U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2018–5108

Prepared in cooperation with the Bureau of Reclamation

Conceptual and Numerical Models of Dissolved Solids in the 

Colorado River, Hoover Dam to Imperial Dam, and Parker Dam 

to Imperial Dam, Arizona, California, and Nevada
Cover. Large picture: Colorado River looking downstream, Imperial National Wildlife Refuge near Cabin Lake, Arizona. 

Small pictures (from upper left to lower right): Hoover Dam, Parker Dam, and Imperial Dam. Photographs by 

Alissa Coes, U.S. Geological Survey.
Conceptual and Numerical Models of 

Dissolved Solids in the Colorado River, 

Hoover Dam to Imperial Dam, and Parker 

Dam to Imperial Dam, Arizona, California, 

and Nevada

By David W. Anning, Alissa L. Coes, and Jon P. Mason

Prepared in cooperation with the Bureau of Reclamation

Scientific Investigations Report 2018–5108

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

RYAN K. ZINKE, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2018

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Suggested citation:

Anning, D.W., Coes, A.L., and Mason, J.P., 2018, Conceptual and numerical models of dissolved solids in the Colorado 

River, Hoover Dam to Imperial Dam, and Parker Dam to Imperial Dam, Arizona, California, and Nevada: U.S. Geological 

Survey Scientific Investigations Report 2018–5108, 34 p., https://doi.org/10.3133/sir20185108.

ISSN 2328-0328 (online)
iii

Acknowledgments

The authors thank Ed Virden, Hong Nguyen-DeCorse, Jon Weiss, and Jim Prairie at the 

Bureau of Reclamation for their input on data compilation and conceptual model development. 

Contents

Abstract...........................................................................................................................................................1

Introduction and Problem Statement .........................................................................................................1

Purpose and Scope ..............................................................................................................................4

Description of Study Area ...................................................................................................................4

Data Compilation............................................................................................................................................5

Measured Data......................................................................................................................................5

Discharge Data.............................................................................................................................5

Dissolved-Solids Data.................................................................................................................7

Data-Quality Assessment..................................................................................................7

Computation of Daily Mean Dissolved-solids Concentrations......................................................7

Method Using Monthly to Bimonthly Dissolved-solids Data................................................8

Method Using Daily Specific Conductance Data...................................................................8

Method Using Monthly to Bimonthly Specific Conductance Data......................................8

Computation of Dissolved-solids Loads and Flow-Weighted Concentrations ...........................9

Conceptual Model..........................................................................................................................................9

Approach................................................................................................................................................9

Results ....................................................................................................................................................9

Numerical Model .........................................................................................................................................19

Approach..............................................................................................................................................19

Parker Dam to Imperial Dam Model .......................................................................................19

Hoover Dam to Imperial Dam Model ......................................................................................22

Results ..................................................................................................................................................23

Parker Dam to Imperial Dam Model .......................................................................................23

Hoover Dam to Imperial Dam Model ......................................................................................26

Comparison of Root-mean Square Error and Measurement Error....................................28

Model Sensitivity and Insight for Running Model Scenarios.............................................29

Summary and Conclusions.........................................................................................................................32

References Cited..........................................................................................................................................33

Appendixes

1. Parker Dam to Imperial Dam and Hoover Dam to Imperial Dam numerical model of 

dissolved-solids concentrations for the Colorado River at Imperial Dam

2. Parker Dam to Imperial Dam numerical model simulation and Hoover Dam to Imperial 

Dam numerical model simulation, and model conversions, statistics, estimations, and 

coefficients

[Available for download at https://doi.org/10.3133/sir20185108]
iv

Figures

1. Monitoring site locations and key hydrologic and land cover features on the Colorado

River, Arizona, California, and Nevada between Hoover Dam and Parker Dam, and

Parker Dam and Imperial Dam ...................................................................................................2

2. Cumulative monthly mean water loss from the Colorado River between Hoover Dam

and Imperial Dam, 1990–2016 ...................................................................................................11

3. Mean monthly discharge for the Colorado River below Hoover Dam, the Colorado

River below Parker Dam, and the Colorado River above Imperial Dam, normalized to

the mean annual discharge of the Colorado River below Hoover Dam, 1990–2016 ........11

4. Cumulative monthly mean dissolved-solids load loss from the Colorado River

between: Hoover Dam and Parker Dam and Parker Dam and Imperial Dam,

1990–2016; and Parker Dam and Imperial Dam .....................................................................12

5. Monthly mean discharge for the Bill Williams River near Parker, monthly mean

dissolved-solids load for the Colorado River below Parker Dam, and daily mean

dissolved-solids load for the Bill Williams River near Parker, 1990–2016 .........................13

6. Cumulative monthly mean dissolved-solids load loss for Basin 2b and monthly mean

discharge for the Colorado River above Imperial Dam, 1990–2016....................................14

7. Mean monthly dissolved-solids loads for the Colorado River below Hoover Dam,

the Colorado River below Parker Dam, and the Colorado River above Imperial Dam,

normalized to the mean annual dissolved-solids load of the Colorado River below

Hoover Dam, 1990–2016.............................................................................................................14

8. Monthly flow-weighted dissolved-solids concentration difference between the

Colorado River below Hoover Dam and below Parker Dam, and between the

Colorado River below Parker Dam and above Imperial Dam, 1990–2016..........................15

9. Monthly flow-weighted dissolved-solids concentration difference between the

Colorado River below Hoover Dam and Parker Dam, and between the Colorado

River below Parker Dam and above Imperial Dam, plotted against monthly

evapotranspiration for the Colorado River and the associated floodplain, 1998–2010...15

10. Monthly flow-weighted dissolved-solids concentration difference between the

Colorado River below Hoover Dam and below Parker Dam, and between the Colorado

River below Parker Dam and above Imperial Dam, plotted against average of monthly

precipitation 1990–2016 .............................................................................................................16

11. Monthly flow-weighted dissolved-solids concentration difference between the

Colorado River below Hoover Dam and below Parker Dam, and between the Colorado

River below Parker Dam and above Imperial Dam, plotted against monthly mean

discharge for the Colorado River, 1990–2016 .........................................................................17

12. Monthly flow-weighted dissolved-solids concentration difference between the

Colorado River below Hoover Dam and below Parker Dam, and the Colorado River

below Parker Dam and above Imperial Dam, 1990–2016 .....................................................18

13. Observed and simulated monthly flow-weighted dissolved-solids concentrations for

the Colorado River above Imperial Dam from 1990 to 2016 from the Parker Dam to

Imperial Dam model and plotted against one another.........................................................23

14. Residual error for the Parker Dam to Imperial Dam model, 1990–2016.............................24

15. Observed and simulated monthly flow-weighted dissolved-solids concentrations for

the Colorado River above Imperial Dam from 1990 to 2016 from the Hoover Dam to

Imperial Dam model and plotted against one another.........................................................27

16. Residual error for the Hoover Dam to Imperial Dam model, 1990–2016............................28
v

Tables

1. Sites and data used to develop conceptual and numerical models, the period of record 

used, and the sources of data used in computations..................................................................6

2. Mean monthly discharge and dissolved-solids load for sites 1–10, expressed as a 

percentage of mean annual discharge and dissolved-solids load at the Colorado River 

below Hoover Dam ...........................................................................................................................10

3. Description of data sources and representation of components of the numerical model......20

4. Constraints for the coefficients of the Hoover Dam to Imperial Dam and Parker Dam to 

Imperial Dam numerical models ....................................................................................................21

5. Additional constraints for Hoover Dam to Imperial Dam numerical model coefficients....22

6. Estimated numerical model coefficients for the Parker Dam to Imperial Dam model and 

for the Hoover Dam to Imperial Dam model ................................................................................22

7. Model diagnostics for the calibration and verification periods for the Parker Dam to 

Imperial Dam model and the Hoover Dam to Imperial Dam model.........................................25

8. Root-mean squared error of model simulations by month for the calibration and 

verification periods of the Parker Dam to Imperial Dam model and the Hoover Dam to 

Imperial Dam model..........................................................................................................................25

9. Sensitivity of simulated concentrations from the Parker Dam to Imperial Dam model to 

the four measured model input variables ....................................................................................30

10. Sensitivity of simulated concentrations from the Hoover Dam to Imperial Dam 

numerical model to the four measured model input variables ................................................31

11. Statistical distribution of the four measured model input variables and of the observed 

monthly flow-weighted dissolved-solids concentration at Imperial Dam, 1990–2016........31
vi

Conversion Factors

[Inch/Pound to International System of Units]

Supplemental Information

Specific conductance is given in microsiemens per centimeter at 25 degrees Celsius (μS/cm at 

25 °C). Concentrations of chemical constituents in water are given in milligrams per liter (mg/L). 

Loads of chemical constituents in water are given in tons per day (ton/d), where one ton equals 

2,000 pounds.

Abbreviations

CNWR Cibola National Wildlife Refuge

CRIR Colorado River Indian Reservation

HNWR Havasu National Wildlife Refuge

NWIS National Water Information System

PVID Palo Verde Irrigation District

RMSE root mean square error

USGS U.S. Geological Survey

Multiply By To obtain

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

acre 0.004047 square kilometer (km2)

square mile (mi2) 2.590 square kilometer (km2)

acre-foot (acre-ft) 1,233 cubic meter (m3)

cubic foot per second (ft3/s) 28.3 liter per second (L/s)

ton per day (ton/d) 0.9072 metric ton per day
Abstract

Conceptual and numerical models were developed to 

understand and simulate monthly flow-weighted dissolved-solids 

concentrations in the Colorado River at Imperial Dam. The ability 

to simulate dissolved-solids concentrations at this location will 

help the Bureau of Reclamation satisfy the binational agreement 

on the volume and salinity of Colorado River water delivered to 

Mexico. A robust spatial- and temporal-resolution dataset that 

consists of river discharge and dissolved-solids concentration 

and load information between January 1990 and September 

2016 for 10 sites on canals, drains, tributaries, and the main 

stem of the Colorado River between Hoover and Imperial Dams 

was generated. Daily mean dissolved-solids concentrations 

were estimated and monthly mean dissolved-solids loads were 

computed for each site. Spatial and temporal load patterns, and 

historical and current controls on loads and concentrations, were 

analyzed in order to develop a conceptual model of dissolvedsolids transport between Hoover and Imperial Dams. Two 

numerical models describing the relations between dissolvedsolids concentrations and components controlling dissolved-solids 

concentrations and loads were developed, calibrated, and verified.

Between January 1990 and September 2016, there was a 

98.8-million-acre-feet loss of water and a 57.0-million-ton loss of 

dissolved-solids load from the Colorado River between Hoover 

and Imperial Dams. Between Hoover and Parker Dams, about 

69.0 million acre-feet of water was lost and 51.1 million tons of 

dissolved solids were lost; between Parker and Imperial Dams, 

about 29.8 million acre-feet of water was lost and 5.9 million tons 

of dissolved solids were lost. Water was removed from the river at 

a relatively consistent rate over the 25-year study period through 

water transfers to California and Arizona, evapotranspiration from 

crop irrigation, transpiration processes of riparian vegetation, 

and evaporation from the river main stem. Dissolved solids were 

removed from the river between Hoover and Parker Dams at a 

relatively constant rate through water transfers to California and 

Arizona, and water pumped from the river for irrigation within 

the Mohave Valley. A small amount of dissolved solids are gained 

by the river from inflow from the Bill Williams River. Between 

Parker and Imperial Dams, however, dissolved solids were not 

removed from the river at a consistent rate over the study period. 

Dissolved solids were generally removed from the river from 

1990 to 2012, then gained by the river from 2012 to 2015, and 

then removed from the river from 2015 through 2016. Dissolved 

solids are assumed to be removed from the river and accumulated 

Conceptual and Numerical Models of Dissolved Solids in the 

Colorado River, Hoover Dam to Imperial Dam, and Parker Dam 

to Imperial Dam, Arizona, California, and Nevada 

By David W. Anning, Alissa L. Coes, and Jon P. Mason

within the floodplain sediments and aquifers during irrigation 

processes; some dissolved solids may also be removed from the 

river through uptake by crops and riparian vegetation. Dissolved 

solids accumulated on the landscape and in the floodplain aquifer 

during irrigation are transported to the river during periods 

when the hydraulic gradient between the floodplain aquifer and 

the river is increased, causing a gain in dissolved solids in the 

river. Dissolved-solids gains in the river occur during periods of 

relatively low river discharge, such as during the winter months 

and during drier climatic conditions.

Two numerical models were developed and coefficients 

were estimated by using data from a May 2008-September 2016 

calibration period. One model simulates concentrations at Imperial 

Dam based on the Colorado River system downstream from 

Parker Dam, and the other model simulates concentrations at 

Imperial Dam based on the Colorado River system downstream 

from Hoover Dam. Both models simulated monthly flowweighted concentrations of dissolved solids for the Colorado 

River at Imperial Dam, which corresponded well with observed 

concentrations for the entire study period. The models are more 

sensitive to input variables of monthly discharge of the Colorado 

River below Parker Dam and monthly flow-weighted dissolvedsolids concentrations of the Colorado River below Hoover Dam 

and Parker Dam than to the rate of change in concentration with 

respect to time and the combined discharge of the Colorado River 

Indian Reservation Main Canal and the Palo Verde Canal. The 

calibrated models can be used to run scenarios of future monthly 

flow-weighted dissolved-solids concentrations in the Colorado 

River at Imperial Dam. Although the models are expected to 

provide concentration estimates within 18 milligrams per liter 

(Parker Dam to Imperial Dam model) to 22 milligrams per liter 

(Hoover Dam to Imperial Dam model), 95 percent of the time, the 

error of future scenarios increases as uncertainty in the estimated 

future input variables increases.

Introduction and Problem Statement

The 1,450-mile-long Colorado River drains 247,000 square 

miles (mi2) of seven U.S. states and two Mexican states (fig. 1). 

The river begins in the southern Rocky Mountains in Colorado 

and flows through the western slopes of the Rocky Mountains; 

the Colorado Plateau regions of Colorado, Utah, and Arizona; 

and the lower Colorado River Valley along the Arizona border 

with Nevada and California before it leaves the United States 

and enters Mexico. 
2 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Study Site (see table 1)

EXPLANATION

Canal or aqueduct

Riparian

Agriculture

Basin drainage boundary

Town

A

HAVASU NATIONAL 

 WILDLIFE REFUGE

LAKE MEAD NATIONAL 

RECREATION AREA

Parker

Bullhead 

 City

Lake Havasu

 City

Needles

PARKER DAM

HEADROCK 

GATE DAM

HOOVER

DAM

Mohave Valley

Site 2

Site 1

Land cover from USGS NLCD datasets, 2006; USGS GAP Land Cover Database, 2011

Palo Verde Drains and Canals, USBR, 2012

Coordinate System: NAD 1983 UTM Zone 12N

Projection: Transverse Mercator

Datum: North American 1983

0 10 20 30 40 MILES

0 10 20 30 40 KILOMETERS

DAVIS

DAM

Site 3

CENTRAL ARIZONA

PROJECT CANAL Site 4

C o l o r a d o R i v e r

COLORADO RIVER AQUEDUCT

Bill Williams River

115°30' 115° 114°30' 114°

36°

35°30'

35°

34°30'

114°W 108°W

42°N

36°N

WYOMING

COLORADO

 BAJA

 DE

 CALIF

UTAH

IDAHO

NEVADA

ARIZONA

 NEW

MEXICO

CALIF

SONORA

 UPPER 

 COLORADO 

 RIVER 

BASIN LOWER

 COLORADO 

 RIVER

 BASIN

Area of map

Dam

 Lee

 Ferry

GLEN 

CANYON 

DAM

MORELOS

DAM

Figure 1. Monitoring site locations and key hydrologic and land cover features on the Colorado River, Arizona, California, and 

Nevada between (A) Hoover Dam and Parker Dam, and (B) Parker Dam and Imperial Dam.
Introduction and Problem Statement 3

Source: Esri, DigitalGlobe, GeoEye, i-cubed, USDA, USGS, AEX,

Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS User

Community

115° 114°30' 114°

34°

33°30'

33°

 COLORADO RIVER 

INDIAN RESERVATION

PICACHO STATE

RECREATION

AREA

Colorado

River

CIBOLA NATIONAL 

WILDLIFE REFUGE

IMPERIAL NATIONAL

WILDLIFE REFUGE

Parker

Poston

Blythe Ehrenberg

Cibola

Martinez 

Lake

PARKER

DAM

HEADROCK GATE

DAM

PALO VERDE

DAM

IMPERIAL

DAM

Parker Strip

Parker Valley

Palo Verde Valley

Cibola

Valley

Site 10

Site 9

Site 6

Site 4

Site 3

Land cover from USGS NLCD datasets, 2006; USGS GAP Land Cover Database, 2011

Palo Verde Drains and Canals, USBR, 2012

Site 5

Site 8

Site 7

0 10 20 30 MILES

0 10 20 30 40 KILOMETERS Coordinate System: NAD 1983 UTM Zone 12N

Projection: Transverse Mercator

Datum: North American 1983

114°W 108°W

42°N

36°N

WYOMING

COLORADO

 BAJA

 DE

 CALIF

UTAH

IDAHO

NEVADA

ARIZONA

 NEW

MEXICO

CALIF

SONORA

 UPPER 

 COLORADO 

 RIVER 

BASIN

LOWER

 COLORADO 

 RIVER

 BASIN

 Lee

 Ferry

Area of map

Study Site (see table 1)

EXPLANATION

Canal or aqueduct

Riparian

Agriculture

Basin drainage boundary

Town

Dam

GLEN 

CANYON 

DAM

MORELOS

DAM

B

Figure 1.—Continued
4 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

The Colorado River is highly regulated with an extensive 

system of dams, reservoirs, and aqueducts that divert about 90 

percent of the river’s water in the United States for agricultural 

and municipal uses. There are 15 dams on the main stem of the 

Colorado River in the United States, which can collectively 

store about 58.3 million acre-feet (acre-ft) of water in 

associated reservoirs. Lee Ferry, located 17 miles downstream 

of Glen Canyon Dam, is the division point between the Upper 

and Lower Colorado River Basins, and the river flow at this 

point is the principal factor in allocating water to the seven 

U.S. states and two Mexican states that have water rights. 

The Utilization of Waters of the Colorado and Tijuana 

Rivers and of the Rio Grande treaty of 1944 guarantees 

that 1.5 million acre-ft of Colorado River water is annually 

delivered from the United States to Mexico. Minute 242 states 

that water delivered to Mexico upstream from Morelos Dam, 

on the United States-Mexico border, must have an average 

annual salinity of no more than 115 parts per million (ppm; 

±30 ppm) over the average annual salinity of Colorado River 

waters at Imperial Dam, located 26.1 river miles upstream 

of Morelos Dam (fig. 1B; International Boundary and Water 

Commission, 1973). For conversion, 1 ppm is approximately 

equivalent to 1 milligram per liter (mg/L) for water with 

dissolved-solids concentrations less than 7,000 mg/L (Hem, 

1992). Salinity is defined as dissolved solids calculated by the 

summation of major constituents.

To meet the binational agreement on the volume and 

salinity of Colorado River water delivered to Mexico, the 

correct volume of groundwater must be added to, or withheld 

from, the river just upstream from Morelos Dam. In order for 

Bureau of Reclamation (referred to herein as Reclamation) 

operators to optimize this volume of groundwater, the river 

salinity must be estimated at Imperial Dam from the current 

month through the end of the calendar year. River salinity 

at Imperial Dam, however, fluctuates on hourly to decadal 

time scales, resulting in uncertainty in groundwater volume 

calculations.

Purpose and Scope

The objective of this study was to provide Reclamation 

with the capability to simulate Colorado River salinity (as 

dissolved solids) at Imperial Dam in order for operators to better 

quantify the volume of groundwater required to be added to, or 

withheld from, the river each month. This report presents (1) 

a conceptual model of the spatial and temporal variability of 

dissolved solids at Imperial Dam and the factors that control 

such variability, and (2) two numerical models that are founded 

on the conceptual model and allow for scenario development 

of dissolved-solids concentrations at Imperial Dam based on 

conditions within the contributing area and at the upstream 

boundaries of the models. The upstream boundary of the first 

numerical model is Parker Dam, and the upstream boundary of 

the second numerical model is Hoover Dam.

Description of Study Area

The study area is located within the Lower Colorado River 

Basin and is defined by the contributing drainage area between 

Hoover Dam and Imperial Dam on the Arizona-California border 

(fig. 1). The study focuses on the river and its floodplain; the 

floodplain is defined as the part of the Colorado River valley that 

was historically inundated by floods prior to the construction 

of dams (Owen-Joyce and Raymond, 1996). Between Hoover 

and Imperial Dams, the Colorado River meanders to divide the 

floodplain into Mohave, Parker, Palo Verde, and Cibola Valleys. 

In 2010, approximately 149,300 acres (51 percent) of 

the floodplain between Hoover Dam and Imperial Dam were 

agricultural and 143,928 acres (49 percent) were riparian 

vegetation (Bureau of Reclamation, 2014). Agricultural acreage 

within the study area is dependent on Colorado River water 

that is either diverted into canals or pumped from the river and 

transported to agricultural areas in the floodplain for irrigation. 

Unused water and irrigation return flows are returned to the river 

through a complex system of wasteways, spillways, and drains.

Between Hoover Dam and Davis Dam, the Colorado River 

is confined by bedrock with small riparian areas at the mouths of 

tributary streams (Owen-Joyce and Raymond, 1996). This reach 

of the river is within the Lake Mead National Recreation Area and 

contains no agricultural acreage.

Mohave Valley begins about 6 miles below Davis Dam and 

lies mostly within Arizona. Land use within the valley was about 

25 percent agricultural acreage in 2010 (Bureau of Reclamation, 

2014). Agricultural areas are irrigated with water pumped from the 

Colorado River; irrigation returns flow through the groundwater 

system. Downstream of Mohave Valley, the Colorado River is 

confined by bedrock within the Havasu National Wildlife Refuge 

(HNWR), which extends to Lake Havasu, above Parker Dam. 

Within HNWR, Colorado River water is diverted to and from 

Topock Marsh through an inlet and an outlet, respectively. Water 

is diverted from the Colorado River just above Parker Dam to 

California through the Colorado River Aqueduct, and to Arizona 

through the Central Arizona Project Canal. The Bill Williams 

River is a regulated tributary that discharges to the Colorado River 

just above Parker Dam.

Parker Dam is the start of the Parker Strip-Parker Valley 

reach of the Colorado River. The Parker Strip is a short, thin 

stretch of the floodplain between Parker Dam and Parker Valley; 

Parker Valley makes up the remainder of this reach. Most of the 

floodplain in Parker Valley lies in Arizona within the Colorado 

River Indian Reservation (CRIR). Land use within the valley 

was about 62 percent agricultural acreage in 2010 (Bureau of 

Reclamation, 2014). Water is diverted from the Colorado River to 

croplands in Arizona at Headgate Rock Dam through the CRIR 
Data Compilation 5

Main Canal. Irrigation return flows are returned to the river near 

the Poston Wasteway and the CRIR Wasteway, which contain 

return flows from both the CRIR Main Canal and the CRIR Upper 

Levee Drain; and just below Palo Verde Dam through the Palo 

Verde Drain and the CRIR Lower Main Drain.

Palo Verde Valley begins below the Palo Verde Dam and lies 

mostly within California. Land use within the valley was about 

94 percent agricultural acreage in 2010 (Bureau of Reclamation, 

2014). At Palo Verde Dam, water is diverted from the river to 

croplands in the valley through the Palo Verde Canal. Irrigation 

return flows are returned to the river through 10 spillways and 

drains, the largest of which is the Palo Verde Irrigation District 

(PVID) Outfall Drain.

Cibola Valley is southeast of Palo Verde Valley, and spans 

both sides of the river within Arizona and California. Colorado 

River water is diverted to and from Cibola Lake, within Cibola 

National Wildlife Refuge (CNWR), through an inlet and an outlet, 

respectively. South of Cibola Valley, the floodplain narrows and 

the river flows through an area dominated by phreatophytes in the 

CNWR, the Imperial National Wildlife Refuge, and Picacho State 

Recreation Area. The downstream boundary of this reach (and of 

the study area) is Imperial Dam. Land use within Cibola Valley 

and downstream to Imperial Dam was 15 percent agricultural 

acreage and 85 percent riparian acreage in 2010 (Bureau of 

Reclamation, 2014). Agricultural areas within Cibola Valley are 

irrigated with water pumped from the Colorado River; irrigation 

returns flow through the groundwater system.

Floodplain sediments comprise the upper water-bearing unit 

of the Colorado River aquifer, called the floodplain aquifer. This 

unit is 0 to 180 feet (ft) thick and is highly permeable (Wilson 

and Owen-Joyce, 1994). A small quantity of direct runoff from 

occasional intense rainfall infiltrates into the floodplain aquifer 

along the edges of the floodplain through tributaries to the 

Colorado River (Wilson and Owen-Joyce, 1994). Most of the 

recharge to the aquifer, however, occurs artificially as diverted 

river water is applied to agricultural fields during irrigation. 

Drainage ditches within the agricultural areas of Parker and Palo 

Verde Valleys intersect the water table of the floodplain aquifer 

and remove excess irrigation water. Drains may also include a 

minor amount of naturally recharged tributary water (Leake and 

others, 2013).

Data Compilation

The approach of this study was to first generate a highspatial- and high-temporal-resolution dataset of discharge 

and dissolved-solids information between January 1990 and 

September 2016 for 10 sites on canals, drains, tributaries, and 

the main stem of the Colorado River between Hoover Dam and 

Imperial Dam (table 1; fig. 1). This dataset included computed 

daily dissolved-solids concentrations and loads for each 

site. Sites were chosen on the basis of their available data and 

relevance to dissolved-solids controls in the study area.

Measured Data

Data used to compute daily dissolved-solids concentrations 

and loads included measured daily mean discharge, measured 

monthly to bimonthly dissolved-solids samples, and measured 

daily specific conductance measurements available from 10 

sites. The period January 1, 1990, through September 30, 2016, 

was used for nine of the sites. At one site (PVID Outfall Drain), 

however, daily mean discharge data were only available from 

January 1, 2005, through September 30, 2016.

Discharge Data

Daily mean discharges and monthly mean discharges 

used for this study were obtained from continuous discharge 

data measured by a network of 10 streamflow gaging stations 

(sites) operated by the U.S. Geological Survey (USGS) and 

Reclamation (table 1). Four sites were located on the main 

stem of the Colorado River, one site was located on a tributary 

to the river, two sites were located on irrigation canals that 

divert water from the river, and three sites were located on 

drains that return water to the river. All USGS discharge 

data are archived in the USGS National Water Information 

System database (NWIS; U.S. Geological Survey, 2017) and 

all Reclamation discharge data are archived at https://doi.

org/10.5066/F7SQ8ZN4. 

Discharge data were not available at the site on the Colorado 

River at Palo Verde Dam (site 6). Discharge at this site was 

computed by summing discharge from the Colorado River below 

Palo Verde Dam (USGS station number 09429100) and from 

the Palo Verde Canal (USGS station number 09429000), then 

subtracting discharge from the CRIR Upper Levee Drain (USGS 

station number 09429030) and the CRIR Lower Main Drain 

(USGS station numbers 09429060 and 09429070).

Discharge at the site on the CRIR Lower Main Drain (site 7) 

was determined from discharge measured at two gaging stations. 

Discharge data were available from January 1, 1990, to September 

30, 2010, at USGS station number 09429060. This station was 

replaced by USGS station number 09429070, for which discharge 

data were available from October 1, 2009, to June 30, 2015. 

Station 09429070 is located about 2.5 miles downstream of station 

09429060, and a large area of irrigated cropland lies between the 

two sites; drainage from irrigation most likely enters the drain 

between the old and new stations. Both stations were in operation 

from October 1, 2009, to September 30, 2010, which allowed for 

comparison of daily mean discharges measured at both stations. 

On average, there was about 66 cubic feet per second (ft3/s) more 

flow at 09429070 compared to 09429060 during this period. To 

compensate for this discrepancy in discharge, 66 ft3/s was added 
6 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Table 1. Sites and data used to develop conceptual and numerical models, the period of record used, and the sources of data used in 

computations. 

[USGS, U.S. Geological Survey; Reclamation, Bureau of Reclamation; CRIR, Colorado River Indian Reservation; PVID, Palo Verde Irrigation District; --, not applicable]

Study site 

number on 

figure 1

Study site name Study site description Period of record 

used

Collecting 

agency for 

discharge 

data

Collecting 

agency for 

dissolvedsolids data

USGS station 

number

Reclamation 

station 

number

1 Colorado River below 

Hoover Dam

Colorado River main 

stem

January 1, 1990–

September 30, 

2016

USGS Reclamation 09421500 09-4215.00

2 Bill Williams River 

near Parker1

Tributary to Colorado 

River

January 1, 1990–

September 30, 

2016

USGS Reclamation, 

USGS

09426620/

09416600

09-4266.20

3 Colorado River below 

Parker Dam

Colorado River main 

stem

January 1, 1990– 

September 30, 

2016

USGS Reclamation, 

USGS

09427520 09-4275.20

4 CRIR Main Canal Irrigation diversion 

from Colorado River

January 1, 1990– 

September 30, 

2016

USGS Reclamation 09428500 09-4285.00

5 Palo Verde Canal Irrigation diversion 

from Colorado River

January 1, 1990– 

September 30, 

2016

USGS Reclamation 09429000 09-4290.00

6 Colorado River at Palo 

Verde Dam2,4

Colorado River main 

stem

January 1, 1990– 

September 30, 

2016

USGS Reclamation 09429000, 

09429100, 

09429030, 

09429060/ 

09429070

09-4290.00

7 CRIR Lower Main 

Drain3

Irrigation return flow to 

Colorado River

January 1, 1990– 

September 30, 

2016

USGS Reclamation 09429060/ 

09429070

09-4280.60

8 CRIR Upper Levee 

Drain

Irrigation return flow to 

Colorado River

January 1, 1990– 

September 30, 

2016

USGS Reclamation 09429030 09-4290.30

9 PVID Outfall Drain Irrigation return flow to 

Colorado River

January 1, 2005– 

September 30, 

2016

Reclamation Reclamation -- 09-4292.20

10 Colorado River above 

Imperial Dam4

Colorado River main 

stem

January 1, 1990– 

September 30, 

2016

USGS Reclamation, 

USGS

09429490 09-4294.90

1

USGS dissolved-solids data was collected at station 09426600. USGS discharge data was collected at station 09426620.

2

Dissolved-solids data from the Palo Verde Canal (site 5) were used for this site. Discharge at this site was computed by summation of discharge at USGS 

station numbers 09429000 and 09429100 and subtraction of discharge at USGS station numbers 09429030 and 09429060/09429070.

3

Discharge at this site was determined using discharge at USGS station number 09429060 from January 1, 1990, to September 30, 2009, and using discharge 

at USGS station number 09420970 from October 1, 2009, to June 30, 2015. An adjustment factor of 66 cubic feet per second was added to each daily mean 

discharge value from 09429060 to account for inflow between the two stations.

4

Site represents conditions immediately upstream from the dam prior to diversions.
Data Compilation 7

to each daily mean discharge at station 09429060 from January 1, 

1990, to September 30, 2009.

Dissolved-Solids Data

Dissolved-solids data (calculated as the sum of the major 

constituents of bicarbonate, calcium, carbonate, chloride, fluoride, 

magnesium, nitrate, potassium, silicon dioxide, sodium, and 

sulfate) were obtained from a network of 10 sampling stations 

operated by the USGS and Reclamation (table 1). USGS and 

Reclamation dissolved-solids data from the same (or proximal) 

stations were combined where data were available from both 

agencies. The frequency of dissolved-solids data at nine of the sites 

varied from bimonthly to monthly; at site 2, dissolved-solids data 

was collected from 2 to 15 times per year on an irregular schedule 

because of its intermittent streamflow. The period of the dissolvedsolids data was the same period as the discharge data. All USGS 

dissolved-solids data are archived in the USGS National Water 

Information System database (U.S. Geological Survey, 2017) and 

all Reclamation dissolved-solids data are archived at https://doi.

org/10.5066/F7SQ8ZN4. 

Data-Quality Assessment

Replicate samples are two or more water samples that are 

collected, prepared, and analyzed in the same manner such that 

they are considered to be essentially identical, and are used to 

estimate the variability of analytical results inherently related 

to the environment, sampling procedures, and lab procedures 

(Mueller and others, 2015). There were 34 dissolved-solids 

replicate sample pairs collected by the USGS and Reclamation at 

the study sites between January 1990 and February 2017 (https://

doi.org/10.5066/F7SQ8ZN4). A bias-corrected log-log regression 

model (Mueller and others, 2015) was used to model the 

variability, as standard deviation, of the replicates. This model is 

based on the approximately linear relation between the logarithms 

of replicate standard deviation and mean concentration, where

 log(SD)=B0+B1log(C) (1)

where

SD is the replicate standard deviation;

B0 is the intercept of the regression line, 

estimated by least-squares;

B1 is the slope of the regression line, estimated 

by least-squares; and

C is the mean replicate concentration, or 

concentration of a single sample.

Standard deviation residuals from the log-log equation are 

then retransformed back to their original units; the mean of the 

retransformed standard deviation residuals is the bias-correction 

factor (bcf). The bcf is multiplied by the estimated standard 

deviations of each replicate in order to express the modeled 

standard deviation (SDM):

 (2)

The variability of the dissolved-solids sample concentrations 

in this study was determined to be

 1.179{10[-1.628+0.735log(C)]}. (3)

Using equation 3, the variability of the measured 

dissolved solids concentrations at the Colorado River above 

Imperial Dam ranges from 2.5 to 4.5 mg/L, with an average 

variability of 3.5 mg/L, or about 0.5 percent.

An additional quality assessment of the monthly to 

bimonthly dissolved-solids data (calculated as the sum of major 

constituents) was completed in order to understand data quality 

and to remove any samples or values deemed erroneous. The 

ratio of cations to anions within each sample was computed. 

Samples with a ratio value that was greater than 1.20 or less than 

0.80 were removed from the dataset. The ratio of dissolved-solids 

concentration calculated as the sum of constituents to dissolvedsolids concentration determined by residue on evaporation 

was computed. Samples with a ratio value that was greater 

than 1.20 or less than 0.80 were removed from the dataset. The 

ratio of dissolved-solids concentration calculated as the sum of 

constituents to specific conductance was computed. Samples 

with ratios having large deviations from the long-term trend were 

removed. In total, 94 samples (1.9 percent of the total samples) 

were removed based on the above data assessments. In addition, 

the quality assessment also determined that 34 pairs of samples 

were incorrectly coded as being collected at identical sites, dates, 

and times; this resulted in the removal of an additional 68 samples 

(1.4 percent of the total samples) because the correct dates and 

times for these samples could not be determined.

For the Colorado River above Imperial Dam (site 10), the 

ratios of the dissolved-solids concentration calculated as the sum 

of constituents to specific conductance generally ranged from 

about 0.60 to 0.64 with no increasing or decreasing trends during 

the period January 1990 through September 2016, with an average 

ratio of 0.619. There is, however, an anomalous period from 

about January 2005 through April 2008 when ratios ranged from 

about 0.54 to 0.62, with an average ratio of 0.582. In March 2008, 

Reclamation conducted an internal review of sample collection, 

handling, and analysis procedures at this site to determine the 

cause of inconsistent water-quality data (Hong Nguyen-DeCorse, 

Bureau of Reclamation, written commun., March 8, 2016). The 

review led to the implementation of minor corrections to field 

sampling and laboratory procedures for water-quality sampling at 

all sites. In addition, the analytical laboratory began using a new 

inductively coupled plasma mass spectrometer for cation analysis.

Computation of Daily Mean Dissolved-solids 

Concentrations

Daily mean dissolved-solids concentrations were estimated 

for nine of the sites using one of three methods depending on the 

type of data available and the numerical model requirements. 

At sites 4–9, a method using discrete monthly to bimonthly 

SD dissolved-solids data was used. At site 10, the Colorado River M = bcf {10

[B0 + B1 log(C)]

}
8 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

above Imperial Dam, a method using daily specific conductance 

data was used. At site 1, the Colorado River at Hoover Dam, and 

site 3, the Colorado River at Parker Dam, a method using monthly 

to bimonthly specific conductance data was used. Daily mean 

dissolved-solids concentrations were not estimated for site 2, the 

Bill Williams River near Parker, because of large periods of time 

with no dissolved-solids data.

Method Using Monthly to Bimonthly Dissolvedsolids Data

Daily mean dissolved-solids concentrations were initially 

estimated for sites 4–9 by using a regression between monthly to 

bimonthly dissolved-solids concentrations and stream discharge 

and also by taking into account the components of time and 

season. The statistical model, Weighted Regression on Time, 

Discharge, and Season (WRTDS) developed by Hirsch and 

others (2010), was applied to the data by using the R package 

Exploration and Graphics for River Trends (EGRET; Hirsch and 

De Cicco, 2015) and the R Project for Statistical Computing (R 

Development Core Team, 2011). The WRTDS model, however, 

did not provide a satisfactory fit between actual and estimated 

dissolved-solids concentrations. Although this estimation approach 

is commonly used in many studies, the lack of fit is not surprising 

when considering the difference between the Colorado River and a 

natural stream. In a natural stream, dissolved-solids concentrations 

are typically a mixture two major sources of water—highconcentration base flow and low-concentration runoff. As a 

result of the contrast in source-water concentrations, natural 

stream concentrations generally decrease as a larger portion of 

the streamflow comes from runoff. For regulated streams such 

as the Colorado River, however, concentrations of higher flows 

from reservoir releases may not be that much different than those 

in base flows and the relative contribution of base flow to total 

streamflow may be small. This is especially true immediately 

downstream from large reservoirs, where stream concentrations 

reflect reservoir contents and exhibit little variation whereas 

reservoir release discharges vary greatly.

Daily mean dissolved-solids concentrations at sites 

4–9 were, therefore, estimated from monthly to bimonthly 

dissolved-solids data using an interpolation approach. Linear 

interpolation between monthly to bimonthly dissolved-solids 

concentrations was done using the R Project for Statistical 

Computing (R Development Core Team, 2011), and a daily time 

step was used to obtain a value that could be used as a daily 

concentration. A LOESS (local regression) smooth function was 

subsequently applied to the linearly interpolated daily data to 

account for uncertainty related to measurement errors, and the 

result was the daily series of concentration data representing 

dissolved-solids conditions for the station.

Method Using Daily Specific Conductance Data

For site 10, the Colorado River above Imperial Dam, 

Reclamation determines daily specific conductance from four 

samples collected at 6-hour intervals. Daily dissolved-solids 

concentrations were computed by Reclamation from this daily 

specific-conductance data by using an interpretation of bimonthly 

dissolved-solids and specific-conductance data at this site.

For the Colorado River at Imperial Dam, Reclamation 

determines the ratio of dissolved-solids concentration to specific 

conductance for each bimonthly sample. The following 2 weeks 

of daily specific-conductance data are then multiplied by this 

ratio to determine the daily dissolved-solids concentrations for 

those 2 weeks. As previously discussed, however, the ratio from 

about January 2005 through April 2008 is biased low because 

of sampling and analytical procedural issues. Therefore, in this 

study, daily mean dissolved-solids concentrations for the Colorado 

River at Imperial Dam were recalculated by using corrected ratios 

and daily specific-conductance data for the periods of January 

1990 through April 2008, and May 2008 through September 

2016. Different ratios were calculated for the periods before and 

after May 2008 to take into account the implementation of new 

field sampling and laboratory procedures. The corrected ratios 

were determined by computing the ratio of dissolved-solids 

concentration (in mg/L) to specific conductance (in microsiemens 

per centimeter [μS/cm]) for each bimonthly sample between 

January 1990 through December 2004, and May 2008 through 

September 2016; the period of January 2005 to April 2008 was 

excluded. The average corrected ratio for January 1990 through 

December 2004 was determined to be 0.625, and each daily 

specific-conductance value prior to May 2008 was multiplied 

by this average corrected ratio to compute daily dissolved-solids 

values. For May 2008 through September 2016, the average 

corrected ratio was determined to be 0.623, and each daily 

specific-conductance value after May 2008 was multiplied by this 

average corrected ratio to compute daily dissolved-solids values. 

Implementing this approach reduced the root mean square error 

(RMSE) of the residuals from preliminary numerical models 

by more than one-third, which validated this recalculation and 

suggests that others may also benefit from using the recalculated 

concentrations.

Method Using Monthly to Bimonthly Specific 

Conductance Data

For sites representing conditions at the upstream boundaries 

of the two numerical models, Colorado River below Hoover 

Dam (site 1) and Colorado River below Parker Dam (site 3), 

Reclamation required the ability to use either specific conductance 

or dissolved solids concentration as an input variable. Daily mean 

specific conductance concentrations at sites 1 and 3 were estimated 

from monthly to bimonthly specific conductance data using an 

interpolation approach. Linear interpolation between monthly to 

bimonthly specific conductance concentrations was done using the 

R Project for Statistical Computing (R Development Core Team, 

2011), and a daily time step was used to obtain a value that could 

be used as a daily concentration. 

To convert daily specific conductance values to 

daily dissolved-solids values, the ratio of dissolved-solids 
Conceptual Model 9

concentration (in mg/L) to specific conductance (in μS/cm) for 

each monthly to bimonthly sample was determined for each 

site. The average ratios for the periods of January 1990 through 

April 2008, and May 2008 through September 2016 were then 

calculated for each site. Different ratios were calculated for 

the periods before and after May 2008 to take into account 

the implementation of new field sampling and laboratory 

procedures. For site 1, the average corrected ratio for January 

1990 through April 2008 was determined to be 0.623, and 

each daily specific-conductance value prior to May 2008 was 

multiplied by this average corrected ratio to compute daily 

dissolved-solids values; for May 2008 through September 

2016, the average corrected ratio was determined to be 0.616, 

and each daily specific-conductance value after May 2008 was 

multiplied by this average corrected ratio to compute daily 

dissolved-solids values. For site 3, the average corrected ratio 

for January 1990 through April 2008 was determined to be 

0.638, and each daily specific-conductance value prior to May 

2008 was multiplied by this average corrected ratio to compute 

daily dissolved-solids values; for May 2008 through September 

2016, the average corrected ratio was determined to be 0.616, 

and each daily specific-conductance value after May 2008 was 

multiplied by this average corrected ratio to compute daily 

dissolved-solids values. 

Computation of Dissolved-solids Loads and 

Flow-weighted Concentrations

Daily mean dissolved-solids loads (Wd, in tons per day 

[ton/d]) were computed for the nine sites with daily mean 

dissolved-solids concentrations (table 1). Discharge and 

dissolved-solids data were often not recorded at the exact 

same site, but from two different stations in proximity to one 

another (table 1). Daily loads were computed by multiplying 

daily mean discharge (Qd, in ft3/s) by daily dissolved-solids 

concentration (Cd, in mg/L), with unit conversions, as follows 

 Wd=Qd×Cd×2.696122e-3 (4)

Daily mean dissolved-solids loads were averaged to 

obtain monthly mean dissolved-solids loads (Wm; in ton/d) 

(https://doi.org/10.5066/F7SQ8ZN4). Monthly mean flowweighted concentrations (Cm, in mg/L) were then computed by 

dividing monthly mean dissolved-solid load (Wm; in ton/d) by 

monthly mean discharge (Qm; in ft3/s) with unit conversions 

using the following equation

 (5)

Monthly mean dissolved-solids loads and flow-weighted 

concentrations were not calculated for site 2, the Bill Williams 

River near Parker, because daily mean dissolved-solids 

concentrations could not be calculated for this site and because 

this site has many months with zero discharge (https://doi.

org/10.5066/F7SQ8ZN4). Instead, daily mean dissolved-solids 

loads were calculated for days with samples by using 

instantaneous dissolved-solids concentrations and daily mean 

discharges (https://doi.org/10.5066/F7SQ8ZN4).

Conceptual Model 

Approach

In order to develop a conceptual understanding of dissolvedsolids transport in the Colorado River between Hoover and 

Imperial Dams, monthly mean discharges and monthly mean 

dissolved-solids loads from January 1990 to September 2016 

were analyzed (https://doi.org/10.5066/F7SQ8ZN4). Dissolvedsolids loads were chosen for the conceptual model instead of 

concentrations because changes in dissolved-solids loads are 

typically related to physical transport processes rather than 

geochemical processes; and, it is easier to identify the effects of 

physical transport processes on loads than on concentrations.

To examine spatial and temporal variability, monthly mean 

discharge and dissolved-solids load data were averaged by month 

(weighted by the number of days in each month) over the January 

1990 to September 2016 period at each of the 10 sites to determine 

mean monthly discharges and dissolved-solids loads (https://doi.

org/10.5066/F7SQ8ZN4). These data were spatially normalized 

by dividing the mean monthly discharges and dissolved-solids 

loads for each site by the mean annual discharge and dissolvedsolids load for the Colorado River below Hoover Dam (site 1) 

and then multiplying by 100 to express the values as a percentage 

(table 2). Such normalization makes it easier to compare the 

data to average conditions at the upstream boundary of the study 

area and makes it easier to track the transport of mass of water 

and dissolved solids through the system. Parker and Palo Verde 

Valleys, as well as their canals and drains to and from the river, 

overlap in the area of Palo Verde Dam (fig. 1). Consequently, a 

new site called site 8a, the Colorado River between Parker and 

Palo Verde Valleys, was established by summing data for sites 6, 7, 

and 8. This site represents the east-west boundary between Parker 

and Palo Verde Valleys and separates all inflows and outflows 

between the river and each valley.

Results

There are five main pathways for Colorado River surface 

water flow and dissolved-solids transport between Hoover and 

Imperial Dams: (1) water and dissolved solids released from 

Hoover Dam that remain within the main stem of the Colorado 

River and travel through to Imperial Dam, (2) water and dissolved 

solids released from Hoover Dam that are pumped from the river 

and are subject to off-river irrigation and return flow processes 

within the Mohave and Cibola Valleys, (3) water and dissolved 

solids released from Hoover Dam that are diverted to California 

through the Colorado River Aqueduct and to Arizona through 

the Central Arizona Project Canal, (4) water and dissolved solids 

that are transported to the Colorado River from the Bill Williams 

Cm = Wm

Qm × 2.696122e−3
10 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

River, and (5) water and dissolved solids released from Hoover 

Dam that are diverted into canals and are subject to off-river 

irrigation and return flow processes within the Parker and Palo 

Verde Valleys. For the purposes of the following discussion, the 

river and associated floodplain valleys is generalized as follows: 

Basin 1 includes the Colorado River from site 1 (below Hoover 

Dam) to site 2 (below Parker Dam) and includes Lake Mead 

National Recreation Area and Mohave Valley; Basin 2 includes 

the Colorado River from site 3 (below Parker Dam) to 10 (above 

Imperial Dam). Basin 2 is further divided into Basin 2a which 

includes the Colorado River from site 3 (below Parker Dam) to 

site 8a (between Parker and Palo Verde Valleys) and includes 

Parker Strip and Parker Valley; and Basin 2b which includes the 

Colorado River from site 8a (between Parker and Palo Verde 

Table 2. Mean monthly discharge and dissolved-solids load for sites 1–10, expressed as a percentage of mean annual discharge and 

dissolved-solids load at the Colorado River below Hoover Dam (site 1).

[CRIR, Colorado River Indian Reservation; PVID, Palo Verde Irrigation District; --, not applicable]

Month

Colorado 

River 

below 

Hoover 

Dam

Bill 

Williams 

River 

near 

Parker

Colorado 

River 

below 

Parker 

Dam

CRIR 

Main 

Canal

Palo Verde 

Canal

Colorado 

River 

at Palo 

Verde 

Dam

CRIR 

Lower 

Main 

Drain

CRIR 

Upper 

Levee 

Drain

Colorado 

River 

between 

Parker and 

Palo Verde 

Valleys1

PVID 

Outfall 

Drain

Colorado 

River 

above 

Imperial 

Dam

Site 1 Site 2 Site 3 Site 4 Site 5 Site 6 Site 7 Site 8 Site 8a Site 9 Site 10

Mean monthly discharge (percent of mean annual discharge at site 1)

January 78.2 0.7 45.0 2.3 3.8 41.2 1.4 0.1 42.7 3.2 42.8 

February 86.9 2.1 62.6 2.5 6.4 55.6 1.6 0.1 57.3 3.2 54.7 

March 115.5 3.0 85.7 2.6 8.3 76.4 1.7 0.1 78.3 3.4 72.8 

April 134.9 0.6 97.7 2.7 10.6 88.0 1.8 0.2 90.0 3.6 82.4 

May 128.6 0.1 89.9 2.8 12.1 78.7 1.9 0.2 80.8 3.9 72.5 

June 120.9 0.1 94.3 2.8 13.3 80.3 1.8 0.2 82.3 4.1 71.5 

July 116.0 0.1 95.1 2.9 13.2 81.6 1.9 0.2 83.7 4.3 73.6 

August 105.0 0.1 81.4 3.0 12.4 70.6 1.9 0.2 72.7 4.4 64.5 

September 84.8 0.1 73.5 3.2 10.3 64.6 1.9 0.2 66.7 4.4 60.5 

October 70.1 0.1 62.2 3.2 7.9 56.5 1.8 0.2 58.5 4.3 55.9 

November 79.5 0.2 49.1 3.3 5.9 45.1 1.7 0.1 47.0 3.9 47.1 

December 79.8 0.1 40.5 3.4 5.6 37.3 1.6 0.1 39.0 3.7 40.1 

Annual 100.0 0.6 73.1 2.9 9.2 64.7 1.8 0.2 66.6 3.9 61.5 

Mean monthly dissolved-solids load (percent of mean annual load at site 1)

January 78.3 -- 46.6 2.4 4.1 43.5 4.2 0.2 47.9 7.7 52.6 

February 87.0 -- 64.4 2.5 6.6 57.9 4.1 0.2 62.2 7.6 64.3 

March 115.5 -- 87.5 2.7 8.5 78.3 4.3 0.3 82.8 8.0 82.6 

April 135.9 -- 101.9 2.8 11.1 91.9 4.6 0.4 96.9 8.5 95.0 

May 130.0 -- 93.8 2.8 12.6 81.9 4.8 0.4 87.0 9.0 85.7 

June 121.4 -- 98.3 2.9 13.8 83.8 4.5 0.4 88.7 9.7 84.4 

July 115.9 -- 98.5 3.0 13.8 85.2 4.7 0.4 90.2 10.1 86.3 

August 104.3 -- 84.6 3.1 12.9 73.7 4.9 0.4 79.1 10.2 77.0 

September 83.8 -- 75.5 3.3 10.8 67.5 4.8 0.4 72.6 10.3 72.2 

October 69.7 -- 64.2 3.3 8.3 59.6 4.7 0.3 64.5 10.1 67.3 

November 78.8 -- 51.0 3.4 6.3 48.2 4.5 0.3 53.0 9.4 58.0 

December 79.6 -- 42.1 3.5 6.0 39.8 4.4 0.2 44.5 8.8 50.6 

Annual 100.0 -- 75.7 3.0 9.6 67.6 4.5 0.3 72.5 9.1 73.0 

1 Mean monthly discharge and dissolved-solids load at this site computed by summing data from sites 6, 7, and 8. 

Valleys) to site 10 (above Imperial Dam), and includes Palo Verde 

Valley, Cibola Valley, and the floodplain downstream of Cibola 

Valley to Imperial Dam.

Between January 1990 and September 2016, there was 

an overall loss of water from the Colorado River of about 

98.8 million acre-ft between Hoover and Imperial Dams 

(fig. 2), which is about a 38.5 percent loss over 26.75 years. 

Approximately 69.8 percent of water removal from the river 

occurs in Basin 1, and 30.2 percent of water removal occurs 

in Basin 2. Water is removed from the river at a relatively 

consistent rate through water transfers to California and Arizona, 

evapotranspiration from crop irrigation, transpiration processes 

of riparian vegetation, and evaporation from the river main stem. 

About 64.1 million acre-ft of water was transferred to California 
Conceptual Model 11

and Arizona between January 1990 and September 2016; this 

accounts for about 92.9 percent of the water lost in Basin 1. 

Independent estimates of evapotranspiration losses for the study 

area averaged 1.60 million acre-ft per year from 1998 to 2010, 

with an average loss from Basin 1 of 0.47 million acre-ft per 

year, and an average loss from Basin 2 of 1.13 million acre-ft 

per year (data furnished by Reclamation). About 1.6 million 

Figure 2. Cumulative 

monthly mean water 

loss from the Colorado 

River between Hoover 

Dam (site 1) and Imperial 

Dam (site 10), 1990–2016. 

Positive cumulative 

monthly mean water 

loss numbers indicate 

the volume of water lost 

from the river.

Figure 3. Mean monthly discharge for the Colorado River below Hoover Dam (site 1), the Colorado River 

below Parker Dam (site 3), and the Colorado River above Imperial Dam (site 10), normalized to the mean 

annual discharge of the Colorado River below Hoover Dam, 1990–2016.

Basin 1—Hoover Dam to Parker Dam

Basin 2—Parker Dam to Imperial Dam

Cumulative monthly mean water loss, 

in millions of acre-feet 40

50

60

70

80

Year

1990 1995 2000 2005 2010 2015

EXPLANATION

10

20

30

0

acre-ft of water was added to the Colorado River from the Bill 

Williams River between January 1990 and September 2016, 

which is about 0.8 percent of the total flow of the Colorado River 

below Parker Dam over 26.75 years. During all months of the 

year, Colorado River discharge at Imperial Dam is, on average, 

less than at Hoover and Parker Dams, though the difference in 

discharge is greatest during the summer months (fig. 3). 

Colorado River below Hoover Dam

Colorado River below Parker Dam

Mean monthly discharge as a percent of mean annual 

discharge of the Colorado River below Hoover Dam

0

20

40

60

80

100

120

140

160

Month

January February March April May June July August September October November December

EXPLANATION

Colorado River above Imperial Dam
12 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Between January 1990 and September 2016, there was an 

overall loss of dissolved-solids load from the Colorado River 

of about 57.0 million tons between Hoover and Imperial Dams 

(fig. 4A), which is about a 27 percent loss over 26.75 years. 

Dissolved solids are assumed to be removed from the river 

primarily through water transfers to California and Arizona and 

secondarily through accumulation of dissolved solids within 

floodplain sediments and aquifers during irrigation processes; 

some dissolved solids may also be removed from the river 

through uptake by crops and riparian vegetation. 

Within Basin 1, there was a loss of about 51.1 million tons 

of dissolved solids from 1990 to 2016; dissolved solids were 

removed from the Colorado River at a relatively constant rate of 

about 2 million tons per year. Most of this loss occurs as water and 

dissolved solids are removed from the river through the Colorado 

River Aqueduct and the Central Arizona Project Canal. Dissolved 

solids are also removed from the river as water is pumped from 

the river for irrigation within the Mohave Valley. A small amount 

Figure 4. Cumulative 

monthly mean dissolvedsolids load loss from the 

Colorado River between: 

(A) Hoover Dam (site 1) and 

Parker Dam (site 3) and 

Parker Dam and Imperial 

Dam (site 10), 1990–2016; 

and (B) Parker Dam and 

Imperial Dam. Positive 

cumulative monthly mean 

dissolved-solids load loss 

numbers indicate the mass 

of dissolved solids lost 

from the river; negative 

cumulative monthly mean 

dissolved-solids load loss 

numbers indicate the mass 

of dissolved solids gained by 

the river.

of dissolved solids are gained by the river within Basin 1 from 

inflow from the Bill Williams River. Typically, daily dissolvedsolids loads in the Bill Williams River are 1 percent or less of the 

dissolved-solids loads in the Colorado River below Parker Dam. 

When discharge on the Bill Williams River is high, however, 

dissolved-solids loads in the Bill Williams River can be 10 percent 

or more of the dissolved-solids loads in the Colorado River below 

Parker Dam (fig. 5).

Within Basin 2, there was a loss of about 5.9 million 

tons of dissolved solids from 1990 to 2016; dissolved solids 

were not removed from the Colorado River at a constant rate. 

Dissolved solids were removed from the river from 1990 

to 2012, gained by the river from 2012 to 2015, and then 

removed from the river from 2015 through 2016 (fig. 4A). 

Within Basin 2a, approximately 6.9 million tons of dissolved 

solids were removed from the river from 1990 to 2016 at a 

relatively constant rate (fig. 4B). Within Basin 2b, however, 

approximately 1.1 million tons of dissolved solids were 

Cumulative monthly mean dissolved-solids 

load loss, in millions of tons

-10

0

10

30

50

60

20

40

Basin 1—Hoover Dam to Parker Dam

Basin 2—Parker Dam to Imperial Dam

EXPLANATION

Basin 2a—Parker Strip and Parker Valley

Basin 2b—Palo Verde Valley, Cibola Valley,

floodplain downstream of Cibola Valley

Cumulative monthly mean dissolved-solids 

load loss, in millions of tons

5

7

8

4

6

B

A

-1

0

1

2

3

-2

1990 1995 2000 2005 2010 2015

Year

EXPLANATION
Conceptual Model 13

10,000

1

10

100

1,000

0.01

0.1

Monthly mean discharge,  in cubic feet per second Dissolved-solids load, in tons

10,000

1

10

100

1,000

0.1

100,000

1990 1995 2000 2005 2010 2015

Year

Bill Williams River near Parker discharge

Colorado River below Parker Dam monthly 

 mean dissolved-solids load

EXPLANATION

Bill Williams River near Parker daily mean 

 dissolved-solids load

Figure 5. Monthly mean discharge for the Bill Williams River near Parker (site 2), monthly mean dissolved-solids load for the 

Colorado River below Parker Dam (site 3), and daily mean dissolved-solids load for the Bill Williams River near Parker (site 2), 

1990–2016.

gained by the river from 1990 to 2016 at an inconsistent rate 

(fig.4B). Within Basin 2b, dissolved solids were gained by the 

river from 1990 to 1992, removed from the river from 1992 

to 1998, neither removed from nor gained by the river from 

1998 to 2011, and gained by the river from 2012 to 2015, 

and then neither removed from nor gained by the river from 

2015 through 2016. The sharp increase in dissolved-solids 

loss from the river from 1995 to 1998 is most likely related 

to increased discharge of the river during this period (fig. 6). 

In this area, increased river discharge, such as from 1995 to 

1998, most likely led to a decreased hydraulic gradient between 

the floodplain aquifer and river, decreasing the rate at which 

agricultural return flows with relatively higher dissolved solids 

concentrations were returned to the river from irrigation. 

On average for the 1990–2016 period, Colorado River 

dissolved-solids loads were greater at Hoover Dam than at 

Imperial Dam throughout the year (fig. 7). The Colorado 

River dissolved-solids loads were greater at Parker Dam than 

at Imperial Dam during February through September, and 

loads were greater at Imperial Dam than at Parker Dam during 

October through January (fig. 7). The seasonal trend in Basin 

2 suggests that dissolved solids, which accumulate in the 

floodplain sediments and aquifer during irrigation, are released 

to the river at a greater rate during winter months when river 

discharge is low (fig. 3) and the hydraulic gradient between the 

floodplain aquifer and the river is increased. During summer 

months, river discharge in Basin 2 is higher (fig. 3) and the 

hydraulic gradient between the floodplain aquifer and the river 
14 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Figure 6. Cumulative monthly mean dissolved-solids load loss for Basin 2b and monthly mean discharge for the 

Colorado River above Imperial Dam (site 10), 1990–2016. Positive cumulative monthly mean dissolved-solids load loss 

numbers indicate the mass of dissolved solids lost from the river; negative cumulative monthly mean dissolved-solids 

load loss numbers indicate the mass of dissolved solids gained by the river.

Basin 2b—Palo Verde Valley, Cibola Valley, floodplain downstream

 of Cibola Valley cumulative dissolved-solids load loss

Colorado River at Imperial Dam discharge

Cumulative monthly mean dissolved-solids load loss, in millions of tons

Monthly mean discharge, in cubic feet per second

Year

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-1.4

1990 1995 2000 2005 2010 2015

EXPLANATION

10

3,000

6,000

9,000

12,000

15,000

18,000

21,000

24,000

27,000

Figure 7. Mean 

monthly dissolvedsolids loads for 

the Colorado River 

below Hoover Dam 

(site 1), the Colorado 

River below Parker 

Dam (site 3), and 

the Colorado River 

above Imperial Dam 

(site 10), normalized 

to the mean annual 

dissolved-solids load 

of the Colorado River 

below Hoover Dam 

(site 1), 1990–2016.

Mean monthly dissolved-solids load as a percent of mean annual 

dissolved-solids load of the Colorado River below Hoover Dam

0

20

40

60

80

100

120

140

160

Month

January February March April May June July August September October November December

Colorado River below Hoover Dam

Colorado River below Parker Dam

EXPLANATION

Colorado River above Imperial Dam
Conceptual Model 15

is decreased, decreasing the rate at which dissolved solids are 

returned to the river. 

Flow-weighted concentrations of dissolved solids in the 

Colorado River were usually higher at Parker Dam than at Hoover 

Dam within Basin 1 and were consistently higher at Imperial Dam 

than at Parker Dam within Basin 2 (fig. 8). The concentration 

Figure 8. Monthly flow-weighted dissolved-solids concentration difference between the Colorado 

River below Hoover Dam (site 1) and below Parker Dam (site 3), and between the Colorado River 

below Parker Dam (site 3) and above Imperial Dam (site 10), 1990–2016.

Figure 9. Monthly flow-weighted dissolved-solids concentration difference between the Colorado River 

below Hoover Dam (site 1) and Parker Dam (site 3), and between the Colorado River below Parker Dam (site 

3) and above Imperial Dam (site 10), plotted against monthly evapotranspiration for the Colorado River and the 

associated floodplain, 1998–2010. Evapotranspiration data furnished by the Bureau of Reclamation.

differences are not strongly related to evapotranspiration (fig.9). 

The concentration differences are also not strongly related 

to precipitation within the study area (fig.10). Infrequent 

extraordinary precipitation events, such as the events in winter 

1993 and the winter of 2005, however, do affect dissolved-solids 

concentrations in the Colorado River (figs. 8 and 10). Within 

Monthly flow-weighted concentration  difference, in milligrams per liter

350

300

250

200

150

100

50

0

1990 1995 2000 2005 2010 2015

-50

-250

-200

-150

-100

Year

Basin 1—Hoover Dam to Parker Dam

Basin 2—Parker Dam to Imperial Dam

EXPLANATION

Monthly flow-weighted concentration  difference, in milligrams per liter

Monthly evapotranspiration, in acre-feet

0

50

100

150

200

250

0 50,000 100,000 150,000 200,000 250,000

-200

-150

-100

-50

-250

Basin 1—Hoover Dam to Parker Dam, against evapotranspiration 

 between Hoover and Parker Dams

Basin 2—Parker Dam to Imperial Dam, against evapotranspiration 

 between Parker and Imperial Dams

EXPLANATION
16 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Basin 1, the precipitation events in the winters of 1993 and 

2005 produced relatively high discharges from the Bill Williams 

River to the Colorado River, which in turn transferred increased 

dissolved-solids loads to the Colorado River; the high flow 

volumes, however, contributed to decreased dissolved-solids 

concentrations in the Colorado River below Parker Dam (fig. 10). 

Within Basin 2, the precipitation event in the winter of 1993 likely 

provided a means to transport dissolved solids to the Colorado 

River that had previously accumulated on the landscape and in 

the floodplain aquifer, as indicated by the highest concentration 

differences being at times with some of the greatest monthly 

precipitation (fig. 10). 

The concentration differences between Hoover and Parker 

Dams are not strongly related to discharge at Parker Dam; the 

concentration difference between Parker and Imperial Dams, 

however, is related to discharge at Imperial Dam (fig. 11). 

Concentration differences are greater when discharge at Imperial 

Dam is low. Lower discharges result in a greater hydraulic gradient 

Figure 10. Monthly flow-weighted dissolved-solids concentration difference between the Colorado River below Hoover 

Dam (site 1) and below Parker Dam (site 3), and between the Colorado River below Parker Dam (site 3) and above Imperial 

Dam (site 10), plotted against average of monthly precipitation 1990–2016. Precipitation data from National Oceanic and 

Atmospheric Administration National Centers for Environmental Information (http://www.ncdc.noaa.gov/cdo-web/).

Monthly flow-weighted concentration difference, in milligrams per liter

0

-100

100

-200

200

-300

300

-400

400

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Average of monthly precipitation, in inches

4.0 4.5 5.0

Basin 1—Hoover Dam to Parker Dam, against precipitation 

 from the Needles, Calif., and Laughlin, Nev., stations

Basin 2—Parker Dam to Imperial Dam, against precipitation 

 from the Yuma, Ariz., Blythe, Calif., and Parker, Ariz., stations

EXPLANATION

Winter 1993 precipitation event

Winter 2005 precipitation event
Conceptual Model 17

Figure 11. Monthly flow-weighted dissolved-solids concentration difference between the Colorado River below Hoover Dam 

(site 1) and below Parker Dam (site 3), and between the Colorado River below Parker Dam (site 3) and above Imperial Dam 

(site 10), plotted against monthly mean discharge for the Colorado River, 1990–2016.

Monthly mean discharge, in cubic feet per second

Monthly flow-weighted concentration difference, in milligrams per liter

EXPLANATION

Basin 1—Hoover Dam to Parker Dam, against discharge 

 for the Colorado River below Parker Dam

Basin 2—Parker Dam to Imperial Dam, against discharge 

 for the Colorado River above Imperial Dam

Winter 1993 precipitation event

Winter 2005 precipitation event

-250

-150

-50

50

150

250

350

450

0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000

between the floodplain aquifer and the Colorado River, which 

promotes increased flow and dissolved-solids transport to the 

river. In addition, lower discharges provide less water for diluting 

dissolved-solids loads in the river and loads from irrigation return 

flows, thereby resulting in a greater concentration difference 

between Parker Dam and Imperial Dam. 

With respect to season, concentration differences between 

Hoover and Parker Dams are not significantly different 

between months (Kruskall-Wallis test p-value of 0.22; TIBCO 

Spotfire S+, 2010; fig. 12A); concentrations differences 

between Parker and Imperial Dams are, however, significantly 

different by month (Kruskall-Wallis test p-value less than 

0.05; TIBCO Spotfire S+, 2010; fig. 12B). Concentration 

differences between Parker and Imperial Dams are higher 

during the months of November-January, when the hydraulic 

gradient between the floodplain aquifer and the river is higher, 

when the river discharge is relatively low, and when relatively 

larger precipitation events tend to occur.
18 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Monthly flow-weighted concentration difference,

in milligrams per liter

-250

-200

-150

-100

-50

0

50

100

-300

A

January

February

April

May

March

August

June

July

November

December

October

September

Monthly flow-weighted concentration difference,

in milligrams per liter

50

100

150

200

250

300

350

400

0

Month

B

Basin 1—Hoover Dam to Parker Dam

Basin 2—Parker Dam to Imperial Dam

Winter 1993 precipitation event

Winter 2005 precipitation event

EXPLANATION

Figure 12. Monthly flow-weighted dissolved-solids concentration difference between (A) the Colorado River below Hoover Dam (site 1) 

and below Parker Dam (site 3), and (B) the Colorado River below Parker Dam (site 3) and above Imperial Dam (site 10), 1990–2016.
Numerical Model 19

Numerical Model 

Approach

Two numerical models were developed to simulate monthly flow-weighted dissolved-solids concentrations in the Colorado 

River above Imperial Dam. One model simulates concentrations above Imperial Dam based on the Colorado River system 

downstream from Parker Dam (Basin 2), and the other model simulates concentrations above Imperial Dam based on the Colorado 

River system downstream from Hoover Dam (Basins 1 and 2). Model users may find one model more advantageous to use than the 

other based on the availability and ease of obtaining input data for the upstream boundary.

Parker Dam to Imperial Dam Model

A numerical model was developed to simulate monthly flow-weighted dissolved-solids concentrations in the Colorado River 

above Imperial Dam. The model is structured to simulate concentrations that result from mixing of the following sources of water 

and dissolved solids in the river from Parker Dam to Imperial Dam: (1) surface water and dissolved solids that are released from 

Parker Dam and remain within the main stem of the Colorado River and travel directly to Imperial Dam; (2) surface water and 

dissolved solids that are diverted from the main stem, used for irrigation, and returned back to the main stem of the river through 

surficial drainage; (3) groundwater and dissolved solids within the floodplain aquifer, largely recharged from irrigation, that 

discharges to the river; and (4) surface water and dissolved-solids tributary inflow to the river from rainfall runoff. Mathematically, 

each source is treated as being separate without mixing until immediately above Imperial Dam, where all are combined into a 

well-mixed system. The mathematical structure of the model is based on the general solution for the mass balance equation for a 

completely mixed system (Chapra, 1997; eqs. 6 and 7): 

 

 (6)

and

 (7)

In these equations, c is concentration, t is time, and dc

dt is the rate of change in concentration with respect to time. W(t) is the 

mass of dissolved-solids loading to the system as a function of time, V(t) is the volume of water in the system as a function 

of time, and Q(t) is the discharge of the system as a function of time. The variables k and v

H are the reaction rate and settling 

rate, respectively, and can be neglected for this application because of the chemically conservative and nonparticulate nature of 

dissolved solids. Combining the above equations, simplifying, and rearranging results in the following equation:

 (8)

In equation 8, τ equals V(t)/Q(t), or the residence time of the system. Although this equation provides the overall structure 

of the concentration model, it was further adapted to accommodate the specific application of determining dissolved-solids 

concentrations at Imperial Dam. Such adaptation includes specifying the load and discharge for each major source of water and 

dissolved solids mixed upstream from Imperial Dam: 

 (9)

The subscripts for W and Q in the above equation indicate the following water sources: CR, Colorado River main stem;

RF, surficial irrigation return flows; FPA, groundwater discharge from the floodplain aquifer; and T, tributary inflow. Although 

both discharge and dissolved solids are monitored at many sites in the Parker Dam to Imperial Dam reach, measured values are 

not available for every component of equation 9. For these cases, surrogates were used and adjusted with empirical coefficients 

(table 3).

Incorporating the representation of components of the numerical model listed as in table 3 into equation 9, and adding an 

error term (e), provides the final form of the numerical model (eq. 10):

 (10)

dc

dt

+ λc(t) = W (t)

V (t)

λ = Q(t)

V (t)

+ k +

v

H

c(t) = [WCR (t) +WRF (t) +WFPA(t) +WT (t)]

[QCR (t) + QRF (t) + QFPA(t) + QT (t)]

− dc

dt

τ

CImperial = [CParker

QMain stem + c3CParkerc2QDiversions + c6c5(c4 − QMain stem ) + c9c8 (P − c7 )]

[c1QMain stem + c2QDiversions + c5(c4 − QMain stem ) + c8 (P − c7 )] − c10

dc

dt

+ c11 + e

c(t) = W (t)

Q(t)

− dc

dt

τ
20 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Model component Representation in model Description and data sources

Modeled monthly flowweighted concentration at 

Imperial Dam, c(t)

c(t) = CImperial CImperial is the observed monthly flow-weighted dissolved-solid concentration at 

Colorado River above Imperial Dam, site 10, in mg/L.

Discharge in the main stem of 

the Colorado River, QCR

QCR(t) = c1QMain stem QMain stem is the monthly discharge measured at Colorado River below Parker 

Dam, site 3, minus the sum of the monthly discharge measured at CRIR 

Main Canal, site 4, and Palo Verde Canal near Blythe, site 5, in ft3/s.

c1 is a coefficient that accounts for discharge loss caused by open water 

evaporation and riparian evapotranspiration within the Parker-Imperial 

reach. Units are dimensionless.

Load in the main stem of the 

Colorado River, WCR

WCR(t) = CParkerQMain stem CParker is the monthly flow-weighted dissolved-solid concentration at Colorado 

River below Parker Dam, site 3, in mg/L. 

QMain stem is defined above but is not adjusted for discharge loss within the 

Parker-Imperial reach like is accounted for in WCR.

Discharge of surficial irrigation 

return flows, QRF

QRF (t) = C2QDiversions QDiversions is the sum of the monthly discharge measured at CRIR Main Canal, 

site 4, and Palo Verde Canal near Blythe, site 5, in ft3/s.

c2 is a coefficient that accounts for discharge loss of irrigation water not 

returned to the Colorado River through surface drainage. Units are 

dimensionless.

Load from surficial irrigation 

return flows, WRF

WRF(t) = c3CParker QRF CParkeris the 1990–2016 average monthly flow-weighted dissolved-solid 

concentration observed for Colorado River below Parker Dam, site 3, in 

mg/L. It is equal to 627 mg/L.

QRF is defined above.

c3 is a coefficient that represents the increase of dissolved-solids 

concentrations in water diverted for irrigation from the point of diversion to 

its return to the Colorado River. Units are dimensionless.

Floodplain aquifer discharge, 

QFPA

QFPA(t) = c5(c4 ˗ QMain stem) QMain stem is defined above.

(c4 ˗ QMain stem) is a surrogate for the head potential between the floodplain 

aquifer and the Colorado River; however, discharge is used rather than 

elevation because these data are more readily available.

c4 is a coefficient representing a discharge threshold, in ft3/s, below which 

aquifer discharges to the Colorado River, and above which the Colorado 

River recharges the floodplain-aquifer. 

c5 is a coefficient that transforms the head potential described by (c4 ˗ QMain stem) 

into discharge from (or to) the floodplain aquifer. Units are dimensionless.

Load from the floodplain 

aquifer discharge, WFPA

WFPA(t) = c6QFPA QFPA is described above.

c6 is a coefficient that reflects the average dissolved-solids concentration, in 

mg/L, of the floodplain aquifer.

Discharge from tributary 

inflow, QT

QT(t) = c8 (P ˗ c7) P is the average of the monthly precipitation recorded at the Parker, Blythe, 

and Yuma stations, in inches.

c7 is a coefficient that reflects the average amount of rainfall, in inches, that occurs 

before runoff is generated. If P < c7, then (P ˗ c7) is assigned to equal zero.

c8 is a coefficient that transforms the effective amount of precipitation described by 

(P ˗ c7) into discharge from tributary runoff. Units are (ft3/s)/inch.

Load from tributary inflow, WT WT(t) = c9QT QT is described above.

c9 is a coefficient representing the average dissolved-solids concentration of 

tributary runoff, in mg/L. 

Rate of change in monthly 

flow-weighted concentration 

with respect to time, dc

dt

 is the difference in flow-weighted concentration between the month of interest 

and the previous month, at Colorado River below Parker Dam, site 3. 

The residence time of the 

system, τ

τ = c10 c10 is a coefficient that represents an estimate of the average residence time in 

months for the water in the Colorado River between Parker and Imperial Dams.

Temporal bias correction c11 c11 is a coefficient that represents an additive bias correction, in mg/L, applied 

to data prior to May 2008.

dc

dt

Table 3. Description of data sources and representation of components of the numerical model.

[mg/L, milligrams per liter; ft3/s, cubic feet per second]

dc

dt
Numerical Model 21

The use of surrogates for the components in the model as 

described in table 3 facilitates model calibration and simulation 

because only four time-varying input variables are needed—the 

monthly flow-weighted dissolved-solids concentrations at the 

Colorado River below Parker Dam; monthly discharges of the 

Colorado River below Parker Dam; combined discharges of the 

CRIR Main Canal and Palo Verde Canal; and average monthly 

precipitation recorded at Yuma, Blythe, and Parker stations. Data 

for these input variables, along with monthly flow-weighted 

concentration data for the Colorado River above Imperial Dam, 

were split into two periods to facilitate model calibration and 

verification. Data for May 2008 through September 2016 were 

used to estimate model coefficients c1–c10, because the protocols 

used to collect and process dissolved-solids samples for that period 

are comparable to those used currently (2016). Data for January 

1990 through April of 2008 were used for additional verification 

of model coefficients c1–c10, and to estimate the relative bias (c11) 

in data for the earlier period relative to the later period. It was 

anticipated that the coefficient c11 would be near but not exactly 

equal to zero because the data collection and laboratory analysis 

methods had changed between periods. 

Model coefficients c1–c10 were estimated using the solver 

function in Microsoft Excel (table 4). Using data for the 

calibration period, the solver function was set to minimize 

the RMSE of equation 10 by changing values of c1–c10 while 

following constraints listed in table 4. To ensure the solver 

function found coefficient values associated with the lowest 

RMSE rather than a local minimum, each coefficient was 

manually changed to higher or lower values and then the 

solver function was then rerun to verify that the original 

coefficient values were attained again. The value for c11

was determined through manual adjustment and using the 

Table 4. Constraints for the coefficients of the Hoover Dam to 

Imperial Dam and Parker Dam to Imperial Dam numerical models. 

[≥, greater than or equal to; >, greater than; ≤, less than or equal to; ft3/s, cubic feet 

per second; mg/L, milligrams per liter]

Coefficient Units Constraints-coefficient must be

c1 Dimensionless ≥0, ≤0.95

c2 Dimensionless ≥0, ≤1

c3 Dimensionless ≥1

c4 ft3

/s ≥7,621

c5 Dimensionless >0

c6 mg/L >0

c7 inches ≥0

c8 (ft3

/s)/inch ≥0

c9 mg/L ≥0

c10 Months ≥0

c11 mg/L No constraint

calibrated model coefficients and data for the verification 

period.

Several constraints were placed on the model coefficient 

values so that they generally made physical sense. For 

example, discharges and dissolved-solids loads from the 

Colorado River main stem, surficial irrigation return flows, 

and tributary inflows, should be nonnegative. Consequently, 

values for c1–c3 and c7–c9 were required to be greater than or 

equal to zero. The estimate of open water evaporation and 

riparian evapotranspiration between Parker Dam and Imperial 

Dam for 1998–2010 was 0.34 million acre-ft per year (data 

furnished by Reclamation), which is about 5.5 percent of the 

average annual observed discharge for QMain stem. Consequently, 

c1 was constrained to be between 0 and 0.95. Likewise, c2

was constrained to be less than one to ensure return flow 

discharge was less than diverted discharge. Coefficient c3 was 

constrained to be greater than one to represent an increase in 

dissolved-solids concentration because of irrigation. If QMain 

stem is larger than c4

, then the Colorado River main stem is 

recharging the floodplain aquifer and both QFPA and WFPA are 

negative. To ensure this is not the case for all observations, c4

was constrained to be greater than the mean discharge observed 

for the Colorado River main stem (7,621 ft3/s). Coefficient c5

was constrained to be greater than zero to ensure that floodplain 

aquifer discharge occurs when the Colorado main stem 

discharge is less than c4. Coefficient c6 was constrained to be 

greater than zero to ensure that the concentration of floodplain 

aquifer discharge is greater than zero. Coefficient c10 was 

constrained to be positive because negative values for residence 

times are nonsensical. Coefficient c11, the relative bias between 

January 1990-April 2008 and May 2008-June 2015, was 

unconstrained but expected to be near zero.
22 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Hoover Dam to Imperial Dam Model

The conceptual model for dissolved solids in the Hoover Dam to Parker Dam reach indicates that concentrations increase an 

average of 22 mg/L between Hoover Dam and Parker Dam. After large precipitation events, however, concentrations briefly decrease 

because of substantial runoff entering above Parker Dam from the Bill Williams River. Given that the 22 mg/L increase from Hoover 

Dam to Parker is a small fraction of the overall average increase of 119 mg/L from Hoover Dam to Imperial Dam, the approach for this 

second model was to make minor modifications to the model for the Parker Dam to Imperial Dam reach (eq. 10). The main adaption 

replaces the term representing the load in the main stem of the Colorado River (WCR) (eq. 9). The adaption replaces the monthly flowweighted concentration observed below Parker Dam with the monthly flow-weighted concentration observed below Hoover Dam and is 

adjusted by two factors:

 (11)

Coefficient c12, is an adjustment to the concentration below Hoover Dam so that it reflects the increase in dissolved-solids 

concentrations that is assumed to occur as a result of evapotranspiration within the Hoover Dam to Parker Dam reach (table 5). It is 

constrained to be equal to or greater than one to ensure concentrations increase in that reach. Coefficient c13, is an adjustment to the 

concentration below Hoover Dam so that it reflects the infrequent decrease that occurs when there is substantial runoff from the Bill 

Williams River (table 5). The model was constructed so that when discharge for the Bill Williams is less than 1,000 ft3/s, then the value 

for c13 is exactly 1.0 and therefore does not adjust the CHoover. When discharge for the Bill Williams is high, greater than 1,000 ft3/s, 

then the calibrated value for c13 is used. The calibrated value for c13 is constrained to be less than one so that CHoover will decrease when 

discharge is greater than 1,000 ft3/s. Incorporating equation 11 into equation 10 yields:

 C (12) Imperial = [c12c13

CHooverQMain stem + c3CParkerc2QDiversions + c6c5(c4 − QMain stem ) + c9c8 (P − c7 )]

[c1

QMain stem + c2QDiversions + c5(c4 − QMain stem ) + c8 (P − c7 )] − c10

dc

dt

+ c11 + e

Table 5. Additional constraints for Hoover Dam to Imperial Dam numerical model coefficients. 

[≥, greater than or equal to; >, greater than; ≤, less than or equal to; ft3/s, cubic feet per second]

Coefficient Units Constraints-coefficient must be

c12 Dimensionless ≥1

c13 Dimensionless ≤1 where discharge from Bill Williams River is >1,000 ft3

/s; 1.0 otherwise

Table 6. Estimated numerical model coefficients for the Parker Dam to Imperial Dam model and for the Hoover Dam to Imperial 

Dam model. 

[NA, coefficient not applicable for that model; ft3/s, cubic feet per second; mg/L, milligrams per liter. The relation of the model coefficients to 

concentrations at Imperial Dam is provided in equations 10 and 12]

Coefficient Estimated value for the Parker Dam to 

Imperial Dam model

Estimated value for the Hoover Dam to 

Imperial Dam model Units

c1 0.950 0.950 Dimensionless

c2 0.70 0.70 Dimensionless

c3 1.187 1.187 Dimensionless

c4 16,860 16,860 ft3

/s 

c5 0.088 0.088 Dimensionless

c6 1,140 1,140 mg/L

c7 0.00 0.00 inches

c8 0.00 0.00 (ft3

/s)/inch

c9 0.00 0.00 mg/L

c10 0.50 0.55 Months

c11 5 through April 2008, and 0 thereafter 20 through April 2008, and 0 thereafter mg/L

c12 NA 1.019 Dimensionless

c13 NA 0.810 Dimensionless

WCR (t) = CParkerQMain stem = c12c13CHooverQMain stem
Numerical Model 23

Results

The Parker Dam to Imperial Dam and Hoover Dam to 

Imperial Dam numerical models (eqs. 10 and 12) can be classified 

as hybrids between mechanistic models and stochastic models 

(appendix 1). Using equations 6 and 7 as the model foundations 

and constraining the coefficients to realistic values give the 

model mechanistic qualities; however, stochastic qualities in 

the models arise through the inclusion of coefficients (c1–c13) 

that are determined by using the solver function to minimize the 

RMSE. The models are not purely statistical because they were 

not calibrated with nonlinear least squares, and the imposed 

constraints result in calibrations with a slightly higher error 

than if they were entirely removed. The models are not purely 

mechanistic either, because surrogates were used for most of the 

source discharge and dissolved solids terms.

Parker Dam to Imperial Dam Model

Model coefficients c1–c10 were successfully estimated 

using data from the May 2008-September 2016 calibration 

period (table 6) for the Parker Dam to Imperial Dam model 

(eq. 10), and c11 was subsequently estimated using data 

from the January 1990-April 2008 verification period. The 

simulated monthly flow-weighted concentrations of dissolved 

solids for the Colorado River above Imperial Dam correspond 

well with observed concentrations for the entire study period 

(1990–2016; fig. 13). 

Observed monthly flow-weighted concentration for the Colorado River at Imperial Dam

Simulated monthly flow-weighted concentration for the Colorado River at Imperial Dam

Verification period, January 1990 to April 2008 Calibration period, May 2008 to September 2016

Monthly flow-weighted dissolved-solids  concentration, in milligrams per liter

500

550

600

650

700

750

800

850

900

950

1,000

1990 1995 2000 2005 2010 2015

Year

A EXPLANATION

B

Calibration data

Verification data

Simulated monthly flow-weighted dissolved-solids 

concentration, in milligrams per liter

500

600

700

800

900

1,000

Observed monthly flow-weighted dissolved-solids 

concentration, in milligrams per liter

500 600 700 800 900 1,000

EXPLANATION Figure 13. Observed and 

simulated monthly flow-weighted 

dissolved-solids concentrations for 

the Colorado River above Imperial 

Dam (site 10) from 1990 to 2016 

from the Parker Dam to Imperial 

Dam model (A) and plotted against 

one another (B).
24 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Model diagnostics and residual analysis indicate that 

the Parker Dam to Imperial Dam model performs well at 

simulating observed values for both the calibration period and 

the verification period (table 7). For the verification period, 

the RMSE for simulated concentrations was 17 mg/L, or 

about 2.4 percent of the mean flow-weighted concentration at 

Imperial Dam for this period. Using this RMSE, a 95-percent 

confidence interval for dissolved-solids concentration estimates 

is ±33 mg/L (1.96 times the RMSE). This error from the 

verification period is likely high because the model input 

data had more uncertainty for this period compared to the 

calibration period, which is visually apparent in the graph 

of the residuals plotted against time (fig. 14). The RMSE of 

the model for the calibration period May 2008-September 

2016 was lower at 9 mg/L, or about 1.3 percent of the mean 

flow-weighted concentration at Imperial Dam for this period. 

Assuming that data collection processes and laboratory methods 

continue to be similar to those of the calibration period, a 

95-percent confidence interval for simulated dissolved-solids 

concentrations is ±18 mg/L. The model RMSE for both periods 

is about one- third of the standard deviation of the observed 

concentrations, indicating that the model improves dissolvedsolids concentration estimates about three times over simply 

using the mean of the observed values. For the calibration 

period, the model RMSE was generally higher for the months of 

September through February than for March, April, and May. In 

the verification period, however, the model RMSE was highest 

in February, March, April, and August (table 8). 

The R-squared value is another measure of performance 

estimation, and for the calibration period it indicates that 93 

percent (and 90 percent for the verification period) of the 

variance in the observed concentrations was explained by 

the model. The mean residual was 0 mg/L for the calibration 

period, which indicates that the model is unbiased towards 

over- or under-estimating concentrations. The coefficient c11

forces the mean residual of the verification period to equal 

zero, and its value of 5 mg/L indicates that only a small 

adjustment (less than 1 percent) is needed to result in unbiased 

concentration estimates. This small bias suggests that the 

model performs well under a wide variety of conditions, not 

just those observed during the calibration period, but for the 

overall study period, and that the model will likely perform 

well for simulating future dissolved-solid concentrations 

where environmental conditions are similar to those occurring 

between 1990 and 2015.

The calibrated value of 16,860 ft3/s for c4, the discharge 

threshold for the floodplain aquifer between Parker Dam 

and Imperial Dam, is greater than the maximum observed 

monthly flow observed in the main stem of the Colorado River 

(16,815 ft3/s). Consequently, the model always predicts occurrence 

of a monthly net discharge (rather than recharge) from the 

floodplain aquifer to the Colorado River, with larger amounts 

delivered to the Colorado when flows in the river are lower. 

This suggests that recharge of the floodplain aquifer is largely 

from nonriver sources, such as irrigation seepage and regional 

aquifer discharge, and that aquifer discharge is delivered to the 

Figure 14. Residual error for the Parker Dam to Imperial Dam model, 1990–2016. Positive numbers indicate that the model 

simulation underestimated flow-weighted concentrations for the Colorado River at Imperial Dam.

Residual error, in 

milligrams per liter

Verification period, January 1990 - April 2008 Calibration period, May 2008 - September 2016

-100

-80

-60

-40

-20

0

20

40

60

1990 1995 2000 2005 2010 2015

Year

80

-120
Numerical Model 25

Table 7. Model diagnostics for the calibration and verification periods for the Parker Dam to Imperial Dam model and the Hoover Dam 

to Imperial Dam model.

Parker Dam to Imperial Dam model Hoover Dam to Imperial Dam model

Milligrams 

per liter

Percent of mean concentration 

at Imperial Dam

Milligrams 

per liter

Percent of mean concentration 

at Imperial Dam

Model calibration period: May 2008-September 2016

Standard deviation of observations 34 4.8 34 4.8

Root-mean squared error of model simulations 9 1.3 11 1.6

Mean residual 0 0 0 0

R-squared 0.93 0.90

Model verification period: January 1990-April 2008

Standard deviation of observations 56 7.7 56 7.7

Root-mean squared error of model simulations 17 2.4 22 3.1

Mean residual 0 0 0 0

R-squared 0.90 0.84

Table 8. Root-mean squared error of model simulations by month for the calibration and verification periods of the Parker Dam to 

Imperial Dam model and the Hoover Dam to Imperial Dam model. 

Month

Root-mean squared error, Parker to Imperial Dam model Root-mean squared error, Hoover Dam to Imperial Dam model

Milligrams per liter Percent of mean concentration 

at Imperial Dam Milligrams per liter Percent of mean concentration at Imperial Dam

Model calibration period: May 2008-September 2016

January 10 1.4 12 1.6

February 10 1.4 11 1.5

March 7 1.0 7 1.0

April 6 0.8 9 1.2

May 5 0.7 6 0.9

June 8 1.2 11 1.6

July 9 1.3 11 1.6

August 9 1.3 12 1.7

September 12 1.6 14 2.0

October 9 1.3 11 1.6

November 11 1.5 12 1.7

December 13 1.9 13 1.9

Model verification period: January 1990-April 2008

January 14 1.9 27 3.7

February 18 2.5 33 4.5

March 32 4.4 45 6.2

April 24 3.2 19 2.6

May 10 1.4 12 1.7

June 13 1.7 13 1.8

July 13 1.8 12 1.7

August 19 2.7 20 2.8

September 15 2.1 18 2.5

October 8 1.1 11 1.5

November 12 1.7 11 1.5

December 13 1.8 16 2.1
26 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Colorado River. The value for c6 indicates the mean dissolvedsolids concentration of the floodplain aquifer discharge to the 

river is about 1,140 mg/L. Dissolved-solids data for the floodplain 

aquifer within the study area are not available in the USGS NWIS 

database. Goldrath and others (2010), however, reported that in the 

Palo Verde Valley, dissolved-solids concentrations in the regional 

aquifer, which underlies the floodplain aquifer, ranged from 637 to 

2,890 mg/L.

The model was not improved with nonzero values for 

c8 and c9—the model coefficients representing the effects of 

tributary runoff. This implies that the effects of rainfall runoff 

within the study area are negligible at the monthly time step, 

which is consistent with Owen-Joyce and Raymond (1996) 

who found that precipitation and tributary inflow is a little 

less than 1 percent of the inflow to the Parker Dam to Imperial 

Dam reach. Such lack of sensitivity to runoff is unlikely to 

occur in other parts of the Colorado River Basin that have 

wetter climates. 

The largest magnitude residual was -110 mg/L, occurring 

in March of 2005, and indicates substantial overestimation 

of concentration (fig. 14). Although there were 2.76 inches 

of average precipitation at the Yuma, Ariz., Blythe, Calif., 

and Parker, Ariz. stations during February 2005, the large 

magnitude of this residual likely resulted from substantial 

changes in dissolved-solids concentrations of reservoir 

releases from Parker Dam. Monthly flow-weighted 

concentrations in the Colorado River below Parker Dam 

were 620 mg/L in February, decreased to 509 mg/L in March, 

and increased to 668 mg/L in April (fig. 11). The decrease 

in concentration was largely a result of substantial reservoir 

inflows of relatively low-concentration runoff from the Bill 

Williams River. Although the model performed better at 

handling a similar decrease in dissolved-solids concentrations 

during the spring of 1993 (figs. 13 and 14), the large March 

2005 residual is a good indication that the model has difficulty 

with uncommon but substantial concentration changes in 

water released from Parker Dam. 

Possibly related to the large March 2005 residual is that, 

ideally, the rate of change in dissolved-solids concentration 

with respect to time ( dc

dt

) should be represented by conditions 

observed above Imperial Dam where water and solutes from 

all sources are well mixed and change with time. In the model, 

however, dc

dt is represented by conditions observed below 

Parker Dam for two reasons. The first is that using data from 

below Parker Dam will greatly facilitate estimating values 

of dc

dt for use in running model scenarios. Second, when 

data from below Parker Dam were used to represent dc

dt , the 

RMSE decreased about 1 mg/L compared to when dc

dt

 was not 

included in the model. When data at Imperial Dam were used, 

however, the RMSE increased compared to when dc

dt was not 

included in the model. The time step of one month may be 

too long for the model to perform well with highly dynamic 

concentration changes, and a shorter time step such as one 

week would result in better correspondence between observed 

and simulated concentrations above Imperial Dam.

Hoover Dam to Imperial Dam Model

For the Hoover Dam to Imperial Dam model, coefficients 

c1–c9 largely represent process below Parker Dam. For that reason, 

estimates of model coefficients c1–c9 for the Parker Dam to 

Imperial Dam model (table 6) were also used as coefficients for 

the Hoover Dam to Imperial Dam model (eq. 12). To complete 

calibration of the Hoover Dam to Imperial Dam model, coefficient 

estimates were determined as follows. Residence time, c10, 

and the Hoover Dam to Parker Dam concentration factor, c12, 

were estimated using data from the May 2008-September 2016 

calibration period. The highest inflows from the Bill Williams 

River occurred prior to 2008 and so the Bill Williams dilution 

factor, c13, was determined using data from the January 1990-April 

2008 verification period. The 1990–2008 bias correction factor 

was determined using data from the verification period as well. 

The simulated monthly flow-weighted concentrations of dissolved 

solids for the Colorado River above Imperial Dam correspond 

well with observed concentrations for the entire study period 

(1990–2016; fig. 15).

The Hoover Dam to Imperial Dam model diagnostics and 

residual analysis indicate that this model performs well at simulating 

observed values for both the calibration period and the verification 

period (table 7). For the verification period, the RMSE for simulated concentrations was 22 mg/L, or about 3.1 percent of the mean 

flow-weighted concentration at Imperial Dam for this period. Using 

this RMSE, a 95-percent confidence interval for dissolved-solids 

concentration estimates is ±43 mg/L (1.96 times the RMSE). Similar 

to the Parker Dam to Imperial Dam model, this error from the 

verification period is likely high because the model input data had 

more uncertainty for this period compared to the calibration period 

(fig. 16). The RMSE of the model for the calibration period May 

2008-September 2016 was lower than for the verification period, at 

11 mg/L, or about 1.6 percent of the mean flow-weighted concentration at Imperial Dam for this period. Assuming that data collection 

processes and laboratory methods continue to be similar to those of 

the calibration period, a 95-percent confidence interval for simulated 

dissolved-solids concentrations is ±22 mg/L. For the calibration 

period, the model RMSE was generally highest for the months of 

August through January, and lowest for the months from March 

through May. In the verification period, however, the model RMSE 

was highest in the months of January through March (table 8). 

The R-squared value for the calibration period indicates that 

90 percent (and 84 percent for the verification period) of the variance in the observed concentrations was explained by the model. 

The mean residual was 0 mg/L for the both periods, which indicates that the model is unbiased towards over- or under-estimating 

concentrations. Similar to the Parker Dam to Imperial Dam model, 

the magnitude of the residuals for the Hoover Dam to Imperial 

Dam model were greatest during the 1993 and 2005 precipitation 

events when the Bill Williams River discharge was large, composing a substantial portion of the total discharge below Parker Dam. 

This confirms that both models have difficulty producing accurate 

simulations when infrequent but highly dynamic concentration 

changes occur in water released from Parker Dam. 
Numerical Model 27

Observed monthly flow-weighted concentration for the Colorado River at Imperial Dam

Simulated monthly flow-weighted concentration for the Colorado River at Imperial Dam

Verification period, January 1990 - April 2008 Calibration period, May 2008 - September 2016

A

Monthly flow-weighted dissolved-solids concentration, in milligrams per liter

500

550

600

650

700

750

800

850

900

950

1990 1995 2000 2005 2010 2015

Year

EXPLANATION

B

Calibration data

Verification data

Simulated monthly flow-weighted dissolved-solids 

concentration, in milligrams per liter

500

600

700

800

900

1,000

Observed monthly flow-weighted dissolved-solids 

concentration, in milligrams per liter

500 600 700 800 900 1,000

EXPLANATION

Figure 15. Observed and simulated 

monthly flow-weighted dissolvedsolids concentrations for the Colorado 

River above Imperial Dam (site 10) 

from 1990 to 2016 from the Hoover 

Dam to Imperial Dam model (A) and 

plotted against one another (B).
28 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Interpretations of model coefficients c1-c9 made for the 

Parker Dam to Imperial Dam model also apply to the Hoover 

Dam to Imperial Dam model, and the values for coefficients 

c10-c13 provide additional hydrologic insight. The value 

for coefficient c10, residence time, is slightly larger for the 

Hoover Dam to Imperial Dam model than the Parker Dam 

to Imperial Dam model, which is expected as that reach is 

longer and residence time therefore should be longer. The 

value for coefficient c11 (20 mg/L, or 2.8 percent of the 

average concentration at Imperial Dam) is 15 mg/L greater 

than that for the Parker Dam to Imperial Dam model, which 

indicates a greater amount of bias for this model that occurs 

between the verification period and the calibration period. If 

the gain in concentration between Hoover Dam and Parker 

Dam is attributed largely to evapotranspiration, then the value 

for coefficient c12 (1.019) suggests that the average rate of 

open water evaporation and riparian evapotranspiration is 

about 1.9 percent of the average flow in that reach. This is a 

slight underestimation compared to Reclamation’s estimate 

that average annual open water evaporation and riparian 

evapotranspiration loss in this reach from 1998–2010 was 

4.0 percent of the average flow below Hoover Dam (data 

furnished by Reclamation). The value for coefficient c13

(0.81) indicates that the concentration in the main stem of the 

Colorado River below Parker Dam is reduced about 19 percent 

when there are high flows entering from the Bill Williams 

Figure 16. Residual error for the Hoover Dam to Imperial Dam model, 1990–2016. Positive numbers indicate that the 

model simulation underestimated flow-weighted concentrations for the Colorado River at Imperial Dam.

Residual error, in 

milligrams per liter

Verification period, January 1990 - April 2008 Calibration period, May 2008 - September 2016

-160

-80

-40

0

40

80

1990 1995 2000 2005 2010 2015

Year

120

-120

River. The 1993 high flow event in the Bill Williams River 

reduced the dissolved solids concentration in the Colorado 

River below Parker Dam by about 14 percent.

Comparison of Root-mean Square Error and 

Measurement Error

 A coarse-scale error analysis indicates that measurement 

error of the model input data is small, but composes a 

substantial component of the RMSE for the Parker Dam to 

Imperial Dam and the Hoover Dam to Imperial Dam models. 

An approximation of the measurement error included in the 

RMSE for the Parker Dam to Imperial Dam model can be 

estimated as the square root of the sum of squared percent 

errors for each measured term in equation 10, assuming such 

errors are not correlated. For the Parker Dam to Imperial Dam 

model, measurement data come from two dissolved-solids 

monitoring stations and from three discharge monitoring 

stations. For this analysis, measurement error of monthly 

dissolved-solids concentrations is assumed to be about equal 

to the previously noted sample concentration variability of 

about 0.5 percent for the Colorado River above Imperial 

Dam. This error neglects the decrease in uncertainty that 

results from using multiple samples to determine the monthly 

concentration estimate, the increase in uncertainty that results 

from using simple mathematical procedures for estimating 
Numerical Model 29

monthly concentrations from the sample concentrations, and the increase in uncertainty that results from bias associated with 

the sampling and analysis of the dissolved-solids data that is not included in the sample variability estimate. For the monthly 

discharge data, it is assumed that the measurement error for each station is about 0.40 percent. This estimate is based on the 

median estimated uncertainty for annual discharge at Colorado River below Parker Dam, 1995–99 (Anning, 2002). The actual 

measurement error for monthly discharges is likely greater because the time period of one month is shorter than one year, 

but this will be neglected given the coarse nature of this analysis. Given these estimates for error in individual measurements 

and combining them in quadrature, the resultant coarse estimate of measurement error contained in the model input data is 

(0.52+0.52+0.42+0.42+0.42)0.5=0.9 percent. Consequently, measurement error within the model input data accounts for a 

substantial portion of the resulting RMSE for the Parker Dam to Imperial Dam model (0.9 of the 1.3 percent for the calibration 

period, and 0.9 of the 2.4 percent for the verification period). Neglecting the contribution of error from the Bill Williams River 

measurements, the coarse estimate of measurement error for the Hoover Dam to Imperial Dam is the same as that for the Parker 

Dam to Imperial Dam model, 0.9 percent. 

Model Sensitivity and Insight for Running Model Scenarios 

The use of the Parker Dam to Imperial Dam model may be preferred because the RMSE of this model is 2 mg/L lower 

than the Hoover Dam to Imperial model, assuming equal ability to obtain model input data. If model input data are more readily 

available, or accurate, for the Hoover Dam to Imperial Dam model, then it is likely worth the small increase in uncertainty for using 

this model over the Parker to Imperial model. Use of either model (eqs. 10 and 12) requires the simulation analyst to assemble input 

datasets (appendix 2). In order to put an appropriate amount of resources into assembling input variable data, the analyst needs to 

understand the sensitivity of the simulated concentrations at Imperial Dam to each input variable. Sensitivity for the Parker Dam 

to Imperial Dam model was assessed by examining the changes in simulated concentrations that result from changing the input 

data across the distribution of their observed values for the study period. Equation 10 describes the model in a form that facilitates 

understanding how concentrations at Imperial Dam are a function of the flows and loads of four water and solute sources. For use in 

simulating concentrations for various scenarios, however, the equation can be simplified by replacing the terms with the calibrated 

model coefficients (table 6) and measured input variables based on relations described in equation 10: 

 (13)

In this simplification, c11 is zero and therefore equation 13 is valid for simulating concentrations after April 2008. 

Equation 13 is not only more convenient than equation 10 for simulating concentrations of future scenarios, but it also benefits 

the sensitivity analysis in that it reduces dependence between variables. In equation 10 there are four input variables; however, these 

are various combinations of five observed values measured in three locations (dissolved-solids concentrations for the current month 

and for the previous month below Parker Dam, discharge below Parker Dam, and discharge diverted into two different canals; 

table 3). Because the model input variables are combinations, they are not completely independent. In addition, there could be 

correlations between some variables, such as discharge below Parker Dam and discharge for diversions into the canals.

For the sensitivity analysis, three of the four input variables (CParker, QParker, QDiversions, and dc

dt of equation 13 were held constant 

at their median value, and dissolved-solids concentrations were simulated while values for the fourth variable was varied from its 

minimum value, 5th percentile, 10th percentile, 25th percentile, 50th percentile, 75th percentile, 90th percentile, 95th percentile, 

and maximum value (table 9). Examining the difference in concentration simulations for the 95th percentile and the 5th percentile 

of input data, the largest magnitude changes in concentration are -121 mg/L for the monthly discharge at the Colorado River 

below Parker Dam, and 114 mg/L for the monthly flow-weighted concentration at the Colorado River below Parker Dam. The 

difference in simulations for the 95th percentile and the 5th percentile of rate of change in concentration data with respect to time, 

and of combined discharge at CRIR Main Canal and Palo Verde Main Canal, are considerably smaller at -12 mg/L and 27 mg/L, 

respectively. 

The sensitivity analysis was repeated but for the Hoover Dam to Imperial Dam model (table 10). Given the similar mathematical structure for the two models, results are very similar. For use in simulating concentrations for various scenarios the 

model can be simplified by replacing the terms with the calibrated model coefficients (table 6) and measured input variables 

based on relations described in equation 12: 

 (14)

In this simplification, c11 is zero and therefore equation 14 is valid for simulating concentrations after April 2008. In addition, the 

model user must supply the value for c13 as equal to 0.81 if discharge for the Bill Williams River exceeds 1,000 ft3/s, or equal to 

1.0 otherwise. 

CImperial = [(CParker −100.32)QParker + (621.29 − CParker )QDiversions +1,691,395]

[(0.862)QParker − (0.162)QDiversions +1,483.68] − 0.50 dc

dt

CImperial = [(1.019C13

CHoover −100.32)QParker + (621.29 −1.019C13CHoover )QDiversions +1,691,395]

[(0.862)QParker − (0.162)QDiversions +1,483.68] − 0.50 dc

dt
30 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

The simulated concentrations using the Hoover Dam 

to Imperial Dam model are most sensitive to monthly flowweighted concentrations at Colorado River below Hoover 

Dam and monthly discharge at Colorado River below Parker 

Dam, and least sensitive to change in concentration data with 

respect to time and combined discharge data at CRIR Main 

Canal and Palo Verde Main Canal (table 10). The model is 

also sensitive to discharge in the Bill Williams River; when 

other input variables have median values and discharge 

exceeds 1,000 ft3/s, the resulting concentration is 97 mg/L less 

than when flow is less than 1,000 ft3/s. 

The calibrated models can be used to run scenarios of 

monthly flow-weighted dissolved-solids concentrations at 

Imperial Dam. Model input data need to be developed in consideration of the scenario objectives and could be developed 

through analysis of recent values and trends for the input data, 

or they might be developed on the basis of historical high, low, 

or typical conditions observed during the study period (table 

11). Although the models should provide concentration estimates within 18 mg/L (Parker Dam to Imperial Dam model) to 

22 mg/L (Hoover Dam to Imperial Dam model), 95 percent of 

the time, the error of future scenario simulations will increase 

as uncertainty in estimated future input variables increases. 

The sensitivity analysis indicates that when the model is used 

to estimate future concentrations, the analysts should be most 

careful determining the input values for monthly discharge 

at Colorado River below Parker Dam and flow-weighted 

dissolved-solids concentrations at the Colorado River below 

Hoover Dam or Colorado River below Parker Dam. If salinity 

Table 9. Sensitivity of simulated concentrations from the Parker Dam to Imperial Dam model (eq. 13) to the four measured model input 

variables. 

Monthly flow-weighted dissolved-solids concentration simulated for Colorado River above Imperial Dam when listed variable is at the 

listed percentile and the other three variables are at their median value (mg/L)

Percentile

Monthly flow-weighted 

dissolved-solids 

concentration, Colorado 

River below Parker Dam 

(site 3), CParker

Monthly discharge, 

Colorado River below 

Parker Dam (site 3), 

QParker

Combined monthly 

discharge, CRIR Main 

Canal and Palo Verde 

Canal (sites 4 and 5),

QDiversions

Rate of change in dissolvedsolids concentration with 

respect to time at Colorado 

River below Parker Dam 

(site 3), dc

dt

Minimum 561 997 685 781

5 658 802 693 714

10 664 777 695 712

25 681 740 699 710

50 (median) 707 707 707 707

75 746 691 715 705

90 766 682 718 703

95 772 680 720 702

Maximum 783 666 724 603

Difference in simulated 

dissolved-solids concentration between 95th and 

5th percentile, mg/L

114 -121 27 -12

[CRIR, Colorado River Indian Reservation; mg/L, milligrams per liter]

input values from either Colorado River below Hoover Dam 

or Colorado River below Parker Dam are readily available 

from specific conductance data (mS/cm), then the analyst 

can multiply them by 0.616 to convert them into dissolvedsolids concentrations in mg/L (see Data Compilation section 

for more details) for use in the models. Analysts should also 

ensure that input data are greater than the minimum values 

and less than the maximum values of the input data used to 

calibrate the model (table 11). Model scenario results generated using input data that have values outside the range used 

to calibrate the model have unknown uncertainties associated 

with them.

When confidence intervals for concentrations are needed 

for scenarios, the following approach is suggested (using the 

Parker Dam to Imperial Dam model as an example). For the 

month of interest, determine the largest and smallest likely 

values for each of the four input variables such that they form 

confidence limits for each variable. For estimating the highest 

likely dissolved-solids concentration for model input, use the 

highest likely values for monthly flow-weighted concentration 

of the Colorado River below Parker Dam and combined 

monthly discharge of CRIR Main and Palo Verde Canals, 

and use the lowest likely values for monthly discharge and 

rate of change in concentration with respect to time of the 

Colorado River below Parker Dam. Then add 18 mg/L to the 

estimated value to account for model error, and the result is 

an estimate of highest likely concentration that might occur. 

For estimating the lowest likely concentration, use the lowest 

likely values for monthly flow-weighted concentration of the 
Numerical Model 31

Table 10. Sensitivity of simulated concentrations from the Hoover Dam to Imperial Dam numerical model (eq. 14) to the four measured 

model input variables. 

[CRIR, Colorado River Indian Reservation; mg/L, milligrams per liter]

Monthly flow-weighted dissolved-solids concentration simulated for Colorado River above Imperial Dam when listed variable is at the 

listed percentile and the other three variables are at their median value (mg/L)

Percentile

Monthly flow-weighted 

dissolved-solids 

concentration, Colorado 

River below Hoover Dam 

(site 1), CHoover

Monthly discharge, 

Colorado River below 

Parker Dam (site 3), 

QParker

Combined monthly 

discharge, CRIR Main 

Canal and Palo Verde 

Canal (sites 4 and 5), 

QDiversions

Rate of change in dissolvedsolids concentration with 

respect to time at Colorado 

River below Hoover Dam 

(site 1), dc

dt

Minimum 568 995 685 781

5 667 807 693 714

10 673 784 695 712

25 690 748 699 710

50 (median) 717 717 707 707

75 756 702 715 705

90 777 693 718 703

95 783 691 720 702

Maximum 794 678 724 603

Difference in simulated 

dissolved-solids concentration between 95th and 

5th percentile, mg/L

117 -116 25 -13

Table 11. Statistical distribution of the four measured model input variables and of the observed monthly flow-weighted dissolvedsolids concentration at Imperial Dam (site 10), 1990–2016. 

Percentile

Monthly flowweighted 

dissolved-solids 

concentration, 

Colorado River 

below Hoover 

Dam (site 1), 

CHoover, mg/L

Monthly 

flow-weighted 

dissolved-solids 

concentration, 

Colorado River 

below Parker 

Dam (site 3), 

CParker, mg/L

Monthly 

discharge, 

Colorado 

River below 

Parker Dam 

(site 3), 

QParker, ft3/s

Combined 

monthly 

discharge, CRIR 

Main Canal and 

Palo Verde Canal 

(sites 4 and 5), 

QDiversions, ft3/s

Rate of change in 

dissolved-solids 

concentration with 

respect to time 

at Colorado River 

below Parker Dam 

(site 3), dc

dt , mg/L 

per month

Colorado River at 

Imperial Dam monthly 

flow-weighted 

dissolved-solids 

concentration, 

Colorado River at 

Imperial Dam (site 10), 

CImperial, mg/L

Minimum 531 440 1,639 119 -147 522

5 543 560 4,583 841 -13 652

10 550 568 5,420 1,024 -8 664

25 569 588 7,292 1,369 -4 684

50 (median) 605 622 10,004 2,122 0 722

75 640 669 12,112 2,770 4 759

90 664 694 13,732 3,031 8 787

95 671 702 14,103 3,173 10 809

Maximum 684 715 17,564 3,508 209 928

[CRIR, Colorado River Indian Reservation; mg/L, milligrams per liter; ft3/s, cubic feet per second]
32 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Colorado River below Parker Dam and combined monthly 

discharge of CRIR Main and Palo Verde Canals, and use the 

highest likely values for monthly discharge and rate of change 

in concentration with respect to time of the Colorado River 

below Parker Dam. Then subtract 18 mg/L from the estimated 

value to attain the lowest likely concentration that might 

occur. A similar approach is suggested for the Hoover Dam to 

Imperial Dam model, with the exception that the uncertainty is 

22 mg/L, and one would use concentrations and concentration 

changes over time from Colorado River below Hoover Dam 

rather than Colorado River below Parker Dam.

On a final note, any major physical changes made within 

the contributing area between Hoover Dam and Imperial Dam, 

especially in the floodplain or the channel of the Colorado 

River, could alter the source and transport characteristics 

of dissolved solids within the reach and may require model 

recalibration. Such changes could include new diversions, 

improvement of drainage systems, substantial channel 

alterations, or widespread changes in agricultural practices. As 

monitoring of model input data continues, the model calibration 

can be verified by entering those data into the model and 

assessing how well the observed and predicted monthly flowweighted concentrations at Imperial Dam correspond over time. 

Consistent and significant deviations of residuals from zero may 

indicate the need to recalibrate the model.

Summary and Conclusions

The objective of this study was to provide the Bureau 

of Reclamation with the capability to simulate salinity (as 

dissolved solids) of the Colorado River at Imperial Dam. 

The ability to simulate dissolved-solids concentrations at 

this location will aid Reclamation in meeting the binational 

agreement on the volume and salinity of Colorado River water 

delivered to Mexico. A robust spatial- and temporal-resolution 

dataset was generated that consists of river discharge and 

dissolved-solids concentration and load information between 

January 1990 and September 2016 for 10 sites on canals, 

drains, tributaries, and the main stem of the Colorado River 

between Hoover and Imperial Dams. Daily mean dissolvedsolids concentrations were estimated, and monthly mean 

dissolved-solids loads were computed for each site. Spatial 

and temporal load patterns, and historical and current controls 

on loads and concentrations, were analyzed in order to develop 

a conceptual model of dissolved-solids transport between 

Hoover and Imperial Dams. Two numerical models describing 

the relations between dissolved-solids concentrations and 

components controlling dissolved-solids concentration loads 

were developed, calibrated, and verified.

Between January 1990 and September 2016, there was a 

98.8-million-acre-ft loss of water and a 57.0-million-ton loss 

of dissolved-solids load from the Colorado River between 

Hoover and Imperial Dams. Between Hoover and Parker 

Dams, about 69.0 million acre-ft of water was lost and 

51.1 million tons of dissolved solids were lost; between 

Parker and Imperial Dams, about 29.8 million acre-ft of water 

was lost and 5.9 million tons of dissolved solids were lost. 

Water was removed from the river at a relatively consistent 

rate over the 25-year study period through water transfers 

to California and Arizona, evapotranspiration from crop 

irrigation, transpiration processes of riparian vegetation, and 

evaporation from the river main stem. Dissolved solids were 

removed from the river between Hoover and Parker Dams at 

a relatively constant rate through water transfers to California 

and Arizona, and water pumped from the river for irrigation 

within the Mohave Valley. The river gains a small amount of 

dissolved solids from inflow from the Bill Williams River. 

Between Parker and Imperial Dams, however, dissolved 

solids were not removed from the river at a consistent rate 

over the study period. Dissolved solids were generally 

removed from the river from 1990 to 2012, then gained by 

the river from 2012 to 2015, and then removed from the 

river from 2015 through 2016. Dissolved solids are assumed 

to be removed from the river and accumulated within the 

floodplain sediments and aquifers during irrigation processes; 

some dissolved solids may also be removed from the river 

through uptake by crops and riparian vegetation. Dissolved 

solids accumulated on the landscape and in the floodplain 

aquifer during irrigation are transported to the river, causing a 

gain in dissolved solids in the river, during periods when the 

hydraulic gradient between the floodplain aquifer and the river 

is increased. Dissolved-solids gains in the river occur during 

periods of relatively low river discharge, such as during the 

winter months and during drier climatic conditions.

Two numerical models were developed to simulate 

monthly flow-weighted dissolved-solids concentrations in 

the Colorado River at Imperial Dam; one for the Parker Dam 

to Imperial Dam reach, and another for the Hoover Dam 

to Imperial Dam reach. Model coefficients of both models 

were estimated using data from the May 2008-September 

2016 calibration period, and the simulated concentrations of 

dissolved solids correspond well with observed concentrations 

for the entire study period (1990–2016). The models are more 

sensitive to monthly discharge at the Colorado River below 

Parker Dam and monthly flow-weighted dissolved-solid 

concentrations of the Colorado River below Parker Dam (or 

Hoover Dam) than to the rate of change in concentration with 

respect to time and the combined discharge of the CRIR Main 

Canal and Palo Verde Canal. The calibrated models can be 

used to run scenarios of monthly flow-weighted dissolvedsolids concentrations at Imperial Dam. The models should 

provide concentration estimates within 18 mg/L (Parker Dam 

to Imperial Dam model) to 22 mg/L (Hoover Dam to Imperial 

Dam model) 95 percent of the time, but future scenario error 

will increase as uncertainty in the estimated future input 

variables increases.
References Cited 33

References Cited

Anning, D.W., 2002, Standard errors of annual discharge and 

change in reservoir content data from selected stations in the 

lower Colorado River streamflow-gaging station network, 1995–

99: U.S. Geological Survey Water-Resources Investigations 

Report 01-4240, 81 p.

Bureau of Reclamation, 2014, Estimates of evapotranspiration and 

evaporation along the Lower Colorado River, calendar year 2010: 

Bureau of Reclamation, 106 p. [Available at http://www.usbr.gov/

lc/region/g4000/4200Rpts/LCRASRpt/2010/report10.pdf.]

Chapra, S.C., 1997, Surface water-quality modeling: Long Grove, 

Ill., McGraw-Hill, 844 p.

Goldrath, D.A., Wright, M.T., and Belitz, K., 2010, Groundwaterquality data in the Colorado River study unit, 2007—Results 

from the California GAMA program: U.S. Geological Survey 

Data Series 474, 66 p., accessed April 20, 2017, at http://pubs.

usgs.gov/ds/474/.

Hem, J.D., 1992, Study and interpretation of the chemical 

characteristics of natural water: U.S. Geological Survey Water 

Supply Paper 2254, 263 p. [Available at http://pubs.usgs.gov/

wsp/wsp2254/pdf/wsp2254a.pdf.]

Hirsch, R.M., and De Cicco, L.A., 2015, User guide to Exploration 

and Graphics for RivEr Trends (EGRET) and dataRetrieval—R 

packages for hydrologic data (ver. 2.0, February 2015): U.S. 

Geological Survey Techniques and Methods, book 4, chap. A10, 

93 p. [Available at http://dx.doi.org/10.3133/tm4A10.]

Hirsch, R.M., Moyer, D.L., and Archfield, S.A., 2010, Weighted 

Regressions on Time, Discharge, and Season (WRTDS), with 

an application to Chesapeake Bay River inputs: Journal of the 

American Water Resources Association, v. 46, no. 5, p. 857–880. 

[Available at http://onlinelibrary.wiley.com/doi/10.1111/j.1752-

1688.2010.00482.x/full.]

International Boundary and Water Commission, 1973, Permanent 

and definitive solution to the international problem of the salinity 

of the Colorado River: International Boundary and Water 

Commission, Minute no. 242, August 30, 1973. [Available at 

http://www.ibwc.gov/Files/Minutes/Min242.pdf.]

Leake, S.A, Owen-Joyce, S.J., and Heilman, J.A., 2013, 

Potential depletion of surface water in the Colorado 

River and agricultural drains by groundwater pumping 

in the Parker-Palo Verde-Cibola area, Arizona 

and California: U.S. Geological Survey Scientific 

Investigations Report 2013–5134, 13 p. [Available at 

https://pubs.er.usgs.gov/publication/sir20135134.]

Mueller, D.K., Schertz, T.L., Martin, J.D., and 

Sandstrom, M.W., 2015, Design, analysis, and 

interpretation of field quality-control data for watersampling projects: U.S. Geological Survey Techniques 

and Methods, book 4, chap. C4, 54 p. [Available at 

http://dx.doi.org/10.3133/tm4C4.]

Owen-Joyce, S.J., and Raymond, L.H., 1996, An 

accounting system for water and consumptive use 

along the Colorado River, Hoover Dam to Mexico: 

U.S. Geological Survey Water-Supply Paper 2407, 94 

p. [Available at https://pubs.er.usgs.gov/publication/

wsp2407.]

R Development Core Team, 2011, R—A language and 

environment for statistical computing: R Foundation 

for Statistical Computing, Vienna, Austria. [Available 

at http://www.R-project.org/.]

TIBCO Spotfire S+, 2010, TIBCO Spotfire S+ 8.2 for 

Windows User’s Guide: TIBCO Software Inc., 568 p., 

accessed April 13, 2017, at https://docs.tibco.com/pub/

spotfire_s_plus/8.2.0_november_2010/pdf/TIB_sf_

s+_8.2.0_win_user_guide.pdf.

U.S. Geological Survey, 2017, National Water 

Information System―Web interface: U.S. Geological 

Survey web page, accessed April 13, 2017, at http://

dx.doi.org/10.5066/F7P55KJN.

Wilson, R.P., and Owen-Joyce, S.J., 1994, Method to 

identify wells that yield water that will be replaced by 

Colorado River water in Arizona, California, Nevada, 

and Utah: U.S. Geological Survey Water-Resources 

Report 94-4005, 19 map sheets, 36 p. [Available at 

https://pubs.er.usgs.gov/publication/wri944005.]
34 Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada

Appendixes

Appendix 1. Parker Dam to Imperial Dam and Hoover Dam to Imperial Dam numerical model of dissolved-solids concentrations for the 

Colorado River at Imperial Dam.

Appendix 2. Parker Dam to Imperial Dam numerical model simulation and Hoover Dam to Imperial Dam numerical model simulation, and 

model conversions, statistics, estimations, and coefficients.

Appendix files are available online only, and may be accessed at https://doi.org/10.3133/sir20185108. 
Menlo Park Publishing Service Center, California

Manuscript approved August 3, 2018

Edited by Katherine Jacques

Layout and design by Cory Hurd
Anning and others—Conceptual and Numerical Models of Dissolved Solids in the Colorado River, Arizona, California, and Nevada—Scientific Investigations Report 2018–5108

ISSN 2328-0328 (online)

https://doi.org/10.3133/sir20185108