.4 9.6 .0010 748 9800
22.7 7.1 15.8 1.6 92.1 .0 44.9 2.9 -.0330 249 27400
58.1 22.7 62.4 4.0 192.3 1.0 198.3 9.7 .0006 724 6900
. 66
I, The ERR values in Tables 17 to 20 are generally smaller than those 
corresponding values in Tables 8 to 11, but the largest values of ERR 
still occur predominately at low ECM values. Although, the values of 
ERR for the simulated common ions in Tables 17 to 20 are not excessively 
large even at low ECM values. ,None of the ERR values for a particular 
site are larger in magnitude than the acceptance limits used in screening 
the original data. For example, the lowest ECM value for the Yellowstone 
River at Billings in Table 17 has a value of 11'8 umho/cm @ 25°c. The 
ERR corresponding to this data point is -.0456, which is about one-half 
the lower acceptance limit of -.0913 used to screen the data.
Table 21 shows a typical set of composite data collected by the 
U. S. Geological Survey and a set of instantaneous data collected by the 
Montana State Water Quality Bureau for the Tongue River at Miles City. 
Table 22 shows the simulated values of the common ions obtained from the 
set of regression equations in Table 15 and the values of ECM and Q from 
Table 21. The simulated data compare favorably with the measured values 
of the common ions.and are quite adequate for the degree to which common 
ion data is essential in characterizing the water chemistry.
The ERR values indicate that no serious errors are occurring for 
the sets of regression equations at high ECM values, as occur at low ECM 
values. High ECM values generally occur at low flow conditions. The 
condition of low Q and high ECM is generally the most critical to the 
water eco-system, and is the condition at which the highest concentrations
of dissolved constituents can occur.
TABLE 21
Composite Data from the U. S. Geological Survey and 
Instantaneous Data from the Montana State W.Q.B. for 
the Tongue River at Miles City
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
CO3
(Mg/L)
SO4
(Mg/L)
Cl
(Mg/L)
ECM
(pmho/cm @ 25°c)
Q
(cfs)
89 68 90 5.6 398 0 349 5.5 1200 143
61 57 85 5.5 311 0 307 8.5 1040 85.2
76 50 56 4.2 3.5 0 243 901 354
50 31 41 5.2 220 0 152 2.5 630 704
38 11 27 4.9 156 0 67 2.6 "396 2800
35 14 13 2.0 148 0 50 1.5 327 1827
53 29 45 4.2 225 0 152 2.9 645 309
74 55 128 2.7 382 0 351 5.8 1240 21.7
100 77 146 7.9 440 0 508 8.4 1540 400
60 45 48 4.2 254 0 216 5.6 798 464
59 40 38 4.2 235 0 190 3.2 721 1139
38 15 13 2.7 153 0 56 2.0 361 3205
96 66 83 5.9 390 0 355 5.6 1200 151
48 26 38 199 2 139 1.7 572 300
Table 21 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
CO3 S04
(Mg/L) (Mg/L)
Cl
(Mg/L)
ECM
(pmho/cm @ 25°c)
Q
(cfs)
71 6l 79 320 4 320 3.0 1060 245
37 17.8 22 151 0 88 1.5 420 1330
67 46 60 266 0 245 4.0 862 66l
72 48 68 280 0 289 4.4 950 428
63 42 60 262 5 228 3.5 840 176
61 45.6 54 5.3 261 0 240 2.5 832 1510
33.7 15.2 18.0 3.3 137 0 70 0.6 362 4510
68 48.5 75 5.2 301 5 280 4.2 956 144
46.9 25.9 32.0 3.0 206 4 120 1.7 540 715
68 56 77 5.4 306 0 300 4.6 IO69 218
67 67 101 5.4 321 0 370 7.0 1135 150
68 58 90 5.5 319 I 315 5.0 1104 185
89 72 155 8.6 434 0 475 7.8 1537 70
TABLE 22
Simulated Common Ion Data for Composite U. S. G. S. and 
Instantaneous W.Q.B. Data for the Tongue River at Miles City
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3 SO4
(Mg/L) (Mg/L)
Cl
(Mg/L) ERR
ECM
(pmho/cm @ 25°c)
Q
(cfs)
82.7 65.0 98.1 6.6 376.8 364.8 5.6 -.0016 1200 143
73-3 53.5 85.1 5.8 345.7 294.4 4.9 -.0019 1040 85
65.8 46.0 65.7 5.3 291.6 249.8 4.2 -.0036 901 354
49.3 29.6 40.9 4.0 215.8 154.0 3.0 -.0067 630 704
34.0 16.9 21.5 2.8 143.0 83.2 1.9 -.0078 396 2800
29.0 13.1 17.5 2.4 125.9 62.9 1.6 -.0203 327 182?
50.0 30.0 44.2 4.0 227.6 155.8 3.1
OV"#OI 645 309
83.8 65.0 114.2 6.5 418.8 363.8 5.8 .0043 1240 22
100.6 86.8 143.1 7.9 481.0 500.8 7.2 .0039 1540 40
59.7 39.7 55.8 4.8 262.9 212.2 3.8 -.0044 798 464
55.3 35.5 46.8 4.5 234.1 188.4 3.4 -.0027 721 1139
31.6 15.1 19.1 2.6 132.5 73.3 1.7 -.0093 361 3205
82.8 65.0 97.7 6.6 376.0 365.3 5.6 -.0017 1200 151
45.3 25.7 38.3 3.6 208.1 131.5 2.7 -.0145 572 300
75.0 56.1 81.7 6.0 335.1 310.8 5.0 -.0028 1060 245
35.5 17.9 24.1 2.9 154.4 88.6 2.0 -.0136 420 1330
Table 22 (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
SO4
(Mg/L)
63.8 44.1 60.0 5.2 274.5 238.6
68.9 49.4 69.2 . 5.6 301.1 270.4
61.8 41.4 63.O 4.9 284.9 222.3
62.4 43.0 54.5 5.1 257.3 232.3
31.7 15.2 18.7 2.6 130.8 74.3
68.7 48.6 74.5 5.5 317.1 265.2
43.5 24.4 33.9 3.5 191.8 124.1
75.5 56.6 83.1 6.1 339.0 313.6
79.1 60.6 91.4 6.3 360.5 337.8
77.4 58.7 87.3 6.2 349.8 326.7
100.9 87.6 138.0 8.0 468.9 506.7
Cl
(Mg/L) ERR
ECM
(>imho/cm @ 25°c)
Q
(cfs)
4.0 -.0032 862 661
4.5 -.0033 950 428
4.0 -.0055 840 176
3.9 -.0017 832 1590
1.7 -.0052 362 4510
4.5 -.0034 956 144
2.6 -.0106 540 715
5.0 -.0026 1069 218
5.3 -.0018 1135 150
5.2 -.0023 1104 185
7.2 .0007 1537 70
71
The Inadequacy of the regression equations at low ECM values, 
which generally occur at high flows, is indicative of a change in the 
relative proportions of the common ions. At low ECM and high Qt the run 
Off component contributes a greater proportion. The common ions in the 
run off component may also exist in some constant relative proportion 
which is quite different from the rest of the water making up the total 
flow. To improve further on the regression equations, i.e., provide 
better predictions at low ECM and high Q conditions, would require know^ 
ledge of the amounts of the different components of the flow and the ECM 
values associated with these various flow components. The flow factor 
is dealt with in Chapter III, where a first level of modeling incorpor­
ating flows is developed.
' I
CHAPTER III
Large Scale Water Quality Model Employing 
Flow and Electrical Conductivity
As was mentioned at the end of Chapter II, the inadequacy of 
the set of regression equations in describing the common ion concentrations 
at low ECM and high Q has lead to the need to partition the flow into 
its various aggregate components such as run off, ground water dis­
charge and instream flow. The set of regression equations are also 
somewhat static in character. That is, they represent the current 
situation "but may not he applicable under a different set of circumstances, 
e.g. different water use policies. The question could be asked, "if a 
different water use policy was employed, how might this affect the 
regression coefficients?". To be able to predict ion concentrations for . 
different water use policies will also require knowledge of the quan­
tities of the different flow components. For example, suppose agricultural 
and industrial uses of water in sub-basin 42KJA increased to the point 
where irrigation return flows became a substantial contributor to the 
total flow in the Yellowstone River. Knowledge of how this might 
affect stream quality is of utmost importance, but the regression con­
stants for the set of regression equations can be different under these 
circumstances. Another example would be the building of a reservoir on 
the Yellowstone River mainstern. The regression constants could be dif­
ferent if the Yellowstone River were regulated. To be able to predict 
these changes requires a large scale model that breaks the flow into its
73
various components as well as appropriating the amounts of the common 
ions associated with the various components.
A state water planning model developed by Boyd and Williams 
(Ref. 2) is currently being employed by the Montana State Department of 
Natural Resources and Conservation as a tool in managing water resources. 
This model represents the various components of flow and has been applied 
with relative success. The state water planning model is an application 
of the macro-to-micro philosophy of modeling large scale systems.
This modeling approach was employed in the Yellowstone River 
sub-basin, 42KJA. Sub-basin 42KJA is a hybrid sub-basin comprising 42KJ 
and 42A. 42KJ is a sub-basin of the Yellowstone River and 42A is the 
Rosebud Creek drainage (see Figure l). Sub-basin 42KJ was previously . 
calibrated (Ref. 3)» "but after a review of the calibration of 42KJ, 42A 
was included. The lack of data for 42A necessitated the change. An annual 
water quantity model was applied to sub-basin 42KJA. Once this model 
was calibrated, it was used to build an annual water quality model for 
42KJA. Both of these models represent the first or most macro level of 
modeling.
Quantity Model
The quantity model is based upon balancing the volume of water 
that arises from the different flow components over a specified time.
The model essentially employs the principle of the conservation of 
matter. . Volumes of water are supplied to and demanded from the various 
sub-systems of the model. Figure 6 shows a schematic of the water quantity
74
Figure 6
Atmosnheric Atmospheric A t m n s n h e r l n
Inflow ^
* -- 4-— ,
OutflowStorage
LX
Precipitation Evaporation
\I
Surface Surface Surface r \
Inflow
Storage Outflow L /
LX
Percolation Discharge
\I
Ground ( \ Ground Ground f\
Inflow
Storage Outflow
Schematic of a Sub-hasin for an Annual Flow Model 
(Ref. 2)
model for a sub-basin at the annual level,.depicting the various sub­
systems . For any of the sub-systems, the supplies and demands must 
balance; i.e., the volume of water entering must equal the volume of 
water leaving, plus storage and loss. For example, consider the surface 
sub-system in Figure 6, surface inflow, precipitation, and ground water 
discharge represent volumes entering (supplies). Surface outflow, 
evaporation and percolation represent volumes leaving (demands). Since 
there can be changes in the amount of storage in the surface water sub­
system, a change in storage can be either a supply or a demand. These 
components combine additively to produce a balance of supply and demand, 
which results in an equation termed a balance equation. Thus, a balance 
equation exists for every sub-system. When there are more unknown com­
ponents than balance equations, regression equations must be supplied. 
Regression equations have the following forms
Xi = S 1 C(RlJ)Xj (15)
Where:
Xi is the parameter determined from the regression equation,
XjtS are the parameters comprising the' independent variables, 
C(RfJ)tS are the coefficients of the XjtS. 
k is the equation number of the sequence.*
Regression equations express the relationships that exist within the 
system. The coefficients of the regression equations are selected on 
the basis of a systems analysis. The systems analysis will often involve
.75
*The numbering sequence is developed in the solution technique (Ref. 2)
76
expert opinion or most likely guesses. These regression equations can 
contain statistical and/or physical relationships. Physical regression 
equations are generally an aggregated form of a balance equation that 
will emerge when a sub-system at a more micro level of modeling is defined.
The quantity model applied to 42KJA was slightly modified from 
that in Figure 6, Only the surface water and ground water sub-systems 
along with the linkages between the surface water and atmospheric water 
sub-systems were considered. Figure ? shows a schematic of this model.
The parameters used in Figure ? are defined as follows;
Xi = Volume of water that leaves the surface water sub-system 
as. outflow in a one-year period,
X2 = Volume of water that enters the surface water sub-system 
as inflow in a one-year period.
X3 = Volume of water stored in the surface water sub-system 
at the beginning of the one-year period.
X4 = Volume of water stored in the surface water sub-system 
at the beginning of the one-year period.
X5 = Volume of water that enters the surface water sub-system 
as precipitation in a one-year period.
X6 = Volume of water that leaves the surface water sub-system 
as evaporation in a one-year period.
X? = Volume of water that leaves the ground water sub-system 
as outflow in a one-year period.
X8 = Volume of water that enters the ground water sub-system 
as inflow in a one-year period.
X9 = Volume available for water storage in the ground water 
sub-system at the beginning of the one-year period.
XlO= Volume available for water storage in the ground water 
sub-system at the end of the one-year period.
77
A
y
Surface Water 
X3 (Storage) X4 
Sub-system
ZN
X12
Surface Water 
-^ j  X9 (Storage) XlO 
Sub-system
-e>
Figure ?• Schematic of Quantity Model used in Sub-basin 42KJA
?8' .
Xll — Volume of water that leaves the surface water sub-system ■ 
entering the ground water sub-system as percolation in 
a one-year period.
X12 = Volume of water that leaves the ground water sub-system 
entering the surface water sub-system as ground water 
discharge in a one-year period.
Applying the balancing condition for the two sub-systems yields the 
following two balance equations.
-Xl + X2 + X5 - X6 - Xll * X12 - (X4 - X3) = 0 (16)
-X? + X8 + Xli - X12 ^ (XIO - X9) = 0 (I?)
Equation (l6) is the surface water sub-system balance equation and equa­
tion (l?) is the ground water sub-system balance equation. The changes' • 
in the amounts of storage in the surface water and ground water sub-systems- 
are explained in Figures 8 and 9 respectively. XlO and X9 of-the ground —  
water storage are much like X4 and X3 of the surface water storage, except 
that- x4 and X3 measure volumes of water stored, and XlO and X9 measure the . 
available volumes for storage of water. This seeming inconsistency, in 
defining XlO and X9 was - an attempt to facilitate their measurement as 
parameters; i.e,, XlO and X9 as defined have the capability of being 
obtained from "specific yield" measurements, "
Equations (l6) and (l?) indicate that 12 parameters are essential 
to define the system. These parameters are classified into, two groups, 
those for which data are available (primary parameters) and those for 
which no data are available (secondary parameters), Data were available ' 
for only 5 of the 12 essential parameters; XI, X2, X3, X4, and X5«- These - 
were the primary parameters. Annual data for XI, X2, and X5 for the- 30
79
X4 - X3 >  O Heams the storage volume increased 
over the time interval, a demand.
X4 - X3 < 0 Means the storage volume decreased 
over the time interval, a supply.
Figure 8. Schematic of Surface Water Storage
80
XlO - X 9 >  0: Means the available volume for
storage increased over the time 
interval, a supply.
XlG - X 9 <  0: Means the available volume for
storage decreased over the time 
interval, a demand.
Figure 9* Schematic of Ground Water Storage
81
year period, 1944 to 1973 were provided by the Montana State Department of 
Natural Besources and Conservation. X3 and X4 were zero for each year.
The remaining 7 parameters; X6, X7, X8, X9, X10, Xll and X12; were the 
secondary parameters. Since'there are two balance equations, 2 of the 7 
secondary parameters can be determined from these equations. This leaves 
five secondary parameters for which regression equations must be supplied. 
The regression coefficients were determined on a subjective basis using 
average' annual values of the parameters as a guide (Ref. 2). The regres­
sion equations and coefficients used in the model for sub-basin 42KJA 
follow quite closely to those used in sub-basins 43Q and 42KJ (Ref. 2, 3)= 
The.evaporation, X6, was determined.from equation (l6), the sur­
face water sub-system balance equation.
The ground water inflow, X8, and the ground water.outflow, X7« .' 
were assumed to follow the alluvium of the surface basin and to consist 
of two components: a., statistical component correlated to the corresponding 
surface flow (X2 or Xl) and a constant flow component. This resulted' in 
the following two regression equations for X7 and X8 respectively:
X? = 0(2, 1)X1 + 0(2, 13)X13 (18)
X8 = o(3, 2)X2 + c(3, 13)X13 (19)
Where X13 is an exogenous constant with unity value as required by the 
solution technique. The sum of the average annual ground water flows,
(X? -lSf- X8) was assumed to be approximately 3 percent of the sum of the 
average annual surface water flows, (XI + X2). From Table I  in Appendix I f 
Xl - 8.317423 Million Acre-Feet (MAF),
82
X2 = 8.223250 MAF.
Thus,
(X? * X8) = .496220 MAF.'
Further, assuming X? equaled about 97.percent of X8,
X? = (.97)X8
then
X8 - .251888 MAF, 
X? = .244332 MAP.
The initial available ground water storage, X9» is equal to the 
final available ground water storage, XlO, of the preceding year. The 
final available ground water storage, XlO, is determined from equation 
(17), the ground water sub-system balance- equations. A stable trend for - 
the average annual available ground water storage was assumed, i.e,, X9 
equals XlO. Thus, an initial value of X9(X90) was specified for the 30- 
year period subject to X9 equaling XlO. An approximation qf the average 
annual final available ground water storage, XlO, was obtained, from the 
sum of the average annual surface water inflow, X2, for the entire Yellow­
stone basin and the average annual precipitation, T5, for sub-basin 42KJA,
A proportion of this sum was allocated to sub-basin 42KJA based upon the 
ratio' 0$© L of the area of 42KJA to the area of the entire Yellowstone Basin, 
X9 = XlO ~)T(X2 entire basin) + (X5 of 42KJA),-.'
XlO = 5.241101 MAF.,
X90  was determined subject to this average annual value.
The percolation was assumed to have the form of a physical 
regression equation.
Xll = (c(5»2)X2 ♦ 0(5,3,4)«(X3 + X4)/2 * G(5,5)X5 * C(5,12)X12)SF
(20)
Where SF is a scaling factor such that on the average the water table 
remains stable. This is consistent with X9 equaling X10. The value of 
SF was determined from average annual values of X2, X3, x4, X5» Xll and 
X12. The following numerical- values were used for the.coefficients in 
the regression equation for Xlli
0(5,2) = 1.0 - 
c(5,3,4) = 0.5" 
o(5,5) = 2.0 
0(5,12) =■ 0,5
The ground water discharge, X12, was also assumed to be given by 
a physical regression equation.
X12 = C(6,13)X13 " C(6;9,10)*(X9 + X10)/2 (2l) .
The average annual ground water discharge, X12, was assumed to be approx­
imately 7 percent of the average annual surface water outflow, XI. Thus,
83
X12 = .582220 MAF
Thus,0(6,13) was assumed to equal twice the value of X12.
0(6,13) = 1.164439 MAF
Then from equation (21) on an average annual "basisj
0(6;9,10) = .11108?
A value.of Xll was obtained from equation (l?) on an average 
annual basis. This enabled a value for SF to be determined:
SF = .0360523
Table 23 summarizes the equations for determining the values of the second­
ary parameters for the quantity model.
The model was balanced (calibrated) by adjusting the value of 
X90 for the. year 1944, until
X9 = HO = 5.241101 MAF - -
Appendix III gives a listing of the simulated secondary parameter values 
and the annual 42KJA computer program.
The validation method for the quantity model is outlined in 
Reference 2. A validation of the quantity model is not shown here since 
values of the various flow components are needed in the quality model, 
and a validation, is presented for the quality model.
Quality Model
By multiplying the various flow components Xj_ by the density of 
water, the volume system could be changed to a mass system. The under­
lying principle would still be the conservation of matter. Suppose now, 
that the concentration of TDS (mass/volume) for the various components
94
85
TABLE 23
Summary of Equations for Quantity Model 
. Secondary Parameter Equation
Evaporation
x6 x6 = -Xl + X2 + X3 - x4 + X5 - xii + xi2
Ground water outflow
X?
Ground water inflow 
X8
Final available ground 
water storage
XlO
Percolation
Xll
Ground water discharge 
X12
x? = ,015x1 + .127127
X8 = .015X2 + .120983
XlO = X7 - x8 + X9 - Xll + X12
Xll = (X2 + .25(X3 + x4) + 2.0X5 + .5X12)• 
.0360523
X12 = 1.164439 - .0555435(X9 + X10)
Initial available ground 
water storage
X9 X9i XlOi _ 1
86’
of flow were known. By multiplying the various volumes of flow for the 
time period by the average TDS concentration for the time period an 
average value of the mass of TDS associated with the various flow com­
ponents over the time period would result. Let Tj_ represent the . 
concentration of TDS associated with flow component Xj_ then,
Li = XieTi (22)
Where
Li is the mass of TDS associated with flow component Xi 
Li is generally termed the load. As an example, consider the balance 
equation for the. surface water sub-system, equation (l6). By multiplying 
the various Xi's by the appropriate Ti a balance equation is obtained for 
the Li's,
-Li + L2 + L5 - L6 - L U  + L12 - (L4 - L3) = 0 (23) .
Equation (23) expresses the balancing (supply versus demand) of the mass 
of TDS arising from the various flow components in the surface water sub­
system, Now if the regression equations for the quantity model were all 
physical regression equations, then the regression equations for the 
quality model would be:
Li = si 0(k,j)XjTj (24)
However, the regression equations for the quantity model are not all 
physical. In fact, only the regression equations for Xll and X12 are 
assumed to have physical character. Problems also arise with individual 
components. For example, X5 in the quantity model gives rise directly to 
run off. In the quality model the concentration that would be associated
8?
with the precipitation and run off are quite different6 Run off has come 
into contact with the soil and as a result becomes highly concentrated 
in TDS as compared to the concentration of TDS in the precipitation. The 
evaporation component (x6) has no associated concentration of TDS; i.e., 
T6 equals zero, and therefore, L6 equals zero, L3 and Ih are analogous 
to X3 and X4 of the quantity model but L9 and LiO differ,
19 = X9’*T9 
LlO = XiO'0TiO
Where X9' and XlO' represent the volume of stored water in the ground 
water sub-system as opposed to X9 and XlO which represent the available 
volume for storage.
Figure 10 shows a schematic of the quality model for sub-basin- 
42KJA on an annual basis. The parameters used in Figure 10 are defined 
as follows:
Li = Mass of TDS that leaves the surface water sub-system via 
the outflow, XI, in a one-year period.
12 = Mass of TDS that enters the surface water sub-system via 
the inflow, X2, in a one-year period.
L3 = Mass of TDS stored in the surface water sub-system at the 
beginning of the period.
Ih - Mass of TDS stored in the surface water sub-system at the 
end of the period.
L5 = Mass of TDS that enters the surface water sub-system in 
one period.
L? = Mass of TDS that leaves the ground water sub-system via 
outflow, X7» in one period.
88
Figure 10. Schematic of Quality Model Used in Sub-basin 42KJA
L8 = Mass of TDS that enters the ground water sub-system via 
inflow, X8, in one period.
L9 - Mass of Tds stored in the ground water sub-system at the 
beginning of the period.
LlO = Mass of TDS stored in the ground water sub-system at the 
end of the period.
L U  = Mass of TDS that is transferred from the surface water to 
the ground water sub-systems in one period.
L12 = Mass of TDS that is transferred from the ground water to 
the surface water sub-systems in one period.
Analogous to the quantity model, there are two sub-systems shown in 
Figure 10 for the quality model. The balance equations for the two sub­
systems are:
-LI + 12 + L5 - L U  + L12 - (L4 - L3) = 0 (23)
-L? + L8 + L U  - L12 - (L10 - L9) = 0 (25)
As has been previously mentioned EGM and TDS values correlate 
highly. That is, for any particular site it has been well established 
that:
ECM = =C(TDS) (26)
Where:
=C is a proportionality constant (Ref. 7)
ECM values were used in place of TDS in the quality model. This sub­
stitution does not present any particular problem; the system is a pseudo 
mass balance, but the conservation, of matter is still the underlying 
principle.
69
Li = XieEi (27)
Where:
Eji is the ECM associated with flow component Xji 
No units are explicitly expressed with the flow weighted ECM» Xji9Ei, 
although it is implicitly understood that the values represent an amount 
of matter. More data were available in the form of ECM than TDS and the 
use of ECM provides the necessary parameter for the use of the regression 
equations for the common ions.
The primary parameters for the water quality model are Li, L2, L3 
and L4. L3 and IA- are zero for each year, since X3 and X4 are zero for 
each year. Average annual values were available for ECM for the 5-year 
period, 1969 to 1973> from the U, S, Geological Survey (Ref. 6). Using 
the regression equations in Table 12 for the Yellowstone River at Billings 
and the Bighorn River at Bighorn,average annual values of ECM were calcu­
lated from Q, the average annual value , of Q. The regression equations in 
Table 12 relate the ECM to instantaneous Q values. The value of Q used 
for the Yellowstone River was increased in proportion to the amount of 
drainage area between Billings and the confluence with the Bighorn River. 
L2 was set equal to the sum of the annual ECM loads (flow weighted ECM) 
for the Yellowstone and Bighorn Rivers near their confluence. Using the 
regression equation in Table 12 for the Yellowstone River at Miles City, 
average annual values of ECM were calculated from the average annual 
values of Q. The values of Q for the Tongue River at Miles City were 
subtracted from the values of Q for the Yellowstone River at Miles City, 
since the values of Q for the Yellowstone River are taken below the
.90 ' ;
91
confluence with the Tongue River and the values of EGM are taken above 
the confluence. LI was set equal to the annual EGM load for the Yellow­
stone River at Miles City, Regression values for both LI and L2 were 
generated using equation (2?) and the regression equations of Table 12, 
for the 25-year period, 1944 to 1968, The quality model was calibrated 
with these 25 years of data and then applied to the 5 years, 1969 to 1973 
for which actual EGM data are available. The values of ECM for LI and L2 
are tabulated in Table J o f  Appendix I.
. Two of the seven secondary parameters can be determined from the 
balance equations (23) and (25). L9 and LlO are assumed to be equal for 
each year; i.e., there is no yearly change in the mass of dissolved 
materials stored in the ground water sub-system. This assumption is. 
based upon the extremely large volume of water stored in the ground water 
sub-system, the presumably small fluctuation that occurs in this volume 
from year to year, and the relatively constant value of ECM associated 
with ground water storage. So X9 minus XlO equals zero for each year. 
This leaves five secondary parameters for which regression equations must 
be supplied.
L5, the run off load is determined from the balance equation for 
the surface water sub-system, equation (23).
L?, the ground water sub-system outflow is determined from 
equation (25), the ground water sub-system balance equation.
More mass on the average is expected to enter the ground water 
sub-system via L U  than L8, The following proportion was assumed:
I92
L8/(L8 + LU) = 1/4
leading to:
L8 = (1/3)LII (28)
L U  was determined "by employing the regression equation for Xll 
from the quantity model.
L U  - (L2 * 2.0L5 + .05L12)SF* (29)
Where SF' is again a scaling factor and is determined from averaging 
values of L2, L$, L U  and L12.
Since the definitions of L9 and LlO correspond to X9* and XlOe 
rather than X9 and XIO1 the physical regression equation X12 in the 
quantity model was not applicable to the quality model. Instead the 
assumption was made that the annual average values of ECM associated with 
L7 and L12 were equal, i.e., E? = E12 for each year. From this assump­
tion the following regression equation was obtained:
L12 = (X12/X?)-L7 (30)
The fraction (X12/X7) varies from year to year and is obtained from the 
quantity model. The ECM values of Rosebud Creek in the fall of the year 
range from 1500 to 2000 urahos/cm @ 25°c (Ref. 13). The water in Rosebud 
Creek at this time of the year was assumed to be primarily due to ground 
water discharge (X12). The concentration associated with this water was 
assumed to be representative of the concentration (E12) due to X12 for 
the entire sub-basin. The average annual volume of water from ground 
water discharge (X12) for the 25 year period, 1944 to 1968, was obtained 
from the quantity model.
XiZ = 579220 MAF
93
This value of ground water discharge was multiplied by the range of ECM 
values for Rosebud Greek to get an estimate of the average annual load 
(L12). L12 ranged from 868.83 to 1158.44. From the quantity model, the
average annual value of the fraction (X12/X?) was determined to be 
2.281907. Using 1000.00 for the value of 112, L7 equaled 438,23. From 
equations (25) and (28) on an average annual basis L U  and L8 were deter­
mined to be 1078.6725 and 359*558 respectively. Using the values of LI 
and L2 along with equation (23) on an average annual basis, L5 was deter­
mined to be IO3O.98, SF1 was then determined to be .165170. An initial 
value of L8 (L80) is assumed such that L8 = (l/3)Lll. Table 24 summarizes 
the equations for determining the .values of the secondary parameters for 
the quality model.
The quality model computer program and data output for the 25-year 
period, 1944-1968, are shown in Appendix TV. The model achieved a reason­
able balance with L80 equal to 380 and SF' equal to .187* The data output 
gives ECM values in ^ imhos/cm @ 25°c for the various flow components. For 
L5» the associated run off flow component (X5*) was assumed to be the dif­
ference between the precipitation, X5» and the percolation, Xll (X5‘ =
X5 - Xll).
The quality model was validated by interchanging the roles of LI 
- and L5; i.e., L5 was assumed to be an exogenous parameter and LI, endoge­
nous, was simulated by the model. The data for the 25-year period, 1944 
to 1968, were used (Appendix IV). The simulated values of the parameters
■94-
vcUPO y,
Summary of Equations for the Quality Model 
Secondary Parameter . Equation
L5 = Li - L2 + L U  - L12 
L? = 18 + L U  - L12 
L8 = 1/3L11
L
L U  = (L2 + 2.065 + 0.5L12)* 1651696 
L12 = (X12/X?)*L7 ■
Run off
L5
Ground water outflow
L7
Ground water inflow 
L8
Percolation
L U
Ground water discharge 
L12
95
using L2 and L5 as exogenous are shown in Table 25. The values of the 
parameters in Table 25 are comparable to the output data shown in Appen­
dix IV. The favorable comparison of these values indicates how well the 
model reiterates the data, that is, its ability to reproduce historical 
record. With the model set up for L5 as an exogenous parameter, the 
values of L5 were perturbed. Increased coal mining and agricultural activ­
ities in the basin could possibly increase the concentration in the run off 
component. The values of L5 were increased by 25 percent. Using the 
perturbed values of L5« the values of the essential parameters simulated 
by the model are shown in Table 26, The values in Table 26 are comparable 
to those in the data output in Appendix IV. As would be expected to occur, 
the values of EGM associated with the outflow, El, show an increase when 
the perturbed values of I>5 were used as exogenous data. The average 
increase in El over the 25-year period was about 5 percent with a high of 
9 percent and a low of 2 percent.
Using LI and L2 as exogenous parameters, the quality model was 
applied to the 30-year period, 1944 to 1973» where available ECM values 
were used for the last five years. The data output for the 30-year per­
iod is also shown in Appendix IV. The simulated secondary parameter 
values of EGM for the last five years (1969 - 1973) are consistent with 
those for the previous 25 years (1944 - 1968).
The values of ECM for the secondary parameters generated by the 
quality model are reasonable, given the aggregating nature of annual aver­
aging on the EGM values associated with the various flow components.
TABLE 25
ECM Values for the Essential Parameters of the Quality Model,
1944 to 1968, Using L2 and L5 as Exogenous
(units = pmhos/cm @ 25°c)
Year El E2 E3 E4 E5 E6 E7 ES E U E12
1944- 612.3 567.8 .0 .0 189.8 .0 2041.2 1485.6 1883.0 2041.2
1945 596.7 542.7 .0 .0 205.4 .0 1880.4 1958.4 2104.9 1880.4
1946 628.9 532.1 .0 .0 209.0 .0 1788.6 1684.6 1899.9 1788.6
194? 575.8 540.2 .0 .0 89.4 .0 1761.3 1439-7 1855.6 1761.3
1948 574.2 493.6 Io .0 279.6 .0 1926.4 1505.1 2088.8 1926.4
1949 638.2 517.4 .0 .0 394.4 .0 1862.8 1920.9 2428.7 1862.8
1950 586.6 496.2 .0 .0 314.2 .0 1958.2 1510.9 2146.6 1958.2
1951 564.6 506.1 .0 .0 259.2 .0 2003.8 1653.1 2109.5 2003.8
1952 599.9 447.4 .0 .0 659.5 .0 2097.0 1789.8 2676.7 2097.0
1953 664.1 491.4 .0 .0 351.0 .0 1950.5 2063.4 2181.1 1950.5
1954 664.1 478.9 .0 .0 439.3 .0 1919.1 1732.9 2376.6 1919.1
1955 683.9 526.5 .0 .0 332.7 .0 1844.4 1831.6 2230.0 1844.4
1956 603.9 448.5 .0 .0 523.9 .0 2110.1 1451.6 2463.1 2110.1
1957 568.2 418.2 .0 .0 447.0 .0 2321.2 1715.5 2222.4 2321.2
1958 636.5 539.2 .0 .0 268.1 .0 1993.5 2177.6 2190.3 1993.5
Table 25 (Cent.)
Year El E2 E3 E4 E5 E6 E7 ES E U E12
1959 639.2 466.4 .0 .0 428.1 .0 2017.2 1611.2 2312.1 2017.2
I960 723.5 501.3 .0 .0 559.6 .0 1903.6 2067.0 2724.7 1903.6
1961 775.9 490.3 • .0 .0 453.2 .0 1816.0 1828.5 2425.9 1816.0
1962 584.1 476.7 .0 .0 311.6 .0 2127.2 1276.5 2049.9 2127.2
1963 611.6 512.9 .0 .0 198.8 .0 2067.4 1868.6 1860.6 2067.4
1964 590.7 522.1 .0 .0 196.0 .0 1994.8 1556.7 1941.9 1994.8
1965 543.6 497.7 .0 .0 184.7 .0 2082.5 1443.0 1908.1 2082.5
1966 705.2 439.3 .0 .0 623.5 .0 2039.5 2232.0 2718.1 2039.5
196? 560.9 464.7 .0 .0 314.6 .0 2136.6 1387.0 2060.4 2136.5
1968 563.9 486.9 .0 .0 226.7 .0 2142.5 1772.6 1900.5 2142.5
TABLE 26
EGM Values for the Essential Parameters of the Quality Model,
rsiC4  to 1968, Employing Perturbed Values of L5
(units = jumho/cm @ 25°c)
Year El E2 E3 E4 E5 E6 E7 ES E U E12
1944 634.8 567.8 .0 .0 237.2 .0 2152.0 1485.6 2012.1 2152.0
1945 6l4.6 542.7 .0 .0 256.7 .0 1987.4 2092.6 2214.6 1987.4
1946 654.9 532.1 .0 .0 261.2 .0 1909.1 1772.4 2037.9 1909.1
194? 584.6 540.2 .0 .0 111.7 .0 1827.4 1544.2 1904.0 1827.4
1948 596.5 493.6 .0 .0 349.5 .0 2042.7 1544.4 2236.1 2042.7
1949 669.I 517.4 .0 .0 493.0 .0 2009.3 2056.4 2627.4 2009.3
1950 615.1 496.2 .0 .0 392.7 .0 2119.6 1634.4 2323.9 2119.6
1951 585.1 506.1 .0 .0 324.0 .0 2140.1 1789.6 2242.8 2140.1
1952 639.3 447.4 .0 .0 824.4 .0 2298.9 1902.9 2963.4 2298.9
1953 709.3 491.4 .0 .0 438.7 .0 2159.1 2284.4 2414.2 2159.1
1954 711.6 478.9 .0 .0 549.1 .0 2131.6 1918.1 2642.7 2131.6
1955 725.0 526.5 .0 .0 415.9 .0 2029.4 2036.7 2444.2 2029.4
1956 646.4 448.5 .0 .0 654.9 .0 2340.0 1590.9 2739.7 2340.0
1957 609.0 418.2 .0 .0 558.7 .0 2583.6 1908.1 2474.0 2583.6
1958 664.4 539.2 .0 .0 335.1 .0 2161.2 2424.2 2345.9 2161.2
Table 26 (Cent.)
Year El E2 E3 E4 E5 E6
1959 683.3 466.4 .0 .0 535.1 .0
I960 778.8 501.3 .0 .0 699.5 .0
1961 647.1 490.3 .0 .0 566.5 .0
1962 617.0 476.7 .0 .0 389.5 .0
1963 636.5 512.9 .0 .0 248.5 .0
1964 610.2 522.1 .0 .0 245.0 .0
1965 558.7 497.7 .0 .0 230.9 .0
1966 767.4 439.3 .0 .0 779.4 .0
196? 589.7 464.7 .0 .0 393.2 .0
1968 587.4 486.9 .0 .0 283.4 .0
E7 ES E U E12
2222.6 1725.6 2568.8 2222.6
2123.4 2296.5 3044.2 2123.4
2053.2 2042.9 2754.2 2053.2
2341.5 1449.3 2239.4 2341.5
2223.5 2041.4 1989.0 2223.5
2116.8 1664.1 2055.8 2116.8
2192.8 1527.7 2005.3 2192.2
2265.6 2345.7 3085.4 2265.6
2340.5 1574.4 2235.2 2340.5
2303.2 1923.1 2036.5 2303.2
100
At this macro level of modeling the regression equations for the 
common ions, which involve instantaneous ECM values, could he applied to 
get annual average concentrations. The assumption made is that for each 
flow component, the common ion regression equations have the same form . 
(e.g. equation 3)• The model must be extended further toward the micro 
to obtain an indication of how the ions are associated with the various 
flow components on a shorter time period. The Montana State Department 
of Natural Resources and Conservation has recently completed calibrating 
a monthly water quantity model for sub-basin 42KJA, which might also be 
used to generate flow for a monthly water quality model.
_ CHAPTER IV
Concluding Remarks.
Current Utility
The regression equations for the common ions for this reach of the 
Yellowstone River are quite useful as they now stand. With the exception 
of high flow when ECM values will be low, excellent continuous values of 
the common ions can be simulated from continuous monitoring of ECM and Q. 
Devises for continuous monitoring of ECM and Q are currently available. 
Continuous flow recording devises have been employed for a number of years 
by the U. S. Geological Survey and the use of continuous recording ECM 
devises is currently increasing. The monitoring of ECM is of less cost 
than the analyzing of water samples for common ion concentrations. This 
aspect could release funds for sampling and analysis of parameters for 
which little data exists. Nutrients are an example of parameters for 
which little data are available. Analyses for nutrients cost consider­
ably more than for common ions and the data which are available do not 
correlate well with either Q or ECM (Ref. 5» 10).
The regression analysis has also, pointed out the possible need 
for better measurements of K and Cl ion concentrations. Better tech­
niques for analysis of K and Cl are available.
In trying to apply the regression equations to the concentration 
of other dissolved constituents, e.g. nutrients and metals, two problems 
are apparent. First, the ability to measure these other constituents is 
hampered by their relatively low concentrations. Thus, they may suffer
-102
from the same problem as K and Cl. Secondly, some constituents, like 
nutrients are nonconservative; i.e., the ionic form of the constituent 
may change in the water solution. For example, • bacterical action on 
ammonia (NH3) can change it to nitrate (NO3).
The regression equations for the common ions could also be used 
for quality control in the laboratory analyses. The values generated by 
the regression equations could provide a check on the chemical analyses.
Recommendation for Future Research
The regression equations involving the ion concentrations as 
expressed in terms of the natural logarithm of EGH seems to reveal some 
underlying physical character of natural- waters. This logarithmic rela­
tionship needs to be tested further. Because data are collected in a 
periodical fashion at the four sites (e.g. monthly, daily), the distri­
bution of the data over the range of flow values is biased. As a 
consequence of periodic sampling, more data are available at median flow 
values. This has a definite effect on the ability of the regression equations 
to predict ion concentrations at high and low flow conditions. The weak­
ness of the ion regression equations at high Q, low ECM conditions 
indicates the need of including Q in the regression equations. The way 
in which Q was introduced into the regression analysis (equation 3) did 
not alleviate the problem at low ECM values. Further investigation is 
needed to determine how the Q dimension could be introduced. An example 
would be to let the regression coefficients of equation 3, M ,  ai, be
103
linear functions of Q; e.g.,
M  « ci‘Q. + di (31)
ai = ei*Q ^ fi (32)
More study is needed on other streams and different reaches of 
the same streams. Streams that cut through different geological strata 
may show different characteristic distributions for the common ion 
concentrations. Different reaches of the same stream may also show a 
different distribution of the common ion concentrations, even though the 
geological features are similar, due to different flow regimes. Invest­
igating the application of the common ion regression equations to 
different streams and reaches of streams, and the various methods of 
introducing a flow dimension may not only increase their usefulness but 
may yield a better understanding of the physical phenomena involved. 
Even.in situations where the given regression equation form is not appli 
cable, further investigation would be useful.
An investigation as to how the regression constants change with 
flow regulation would be of value. For example, by studying a stream 
for which sufficient data exists before and after the implementation of 
a reservoir, the change in the regression coefficients due to regulation 
could be determined. Thus, an indication of how the distribution of 
common ions may change for a similar stream might be predicted.
In this study, the first level of a large-scale, water quality 
model was investigated as a possible method of obtaining information on 
the amounts of ECM associated with the various flow components, The
model produced reasonable results in the first level, but application of 
the common ion regression equations would yield only annual average com­
mon ion concentrations, Since the common ion regression equations 
involve instantaneous values of ECM and Q, they are at a much more micro 
level. By developing the model to a more micro level, the application of 
the common ion regression equations would yield more useful results.
Since the monthly quantity model for 42KJA has been calibrated and is 
available, the development of a monthly quality model would be an appro­
priate next step. At the monthly level, the application of the regression 
equations for the common ion concentrations would yield monthly average 
values„ Monthly average values of the common ion concentrations would be 
more useful to water quality planners and managers.
After the monthly water quality model for sub-basin 42KJA, the 
next more micro level of modeling might focus on a smaller geological 
unit. For example, the modeling of a drainage area of similar geologic 
features may be appropriate. This level of modeling might be coupled 
with an investigation of the common ion regression equations to the same 
drainage area.
REFERENCES
References
1. Federal Water Pollution Control Act. Amendments of 1972, 70 Stat.
498; 84 Stat. 91, 33 USC 1151.
2. Boyd, D. W. and T. T. Williams. Development of a State Water-
Planning Model, Part I and Part 3: Methodology and Peripheral
Models of Sub-basin 430, of the Yellowstone Basin. Montana 
State University Water Resources Research Center, Montana State 
University. Report Nos. 15 and 32. Bozeman, Montana, Second 
Printing, April and December 1972.
3« Weaver, James T., et. al. Calibration of the State Water-Planning 
Model for the Sub-basin 42KJ of the Yellowstone River ,Basin. 
Montana State University Water Resources Research Center, 
Montana State University, Report No. 73» Bozeman, Montana, 
September 1975»
4. Climatic Summary of the United States, United States Department of 
Commerce, Water Bureau, Montana, 1975»
5» Frost, Leonard R., Jr. Evaluation and Simulation of Chemical-
Quality Data for Five Montana Sampling Stations. United States 
Department of the Interior, Geological Survey, Helena, Montana, 
1974.
6» Water Resources Data for Montana 1966-1974, Part 2, Water Quality 
Records 1967-1975» United States Department of the Interior, 
Geological Survey.
7« Hem, John D. Study and Interpretation of the Chemical Character­
istics of Natural Water. United States Department of the 
Interior, Geological Survey (WSP-1473), Second Edition, 1970.
8 . Hall, F. R. "Dissolved Solids-Discharge Relationships, I, Mixing
Models." Water Resources'Research, 6:845-850, 1970.
9. Hall, F. R. "Dissolved Solids-Discharge Relationships, 2, Applica­
tion to Field Data." Water Resources Research, 7:591-601,
1971.
IOe Steele, T. D. and M. E. Jennings. "Regional Analysis of Stream-, 
flow Chemical Quality in Texas." Water Resources Research, 
8:460, 1972.
11. Ledbetter, J . 0., and E. F. Gloyna M. "Predictive Techniques for 
Water Quality Inorganics." Journal Sanitary Engineering Div­
ision, ASCE, 90:127-151, February, 1964.
10?,
12o Karp, Richard H. and M. K. Botz. Water Quality Inventory and
Management Plan Middle Yellowstone River Basin. .Montana Depart­
ment of Health and Environmental Sciences, Water Quality Bureau,
1975.
13« Unpublished Water Quality Data, Montana Department of Health and 
Environmental Sciences, Water Quality Bureau,
APPENDICES
-APPENDIX I
Raw Data
Contents
Tables of raw data points for the Yellowstone River at Billings 
the Bighorn River at Bighorn, the Tongue River at Miles City, and the 
Yellowstone River at Miles City. This data was obtained from U. S. 
Geological Survey annual report of quality of surface waters in Montana 
(Ref. 6).
Table of annual inflow and outflow and precipitation for sub­
basin 42KJA.
Cross Reference
Pages Iv, 13, 43, 44, and 51
TABLE A
Raw Data for the Yellowstone River at Billings
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(MgA)
S04
(Mg/L)
Cl
(Mg/L)
CO/3
(Mg/L)
ECM
(pmho/cm @ 25°c)
Q
(cfs)
37.0 13.0 23.0 2.8 155 77 3.7 0.0 393 6090
37.0 12.0 23.0 2.5 150 68 5.0 0.0 389 4640
38.0 13.0 24.0 3.1 151 77 6.6 0.0 398 3700
45.0 15.0 26.0 3.2 171 89 10.0 0.0 465 2180
44.0 14.0 25.0 3.5 170 87 10.0 0.0 438 2970
45.0 14.0 35.0 3.5 172 97 8.6 0.0 479 5000
39.0 13.0 27.0 3.1 151 74 7.8 0.0 405 3690
30.0 9.9 19.0 3.6 118 51 5.7 0.0 315 6230
14.0 3-3 5.8 1.3 56 12 1.1 0.0 118 41300
18.0 6.0 11.0 1.9 79 24 2.7 0.0 185 16500
29.0 10.0 19.0 2.5 130 46 4.4 0.0 304 7020
31.0 11.0 20.0 4.2 138 58 4.5 0.0 345 6510
38.0 13.0 25.0 3.2 156 67 5.9 0.0 386 4930
37.0 13.0 28.0 3.1 153 76 6.4 0.0 399 4180
38.0 12.0 25.0 2.8 14? 76 6.3 0.0 386 4160
36.0 10.0 25.0 4.1 134 49 6.4 0.0 340 4840
41.0 11.0 26.0 3.5 155 58 6.3 0.0 392 3110
CTx
 
O
Table A (Cent.)
, Q 
(cfs)
Ca
(Mg/L)
43.0
43.0
24.0
16.0
15.0
24.0
30.0
30.0
34.0
38.0 
.0 
.0
39.0
41.0
46.0
21.0
17.0
35.0
37.0
Mg Na K HOO3 SO4 Cl CO3 ECM
(Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (umho/cm @ 25°c)
15.0 28.0 3.7 171 70 8.5 0.0 440 2940
14.0 30.0 3.6 160 88 7.9 0.0 429 3330
7.1 11.0 3.3 104 22 3.6 0.0 225 13700
5.0 7.3 1.2 74 19 2.0 0.0 161 24400
5.2 10.0 1.7 76 22 3.0 0.0 179 27000
8.3 16.0 2.4 119 39 3.7 0.0 263 8890
10.0 19.0 2.7 146 46 4.8 0.0 318 7440
11.0 20.0 2.6 128 59 5.7 0.0 326 6460
12.0 21.0 ' 2.6 140 64 6.0 0.0 372 5370
13.0 25.0 2.8 151 77 6.0 0.0 398 4440
17.0 31.0 3.9 186 96 7.9 0.0 485 2140
13.0 25.0 3.5 161 84 7.9 0.0 429 3160
13.0 27.0 3.4 153 84 8.1 0.0 422 3470
14.0 29.0 3.2 162 79 6.6 0.0 439 3970
18.0 46.0 3.3 197 120 7.2 0.0 549 8500
6.5 13.0 1.8 96 33 3.7 0.0 217 15200
5.7 9.7 1.6 75 22 2.8 0.0 177 18500
13.0 26.0 3.0 138 62 5.7 0.0 386 3650
13.0 25.0 2.9 157 69 5.2 0.0 382 5970
Table A (Cent.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HOO3 SO4
(Mg/L) (Mg/L)
Cl GO3
(Mg/L) (Mg/L)
ECM
(uraho/cm @ 25°c)
, Q
(cfs)
46.0 16.0 31.0 4.1 178 88 7.3 0.0 477 3030
50.0 20.0 40.0 6.1 202 118 9.0 0.0 602 1330
36.0 15.0 25.0 4.2 152 77 4.4 0.0 439 3030
39.0 14.0 31.0 4.5 158 86 10.0 0.0 444 2970
41.0 14.0 31.0 4.6 162 88 7.4 0.0 446 3560
38.0 16.0 37.0 7.7 174 97 8.8 0.0 499 4240
22.0 6.3 12.0 1.7 94 27 2.4 0.0 225 24200
18.0 5.7 11.0 2.1 78 24 7.9 0.0 189 17600
22.0 8.4 18.0 '2.5 108 39 4.3 0.0 276 9000
36.0 13.0 26.0 3.6 14? 68 6.6 0.0 367 3890
38.0 13.0 • 24.0 3.2 153 65 6.8 0.0 400 3830
37.0 12.0 26.0 3.6 146 74 7.2 0.0 402 3720
38.0 16.0 30.0 4.1 173 77 6.0 0.0 457 3140
21.0 5.5 10.0 1.8 94 24 2.1 0.0 199 33400
40.0 15.0 3l.o 3.1 156 94 6.5 3.0 438 4610
43.0 15.0 29.0 3.2 167 70 7.6 0.0 436 4450
49.0 17.0 28.0 3.8 180 86 8.7 0.0 492 2680
48.0 16.0 30.0 3.6 171 98 7.6 0.0 476 2730
4?.0 16.0 32.0 3.9 165 91 7.7 2.0 483 2550
Table A (Cent.)
Ga
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HOO3 SO4
(Mg/L) (Mg/L)
Cl OO3
(Mg/L) (Mg/L)
ECM
(pmho/cm @ 25°c) (cfs)
39.0 13.0 28.0 3.4 149 78 6.4 0.0 411 3770
30.0 9.7 24.0 2 .8 114 63 7.3 0.0 295 6440
14.0 4.1 6.8 1.4 67 13
CO 0.0 148 38600
10.0 5.3 8.8 1.7 67 21 2.0 0.0 152 29800
25.0 8.1 16.0 2.5 103 42 3.5 0.0 252 9210
36.0 13.0 33.0 3.1 143 63 5.6 2.0 373 5040
41.0 16.0 27.0 2.7 169 79 6.0 0.0 439 3950
38.0 14.0 25.0 3.0 158 67 6.6 0.0 406 3150
38.0 15.0 30.0
CMcA 164 75 5.6 0.0 428 4160
38.0 14.0 2?.0 3.4 149 83 6.6 0.0 421 3850
40.0 13.0 ' 27.0 2.8 162 76 6.6 0.0 417 3320
21.0 5.5 10.0 1.8 94 24 2.1 0.0 199 33400
29.0 10.0 19.0 2.4 131 49 4.5 0.0 317 8250
TABLE B
Raw Data for the Bighorn River at Bighorn
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HOO3
(Mg/L)
SO4
(Mg/L)
Cl
(Mg/L)
CO3
(Mg/L)
ECM
(jumho/cm @ 25°c)
, Q 
(cfs)
77 25 82 4.7 210 310 11.0 0 982 5970
77 27 81 3.6 207 290 11.0 0 906 5830
87 32 86 3.9 206 330 12.0 8 988 1900
92 26 95 3.8 211 320 14.0 I 957 3500
81 29 84 4.1 216 300 11.0 0 950 3650
75 28 82 3.9 218 290 11.0 0 961 4960
86 31 85 4.3 210 300 12.0 6 970 5230
85 30 81 3.9 231 300 12.0 0 946 6950
80 28 72 5.2 233 280 14.0 0 890 7900
77 27 66 3.3 214 250 11.0 0 840 6270
62 23 61 3.6 180 210 7.6 0 740 3740
48 17 46 3.0 149 160 6.1 0 580 3470
65 26 74 3.2 173 250 10.0 4 910 2900
63 22 66 3.3 183 230 9.3 0 752 3310
76 26 80 3.9 202 290 10.0 0 895 1280
72 25 80 3.7 200 260 11.0 0 747 4300
71 25 75 3-3 195 270 10.0 0 836 4260
TatxLe B (Cent.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
S04
(Mg/L)
73 24 73 3.5 203 270
81 28 81 3.7 221 290
80 29 85 3.3 223 280
91 41 120 5.0 212 450
73 25 62 3-6 21? 230
73 26 71 3.6 216 260
76 29 86 3.8 215 300
70 24 72 4.0 195 260
72 25 76 '3.4 200 290
64 22 76 3.1 O
O
V
i 250
70 25 ' 76 3.5 197 250
68 22 70 3.5 183 240
78 27 81 4.2 216 290
75 25 71 4.3 204 270
72 25 76 4.2 20? 270
77 26 81 4.0 202 280
79 29 95 4.2 204 340
77 27 94 3-3 215 320
74 28 84 2.3 209 280
Cl CO3 ECM Q
(Mg/L) (Mg/L) (pmho/cm @ 25°c) (cfs)
10.0 0 823 3740
11.0 0 902 3240
10.0 0 921 3320
11.0 0 1170 7220
10.0 0 793 7260
12.0 0 858 3760
12.0 0 914 1490
10.0 0 823 3540
7.3 0 878 5500
8.8 0 806 5280
10.0 0 859 3360
9.1 0 762 4060
12.0 0 926 4000
11.0 0 854 7070
12.0 0 871 7170
12.0 0 901 6750
11.0 0 936 2760
11.0 0 946 2970
8.4 0 923 3100
115
Table B (Coni.)
Ga
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
sou
(Mg/L)
Cl CO3
(Mg/L) (Mg/L)
ECM
(jimho/cm @ 25°c) (cfs)
70 24 69 3.9 191 250 8.1 0 740 3930
72 23 70 4.2 191 260 9.1 0 747 6500
83 32 88 3.9 226 300 12.0 0 983 4820
83 29 85 3.9 230 320 11.0 0 943 6420
85 30 81 6.1 226 290 13.0 0 954 6340
75 28 90 3.4 213 270 9-8 0 926 9160
58 22 33 3.1 176 150 7.6 0 599 6300
49 16 52 2.5 160 170 5.5 0 598 5100
50 18 52 3.0 160 180 6.9 0 609 3030
80 28 98 4.3 220 329 11.0 0 986 3340
75 32 93 4.7 226 326 11.0 0 994 3900
80 31 92 4.6 224 316 11.0 0 963 3720
83 31 100 5.1 235 334 13.0 0 1010 5040
88 45 123 5.5 279 418 10.0 0 1180 2040
84 34 HO 5.7 236 357 10.0 0 1050 2880
88 31 81 3.7 246 302 7.4 0 933 5610
85 29 81 4.4 230 284 13.0 0 899 11100
71 28 84 3.9 216 277 11.0 0 867 3080
64 24 72 4.0 187 237 7.2 0 761 2060
Table B (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCOi
(Mg/L)
SO4
(Mg/L)
Cl
(Mg/L)
CO3
(Mg/L)
ECM
Oumho/cm @ 25°c)
, Q 
(cfs)
73 26 84 3.6 212 288 13.0 0 900 4230
80 27 83 4.2 219 295 11.0 0 914 5030
70 24 84 4.1 188 284 9.4 0 866 3750
68 22 73 3.9 190 260 8.0 0 892 5800
78 29 87 4.1 229 301 11.0 0 934 4500
73 26 79 4.0 210 271 9.6 0 878 12800
60 21 64 ' 3.4 172 220 7.7 0 723 3420
96 34 117 3.9 228 400 15.0 0 1160 1310
92 37 106 4.5 246 373 14.0 0 1120 2860
52 17 6l 3.2 150 18? 5.0 0 637 2630
87 35 100 4.4 229 349 14.0 0 1070 840
TABLE C
Raw Data for the Tongue River at Miles City
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
SO4
(Mg/L)
Cl CO3
(Mg/L) (Mg/L)
ECM
()imho/cm @ 25°c)
, Q 
(cfs)
58 42 47 5.2 262 190 4.1 0 755 444
65 45 49 4.0 279 200 3.5 5 802 539
85 62 83 4.7 396 310 8.2 0 1120 188
95 69 88 6.1 448 350 6.0 0 1230 222
77 52 70 5.5 325 270 4.4 0 988 340
63 47 68 4.6 300 260 4.3 0 955 389
77 62 92 6.1 334 350 5.4 3 1140 245
71 50 55 5.1 280 280 5.0 0 914 569
46 30 40 3.6 208 160 2.6 0 625 815
33 16 17 1.9 139 68 1.4 0 365 1310
46 26 43 3.6 230 130 3.4 0 628 173
52 31 46 4.3 239 160 3.0 0 680 270
57 41 60 4.9 266 210 4.2 3 980 140
65 45 58 4.4 277 230 4.1 0 849 372
65 49 61 4.5 300 240 4.5 0 877 395
86 64 92 6.3 323 370 6.3 0 1030 78
76 54 77 5.1 372 280 4.8 0 1130 195
Table C (Coni.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
SO4
(Mg/L)
Cl
(Mg/L)
CO3
(Mg/L)
ECM
(jumho/cm @ 25°c)
, Q 
(cfs)
78 57 75 5.2 372 280 5.3 O 1050 270
64 48 90 5.5 299 320 4.9 O 1040 430
73 59 100 6.4 343 340 5.4 O 1170 179
68 51 54 4.8 282 260 4.4 O 886 983
36 20 26 2.8 157 89 2.8 O 445 1950
41 22 28 3.6 188 100 3.5 O 490 554
52 30 46 4.4 242 160 3.4 O 599 257
45 25 68 5.0 226 170 3.0 O 704 361
44 27 70 ‘4.9 219 210 2.3 O 724 410 .
69 50 82 5.2 318 290 3.4 O 1030 341
75 55 ' 88 5.3 334 330 5.3 O 1080 345
80 58 88 4.8 363 330 6.3 O 1120 128
84 59 78 5.5 388 310 5.2 O 1100 198
31 17 39 5.2 139 120 2 .8 O 463 2800
70 47 64 6.0 263 270 4.7 O 899 1230
68 51 H O 8.4 283 360 5.7 O 1100 604
45 25 30 3.2 169 130 2.4 O 535 1660
64 43 59 5.1 272 230 4.8 O 822 269
68 47 65 4.9 304 250 4.6 O 902 272
Table C (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HOOri SO/.
(Mg/L) (Mg/L)
Cl CO3
(Mg/L) (Mg/L)
ECM
(nmho/cm @ 25°c)
, Q
(cfs)
75 51 74 5.7 346 280 6.0 O 988 227
91 54 72 5.4 369 280 4.5 O 966 172
65 37 42 6.0 255 190 4.4 O 725 1070
78 57 94 8.2 293 360 4.4 O 1140 930
72 51 67 7.8 259 300 5.0 O 922 1040
35 20 33 3.0 152 H O 5.4 O 467 1840
51 31 46 3.7 231 160 2.7 O 653 300
57 37 79 4.5 287 230 3.3 O 881 69
53 34 59 '4.7 238 190 2 .8 O 759 44?
62 65 121 6 .8 301 396 5.8 O 1170 125
85 70 ' 87 6.3 389 337 4.2 O 1150 150
70 54 74 5.8 334 282 4.2 O 1000 265
68 49 74 6.0 315 260 4.6 O 939 350
73 53 71 6.2 310 278 4.2 O 958 526
64 47 66 6.6 268 265 4.2 O 88? 596
49 23 25 3.3 196 114 0.8 O 503 3360
48 27 43 4.2 220 145 3.0 O 610 363
45 26 40 3.5 213 134 2.6 O 580 243
57 30 52 5.2 252 276 2.9 O 719 235
Table C (Cont.)
Ca
(Mg/L)
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCO3
(Mg/L)
SO4
(Mg/L)
Cl CO3
(Mg/L) (Mg/L)
ECM
Oumho/cm @ 25°c) (cfs)
76 54 73 5.1 327 276 4.8 9 997 267
70 48 64 4.5 330 243 4.4 0 924 200
63 41 62 5.9 227 211 3.4 0 813 184
63 45 65 5.0 288 234 5.2 0 859 380
65 54 70 5.3 303 267 4.6 9 963 567
53 35 36 4.1 228 171 2.4 0 667 1270
53 31 34 3.5 237 143 2.2 0 627 575
76 54 100 9.9 339 345 5.0 0 1020 33
65 49 65 5.0 266 283 3.6 7 969 373
58 38 56 4.7 250 199 2.2 0 759 150
60 42 79 5.6 274 245 3.5 4 900 52
e
 
%
TABLE D
Raw Data for the Yellowstone River at Miles City
Ca Mg Na K HCO3 SO/. Cl CO3 ECM Q
(Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (Mg/L) (jumho/cm @ 25°c) (cfs)
22.0 55
23.0 63
65 25.0 74
72 27.0 76
65 24.0 68
53 18.0 43
40 7.9 25
28 7 .6 17
33 12.0 • 27
38 16.0 39
50 19.0 52
66 25.0 70
56 20.0 57
33 12.0 28
54 22.0 62
52 19.0 60
23 7.0 15
3.6 183 180 9.2 7
3.5 193 210 10.0 0
3.9 205 250 9.3 0
4.3 203 250 12.0 2
4.0 201 230 10.0 0
3.1 162 140 7.5 0
2.3 116 85 4.8 0
1.7 90 49 3.8 0
2.7 116 82 4.0 0
2.7 153 H O 5.8 0
3.3 162 150 7.7 5
3.7 200 220 11.0 0
3.4 184 170 10.0 0
2.1 119 81 5.6 0
4.2 18? 190 9.5 0
4.2 163 200 9.1 0
1.5 83 47 2 .8 0
681 7460
762 8530
828 7880
857 8000
800 8680
566 14800
365 40200
274 58400
385 19900
480 10600
660 7860
824 6900
671 9000
389 28600
701 10700
661 22000
247 48400
Table D (Cent.)
Ca
( g/D
Mg
(Mg/L)
Na
(Mg/L)
K
(Mg/L)
HCOi
(MgA)
SO4
(Mg/L)
Cl
(Mg/L)
COi
(Mg/L)
ECM
(pmho/cm @ 25°c)
Q
(cfs)
38 15-0 37 2.7 151 H O 5.8 O 475 10400
58 21.0 55 4.1 192 190 9.8 O 668 6840
64 30.0 62 4.4 189 230 7.9 O 746 8100
58 24.0 59 4.1 204 200 9.5 O 703 14400
36 12.0 29 1.9 123 98 3.9 O 403 32000
41 15.0 46 3.4 167 130 8.1 O 534 10300
61 25.0 69 4.4 196 223 9.0 O 764 6800
68 26.0 80 5.3 196 266 11.0 O 883 8600
39 14.0 30 2.6 142 91 3.2 O 439 31100
51 18.0 54 4.1 178 166 9.2 O 625 6610
50 16.0 45 2.9 160 152 7.0 O 571 10700
57 22.0 6l 3.9 189 202 9.6 O 702 7640
63 22.0 64 4.5 196 220 11.0 O 748 9800
23 7.5 16 1.9 91 46 2 .8 O 249 27400
59 23.0 70 4.3 196 210 6.8 O 724 6900
TABLE E
Common Ion Data for the Yellowstone River 
at Billings in Meq/L
Ca Mg Na K ALK SO4 Cl ERR
1.8463 1.0694 1.0004 .0716 2.5402 1.6031 .1044 -.0631
1.8463 .9872 1.0004 .0639 2.4583 1.4157 .1410 -.0296
1.8962 1.0694 1.0439 .0793 2.4746 1.6031 .1862 -.0419
2.2455 1.2340 1.1309 .0818 2.8024 1.8529 .2821 -.0509
2.1956 1.1517 1.0874 .0895 2.7860 1.8113 .2821 -.0755
2.2455 1.1517 1.5224 .0895 2.8188 2.0195 .2426 -.0142
1.9461 1.0694 1.1744 .0793 2.4746 1.5406 .2200 .0080
1.4970 .8144 .8264 .0921 1.9338 1.0618 .1608 .0230
.6986 .2715 .2523 .0332 .9177 .2498 .0310 .0464
.8982 .4936 .4785 .0486 1.2947 .4997 .0762 .0255
1.4471 .8226 .8264 .0639 2.1305 .9577 .1241 -.0164
1.5469 .9049 .8699 .1074 2.2616 1.2075 .1269 -.0475
1.8962 1.0694 1.0874 .0818 2.5566 1.3949 .1664 .0041
1.8463 1.0694 1.2179 .0793 2.5074 1.5823 .1805 -.0135
1.8962 .9872 1.0874 .0716 2.4091 1.5406 .1777 -.0208
1.7964 .8226 1.0874 .1049 2.1960 1.0202 .1805 .1150
%
Table E (Coni.)
Ca Mg Na K ALK SO4 Cl ERR
2.0459 .9049 1.1309 .0895 2.5402 1.2075 .1777 .0607
2.1457 1.2340 1.2179 .0946 2.8024 1.4574 .2398 .0419
2.145? 1.1517 • 1.3049 .0921 2.6221 1.8321 .2228 .0037
1.1976 .5841 .4785 .0844 1.7044 .4580 .1015 .0350
.7984 .4113 .3175 .0307 1.2127 .3956 .0564 -.0663
.7485 .4278 .4350 .0435 1.2455 .4580 .0846 -.0775
1.1976 .6828 .6960 .0614 1.9502 .8120 .1044 -.0831
1.4970 .8226 .8264 .0691 2.3927 .9577 .1354 -.0808
1.4970 • 9049 .8699 .0665 2.0977 1.2283 .1608 -.0435
I.6966 .9872 .9134 .0665 2.2944 1.3324 .1692 -.0355
I.8962 1.0694 1.0874 .0716 2.4746 1.6031 .1692 -.0292
2.4950 1.3985 1.3484 .0997 3.0482 1.9987 .2228 .0136
2.2954 1.0694 1.0874 .0895 2.6385 1.7488 .2228 -.0149
1.9461 1.0694 1.1744 .0870 2.5074 1.7488 .2285 -.0474
2.0459 1.1517 1.2614 .0818 2.6549 1.6447 .1862 .0122
2.2954 1.4808 2.0009 .0844 3.2285 2.4983 .2031 -.0116
1.0479 .5347 .5655 .0460 1.5733 .6870 .1044 -.0748
.8483 .4689 .4219 .0409 1.2291 .4580 .0790 .0078
1.7465 1.0694 1.1309 .0767 2.2616 1.2908 .1508 .0802
1.8463 1.0694 1.0874 .0742 2.5730 1.4365 .1467 -.0192
>-*>
PO
Vx
Table E (Cont.)
Ga Mg Na K ALK SO4 Cl ERR
2.2954 1.3162 1.3484 .1049 2.9171 1.8321 .2059
ON1HS
2.4950 1.6453 1.7399 .1560 3.3104 2.4567 .2539 .0025
1.7964 1.2340 1.0874 .1074 2.4910 1.6031 .1241 .0017
1.9461 1.1517 1.3484 .1151 2.5894 1.7905 .2821 -.0218
2.0459 1.1517 1.3484 .1176 2.6549 1.8321 .2087 -.0069
1.8962 1.3162 1.6094 .1969 2.8516 2.0195 .2482 -.0198
1.0978 .5183 .5220 .0435 1.5405 .5621 .0677 .0051
.8982 .4689 .4785 .0537 1.2783 .4997 .2228 -.0521
1.0978 .6910 .7829 .0639 1.7699 .8120 .1213 -.0253
1.7964 1.0694 1.1309 .0921 2.4091 1.4157 .1862 .0192
1.8962 1.0694 1.0439 .0818 2.5074 1.3533 .1918 .0096
1.8463 .9872 1.1309 .0921 2.3927 1.5406
v-4ON -.0195
1.8962 1.3162 1.3049 .1049 2.8352 1.6031 .1692 .0032
1.0479 .4525 .4350 .0460 1.5405 .4997 .0592 -.0579
1.9960 1.2340 1.3484 .0793 2.6566 1.9570 .1833 -.0295
2.1457 1.2340 1.2614 .0818 2.7369 1.4574 .2144 .0688
2.4451 1.3985 1.2179 .0972 2.9499 1.7905 .2454 .0341
2.3952 1.3162 1.3049 .0921 2.8024 2.0403 .2144 .0101
2.3453 1.3162 1.3919 .0997 2.7707 1.8946 .2172 .0539
1.9461 1.0694 1.2179 .0870 2.4419 1.6239 .1805 .0173
>-*
Table E (Cont.)
Ca Mg Na K ALK S04 Cl ERR
1.4970 .7980 1.0439 .0716 1.8683 1.3116 .2059 .0073
.6986 .3373 .2958 .0358 1.0980 .2707 .0508 -.0373
.4990 .4360 • .3828 .0435 1.0980 .4372 .0564 -.1560
1.2475 .6663 .6960 .0639 1.6880 .8744 .0987 .0047
1.7964 1.0694 1.4354 .0793 2.4102 1.3116 .1580 .1212
2.0459 1.3162 1.1744 .0691 2.7696 1.6447 .1692 .0048
1.8962 1.1517 1.0874 .0767 2.5894 1.3949 .1862 .0099
1.8962 1.2340 1.3049 .0818 2.6877 1.5615 .1580 .0246
1.8962 1.1517 1.1744 .0870 2.4419 1.7280 .1862 -.0108
1.9960 1.0694 1.1744 .0716 2.6549 1.5823 .1862 -.0256
1.0479 .4525 .4350 .0460 1.5405 .4997 .0592 -.0579
1.4471 .8226 .8264 .0614 2.1469 1.0202 .1269 -.0423
ERR - -.0092 
sERR “ .0466
TABLE F
Common Ion Data for the Bighorn River 
at Bighorn in Meq/L
Ca Mg Na K ALK SO4 Cl ERR
3.8423 2.0566 4.0017 .1202 3.4416 6.4540 .3103 -.0183
3.8423 2.2211 3.5233 .0921 3.3924 6.0376 .3103 -.0063
4.3413 2.6324 3.7408 .0997 3.6426 6.8704 .3385 -.0034
4.5908 2.1389 4.1322 .0972 3.4913 6.6622 .3949 .0382
4.0419 2.3857 3.6538 .1049 3.5399 6.2458 .3103 .0089
3.7425 2.3034 3•5668 .0997 3.5727 6.0376 .3103 -.0212
4.2914 2.5502 3.6973 .1100 3.6415 6.2458 .3385 .0405
4.2415 2.4679 3.5233 .0997 3.7857 6.2458 .3385 -.0036
3.9920 2.3034 3.1318 .1330 3.8185 5.8294 .3949 -.0492
3.8423 2.2211 2.8708 .0844 3.5071 5.2049 .3103 -.0004
3.0938 1.8921 2.6533 .0921 2.9499 4.3721 .2144 .0255
2.3952 1.3985 2.0009 .0767 2.4419 3.3311 .1721 -.0125
3.2435 2.1389 3.2188 .0818 2.9685 5.2049 .2821 .0266
3.1437 1.8098 2.8708 .0844 2.9991 4.7885 .2623 -.0177
3.7924 2.1389 3.4798 .0997 3.3104 6.0376 .2821 -.0125
3.5928 2.0566 3.4798 .0946 3.2777 5.4131 .3103 .0245
Table F (Cent.)
Ca Hg Na K ALK SO4 Cl ERR
3.5429 2.0566 3.2623 .0844 3.1957 5.6213 .2821 -.0169
3.642? 1.9743 3.1753 .0895 3.3268 5.6213 .2821 -.0385
4.0419 2.3034 3.5233 .0946 3.6218 6.0376 .3103 -.0007
3.9920 2.385? 3.6973 .0844 3.6546 5.8294 .2821 .0395
4.5409 3.3728 5.219? .1279 3.4743 9.3688 .3103 .0082
3.642? 2.0566 2.6968 .0921 3.5563 4.7885 .2821 -.0162
3.6427 2.1389 3.0883 .0921 3.5399 5.4131 • 3385 -.0361
3.7924 2.3857 3.7408 .0972 3.5235 6.2458 • 3385 -.0091
3.4930 1.9743 3.1318 .1023 3.1957 5.4131 .2821 -.0215
3.5928 2.0566 3.3058 .0870 3.2777 6.0376 .2059 -.0516
3.1936 1.8098 3.3058 .0793 3.0318 5.2049 .2482 -.0114
3.4930 2.0566 3.3058 .0895 3.2285 5.2049 .2821 .0260
3.3932 1.8098 3.0448 .0895 2.9991 4.9967 .2567 .0102
3.8922 2.2211 3.5233 .1074 3.5399 6.0376 .3385 -.0175
3.7425 2.0566 3.0883 .1100 3.3432 5.6213 .3103 -.0304
3-5928 2.0566 3.3058 .1074 3.3924 5.6213 .3385 -.0314
3.8423 2.1389 3.5233 .1023 3.3104 5.8294 .3385 .0135
3.9421 2.385? 4.1322 .1074 3.3432 7.0786 .3103 -.0155
3.8423 2.2211 4.0887 .0844 3.5235 6.6622 .3103 -.0250
Table F (Cent.)
Ca Mg Na K ALK SO4 Cl ERR
3.6926 2.3034 3.6538 .0588 3.4252 5.8294 .2369 .0226
3.4930 1.9743 3.0013 .0997 3.1302 5.2049 .2285 .0006
3.5928 1.8921 3.0448 .1074 3.1302 5.4131 .2567 -.0187
4.141? 2.6324 3.8278 .0997 3.7038 6.2458 .3385 .0394
4.141? 2.3857 3.6973 .0997 3.7693 6.6622 .3103 -.0396
4.2415 2.4679 3.5233 .1560 3.7038 6.0376 .3667 .0274
3.7425 2.3034 3.9147 .0870 3.4907 5.6213 .2764 .0678
2.8942 1.8098 1.4354 .0793 2.8843 3.1229 .2144 -.0005
2.4451 1.3162 2.2619 .0639 2.6221 3.5393 .1551 -.0370
2.4950 1.4808 2.2619 .0767 2.6221 3.7475 .1946 -.0388
3.9920 2.3034 4.2627 .1100 3.6054 6.8496 .3103 -.0091
3.7425 2.6324 4.0452 .1202 3.7038 6.7871 .3103 -.0244
3.9920 2.5502 4.0017 .1176 3.6710 6.5789 .3103 .0096
4.1417 2.5502 4.3497 .1304 3.8513 6.9537 .3667 .0000
4.3912 3.7019 5.3502 .1407 4.5723 8.7025 .2821 .0020
4.1916 2.7970 4.7847 .1458 3.8676 7.4325 .2821 .0287
4.3912 2.5502 3.5233 .0946 4.0315 6.2875 .2087 .0030
4.2415 2.3857 3.5233 .1125 3.7693 5.9127 .3667 .0211
3.5429 2.3034 3.6538 .0997 3-5399 5.7670 .3103 -.0018
3.1936 1.9743 3.1318 .1023 3.0646 4.9342 .2031 .0241
Table F (Cent.)
Ga Mg Na K ALK SO4 Cl ERR
3.642? 2.1389 3.6538 .0921 3.4743 5.9960 .3667 -.0320
3.9920 2.2211 3.6103 .1074 3.5890 6.1417 .3103 -.0110
3.4930 1.9743 3.6538 .1049 3.0810 5.9127 .2651 -.0036
3.3932 1.8098 3.1753 .0997 3.1138 5.4131 .2257 -.0319
3.8922 2.3857 3.7843 .1049 3.7529 6.2667 .3103 -.0159
3.642? 2.1389 3.4363 .1023 3.4416 5.6421 .2708 -.0037
2.9940 1.7275 2.7838 .0870 2.8188 4 .5803 .2172 -.0031
4.7904 2.7970 5.0892 .0997 3.7365 8.3278 .4231 .0229
4 .5908 3.0438 4.6107 .1151 4.0315 7.7657 .3949 .0137
2.5948 1.3985 2.6533 .0818 2.4583 3.8932 .1410 .0357
4.3413 2.8792 4.3497 .1125 3.7529 7.2660 .3949 .0233
ERR -  -.0020  
s ERR °  .0251
TABLE G
Common Ion Data for the Tongue River 
at Miles City in Meq/L
Ca Mg Na K ALK SO4 Cl ERR
2.8942 3.4551 2.0444 .1330 4.2937 3.9557 .1156 .0191
3.2435 3.7019 2.1314 .1023 4.7390 4.1639 .098? .0195
4.2415 5.1004 3.6103 .2225 6.4898 6.4540 .2313 -.0000
4.7405 5.6762 3.8278 .1560 7.3420 7.2868 .1692 -.0272
3.8423 4.2777 3.0446 .140? 5.3262 5.6213 .1241 .0209
3.1437 3.8664 2.9578 .1176 4.9165 5.4131 .1213 -.0356
3.8423 5.1004 4.0017 .1560 5.5737 7.2868 .1523 .0067
3-5429 4.1132 2.3923 .1304 4.588? 5.8294 .1410 -.0367
2.2954 2.4679 1.7399 .0921 3.4088 3.3311 .0733 -.0325
1.646? 1.3162 .7395 .0486 2.2780 1.4157 .0395 .0048
2.2954 2.1389 1.8704 .0921 3.7693 2.7065 .0959 -.0270
2.5948 2.5502 2.0009 .1100 3.9168 3.3311 .0846 -.0105
2.8443 3.3728 2.6098 .1253 4.4593 4.3721 .1185 .0003
3.2435 3.7019 2.5228 .1125 4.5396 4.7885 .1156 .0144
3.2435 4.0309 2.6533 .1151 4.9165 4.9967 .1269 .0003
4.2914 5.2649 4.0017 .1611 5.2934 7.7032 .1777 .0405
Table G (Coni.)
Ca Mg Na K AIK SOi4. Cl ERR
3.7924 4.4423 3.3493 .1304 6.0965 5.8294 .1354 -.0292
3.8922 4.6890 3.2623 .1330 6.0965 5.8294 .1495 -.0082
3.1936 3.948? • 3.9147 .140? 4.9001 6.6622 .1382 -.0439
3.642? 4.8536 4.3497 .1637 5.6212 7.0786 .1523 .0122
3.3932 4.1955 2.3488 .1228 4.6215 5.4131 .1241 -.0097
1.7964 1.6453 1.1309 .0716 2.5730 1.8529 .0790 .0305
2.0459 1.8098 1.2179 .0921 3.0810 2.0819 .0987 -.0184
2.5948 2.4679 2.0009 .1125 3.9660 3.3311 .0959 -.0298
2.2455 2.0566 2.9578 .1279 3.7038 3.5393 .0846 .0082
2.1956 2.2211 3.0448 .1253 3.5890 4.3721 .0649 -.0563
3.4431 4.1132 3.5668 .1330 5.2115 6.0376 .0959 -.0079
3.7425 4.5245 3.8278 .1355 5.4737 6.8704 .1495 -.0213
3.9920 4.7713 3.8278 .1228 5.9490 6.8704 .1777 -.0220
4.1916 4.8536 3.3928 .140? 6.3587 6.4540 .146? -.0298
1.5469 1.3985 1.6964 .1330 2.2780 2.4983 .0790 -.0167
3.4930 3.8664 2.7838 .1534 4.3101 5-6213 .1326 .0229
3.3932 4.1955 4.784? .2148 4.6379 7.4950 .1608 .0237
2.2455 2.0566 1.3049 .0818 2.7696 2.7065 .0677 .0258
3.1936 3.5373 2.5663 .1304 4.4576 4.7885 .1354 .0049
Table G (Cont.)
Ca Mg Na K ADC SO4 Cl ERR-
3.3932 3.8664 2.8273 .1253 4.9821 5.2049 .1297 -.0102
3.7425 4.1955 3.2188 .1458 5.6704 5.8294 .1692 -.0319
4.5409 4.4423 3-1318 .1381 6.0473 5.8294 .1269 .0206
3.2435 3.0438 1.8269 .1534 4.1790 3.9557 .1241 .0011
3.8922 4.6890 4.0887 .2097 4.8018 7.4950 .1241 .0363
3.5928 4.1955 2.9143 .1995 4.2446 6.2458 .1410 .0251
1.7465 1.6453 1.4354 .0767 2.4910 2.2901 .1523 -.0060
2.5449 2.5502 2.0009 .0946 3.7857 3.3311 .0762 -.0003
2.8443 3.0438 3.4363 .1151 4.7035 : 4.7885 .0931 -.0153
2.644? 2.7970 2.5663 .1202 3.9004 3.9557 .0790 .0240
3.0938 5.3472 5.2632 .1739 4.9329 8.2445 .1636 .0395
4.2415 5.7585 3.7843 .1611 6.3751 7.0162 .1185 .0317
3.4930 4.4423 3.2188 .1483 5.4737 5.8711 .1185 -.0141
3.3932 4.0309 3.2188 .1534 5.1623 5.4131 .1297 .0085
3.6427 4.3600 3.0883 .1586 5.0804 5.7878 .1185 .0236
3.1936 3.8664 2.8708 .1688 4.3921 5.5172 .1185 .0071
2.4451 1.8921 1.0874 .0844 3.2121 2.3734 .0226 -.0178
2.3952 2.2211 1.8704 .1074 3.6054 3.0188 .0846 -.0173
2.2455 2.1389 1.7399 .0895 3.4907 2.7898 .0733 -.0223
Table G (Cont.)
Ga Mg Na K ALK S04 Cl ERR
2.8443 2.4679 2.2619 .1330 4.1299 3.6642 .0818 -.0217
3.7924 4.4423 3.1753 .1304 5.6589 5.7462 .1354 -.0000
3.4930 3.948? 2.7838 .1151 5.4082 5.0591 .1241 -.0240
3.1437 3.3728 2.6968 .1509 3.7202 4.3929 .0959 .1315
3.1437 3.7019 2.8273 .1279 4.7198 4.8718 .1467 .0064
3.2435 4.4423 3.0448 .1355 5.2656 5.5588 .1297 -.0081
2.6447 2.8792 1.5659 .1049 3.7365 3.5601 .0677 -.0233
2.644? 2.5502 1.4789 .0895 3.8840 2.9772 .0621 -.0234
3.7924 4.4423 4.3497 .2532 5.5556 7.1827 .1410 -.0033
3.2435 4.0309 2.8273 .1279 4 .5926 5.8919 .1015 -.0342
2.8942 3.1260 2.4358 .1202 4.0971 4.1431 .0621 .0325
2.9940 3.4551 3.4363 .1432 4.6237 5.1008 .0987 .0207
ERR - -.0011
LO00 b k0282
TABLE H
Common Ion Data for the Yellowstone River 
at Miles City in Meq/L
Ca Mg Na K ALK SO4 Cl ERR
2.8942 1.8098 2.3923 .0921 3.2324 3.7475 .2595 -.0071
3.1936 1.8921 2.7403 .0895 3.1629 4.3721 .2821 .0125
3.2435 2.0566 3.2188 .0997 3.3596 5.2049 .2623 -.0239
3.5928 2.2211 3.3058 .1100 3.3935 5.2049 .3385 .0322
3.2435 1.9743 2.9578 .1023 3.2941 4.7885 .2821 -.0104
2.644? 1.4808 1.8704 .0793 2.6549 2.9147 .2115 .0496
1.9960 .6499 1.0874 .0588 1.9010 1.7697 .1354 -.0037
1.3972 .6252 .7395 .0435 1.4750 1.0202 .1072 .0751
1.6467 .9872 1.1744 .0691 1.9010 1.7072 .1128 .0411
1.8962 1.3162 I.6964 .0691 2.5074 2.2901 .1636 .0034
2.4950 1.5630 2.2619 .0844 2.8215 3.1229 .2172 .0386
3.2934 2.0566 3.0448 .0946 3.2777 4.5803 .3103 .0386
2.7944 1.6453 2.4793 .0870 3-0155 3.5393 .2821 .0244
1.6467 .9872 1.2179 .0537 1.9502 1.6864 .1580 .0288
2.6946 1.8098 2.6968 .1074 3.0646 3.9557 .2680 .0028
2.5948 1.5630 2.6098 .1074 2.6713 4.1639 .2567 -.0310
%
Table H (Cont.)
Ca Mg Na K ADC SO4 Cl ERR
1.147? .5758 .6525 .0384 1.3602 .9785 .0790 -.0014
1.8962 1.2340 1.6094 .0691 2.4746 2.2901 .1636 -.0246
2.8942 1.7275 2.3923 .1049 3.1466 3.9557 .2764 -.0358
3.1936 2.4679 2.6968 .1125 3.0974 4.7885 .2228 .0437
2.8942 1.9743 2.5663 .1049 3.3432 4 .1629 .2680 -.0307
1.7964 .9872 1.2614 .0486 2.0158 2.0403 .1100 -.0176
2.0459 1.2340 2.0009 .0870 2.7369 2.7065 .2285 -.0551
3.0439 2.0566 3.0013 .1125 3.2121 4.6427 .2539 .0129
3.3932 2.1389 3.4798 .1355 3.2121 5.5380 .3103 .0096
1.9461 1.1517 1.3049 ,0665 2.3271 1.8946 .0903 .0358
2.5449 1.4808 2.3488 .1049 2.9171 3.4560 .2595 -.0234
2.4950 1.3162 1.9574 .0742 2.6221 3.1646 .1974 -.0239
2.8443 1.8098 2.6533 .0997 3.0974 4.2055 .2708 -.0222
3.1437 1.8098 2.7838 .1151 3.2121 4.5803 .3103 -.0314
1.1477 .6170 .6960 .0486 1.4913 .9577 .0790 -.0075
2.9441 1.8921 3.0448 .1100 3.2121 4.3721 .1918 .0273
ERR = .0019 
sERR " »0211
TABLE I
Primary Flow Data 30 Years (1944 - 1973) 
in Million Acre Feet (MAF)
Year
Yellowstone
Bighorn
Bighorn
Bighorn
Yellowstone 
Miles City Precipitation
1944 4.429432 4.558000 9.662717 5.976344
1945 4.903299 3.401000 8.537485 3.412980
1946 4.536863 2.856000 7.488556 4.494581
1947 5.598576 3.983000 9.333501 3.853739
1948 6.032406 3.284000 9.396159 3.923335
1949 4.556255 2.600000 7.219993 2.763441
1950 5.628138 3.034000 8.911475 3.987130
1951 6.012505 3.543000 9.80178? 3.752250
1952 5.988362 2.340000 8.426226 2.656153
1953 4.391895 2.030000 6.538806 3.876938
1954 4 .567656 1.958000 6.538840 3.381083
1955 3.870799 2.178000 6.076215 3.439079
1956 5.892619 2.302000 8.287723 3.427480
1957 7.134494 2.435000 9.648042 4.378593
1958 4.337206 2.813000 7.267318 3.279590
1959 5.077166 2.098000 7.191112 3.702952
I960 3.639390 1.614000 5.280476 2.461868
1961 3.237737 1.175000 4.434866 3.241898
1962 6.013068 2.917000 9.004552 4.627968
1963 5.151859 3.000000 8.029416 4.593173
1964 5.400421 3.397000 8.755631 4.204611
1965 6.816616 3.967000 10.775285 4.419186
1966 4.327211 1.240000 5.629292 2.818538
1967 6.784331 3.184000 9.962166 4.410489
139
Table I (Cent.)
Yellowstone Bighorn Yellowstone
Year Bighorn Bighorn Miles City Precipitation
1968 6.334088 3.359000 9.831044 4.857047
1969 5.623566 2.881000 8.635795 4.123416
1970 6.349800 2.851000 9.349343 4.384390
1971 7.077032 3.634000 10.880866 3.804442
1972 6.342625 3.667000 10.210567 4.857046
1973 5.163317 3.180000 8.417696 4.558376
140
Simulated Values of ECM for the 25 Years 
1944 to 1968 Associated with LI and L2
TABLE J
Year BO EI
1944 612.3 567.8
1945 596.7 542.7
1946 628.9 532.1
1947 575.8 540.2
1948 574.2 493.6
1949 638.2 517.4
1950 586.6 496.2
1951 564.6 506.1
1952 599.9 447.4
1953 664.1 491.4
1954 664.1 478.9
1955 683.9 526.5
1956 603.9 448.5
1957 568.2 418.2
1958 636.5 539.2
1959 639.2 466.4
I960 723.5 501.3
1961 775.9 490.3
1962 584.1 476.7
1963 611.6 512.9
1964 590.7 522.1
1965 543.6 497.7
1966 705.2 439.3
1967 560.9 464.7
1968 563.9 486.9
APPENDIX II
Contents
Chi-square goodness of Fit Tests for the error of "balance, ERR, 
for common ion data for the specified sites; Yellowstone River at Bil­
lings, Bighorn River at Bighorn, Tongue River at Miles City, and 
Yellowstone River at Miles City.
Cross Reference
Pages iv and 15
n = 68
Goodness of Fit for ERR for Yellowstone River at Billings
Hypothesis; ERR/^ N(ju, (S^ ) with I~) 
r (Ohs-? - Expi)^
X2j = S . ----- 3- ------ 3—
0=1 Expj
(ERR - p)
Z = -----------
0 R^R-
= 0, = S§pp = ,0022, oL -  .05 ,
Where;
r = number of intervals 
Obs = observed frequency 
Exp = expected frequency
ERR
Interval
Z
Interval ObSj I O. X2j
Oi$Si O  -1.502 6 4.522 .483
XOi
0I1O-O -1.502 -1.073 6 •5.IO?1 .156
-.05 ^  -«03 -1.073 - .644 9 8.036 .116
- .0 3  - .0 1 - .644 6  - .215 14 10,545 1.132
—.01 .01 - .215 <— *..215 12 11.580 .015
.01 .03 .215 ? .644 11 10.545 .020
.03 <-> 0 0 .644 00 10 17.665 3.326
k = the number of sample statistics used = 5»248
d.o.f. = r - k - I here k = !(S^) so d.o.f. = 5
X2 = 11.0? so X2C X 2^. and the hypothesis can not be rejected
Z-Test of MERR for the Yellowstone River at Miles City
143
oxi jO00 D x
Hi* PERR 5V 0
<*•= ,05
ERR = ,0019; ix\\ = .0211
Since n = 32>30 LO00 was used to approximate OgRR
ERR -.MERR
Z = ----------  = .509
oisR/Vn"'
Z^y2 = *1.96
Z < 2 ,^ /2 so the hypothesis can not .he rejected
APPENDIX III
Quantity Model
1944 - 1973
Contents
Primary data and simulated secondary data output for the 30 year 
period 1944 - 1973 annual 42KJA computer program.
Cross Reference
Pages v and 84
COOaOO************************************************************************** 
C00» 05'AP PLICAT ION'jIl******************************** **************************** 
C 42KJASUB-8ASIN# ANNUAL; SOLUTION FOR YEARS 1944 THROUGH 1973'
C SIMULATE SECONDARY DATA*
C OUTPUT: PRIMARY AND SECONDARY DATA,
COOt10 I DIMENS I ONS'¥¥*¥**********¥*♦¥******************************************** 
C NP - NUMBER OF PERIODS#
NR-6J NC-13j NP-30
C DIMENSION INSTRUCTIONS:D(NR*NC)'Z(NC-NR),B(N%),<(NP),
C wl<NR)#w2tNR)*Y(NR)#A(NR*NR)>L(NC)#S(NC)#C< NP* NC)
DIMENSION D«78I*Z<7I/3(6)#K(6)#W1(6)*W2(6)*Y(6)#A(36)#L(13)*S(13I* 
*C(6*13)*FY(30)*FB(30)
INTEGER P
COltOO************************************************************************** c INITIALIZE Z AND C*
Jl-NC-NRJ J2-NC*NR 
DO 10 J-ItJl
10 Z(J)-It I
EQUIVALENCE (DtC) <
DO 15 J-ItJ2 
15 D(J)-Ot
COgtOO************************************************************************** 
C .wja' SOLUTION SUBSCRlPTSt INITIAL EXOGENOUS VALJEgt AND LFL'MG qdHL'■H'Ll
NAMELIST KtZtC 
INPUT 
INPUT 
INPUT
C03*00********************************************** **************************** 
c ESTABLISH SOLUTION CODE VECTOR,
CALL CODElKtNRtNCtL)
CO4'00********************************************** **************************** 
c ESTABLISH EXOGENOUS VECTOR,
CALL EXOGENOUSILtZtNCtS)
C04'05'INITIAL VALUES'********************************************************** 
SUM7-0IJ SUM8-0* J SUMl1-3'< SUMlO-Ot JSUMS-OiJSJMl2-3'
42 X9I-Z16)JS19)-Z(6II HOLD-O;SF-•0360523
' Ct5#lll— IeVSF 
DO 91 J"1'NP
Wx,iV B :G c" O O K B z O w B I 1 qo■H9MLiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii 
NM■7Sh6rxaAQFS JA­Q-S JA­LSr A­LShA
S I 2 ) - F Y ( J I + F B I J )  ___  ______
4100 FORMAT!4X#F9.6#3FlOe6)
I F ! J ' G T » 1 ) S I 9) -Y< 4)
S ( 4 ) eO I J SI 3 ) "HOLD;HObO-Sl4)
Z Q S  t O O + * * *  + * * * * * * *  i-****** + * * * *  + * * *  + ** + * * * * * * * *  + + ** * * * * * *  * * * * * * * * * * * * * * * * * * * * * * * *
C ESTABLISH ENDOGENOUS C=TFCNT MATRIX#
CALL DELElE COLUMN(0#u# NR#NC# A)
Z O b t Q O * * * * * * * * * * * + * * * * * * * * * * * * * * * + * * * * * * * * * * + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
F FigureI —SkItcI ih e mI FahFSe gpet—so
CALL INVERSE!A,NR*dl 'W2)
ZQ7 tOQifit + * + + ***** * + * + + ** + * + * + + * *** + * + + *** + * + ** + * + + + + * + **** + + + *** + ** + **** + ******* 
Z COMPUTE RIGHT-HAND-SIDE COLUMN VECTOR*
CALL MULTIPLY! D * S * NR*NC*I  * B )
DO 70 JJ-1,NR 
70 B ( J J ) - - B t J U )
Z Q £ tQQ*+*+*+*+*+**++***++*+*++***+*+******+*++**++******************************
Z 3 pc 4 U p O w 2 B z O w B I 1 1 31 vO G 1 B P I v V B w i
wn FpMM grMe—uMdlpoYR StoStovodB
C08*05'TEST FOR O U T P U T  * * + * + + + + + + * * *  + * + * + * + * + ** + + + + * + + + * * * * *  + + * + + * * *  + + * + * * * *  + + ** 
ZQ9t00>t>t* + ** +*******+ *** + * + * + + + * + ** + * + *** + + + + ** + ** ** + *************************** 
Z ADJUST ARGUMENTS F DR DESIRED OUTPUT APPLICATION*
90 CALL OUTPUT!Y*S*J)
C
C09*05I TEST FOR FINAL ITERATION * * * * * * * * *  + + + ** + * + ** + + + * + * * *  + + ** + * + * + + + * + * + + * + + * + 
SUM6-SUM6 + Y 11 ) ISUM7-SJM7FY!2)j SUM8-SUM8 + Y (3) I SJM10-SUM10 + Y ( 4)
S U M l l - s U M l l t Y ! 5 ) ) SJM12-SUM12 + Y! 6)
91 CONTINUE
C09*15 I FOOTNOTE REpORT * **** + **** + ** + ****** + ** + + * + + + + + + * + ** + * + ***** + + + ** + + ******* 
97 OUTPUT SUM 6 ,S UM 7 / 3 0 . , S UM 8 / 3 0 * , S UM lO / 3 0 * * S UM l l / 3 0 * * S UM l 2 / 3 0 ,
END
SUBROUTINE COOE(K#NR#NC#L)
[."SOLUTION CODE VECTORS •I•"EXOGENOUS# «0*"ENDOGENOUS,
DIMENSION K(DiLd)
I-I
DO 10 J-IiNC 
L ( J ) "I'
IF(K( I ) ,EO.J) L(J)-O. _
IF (I.EQ.NN) QO TO 10 
I F I K ( I )  ,E Q, J) I"I + 1 
10 CONTINUE 
RETURN 
END
SUBROUTINE DELETE COLUMN*O'L'NR'NC*A)
c eO w 2 B z O w B I 1 W 8 8 W w v G c v 0 V K B U v c V w O 2 U3 2MP MvVw1 MK B z O w B I 1 W B P I G w 1 i
DIMENSION AIDiD(DiLd)
M-OJN-O
DO 20 I-IiNC
IF< LI I 1 tNEeO.I GO TO 20
DO 10 J-IiNR
N-N + l
10 A(N)-D(M^J)
20 M-M+NR 
0 O v I 0 w
END
LaXNda'.HM MKd9MHdJLSY­ZiHqiLA
FORMULATE S FROM Li I ONES I REPLACED BY V w V v V c P M;d9MHdaL u■YaMLl 
LpM;d9MHdaL uMq'dNi
DIMENSION Hl I iS(DiZd)
I-O
DO 10 J-IiNC 
S(J)-L(J)
IF(SIJ)«E0'1*) 1-1+1 
IF(SU)'EQd.) S(J)-Z(I)
10 CONTINUE 
RETURN 
END
SUBROUTINE INVERSE!A#N#UM)
REAL LiM
DIMENSION AtDiL(DiMll)
1D - D O
NK--N __
DO 80 K-IiN
NK-NKtN
L(K)-K
M(K)-K
KK-NKtK
BIQA-A(KK)
DO 20 J-KiN 
IZeNt(J-I)
DO 20 I-KiN
IJ-IZtI
10 IF I ABSfBlQA)- ABslA(IJ))) 15i20,20 
15 BIQA-Al IJ)
L(K)-I
M(K)-J
20 CONTINUE 
J-L(K)
IF(J-K) 35i35i25 
25 KI-K-N
DO 30 I-IiN 
KI-KItN 
HOLD--AIKl I 
JI-KI-KtJ 
A(KII-AlJI)
30 A(JI) -HOLD 
35 I-M(K)
IF(I-K) 45i45i38
38 JP-N*(I-I)
DO 40 J-IiN
JK-NKtJ
JI-JPtJ
INVER 2
INVER 3
INVER 4
INVER 5
INVER 6
INVER 7
INVER 8
INVER 9
INVER 10
INVER 11
INVER 12
INVER 13
INVER 14
INVER 15
INVER 16
INVER 17
INVER 18
INVER 19
INVER 20
INVER 21
INVER 22
INVER 23
INVER 24
INVER 25
INVER 26
INVER 27
INVER 28
INVER 29
INVER 30
INVER 31
INVER 32
INVER 33
INVER 34
INVER 35
148
HOLD--A(JK) INVER 36
A(JK)-AUl ) INVER 37
40 A(JI) -HOLD INVER 38
45 IF(BIQA) 4 «$>46*48 INVER 39
46 0*0.0 INVER 40
WRITEt108*1000 I
1000 FORMAT! I THE DETERMINANT OF''A''IS ZERO* THEREFORE EXECUTION WAS
' ,HALTED* CHECK'/'FOR ALL ZERO ROW OR COLUMN AND FOR P V w O c 0 2M0Mw2Ow
IkCE U O v • O O w O T I c v V B w 1 elA
CALL EXIT
48 DO 55 I-I * N INVER 42
IF(I-K) 50,55*50 INVER 43
50 IK-NK+I INVER 44
A(IK )■A(I K )/(-BIQA) INVER 45
55 CONTINUE INVER 26
DO 65 I■I * N INVER 47
IK-NK+I INVER 48
IJ-I-N INVER 49
DO 65 J-I * N INVER 50
IJ-IJ+N INVER 51
IF(I-K) 60,65*60 INVER 52
60 IF(J-K) 62>65*62 INVER 53
62 " C pV CpV5" INVER 54
A(IJI-AIIK),A(KJ»+A(IJ) INVER 55
65 CONTINUE INVER 56
KJ-K-N INVER 57
DO 75 J-I * N INVER 58
KJ-KJ+N INVER 59
IF(J-K) 70> 75* 70 INVER 60
70 A(KJ)-A(KJT/BIQA INVER 61
75 CONTINUE INVER 62
C D- VALUE Op THE DETERMINANT
D-DtBIGA INVER 63
A(KK)-I.OZBIQA INVER 64
80 CONTINUE INVER 65
K-N
100 K-(K-I)
IF(K) 1501150*105 
105 I-L(K)
IF(I-K) 12U,120,10g
108 JQ-N*(K-I)
JR»N*(I-I)
DO H O  J-IiN 
JKeJO^J 
HOLD-AIJK I 
JI-JRtJ 
A(JK)--A(JI)
H O  A(JI) -HOLO 
120 J-M(K)
IF(J-K) 100,100,125
125 KI-K-N
DO 130 I■I,N 
KI-KItN 
HOLDeAlKI I 
JI-KI-KtJ 
A(KI)--A(JI)
130 A(JI) -HOLD 
QO TO 100 
150 RETURN 
END
SUBROUTINE MULTIPLY!A*3,N,M#L*R) 
DIMENSION A(i),BtlMR(I)
IR-O
IK--M
DO 10 K-I * L 
IK-IKtM 
DO 10 J-I,N
IR-IRtl
JI-J-N
IB-IK
INVER 66 
INVER 67 
INVER 68 
INVER 69 
INVER 70
INVER 71 
INVER 72 
INVER 73 
INVER 74 
INVER 75 
INVER 76 
INVER 77 
INVER 78 
INVER 79 
INVER 80 
INVER 81 
INVER 82
INVER 83 ^ 
INVER 84 vS 
INVER 85 
INVER 86 
INVER 87 
INVER 88 
INVER 89 
INVER 90
n
 o
 o
 o
R(IR)-O 
Do 10 I-1#M 
JI-JH-N 
IB-IBtl
10 R(IR)-RlIK)*A(JI)*3(IB ) 
^ ER - ^ u
END
SUBROUTINb OUTPUT(Y* S# J )
C OUTPUT PRIMARY ANU SECONDARY VALUES'
DIMENSION Y(DiSll)
M-1943+J
WRITE(108,100)M,S(I),SI2),S(3),S(4)fY(l)iS(&)'Yl2), 
*SI8)iS(9)iSI 10)i Y(3)i Y(4)
100 FORMAT!14i 2F8•I i 2F4•I i F8•I i F4•I i 2F8•I i 2F4. 1i2F3•I I 
RETURN 
END
2
SUBROUTINE CONDUCT(Qi Si Y)
C CALCULATION OF CONDUCTIVITIES 
DIMENSION S(IIiYIlIiQIl)
Q ( S ) - O ( S ) - Q ( I l )
SI  I I - S l  I )/U(I U  SI  2)" S I 2)/Q(2)I Y(I)-Y(I)ZQ(S)
Y(2)-Y (2)/Q (7) )S(8)-S( 8) ZQ (8) IYOl-Y (3) ZQ (11 I I Yltf-Y (4) ZQ (12)
RETURN
END
Xl
Year X?
lf144 9.66n7i7
•27206%
1^45 8.537495
•255189
1946 7.488556
•239455
1^47 9.3336 'll 
•267130
1948 9.396169
• 2 6 8 6  9
1949 7 .2 1 <0993 
• 2 3 5 4 ? 7
1950 8•Q l14 75 
.2^0799
lr,5l 9 •%0 1 737 
•274154
1952 ' 8 •4p6?p6 
•253520
1953 6 .538806
•225?oq
1954 6 .538840
.225210
1955 6*076215
• 2 18 ? 7 O
1956 8.287723
,25i443
1957 9 • 648( '4 2 
• 2 7 1848
1953 7.267318 
• 2 3 6 1 3 7
I q59 7 • I 9 1 1 1 2 
.234994
X2 X3
X8 X9
8.987432 • 000000
•255794 4*958022
8.304299 • 000000
•245547 4*828998
7.392863 •000000
.231876 4*905658
9,581576 • 000000
»264707 4*929637
9.3)6406 * 000000
.260729 4.915237
7.156255 «000000
.228327 4 . 9 H 2 9 6
8.662138 •000000
k yhxsrU 5*060693
9.555505 ♦000000
.264316 5*062124
8 .398362 •000000
»245908 5«Q48882
6.4?)895 •000000
*217311 5.151797
6.5?565h •000000
•2)8868 5*226o43
6*048799 • 000000
•211.715 5.32149Q
8.194619 •000000
•243902 5» 419604
9.569494 •000000
*264525 5*435936
7.150206 • 000000
•228236 5*338334
7.175166 * 000000
•228610 5*409232
X4
XlO
*000000 
4*828998 
*000000 
4*905658 
•000000 
4*929637 
• 0n0C00 
4*915237 
•QO0000 
4*911296 
•000000 
5 * 0^*0693 
♦ 000000 
5*062124 
•000000 
5*048822 
•000000 
5*1^1797 
•0O0000 
5*226o43 
•000000  
5*321490 
♦000000 
hi ,rstT , 
•000000  
5*435936 
•000000  
5*338334 
*000000 
5*40923? 
•000000  
5.441556
X5
Xll
5.976344 
*766131 
3.41298Q 
.556724 
4.494581 
.6Q1753 
3.853739 
.634443 
3.923335 
.629920 
2.763441 
.468262 
3.987130 
.610635 
3.75225Q 
.625919 
2.656153 
•5Q2555 
3.876938 
.521669 
3.38ioS3 
.489486 
3•439079 
(,)ty/- 
-(,y),/B 
ehhyts- 
,e-)/hs- 
et)^syy
3.?79590 
* 504484 
3.7Q2952 
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x6
X12
5*155762
•620834
3*246812
•623742 
4•415?38
• 618162
4*08499i
• 6 1)tyT 
-l/-y-xr
•618639 
2*842000 
•610560 
-i)ys-,T 
•602182 
3*482887 
•6o2838 
2•653592 
khs)/h/ 
-i/yt-)t 
kh//^r) 
- k ,h)T xh 
kh)/hs& 
-lh0322l 
•567841 
3* 34316?
• 561484 
4'I 9512?
•565998 
3*225475 
•ht),/V 
3.712947 
•561748
H*
•>S
I960 5*280476 5.753390 •OOOOOO
*206334 .199784 5*44)556
. sSCr 4 .434866 4.4)2737 •OOOOOO
*193650 • l87i74 5.621256
I q62 9.004552 8.930(,6R •OOOOOO
• 26?)q5 * 254934 5.7577Q1
1963 8.0294)6 • 8.151859 •OOOOOO
*24796% •24326) 5*631567
I q64 8 * 7 5 51- 31 8*797421 •OOOOOO
*258461 .252Q 44 5 . 544664
1965 10*775285 10*783616 •OOOOOO
.788756 *282737 5*472395
1966 5.6292C|2 5*567211 •oooooo
•?1)566 * 2 O 4 4 91 5.325504
1967 9.q67166 0.968931 •OOOOOO
.276559 *270508 5*482589
lq68 9 .8 3 )n44 9.693088 •OOOOOO
*?745q3 *266379 5*363141
1969 8.635795 8*504566 •oooooo
* 256664 *248551 5*236975
197C 9.349343 9.200800 •OOOOOO
*967367 .75899% 5*214559
1371 1 0 * 839860 10*711032 •OOOOOO
*290340 ,281643 5*153062
1972 10*21o567 10* 009625 'OOOOOO
*280286 *271)27 5*0*6252
1973 8.4i76q6 3.343317 •OOOOOO
*753392 .746)33 4.978302
•oooooo
5'621256 
•000000  
5.757701 
•oooooo
5•631b67 
•OOOOOO 
5*544664 
•OOOOOO 
5'472395 
•OOOOOO 
5*325504
•oooooo
5*482539
•oooooo
5*363141
•OOOOOO
5*236975
•oooooo
5*214559
•OOOOOO
5*153062
•OOOOOO
5,0%b252
•oooooo
4*978802 
.000000  
,esh/T ht
2.461868
.376823
3.241898
.402442
4.627968
.665234
4.593)73
. 634882
4.204611
*630299
4.419186
.717597
2.818538
,414109
4.410489
.687529
4.857Q47
.710051
4.123416
.614452
4.384390
.658455
3.804442
.671213
4.857046
.721998
4.558376
(t,T hr)
y= tT )s-A
♦ 549972 
3*349738
•532412
4*420089
•531839
4*624406
*543672
4*168615
•552513
4*274606
• 564686 
2*9o6467
•564120
4*291154 
'562029 
4 * 584712 
*575672 
3*961659
'583924 
4*165977 
•588585 
3 * 559106 
*595712 
4*539496 
•6o539i 
4.455991 
•612511
APPENDIX IV
Quality Model 
1 9 #  - 1968 
1 9 #  - 1973
Contents
Primary data and secondary output for the 25 year period 19#-19^8 
and the 30 year period 19#-1973« Computer program for sub-basin #KJ.A.
Cross Reference
Pages v, 93 and 95
O
O
O
O
C00+00************************************************************************** 
[00*05#APPLICATION'************************************************************* 
42KJA SB-BASIN#L0A3 MODEL 
ANNUAL SOLUTION FOR THE YEARS 1944-1968 
SIMULATE SECONDARY DATA*
da'ja'@ jN.6■NF cw2 LMqdH7■NF 2 c v c i
[OOilO'DlMENSIONS#**************************************************************
C NP - NUMBER OF PERIODS*
NR«4j NC-12)NP-30
C DIMENSION INSTRUCTIONS:D (NR*NC),Z(NC-NR),B(NR),K(NR),
C Wl(NR),W2(NR),Y(NR),A(NR¥NR),L(NC)>S(NC)>C(NR,NO
DIMENSION 0(48),Z(8),3(4),K(4),W1(4),W2(4),Y14),A(16),L(12),S(12), 
*C(4,12),EI(30)#E0(30)#3(12)
V w v O z O 0 j
[01 • 00¥¥¥¥¥¥¥¥¥<l**¥<|l¥1|l**y*******¥******************* **************************** 
c V w V v V c P V ° O ° cw2 W =
Jl-NC-NRJ J2.NC+NR
DO lO J-l<jl ,
10 Z(J)-I'
EQUIVALENCE (DiC)
DO 15 J-I,J2 
15 D(J)-O*
C02,00¥,«l¥¥¥1'l¥¥¥¥J,lJ'M|l¥********* *************************************************** 
C INpUT SOLUTION SUaSCRlpTS, INITIAL EXOGENOUS VALUaS' AND LfL'MG qdHL'■H'Ll
NAMELIST K,Z'C 
INPUT
&
INPUT
INPUT
Q03«00¥¥¥*******************¥*********************** ****************************
c O 1 v c U P V 1 m LdYa'.dH qd7M uMq'dNp
CALL CODE(K,NR,NC'U )
C04'00»¥*****************,|l***¥**************¥**'11**** **************************** 
c ESTABLISH EXOGENOUS VECTOR-
q■YY O K B z O w B I 1 S P ( z ,  w B  s )
C04 * 05 ' INITIAL VALUES'***¥¥**¥¥¥¥¥¥¥¥*¥¥¥¥¥¥¥¥¥¥¥***¥*************************** 
CP-•I 87
SUMC-OiJ SUMl-O•J SUM2-0.J SUMS-OiiSUM7-0ijSUMll-O''SUMlZ-Oi 
SUMS-Oi
Cl 3*11)""l(/5FjSI 6)-OiS(S)-S(IO)-Oi 
42 XSl-ZI6)JS(B)-ZI6)iHOLO-Oi 
DO 91 J-IiNP
qx,Vrx>G ■" O MK B z O w B I 1 W m c w z O 1 k m * * * * * * * * * * * * *  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  
READ(Si 41OO) (0(1 I,I-Ii12)
READ(15i4105)E0(JIiEI(J)
S(I)-O(I)^EO(J) .
S(2)"Q(2)*EI(J)
C(4i7)-QH2)/Q(7)
4100 FORMATISXi6F12i6/5Xi6F12i6)
4105 FORMAT!2F7.1 I
IF(J.GTil) SI Sl-Yl3)/3 
SI 4 I-Oi;SI 3)-HOLDlHObD-SI4)
»00*********************************************i‘*,,,*‘,‘¥*****,f*******¥***,t-*¥**,,‘ 
C ESTABLISH ENDOGENOJS CFFCNT MATRIX*
CALL DELETE COLUMN(DiLiNRiNCiA)
^0(,t OO********************************************  ^ * ******** ******** ****** ******
q qd6ja'M .HuMNLM dQ 'm M CrFCNT MATRIX'
CALL INVERSE(AiNRiWlidS)
CO? *00^********************************************* **************************** 
C COMPUTE RlGHT-HAND-SIDr COLUMN VECTOR*
CALL MULTIpLY(DiSiNRiNCiliB)
DO 70 JJ-IiNR 
70 B(JJ)--B(JJ)
C0&*OQ********I*****************************+***********************************
C Y"A*B" ENDOGENOUS SYSTEM SOLUTION*
SO CALL MULTIPLY!AiB'NRiNRiliY)
COSiOS•TEST FOR OUTPUT'*****************#*******************************#****** 
C09 *00* f-****^** ********* ^ * ^ *****1* ^ ****************** *********** ***************** 
C ADJUST ARGUMENTS FDR DESIRED OUTPUT APPLICATION*
CALL CONDUCT(Q*Si Y )
90 CALL OUTPUT (YiSiJ)
C09•05'TEST FOR FINAL ITERATION '*********************************************** 
8UM5*SUM5*T(1I<SUM7-SJN7*Y(2)jSUMll-SUMll+YI3)JSJM12.SUM12+Y(4)
SUMC"SUMC + C(4*7)iSJMl * SUMl+ S(I )ISUM2-SUM2 + S(2)J SJM8"SUM8 + S(8)
S(B)-S (8) *Q(8MY(3)«Y(3)vQ(ll)
91 CONTINUE
C09»15'FOOTNOTE REPORT'*****#***********************************************$*** 
97 OUTPUT SUM5/30.,SUl7/30« # SUMll/30« > SUMl2/30•» SJNl/30, ,
*SUM2/30.,SUM8/30,
END
Year El E2
1=944 612*3 567.
1945 596*7 54?.
1946 628.9 532*
1=H7 575.8 54o*
1348 574.2 493»
194Q 6 38 . 2 517.
)9Sn 586.6 496»
19S1 564*6  ^06 *
19b? 599.9 4 47*
1953 66 4. I 4? I »
1954 664*1 478»
1955 683 *q 526*
1956 6 0 3 • 9 448.
1957 568.2 4 18.
1958 636*5 539.
1959 639.2 4 6 6.
1960 723.5 501 •
1961 775.9 ,sT e
3 96? 584.1 476«
1963 6 n . 6 512*
I 964 590 * 7 52?«
I 965 543.6 497.
I 966 7(;i5 • ? 439.
1967 56o • ° 464.
1968 563*9 4 8 6 »
) E4 E5 Ot
0 »0 189,8 •0
0 •0 205*4 '0
0 •0 209.0 *0
0 •0 39.4 '0
0 •0 279,6 •0
0 •0 394.4 •0
0 •0 314.2 •0
0 •0 259,2 •0
0 •0 659.5 •0
0 •0 351.0 •0
C •0 439.3 •0
0 •0 332*7 •0
0 «0 523.9 •0
0 •0 447«o ♦0
C •0 ?68 .1 •0
0 •0 428,1 •0
0 ' •0 559,6 •0
0 •0 453.2 •0
0 •0 311.6 •0
0 *0 198,8 •0
3 •0 l96.o •0
0 •0 184,7 •0
0 •0 623.5 •0
3 •0 314.6 «n
0 •c 226.7 '0
8
7
I
2
6
4
2
I
4
4
9
5
5
2
2
4
3
3
7
9
I
7
3
7
9
E?
2041*2  
188f)* 3 
1783.5
1761.3 
1=926.3 
l 8 6 ? .7  
1=958.2 
2003*8
2097 .0
19 5 . 4
1919.1
1844.3 
2110*2 
2321*2  
1993*5
2017.1  
1903* 5 
I3l6 • 0
2127.3
2067.4  
1994«8
2089.5  
2039*5
2136.6  
2142*6
E8
1485.6 . 0 . 0
1958.3 •0 •0
1684.5 •0 «0
1439.6 *0 •0
1505 . 0 • 0 •0
1920*8 •0 •0
1510.8 *0 •0
1653.1 •0 •0
1789,8 «0 *0
2o63.4 . 0 '0
1732.8 *0 *0
1.831.5 •0 *0
1451.5 •0 '0
17l5,5 ♦0 *0
2177.6 »0 *0
1611.? •0 *0
2066.9 • 0 *0
1828.4 • 0 »0
1276.5 • 0 '0
1868 ,6 •0 *0
1556,7 • 0 '0
1443,1 •0 •0
2232.I *0 •0
1387,0 . 0 •0
177?.6 *0 • 0
Eii E12
1882*9 2041*2
2104*8 1880.3
1899.8 1783.5
1855.5 1761.3
2088.7 1926.3
24?8.6 1862.7
2146*6 1958.2
Pl09 •5 20Q3.8
2676*7 2097.Q
2181*0 l95o» 4
2376*6 l9l9.i
2229*9 1844.3
2463*2 2110*2
2222*4 2321*2
219o*3 1993.5
2312*0 2017.1
2724*6 rsT - •5
2425*9 l3l6.o me
2050*0 2127.3 'S
i860* 6 2067,4
19^2•0 1994.8
Is T / i y 2082.5
2718•I 2039.5
2060*3 2136.6
1900*6 2142*6
Year El W
1^ 44 612*3 567*596.7 542*
1946 628*9 532'
1947 575*8 540.
IcHS 574*2 493*
1949 638*2 517*
195n 586*6 496*
1951 564*6 506 •
1952 599.9 447 •
1953 664» I ,sV i
I Otj 4 664*1 473.
1955 683.9 526»
1956 6o3.9 448.
1957 568.2 418.
1958 6 36.5 539.
1959 639.2 4 6 6»
I960 7 ? 3 * 5 501«
I 9 (i I 775.9 4 90 •
1962 584.1 4^ 6«
196 3 611.6 ' 512"
19(4 hsT k) 52?.
I 9(.5 543.6 497*
1966 7fj5 • 2 439*
1967 56c *9 4 6 4.
1968 563.9 4O6 *
1969 622*2 53?.
19 7n 66 8.5 556*
1971 55?.n 508 ♦
197? 577.0 507*
1973 633.0 S6?.
) E4 E5 E6
0 •0 189.8 •0
0 •0 y T hi , »0
0 »0 209*0 *0
0 "0 89*4 •0
0 •0 279.6 •0
0 •0 394.4 •0
0 •0 314.2 •0
0 »0 259*2 »0
0 »0 659.5 •0
0 •0 35i.o •0
0 .0 439.3 . 0
0 •0 332.7 •0
0 •0 523.9 *0
0 •0 4 47*0 •0
0 •0 268 .1 •0
0 •0 428.1 '0
0 *0 559.6 •0
0 •0 453.2 »0
0 •0 311*6 •0
0 *0 198.8 *0
0 *0 196 . 0 •0
0 *0 184.7 »0
0 »0 6?3 * 5 •0
0 »0 314.6 "0
0 •0 226.7 •0
0 •0 ?65» 8 •0
0 •0 356.4 •0
0 •0 333.7 •0
0 *0 219.8 «0
0 •0 169.2 •0
3
7
I
?.
6
4
2
I
4
4
9
5
5
2
2
4
3
3
7
9
I
7
3
7
9
4
I
6
4
7
E? ES Eli E12
2041*2 1485.6 *0 •0 1882*9 2041*2
I880 * 3 1958.3 •0 •0 21Q4.8 1880*3
1788*5 1684*5 •0 •0 1899*8 1788,5
l76i•3 1439.6 *0 •0 1855*5 1761.3
1926*3 .LT L . 0 *0 *0 2088*7 1926*3
I 86?*7 1920*8 • 0 '0 2428*6 1862*7
1953*2 1510*8 *0 •0 2146*6 1958,2
2003*8 1653*1 *0 *0 2109*5 20o3»8
2097*0 1789.8 *0 »0 2676*7 2097*0
l950 .5 2o63,4 • 0 »0 2181*0 1950*5
1919»I 3732.8 .0 *0 2376*6 1919,1
1844*3 1831,5 «0 •0 2229*9 1844.3
2110’2 1451,5 • 0 »0 2463*2 2110*2
2321*2 1715.5 • 0 »0 2222*4 2321*2
1993*5 2177.6 *0 *0 2190*3 1993*5
2017* I 1611.2 *0 *0 ?3i2*0 2017.1
I 9 0 3 • 5 2066.9 *0 •0 2724*6 . sT -eh G e
I 8 1 6 * 0 1828,4 *0 »0 2425*9 1816.0
2127*3 1276,5 •0 •0 2050*0 2127.3
2067,5 1868 ,6 *0 *0 1860*6 2067,5
1994.9 1556.7 •0 •0 1942*0 1994.9
?082*5 1443.1 *0 »0 . s T / ky 2082.5
2039.5 2232.1 *0 •0 y) V /ir 2039.5
2136*6 1387,o »0 •0 206o•3 2136.6
214?*6 1772•6 •0 •0 - sT x i ) 2142.6
2093*4 1809*9 •0 '0 2131*8 2093.4
2359*8 1685.8 •0 •0 24o4'5 2359.8
2333*4 1373.8 •0 «0 2294*0 2333.4
2174*6 1893.0 . 0 .0 1956•7 2174.6
1974*7 1913.3 • 0 »0 I 934•4 1974.7
1762
cop. 2
Karp, Richard W
A model for predicting 
ion concentrations in 
the Yellovstone River 
between Billings and 
Miles City