U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2007–5153

Prepared in cooperation with the North Dakota State Water Commission

Continuous Water-Quality Monitoring and Regression 

Analysis to Estimate Constituent Concentrations and 

Loads in the Sheyenne River, North Dakota, 1980–2006
Continuous Water-Quality Monitoring 

and Regression Analysis to Estimate 

Constituent Concentrations and Loads 

in the Sheyenne River, North Dakota, 

1980–2006

By Karen R. Ryberg

Prepared in cooperation with the North Dakota State Water Commission

Scientific Investigations Report 2007–5153

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2007

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Suggested citation:

Ryberg, K.R., 2007, Continuous water-quality monitoring and regression analysis to estimate constituent 

concentrations and loads in the Sheyenne River, North Dakota, 1980–2006: U.S. Geological Survey Scientific 

Investigations Report 2007–5153, 22 p.
iii

Contents

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

Introduction.....................................................................................................................................................1

Purpose and Scope ..............................................................................................................................2

Description of Study Area ...................................................................................................................2

Previous Studies ...................................................................................................................................4

Methods...........................................................................................................................................................4

Streamflow Measurements ................................................................................................................4

Manual Water-Quality Measurements, Sample Collection, and Analysis..................................4

Development of Regression Equations to Estimate Constituent Concentrations ......................5

Calculation of Measured and Estimated Constituent Loads .........................................................7

Continuous Water-Quality Monitoring........................................................................................................9

Summary Statistics......................................................................................................................................10

Results of Regression Analysis .................................................................................................................10

Hardness ..............................................................................................................................................10

Dissolved Solids..................................................................................................................................10

Calcium.................................................................................................................................................14

Magnesium ..........................................................................................................................................14

Sodium..................................................................................................................................................14

Sulfate ..................................................................................................................................................15

Measured and Estimated Constituent Loads ..........................................................................................19

Summary........................................................................................................................................................19

References Cited..........................................................................................................................................20

Figures

1. Map showing location of the seven sampling sites on the Sheyenne River, N. Dak ........3

2–5. Graphs showing:

2. Comparison of measured and estimated sodium concentrations in the 

Sheyenne River, N. Dak., at site 1 .....................................................................................8

3. Predicted sulfate concentration, 90-percent prediction interval, and 

individual samples used in regression analyses for sites 2 and 3 on the 

Sheyenne River, N. Dak., 2005–06 ...................................................................................17

4. Predicted sulfate concentration, 90-percent prediction interval, and 

individual samples used in regression analyses for sites 5, 6, and 7 on 

the Sheyenne River, N. Dak., 2000–2006 ........................................................................18

5. Comparison of measured and estimated sulfate loads for site 7 on the 

Sheyenne River, N. Dak., 1997–2006 ...............................................................................19
iv

Tables

1. Sites used for regression analysis, Sheyenne River, N. Dak., 1980–2006 ...........................2

2. Constituents estimated using regression equations for the Sheyenne River, 

N. Dak., 1980–2006 ........................................................................................................................6

3. Possible surrogate physical properties, or explanatory variables, used to 

develop regression relations to estimate constituent concentrations in 

Sheyenne River, N. Dak., 1980–2006 ..........................................................................................7

4. Conversion factors used in calculation of measured and estimated loads .......................9

5. Summary of statistics for water-quality sampling sites on the Sheyenne 

River, N. Dak., 1980–2006 ...........................................................................................................11

6. Regression equations for estimates of hardness in the Sheyenne River, 

N. Dak., 1980–2006 ......................................................................................................................13

7. Regression equations for estimates of dissolved solids in the Sheyenne 

River, N. Dak., 1980–2006 ...........................................................................................................13

8. Regression equations for estimates of calcium in the Sheyenne River, 

N. Dak., 1980–2006 ......................................................................................................................14

9. Regression equations for estimates of magnesium in the Sheyenne River, 

N. Dak., 1980–2006 ......................................................................................................................15

10. Regression equations for estimates of sodium in the Sheyenne River, 

N. Dak., 1980–2006 ......................................................................................................................16

11. Regression equations for estimates of sulfate in the Sheyenne River, 

N. Dak., 1980–2006 ......................................................................................................................16
Conversion Factors, Abbreviations, and Datum

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

yard (yd) .9144 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Mass

pound, avoirdupois (lb) 0.4536 kilogram (kg) 

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F=(1.8×°C)+32.

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8.

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).

Horizontal coordinate information is referenced to North American Datum of 1927 (NAD 27).
Continuous Water-Quality Monitoring and Regression 

Analysis to Estimate Constituent Concentrations and 

Loads in the Sheyenne River, North Dakota, 1980–2006

By Karen R. Ryberg

Abstract

This report presents the results of a study by the U.S. 

Geological Survey, done in cooperation with the North Dakota 

State Water Commission, to estimate water-quality constituent concentrations at seven sites on the Sheyenne River, 

N. Dak. Regression analysis of water-quality data collected in 

1980–2006 was used to estimate concentrations for hardness, 

dissolved solids, calcium, magnesium, sodium, and sulfate. 

The explanatory variables examined for the regression 

relations were continuously monitored streamflow, specific 

conductance, and water temperature. For the conditions 

observed in 1980–2006, streamflow was a significant explanatory variable for some constituents. Specific conductance 

was a significant explanatory variable for all of the constituents, and water temperature was not a statistically significant 

explanatory variable for any of the constituents in this study.

The regression relations were evaluated using common 

measures of variability, including R2, the proportion of 

variability in the estimated constituent concentration explained 

by the explanatory variables and regression equation. R2 values 

ranged from 0.784 for calcium to 0.997 for dissolved solids. 

The regression relations also were evaluated by calculating 

the median relative percentage difference (RPD) between 

measured constituent concentration and the constituent 

concentration estimated by the regression equations. Median 

RPDs ranged from 1.7 for dissolved solids to 11.5 for sulfate. 

The regression relations also may be used to estimate daily 

constituent loads.

The relations should be monitored for change over time, 

especially at sites 2 and 3 which have a short period of record. 

In addition, caution should be used when the Sheyenne River 

is affected by ice or when upstream sites are affected by 

isolated storm runoff. Almost all of the outliers and highly 

influential samples removed from the analysis were made 

during periods when the Sheyenne River might be affected by 

ice.

Introduction

The U.S. Geological Survey (USGS) continuously 

records gage height, specific conductance, and water temperature at seven gaging sites on the Sheyenne River (fig. 1; 

table 1) in cooperation with several State and Federal agencies. 

Data from the continuous real-time water-quality monitors 

on the Sheyenne River are available on the Internet from 

the USGS National Water Information System (NWISWeb, 

http://waterdata.usgs.gov/nd/nwis). Continuous monitoring 

provides valuable information and may identify important 

variability that may not be evident in discrete water-quality 

sampling schemes. This information could be used to estimate 

concentrations of other water-quality constituents of interest, 

such as sulfate.

Devils Lake, in the adjoining Devils Lake Basin 

(fig. 1), began flooding in the 1990s and continues to the 

present (2007). According to the North Dakota State Water 

Commission’s Devils Lake Quick Facts [n.d.], Devils Lake 

flooding “has destroyed hundreds of homes and businesses 

and inundated thousands of acres of productive farmland. The 

State of North Dakota and the U.S. Government have spent 

over $450 million dollars in flood mitigation efforts including moving roads, rail and power lines, and building dikes.” 

The State of North Dakota has constructed an outlet to allow 

water from Devils Lake to flow into the Sheyenne River. 

The maximum outlet discharge is based in part on the sulfate 

concentration at the Sheyenne River downstream from the 

outlet insertion point (North Dakota State Water Commission 

[n.d.], Devils Lake Flooding—Outlet). Many water-quality 

stakeholders are concerned about the effect that water from 

Devils Lake might have on the Sheyenne River. The Sheyenne 

River is a tributary of the Red River of the North (Red River), 

which flows into Canada, therefore, the water quality of the 

Sheyenne River is of interest to users of the Red River in 

North Dakota, Minnesota, and Canada.
Table 1. Sites used for regression analysis, Sheyenne River, N. Dak., 1980–2006.

[--, not applicable]

Site number

(figure 1)

U.S. Geological 

Survey 

station number

North Dakota State 

Water Commission 

location number

U.S. Geological Survey station name Latitude Longitude

1 05054500 -- Sheyenne River above Harvey, N. Dak. 47°42’10” 99°56’55”

2 05055300 15106807 Sheyenne River above Devils Lake State outlet 

near Flora, N. Dak.

47°54’28” 99°24’57”

3 05055400 15006708 Sheyenne River below Devils Lake State outlet 

near Bremen, N. Dak.

47°49’17” 99°16’34”

4 05056000 -- Sheyenne River near Warwick, N. Dak. 47°48’20” 98°42’57”

5 05057000 -- Sheyenne River near Cooperstown, N. Dak. 47°25’58” 98°01’38”

6 05058000 -- Sheyenne River below Baldhill Dam, N. Dak. 47°02’02” 98°05’00”

7 05059300 -- Sheyenne River above Sheyenne River Diversion 

near Horace, N. Dak.

46°45’01” 96°55’35”

For the study described in this report, the USGS, in 

cooperation with the North Dakota State Water Commission 

(NDSWC), analyzed the data collected at the seven sites on 

the Sheyenne River to determine whether streamflow and the 

continuously recorded physical properties could act as surrogates to estimate water-quality constituents that are important 

indicators of surface-water quality. The methods used in this 

study replicate those previously used in North Dakota for the 

Red River at Fargo (Ryberg, 2006) and in Kansas (Christensen 

and others, 2000; Christensen, 2001; Rasmussen and others, 

2005; and Christensen and others, 2006) and may be replicated for other sites in North Dakota and the Nation to monitor 

water quality.

Estimation of water-quality constituents on the basis 

of surrogates provides several benefits. Although periodic 

Sheyenne River water-quality samples are collected manually, 

the delay associated with laboratory analysis does not permit 

immediate identification of undesirable levels of constituents. 

A relation between manually collected water-quality samples 

and continuously recorded water-quality physical properties 

allows immediate identification of potential water-quality 

problems. Examination of streamflow and physical properties of water that act as surrogates for constituents of interest 

also helps optimize visits for the collection of water-quality 

samples. For example, if specific conductance is used as 

a surrogate for estimating sulfate, it is desirable to collect 

water-quality samples representative of the range of specificconductance values possible for the Sheyenne River at each 

site. In addition, the continuous estimation of water-quality 

information is useful in tracking changes in water quality as 

they occur.

Purpose and Scope

This report presents the results of regression analysis 

of water-quality constituents for seven sites on the Sheyenne 

River, N. Dak., using manually collected water-quality data 

and continuously recorded streamflow and physical property 

data from seven sites on the Sheyenne River. The data were 

collected during 1980–2006. The report provides regression 

equations that can be used to estimate concentrations and 

loads for hardness, dissolved solids, calcium, magnesium, 

sodium, and sulfate.

Description of Study Area

The Sheyenne River (fig. 1) originates in Sheridan 

County in central North Dakota and flows through southeastern North Dakota, ultimately emptying into the Red River, 

north of Fargo. The Sheyenne River has a drainage area of 

about 6,910 square miles, not including the closed Devils Lake 

Basin (Emerson, 2005).

At times there is no flow in the upper reaches of the 

Sheyenne River. Flows in lower reaches are regulated partly by 

release from Baldhill Dam, approximately 12 miles north of 

Valley City. Baldhill Dam, completed in 1951, provides water 

supply, flood control, recreation, and wildlife habitat (U.S. 

Army Corps of Engineers, 2005).

Site 1 is USGS gaging station 05054500 Sheyenne River 

above Harvey, N. Dak. and is located 2 miles upstream from 

an unnamed tributary, and 4.5 miles south of Harvey. The 

recording gage was established in September 1955. Periodic 

water-quality sampling began at this site in 1971. Specific 

   Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Figure 1. Location of the seven sampling sites on the Sheyenne River, N. Dak.

Sheyenne

River

Creek

Pipestream

River

Sheyenne

CANADA

Map area

NORTH DAKOTA

0 20 40 MILES 100 20

30

30 40 KILOMETERS 10

Towner Rugby

PIERCE

COUNTY BENSON

COUNTY

WELLS

COUNTY

EDDY COUNTY

RAMSEY COUNTY WALSH COUNTY

GRAND FORKS

COUNTY NELSON

COUNTY

GRIGGS

COUNTY

STEELE

COUNTY

TRAIL

COUNTY

BURLEIGH

COUNTY KIDDER

COUNTY

STUTSMAN

COUNTY

LOGAN COUNTY

EMMONS COUNTY

MORTON CO SIOUX CO MCINTOSH COUNTY LAMOURE COUNTY DICKEY COUNTY

BARNES COUNTY

CASS

COUNTY

RANSOM COUNTY

RICHLAND

COUNTY

SARGENT COUNTY

FOSTER

COUNTY

SHERIDAN COUNTY

MCLEAN

CO

Devils Lake

Minnewaukan Lakota

McClusky

Fessendon

NewRockford

Cooperstown

Finley

Grand Forks

Fargo

Hillsboro

Valley City

Ransom

LaMoure

Steele

Linton

Carrington

Jamestown

Grafton

97° 98° 99° 100°

48°47°

1

2

3 4

5

6

7

Base from Redistricting Census 2000 TIGER/Line Files, U.S. Census Bureau, 2001 Roads modified from Federal Highway Administration digital data, 2002Hydrography modified from U.S. Geological Survey digital data, 2004Albers Equal-Area Conic projection, standard parallels 45°30’N and 49°30‘N, central meridian 99°00’W

Big Coulee

North

Fork

Souris River

Oahe Reservoir

Lake

Long

Spirit LakeIndianReservation

Lonetree Wildlife

Management Area

29

29

94 94

94

29

281

83

52

52

52

2 2

2

281

281

State outlet

Stump Lake

Devils Lake

Morison Lake

MINNESOTA

NORTH DAKOTA

LakeAshtabula

Maple

River

Forest

North

Red River

River

James

River

James

ArrowwoodLake

River

Goose

Goose

River

River

Rush

River

Elm

River

Maple

Jim Lake

Creek

Beaver

Alkaline

Lake

EXPLANATION

River basin or sub-basin

Red River of the NorthDevils LakeSheyenne RiverJames RiverMissouri RiverSouris River

Sampling site and site identification number

5

River

River

Turtle

North

Branch

Cottonwood Creek

Bear

Maple

Creek

Baldhill

MCHENRY

COUNTY

Wintering

River

Creek

Apple

Creek

Beaver

keer C

Little Goose River of the Jamestown Reservoir

River

Harvey Warwick

Flora

Bremen

Horace

Baldhill Dam

Red River of the North

Introduction 
conductance and water temperature have been continuously 

recorded since May 2006. This gage records little or no flow 

most winters and is heavily affected by ice (L.A. Cottengaim, 

USGS, station description, unpub. data, 2007).

Site 2 is USGS gaging station 05055300 Sheyenne 

River above Devils Lake State outlet near Flora, N. Dak. 

The gage began recording stage, specific conductance, and 

water temperature in October 2004, 3.5 mi southeast of 

Flora (R.A. Nustad, USGS, station description, unpub. data, 

2006). Periodic water-quality sampling by the USGS and the 

NDSWC began at this site in 2005. Gage height is affected by 

ice and by growth of vegetation in the channel during periods 

of low flow (P.M. Scarpari, USGS, station analysis, unpub. 

data, 2006).

Site 3 is USGS gaging station 05055400 Sheyenne River 

below Devils Lake State outlet near Bremen, N. Dak. The 

seasonal (operated April through November) recording gage 

was established in March 2005, 10 mi northeast of Bremen. 

Specific conductance and water temperature have been continuously recorded, seasonally, since April 2005. Periodic waterquality sampling by the USGS and the NDSWC began at this 

site in 2005 (R.A. Nustad, USGS, station description, unpub. 

data, 2007).

Site 4 is USGS gaging station 05056000 Sheyenne River 

near Warwick, N. Dak. The recording gage was installed 

November 1949, 0.25 mi west and 3.3 mi south of Warwick. 

Periodic water-quality sampling at this site began in 1951. 

Specific conductance and water temperature have been continuously recorded since May 4, 2006 (P.M. Scarpari, USGS, 

station description, unpub. data, 2007).

Site 5 is USGS gaging station 05057000 Sheyenne River 

near Cooperstown, N. Dak. A recording gage was established 

in 1950 and moved to the current location in October 1985, at 

Ueland Dam, 5 mi east of Cooperstown. Periodic water-quality 

sampling at this site began in 1959. Specific conductance and 

water temperature have been continuously recorded since 

June 1997. Records may be poor during winter periods (R.A. 

Nustad, USGS, station description, unpub. data, 2006).

Site 6 is USGS gaging station 05058000 Sheyenne River 

below Baldhill Dam, N. Dak. The gage was established in 

1949 and has been moved several times. Since September 

2000, the gage has been located 0.1 mi downstream from 

Baldhill Dam, approximately 0.25 mi upstream from a bridge. 

Periodic water-quality sampling at this site began in 1959. 

Specific conductance and water temperature have been continuously recorded since April 1997. The station is generally not 

affected by ice because of releases from Baldhill Dam (R.A. 

Nustad, USGS, station description, unpub. data, 2007).

Site 7 is USGS gaging station 05059300 Sheyenne 

River above Sheyenne River Diversion near Horace, N. Dak. 

The recording gage was established in October 1992, 1 mi 

southwest of Horace, upstream from the diversion structure. 

Periodic water-quality sampling began at this site in 1992. 

Specific conductance and water temperature have been continuously recorded since June 1997. Gage height is affected by 

ice through most of the winter months and flow is regulated 

to a large degree by Baldhill Dam, approximately 230 mi 

upstream (P.M. Scarpari, USGS, station description, unpub. 

data, 2007).

Additional information about these sites is available 

online at http://waterdata.usgs.gov/nd/nwis/nwisman/?site_

no=XXXXXXXX, where XXXXXXXX is the 8-digit USGS 

station number listed in table 1.

Previous Studies

Regression relations have been developed previously 

for sites on the Sheyenne River to estimate monthly mean 

constituent concentrations (Guenthner, 1991; Williams-Sether, 

2004). Guenthner’s study (1991) preceded continuous recording of specific conductance and water temperature; therefore, 

the only continuously recorded variable available for regression analysis was streamflow. The only continuously recorded 

variable used by Williams-Sether (2004) was streamflow. 

Regression analysis to estimate constituent concentrations and 

loads was used by Ryberg (2006) for the Red River at Fargo, 

N. Dak., and has been used for numerous sites in Kansas 

(Christensen and others, 2000; Christensen, 2001; Rasmussen 

and others, 2005; and Christensen and others, 2006).

Methods

Data collection and analysis methods have varied over 

time and from site to site. This section describes how streamflow was measured, water-quality samples were collected and 

analyzed, and data were interpreted for this study.

Streamflow Measurements

Stage, or gage height, was measured to the nearest 0.01 ft 

at all stations. The data for all stations are stored and publicly 

available in the National Water Information System: Web 

Interface (NWISWeb), http://waterdata.usgs.gov/nd/nwis.

Methods used to determine streamflow are described in 

Buchanan and Somers (1969). Streamflow measurements were 

made periodically at each station. A stage-streamflow relation 

was developed on the basis of streamflow measurements and 

the stage of the stream at the time of measurement (Kennedy, 

1984), and this relation was used to compute a continuous 

record of streamflow (Kennedy, 1983).

Manual Water-Quality Measurements, Sample 

Collection, and Analysis

During the collection of water-quality samples, physical 

properties of water, including specific conductance and water 

temperature, were measured. USGS water-quality samples 

were collected manually according to methods described in 

USGS techniques manuals (U.S. Geological Survey, variously 

 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
dated). After the manually measured physical property and 

water-quality data were checked and approved by the USGS, 

they were made publicly available in NWISWeb at http://nwis.

waterdata.usgs.gov/nd/nwis/qwdata.

Water-quality samples collected by the USGS were 

analyzed by the NDSWC Laboratory or the North Dakota 

Department of Health Laboratory. The NDSWC Laboratory and the North Dakota Department of Health Laboratory 

followed U.S. Environmental Protection Agency approved 

methods and procedures. The NDSWC Laboratory closed 

July 2003. Water-quality samples were analyzed by the North 

Dakota Department of Health Laboratory using standardized 

methods (North Dakota Department of Health, 2003b) and 

quality assurance procedures (North Dakota Department of 

Health, 2003a). External agencies and customer organizations 

audited the North Dakota Department of Health Laboratory 

to assure the laboratory analytical methods and qualityassurance/quality-control procedures. The U.S. Environmental 

Protection Agency reviewed the laboratory procedures about 

every 3 years, and the Branch of Quality Systems of the USGS 

periodically reviewed the laboratory procedures.

Water-quality samples collected by the NDSWC were 

analyzed by the North Dakota Department of Health Laboratory using standardized methods (North Dakota Department 

of Health, 2003b) and quality-assurance procedures (North 

Dakota Department of Health, 2003a). Data are stored 

in the NDSWC’s Ground and Surface Water database 

available online at http://www.swc.nd.gov/4dlink2/4dcgi/

wellsearchform/Map%20and%20Data%20Resources. In the 

NDSWC database, the NDSWC identifies site 2, Sheyenne 

River above Devils Lake State outlet near Flora, N. Dak., with 

the location number 15106807, and site 3, Sheyenne River 

below Devils Lake State outlet near Bremen, N. Dak., with 

the location number 15006708. Only data from the NDSWC 

database that had field specific-conductance measurements 

were used for this report. The NDSWC’s Ground and Surface 

Water database does not provide streamflow with the waterquality data. Therefore, the USGS automated data processing system (ADAPS, Revision NWIS–4.6.0–40) was used to 

obtain interpolated unit values for streamflow. ADAPS stores 

the streamflow values for the USGS Sheyenne River gaging 

sites in one-hour increments and interpolates a streamflow 

value based on the Water Commission’s specified sampling 

time and the streamflow for the closest hour before and the 

closest hour after the sampling time.

Water-quality data were collected as long ago as 1951. 

However, using data from such a long period of time introduces many sources of variability. Field and laboratory 

methods change over time and three different laboratories 

were used for some sites with long periods of record (sites 1, 

4, 5, and 6; fig. 1). In examining scatterplots and regression 

diagnostic plots, an inordinate number of outliers appeared 

in the data from the 1970s (data not shown). Many of the 

potential outliers may have been transposed during data 

entry, but the original data are no longer available to check. 

Water temperature was identified as a statistically significant 

explanatory variable for many constituents, but this was an 

unexpected finding and did not follow from the results of 

previous studies (Christensen and others, 2000; Christensen, 

2001; Ryberg, 2006). By limiting water-quality data to 

samples collected beginning in 1980 through 2006, some of 

the variability in laboratories, two rather than three analyzing 

laboratories, was eliminated, the number of potential outliers 

was reduced, and water temperature was no longer statistically 

significant as an explanatory variable for constituent concentration.

Data were removed from the regression analysis based 

on streamflow to eliminate samples collected from pooled and 

stagnant water. For sites 4 and 5, only water-quality samples 

with an associated streamflow of at least 5.00 ft3/s were used, 

based on structures at the gage and river characteristics (R.F. 

Lundgren and W.C. Damschen, USGS, oral commun., 2007). 

Site 1, generally having streamflow or being dry, is not as 

prone to pooling as sites 4 and 5 (R.F. Lundgren and W.C. 

Damschen, USGS, oral commun., 2007); however, the point 

of zero flow at site 1 is a gage height of 3.50 ± 0.05 ft (L.A. 

Cottengaim, USGS, station description, unpub. data, 2007), 

and 3.55 ft is equivalent to a streamflow of 0.34 ft3/s (U.S. 

Geological Survey, stage-discharge rating 10.0, November 21, 

2006). Therefore, only water-quality samples with an associated streamflow greater than 0.34 ft3

/s were used for site 1. 

For sites 2 and 3, only samples with an associated streamflow 

of at least 4.00 ft3/s were used. No data were used that were 

collected after July 13, 2006, for site 2 and July 7, 2006, 

for site 3 because streamflow was less than 4.0 ft3/s or was 

affected by ice for the remainder of 2006. Sites 6 and 7 have 

higher minimum streamflow caused by regulation at Baldhill 

Dam and tributary inflow; therefore, data for these sites were 

not removed based on streamflow.

The frequency of water-quality sampling varied from 

site to site and with time, resulting in some instances of serial 

correlation, especially with data collected in the 1980s. To 

reduce serial correlation, which may result in residuals that 

are not independent, the data were thinned (Helsel and Hirsch, 

1995). After removal of data based on streamflow, the data 

were thinned for sites 1, 4, 5, 6, and 7 (fig. 1; table 1) so that 

no samples were closer than 4 weeks apart. Because of the 

shorter period of record, data for sites 2 and 3 (fig. 1; table 1) 

were thinned so that no samples were closer than 3 weeks 

apart.

Development of Regression Equations to 

Estimate Constituent Concentrations

All available water-quality constituents were considered 

for estimation using regression equations. However, some of 

them were removed from consideration due to various data 

issues, such as no data since the early 1980s (bicarbonate), not 

enough data (total phosphorus), and the data being too coarse, 

or discrete, for regression analysis (fluoride). Other constituents had enough data, but relations useful for prediction were 

Methods 
Table 2. Constituents estimated using regression equations for the Sheyenne River, N. Dak., 1980−2006.

Constituent Unit of measurement U.S. Geological Survey 

parameter code

Hardness Milligrams per liter as calcium carbonate P00900

Dissolved solids, calculated Milligrams per liter P70301

Calcium Milligrams per liter P00915

Magnesium Milligrams per liter P00925

Sodium Milligrams per liter P00930

Sulfate Milligrams per liter P00945

not found (potassium and chloride at sites 1, 4, 5, 6, and 7, 

and dissolved nitrite plus nitrate at site 1). The water-quality 

constituents selected for estimation are listed in table 2.

The rationale and methodology for expressing waterquality constituent concentrations in terms of other surrogate 

constituents or physical properties in a regression equation 

are explained in Helsel and Hirsch (1995). Helsel and Hirsch 

(1995) also detail the computations for regression estimation 

and identify measures commonly used to evaluate regression 

relations, including R2, the coefficient of determination. They 

explain the benefit of log transformation of variables, which 

was a technique used for some of the regression relations in 

this study.

R2 is calculated as follows:

R SSE

SSy

2 = −1 (1)

where

	 SSE	 is the error sum of squares, and

	 SSy	 is the sums of squares y, or total sums of 

squares, defined in Helsel and Hirsch 

(1995).

R2 is a number, 0 through 1, that when multiplied by 

100 is interpreted as the percentage of the variability in the 

response variable explained by the explanatory variables and 

the regression equation. Generally, the higher the R2, the better 

the regression relation. However, this does not guarantee the 

regression equation is useful (Neter and others, 1996). For 

example, if estimates require extrapolation outside the ranges 

of observed variables, the regression equation may not provide 

accurate estimates.

All-subset regression by leaps and bounds, a procedure that attempts to find the best regression relations using 

subsets of the given explanatory variables (Insightful Corporation, 2005), was used to select the best regression model 

for estimating a particular constituent. The possible explanatory variables were streamflow, specific conductance, and 

water temperature (table 3) as well as transformations of 

streamflow and specific conductance, log ( ) 10 streamflow , and 

log ( ) 10 specific conductance . 

The criteria for determining the best regression relation were 

the adjusted coefficient of multiple determination 

( Ra

2 ) and Mallow’s Cp (Cp ). R2 increases with the number 

of explanatory variables in the regression model, but Ra

2

allows for the comparison of models that have differing 

numbers of explanatory variables by penalizing models that 

have additional coefficients (Helsel and Hirsch, 1995). The 

Cp criterion is a measure of the total mean squared error 

and an indicator of model bias (Neter and others, 1996). In 

comparing models, the models with the lowest Cp values are 

considered those with the least bias.

Potential models with relatively high Ra

2 and low Cp

were further examined using standard diagnostics for regression (Neter and others, 1996). The most common problems 

in the diagnostic residual plots were non-normality and 

heteroscedasticity, or non-constant variance, both violations 

of the assumptions underlying parametric regression. Transformations of the response variable (constituent concentration 

in this study) are effective fixes for both of these problems, 

which often occur together (Neter and others, 1996). Logarithmic transformations of the explanatory and response variables 

resulted in residuals that were approximately normally distributed. An explanatory variable was considered statistically 

significant and selected for regression relations if the p-value 

(attained significance level) for the variable was less than 0.05.

A resampling technique, jackknife after bootstrap, was 

used to examine the influence of individual observations in 

the development of the regression relations. “In bootstrap 

resampling, B [where B is a large number, such as 1,000] new 

samples, each of the same size as the observed data, are drawn 

with replacement from the observed data. The statistic is first 

calculated using the observed data and then recalculated using 

each of the new samples, yielding a bootstrap distribution. 

The resulting replicates are used to calculate the bootstrap 

estimates of bias, mean, and standard error for the statistic” 

(Insightful Corporation, 2001a). In this study the bootstrap 

statistics were the regression parameter estimates, the coefficients of the explanatory variables, and the intercept term. The 

jackknife after bootstrap was helpful in identifying very influential observations that were removed as outliers and indicated 

   Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Table 3. Possible surrogate physical properties, or explanatory variables, used to develop regression 

relations to estimate constituent concentrations in Sheyenne River, N. Dak., 1980−2006.

Physical properties Unit of measurement U.S. Geological Survey 

parameter code

Streamflow Cubic feet per second P00060

Specific conductance Microsiemens per centimeter at 25° Celsius P00095

Temperature, water Degrees Celsius P00010

that the regression parameters estimated in this study were 

relatively unbiased in their mean and standard error.

An additional measure of influence, Cook’s distance, was 

used along with residual plots to determine outliers or highly 

influential samples. Cook’s distance quantifies the influence of 

individual samples on the predicted values (Neter and others, 

1996).

In building a regression model, the final step of the 

process is validation of the selected regression model. Validation generally is done in one of three ways: (1) collection of 

new data to check the model and its predictions; (2) comparison of results with earlier observations, simulations, theoretical experiments, or physical theory; or (3) use of data that 

were held out of the model-building process to check the 

model and its prediction accuracy (Neter and others, 1996). 

Collection of new data is time-consuming and expensive and 

several years of data may be necessary to obtain samples 

representative of high and low streamflows, specific conductance, and constituent concentrations. Results of continued 

water-quality sampling should be compared with the regression relations presented in this report as hydrologic, climatologic, or human factors may change the relations. The 

regression relations were compared to previous studies and 

examined for scientific validity. Some data were withheld 

from the model building process and used to check the predictions made by regression relations.

As an indicator of the ability of the regression relations to 

estimate constituent concentrations, the measured concentrations were compared to the concentrations estimated by the 

regression relations by calculating relative percentage differences (RPDs) using the following equation:

RPD E M

M = X − 100 , (2)

where

E is the constituent concentration estimated 

from the regression equation, and

M is the measured constituent concentration.

M is assumed to be correct and the RPD is the relative difference of E from M, expressed as a percentage.

Logarithmic transformations of the explanatory and 

response variables resulted in improved RPDs over untransformed variables and in a better linear relation between the 

measured and estimated concentrations. The improvement in 

the RPD when the explanatory and response variables were 

logarithmically transformed is shown in figure 2 for sodium 

concentrations at site 1. The top graph shows estimated and 

measured sodium concentrations with an RPD of 14.3. When 

the variables were logarithmically transformed, the RPD

decreased to 7.5. In figure 2A, the relation between estimated 

and measured concentrations appears slightly curvilinear, 

with measured concentrations generally above the estimated 

concentration line for low concentrations, measured concentrations scattered on both sides of the line for concentrations 

in the middle, and high measured concentrations below the 

estimation line. In figure 2B the data generated with logarithmically transformed explanatory and response variables 

shows a much more linear relation between the measured and 

estimated concentrations.

Measured and estimated concentrations were compared 

graphically to evaluate the usefulness of the regression 

relations for prediction using continuously recorded streamflow and specific conductance data at each gage. The daily 

mean of the continuously recorded explanatory variables at 

each site and the regression relations for sulfate were used to 

predict sulfate concentrations along with 90-percent prediction 

intervals (Neter and others, 1996). The prediction intervals 

have a 90-percent probability of containing the true sulfate 

population (Helsel and Hirsch, 1995).

Calculation of Measured and Estimated 

Constituent Loads

Daily load is the total mass of a constituent that is 

transported past a gaging station in 1 day. Measured constituent loads were calculated by multiplying measured constituent concentrations by streamflow at the time the constituent 

concentrations were measured and multiplying by the conversion factor listed in table 4. Estimated mean daily constituent 

loads were calculated by multiplying the daily constituent 

concentrations estimated using the regression equations by 

daily streamflow and by a conversion factor (table 4).

For regression relations developed in terms of a logarithmically transformed constituent concentration, retransformation to the original units can cause an underestimation of 

chemical loads when adding individual load estimates over 

Methods 
Figure 2. Comparison of measured and estimated sodium concentrations in the Sheyenne River, N. Dak., at 

site 1, A is a comparison using untransformed explanatory and response variables and B is a comparison using 

logarithmically transformed explanatory and response variables.

0 100 200 300 400 500 600

0

100

200

300

400

500

600

RPD=14.3

Estimated concentration

Measured concentration

RPD

MEASURED SODIUM CONCENTRATION, IN MILLIGRAMS PER LITER

0 100 200 300 400 500 600

0

100

200

300

400

500

600

RPD=7.5

A

B

ESTIMATED SODIUM CONCENTRATION, IN MILLIGRAMS PER LITER

Median relative percentage difference between

instantaneous measured concentrations

and instantaneous estimated concentrations

Estimated concentration

Measured concentration

RPD Median relative percentage difference between

instantaneous measured concentrations

and instantaneous estimated concentrations

   Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Table 4. Conversion factors used in calculation of measured and estimated loads.

[ft3/s, cubic feet per second]

Multiply By By To obtain

milligrams per liter streamflow, in ft3/s 5.39 pounds per day

a long period of time (Christensen, 2001). Multiplying the 

calculated load by a bias correction factor (BCF; Duan, 1983) 

corrects for this underestimation. Calculation of the BCF is 

shown in the equation below:

BCF

n

e

i

n

i

= =

∑10

1

(3)

where

ei is the regression residual, in log units, and

n is the number of samples used to develop 

the regression relation.

Continuous Water-Quality Monitoring

Continuous monitoring of specific conductance and water 

temperature began at site 1 (fig. 1; table 1) in May 2006 using 

a Campbell Scientific (CS) 547A conductivity/temperature 

probe. At site 2 (fig. 1; table 1), continuous monitoring of 

specific conductance and water temperature began in October 

2004 with a CS 547 conductivity/temperature probe. On 

May 9, 2006, the probe was replaced with a YSI Incorporated 

(YSI) 600R conductivity/temperature probe. At site 3 (fig. 1; 

table 1), continuous monitoring of specific conductance and 

water temperature began in April 2005 with a CS 547 conductivity/temperature probe. On May 8, 2006, the probe was 

replaced with a YSI 600R conductivity/temperature probe.

At site 4 (fig. 1; table 1), continuous monitoring of 

specific conductance and water temperature began in May 

2006, with a CS 547 conductivity/temperature probe. The 

probe was installed on the left edge of the water on the same 

side as a natural spring that discharges to the Sheyenne River 

60–70 ft upstream from the probe. Based on field readings on 

October 24, 2006, the specific conductance in the spring may 

be lower than the specific conductance in the river and the 

spring may affect the specific conductance recorded by the 

probe. A hydrologic comparison of specific conductance at 

sites 2, 3, 4, and 5 showed that specific conductance at site 4 

follows the same trend as that at site 5 but may be influenced 

by the spring during periods of low streamflow (1–10 ft3/s) 

(P.M. Scarpari, USGS, station analysis, unpub. data, 2007).

At site 5 (fig. 1; table 1), continuous monitoring of 

specific conductance and water temperature began in June 

1997, with a CS 247 conductivity/temperature probe. At 

site 6 (fig. 1; table 1), continuous monitoring of specific 

conductance and water temperature began in April 1997, with 

a CS 547 conductivity/temperature probe. The probe was 

initially installed 0.3 mi downstream from the current position. 

The initial location was downstream from Baldhill Dam and 

downstream from return flow from the Baldhill Dam National 

Fish Hatchery. In the new location, since September 18, 2000, 

only water released from the dam is in contact with the probe 

(R.A. Nustad, USGS, station description, unpub. data, 2007). 

At site 7 (fig. 1; table 1), continuous monitoring of specific 

conductance and water temperature began in June 1997, with a 

CS 547 conductivity/temperature probe.

The CS probes were installed in buried, flexible polypipe intakes which extended from the gage houses to the river. 

The pipes were perforated at the end to allow river water 

to come in contact with the probes. In the CS probes, the 

electrical conductance sensor “consists of three stainless steel 

rings mounted in an epoxy tube...Resistance of water passing 

through the tube is measured by excitation of the center 

electrode with positive and negative voltage...Temperature is 

measured with a thermistor in a three wire half bridge configuration” (Campbell Scientific, Inc., 1996, 2000, and 2001). 

The accuracy of the conductivity sensor in standard ionized 

solutions is ±5 percent of reading in the range of 440 to 

7,000 μS/cm at 25°C and ±10 percent of reading in the range 

of 5 to 440 μS/cm at 25°C (Campbell Scientific, Inc., 1996, 

2000, and 2001). The accuracy of the temperature sensor is 

±0.4°C in the range of –24°C to 48°C (Campbell Scientific, 

Inc., 1996, 2000, and 2001).

The YSI probes were installed in buried galvanized steel 

pipe. The pipes were perforated at the end to allow river water 

to come in contact with the probes. The YSI probes utilize 

“a cell with four pure nickel electrodes for the measurement 

of solution conductance. Two of the electrodes are current 

driven, and two are used to measure the voltage drop” (YSI 

Incorporated, 2006). For temperature measurement, the probes 

utilize “a thermistor of sintered metallic oxide that changes 

predictably in resistance with temperature variation” (YSI 

Incorporated, 2006). The accuracy of the conductivity sensor 

is ±0.5 percent of reading plus 1 μS/cm in the range of 0 to 

Continuous Water-Quality Monitoring   
100,000 μS/cm and the accuracy of the temperature sensor is 

±0.15°C in the range of –5 to 50°C (YSI Incorporated, 2006).

All probes were periodically cleaned and calibrated. For 

the CS 247, 547, and 547A models, calibration drift corrections were made when the specific conductance readings 

differed by more than 5 percent of the specific conductance of 

the standard solution used for calibration. Fouling corrections 

are made when readings before and after cleaning differed by 

more than 5 percent (L.A. Cottengaim, USGS, station analysis, unpub. data, 2007). The YSI 600R probes were calibrated 

and data were corrected according to criteria described by 

Wagner and others (2000).

The data for all sites were transmitted through satellite 

telemetry and downloaded into the USGS Automated Data 

Processing System (ADAPS), and then the data were made 

available on the Internet from NWISWeb.

Summary Statistics

Summary statistics of the manually measured physical 

properties and water-quality constituents are listed in table 5. 

The data collected by the USGS are stored in the USGS NWIS 

database and are available online at http://nwis.waterdata.

usgs.gov/nd/nwis/qwdata?site_no=XXXXXXXX, where 

XXXXXXXX is the eight digit USGS station number identified in table 1. The data collected by the NDSWC are stored 

in the NDSWC’s Ground and Surface Water database and 

are available online at http://www.swc.nd.gov/4dlink2/4dcgi/

wellsearchform/Map%20and%20Data%20Resources.

Results of Regression Analysis

Regression relations between constituents and surrogate physical properties were examined, and a regression 

relation was developed for each constituent using one or more 

surrogate variables. A discussion of each constituent and the 

associated regression relations follows. Also included in the 

discussion are the samples, if any, that were removed from the 

regression analysis. Almost all of the samples removed were 

from February, March, and November with the majority being 

from March. These are periods when the sites may be affected 

by ice, especially sites 1–5 and 7; therefore, caution should be 

used in using the regression relations for concentration estimation during periods of ice.

Hardness

Hardness is a property of water resulting from the 

presence of calcium and magnesium that is associated with 

the reaction of water with soap (Hem, 1985). The higher the 

hardness, the more soap is required for cleansing, while water 

with hardness less than 30–50 mg/L may be corrosive to pipes, 

depending on pH, alkalinity, and dissolved oxygen (North 

Dakota Department of Health, 2006). The hardness data used 

in this study were expressed in terms of calcium carbonate 

concentration.

Specific conductance “provides general indication of the 

content of dissolved matter for water that is not too saline or 

too dilute” (Hem, 1985); therefore, specific conductance may 

indicate the presence of calcium and magnesium. Specific 

conductance was a statistically significant predictor variable 

for hardness in the Sheyenne River at sites 4, 5, 6, and 7 

(table 6). Site 1 showed nonconstant variance in the residual 

plots and had a high median RPD (greater than 20). Hardness 

was not analyzed for a regression relation at sites 2 and 3 

because of insufficient data.

Regression diagnostics plots, Cook’s distance (Neter 

and others, 1996), and jackknife after bootstrap (Insightful 

Corporation, 2001b), were all used to examine regression 

results for outliers and highly influential samples. For site 5, 

jackknife after bootstrap and Cook’s distance indicated that the 

March 31, 2004, sample was highly influential in the estimation of regression parameters and therefore was removed from 

the regression analysis. For site 7, jackknife after bootstrap 

indicated that the February 6, 2006, sample was highly influential in the estimation of regression parameters. This sample 

also stood out in regression diagnostic plots of the residuals; 

therefore, the February 6, 2006, sample was removed from the 

regression analysis.

Dissolved Solids

Dissolved-solids concentration is used to evaluate water 

quality and to compare water from one location to the next. 

Dissolved-solids concentration may be determined in two 

ways: (1) by the weight of the dry residue remaining after 

evaporation of the water samples; or (2) by summing the 

concentrations of other dissolved constituents, if the concentrations of major ions are known (Hem, 1985). According 

to Hem, the computed value for dissolved solids, which was 

used in this study, “may give a more useful indication of total 

dissolved-ion concentration than the residue left by evaporation.” High concentrations of dissolved solids may have an 

adverse effect on taste and make the water appear cloudy 

(North Dakota Department of Health, 2006).

Specific conductance has been an effective predictor 

of dissolved solids in previous studies (Ryberg, 2006; and 

Christensen and others, 2000). Streamflow also may affect 

dissolved solids, but the relation is not a simple one (Allan, 

1995). For the Sheyenne River, specific conductance was 

a significant explanatory variable for estimating dissolved 

solids at sites 1, 4, 5, 6, and 7; streamflow was not (table 7). 

Dissolved solids was not analyzed for a regression relation at 

sites 2 and 3 because of insufficient data.

Regression diagnostics plots, Cook’s distance (Neter 

and others, 1996), and jackknife after bootstrap (Insightful 

Corporation, 2001b), were used to examine regression results 

10 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Table 5. Summary of statistics for water-quality sampling sites on the Sheyenne River, N. Dak., 1980–2006.—Continued

[Samples for sites 2 and 3 collected by North Dakota State Water Commission, samples for all other sites collected by U.S. Geological Survey; streamflow in 

cubic feet per second; specific conductance in microsiemens per centimeter at 25 degrees Celsius; temperature in degrees Celsius; hardness in millligrams per 

liter as calcium carbonate; other constituents in milligrams per liter]

Water-quality 

physical property 

or constituent

Descriptive statistics Percent of samples in which values were less than 

or equal to those shown

Sample 

size Maximum Minimum Mean 0.95 0.75

(Median) 

0.50 0.25 0.05

Site 1, Sheyenne River above Harvey, N. Dak.

Streamflow 224 498 0.19 36 199 28 6.6 2.1 0.79

Specific conductance 215 2,300 280 1,270 1,860 1,540 1,320 1,110 435

Temperature, water 221 28.2 0 9.5 24.5 17.3 8.0 .5 0

Hardness 147 550 61 220 440 290 190 140 87

Dissolved solids, 

calculated

146 1,610 192 869 1,240 1,010 892 793 327

Calcium 146 74 14 36 62 45 34 27 20

Magnesium 147 92.6 5.6 30.1 65.8 42 23 16 8.6

Sodium 147 480 20 231 370 300 240 170 57.2

Sulfate 147 610 44 240 460 280 220 180 85

Chloride 146 32 3.6 17 28 20 16 13 6.0

Site 2, Sheyenne River above Devils Lake State outlet near Flora, N. Dak.

Streamflow 125 600 4.0 59 264 67 20 8.3 4.4

Specific conductance 125 1,880 570 1,550 1,830 1,700 1,600 1,530 806

Sulfate 125 520 110 370 490 420 380 330 200

Site 3, Sheyenne River below Devils Lake State outlet near Bremen, N. Dak.

Streamflow 125 579 4.0 61 171 84 30 13 5.5

Specific conductance 125 1,880 625 1,560 1,810 1,710 1,620 1,530 940

Sulfate 125 540 160 380 500 440 380 350 220

Site 4, Sheyenne River near Warwick, N. Dak.

Streamflow 247 3,160 0.29 206 997 156 29 11 2.2

Specific conductance 238 1,900 218 903 1,420 1,130 909 655 440

Temperature, water 243 30.0 0 9.9 25.9 18.0 8.0 1.0 0

Hardness 60 430 87 260 380 320 270 190 100

Dissolved solids, 

calculated

60 911 184 530 851 713 527 363 223

Calcium 60 79 20 48 68 56 49 38 23

Magnesium 60 59 9.0 33 52 43 33 22 12

Sodium 60 190 14 92 160 130 81 51 25

Sulfate 60 310 37 140 280 210 130 80.5 46.8

Chloride 60 37 3.5 14 22 17 14 10 5.5

Results of Regression Analysis 11
Table 5. Summary of statistics for water-quality sampling sites on the Sheyenne River, N. Dak., 1980–2006.—Continued

[Samples for sites 2 and 3 collected by North Dakota State Water Commission, samples for all other sites collected by U.S. Geological Survey; streamflow in 

cubic feet per second; specific conductance in microsiemens per centimeter at 25 degrees Celsius; temperature in degrees Celsius; hardness in millligrams per 

liter as calcium carbonate; other constituents in milligrams per liter]

Water-quality 

physical property 

or constituent

Descriptive statistics Percent of samples in which values were less than 

or equal to those shown

Sample 

size Maximum Minimum Mean 0.95 0.75

(Median) 

0.50 0.25 0.05

Site 5, Sheyenne River near Cooperstown, N. Dak.

Streamflow 327 5,290 0.4 395 1,810 321 88.8 34.7 7.64

Specific conductance 314 2,000 280 921 1,360 1,110 948 686 410

Temperature, water 321 28.0 0 9.6 24.5 17.2 8.0 1.0 0

Hardness 102 510 99 290 430 350 310 220 130

Dissolved solids, 

calculated

103 1,130 182 558 801 686 587 401 261

Calcium 102 110 24 61 97 72 64 46 29

Magnesium 102 58 9.5 33 50 40 33 23 13

Sodium 102 190 19 86 140 110 89 57 31

Sulfate 103 360 51 160 270 180 150 120 70

Chloride 112 39 5.2 15 24 17 15 11 6.6

Site 6, Sheyenne River below Baldhill Dam, N. Dak.

Streamflow 285 5,510 0.05 450 2,360 346 107 33 10

Specific conductance 280 1,630 366 866 1,270 1,020 861 684 529

Temperature, water 283 26.2 0 10.5 24.4 19.3 7.5 3.0 1.0

Hardness 77 430 150 290 410 360 280 230 200

Dissolved solids, 

calculated

79 842 262 566 781 685 542 455 382

Calcium 77 78 33 58 78 69 58 47 39

Magnesium 77 60 17 36 55 46 35 27 23

Sodium 77 130 27 83 120 100 78 65 49

Sulfate 79 310 78 180 300 230 160 130 100

Chloride 89 31 8.2 16 24 18 15 13 9.3

Site 7, Sheyenne River above Sheyenne River Diversion near Horace, N. Dak.

Streamflow 175 4,080 40 857 3,110 1,030 402 209 78

Specific conductance 168 1,510 418 908 1,270 1,050 945 749 490

Temperature, water 169 28.5 0 11.8 25.0 21.0 12.0 1.0 0

Hardness 32 500 170 350 440 400 370 300 190

Dissolved solids, 

calculated

34 810 291 608 795 692 642 524 320

Calcium 32 110 40 76 99 88 76 65 43

Magnesium 32 54 18 39 50 45 40 32 20

Sodium 32 110 31 75 100 91 80 59 33

Sulfate 34 300 98 220 290 250 220 190 110

Chloride 34 40 2.9 23 35 28 22 18 11

12 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Table 6. Regression equations for estimates of hardness in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; CaCO3, hardness, in milligrams per liter as calcium carbonate; SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2

, coefficient of multiple determination; median RPD, median relative percentage 

difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable

R2 Median 

RPD BCF

1 -- -- -- -- -- --

2 -- -- -- -- -- --

3 -- -- -- -- -- --

4 45 log . log . 10 3 10 (CaCO ) = 0 991 ( ) SC − 0 501 SC: 311–1,372 0.915 7.7 1.0081

5 64 log . log . 10 3 10 (CaCO ) = 1 040 ( ) SC − 0 601 SC: 295–1,600 .947 6.1 1.0041

6 65 log . log . 10 3 10 (CaCO ) = 1 003 ( ) SC − 0 489 SC: 438–1,320 .936 3.9 1.0020

7 30 log . log . 10 3 10 (CaCO ) = 1 016 ( ) SC − 0 482 SC: 449–1,293 .942 5.8 1.0020

Table 7. Regression equations for estimates of dissolved solids in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; DS, dissolved solids concentration, in milligrams per liter; SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2

, coefficient of multiple determination; median RPD, median relative percentage 

difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable

R2 Median 

RPD BCF

1 127 log . log . 10 10 ( ) DS = 1 054 ( ) SC − 0 352 SC: 320–2,300 0.986 2.2 1.0011

2 -- -- -- -- -- --

3 -- -- -- -- -- --

4 45 log . log . 10 10 ( ) DS = 1 098 ( ) SC − 0 488 SC: 311–1,372 .993 1.7 1.0008

5 64 log . log . 10 10 ( ) DS = 1 065 ( ) SC − 0 386 SC: 320–2,300 .991 2.2 1.0007

6 65 log . log . 10 10 ( ) DS = 1 046 ( ) SC − 0 328 SC: 295–1,600 .997 2.6 1.0007

7 30 log . log . 10 10 ( ) DS = 1 032 ( ) SC − 0 290 SC: 449–1,380 .966 2.7 1.0012

for outliers or highly influential points. For site 5, the sample 

from March 28, 1991, appeared to be an outlier in regression 

diagnostic plots and the sample from March 31, 2004, was 

highly influential in the estimation of regression parameters 

based on Cook’s distance and jackknife after bootstrap; both 

samples were removed. For site 6, the sample from November 21, 2005, appeared to be an outlier in residual plots 

and had a large Cook’s distance; therefore, the sample was 

removed from the regression analysis.

Results of Regression Analysis 13
Table 8. Regression equations for estimates of calcium in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; Ca, calcium concentration, in milligrams per liter; Q, streamflow, in cubic feet per second; SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2

, coefficient of multiple determination; median RPD, median 

relative percentage difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable(s)

R 2 Median 

RPD BCF

1 -- -- -- -- -- --

2 -- -- -- -- -- --

3 -- -- -- -- -- --

4 45 log . log . 10 10 ( ) Ca = 0 770 ( ) SC − 0 585 SC: 311–1,372 0.784 9.5 1.0148

5 65

log . log . log

.

( ) 10 10 0 040 0 818

0 548

Ca 10 ( ) = − Q + ( ) SC

−

Q: 7.6–4,260

SC: 280–1,600 .909 7.6 1.0065

6 65

log . log . log

.

( ) 10 10 0 014 0 856

0 790

Ca 10 ( ) = Q + ( ) SC

−

Q: 8.2–3,050

SC: 438–1,362 .839 5.7 1.0040

7 27

log . log . log

.

( ) 10 10 0 061 0 708

0 080

Ca 10 ( ) = − Q + ( ) SC

−

Q: 60.2–3,810

SC: 449-1,380 .884 5.3 1.0037

Calcium

Calcium contributes to hardness, is “a major component 

of the solutes in most natural waters,” and it is generally “the 

predominant cation in river water” (Hem, 1985). Previous 

studies have shown a positive linear relation between calcium 

and specific conductance (Christensen and others, 2003; 

Rasmussen and others, 2005; Christensen and others, 2006). 

Specific conductance was a statistically significant predictor variable for calcium in the Sheyenne River at sites 4, 5, 6, 

and 7 (table 8). Streamflow was also a statistically significant 

predictor variable for calcium in the Sheyenne River at sites 5, 

6, and 7. Despite logarithmic transformation of the explanatory and response variables, site 1 had a nonlinear regression 

relation; therefore, a relation is not reported for calcium at 

site 1. This supports the lack of relation found for hardness 

as well. Calcium was not analyzed for a regression relation at 

sites 2 and 3 because of insufficient data.

Magnesium

Magnesium contributes to hardness and is “a common 

element essential in plant and animal nutrition” (Hem, 

1985). Previous studies have shown a positive linear relation 

between magnesium and specific conductance (Rasmussen 

and others, 2004; Christensen and others, 2006). Specific 

conductance was a statistically significant predictor variable 

for magnesium in the Sheyenne River at sites 4, 5, 6, and 

7 (table 9). Streamflow was also a statistically significant 

predictor variable for magnesium in the Sheyenne River at site 

5. Despite logarithmic transformation of the explanatory and 

response variables, site 1 showed nonconstant variance in the 

residuals; therefore a relation is not reported for magnesium at 

site 1. This supports the lack of relation found for hardness as 

well. Magnesium was not analyzed for a regression relation at 

sites 2 and 3 because of insufficient data.

Sodium

Sodium in drinking water may be a concern for individuals on sodium restricted diets and high concentrations of 

sodium may make water unsuitable for irrigation (North 

Dakota Department of Health, 2006). Human activities, 

such as deicing of highways in the winter, may influence the 

concentrations of sodium in ground and surface water (Hem, 

1985).

Previous studies have shown a positive linear relation 

between sodium and specific conductance, as well as multiple regression relations that include pH and streamflow in 

addition to specific conductance (Christensen and others 2003; 

Rasmussen and others, 2004; Christensen and others, 2006). 

Specific conductance was a statistically significant predictor 

variable for sodium in the Sheyenne River at sites 1, 4, 5, 6, 

and 7 (table 10). Streamflow was also a statistically significant 

predictor variable for sodium in the Sheyenne River at sites 1 

and 6. Sodium was not analyzed for a regression relation at 

sites 2 and 3 because of insufficient data.

14  Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Table 9. Regression equations for estimates of magnesium in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; Mg, magnesium concentration, in milligrams per liter; Q, streamflow, in cubic feet per 

second; SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2, coefficient of multiple determination; median RPD, median 

relative percentage difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable(s)

R 2 Median 

RPD BCF

1 -- -- -- -- -- --

2 -- -- -- -- -- --

3 -- -- -- -- -- --

4 45 log . log . 10 10 ( ) Mg = 1 205 ( ) SC − 2 016 SC: 311–1,372 0.953 6.6 1.0062

5 65

log . log . log

.

( ) 10 10 0 038 1 232

2 181

Mg 10 ( ) = Q + ( ) SC

−

Q: 7.6–4,260

SC: 280–1,600 .949 6.8 1.0054

6 65 log . log . 10 10 ( ) Mg = 1 144 ( ) SC −1 814 SC: 438–1,320 .944 4.9 1.0023

7 28 log . log . 10 10 ( ) Mg = 1 079 ( ) SC −1 627 SC: 449–1,380 .955 4.9 1.0018

Sulfate

Sulfate sources include the weathering of rocks, agricultural runoff, fuel combustion, municipal and industrial effluent, and precipitation (Hem, 1985; Allan, 1995). The presence 

of too much sulfate has three main undesirable effects in water 

used by humans: (1) sulfate can have a laxative effect with 

excessive intake, (2) water with high sulfate concentrations 

can form hard scales in boilers, and (3) sulfate can negatively 

affect taste (North Dakota Department of Health, 2006). 

Sulfate is negatively charged and increased sulfate concentrations increase specific conductance (Hem, 1985). Streamflow 

was statistically significant at sites 1, 4, and 5 only (table 11).

Jackknife after bootstrap, Cook’s distance, and regression plots identified the March 30, 2005, sample at site 1, the 

November 21, 2005, sample at site 6, and the September 16, 

1993, sample at site 7 as highly influential points. They were 

removed from the regression analyses.

For sites 2 and 3, the period of record is small compared 

to the other sites and it is a period of record with relatively 

low flow and high specific conductance. Additional high flow 

data might result in the inclusion of the streamflow term in 

the regression equation, which is present in the equations for 

the upstream and downstream sites (1 and 4). The equations 

presented in table 11 should not be extrapolated beyond the 

range of explanatory variables used to develop the regression 

equations.

Additional high flow data might include concentrations 

affected by storm runoff and sites 1, 2, and 3 may be more 

affected by storm runoff than sites farther downstream. The 

storm effect would depend on the location of the storm. The 

north fork of the Sheyenne River, which joins the Sheyenne 

River between Harvey and Flora (fig. 1) is less sulfatic than 

the Sheyenne River (W.M. Schuh, North Dakota State Water 

Commission, written commun., 2007). Therefore, increased 

streamflow at sites 2 and 3 caused by storm runoff may have 

varying effects on sulfate because of the location of the storm.

In addition to the numeric measurements used to evaluate 

the regression relations, measured and estimated concentrations were compared graphically to evaluate the usefulness 

of the regression relations for prediction using continuously 

recorded streamflow and specific conductance data at each 

gage. The permit to release water from the Devils Lake outlet 

into the Sheyenne River is based in part on sulfate concentration; therefore, there is a great deal of interest in sulfate in the 

Sheyenne River Basin and sulfate was used to illustrate the 

graphical comparison of measured and estimated concentrations in figures 3 and 4. For sites 1 and 4, continuous recording of specific conductance began in May 2006, resulting in 

little data for continuous prediction in 2006; therefore, sites 1 

and 4 are not compared graphically. For sites 2 and 3, the 

daily mean of the continuously recorded specific conductance and streamflow data at each site from January 1, 2005, 

through December 31, 2006, and the regression relations 

for sulfate were used to predict sulfate concentrations along 

with 90-percent prediction intervals, representing the error in 

the prediction of sulfate. The sample concentrations used to 

develop the regression relations were plotted as points as well 

as the sample concentrations that were removed when the data 

were thinned. The regression relations for sites 2 and 3 were 

developed using samples collected by the NDSWC; however, 

the USGS also collected samples at these sites as an independent verification of the NDSWC’s sampling program and the 

USGS sample concentrations also are plotted as points.

Results of Regression Analysis 15
Table 10. Regression equations for estimates of sodium in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; Na, sodium concentration, in milligrams per liter; Q, streamflow, in cubic feet per second; 

SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2, coefficient of multiple determination; median RPD, median relative 

percentage difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable(s)

R2 Median 

RPD BCF

1 125

log . log . log

.

( ) 10 10 0 120 1 168

1 217

Na 10 ( ) = − Q + ( ) SC

−

Q: 0.35–362

SC: 320–2,300 0.952 7.5 1.0076

2 -- -- -- -- -- --

3 -- -- -- -- -- --

4 43 log . log . 10 10 ( ) Na = 1 379 ( ) SC − 2 069 SC: 311–1,372 .958 9.2 1.0072

5 65 log . log . 10 10 ( ) Na = 1 282 ( ) SC −1 857 SC: 280–1,600 .938 9.0 1.0083

6 64

log . log . log

.

( ) 10 10 0 029 1 235

1 671

Na 10 ( ) = − Q + ( ) SC

−

Q: 8.2–3,050

SC: 438–1,362 .886 7.5 1.0058

7 28 log . log . 10 10 ( ) Na = 1 242 ( ) SC −1 827 SC: 449–1,380 .876 8.8 1.0075

Table 11. Regression equations for estimates of sulfate in the Sheyenne River, N. Dak., 1980−2006.

[n, numbers of samples used to develop regression equation; SO4, sulfate concentration, in milligrams per liter; Q, streamflow. in cubic feet per second; 

SC, specific conductance, in microsiemens per centimeter at 25 degrees Celsius; R2, coefficient of multiple determination; median RPD, median relative 

percentage difference; BCF, bias correction factor; --, not applicable]

Site 

number n Equation

Range of 

explanatory 

variable(s)

R2 Median RPD BCF

1 125 log . log . log

.

( ) 10 4 10 10 0 161 1 492

2 414

(SO ) = Q + ( ) SC

−

Q: 0.35–362

SC: 320–2,300

0.894 8.0 1.0108

2 11 log . log . 10 4 10 (SO ) = 1 251 ( ) SC −1 434 SC: 575–1,760 .922 8.9 1.0059

3 11 log . log . 10 4 10 (SO ) = 1 021 ( ) SC − 0 692 SC: 625–1,812 .850 8.2 1.0072

4 45 log . log . log

.

( ) 10 4 10 10 0 126 1 475

2 420

(SO ) = Q + ( ) SC

−

Q: 9–2,120

SC: 311–1,372

.898 7.9 1.0145

5 66 log . log . log

.

( ) 10 4 10 10 0 096 1 308

1 848

(SO ) = Q + ( ) SC

−

Q: 7.6–4,260

SC: 280–1,600

.893 9.8 1.0108

6 65 log . log . 10 4 10 (SO ) = 1 318 ( ) SC −1 635 SC: 438–1,362 .820 11.5 1.0107

7 28 log . log . 10 4 10 (SO ) = 1 026 ( ) SC − 0 722 SC: 449–1,380 .928 5.2 1.0026

16 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Figure 3. Predicted sulfate concentration, 90-percent prediction interval, and individual samples used in regression analyses for sites 2 and 3 on the Sheyenne River, 

N. Dak., 2005–06.

 0

 800

 600

 400

 200

 1,000

 0

 800 600

 400

 200

 1,000

D N O S A J J M A M F J D N O S A J J M A M F J

Estimated sulfate concentration90-percent prediction interval

U.S. Geological Survey sample

SULFATE CONCENTRATION, IN MILLIGRAMS PER LITER

Sample used to develop regression relationSample removed when data thinned

Site 2Site 3

EXPLANATION

Results of Regression Analysis 17
Figure 4. Predicted sulfate concentration, 90-percent prediction interval, and individual samples used in regression analyses for sites 5, 6, and 7 on the Sheyenne 

River, N. Dak., 2000–2006.

 0

 800 700 600 500 400 300 200 100

 0

 800 700 600 500 400 300 200 100

 0

 800 700 600 500 400 300 200 100

2000 2001 2002 2003 2004 2005 2006 2007

Estimated sulfate concentration

EXPLANATION

90-percent prediction intervalSample used to develop regression relationSample removed when data thinned

SULFATE CONCENTRATION, IN MILLIGRAMS PER LITER

Site 5Site 6Site 7

18 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Figure 5. Comparison of measured and estimated sulfate loads for site 7 on the Sheyenne River, N. Dak., 1997–2006.

 10,000

 100,000

 1,000,000

 10,000,000

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

SULFATE LOAD, IN POUNDS PER DAY

Estimated load

Measured load, sample used to develop regression relation

Measured load, sample removed when data thinned

For sites 5, 6, and 7, the daily mean of the continuously 

recorded specific conductance and streamflow data from 

January 1, 2000, through December 31, 2006, and the regression relations for sulfate were used to predict sulfate concentrations at each site along with 90-percent prediction intervals, 

representing the error in the prediction of sulfate. The sample 

concentrations used to develop the regression relations were 

plotted as points, as well as the sample concentrations that 

were removed when the data were thinned. Breaks in the 

prediction lines (figs. 3 and 4) indicate periods in which one 

or more of the predictor variables were unavailable or periods 

in which streamflow fell outside the bounds defined in the 

“Manual Water-Quality Measurements, Sample Collection, 

and Analysis” section of this report.

Measured and Estimated Constituent 

Loads

Because the explanatory variables used in the regression 

equations (tables 6−11) are continuously monitored properties, the regression relations developed in this study may be 

used to estimate daily constituent loads in the Sheyenne River. 

As an example of this application of the relations, the daily 

load for sulfate at site 7 is shown in figure 5. Measured daily 

load was computed and the daily load was estimated using 

the regression equation (table 11) and the daily mean streamflow and specific conductance. Breaks in the line representing 

estimated daily load indicate periods in which one or more of 

the explanatory variables was unavailable.

The estimated loads and measured loads compare well 

and the estimated load illustrates the high degree of variability 

in daily sulfate load. The peak load generally occurs in the 

second quarter of the year during spring runoff, April–June.

Summary

The U.S. Geological Survey, in cooperation with the 

North Dakota State Water Commission (NDSWC), analyzed 

data collected at seven sites on the Sheyenne River to determine whether streamflow and the continuously recorded 

physical properties could act as surrogates to estimate waterquality constituents that are important indicators of surfacewater quality. Estimation of water-quality constituents on 

Summary 19
the basis of surrogates provides several benefits. Although 

periodic Sheyenne River water-quality samples are collected 

manually and analyzed, the delay associated with laboratory analysis does not permit immediate identification of 

undesirable concentrations of constituents. A relation between 

manually collected water-quality samples and real-time 

water-quality measurements allows immediate identification 

of potential water-quality problems. Examination of streamflow and physical properties of water that act as surrogates 

for constituents of interest also helps optimize visits for the 

collection of water-quality samples.

Data collection and analysis methods, as well as the 

periods of record for manual and continuous water-quality 

monitoring, vary among the sites. This variability was 

documented as part of the study and should be considered 

when applying the regression relations.

Constituents for which regression relations were developed were hardness, dissolved solids, calcium, magnesium, 

sodium, and sulfate. Potential explanatory variables were the 

continuously recorded water properties streamflow, specific 

conductance, and water temperature. Each regression relation 

was evaluated for usefulness based on the R2 and median 

relative percentage differences (RPDs).

The base-10 logarithm of hardness was a function of the 

base-10 logarithm of specific conductance for sites 4–7. The 

hardness relations were characterized by R2 of 0.915–0.947 

and median RPDs of 3.9–7.7.

The base-10 logarithm of dissolved solids was a function 

of the base-10 logarithm of specific conductance for sites 1 

and 4–7. The dissolved solids relations were characterized by 

R2 of 0.966–0.997 and median RPDs of 1.7–2.7.

The base-10 logarithm of calcium was a function of 

the base-10 logarithm of specific conductance for site 4 and 

a function of the base-10 logarithm of streamflow and the 

base-10 logarithm of specific conductance for sites 5–7. The 

calcium relations were characterized by R2 of 0.784–0.909 and 

median RPDs of 5.3–9.5.

The base-10 logarithm of magnesium was a function of 

the base-10 logarithm of specific conductance for sites 4, 6, 

and 7 and a function of the base-10 logarithm of streamflow 

and the base-10 logarithm specific conductance for site 5. The 

magnesium relations were characterized by R2 of 0.944–0.955 

and median RPDs of 4.9–6.8.

The base-10 logarithm of sodium was a function of the 

base-10 logarithm of streamflow and the base-10 logarithm 

of specific conductance for sites 1 and 6 and a function of the 

base-10 logarithm of specific conductance for sites 4, 5, and 7. 

The sodium relations were characterized by R2 of 0.876–0.958 

and median RPDs of 7.5–9.2.

The base-10 logarithm of sulfate was a function of the 

base-10 logarithm of streamflow and the base-10 logarithm 

of specific conductance for sites 1, 4, and 5 and a function 

of the base-10 logarithm of specific conductance for sites 2, 

3, 6, and 7. The sulfate relations were characterized by R2 of 

0.820–0.928 and median RPDs of 5.2–11.5.

The regression relations may be used to continuously 

estimate constituent concentrations in the Sheyenne River and 

these estimates may be used to continuously estimate concentration loads. The relations should be monitored for change 

over time, especially at sites 2 and 3 which have a short 

period of record. In addition, caution should be used when the 

Sheyenne River is affected by ice and when the upstream sites 

(sites 1, 2, and 3) are affected by isolated storm runoff. Almost 

all of the outliers and highly influential samples removed 

from the analysis were from periods when the Sheyenne River 

might be affected by ice.

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Prepared by the Helena Publishing Service Center.

For more information concerning the research in this report, contact:

Director, U.S. Geological Survey

North Dakota Water Science Center

821 East Interstate Avenue

Bismarck, North Dakota 58503

(701) 250-7400

http://nd.water.usgs.gov/

22 Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006
Ryberg—Continuous Water-Quality Monitoring and Regression Analysis, Sheyenne River, 1980–2006—Scientific Investigations Report 2007–5153

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