U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2006-5238

Prepared in cooperation with the 

North Dakota State Water Commission, Devils Lake Basin Joint Water Resource Board, 

and Red River Joint Water Resource District

Water-Quality Trend Analysis and Sampling Design for the 

Devils Lake Basin, North Dakota, January 1965 Through 

September 2003
Water-Quality Trend Analysis and 

Sampling Design for the Devils Lake Basin, 

North Dakota, January 1965 Through 

September 2003

By Karen R. Ryberg and Aldo V. Vecchia

Prepared in cooperation with the 

North Dakota State Water Commission, Devils Lake Basin Joint Water Resource 

Board, and Red River Joint Water Resource District

Scientific Investigations Report 2006–5238

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2006

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Suggested citation:

Ryberg, K.R., and Vecchia, A.V., 2006, Water-quality trend analysis and sampling design for the Devils Lake 

Basin, North Dakota, January 1965 through September 2003: U.S. Geological Survey Scientific Investigations 

Report 2006-5238, 64 p.
iii

Contents

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

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

Description of Devils Lake Basin.................................................................................................................5

Streamflow and Concentration Data Used for Water-Quality Trend Analysis ....................................7

Time-Series Model Used for Water-Quality Trend Analysis.................................................................11

Analysis of Streamflow-Related Variability in Concentration Data.....................................................13

Water-Quality Trend Analysis ....................................................................................................................26

Dissolved Major Ions..........................................................................................................................26

Nutrients...............................................................................................................................................29

Trace Element......................................................................................................................................29

Sampling Design ..........................................................................................................................................34

Summary........................................................................................................................................................39

References Cited..........................................................................................................................................43

Appendix........................................................................................................................................................45

Parametric Flow-Adjustment Model ...............................................................................................46

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen...........................................47

Variogram Analysis of Flow-Adjusted Concentrations.................................................................58

Model Robustness..............................................................................................................................60

Figures

1–2. Maps showing:

1. Location of the Devils Lake Basin in North Dakota. ......................................................2

2. Locations of stations used for water-quality trend analysis. .......................................3

3–22. Graphs showing:

3. Elevation of Devils Lake, North Dakota, June 30, 1867, through September 30, 2003. ..........................................................................................................................4

4. Daily streamflow for 1960 through September 2003 for the Mauvais Coulee 

near Cando, North Dakota, and Edmore Coulee near Edmore, North Dakota, 

stations. .................................................................................................................................6

5. Calcium concentrations for 1965 through September 2003 for the Lake Irvine 

near Churchs Ferry, North Dakota, station....................................................................11

6. Streamflow anomalies for the Mauvais Coulee near Cando, North Dakota, and 

Edmore Coulee near Edmore, North Dakota, stations.................................................12

7. Measured and fitted dissolved sulfate concentrations for 1970 through 

September 2003 for the Mauvais Coulee near Cando, North Dakota, station.........14

8. Flow-adjusted dissolved sulfate concentrations for 1970 through 

September 2003 for the Mauvais Coulee near Cando, North Dakota, station.........14

9. Partial residuals of dissolved sulfate concentrations for the Mauvais Coulee 

near Cando, North Dakota, station in relation to streamflow anomalies. ................16

10. Measured and fitted dissolved ammonia concentrations for 1985 through 

September 2003 for the Edmore Coulee near Edmore, North Dakota, station. .......17
iv

11. Flow-adjusted dissolved ammonia concentrations for 1985 through 

September 2003 for the Edmore Coulee near Edmore, North Dakota, station. .......17

12. Partial residuals of dissolved ammonia concentrations for the Edmore Coulee 

near Edmore, North Dakota, station in relation to streamflow anomalies...............18

13. Measured and fitted dissolved nitrite plus nitrate concentrations and 

censored values for 1980 through September 2003 for the Mauvais Coulee 

near Cando, North Dakota, station .................................................................................19

14. Flow-adjusted dissolved nitrite plus nitrate concentrations and censored 

values for 1985 through September 2003 for the Mauvais Coulee near Cando, 

North Dakota, station ........................................................................................................19

15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality trend analysis.........................................20

16. Fitted trends for flow-adjusted dissolved calcium concentrations ..........................30

17. Fitted trends for flow-adjusted dissolved sulfate concentrations ............................31

18. Fitted trends for flow-adjusted dissolved chloride concentrations..........................32

19. Fitted trends for flow-adjusted dissolved nitrite plus nitrate concentrations.........33

20. Fitted trends for flow-adjusted dissolved ammonia concentrations ........................35

21. Fitted trends for flow-adjusted total phosphorus concentrations ............................37

22. Fitted trends for flow-adjusted dissolved strontium concentrations........................38

A1. Dissolved nitrite plus nitrate concentrations for the Lake Alice near Churchs 

Ferry, North Dakota, station.............................................................................................55

A2. Dissolved nitrite plus nitrate concentrations for the Mauvais Coulee near 

Cando, North Dakota, station...........................................................................................55

A3. Dissolved nitrite plus nitrate concentrations and censored values for the 

Mauvais Coulee near Cando, North Dakota, station ...................................................56

A4. Fourth root of squared residuals for flow-adjusted dissolved sulfate concentrations for the Starkweather Coulee near Webster, North Dakota, station ...........59

A5. Fourth root of squared residuals for flow-adjusted dissolved chloride concentrations for the Edmore Coulee near Edmore, North Dakota, station .......................59

A6. Estimated variogram for flow-adjusted dissolved sulfate concentrations for the 

Big Coulee near Churchs Ferry, North Dakota, station for season 1 (March 

through May)......................................................................................................................63

A7. Estimated variogram for flow-adjusted dissolved sulfate concentrations for 

the Big Coulee near Churchs Ferry, North Dakota, station for season 2 (June 

through November) ...........................................................................................................63

A8. Estimated variogram for flow-adjusted total phosphorus concentrations for 

the Channel A near Penn, North Dakota, station for season 1 (March through 

May).....................................................................................................................................64

A9. Estimated variogram for flow-adjusted total phosphorus concentrations for 

the Channel A near Penn, North Dakota, station for season 2 (June through 

November) ..........................................................................................................................64
Tables

1. Stations used for water-quality trend analysis........................................................................8

2. Constituents used for water-quality trend analysis................................................................8

3. Stations and constituents used for water-quality trend analysis and number of 

samples for concentration data .................................................................................................9

4. Stations and constituents used to test significance of Channel A step trend and 

linear trend from 1980 through September 2003....................................................................27

5. Number of samples required to obtain specified error tolerance for estimated median 

concentrations ....................................................................................................................................41

A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations 

(sites 3, 5, and 6)..........................................................................................................................49

A2. Fitted survival regression model coefficients for dissolved nitrite plus nitrate for 

stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).......................57

A3. Estimated seasonal standard deviations and correlation ranges for flow-adjusted 

concentrations ............................................................................................................................61

A4. Results of test of significance for Channel A step trend and linear trend from 1980 

through September 2003 for the Big Coulee near Churchs Ferry, North Dakota, 

station ...........................................................................................................................................62

Conversion Factors, Abbreviations, and Datum

Multiply By To obtain

Length

foot (ft) 0.3048 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)

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

°F = (1.8 x °C) + 32

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

°C = (°F - 32) / 1.8

Vertical coordinate information is referenced to the National Geodetic Vertical Datum of 1929 

(NGVD 29).

Horizontal coordinate information is referenced to the North American Datum of 1927 (NAD 27).

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 either in milligrams per liter (mg/L) 

or micrograms per liter (μg/L).
Water-Quality Trend Analysis and Sampling Design for the 

Devils Lake Basin, North Dakota, January 1965 Through 

September 2003

By Karen R. Ryberg and Aldo V. Vecchia

Abstract

This report presents the results of a study conducted by 

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

Dakota State Water Commission, the Devils Lake Basin 

Joint Water Resource Board, and the Red River Joint Water 

Resource District, to analyze historical water-quality trends in 

three dissolved major ions, three nutrients, and one dissolved 

trace element for eight stations in the Devils Lake Basin in 

North Dakota and to develop an efficient sampling design to 

monitor the future trends.

A multiple-regression model was used to detect and 

remove streamflow-related variability in constituent concentrations. To separate the natural variability in concentration as 

a result of variability in streamflow from the variability in concentration as a result of other factors, the base-10 logarithm 

of daily streamflow was divided into four components—a 

5-year streamflow anomaly, an annual streamflow anomaly, a 

seasonal streamflow anomaly, and a daily streamflow anomaly. The constituent concentrations then were adjusted for 

streamflow-related variability by removing the 5-year, annual, 

seasonal, and daily variability. Constituents used for the waterquality trend analysis were evaluated for a step trend to examine the effect of Channel A on water quality in the basin and a 

linear trend to detect gradual changes with time from January 

1980 through September 2003.

The fitted upward linear trends for dissolved calcium concentrations during 1980-2003 for two stations were significant. 

The fitted step trends for dissolved sulfate concentrations for 

three stations were positive and similar in magnitude. Of the 

three upward trends, one was significant. The fitted step trends 

for dissolved chloride concentrations were positive but insignificant. The fitted linear trends for the upstream stations were 

small and insignificant, but three of the downward trends that 

occurred during 1980-2003 for the remaining stations were 

significant. The fitted upward linear trends for dissolved nitrite 

plus nitrate as nitrogen concentrations during 1987-2003 for 

two stations were significant. However, concentrations during recent years appear to be lower than those for the 1970s 

and early 1980s but higher than those for the late 1980s and 

early 1990s. The fitted downward linear trend for dissolved 

ammonia concentrations for one station was significant. The 

fitted linear trends for total phosphorus concentrations for two 

stations were significant. Upward trends for total phosphorus concentrations occurred from the late 1980s to 2003 for 

most stations, but a small and insignificant downward trend 

occurred for one station. Continued monitoring will be needed 

to determine if the recent trend toward higher dissolved nitrite 

plus nitrate as nitrogen and total phosphorus concentrations 

continues in the future.

For continued monitoring of water-quality trends in the 

upper Devils Lake Basin, an efficient sampling design consists 

of five major-ion, nutrient, and trace-element samples per 

year at three existing stream stations and at three existing 

lake stations. This sampling design requires the collection of 

15 stream samples and 15 lake samples per year rather than 

16 stream samples and 20 lake samples per year as in the 

1992-2003 program. Thus, the design would result in a program that is less costly and more efficient than the 1992-2003 

program but that still would provide the data needed to monitor water-quality trends in the Devils Lake Basin.

Introduction

The Devils Lake Basin in North Dakota is a 3,810-

square-mile closed subbasin in the Red River of the North 

(Red River) Basin (fig. 1). About 3,320 square miles of the 

total 3,810 square miles is tributary to Devils Lake, and the 

remainder is tributary to East and West Stump Lakes (fig. 2). 

The Devils Lake Basin contributes naturally to the Red River 

Basin only when the elevation of Devils Lake is greater than 

1,459 feet. At an elevation of about 1,447 feet, Devils Lake 

begins to spill into the Stump Lakes; and at an elevation of 

about 1,459 feet, the combined lakes begin to spill through 

Tolna Coulee into the Sheyenne River, a tributary to the Red 

River. Devils Lake is characterized by large fluctuations in 

elevation (fig. 3) and in concentrations of dissolved chemical 

constituents. From 1867 through September 2003, the elevation ranged from 1,400.9 feet in October 1940 to 1,448.3 feet 

in July 2001. The elevation recorded in July 2001 was about 

24.7 feet higher than the elevation recorded in July 1993.
Figure 1. Location of the Devils Lake Basin in North Dakota.

0 40 80 120 MILES

Base from U.S. Geological Survey

1:2,000,000, 1972

103°

101° 99° 97°

95°

93°

91°

49°

47°

45°

43°

43°

45°

47°

49°

51°

91°

95° 93°

97°

99° 101° 103°

51°

Lake

of the

Woods

 Upper

Red

Lake Devils

 Lake

Mille

Lacs

Lake

Sakakawea

Lake

Oahe

Lower

Red

Lake 

Lake

River

Souris

Red

River

of

the

North

Minnesota

River

River

Mississippi

Missouri

River

River

Missouri

James

River

Assiniboine

River

Big

Sioux

River

CANADA

UNITED STATES

Manitoba

Winnipeg

Lake Lake

Winnipegosis

Lake

Minot

BISMARCK Jamestown Moorhead Fargo

Wahpeton

Grand

Forks

Valley

 City

Emerson

WINNIPEG

Halstad

 EXPLANATION

Red River of the North Basin

(excluding Assiniboine Basin)

Devils Lake Basin

MANITOBA

ONTARIO

NORTH DAKOTA

SOUTH DAKOTA

MINNESOTA

SASKATCHEWAN

   Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 2. Locations of stations used for water-quality trend analysis.

Channel A

St. Joe Coulee

East

Stump

Lake

Lake

Laretta

Lake Mary

McHugh

Rose Lake

Slough

Swan

Lake

Tolna

Coulee

West

Stump

 Lake

Sixmile Bay

Creel Bay

Big Coulee

Little

Coulee

Pelican Lake

Coulee

Comstock

Peterson

Coulee

DEVILS

SHEYENNE

RIVER

LAKE

 East Devils Lake

Bay

East

Main Bay

West

Bay

Mission

Bay

Fort

Totten

Bay

BlackTiger Bay

Ziebach

Pass

LakeIrvine

Ibsen Lake

LakeAlice

Chain Lake

Mikes Lake

Coulee

Starkweather

Webster Coulee

Morrison Lake

Cavanaugh Lake

Sweetwater Lake

 DryLake

Mauvais

Mauvais

Coulee

Coulee

Tributary

Tributary #3

Edmore

Edmore

Calio

Coulee

Coulee

Coulee

Penn

Sweetwater

Lakota

Michigan

Churchs

Ferry

Cando

Edmore

Leeds

Brinsmade

Oberon

Josephine

Webster

Sheyenne

Warwick

FortTotten

Minnewaukan

Devils Lake

8 7

6

5 4

3

2

1

Woods Rutten Road

Grahams Island Road

281

281

57

20

2

2

19

20

Levee

99° 98°30'

48°

48°30'48°15'

47°45'

 EXPLANATIONU.S. Geological Survey station and sitenumber (given in table 1)

SOUTH DAKOTA

MONTANA

MINNESOTA

SASKATCHEWAN MANITOBA

RED RIVER OF THE NORTH

MISSOURI

RIVER

River

River

Sheyenne

James

River

Souris

99° 101° 103°

99° 101° 103°

47° 47°

97°

49° 49°

FARGO

BISMARCK

Devils Lake

Grand

Forks

Minot

Index Map

StudyArea

Levee

0 1 2 3 4 5 10 MILES

2

State

Outlet

Introduction   
Figure 3. Elevation of Devils Lake, North Dakota, June 30, 1867, through September 30, 2003.

1,400

1,405

1,410

1,415

1,420

1,425

1,430

1,435

1,440

1,445

1,450

1860 1880 1900 1920 1940 1960 1980 2000 2020

ELEVATION, IN FEET ABOVE THE NATIONAL GEODETIC VERTICAL DATUM OF 1929

 EXPLANATION

Continuous measurements

(yearly, monthly, and then daily)

Sporadic measurements

 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Maintaining the quality of surface waters in the Devils 

Lake Basin is important for the protection of agricultural 

resources, fisheries, waterfowl and wildlife habitat, and recreation in the basin and for the operation of the State of North 

Dakota Devils Lake outlet. To maintain the quality of surface 

waters in the Devils Lake Basin, information was needed on 

historical water-quality trends in the basin, and an efficient 

sampling design was needed to monitor the future trends. 

Therefore, the U.S. Geological Survey (USGS), in cooperation with the North Dakota State Water Commission, the 

Devils Lake Basin Joint Water Resource Board, and the Red 

River Joint Water Resource District, conducted this study to 

analyze historical water-quality trends in selected constituents 

and selected streams and lakes in the Devils Lake Basin and 

to develop an efficient sampling design to monitor the future 

trends. The Devils Lake outlet transfers water from the Devils 

Lake Basin to the Sheyenne River (fig. 2) to mitigate flooding 

in the Devils Lake Basin. The maximum outlet discharge is 

based, in part, on the sulfate concentration, in milligrams per 

liter, in the Sheyenne River downstream from the outlet insertion point (North Dakota State Water Commission, accessed 

October 17, 2005).

This report, which focuses on upper-basin streams and 

lakes that are tributary to Devils Lake, presents the results 

of the water-quality trend analysis and presents a sampling 

design to monitor future trends in the upper Devils Lake 

Basin. The results presented in the report are based on 

streamflow data for January 1960 through September 2003 

and on concentration data for January 1965 through September 2003. The constituents evaluated for the report include 

three dissolved major ions (calcium, sulfate, and chloride), 

three nutrients (dissolved nitrite plus nitrate as nitrogen, dissolved ammonia, and total phosphorus), and one dissolved 

trace element (strontium). The constituents were evaluated for 

eight stations in the Devils Lake Basin, including three on the 

upstream tributaries, three on the chain of lakes, one on Big 

Coulee, and one on Channel A (fig. 2). All of the constituents 

were evaluated for the stream stations. However, for the lake 

stations, few data are available for some of the constituents. 

Therefore, for those stations, constituents were evaluated on 

the basis of data availability.

Description of Devils Lake Basin

The surface-water drainage system of the Devils Lake 

Basin is a complex system of interconnected tributaries and 

lakes (Wiche and others, 1986). Most surface runoff from the 

upper basin flows through several major tributaries (Mauvais, 

Calio, Starkweather, and Edmore Coulees; fig. 2) into the 

interconnected chain of lakes north of Devils Lake (Sweetwater, Morrison, Dry, Mikes, and Chain Lakes and Lakes Alice 

and Irvine; fig. 2), and outflow from the upstream chain of 

lakes flows into Devils Lake through either Big Coulee or 

Channel A. Before 1979, all outflow from the upstream chain 

of lakes flowed through Big Coulee into Pelican Lake and 

eventually into Devils Lake. However, in 1979, the Ramsey 

County and Cavalier County Water Management Boards 

constructed Channel A, which connects Dry Lake to Sixmile 

Bay of Devils Lake (fig. 2), and a levee was constructed across 

the natural outlet of Dry Lake to Mikes Lake. Thus, since 

1979, outflow from Sweetwater, Morrison, and Dry Lakes has 

flowed through Channel A into Devils Lake and outflow from 

the remaining lakes has flowed through Big Coulee into Devils 

Lake. During extremely large floods, Dry Lake has overtopped 

the levees across the natural outlet, causing some flow to enter 

the natural watercourse.

Since the end of glaciation about 10,000 years ago, 

Devils Lake has fluctuated between the extremes of spilling into the Sheyenne River and being dry. Research by the 

North Dakota Geological Survey indicates Devils Lake has 

overflowed into the Sheyenne River at least twice during the 

past 4,000 years and has spilled into the Stump Lakes several 

times. The North Dakota State Geologist concluded that the 

natural condition for Devils Lake is either rising or falling, and 

the lake should not be expected to remain at any elevation for 

a long period (Bluemle, 1991; Murphy and others, 1997). This 

natural condition is illustrated, in part, in figure 3. Because 

Devils Lake consists of a series of bays connected by bridge 

openings or culverts, the water quality between the bays can 

differ substantially (Sether and others, 1999).

The elevation of Devils Lake declines during dry conditions and rises during wet conditions because surface runoff 

from the Devils Lake Basin drains into Devils Lake. Except 

for two distinct periods of drought from the late 1950s to the 

late 1960s and from the late 1980s to the early 1990s, conditions from 1950 through September 2003 generally were 

wetter than conditions from 1870 through 1949. The rise in 

elevation during the 1970s and 1980s culminated in August 

1987 when the elevation of Devils Lake reached 1,428.8 feet, 

the highest elevation since the 1880s (fig. 3).

Daily streamflow in Mauvais and Edmore Coulees was 

high during 1970-87 except for occasional dry years (fig. 4). 

A severe drought began late in the summer of 1987, and the 

elevation of Devils Lake declined to 1,422.6 feet by February 

1993 (fig. 3). During the drought, the volume of water in Devils Lake decreased about 37 percent, from 884,000 acre-feet 

in August 1987 to 558,400 acre-feet in February 1993 (Wiche 

and others, 2000). Daily streamflow in Mauvais and Edmore 

Coulees was at or near zero many times during the drought 

(fig. 4).

The drought also caused elevation declines on other 

lakes in the Devils Lake Basin, including on the chain of 

lakes north of Devils Lake. According to Wiche and others 

(2000), “Dry Lake was below the outlet elevation (1,445 feet 

above sea level) to Channel A during most of 1988-92. Crops 

were planted in the lakebed of Lake Irvine during part of the 

drought.... Much-below-average streamflow into the chain of 

lakes and above-average evaporation from the lake surfaces 

during most of the drought resulted in little flow out of the 

chain of lakes into Devils Lake.”

Description of Devils Lake Basin   
Figure 4. Daily streamflow for 1960 through September 2003 for the Mauvais Coulee near Cando, North Dakota, and Edmore Coulee 

near Edmore, North Dakota, stations. STREAMFLOW, IN CUBIC FEET PER SECOND

Mauvais Coulee near Cando, North Dakota

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Edmore Coulee near Edmore, North Dakota

0

-500

500

1,000

1,500

2,000

0

-500

500

1,000

1,500

2,000

2,500

3,000

   Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Since 1993, severe flooding has occurred in the Devils 

Lake Basin. Devils Lake began to spill into the Stump Lakes 

in 2001 and continued to do so through 2003. The elevations 

of Devils Lake and the Stump Lakes were higher in the early 

2000s than at any other time since the 1800s (Wiche and others, 2000).

Streamflow and Concentration Data 

Used for Water-Quality Trend Analysis

From 1957 through September 2003, the USGS collected 

water-quality samples from streams and lakes in the Devils 

Lake Basin in cooperation with the Devils Lake Basin Joint 

Water Resource Board, the North Dakota State Water Commission, the North Dakota Department of Health, the U.S. 

Army Corps of Engineers, and other local, State, and Federal 

agencies. Sample-collection techniques followed published 

USGS protocols (Wilde and others, various dates). The waterquality samples were analyzed by the USGS National Water 

Quality Laboratory, the North Dakota State Water Commission Laboratory, or the North Dakota Department of Health 

Laboratory. The USGS National Water Quality Laboratory 

followed strict, well-documented protocols (U.S. Geological 

Survey, various dates). The North Dakota State Water Commission Laboratory and the North Dakota Department of 

Health Laboratory followed U.S. Environmental Protection 

Agency approved methods and procedures. External agencies and customer organizations audit the State laboratories 

to assure the laboratory analytical methods and quality-assurance/quality-control procedures. The U.S. Environmental Protection Agency reviews the laboratory procedures about every 

third year, and the Board of Quality Systems of the USGS 

reviews the laboratory procedures periodically.

The streamflow and concentration data used for the 

water-quality trend analysis were obtained from the USGS 

National Water Information System (NWIS; http://waterdata.usgs.gov/nd/nwis/). Ryberg and others (2005) described 

much of the data for the study area and presented a graphical user interface for graphical analysis of the data. During 

development of the graphical user interface, the large amount 

of historical data available from the USGS was reduced 

by selecting the stations and constituents that provided the 

most useful and reliable information for the assessment of 

water quality in the Devils Lake Basin. The most recent data 

included were for water year 20031. The amount of historical 

data then was reduced a second time by selecting the stations 

and constituents for which the period of record and the number 

of concentrations was sufficient for trend analysis. The eight 

stations used for the trend analysis are given in table 1, and the 

locations of the stations are shown in figure 2. The constitu1

A water year is defined as the 12-month period from October 1 through 

September 30 and is designated by the calendar year in which it ends. Thus, 

the year ending September 30, 2003, is called water year 2003.

ents used are given in table 2. The number of samples for each 

constituent at each site, the number of samples for specific 

periods during 1965 through September 2003, and the percentage of censored values (values that are known to be less than 

the laboratory reporting level but for which an exact value is 

not known) are given in table 3. Because of the drought of 

1988-92, few or no data are available for many stations from 

1987 through 1992. For example, the scatter plot of calcium 

concentrations for the Lake Irvine near Churchs Ferry station 

(site 6) (fig. 5) shows a gap in the data because no waterquality samples were collected from Lake Irvine between 

October 15, 1986, and March 16, 1993.

Streamflow is important in the analysis of water-quality 

trends because much of the variability in concentration is 

caused by variability in streamflow (Vecchia, 2003). Therefore, for this trend analysis, all available streamflow data for 

the Devils Lake Basin were examined for use as possible 

predictors of water quality. Daily mean streamflow data for the 

Mauvais Coulee near Cando station (site 1) were used for the 

Lake Alice near Churchs Ferry, Lake Irvine, and Big Coulee 

near Churchs Ferry stations (sites 5, 6, and 7, respectively) 

(table 1, fig. 2). Streamflow at those stations is contributed to 

Devils Lake through Big Coulee. Daily mean streamflow data 

for the Edmore Coulee near Edmore station (site 2) were used 

for the Sweetwater Lake at Sweetwater, Starkweather Coulee 

near Webster, and Channel A near Penn stations (sites 3, 4, 

and 8, respectively). Streamflow at those stations was contributed to Devils Lake through Big Coulee until 1979 and now is 

contributed to Devils Lake through Channel A. Streamflows 

for the Mauvais Coulee and Edmore Coulee stations (sites 

1 and 2, respectively) were selected as being representative 

of streamflows in the Devils Lake Basin because of the long 

period of record and geographic location of each station.

The Mauvais Coulee station (site 1) data set contains 

streamflow values for June 1956 through September 2003 with 

seasonal records only since 1982 except for water year 1993 

for which the record is complete. The seasonal records contain 

values for March 1 through September 30 except for 1985 

when data were collected through October 30. The Edmore 

Coulee station (site 2) data set contains streamflow values for 

April through June 1956 and for July 1957 through September 2003 with seasonal records only since 1982 except for 

water year 1993 for which the record is complete. Again, the 

seasonal records contain values for March 1 through September 30 except for 1985 when data were collected through 

October 30. The streamflow data used for the trend analysis 

are for January 1, 1960, through September 30, 2003. The 

streamflow values used are shown in figure 4.

Strontium was the only trace element included in the 

water-quality trend analysis. Although many trace elements 

were considered for inclusion, most were not included because 

of a large percentage of censored values, a small number of 

samples, or both. Dissolved arsenic initially was included but, 

because most of the measured arsenic concentrations were 

reported by the analyzing laboratory as whole numbers, in 

micrograms per liter, the data were coarse and had few unique 

Streamflow and Concentration Data Used for Water-Quality Trend Analysis 
values. Dissolved iron also was included initially but most 

of the measured iron concentrations were reported by the 

analyzing laboratory to two decimal places, in milligrams per 

liter, and then entered into the USGS water-quality database as 

micrograms per liter. The conversion from milligrams per liter 

to micrograms per liter moved the decimal point three places 

to the right and caused the iron data to be coarse and overly 

discrete (for example, 60, 70, 80, 110, and 140). Therefore, 

because use of continuous distribution methods is not appropriate for coarse or overly discrete data, dissolved arsenic and 

dissolved iron concentrations were not included in the trend 

analysis.

Table 1. Stations used for water-quality trend analysis.

Site number 

(figure 2)

U.S. Geological 

Survey station 

number

U.S. Geological Survey station name

Approximate 

contributing drainage area 

(square miles)

Latitude Longitude

1 05056100 Mauvais Coulee near Cando, North Dakota 377 48°26’53” 99°06’08”

2 05056200 Edmore Coulee near Edmore, North Dakota 282 48°20’12” 98°39’36”

3 05056220 Sweetwater Lake at Sweetwater, North Dakota 380 48°12’37” 98°52’15”

4 05056239 Starkweather Coulee near Webster, North Dakota 210 48°19’14” 98°56’25”

5 05056250 Lake Alice near Churchs Ferry, North Dakota 1,600 48°19’33” 99°07’16”

6 05056260 Lake Irvine near Churchs Ferry, North Dakota 1,620 48°16’57” 99°10’25”

7 05056400 Big Coulee near Churchs Ferry, North Dakota 1,462 48°10’40” 99°13’15”

8 05056410 Channel A near Penn, North Dakota 790 48°10’00” 98°58’47”

Table 2. Constituents used for water-quality trend analysis.

Constituent Unit

U.S. Geological Survey

National Water Information System

parameter code

Major ions

Calcium, dissolved Milligrams per liter P00915

Sulfate, dissolved Milligrams per liter P00945

Chloride, dissolved Milligrams per liter P00940

Nutrients

Nitrite plus nitrate as nitrogen, dissolved Milligrams per liter P00631

Ammonia, dissolved Milligrams per liter P00608

Phosphorus, total Milligrams per liter P00665

Trace element

Strontium, dissolved Micrograms per liter P01080

 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Table 3. Stations and constituents used for water-quality trend analysis and number of samples for concentration data.

Site

number

(figure 2)

U.S. Geological Survey station name

Number of samples for Percent of

censored

values

for 1965

through

September

2003

1965 through 

September 

2003

1965-79 1980-89

1990 through

September

2003

Calcium, dissolved

1 Mauvais Coulee near Cando, North Dakota 106 17 29 60 0

2 Edmore Coulee near Edmore, North Dakota 96 12 23 61 0

3 Sweetwater Lake at Sweetwater, North Dakota 83 45 10 28 0

4 Starkweather Coulee near Webster, North Dakota 90 0 29 61 0

5 Lake Alice near Churchs Ferry, North Dakota 73 35 10 28 0

6 Lake Irvine near Churchs Ferry, North Dakota 75 35 10 30 0

7 Big Coulee near Churchs Ferry, North Dakota 198 128 24 46 0

8 Channel A near Penn, North Dakota 75 0 17 58 0

Sulfate, dissolved

1 Mauvais Coulee near Cando, North Dakota 106 17 29 60 0

2 Edmore Coulee near Edmore, North Dakota 96 12 23 61 0

3 Sweetwater Lake at Sweetwater, North Dakota 83 45 10 28 0

4 Starkweather Coulee near Webster, North Dakota 90 0 29 61 0

5 Lake Alice near Churchs Ferry, North Dakota 73 35 10 28 0

6 Lake Irvine near Churchs Ferry, North Dakota 75 35 10 30 0

7 Big Coulee near Churchs Ferry, North Dakota 198 128 24 46 0

8 Channel A near Penn, North Dakota 77 0 17 60 0

Chloride, dissolved

1 Mauvais Coulee near Cando, North Dakota 106 17 29 60 0

2 Edmore Coulee near Edmore, North Dakota 96 12 23 61 0

3 Sweetwater Lake at Sweetwater, North Dakota 83 45 10 28 0

4 Starkweather Coulee near Webster, North Dakota 90 0 29 61 0

5 Lake Alice near Churchs Ferry, North Dakota 73 35 10 28 0

6 Lake Irvine near Churchs Ferry, North Dakota 75 35 10 30 0

7 Big Coulee near Churchs Ferry, North Dakota 198 128 24 46 0

8 Channel A near Penn, North Dakota 77 0 17 60 0

Streamflow and Concentration Data Used for Water-Quality Trend Analysis   
Site

number

(figure 2)

U.S. Geological Survey station name

Number of samples for Percent of

censored

values

for 1965

through

September

2003

1965 through 

September 

2003

1965-79 1980-89

1990 through

September

2003

Nitrite plus nitrate as nitrogen, dissolved

1 Mauvais Coulee near Cando, North Dakota 65 0 12 53 48

2 Edmore Coulee near Edmore, North Dakota 67 0 11 56 39

3 Sweetwater Lake at Sweetwater, North Dakota 67 30 10 27 40

4 Starkweather Coulee near Webster, North Dakota 61 0 12 49 39

5 Lake Alice near Churchs Ferry, North Dakota 57 21 10 26 26

6 Lake Irvine near Churchs Ferry, North Dakota 58 21 10 27 36

7 Big Coulee near Churchs Ferry, North Dakota 118 61 10 47 32

8 Channel A near Penn, North Dakota 67 0 12 55 37

Ammonia, dissolved

1 Mauvais Coulee near Cando, North Dakota 65 0 12 53 12

2 Edmore Coulee near Edmore, North Dakota 66 0 10 56 12

4 Starkweather Coulee near Webster, North Dakota 61 0 12 49 13

7 Big Coulee near Churchs Ferry, North Dakota 57 0 10 47 7

8 Channel A near Penn, North Dakota 65 0 12 53 5

Phosphorus, total

1 Mauvais Coulee near Cando, North Dakota 57 0 3 54 0

2 Edmore Coulee near Edmore, North Dakota 59 0 3 56 0

4 Starkweather Coulee near Webster, North Dakota 54 0 3 51 0

7 Big Coulee near Churchs Ferry, North Dakota 49 0 2 47 0

8 Channel A near Penn, North Dakota 57 0 3 54 0

Strontium, dissolved

1 Mauvais Coulee near Cando, North Dakota 81 0 22 59 0

2 Edmore Coulee near Edmore, North Dakota 80 0 19 61 0

3 Sweetwater Lake at Sweetwater, North Dakota 69 36 5 28 0

4 Starkweather Coulee near Webster, North Dakota 79 0 20 59 0

5 Lake Alice near Churchs Ferry, North Dakota 58 25 5 28 0

6 Lake Irvine near Churchs Ferry, North Dakota 60 25 5 30 0

7 Big Coulee near Churchs Ferry, North Dakota 87 20 20 47 0

8 Channel A near Penn, North Dakota 78 0 17 61 0

Table 3. Stations and constituents used for water-quality trend analysis and number of samples for concentration data.—Continued

10 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
0

50

100

150

200

250

1965 1970 1975 1980 1985 1990 1995 2000 2005

CONCENTRATION, IN MILLIGRAMS PER LITER

Figure 5. Calcium concentrations for 1965 through September 2003 for the Lake Irvine near Churchs Ferry, North Dakota, station.

Time-Series Model Used for WaterQuality Trend Analysis

For this report, a parametric multiple-regression model 

was used to detect and remove the streamflow-related variability in constituent concentrations. The approach used is 

similar to the approach used by Vecchia (2000, 2003, 2005). 

The methods used to fit the regression model are described in 

the appendix.

To separate the natural variability in concentration as a 

result of variability in streamflow from the variability in concentration as a result of other factors, the base-10 logarithm of 

daily streamflow for each station was divided into four components—a 5-year streamflow anomaly, an annual streamflow 

anomaly, a seasonal streamflow anomaly, and a daily streamflow anomaly. The four components were defined as follows:

X t  Cx Lx t Ax t Sx t Dx = + + + +  t

where

X(t) is the base-10 logarithm of the maximum of 

0.1 or the daily streamflow, in cubic feet 

per second, for time t;

t is time, in decimal years;

Cx is equal to mean{X(u), 1960 ≤ u ≤ 2003.75};

Lx(t) is the 5-year streamflow anomaly 

(dimensionless) for time t and is equal to 

mean{X(u), t – 5 < u ≤ t} – Cx;

Ax(t) is the annual streamflow anomaly 

(dimensionless) for time t and is equal to 

mean{X(u), t – 1 < u ≤ t} – mean{X(u), 

t – 5 < u ≤ t};

Sx(t) is the seasonal streamflow anomaly 

(dimensionless) for time t and is equal to 

mean{X(u), t – 0.25 < u ≤ t} – mean{X(u), 

t – 1 < u ≤ t};

and

Dx(t) is the daily streamflow anomaly 

(dimensionless) for time t and is equal to 

X(t) – mean{X(u), t – 0.25 < u ≤ t}.

The 5-year anomaly, Lx(t), represents the 5-year variability 

in streamflow from the long-term mean; the annual anomaly, 

Ax(t), represents the annual variability in streamflow from 

the 5-year mean; the seasonal anomaly, Sx(t), represents the 

seasonal variability in streamflow from the annual mean; and 

the daily anomaly, Dx(t), represents the daily variability in 

streamflow from the seasonal mean. Vecchia (2003) used a 

similar equation to identify natural variability in constituent 

concentrations as a result of variability in streamflow.

To calculate the 5-year anomaly, 5 years of data, up to 

and including time t, were needed. Therefore, the complete 

set of anomalies was computed for January 1, 1965, through 

September 30, 2003. That time period also was used for the 

remainder of the water-quality trend analysis.

The 5-year, annual, seasonal, and daily anomalies for all 

stations used in the water-quality trend analysis are closely 

related. The anomalies for the Mauvais Coulee near Cando 

and Edmore Coulee near Edmore stations (sites 1 and 2, 

respectively) are shown in figure 6. Streamflows for those 

stations are subject to 5-year variability and a high degree of 

interannual variability. The seasonal anomalies indicate the 

highest streamflows generally occurred in June or July except 

during 1992 when the highest streamflows occurred in March 

or April.

Time-Series Model Used for Water-Quality Trend Analysis 11

(1)
Figure 6. Streamflow anomalies for the Mauvais Coulee near Cando, North Dakota, and Edmore Coulee near Edmore, North Dakota, stations.

-1-2

210

-1-2

210

-1-2-3

43210

-1-2-3

543210

1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 20051965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 20051965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005

STREAMFLOW ANOMALIES, AS BASE-10 LOGARITHM OF CUBIC FEET PER SECOND

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

Mauvais Coulee near Cando, North Dakota Edmore Coulee near Edmore, North Dakota

5-year anomaly

Annual anomaly

Seasonal anomaly

Daily anomaly

5-year anomaly

Annual anomaly

Seasonal anomaly

Daily anomaly

12 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
The seasonal anomalies indicated by the straight lines in 

January, February, October, November, and December are for 

years in which little or no streamflow occurred during the winter months. The seasonal anomalies indicated by the lines that 

begin March 1 are for years in which the station was operated 

seasonally and data collection began March 1.

The daily anomalies show a greater degree of highfrequency variability than the remaining anomalies. Individual 

meteorological events, temperature changes, and other factors 

contribute to this variability. The breaks in the daily anomaly 

plots in figure 6 indicate dates for which data are not available 

because of seasonal operation of the station.

Analysis of Streamflow-Related 

Variability in Concentration Data

For a water-quality trend analysis, streamflow-related 

variability generally is considered nuisance variability that 

needs to be removed before the trends are analyzed. If the 

streamflow-related variability is not properly fitted and 

removed, trends might be detected when no trends exist. Flowadjusted concentrations, defined as the residuals from the 

flow-adjustment model plus the intercept (see appendix), are 

estimates of what the actual concentrations would have been if 

no streamflow-related variability occurred (that is, if streamflow conditions were constant) during the entire trend-analysis 

period.

The measured dissolved sulfate concentrations for the 

Mauvais Coulee near Cando station (site 1) and the fitted 

concentrations from the stream-station model for dissolved 

sulfate (table A1) are shown in figure 7. Most of the variability 

in the fitted concentrations (R2 = 76.3 percent; table A1) can 

be attributed to streamflow-related variability. Both long-term 

(interannual) and short-term (seasonal and daily) variability are evident. However, as indicated by the flow-adjusted 

concentrations (fig. 8), no obvious trends exist. Therefore, the 

interannual variability shown in figure 7 for 1988-2003 probably is an artifact of the severe drought that occurred during 

1988-92 and the extreme wet conditions that followed the 

drought.

The relation between concentration and each of the 

streamflow anomalies (eq. 1) can be analyzed by examining partial residual plots. The partial residual plot for a given 

streamflow anomaly shows the fitted relation between the 

log-transformed concentration and the given anomaly after 

the variability caused by the remaining anomalies is removed 

(see appendix). For example, to determine the partial residuals of dissolved sulfate concentrations for the Mauvais Coulee 

station (site 1) for the annual streamflow anomaly, the variability caused by the 5-year, seasonal, and daily streamflow 

anomalies was subtracted from the measured concentrations. 

The resulting residuals are shown in figure 9. The lines in the 

figure correspond to the fitted relation between the partial 

residuals and the given anomaly. For this example, annual 

wet conditions (conditions for the present year are wetter than 

average conditions for the previous 5 years; indicated by a 

positive annual anomaly) tend to result in lower concentrations than do dry conditions (indicated by a negative annual 

anomaly). However, extended wet conditions (indicated by 

a positive 5-year anomaly) tend to have little effect on the 

concentrations. The slope for the 5-year anomaly is 0.028 

(table A1), indicating only a slight increase for the high 5-year 

anomalies. However, the 95-percent confidence interval 

for the slope is (-0.030, 0.086). Therefore, because zero is 

contained in the confidence interval, the 5-year anomaly is 

not significant. Wet conditions on short time scales (less than 

1 year) (indicated by positive seasonal and daily anomalies) 

tend to result in substantially lower concentrations than do dry 

conditions (indicated by negative seasonal and daily anomalies). The negative slopes indicate both the seasonal and daily 

anomalies are highly significant.

The measured dissolved ammonia concentrations for 

the Edmore Coulee near Edmore station (site 2) and the fitted 

concentrations from the stream-station model for dissolved 

ammonia (table A1) are shown in figure 10. Much of the 

variability in the fitted concentrations (R2 = 50.7 percent; 

table A1) can be attributed to streamflow-related variability. 

However, the variability in the concentrations for this station is in sharp contrast to the variability in dissolved sulfate 

concentrations for the Mauvais Coulee station (site 1) (fig. 7). 

For this example, concentrations were high at the end of the 

severe drought (1992) and then decreased after the onset of the 

extreme wet conditions that followed the drought. However, 

as indicated by the flow-adjusted concentrations (fig. 11), no 

obvious trends exist. Therefore, the interannual variability 

shown in figure 11 for 1988-2003 probably is an artifact of the 

severe drought that occurred during 1988-92 and the extreme 

wet conditions that followed the drought.

The partial residuals of dissolved ammonia concentrations for the Edmore Coulee station (site 2) are shown in 

figure 12. Extended wet conditions tend to result in lower 

concentrations than do dry conditions. The slope for the 5-year 

anomaly is -0.322 (table A1), and the 95-percent confidence 

interval is (-0.506, -0.138). Annual wet conditions tend to have 

little effect on the concentrations. The zero slope indicates the 

annual anomaly is not significant. Seasonal wet conditions 

tend to result in substantially lower concentrations than do 

dry conditions. The slope for the seasonal anomaly is -0.295 

(table A1), and the 95-percent confidence interval is (-0.443, 

-0.146). Daily wet conditions tend to have little effect on the 

concentrations. The slope for the daily anomaly is -0.026 

(table A1), indicating only a slight decrease for the high daily 

anomalies. However, the 95-percent confidence interval for the 

slope is (-0.217, 0.165). Therefore, because zero is contained 

in the confidence interval, the daily anomaly is not significant.

The measured dissolved nitrite plus nitrate concentrations for the Mauvais Coulee station (site 1) and the fitted 

concentrations from the stream-station model for dissolved 

nitrite plus nitrate (table A2) are shown in figure 13. Much 

of the variability in the fitted concentrations probably can be 

a

a

Analysis of Streamflow-Related Variability in Concentration Data 13
Figure 8. Flow-adjusted dissolved sulfate concentrations for 1970 through September 2003 for the Mauvais 

Coulee near Cando, North Dakota, station.

Figure 7. Measured and fitted dissolved sulfate concentrations for 1970 through September 2003 for the Mauvais Coulee 

near Cando, North Dakota, station. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1970 1975 1980 1985 1990 1995 2000 2005

1.5

2

2.5

3

3.5

Fitted concentrations

Measured concentrations

CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1970 1975 1980 1985 1990 1995 2000 2005

1.5

2

2.5

3

3.5

14  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
attributed to streamflow-related variability. However, because 

of the large percentage of censored values for this constituent 

(table 3), an alternative technique known as survival regression 

was used to fit the model (see appendix) and the fitted model 

was difficult to evaluate for low concentrations. The flowadjusted concentrations, which in this case were defined as the 

residuals from the survival regression model plus the intercept, 

are shown in figure 14. Although these concentrations also 

were difficult to evaluate, no obvious trends exist.

The effects of adjusting concentrations for streamflowrelated variability are shown in figure 15. The first set of box 

plots shows measured dissolved calcium concentrations for 

1980 through September 2003 for the eight stations used for 

the water-quality trend analysis, and the second set of box 

plots shows flow-adjusted dissolved calcium concentrations. 

The second set of box plots indicates a decrease in streamflow-related variability for all of the stations. Because data are 

not available for some stations before 1980 (table 3), all data 

collected before that year were excluded to ensure a consistent 

period of record for the box plots. In some cases, the number 

of samples (fig. 15) is smaller for the flow-adjusted concentrations than for the measured concentrations because streamflow 

was not always measured when samples were collected.

In addition to reducing the streamflow-related variability 

in the constituent concentrations, the flow-adjustment process 

also may change the location of the distribution of the concentrations. For example, the median measured dissolved calcium concentrations for all of the stations are similar, but the 

median flow-adjusted concentrations for the Sweetwater Lake 

at Sweetwater and Starkweather Coulee stations (sites 3 and 4, 

respectively) are substantially lower than the median flowadjusted concentrations for the remaining stations (fig. 15). 

The median measured concentrations may be unbiased estimates of the actual long-term median concentrations because 

sample-collection times do not represent random sampling 

dates during the year or during a full range of streamflow 

conditions. The flow-adjustment process accounts for the bias 

by adjusting the measured concentrations for seasonality and 

for streamflow-related variability.

The box plots for flow-adjusted dissolved sulfate concentrations (fig. 15) indicate large reductions in the variability of 

the distribution of the flow-adjusted concentrations in relation 

to the distribution of the measured concentrations for most 

stations. As for calcium, the median flow-adjusted sulfate 

concentrations were lower for the Starkweather Coulee station 

(site 4) than for the remaining stations.

The box plots for flow-adjusted dissolved chloride concentrations (fig. 15) indicate a decrease in streamflow-related 

variability and in bias in the measured concentrations for most 

stations. The median flow-adjusted concentrations for the 

Edmore Coulee and Starkweather Coulee stations (sites 2 and 

4, respectively) were most affected by the flow-adjustment 

process. The median flow-adjusted concentrations for the 

Edmore Coulee station (site 2) were considerably higher than 

the median measured concentrations for that station. However, for the Starkweather Coulee station (site 4), the median 

flow-adjusted concentrations were considerably lower than 

the median measured concentrations. The low flow-adjusted 

concentrations for the Starkweather Coulee station (site 4) 

indicate that, for typical streamflow conditions, dissolved 

chloride concentrations for that station are much lower than 

concentrations for the remaining stations.

The box plots for flow-adjusted dissolved ammonia 

concentrations (fig. 15) indicate only a small decrease in 

streamflow-related variability. However, the flow-adjusted 

concentrations for the Big Coulee near Churchs Ferry and 

Channel A near Penn stations (sites 7 and 8, respectively), 

were much more symmetrically distributed than the measured 

concentrations for those stations. Thus, the flow-adjustment 

process also removes potential skewness in log-transformed 

concentrations. The concentrations for the Edmore Coulee 

station (site 2) are a particularly useful example of the effects 

of the flow-adjustment process. Most of the water-quality 

samples for that station were collected from May through 

October and the concentrations generally were low. However, 

the concentrations for February through April, when only 

a few water-quality samples were collected, generally were 

high. Weighting the sampling more heavily toward the May 

through October period caused a downward bias in the median 

measured concentrations, and the median flow-adjusted 

concentrations generally were higher than the median measured concentrations. The Sweetwater Lake, Lake Alice near 

Churchs Ferry, and Lake Irvine near Churchs Ferry stations 

(sites 3, 5, and 6, respectively) were not included in the evaluation of dissolved ammonia concentrations because too few 

data are available for those stations.

The box plots for flow-adjusted total phosphorus concentrations (fig. 15) indicate a decrease in streamflow-related 

variability and in bias in the measured concentrations for most 

stations included in the water-quality trend analysis. The low 

flow-adjusted concentrations for the Starkweather Coulee 

and Channel A stations (sites 4 and 8, respectively) indicate 

that, for typical streamflow conditions, concentrations for 

those stations are much lower than for the remaining stations. 

The Sweetwater Lake, Lake Alice, and Lake Irvine stations 

(sites 3, 5, and 6, respectively) were not included in the evaluation of total phosphorus concentrations because too few data 

are available for those stations.

The box plots for flow-adjusted dissolved strontium concentrations (fig. 15) indicate a decrease in streamflow-related 

variability for all stations.

Analysis of Streamflow-Related Variability in Concentration Data 15
Figure 9. Partial residuals of dissolved sulfate concentrations for the Mauvais Coulee near Cando, North Dakota, station in relation 

to streamflow anomalies. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

5-year anomaly Annual anomaly

Seasonal anomaly Daily anomaly

STREAMFLOW ANOMALY STREAMFLOW ANOMALY

1.5

2

2.5

3

3.5

-2 -1 0 1 2 3 4

1.5

2

2.5

3

3.5

-2 -1 0 1 2 3 4

 EXPLANATION

Fitted relation between partial

residuals and given anomaly

Partial residuals

16 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1985 1990 1995 2000 2005

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Fitted concentrations

Measured concentrations

Figure 10. Measured and fitted dissolved ammonia concentrations for 1985 through September 2003 for the Edmore 

Coulee near Edmore, North Dakota, station.

Figure 11. Flow-adjusted dissolved ammonia concentrations for 1985 through September 2003 for the Edmore 

Coulee near Edmore, North Dakota, station. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1985 1990 1995 2000 2005

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Analysis of Streamflow-Related Variability in Concentration Data 17
Figure 12. Partial residuals of dissolved ammonia concentrations for the Edmore Coulee near Edmore, North Dakota, station in 

relation to streamflow anomalies. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-2

-1.5

-1

-0.5

0

0.5

-2

-1.5

-1

-0.5

0

0.5

-3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4

STREAMFLOW ANOMALY STREAMFLOW ANOMALY

5-year anomaly Annual anomaly

Seasonal anomaly Daily anomaly

 EXPLANATION

Fitted relation between partial

residuals and given anomaly

Partial residuals

18  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 13. Measured and fitted dissolved nitrite plus nitrate concentrations and censored values for 1980 through 

September 2003 for the Mauvais Coulee near Cando, North Dakota, station. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1985 1990 1995 2000 2005

-4

-3

-2

-1

0

1

2

Fitted concentrations

Measured concentrations

Censored values

Figure 14. Flow-adjusted dissolved nitrite plus nitrate concentrations and censored values for 1985 through September 

2003 for the Mauvais Coulee near Cando, North Dakota, station. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1985 1990 1995 2000 2005

-4

-3

-2

-1

0

1

2

Measured concentrations

Censored values

Analysis of Streamflow-Related Variability in Concentration Data 19
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Measured dissolved calcium concentrations

Flow-adjusted dissolved calcium concentrations

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

(72)

(89) (84) (38) (90) (38) (40) (72) (78)

(89) (84) (56) (90) (53) (57) (72) (80)

20 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis.—Continued CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1

0

2

3

4

1

0

2

3

4

Measured dissolved sulfate concentrations

Flow-adjusted dissolved sulfate concentrations

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

0.5

1.5

2.5

3.5

0.5

1.5

2.5

3.5

(89) (84) (38) (90) (38) (40) (72) (80)

(89) (84) (56) (90) (53) (57) (72) (82)

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

(72)

Analysis of Streamflow-Related Variability in Concentration Data 21
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis.—Continued CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-1

-0.5

0.5

0

1

1.5

2.5

-0.5

0.5

1.5

2.5

2

3

-1

0

1

2

3

Measured dissolved chloride concentrations

Flow-adjusted dissolved chloride concentrations

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

(89) (84) (38) (90) (38) (40) (72) (80)

(89) (84) (56) (90) (53) (57) (72) (82)

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

(72)

22 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis.—Continued CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-2

-3

-1

0

1

-2

-3

-1

0

1

Measured dissolved ammonia concentrations

Flow-adjusted dissolved ammonia concentrations NOT INCLUDED IN EVALUATION

NOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATION

NOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATION

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

-2.5

-1.5

-0.5

0.5

-2.5

-1.5

-0.5

0.5

(65) (66) (61) (58) (67)

(65) (66) (61) (58) (69)

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

(58)

Analysis of Streamflow-Related Variability in Concentration Data 23
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis.—Continued CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-1

-2

1

0

2

-1

-2

1

0

2

Measured total phosphorus concentrations

Flow-adjusted total phosphorus concentrations

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

-0.5

0.5

1.5

-0.5

-1.5

0.5

1.5

-1.5

(57) (59) (54) (50) (57)

(57) (59) (54) (50) (59)

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

(57)

NOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATION

NOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATIONNOT INCLUDED IN EVALUATION

24  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 15. Measured and flow-adjusted concentrations for 1980 through September 2003 for stations used in water-quality 

trend analysis.—Continued CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

3

2

1

0

4

3

2

1

0

0.5

1.5

2.5

3.5

0.5

1.5

2.5

3.5

4

Measured dissolved strontium concentrations

Flow-adjusted dissolved strontium concentrations

Mauvais Coulee near

Cando, North Dakota

Edmore Coulee near

Edmore, North Dakota

Starkweather Coulee near

Webster, North Dakota

Lake Alice near Churchs

Ferry, North Dakota

Lake Irvine near Churchs

Ferry, North Dakota

Big Coulee near Churchs

Ferry, North Dakota

Channel A near

Penn, North Dakota

Sweetwater Lake at

Sweetwater, North Dakota

(81) (80) (33) (79) (33) (35) (68) (78)

(81) (80) (51) (79) (48) (52) (68) (80)

 EXPLANATION

Number of samples

Percentile--Percentage of analyses

equal to or less than indicated values

Maximum

75th percentile

50th percentile (median)

25th percentile

Minimum

(78)

Analysis of Streamflow-Related Variability in Concentration Data 25
Water-Quality Trend Analysis

The constituents used for the water-quality trend analysis were evaluated for two specific trends, a step trend and a 

linear trend. The step trend was used to examine the effect of 

Channel A on water quality in the basin, and the linear trend 

was used to detect gradual changes with time from January 

1980 through September 2003. Operation of Channel A began 

in 1979 to reduce spring-runoff-induced flooding around Dry 

Lake and around other lakes adjacent to Dry Lake by lowering 

the lake levels more rapidly than what would have occurred 

through Dry Lake’s natural outlet (U.S. Army Corps of Engineers, 2003). Channel A connects Dry Lake to Sixmile Bay of 

Devils Lake, thus creating a shorter path to Devils Lake than 

through the upstream chain of lakes, Big Coulee, and Pelican 

Lake (fig. 2).

The constituents used for the water-quality trend analysis 

were evaluated by adding two trend variables to the flowadjustment model—one for the step trend and one for the 

linear trend. The step trend variable represented whether the 

sample was collected before or after operation of Channel A 

began and was used only for those stations for which data are 

available before 1980. The step trend variable was included 

both for stations (such as Big Coulee near Churchs Ferry) that 

potentially were affected by Channel A and for stations (such 

as Edmore Coulee near Edmore) that were not affected by 

Channel A to evaluate whether trends unrelated to Channel A 

were confounded with the effect of Channel A.

Results of the trend analysis are given in table 4. For 

discussion purposes, a trend will be called significant if the 

p-value (attained significance level) for the trend is less than 

0.05.

Dissolved Major Ions

The fitted step trends for flow-adjusted dissolved calcium 

concentrations (fig. 16, table 4) were small and insignificant. 

Therefore, the concentrations probably were not affected by 

Channel A or any other factor. The fitted linear trends for the 

upstream stream stations (sites 1, 2, and 4) also were small 

and insignificant. The upward trends that occurred for one lake 

station (site 5) and for the Big Coulee near Churchs Ferry and 

Channel A near Penn stations (sites 7 and 8, respectively) were 

small and insignificant, but the upward trends that occurred 

during 1980-2003 for the remaining lake stations (sites 3 and 

6) were significant. Because no data are available for the Big 

Coulee station (site 7) for 1999-2003, some trends for that 

station may not have been detected. During 1999-2003, the 

concentrations for the Lake Irvine near Churchs Ferry station 

(site 6) were higher than the concentrations for the remaining stations. The differences among the fitted linear trends for 

the upstream stream stations and for the remaining stations 

indicate chemical or hydrologic processes occurring in the 

lakes probably had a substantial effect on the concentrations. 

During the typical streamflow conditions that occurred during 

the 1970s and early 1980s, the concentrations for the Sweetwater Lake at Sweetwater and Lake Irvine stations (sites 3 and 

6, respectively) generally were smaller than the concentrations 

for the Mauvais Coulee near Cando and Edmore Coulee near 

Edmore stations (sites 1 and 2, respectively). However, during 

the wet conditions from the mid-1990s to 2003, the concentrations for the lake stations were similar to the concentrations 

for the upstream stream stations.

The fitted step trends for flow-adjusted dissolved sulfate 

concentrations for the Lake Alice near Churchs Ferry, Lake 

Irvine, and Big Coulee stations (sites 5, 6, and 7, respectively) 

(fig. 17, table 4) were positive and similar in magnitude. Of 

the three upward trends, one, that for the Lake Irvine station (site 6), was significant. The upward trends for the three 

stations, along with the insignificant trends for the Mauvais 

Coulee, Edmore Coulee, and Sweetwater Lake stations 

(sites 1, 2, and 3, respectively), may indicate the concentrations for the Lake Alice, Lake Irvine, and Big Coulee stations 

(sites 5, 6, and 7, respectively) were affected by Channel A. 

The concentrations for the Edmore Coulee and Starkweather 

Coulee near Webster stations (sites 2 and 4, respectively) were 

considerably lower than the concentrations for the Mauvais 

Coulee station (site 1) so the lack of flow from the east to the 

west after 1979 probably caused the upward trends for sulfate. 

Downward linear trends occurred during 1980-2003 for most 

stations; however, the trends were not statistically significant. 

Because abundant sources of sulfur exist in soils and wetlands 

in the Devils Lake Basin, the decreasing concentrations may 

be related to changes in land use that may have reduced the 

amount of naturally occurring sulfate that reached the streams 

and lakes in the basin.

The fitted step trends for flow-adjusted dissolved chloride 

concentrations (fig. 18, table 4) were positive but insignificant. 

The upward trends for the Lake Alice and Big Coulee stations 

(sites 5 and 7, respectively) may indicate the concentrations 

for those stations were affected by Channel A. However, an 

upward trend also occurred for the Sweetwater Lake station 

(site 3) and concentrations for that station were not affected by 

Channel A. Therefore, although the results are inconclusive, 

the upward trends in 1979 may have been caused, in part, 

by factors other than Channel A. As for calcium, the fitted 

linear trends for the upstream stream stations (sites 1, 2, and 

4) were small and insignificant. However, three of the downward trends that occurred during 1980-2003 for the remaining stations (those for sites 3, 7, and 8) were significant. The 

contrasting trends for calcium and chloride for the lake and 

downstream stream stations may indicate the chemical or 

hydrologic processes that caused increasing calcium concentrations also may have caused decreasing chloride concentrations. However, without a more detailed chemical, water-massbalance model, the specific cause of the contrasting trends is 

difficult to determine.

26 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Table 4. Stations and constituents used to test significance of Channel A step trend and linear trend from 1980 through September 

2003.

[--, not tested for trend; <, less than; a p-value of less than 0.05 is significant]

Site

number

(figure 2)

U.S. Geological Survey station name

Coefficient

for

Channel A

step trend

p-value for

Channel A

step trend

Coefficient for

linear trend

from 1980

through

September

2003

p-value for

linear trend

from 1980

through

September

2003

Calcium, dissolved

1 Mauvais Coulee near Cando, North Dakota 0.020 0.622 0.001 0.751

2 Edmore Coulee near Edmore, North Dakota -.069 .170 0 .924

3 Sweetwater Lake at Sweetwater, North Dakota .018 .666 .011 .003

4 Starkweather Coulee near Webster, North 

Dakota

-- -- -.003 .360

5 Lake Alice near Churchs Ferry, North Dakota .050 .240 .004 .279

6 Lake Irvine near Churchs Ferry, North Dakota -.044 .292 .014 <.001

7 Big Coulee near Churchs Ferry, North Dakota .012 .754 .005 .117

8 Channel A near Penn, North Dakota -- -- .003 .608

Sulfate, dissolved

1 Mauvais Coulee near Cando, North Dakota -0.072 0.188 0.004 0.279

2 Edmore Coulee near Edmore, North Dakota -.034 .670 -.008 .071

3 Sweetwater Lake at Sweetwater, North Dakota .076 .102 -.005 .182

4 Starkweather Coulee near Webster, North 

Dakota

-- -- .001 .852

5 Lake Alice near Churchs Ferry, North Dakota .113 .075 -.002 .719

6 Lake Irvine near Churchs Ferry, North Dakota .090 .040 -.007 .062

7 Big Coulee near Churchs Ferry, North Dakota .104 .053 -.005 .291

8 Channel A near Penn, North Dakota -- -- -.002 .773

Chloride, dissolved

1 Mauvais Coulee near Cando, North Dakota 0.062 0.377 -0.002 0.697

2 Edmore Coulee near Edmore, North Dakota .046 .603 -.005 .286

3 Sweetwater Lake at Sweetwater, North Dakota .059 .090 -.010 .001

4 Starkweather Coulee near Webster, North 

Dakota

-- -- .004 .478

5 Lake Alice near Churchs Ferry, North Dakota .125 .066 -.012 .064

6 Lake Irvine near Churchs Ferry, North Dakota .062 .249 -.007 .167

7 Big Coulee near Churchs Ferry, North Dakota .105 .067 -.015 .001

8 Channel A near Penn, North Dakota -- -- -.018 .002

Water-Quality Trend Analysis 27
Site

number

(figure 2)

U.S. Geological Survey station name

Coefficient

for

Channel A

step trend

p-value for

Channel A

step trend

Coefficient for

linear trend

from 1980

through

September

2003

p-value for

linear trend

from 1980

through

September

2003

Nitrite plus nitrate as nitrogen, dissolved

1 Mauvais Coulee near Cando, North Dakota -- -- 0.056 0.001

2 Edmore Coulee near Edmore, North Dakota -- -- .016 .465

3 Sweetwater Lake at Sweetwater, North Dakota 0.078 0.734 .003 .867

4 Starkweather Coulee near Webster, North 

Dakota

-- -- -.026 .263

5 Lake Alice near Churchs Ferry, North Dakota .422 .146 -.035 .092

6 Lake Irvine near Churchs Ferry, North Dakota .110 .672 -.007 .757

7 Big Coulee near Churchs Ferry, North Dakota -.115 .662 -.030 .100

8 Channel A near Penn, North Dakota -- -- .082 .009

Ammonia, dissolved

1 Mauvais Coulee near Cando, North Dakota -- -- 0.015 0.362

2 Edmore Coulee near Edmore, North Dakota -- -- -.013 .278

4 Starkweather Coulee near Webster, North 

Dakota

-- -- -.039 .005

7 Big Coulee near Churchs Ferry, North Dakota -- -- .026 .251

8 Channel A near Penn, North Dakota -- -- .024 .466

Phosphorus, total

1 Mauvais Coulee near Cando, North Dakota -- -- 0.053 0.030

2 Edmore Coulee near Edmore, North Dakota -- -- .019 .203

4 Starkweather Coulee near Webster, North 

Dakota

-- -- .036 .159

7 Big Coulee near Churchs Ferry, North Dakota -- -- -.013 .617

8 Channel A near Penn, North Dakota -- -- .075 .003

Strontium, dissolved

1 Mauvais Coulee near Cando, North Dakota -- -- 0.005 0.128

2 Edmore Coulee near Edmore, North Dakota -- -- .005 .128

3 Sweetwater Lake at Sweetwater, North Dakota -0.029 0.482 .005 .202

4 Starkweather Coulee near Webster, North 

Dakota

-- -- .007 .098

5 Lake Alice near Churchs Ferry, North Dakota .058 .432 .006 .313

Table 4. Stations and constituents used to test significance of Channel A step trend and linear trend from 1980 through September 

2003.—Continued

[--, not tested for trend; <, less than; a p-value of less than 0.05 is significant]

28  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Site

number

(figure 2)

U.S. Geological Survey station name

Coefficient

for

Channel A

step trend

p-value for

Channel A

step trend

Coefficient for

linear trend

from 1980

through

September

2003

p-value for

linear trend

from 1980

through

September

2003

Strontium, dissolved—Continued

6 Lake Irvine near Churchs Ferry, North Dakota -0.019 0.773 0.010 0.036

7 Big Coulee near Churchs Ferry, North Dakota .073 .214 .006 .134

8 Channel A near Penn, North Dakota -- -- -.005 .310

Table 4. Stations and constituents used to test significance of Channel A step trend and linear trend from 1980 through September 

2003.—Continued

[--, not tested for trend; <, less than; a p-value of less than 0.05 is significant]

Nutrients

The fitted step trends for flow-adjusted dissolved nitrite 

plus nitrate concentrations (fig. 19, table 4) were insignificant. 

Therefore, either Channel A had no effect on the concentrations or the effect was too small to be detected. The fitted 

linear trends for the Mauvais Coulee near Cando and Channel A near Penn stations (sites 1 and 8, respectively) were 

significant. Upward trends occurred for those stations, but, 

because no data are available before 1987, the trends are for 

1987-2003 rather than for 1980-2003. Most of the stations for 

which data are available before the mid-1980s had downward 

trends. Therefore, concentrations during recent years appear to 

be lower than those for the 1970s and early 1980s but higher 

than those for the late 1980s and early 1990s. Continued monitoring will be needed to determine if the recent trend toward 

higher concentrations continues in the future.

The trend results for flow-adjusted dissolved ammonia 

concentrations are shown in figure 20 and in table 4. Because 

no dissolved ammonia data are available before 1980, step 

trends were not included for that constituent. The fitted linear 

trends for the Mauvais Coulee, Big Coulee near Churchs 

Ferry, and Channel A stations (sites 1, 7, and 8, respectively) 

for 1980-2003 were small and insignificant. Upward trends 

occurred for those stations during 1980-2003, but downward 

trends occurred for the Edmore Coulee near Edmore and 

Starkweather Coulee near Webster stations (sites 2 and 4, 

respectively). The trend for the Starkweather Coulee station (site 4) was significant. From the mid- to late 1980s, the 

concentrations for the Channel A station (site 8) were substantially lower than the concentrations for the Edmore Coulee 

and Starkweather Coulee stations (sites 2 and 4, respectively). 

However, from the mid-1990s through 2003, the concentrations for the Channel A station (site 8) were similar to the 

concentrations for the Edmore Coulee and Starkweather 

Coulee stations (sites 2 and 4, respectively). Therefore, chemical processes occurring in the lakes may have caused decreasing concentrations in Channel A during the late 1980s but 

not during the latter years of record. The Edmore Coulee and 

Starkweather Coulee stations (sites 2 and 4, respectively) are 

located upstream from the Channel A station (site 8).

The trend results for flow-adjusted total phosphorus 

concentrations are shown in figure 21 and in table 4. Because 

no total phosphorus data are available before 1989, step trends 

were not included for that constituent. The fitted linear trends 

for total phosphorus concentrations (fig. 21) for the Mauvais 

Coulee and Channel A stations (sites 1 and 8, respectively) 

were significant. Upward trends occurred from the late 1980s 

to 2003 for most stations, but a small and insignificant downward trend occurred for the Big Coulee station (site 7). However, few data are available for that station before 1993 and 

no data are available after 1998. Therefore, an upward trend 

may have occurred for that station, but the trend could not be 

detected. Continued monitoring will be needed to determine if 

the recent trend toward higher concentrations continues in the 

future.

Trace Element

The fitted step trends for flow-adjusted dissolved strontium concentrations (fig. 22, table 4) were small and insignificant. Therefore, the concentrations probably were not affected 

by Channel A. The fitted linear trends were insignificant 

except for the Lake Irvine near Churchs Ferry station (site 6). 

The upward trend for that station (site 6) was significant.

Water-Quality Trend Analysis 29
Figure 16. Fitted trends for flow-adjusted dissolved calcium concentrations.

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Mauvais Coulee near Cando, North Dakota

Sweetwater Lake at Sweetwater, North Dakota

Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota

Lake Alice near Churchs Ferry, North Dakota

Big Coulee near Churchs Ferry, North Dakota

Lake Irvine near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Step trend for 1979 Linear trend for 1980 through September 2003 Flow-adjusted concentrations

30 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 17. Fitted trends for flow-adjusted dissolved sulfate concentrations.

1

2

1.5

2.5

3

3.5

1

2

1.5

2.5

3

3.5

1

2

1.5

2.5

3

3.5

1

2

1.5

2.5

3

3.5

CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Mauvais Coulee near Cando, North Dakota

Sweetwater Lake at Sweetwater, North Dakota

Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota

Lake Alice near Churchs Ferry, North Dakota

Big Coulee near Churchs Ferry, North Dakota

Lake Irvine near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Step trend for 1979 Linear trend for 1980 through September 2003 Flow-adjusted concentrations

Water-Quality Trend Analysis 31
Figure 18. Fitted trends for flow-adjusted dissolved chloride concentrations.

-0.5

0

1

0.5

1.5

2

2.5

-0.5

0

1

0.5

1.5

2

2.5

CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-0.5

0

1

0.5

1.5

2

2.5

-0.5

0

1

0.5

1.5

2

2.5

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Mauvais Coulee near Cando, North Dakota

Sweetwater Lake at Sweetwater, North Dakota

Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota

Lake Alice near Churchs Ferry, North Dakota

Big Coulee near Churchs Ferry, North Dakota

Lake Irvine near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Step trend for 1979 Linear trend for 1980 through September 2003 Flow-adjusted concentrations

32 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Figure 19. Fitted trends for flow-adjusted dissolved nitrite plus nitrate concentrations. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

Mauvais Coulee near Cando, North Dakota

Sweetwater Lake at Sweetwater, North Dakota

Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota

Lake Alice near Churchs Ferry, North Dakota

Big Coulee near Churchs Ferry, North Dakota

Lake Irvine near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Step trend for 1979 Flow-adjusted concentrations

Linear trend for 1980 through September 2003 Censored values

Water-Quality Trend Analysis 33
Sampling Design

The need for continued monitoring of water-quality 

trends in the upper Devils Lake Basin must be weighed against 

the cost of obtaining the monitoring data. Therefore, various 

sampling designs were evaluated with respect to two related 

concepts—sensitivity and efficiency. Sensitivity measures the 

ability of a design to detect a trend—the smaller the trend that 

can be detected, the more sensitive the design. An efficient 

design maximizes the sensitivity to detect a trend for a given 

cost, which for this report was measured in terms of the number of samples collected per year.

To evaluate sampling designs, the number of samples 

collected (on average) each year needs to be considered along 

with the sampling dates and whether the number of samples 

or sampling dates should be fixed (the same year after year) or 

variable. Because trends can occur anywhere, at any time, and 

the causes of the trends generally are not known in advance, 

the most efficient designs for monitoring trends in concentrations of multiple constituents generally are fixed designs in 

which the sampling dates and sampling locations are approximately the same year after year (Vecchia, 2005). Therefore, 

only fixed designs were considered for this report. The designs 

were evaluated with respect to their sensitivity to detect trends 

during two seasons—March through May, which generally 

includes the spring breakup and snowmelt season, and June 

through November, which generally includes the summer and 

fall low-flow season. Chemical source and transport mechanisms tend to differ for the two seasons, and, thus, trends that 

occur in one season may not necessarily occur in the other. 

For example, nutrients that accumulate during the winter, such 

as wastes from confined livestock operations, tend to reach 

streams in early spring. However, nutrients that are a result 

of fertilizer application or soil erosion tend to reach streams 

in late spring or early summer when streamflow consists 

mostly of rainfall runoff. Other constituents, such as dissolved 

chloride or dissolved nitrite plus nitrate, may originate from 

ground-water sources and, thus, may require monitoring during summer and fall low-flow conditions.

For this report, a trend in concentration was defined as 

a change in the mean of the log-transformed concentrations. 

Using the null hypothesis that the mean of log-transformed 

concentrations is constant, the log-transformed concentrations 

were expressed as follows:

logC(t) = μ + E[logC(t) – μ logQ(u), u ≤ t] + ε(t)

where

log denotes the base-10 logarithm;

C(t) is the concentration, in milligrams or 

micrograms per liter, for time t, in decimal 

years;

μ is the mean (expected value) of the logtransformed concentrations, as the base-10 

logarithm of milligrams or micrograms per 

liter;

E[ ] is the conditional expectation of logC(t) –

μ given log-transformed streamflow for 

all times μ up to and including time t

(dimensionless);

Q is streamflow, in cubic feet per second, for 

time u;

u is time, in decimal years;

and

ε(t) is the residual, which, by definition, is 

uncorrelated with current and past 

values of log-transformed streamflow 

(dimensionless).

Flow-adjusted concentrations were expressed as follows:

FAC t  logC t  E C t 

– P log , Q u  u t d

>log

@

–

P H +  t

=

=

where

FAC(t) is the flow-adjusted concentration, as the 

base-10 logarithm of milligrams or 

micrograms per liter for time t, in decimal 

years.

Because the mean of the log-transformed concentrations is 

identical to the mean of the flow-adjusted concentrations, the 

trends in log-transformed concentrations are equivalent to the 

trends in flow-adjusted concentrations. Trends are easier to 

detect using flow-adjusted concentrations rather than logtransformed concentrations because flow-adjusted concentrations generally have less variability than log-transformed 

concentrations.

For this report, the following statistic was used to determine potential sampling designs:

Xn, d Ave FAC uj = { ( ), j = 1, 2, ... nd}

where

X is a statistic used to detect trends in 

concentrations;

n is the number of samples per year;

d is the duration, in years, of the time interval 

used to compute the statistic (the beginning 

and ending times of the interval are not 

important);

Ave is the average of the values in braces; 

and

u1, u2, ..., und are the specific times of sample collection.

34  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

(2)

(3)
Figure 20. Fitted trends for flow-adjusted dissolved ammonia concentrations. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

-2.5

-2

-1.5

-1

-0.5

0

0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1980 1985 1990 1995 2000 2005

Mauvais Coulee near Cando, North Dakota Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota Big Coulee near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Linear trend for 1980 through September 2003

Flow-adjusted concentrations

1980 1985 1990 1995 2000 2005

An increase in d increases the sensitivity of a sampling design 

but not the efficiency of the design. Therefore, d = 3 was 

selected as a benchmark to compare the efficiency of various 

designs. Substituting equation 2 into equation 3 and setting 

d = 3 yielded the following equation:

Xn, 3 μ En, 3 = +

where

 En,3 = Ave {ε(uj), j = 1, 2, ... 3n is the estimation error.

The sensitivity of a design for detecting a trend depends 

on the variance of the estimation error, which, in turn, depends 

on the variance and serial correlation of the errors in the flowadjusted concentrations. Therefore, variogram analyses were 

used to explore the variance and serial correlation structure 

of the errors in flow-adjusted concentrations (see appendix). 

The errors generally were uncorrelated for time lags of at 

least 2 weeks for season 1 and 6 weeks for season 2. Furthermore, the error variance was constant within each of two 

seasons—March through May and June through November. 

However, for some stations and some constituents, substantial 

differences occurred in the error variance between the two 

seasons. Therefore, provided that water-quality samples are 

collected at least 2 weeks apart for season 1 and 6 weeks apart 

for season 2,

Var En, 3 { }

n1v1 n2v2 +

3 n1 n2 ( + )

2 = ----------------------------

Sampling Design 35

(4)

(5)
where

Var{ } is the variance of the term in braces,

n1 is the number of samples per year for season 1 

(March through May),

v1 is the variance of ε(t) for t in season 1,

n2 is the number of samples per year for season 2 

(June through November), 

and

v2 is the variance of ε(t) for t in season 2.

For the sample design analysis, design efficiencies were evaluated separately for each season to monitor trends during both 

the snowmelt-runoff season (season 1) and the summer and 

fall low-flow season (season 2). Design efficiencies were not 

evaluated for December through February because too few 

data are available for those months. Evaluating equation 5 

separately for each season and omitting the subscripts yielded 

the following equation:

Var(E) v

3n = -----

where

E is the estimation error for either season 1 or 

season 2,

v is the variance of the flow-adjusted 

concentrations for either season 1 or 

season 2, 

and

n is the number of samples per year for either 

season 1 or season 2.

For equation 6, samples were assumed to be collected at least 

2 weeks apart for season 1 and 6 weeks apart for season 2. 

At most, 7 equally spaced samples per year for season 1 

and 6 equally spaced samples per year for season 2 can be 

collected to satisfy this assumption. If more samples are 

collected, equation 6 should not be used to evaluate design 

efficiencies.

For future water-quality monitoring, a goal was set to 

have the estimated median concentration for any 3-year interval be within 25 percent of the actual median without increasing sampling costs. Using equation 4 and the log-transformed 

concentrations, the estimated median concentration was 10μ + E

and the actual median concentration was 10μ. Therefore, the 

probability that the estimated median was within 25 percent of 

the actual median was given by the following equation:

Prob 25 100 10μ + E 10

μ –

10μ – < ------------------------------- < 25

Prob 25 100 10E = {– < ( – 1) < 25}

where

Prob{ } is the probability of the event in braces.

Assuming E is approximately normally distributed and 

substituting equation 6 into equation 7 yielded the following 

equation:

Prob{–25 < PE < 25}

Prob Z (0.75) 3n

v

-----

1

2

--

– < log

Prob Z (1.25) 3n

v

-----

1

2

--

= < log

where

PE is the percent error, 

and

Z is a standard normal random variable.

For this report, the smallest integer value for n was 

determined such that the probability (eq. 8) was at least 0.8 

or 0.6—that is, for which at least an 80 or 60 percent chance 

existed that the estimated median was within the error tolerance. A separate determination for n was made for season 1 

(that is, assuming all samples were collected from March 

through May) and for season 2 (that is, assuming all samples 

were collected from June through November) so the same 

error tolerance was achieved for both seasons.

Because the sampling specifications used to develop 

the sampling designs for this report are lenient, the sampling 

designs may not be sensitive enough for some potential applications. For example, a study to evaluate the mass balance 

of certain chemical constituents may require a larger number 

of samples (thus increasing the cost) than indicated in this 

report. Similarly, a study to evaluate the changes in and causes 

of the complex relation between antecedent streamflow and 

water quality would require a larger number of samples than 

indicated in this report. Therefore, given the constraint that 

sampling costs remain at or below those in the 1992-2003 

sampling program, an 80- or 60-percent chance that the estimated median was within 25 percent of the actual median was 

a realistic goal.

The number of samples required per year for each constituent is given in table 5. Dissolved nitrite plus nitrate was 

not included because of the large percentage of censored values for that constituent. Rather, dissolved ammonia was used 

as a surrogate for dissolved nitrite plus nitrate in the design 

analysis. Design results for the Lake Irvine near Churchs Ferry 

station (site 6) also were not included because those results 

were similar to the results for the Lake Alice near Churchs 

Ferry station (site 5).

36 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

(6)

(7)

(8)
Figure 21. Fitted trends for flow-adjusted total phosphorus concentrations. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1980 1985 1990 1995 2000 2005

Mauvais Coulee near Cando, North Dakota Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota Big Coulee near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

-2.5

-2

-1.5

-1

-0.5

0

0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

 EXPLANATION

Linear trend for 1980 through September 2003

Flow-adjusted concentrations

1980 1985 1990 1995 2000 2005

The smallest numbers of samples required per year were 

for dissolved calcium and dissolved strontium (table 5). To 

obtain at least an 80-percent chance of being within the error 

tolerance for those constituents, one sample per year per season generally was sufficient. The numbers of samples required 

per year for dissolved sulfate and dissolved chloride were 

similar. To obtain at least an 80-percent chance of being within 

the error tolerance, two or three samples per year per season 

generally were sufficient. However, for the Starkweather Coulee station (site 4), three samples were required per year for 

season 2 to obtain at least a 60-percent chance of being within 

the error tolerance for dissolved sulfate. The largest numbers 

of samples required per year were for dissolved ammonia. To 

obtain at least an 80-percent chance of being within the error 

tolerance for that constituent, 4 to 11 samples were required 

per year for season 1, and 3 to 14 samples were required 

per year for season 2. If the chance of being within the error 

tolerance is lowered to 60 percent, then two to five samples 

were required per year for season 1, and two to six samples 

were required per year for season 2. To obtain at least an 

80-percent chance of being within the error tolerance for total 

phosphorus, two samples were required per year for season 1 

and one sample generally was required per year for season 2. 

Exceptions were for the Starkweather Coulee near Webster 

station (site 4), for which four samples were required per year 

for season 2, and the Channel A near Penn station (site 8), for 

which three samples were required per year for season 2. For 

those stations, two samples were required per year for season 2 to obtain at least a 60-percent chance of being within the 

error tolerance.

Sampling Design 37
CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER 

1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005

3

2.5

2

1.5

3.5

3

2.5

2

1.5

3.5

3

2.5

2

1.5

3.5

3

2.5

2

1.5

3.5

Mauvais Coulee near Cando, North Dakota

Sweetwater Lake at Sweetwater, North Dakota

Edmore Coulee near Edmore, North Dakota

Starkweather Coulee near Webster, North Dakota

Lake Alice near Churchs Ferry, North Dakota

Big Coulee near Churchs Ferry, North Dakota

Lake Irvine near Churchs Ferry, North Dakota

Channel A near Penn, North Dakota

 EXPLANATION

Step trend for 1979 Linear trend for 1980 through September 2003 Flow-adjusted concentrations

Figure 22. Fitted trends for flow-adjusted dissolved strontium concentrations.

38  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
The collection of water-quality samples can be costly. 

Therefore, an efficient plan is needed for the collection of samples that are representative of various seasonal and hydrologic 

conditions in the basin. Results of this study indicate a good 

overall design for the stations used in the water-quality trend 

analysis might consist of six samples per year per station, with 

three samples collected during season 1 and three samples collected during season 2. Except for dissolved ammonia for the 

Big Coulee near Churchs Ferry and Channel A stations (sites 7 

and 8, respectively), this six-sample design generally resulted 

in at least a 60-percent chance, and in most cases at least an 

80-percent chance, of being within the error tolerance for all 

constituents. For dissolved ammonia for the Big Coulee station 

(site 7), four samples were required per year for season 1 to 

obtain a 60-percent chance of being within the error tolerance. 

For dissolved ammonia for the Channel A station (site 8), five 

samples were required per year for season 1 and six samples 

were required per year for season 2 to obtain a 60-percent 

chance of being within the error tolerance.

Although constituents were evaluated separately for each 

season, in any given year a smooth transition occurs from season 1 to season 2 when, in late May or early June, streamflows 

change from primarily snowmelt to primarily rainfall runoff. 

Therefore, a sample collected in late May or early June generally could be considered “shared” between the two seasons, 

thus reducing the number of samples required per year from 

six to five. To minimize the potential effects of serial correlation, the remaining samples should be collected as far apart as 

possible within each season. Therefore, a five-sample design 

with samples collected in about early April, May, and June; 

late July or early August; and late September or early October 

would be a reasonable design for all stations and constituents. 

Except for dissolved ammonia and probably dissolved nitrite 

plus nitrate, this five-sample design generally would result in 

at least an 80-percent chance of being within the error tolerance for most stations. For dissolved ammonia and dissolved 

nitrite plus nitrate, the five-sample design generally would 

result in at least a 60-percent chance of being within the error 

tolerance for most stations.

Assuming a five-sample design, as described previously, 

sampling possibly could be discontinued at some of the existing stream or lake stations or initiated at additional stations. 

In the USGS lake-sampling program for 1992-2003, four 

major-ion, nutrient, and trace-element samples were collected 

per year for each of five lake stations (Sweetwater Lake, Morrison Lake, Dry Lake, Lake Alice, and Lake Irvine) (Ryberg 

and others, 2005). In addition, four major-ion, nutrient, and 

trace-element samples were collected per year for each of 

three stream stations (Mauvais Coulee, Edmore Coulee, and 

Starkweather Coulee), and two major-ion and trace-element 

samples were collected per year for two stream stations 

(Edmore Coulee Tributary and Mauvais Coulee Tributary) 

(Ryberg and others, 2005). For two of the lake stations (Morrison Lake and Dry Lake), no data are available before 1992. 

Therefore, those stations were not included in the trend analysis but were included in the comparison of data to determine 

at which stations sampling possibly could be discontinued or 

initiated. Concentration data for the five stream stations and 

the five lake stations were compared using the graphical user 

interface described by Ryberg and others (2005). Results of 

the comparisons indicate sampling could be discontinued at 

two lake stations (Sweetwater Lake and Lake Irvine) because 

of similarities with other lake stations (Morrison Lake and 

Lake Alice, respectively). In addition, sampling could be 

discontinued at two stream stations (Edmore Coulee Tributary 

and Mauvais Coulee Tributary) because of redundancy of data 

with data for other stream and/or lake stations. For example, 

concentration data for the Edmore Coulee station and the Morrison Lake station can be used to determine if water-quality 

trends occur for Edmore Coulee Tributary (fig. 2). In 1997, 

the long-term stream gage for the Big Coulee station (site 7) 

was inundated with water from Devils Lake, and, in 1979, 

the long-term stream gage for the Channel A station (site 8) 

was inundated. Therefore, water-quality sampling has been 

discontinued at those stations. However, concentration data for 

Big Coulee were closely correlated with concentration data for 

Lake Alice, and concentration data for Channel A were closely 

correlated with concentration data for Dry Lake.

The previous discussion indicates that, for continued 

monitoring of water-quality trends in the upper Devils Lake 

Basin, an efficient sampling design consists of five major-ion, 

nutrient, and trace-element samples per year (collected in 

about early April, May and June; late July or early August; 

and late September or early October) at three existing stream 

stations (Mauvais Coulee, Edmore Coulee, and Starkweather 

Coulee) and at three existing lake stations (Morrison Lake, 

Dry Lake, and Lake Alice). This sampling design requires 

the collection of 15 stream samples and 15 lake samples per 

year rather than 16 stream samples and 20 lake samples per 

year as in the 1992-2003 program. Thus, the design would 

result in a program that is less costly and more efficient than 

the 1992-2003 program but that still would provide the data 

needed to monitor water-quality trends in the Devils Lake 

Basin.

Summary

This report presents the results of a study conducted by 

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

Dakota State Water Commission, the Devils Lake Basin 

Joint Water Resource Board, and the Red River Joint Water 

Resource District, to analyze historical water-quality trends 

in three dissolved major ions (calcium, sulfate, and chloride), 

three nutrients (dissolved nitrite plus nitrate as nitrogen, dissolved ammonia, and total phosphorus), and one dissolved 

trace element (strontium) for eight stations in the Devils Lake 

Basin in North Dakota and to develop an efficient sampling 

design to monitor the future trends. Streamflows for the 

Mauvais Coulee near Cando, North Dakota, and Edmore 

Coulee near Edmore, North Dakota, stations were selected as 

Summary 39
being representative of streamflows in the Devils Lake Basin 

because of the long period of record and geographic location 

of each station.

A multiple-regression model was used to detect and 

remove streamflow-related variability in constituent concentrations. To separate the natural variability in concentration 

as a result of variability in streamflow from the variability in 

concentration as a result of other factors, the base-10 logarithm of daily streamflow for each station was divided into 

four components—a 5-year streamflow anomaly, an annual 

streamflow anomaly, a seasonal streamflow anomaly, and a 

daily streamflow anomaly. The constituent concentrations then 

were adjusted for streamflow-related variability by removing 

the 5-year, annual, seasonal, and daily variability.

Much of the variability in the fitted dissolved nitrite 

plus nitrate concentrations probably can be attributed to 

streamflow-related variability. However, because of the large 

percentage of censored values for this constituent, a survival 

regression technique was used to fit the model rather than the 

multiple-regression model used for the remaining constituents.

Constituents used for the water-quality trend analysis were evaluated for a step trend to examine the effect of 

Channel A on water quality in the basin and a linear trend to 

detect gradual changes with time from January 1980 through 

September 2003. The fitted upward linear trends for dissolved 

calcium concentrations during 1980-2003 for two stations 

were significant. The fitted step trends for dissolved sulfate 

concentrations for three stations were positive and similar in 

magnitude. Of the three upward trends, one was significant. 

The fitted step trends for dissolved chloride concentrations 

40 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

were positive but insignificant. The fitted linear trends for the 

upstream stations were small and insignificant, but three of 

the downward trends that occurred during 1980-2003 for the 

remaining stations were significant. The fitted upward linear 

trends for dissolved nitrite plus nitrate as nitrogen concentrations during 1987-2003 for two stations were significant. 

However, concentrations during recent years appear to be 

lower than those for the 1970s and early 1980s but higher than 

those for the late 1980s and early 1990s. The fitted downward 

linear trend for dissolved ammonia concentrations for one station was significant. The fitted linear trends for total phosphorus concentrations for two stations were significant. Upward 

trends occurred from the late 1980s to 2003 for most stations, 

but a small and insignificant downward trend occurred for one 

station. The fitted step trends for dissolved strontium concentrations were small and insignificant. The fitted linear trends 

were insignificant except for one station. Continued monitoring will be needed to determine if the recent trend toward 

higher dissolved nitrite plus nitrate as nitrogen and total phosphorus concentrations continues in the future.

For continued monitoring of water-quality trends in the 

upper Devils Lake Basin, an efficient sampling design consists 

of five major-ion, nutrient, and trace-element samples per 

year at three existing stream stations and at three existing 

lake stations. This sampling design requires the collection of 

15 stream samples and 15 lake samples per year rather than 

16 stream samples and 20 lake samples per year as in the 

1992-2003 program. Thus, the design would result in a program that is less costly and more efficient than the 1992-2003 

program but that still would provide the data needed to monitor water-quality trends in the Devil Lake Basin.
Table 5. Number of samples required to obtain specified error tolerance for estimated median concentrations.

Probability that

estimated median

for a 3-year

interval is within

25 percent

of actual median

Season1 Site number U.S. Geological Survey station name

Number of samples required per year for

Dissolved

calcium

Dissolved

sulfate

Dissolved

chloride

Dissolved

ammonia

Total

phosphorus

Dissolved

strontium

0.8 1 1 Mauvais Coulee near Cando, North Dakota 1 1 2 5 2 1

1 2 Edmore Coulee near Edmore, North Dakota 1 2 2 4 2 11 3 Sweetwater Lake at Sweetwater, North Dakota 1 1 1 -- -- 11 4 Starkweather Coulee near Webster, North Dakota 1 2 2 6 2 11 5 Lake Alice near Churchs Ferry, North Dakota 1 1 2 -- -- 11 7 Big Coulee near Churchs Ferry, North Dakota 1 1 2 92 2 11 8 Channel A near Penn, North Dakota 1 2 2 11 2 2 12 1 Mauvais Coulee near Cando, North Dakota 1 1 1 3 1 12 2 Edmore Coulee near Edmore, North Dakota 1 2 1 3 1 12 3 Sweetwater Lake at Sweetwater, North Dakota 1 1 1 -- -- 12 4 Starkweather Coulee near Webster, North Dakota 1 6 3 3 4 12 5 Lake Alice near Churchs Ferry, North Dakota 1 1 1 -- -- 12 7 Big Coulee near Churchs Ferry, North Dakota 1 2 3 6 1 12 8 Channel A near Penn, North Dakota 1 2 1 14 2 3 1

0.6 1 1 Mauvais Coulee near Cando, North Dakota 1 1 1 3 1 1

1 2 Edmore Coulee near Edmore, North Dakota 1 1 1 2 1 11 3 Sweetwater Lake at Sweetwater, North Dakota 1 1 1 -- -- 11 4 Starkweather Coulee near Webster, North Dakota 1 1 1 3 1 11 5 Lake Alice near Churchs Ferry, North Dakota 1 1 1 -- -- 11 7 Big Coulee near Churchs Ferry, North Dakota 1 1 1 4 1 11 8 Channel A near Penn, North Dakota 1 1 1 5 1 1

Summary  41
Probability that

estimated median

for a 3-year

interval is within

25 percent

of actual median

Season1 Site number U.S. Geological Survey station name

Number of samples required per year for

Dissolved

calcium

Dissolved

sulfate

Dissolved

chloride

Dissolved

ammonia

Total

phosphorus

Dissolved

strontium

2 1 Mauvais Coulee near Cando, North Dakota 1 1 1 2 1 12 2 Edmore Coulee near Edmore, North Dakota 1 1 1 2 1 12 3 Sweetwater Lake at Sweetwater, North Dakota 1 1 1 -- -- 12 4 Starkweather Coulee near Webster, North Dakota 1 3 1 2 2 12 5 Lake Alice near Churchs Ferry, North Dakota 1 1 1 -- -- 12 7 Big Coulee near Churchs Ferry, North Dakota 1 1 2 3 1 12 8 Channel A near Penn, North Dakota 1 1 1 6 2 1

1Season 1 is March through May, and season 2 is June through November.2Minimum spacing between samples is less than 2 weeks for season 1 and less than 6 weeks for season 2.

42 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota Table 5. Number of samples required to obtain specified error tolerance for estimated median concentrations.—Continued
References Cited

Bluemle, J.P., 1991, Radiocarbon dating of beaches and outlets 

of Devils Lake: North Dakota Geological Survey Miscellaneous Series 75, 10 p.

Cressie, N., 1991, Statistics for spatial data: Wiley, New York, 

900 p.

Cressie, N., and Hawkins, D.M., 1980, Robust estimation of 

the variogram: Journal of the International Association for 

Mathematical Geology, v. 12, p. 115-125.

Helsel, D.R., 2005, Nondetects and data analysis: Hoboken, 

New Jersey, John Wiley & Sons, Inc., 250 p.

Helsel, D.R., and Hirsch, R.M., 1995 (3rd printing), Studies 

in environmental science 49—Statistical methods in water 

resources: Amsterdam, Elsevier Science B.V., 529 p.

Insightful Corporation, 2001, Overview of survival analysis, 

in S-PLUS 6 for Windows guide to statistics, volume 2: 

Seattle, Washington, Insightful Corporation, p. 297-309.

Murphy, E.C., Fritz, A.K., and Fleming, R.F., 1997, The 

Jerusalem and Tolna outlets in the Devils Lake Basin, North 

Dakota: North Dakota Geological Survey Report of Investigation No. 100, 36 p.

Neter, J., Kutner, M.H., Nachtsheim, C.J., and Wasserman, 

W., 1996, Applied linear statistical models, Fourth Edition: 

Boston, WCB McGraw-Hill, 1408 p.

North Dakota State Water Commission, n.d., Outlet quick 

facts: accessed October 17, 2005, at http://www.swc.state.

nd.us/projects/newdevilslake/pdfs/Outlet_Quick_Facts.pdf

Ryberg, K.R., Vecchia, A.V., and Damschen, W.C., 2005, 

Graphical user interface for accessing water-quality data for 

the Devils Lake Basin, North Dakota, 2005: U.S. Geological Survey Open-File Report 2005-1419, online at 

http://pubs.water.usgs.gov/ofr20051419

Sether, B.A., Vecchia, A.V., and Berkas, W.R., 1999, Spatial 

and temporal variability of dissolved sulfate in Devils Lake, 

North Dakota, 1998: U.S. Geological Survey Fact Sheet 

FS-096-99, 4 p., online at http://nd.water.usgs.gov/pubs/fs/

fs09699/index.html

U.S. Army Corps of Engineers, 2003, Final - Devils Lake, 

North Dakota Integrated Planning Report and Environmental Impact Statement, Volume 1, Chapter 3, online at 

http://www.health.state.nd.us/PublicComment/DevilsLakeO

utletDischargePermit/DevilsLakeEIS/

U.S. Geological Survey, various dates, Water analysis: U.S. 

Geological Survey Techniques of Water-Resources Investigations, book 5, chaps. A1-A6, variously paginated, 

accessed February 7, 2006, at http://pubs.usgs.gov/twri/

Vecchia, A.V., 2000, Water-quality trend analysis and sampling design for the Souris River, Saskatchewan, North 

Dakota, and Manitoba: U.S. Geological Survey WaterResources Investigations Report 00-4019, 77 p., online at 

http://nd.water.usgs.gov/pubs/wri/wri004019/index.html

Vecchia, A.V., 2003, Water-quality trend analysis and sampling design for streams in North Dakota, 1971-2000: U.S. 

Geological Survey Water-Resources Investigations Report 

03-4094, 73 p., online at http://nd.water.usgs.gov/pubs/wri/

wri034094/index.html

Vecchia, A.V., 2005, Water-quality trend analysis and sampling design for streams in the Red River of the North 

Basin, Minnesota, North Dakota, and South Dakota, 1970-

2001: U.S. Geological Survey Scientific Investigations 

Report 2005-5224, 54 p., online at http://pubs.water.usgs.

gov/sir2005-5224/

Wiche, G.J., Hoetzer, S.M., and Rankl, J.G., 1986, Hydrology of the Devils Lake Basin, northeastern North Dakota: 

North Dakota State Water Commission Water-Resources 

Investigations Report 3, 86 p., online at http://nd.water.usgs.

gov/pubs/wri/wri3/index.html

Wiche, G.J., Vecchia, A.V., Osborne, Leon, Wood, C.M., and 

Fay, J.T., 2000, Climatology, hydrology, and simulation of 

an emergency outlet, Devils Lake Basin, North Dakota: U.S. 

Geological Survey Water-Resources Investigations Report 

00-4174, 16 p., online at http://nd.water.usgs.gov/pubs/wri/

wri004174/index.html

Wilde, F.D., Radtke, D.B., Gibs, J., and Iwatsubo, R.T., eds., 

various dates, National field manual for the collection of 

water-quality data: U.S. Geological Survey Techniques 

of Water-Resources Investigations, book 9, chap. A1-A9, 

variously paginated, online at http://water.usgs.gov/owq/

FieldManual/

Heading 1  43
Appendix
Parametric Flow-Adjustment Model

To model streamflow-related variability, concentrations 

for each constituent were expressed using the following equation:

Y(t) α0 α1Lx(t) α2Ax(t) α3Sx(t) α4Dx(t)

α5 cos(2πt) α6 sin(2πt) α7 (4πt) α8 sin(4πt)

α9Dx(t) cos(2πt) α10Dx(t)sin(2πt)

α11Dx(t) cos(4πt) α12Dx(t)sin(4πt) U(t)

+

+ +

+ + +

cos

+ + + +

+ + +

=

where

Y(t) is the base-10 logarithm of concentration,

in milligrams or micrograms per liter,

for time t, in decimal years;

αo, α1, ... , and α12 are regression parameters to be 

estimated;

Lx(t) is the 5-year streamflow anomaly 

(dimensionless) for time t;

Ax(t) is the annual streamflow anomaly 

(dimensionless) for time t;

Sx(t) is the seasonal streamflow anomaly 

(dimensionless) for time t;

Dx(t) is the daily streamflow anomaly 

(dimensionless) for time t;

cos(2�t), sin(2�t),

cos(4�t), and

sin(4�t) are periodic functions for describing

seasonal variability not explained by

 variability in streamflow;

and

U(t) is the residual for time t.

Terms 5 through 12 in equation A1 were selected through 

an exploratory analysis of the residuals computed from the 

model that had only the first four terms. Terms 5 through 8 

model seasonal variability in concentration that seemingly is 

unrelated to seasonal variability in streamflow. For example, 

fertilizer application may cause an increase in dissolved 

ammonia concentrations during some months, irrespective of 

streamflow conditions. Terms 9 through 12 model seasonal 

variability in the slope of the line that relates Y(t) to Dx(t). 

For example, a high daily streamflow anomaly that occurs 

during the early spring (when soils are frozen) may cause a 

decrease in total phosphorus concentrations whereas a high 

daily streamflow anomaly that occurs during the summer may 

cause an increase in total phosphorus concentrations. The various terms in equation A1 are necessary to model the complex 

relations between concentration, streamflow, and time of year 

for most constituents. In some cases, additional terms, such 

as those that model interactions between Lx(t) and Dx(t) or 

between time and the various cosine and sine terms, could 

improve the model. However, such cases tend to occur only for 

certain isolated stations or constituents.

For constituents for which less than 20 percent of the 

values were censored, an exhaustive (all-subsets) regression 

procedure was performed to determine the best potential models for each constituent at each station. The procedure examined subsets of the 12 potential explanatory variables from 

size 1 to size 11 and the entire set of 12 explanatory variables. 

The procedure returned as many as 100 of the best potential 

models for each possible number of explanatory variables 

(1 through 12). The models were ranked by standard error, 

adjusted coefficient of multiple determination (R2), Mallow’s 

Cp, and the predicted error sum of squares (PRESS). R2 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). The PRESS criterion is 

a measure of how well the fitted values from a potential model 

predict the measured values (Neter and others, 1996).

For lake stations, changes in inflows to the lake can 

take several days to cause substantial changes in lake volumes because of the large volume of water stored in the 

lakes. Therefore, to determine if time delays could improve 

the model, potential lake-station models were examined for 

all constituents at all lake stations with time lags of 7 days 

(t – 7 days) and 14 days (t – 14 days).

All potential models were examined to determine one 

stream-station model for each constituent and one lake-station 

model for each constituent. Selecting a common model for 

each station grouping (streams versus lakes) was beneficial for 

comparing trends among the various stations. The number of 

potential models was reduced from the initial list created by 

the all-subsets regression procedure by examining the models that had R2

 values that were greater than a limiting value 

(large R2 values were desirable) and models that had Cp and 

PRESS values that were less than limiting values (small Cp

and PRESS values were desirable). The limits for the criterion 

varied depending upon how closely the constituents were 

estimated by the model. The number of potential models also 

was limited by requiring that the cosine (cos) and sine (sin) 

terms were included as pairs. For example, if a potential 

model included cos(2�t), the model also was required to 

include sin(2�t). Including pairs of cos/sin terms may result in 

models where one member of the pair is significant while the 

other is not. However, using only one member of the cos/sin 

pair forces an arbitrary phase shift rather than a phase shift 

determined by the data (Helsel and Hirsch, 1995). Finally, 

the number of potential models was reduced by requiring the 

individual terms in the interaction terms to be included in the 

model if the interaction terms were used. For example, including Dx

(t)cos(2�t) meant that the individual terms Dx(t) and 

cos(2�t) also had to be in the model and that the corresponding sine terms, Dx

(t)sin(2�t) and sin(2�t), had to be in the 

model.

The fitted model coefficients for each constituent for 

stream stations and for each constituent for lake stations are 

a

a

a

a

46 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

(A1)
given in table A1. The lake-station models are for explanatory 

variables lagged by 7 days because that lag was determined 

to result in the best overall lake-station models. Because lakes 

are the primary sources of streamflow for the Big Coulee near 

Churchs Ferry and Channel A near Penn stations (sites 7 and 

8, respectively), the explanatory variables also were lagged by 

7 days for those stations. R2, which adjusts for the number of 

predictor variables, and the p-value for the F-test for a regression relation between the response variable and the set of 

predictor variables also are given in table A1.

At an α-level, or significance level, of 0.05, all of the 

models given in table A1 are statistically significant. That is, 

the p-value, based on an F-test for the regression relation, is 

less than 0.05 in all of the models, indicating the probability of 

obtaining the computed F-test statistic if the models were not 

statistically significant is less than 0.05 in all cases. A p-value 

of less than 0.05 indicates that, for each model, a significant 

regression relation exists between the base-10 logarithm of 

concentration and the model predictor variables. However, a 

p-value of less than 0.05 does not ensure that useful predictions can be made using the model. The model removes as 

much streamflow-related variability in measured concentrations as possible but does not predict actual concentrations.

At an α-level of 0.01, the model for total phosphorus for 

the Starkweather Coulee near Webster station (site 4) is the 

only model that is not significant—the p-value for that model 

is 0.0240.

R2, which indicates the proportionate reduction in the 

variability of the base-10 logarithm of concentration obtained 

using the model predictor variables and which is adjusted for 

the number of predictor variables to allow for the comparison of models, can be used to compare the effectiveness of 

the models in accounting for the variability in the measured 

concentrations. For the stream stations, especially the Mauvais 

Coulee near Cando and Edmore Coulee near Edmore stations 

(sites 1 and 2, respectively), R2 values generally were higher 

for the major ions than for the nutrients and trace element. For 

the lake stations, R2 values were higher for the major ions than 

for the trace element. Because too few nutrient data are available for the lake stations, multiple-regression models could not 

be determined for nutrients.

The highest R2 value for the stream stations was 76.3 

for dissolved sulfate for the Mauvais Coulee station (site 1). 

The highest R2 value for the lake stations was 70.0 for dissolved sulfate for the Lake Irvine near Churchs Ferry station 

(site 6). The lowest R2 value for the stream stations was 18.3 

for dissolved strontium for the Big Coulee near Churchs Ferry 

station (site 7). The lowest R2 value for the lake stations was 

33.6 for dissolved strontium for the Lake Alice near Churchs 

Ferry station (site 5).

Survival Regression for Dissolved 

Nitrite Plus Nitrate as Nitrogen

Dissolved nitrite plus nitrate concentrations (fig. A1) 

were highly censored for the stations used for the waterquality trend analysis (table 3). The censored values (more 

than 20 percent of the concentrations) indicate nitrite plus 

nitrate was present in the water-quality samples and the 

concentrations were between zero and a particular laboratory 

reporting level. The laboratory reporting levels were 0.05, 

0.06, and 0.10. The multiple reporting levels are a result of 

different laboratories analyzing the samples with time and of 

changes in laboratory equipment.

Because parametric multiple regression does not work 

well for highly censored data, a parametric regression method 

called survival regression was used to estimate the parameters 

for the regression equation (eq. A1). Survival regression was 

developed for the analysis of time to death for patients in 

medical studies in which some patients lived beyond the cutoff 

point of the study, such as 10 years (Insightful Corporation, 

2001). The times to death for patients living beyond 10 years, 

in a 10-year study, are right censored at 10 (time to death > 10) 

because the true times of death are unknown. Survival analysis 

methods have since been applied to other fields of study, such 

as water-quality monitoring, and the term survival analysis has 

expanded to apply to problems other than time to death or time 

to failure. In this study, censored data are left censored—that 

is, the data are less than a particular laboratory reporting level.

For nitrite plus nitrate, one model was selected for the 

stream stations and one model was selected for the lake stations. The models were chosen by examining potential models 

such as those used for other constituents, using analysis 

of variance (anova) to test the significance of terms in the 

potential models, and visually inspecting plots of the measured 

concentrations and the concentrations predicted by the potential models. As with the multiple-regression models, the cosine 

and sine terms were required to be included as pairs and the 

individual terms in the interaction terms were required to be 

included in the model if the interaction terms were used.

Parametric survival regression uses maximum likelihood 

estimation to compute parameter estimates based on measured 

concentrations and the observed proportions of data below one 

or more censoring levels (Helsel, 2005). Maximum likelihood 

estimation assumes that the measured concentrations are from 

a particular underlying distribution. For this study, the assumption was that the base-10 logarithm of concentration followed 

a normal distribution. This assumption was checked before 

performing survival regression by examining probability plots 

that check the similarity of the measured concentrations to the 

a

a

a

a

a

a

a

a

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen  47
specified distribution. In the probability plots, the assumed 

distribution is represented by a straight line and the concentrations are plotted individually. Departures from the straight 

line show how the data differ from the assumed distribution. 

Concentrations that closely follow the line are assumed to be 

from the specified distribution (Helsel, 2005). The normal distribution assumption was reasonable for the measured concentrations, as shown for the Mauvais Coulee near Cando station 

(site 1) (fig. A2), and the distribution used for the survival 

regression was the normal distribution.

Another important consideration in maximum likelihood 

estimation is the number of samples collected. Maximum 

likelihood estimation methods in survival analysis work poorly 

for a small number of samples. According to Helsel (2005), 

the method works best when n > 50. The number of samples 

for dissolved nitrite plus nitrate was greater than 50 for all of 

the stations used for this study (table 3).

After survival regression was performed, the residuals 

were checked using probability plots to determine whether the 

residuals followed the assumed normal distribution. A normal 

distribution was suitable for survival regression of nitrite plus 

nitrate for the Mauvais Coulee station (site 1) (fig. A3).

The fitted survival regression model coefficients for 

nitrite plus nitrate for each stream station and for each lake 

station are given in table A2. As with the multiple-regression analysis, the explanatory variables for the Big Coulee 

near Churchs Ferry and Channel A near Penn stations (sites 7 

and 8, respectively) and for the lake stations were lagged by 

7 days. Blank cells in the table indicate explanatory variables 

that were not significant and, thus, were not used in the model. 

The model coefficients obtained from survival regression 

describe the distribution with the maximum likelihood of 

having produced the measured concentrations and the proportions of censored values below each censoring level.

The p-value for the chi-squared test of the overall significance of the regression model also is given in table A2. The 

test determines whether the entire model is a statistically significant improvement over the null model, the model in which 

only the intercept term is given and all other coefficients are 

equal to zero.

At an α-level of 0.01, the model for nitrite plus nitrate 

was significant for all stream stations. That is, the p-value, 

based on a chi-squared test for regression relation, is less than 

0.01 for each station, indicating the probability of obtaining 

the computed chi-squared test statistic if the model was not 

statistically significant is less than 0.01. A p-value of less than 

0.01 indicates a significant regression relation exists between 

the base-10 logarithm of concentration and the model predictor variables for stream stations. However, a p-value of less 

than 0.01 does not ensure that useful predictions can be made 

using the model.

At an α-level of 0.01, the model was significant for the 

Lake Alice near Churchs Ferry and Lake Irvine near Churchs 

Ferry stations (sites 5 and 6, respectively). The p-values for 

those models indicate a significant regression relation exists 

between the base-10 logarithm of concentration and the model 

predictor variables for those stations. The Sweetwater Lake 

at Sweetwater station (site 3) had a high p-value of 0.720, indicating the explanatory variables in the model do not provide a 

statistically significant improvement for predicting the base-10 

logarithm of nitrite plus nitrate concentration over the null 

model.

Partial significance tests were performed for each of the 

coefficients in the survival regression. The p-values for those 

tests also are given in table A2.

48  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Calcium, dissolved

1 1.960

(<.001)

0.002

(.917)

0.008

(.734)

-0.099

(<.001)

-0.060

(.223)

-0.027

(.673)

-0.051

(.069)

0.057

(.111)

-0.068

(.021)

0.011

(.879)

-0.022

(.474)

0.027

(.437)

0.016

(.611)

67.2 <0.001

2 1.972

(<.001)

.099

(.001)

0

(.999)

-.033(.187)

-.004(.946)

.219

(.120)

-.105(.132)

.092

(.099)

-.158(.003)

.118

(.129)

-.091(.073)

.007

(.841)

-.057(.174)

61.9 <.001

3 1.700

(<.001)

.184

(<.001)

.066

(.027)

-.018(.539)

-.023(.339)

.092

(.004)

.098

(<.001)

-.006(.838)

.082

(<.001)

-.002(.963)

.045

(.056)

-- -- 56.6 <.001

4 1.342

(<.001)

.087

(.008)

.042

(.128)

.028

(.243)

.166

(.002)

-.689

(<.001)

-.133(.001)

-.340

(<.001)

-.151

(<.001)

.309

(.002)

-.018(.564)

.135

(.014)

.006

(.862)

48.6 <.001

5 1.997

(<.001)

.011

(.694)

-.061(.046)

-.080(.004)

.005

(.858)

.162

(<.001)

.028

(.361)

.056

(.095)

.054

(.048)

.031

(.468)

-.028(.246)

-- -- 53.7 <.001

6 1.925

(<.001)

.022

(.430)

-.074(.020)

-.047(.096)

-.012(.721)

.116

(.014)

.057

(.067)

-.018(.584)

.049

(.067)

.036

(.451)

-.063(.016)

-- -- 43.9 <.001

7 1.857

(<.001)

-.034(.323)

-.054(.023)

.054

(.002)

.017

(.323)

.163

(<.001)

-.035(.080)

.085

(<.001)

.004

(.816)

-.004(.871)

.003

(.862)

-.012(.538)

.010

(.551)

31.4 <.001

8 1.910

(<.001)

-.109(.021)

-.110(.009)

-.021(.528)

.020

(.774)

.141

(.248)

.051

(.503)

.091

(.109)

.034

(.581)

.033

(.713)

.058

(.230)

.042

(.342)

.050

(.191)

23.5 .002

a

a

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen  49
Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).—Continued

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

a

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Sulfate, dissolved

1 2.494

(<.001)

0.028

(.345)

-0.126

(<.001)

-0.241

(<.001)

-0.124

(<.001)

-0.022

(.723)

-0.078

(.009)

0.173

(<.001)

-0.072

(.010)

-- -- -- -- 76.3 <0.001

2 2.179

(<.001)

.273

(<.001)

.044

(.323)

-.143

(<.001)

-.144

(<.001)

-.073(.532)

.143

(.016)

.150

(.030)

-.085(.071)

-- -- -- -- 67.9 <.001

3 2.330

(<.001)

.171

(<.001)

-.133

(<.001)

-.085(.006)

-.005(.856)

.036

(.277)

.047

(.019)

-- -- 0.062

(.135)

-0.013

(.583)

-- -- 55.8 <.001

4 1.818

(<.001)

.277

(<.001)

.099

(.064)

-.051(.298)

0

(.992)

-.246(.222)

-.103(.136)

.004

(.966)

-.173(.010)

-- -- -- -- 39.3 <.001

5 2.394

(<.001)

.020

(.611)

-.189

(<.001)

-.195

(<.001)

.078

(.063)

.009

(.858)

-.044(.233)

-- -- .160

(.010)

-.045(.173)

-- -- 56.1 <.001

6 2.351

(<.001)

.052

(.059)

-.240

(<.001)

-.088(.001)

.070

(.033)

.047

(.201)

-.038(.141)

-- -- .180

(<.001)

-.054(.024)

-- -- 7.0 <.001

7 2.343

(<.001)

.053

(.285)

-.257

(<.001)

.040

(.103)

.011

(.518)

.167

(<.001)

-.052(.048)

.043

(.091)

.001

(.968)

-- -- -- -- 34.0 <.001

8 2.436

(<.001)

-.046(.433)

-.284

(<.001)

0

(.996)

-.074(.006)

.077

(.337)

.225

(.001)

.074

(.152)

.111

(.030)

-- -- -- -- 54.2 <.001

a

50 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen 51

Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).—Continued[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

a

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Chloride, dissolved

1 1.447

(<.001)

-0.003

(.944)

-- -0.226

(<.001)

-0.011

(.890)

-0.043

(.697)

-0.123

(.011)

0.076

(.216)

-0.080

(.113)

0.114

(.344)

-0.061

(.250)

0.055

(.352)

0.020

(.697)

61.4 <0.001

2 1.741

(<.001)

.196

(<.001)

-- -.112

(.010)

.071

(.460)

.616

(.013)

-.216(.062)

.277

(.006)

-.280(.002)

.391

(.004)

-.165(.064)

.042

(.465)

-.225(.003)

55.9 <.001

3 1.470

(<.001)

.048

(.030)

-0.129

(<.001)

-.054(.026)

-.019(.325)

.065

(.012)

.037

(.036)

.024

(.264)

.024

(.128)

.029

(.344)

-.014(.454)

-- -- 55.2 <.001

4 .586

(.003)

.227

(<.001)

-- -.014

(.765)

.214

(.038)

-.890(.006)

-.169(.021)

-.434(.006)

-.158(.031)

.442

(.018)

.012

(.833)

.184

(.077)

-.023(.721)

32.3 <.001

5 1.500

(<.001)

-.127(.004)

-.173

(<.001)

-.200

(<.001)

-.001(.976)

-.014(.837)

-.059(.223)

.006

(.900)

-.001(.978)

.092

(.171)

-.075(.050)

-- -- 51.8 <.001

6 1.442

(<.001)

-.055(.100)

-.209

(<.001)

-.115(.001)

.004

(.919)

-.014(.798)

-.060(.096)

-.061(.118)

-.019(.541)

.106

(.064)

-.102(.001)

-- -- 59.2 <.001

7 1.408

(<.001)

.019

(.722)

-- .030

(.254)

.026

(.331)

.179

(<.001)

-.098(.002)

.038

(.199)

.040

(.173)

.040

(.320)

-.009(.764)

-.015(.615)

.035

(.198)

24.7 <.001

8 1.226

(<.001)

.096

(.071)

-- -.015

(.678)

-.060(.470)

-.013(.930)

.251

(.003)

.066

(.339)

.134

(.064)

.044

(.690)

.126

(.041)

.037

(.500)

.089

(.065)

27.2 <.001

a
Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).—Continued

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

a

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Ammonia, dissolved

1 -1.068

(<.001)

-0.167

(.106)

-- -0.089

(.249)

0.111

(.415)

0.461

(.093)

0.146

(.380)

0.300

(.098)

0.194

(.178)

0.234

(.235)

-.002(.984)

-- -- 20.5 0.008

2 -.806

(<.001)

-.322(.001)

-- -.295

(<.001)

-.026(.786)

.628

(.007)

.254

(.094)

.299

(.026)

.340

(.010)

-.001(.996)

-.003(.972)

-- -- 50.7 <.001

4 -.817

(.014)

-.170(.106)

-- .048

(.540)

.133

(.398)

.873

(.091)

.201

(.304)

.479

(.049)

.072

(.680)

.281

(.099)

.012

(.930)

-- -- 39.4 <.001

7 -.713

(<.001)

.123

(.470)

-- -.245

(.009)

-.007

(0.950)

.368

(.089)

-.014(.917)

.086

(.561)

.288

(.032)

.141

(.316)

-.088(.387)

-- -- 28.3 .002

8 -1.169

(<.001)

.234

(.183)

-- -.226

(.058)

-.217(.176)

-.053(.852)

.036

(.855)

-.054(.704)

.319

(.103)

0

(.998)

.198

(.138)

-- -- 23.9 .003

a

52 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).—Continued

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

a

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Phosphorus, total

1 -0.702

(.006)

0.032

(.800)

-0.093

(.583)

-- 0.375

(.046)

0.487

(.165)

-0.357

(.145)

0.451

(.055)

0.029

(.886)

0.537

(.064)

-0.147

(.278)

-- -- 22.8 0.010

2 -.676

(.005)

.091

(.385)

-.293(.013)

-- .088

(.503)

.534

(.085)

-.114(.615)

.386

(.033)

.325

(.090)

.181

(.349)

.122

(.232)

-- -- 34.8 <.001

4 -2.156

(.002)

.202

(.279)

-.154(.400)

-- .333

(.331)

-1.068

(.296)

.115

(.797)

-.931(.053)

.232

(.565)

.491

(.222)

-.232(.440)

-- -- 19.6 .024

7 -.671

(<.001)

-.400(.006)

-.134(.350)

-- .278

(.211)

.396

(.130)

-.242(.494)

.228

(.228)

.242

(.019)

.298

(.265)

-.037(.902)

-- -- 37.8 .001

8 -.835

(<.001)

.228

(.004)

.102

(.213)

-- .064

(.472)

-.179(.222)

-.083(.523)

-.095(.216)

.108

(.359)

-.013(.920)

.038

(.582)

-- -- 37.8 <.001

a

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen 53
Table A1. Fitted model coefficients for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).—Continued

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; 

R2, adjusted coefficient of multiple determination; <, less than; numbers in parentheses indicate p-values for individual model coefficients; --, 

indicates regression terms not used for a particular site]

a

Site number (figure 2)

Regression term

R 2

p-value

Intercept

Lx(t)

Ax(t)

Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx

(t ) 

cos(2

πt )

Dx

(t ) 

sin(2

πt )

Dx

(t ) 

cos(4πt )

Dx

(t ) 

sin(4πt )

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Strontium, dissolved

1 2.631

(<.001)

0.003

(.930)

-0.041(0.195)

-0.092

(<.001)

-0.053

(.169)

-0.012

(.882)

-0.064

(.166)

0.031

(.538)

-0.050

(.204)

0.018

(.737)

-0.066

(.031)

-- -- 61.1 <0.001

2 2.505

(<.001)

.111

(.004)

-.017(.605)

.008

(.788)

.027

(.503)

.042

(.657)

-.183(.005)

.004

(.943)

-.218

(<.001)

.146

(.009)

-.073(.027)

-- -- 54.9 <.001

3 2.465

(<.001)

.095

(<.001)

-.024(.304)

-- -.015

(.271)

.073

(<.001)

.065

(<.001)

-.014(.497)

.065

(<.001)

-- -- -0.019

(.478)

0.024

(.142)

47.1 <.001

4 2.214

(<.001)

.119

(.005)

.029

(.387)

.040

(.188)

.086

(.203)

-.289(.167)

-.194(.031)

-.183(.053)

-.182(.018)

.153

(.037)

-.085(.158)

-- -- 36.5 <.001

5 2.617

(<.001)

.038

(.283)

-.084(.075)

-- .033

(.168)

.222

(<.001)

.046

(.255)

.066

(.135)

.064

(.088)

-- -- -.043

(.342)

.051

(.088)

33.6 <.001

6 2.570

(<.001)

.022

(.456)

-.108(.007)

-- -.007

(.746)

.128

(.001)

.061

(.052)

-.037(.302)

.069

(.022)

-- -- -.036

(.336)

.076

(.005)

37.3 <.001

7 2.496

(<.001)

.005

(.917)

-.080(.026)

.054

(.028)

.014

(.549)

.123

(.017)

-.064(.040)

.083

(.022)

.025

(.406)

.039

(.256)

.036

(.142)

-- -- 18.3 <.001

8 2.724

(<.001)

-.157(.001)

-.172

(<.001)

-.019(.548)

.037

(.427)

.190

(.020)

-.033(.633)

.058

(.144)

-.034(.538)

.141

(.016)

-.029(.430)

-- -- 43.9 <.001

54  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

a
0.001

0.01

0.1

1

10

1970 1975 1980 1985 1990 1995 2000 2005

CONCENTRATION, IN MILLIGRAMS PER LITER AS NITROGEN

Measured concentrations

Censored values (three laboratory reporting

levels were used during the period of record)

Figure A1. Dissolved nitrite plus nitrate concentrations for the Lake Alice near Churchs Ferry, North Dakota, 

station.

Figure A2. Dissolved nitrite plus nitrate concentrations for the Mauvais Coulee near Cando, North Dakota, station. CONCENTRATION, AS BASE-10 LOGARITHM OF MILLIGRAMS PER LITER

-3 -2 -1 0 1 2 3

-3

-2

-1

1

0

2

QUANTILES OF STANDARD NORMAL

Normal distribution

Measured concentrations

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen 55
Figure A3. Dissolved nitrite plus nitrate concentrations and censored values for the Mauvais Coulee near Cando, North Dakota, 

station. RESIDUALS FROM SURVIVAL REGRESSION

-3 -2 -1 0 1 2 3

-3

-2

-1

1

0

2

QUANTILES OF STANDARD NORMAL

Normal distribution

Measured concentrations

Censored values

56 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
Table A2. Fitted survival regression model coefficients for dissolved nitrite plus nitrate for stream stations (sites 1, 2, 4, 7, and 8) and lake stations (sites 3, 5, and 6).

[Lx

(t), 5-year streamflow anomaly for time t; Ax

(t), annual streamflow anomaly for time t; Sx

(t), seasonal streamflow anomaly for time t; Dx

(t), daily streamflow anomaly for time t; cos(2�t), sin(2�t), cos(4�t), 

and sin(4�t), periodic functions for describing seasonal variability; --, indicates regression terms not used for a particular site; <, less than; numbers in parentheses indicate p-values for individual model coefficients]

Site number (figure 2)

Regression term

p-value for chi-squared

test of model significance

Intercept

Lx

(t ) Ax

(t ) Sx

(t ) Dx

(t ) cos(2

πt ) sin(2

πt ) cos(4πt ) sin(4πt ) Dx(t) cos(2

πt)

Dx(t)

sin(2

πt)

Dx(t)

cos(4πt)

Dx(t)

sin(4πt)

Model coefficient

B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

Nitrite plus nitrate as nitrogen, dissolved

1 -1.463

(<.001)

-- -- -0.052

(.630)

-0.173

(.524)

0.864

(.080)

0.686

(.099)

0.815

(.020)

0.885

(.004)

-0.518

(.160)

0.772

(.001)

-- -- <0.001

2 -1.243

(.002)

-- -- -.013

(.920)

-.057(.782)

.831

(.126)

1.064

(.008)

.686

(.032)

.933

(.004)

-.357(.234)

.468

(.010)

-- -- <.001

3 -1.416

(<.001)

-0.086

(.489)

-0.072

(.650)

-- -.314

(.082)

-.055(.591)

.087

(.437)

-.006(.963)

-.072(.415)

-.428(.108)

-.111(.340)

-- -- .720

4 -1.659

(.490)

-- -- -.067

(.570)

-.514(.191)

.836

(.778)

2.513

(.220)

1.262

(.048)

1.406

(.242)

-.918(.012)

.808

(.030)

-- -- <.001

5 -1.253

(<.001)

-.363(.029)

-.222(.297)

-- .011

(.956)

-.139(.649)

-.599(.023)

.210

(.288)

-.384(.121)

.137

(.662)

.176

(.288)

-- -- .008

6 -1.183

(<.001)

-.583

(<.001)

-.667(.001)

-- -.504

(.015)

.171

(.506)

-.368(.087)

.404

(.027)

-.540(.012)

-.460(.150)

.302

(.069)

-- -- <.001

7 -.636

(<.001)

-- -- -.024

(.823)

-.005(.968)

.469

(.004)

.098

(.448)

.328

(.008)

.112

(.389)

.174

(.318)

.313

(.005)

-- -- <.001

8 -1.506

(<.001)

-- -- -.591

(.001)

-.144(.557)

-.362(.384)

-.188(.574)

-.120(.587)

.597

(.062)

-.356(.278)

.245

(.237)

-- -- <.001

Survival Regression for Dissolved Nitrite Plus Nitrate as Nitrogen 57
Variogram Analysis of Flow-Adjusted 

Concentrations

Techniques used to estimate the variance and serial correlation of the flow-adjusted concentrations are described in this 

section. Standard time-series techniques such as correlograms 

are not readily applied to unequally spaced data. Therefore, 

variograms, which are easily applied to unequally spaced data, 

were used to analyze the flow-adjusted concentrations.

Given the flow-adjusted concentration residuals, ε(t) 

(eq. 2), a determination needed to be made as to whether the 

variance of the residuals changed depending on the time of 

year the water-quality sample was collected. Therefore, graphs 

of the fourth root of the squared residuals in relation to the 

time of year were used to detect seasonality in the residual 

variance (see Cressie and Hawkins, 1980, for the rationale for 

using such plots). The residuals for the flow-adjusted sulfate 

concentrations for the Starkweather Coulee near Webster 

station (site 4) are shown in figure A4. An abrupt increase 

occurred in the central tendency of the transformed residuals 

from late May to early June, and the mean of the values for 

season 2 (June through November) was significantly (p-value 

of less than 0.001) higher than the mean for season 1 (March 

through May). The residuals for the flow-adjusted dissolved 

chloride concentrations for the Edmore Coulee near Edmore 

station (site 2) (fig. A5) indicated a significant (p-value of 

0.039) decrease in variability from season 1 to season 2.

Results of the variability analysis for all stations and all 

constituents are given in table A3. The estimated seasonal 

standard deviations given in table A3 are robust estimates 

obtained by squaring the means of the fourth root of the 

squared residuals for each season and applying a bias correction factor (Cressie and Hawkins, 1980). The p-values were 

obtained using a two-sided, two-sample t-test for the difference between the means of the fourth root of the squared 

residuals for seasons 1 and 2. As indicated in table A3, only 

sulfate for the Starkweather Coulee station (site 4) and chloride for the Mauvais Coulee near Cando and Edmore Coulee 

stations (sites 1 and 2, respectively) indicated a significant difference between the standard deviations for seasons 1 and 2. 

However, in the sample design analysis, the seasonal standard 

deviations given in table A3 were used for each constituentstation combination regardless of the significance of the difference between the two seasons. The seasonal standard deviations were used in the design analysis for the following 

reasons—first, differences between season 1 and season 2 

were expected because streamflow for season 1 consisted 

primarily of snowmelt or precipitation runoff from frozen soils 

but streamflow for season 2 consisted primarily of subsurface 

flow or ground-water discharge; and second, the variogram 

analysis indicated seasonal differences existed in the serial 

correlation structure for many constituent-station combinations.

To detect potential serial correlation in the flow-adjusted 

concentration residuals, the residuals were standardized by 

dividing the residuals by the seasonal standard deviations 

given in table A3. The (semi-)variogram of the standardized 

residuals was defined as

γ(h) 1

2

--EV [ε*(t + h) – ε*(t)]

2 = { }

where

γ is the variogram;

h is the time lag, in weeks;

EV denotes the expected value;

ε* is the standardized residual for the flowadjusted concentration; 

and

t is time, in decimal years.

The variogram and correlogram of the standardized 

residuals were related through the following equation (Cressie, 

1991):

ρ(h) = 1 – γ(h)

where

ρ(h) is the correlogram [the correlation between 

ε*(t + h) and ε*(t)].

A robust estimator of the variogram (Cressie, 1991, 

eq. 2.4.12) was computed for each constituent-station pair. 

To detect potential differences between the two seasons, two 

estimated variograms were computed. For the first estimated 

variogram, only residual pairs with at least one of the residuals from season 1 were used in the computations, and, for the 

second estimated variogram, only residual pairs with at least 

one residual from season 2 were used. The estimated seasonal variograms for dissolved sulfate for the Big Coulee near 

Churchs Ferry station (site 7) are shown in figures A6 and A7. 

The points in the graph were computed using pairs of residuals with time lags binned into 2-week intervals—for example, 

the first point was computed using residual pairs separated by 

less than 2 weeks, the second using residual pairs separated by 

2 to 4 weeks, etc. As indicated in figure A6, for season 1, the 

variogram was approximately constant at 1.0, which indicates no discernible serial correlation existed among residuals separated by 2 weeks or more. In contrast, as indicated in 

figure A7, the variogram for season 2 was substantially less 

than 1.0 for small lags and increased gradually until reaching 

a sill of 1.0 at about a 12-week lag. Therefore, in season 2, 

serial correlation existed between neighboring residuals up 

to a lag of about 12 weeks. Most of the flow for Big Coulee 

is from Lakes Alice and Irvine (fig. 2) and the flow-adjusted 

dissolved sulfate concentrations for those lakes were expected 

to be serially correlated, especially during the summer months. 

Because too few concentration data are available for the lakes, 

the variogram for time lags of less than 12 weeks could not be 

estimated. Therefore, the variograms for the lake data could 

58  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

(A2)

(A3)
Figure A4. Fourth root of squared residuals for flow-adjusted dissolved sulfate concentrations for the 

Starkweather Coulee near Webster, North Dakota, station.

Figure A5. Fourth root of squared residuals for flow-adjusted dissolved chloride concentrations for the Edmore 

Coulee near Edmore, North Dakota, station.

MAR APR MAY JUNE JULY AUG SEPT OCT NOV DEC JAN FEB

0

0.2

0.4

0.6

0.8

FOURTH ROOT OF SQUARED RESIDUAL

Seasonal mean

Transformed residuals

MAR APR MAY JUNE JULY AUG SEPT OCT NOV DEC JAN FEB

0

0.2

0.4

0.6

0.8

FOURTH ROOT OF SQUARED RESIDUAL

Seasonal mean

Transformed residuals

Variogram Analysis of Flow-Adjusted Concentrations 59
not be compared easily to the variogram for the Big Coulee 

station (site 8).

The estimated variograms for total phosphorus concentrations for the Channel A near Penn station (site 8) (figs. A8 and 

A9) were similar to the variograms for dissolved sulfate. No 

discernible serial correlation existed for season 1, but some 

serial correlation may have occurred for season 2 for time lags 

of less than about 8 weeks. Estimated variograms for dissolved 

calcium, sulfate, and chloride for most stations were similar 

to the variograms shown in figures A6 and A7 as were the 

estimated variograms for total phosphorus for the Big Coulee 

station (site 7). For total phosphorus for the remaining stations 

and for dissolved ammonia and strontium for all stations, no 

discernible serial correlation existed in either season.

Plots of the estimated variograms were examined as 

described previously to select a correlation range (an approximate time lag beyond which the flow-adjusted concentrations 

could be assumed to be uncorrelated). The correlation ranges, 

which were used in the design analysis, are given in table A3.

Model Robustness

The fitted trends and p-values given in table 4 were 

based on the assumptions that the residuals from the regression model, with both the flow-adjustment variables and the 

trend variables included, were uncorrelated and had constant 

variance. However, as indicated previously, in a few cases, 

serial correlation existed among neighboring residuals and/or 

a significant difference existed between the residual variances 

for the two seasons (table A3).

The potential difference between the residual variances 

for the two seasons was expected to have a negligible effect on 

the fitted trends and p-values. In all but a few cases, the difference between the residual variances for the two seasons was 

small and insignificant (table A3). In cases when a significant 

difference did occur, time-series plots of the residuals did not 

indicate a systematic pattern of change in the residual variances from year to year (only from season to season) because 

the apportionment of samples among seasons did not change 

appreciably with time.

Unlike the seasonal residual variances, serial correlation 

in the residuals was expected to have an effect on the fitted 

trends and p-values in some cases. Ignoring serial correlation 

can inflate the error variance and thus reduce the significance 

of a fitted trend or inflate the effective number of samples 

and thus artificially increase the significance of a fitted trend. 

The effects of serial correlation can be removed by estimating 

60 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

the correlation matrix of the residuals along with the regression parameters and generalized least squares used to fit the 

regression model or by “thinning” the data so the residuals are 

spaced far enough apart to be uncorrelated. Using the correlation ranges given in table A3, original data needed to be 

thinned only for the Big Coulee near Churchs Ferry station 

(site 7). Therefore, that station likely would be most affected 

by serial correlation.

The trend results for dissolved chloride and dissolved 

nitrite plus nitrate for the Big Coulee station (site 7), both with 

and without data thinning, are given in table A4. For dissolved 

chloride, the data set contained 198 measurements before 

thinning and 103 measurements after thinning. The estimated 

coefficient for the Channel A step trend increased from 0.105 

without thinning to 0.160 with thinning, and the p-value 

decreased from 0.067 to 0.025. The estimated coefficient for 

the linear trend essentially was unchanged, but the p-value of 

the trend increased from 0.001 to 0.014. Thus, the step trend 

became more significant and the linear trend less significant 

as a result of data thinning. For dissolved chloride, the large 

group of low concentrations that occurred from 1992-97 had a 

substantial effect on the fitted trends (see figure 18). Removing some of the low concentrations probably resulted in the 

increased significance of the step trend and the decreased 

significance of the linear trend. However, the thinning did not 

have a drastic effect on the fitted trends.

For dissolved nitrite plus nitrate, the data set contained 

118 measurements before thinning and 63 measurements 

after thinning. For both the step trend and the linear trend, the 

sign of the trends was unchanged but the magnitude of the 

trends decreased and the p-values increased as a result of the 

thinning. As for dissolved chloride, the differences probably 

resulted from thinning of the low concentrations that occurred 

at the end of the period of record (see figure 19). However, 

because few high concentrations occurred during 1980-90 to 

counter the effects of the low concentrations, both the step 

trend and the linear trend became less significant as a result of 

the data thinning rather than more significant as for dissolved 

chloride.

As shown in the previous examples, long sampling gaps 

or highly variable sampling frequencies from year to year 

probably have more of an effect on trend results than potential 

serial correlation. Also, as mentioned previously, fixed sampling designs in which sampling frequencies remain constant 

from year to year generally are more efficient for analyzing 

long-term trends than variable sampling designs in which the 

sampling frequencies change. Sampling at too high a frequency during some years may introduce redundancy and thus 

decrease the efficiency of a design without drastically changing the direction or significance of the fitted trends.
Variogram Analysis of Flow-Adjusted Concentrations 61

Table A3. Estimated seasonal standard deviations and correlation ranges for flow-adjusted concentrations.

U.S. Geological Survey station name

Season 1 

(March through May)

Season 2 

(June through November) p-value for

difference

in seasonal

standard

deviations

Standard

deviation

Correlation

range

Standard

deviation

Correlation

range

Calcium, dissolved

Mauvais Coulee near Cando, North Dakota 0.094 2 0.062 4 0.056

Edmore Coulee near Edmore, North Dakota .108 2 .076 4 .117

Sweetwater Lake at Sweetwater, North Dakota .119 2 .075 4 .106

Starkweather Coulee near Webster, North Dakota .104 2 .107 2 .899

Lake Alice near Churchs Ferry, North Dakota .093 2 .069 6 .342

Big Coulee near Churchs Ferry, North Dakota .108 2 .091 6 .395

Channel A near Penn, North Dakota .139 2 .090 6 .059

Sulfate, dissolved

Mauvais Coulee near Cando, North Dakota 0.135 2 0.116 2 0.458

Edmore Coulee near Edmore, North Dakota .188 2 .183 4 .893

Sweetwater Lake at Sweetwater, North Dakota .126 2 .147 4 .522

Starkweather Coulee near Webster, North Dakota .185 2 .357 8 < .001

Lake Alice near Churchs Ferry, North Dakota .121 2 .106 12 .686

Big Coulee near Churchs Ferry, North Dakota .148 2 .193 12 .180

Channel A near Penn, North Dakota .207 2 .155 4 .158

Chloride, dissolved

Mauvais Coulee near Cando, North Dakota 0.177 2 0.118 6 0.047

Edmore Coulee near Edmore, North Dakota .210 2 .136 4 .039

Sweetwater Lake at Sweetwater, North Dakota .098 2 .116 4 .448

Starkweather Coulee near Webster, North Dakota .201 2 .226 6 .554

Lake Alice near Churchs Ferry, North Dakota .181 2 .100 6 .097

Big Coulee near Churchs Ferry, North Dakota .171 2 .233 6 .148

Channel A near Penn, North Dakota .181 2 .124 4 .083

Ammonia, dissolved

Mauvais Coulee near Cando, North Dakota 0.327 2 0.246 2 0.327

Edmore Coulee near Edmore, North Dakota .276 2 .232 2 .462

Starkweather Coulee near Webster, North Dakota .348 2 .243 2 .119

Big Coulee near Churchs Ferry, North Dakota .437 2 .361 2 .345

Channel A near Penn, North Dakota .486 2 .534 2 .551
Table A3. Estimated seasonal standard deviations and correlation ranges for flow-adjusted concentrations.—Continued

U.S. Geological Survey station name

Season 1 

(March through May)

Season 2 

(June through November) p-value for

difference

in seasonal

standard

deviations

Standard

deviation

Correlation

range

Standard

deviation

Correlation

range

Phosphorus, total

Mauvais Coulee near Cando, North Dakota 0.162 2 0.105 2 0.178

Edmore Coulee near Edmore, North Dakota .151 2 .122 2 .417

Starkweather Coulee near Webster, North Dakota .171 2 .276 2 .096

Big Coulee near Churchs Ferry, North Dakota .148 2 .118 8 .323

Channel A near Penn, North Dakota .188 2 .230 8 .384

Strontium, dissolved

Mauvais Coulee near Cando, North Dakota 0.113 2 0.079 2 0.098

Edmore Coulee near Edmore, North Dakota .125 2 .097 2 .225

Sweetwater Lake at Sweetwater, North Dakota .065 2 .086 2 .350

Starkweather Coulee near Webster, North Dakota .113 2 .119 2 .809

Lake Alice near Churchs Ferry, North Dakota .103 2 .101 2 .940

Big Coulee near Churchs Ferry, North Dakota .114 2 .101 2 .601

Channel A near Penn, North Dakota .120 2 .102 2 .406

62 Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota

Table A4. Results of test of significance for Channel A step trend and linear trend from 1980 through September 2003 for the Big 

Coulee near Churchs Ferry, North Dakota, station.

[Numbers in parentheses are for data that were thinned to remove serial correlation.]

Site

number

(figure 2)

U.S. Geological Survey

station name

Coefficient for

Channel A

step trend

p-value for

Channel A

step trend

Coefficient for

linear trend

from 1980

through

September 2003

p-value for

linear trend

from 1980

through

September 2003

Chloride, dissolved

7 Big Coulee near Churchs Ferry, North Dakota 0.105

(.160)

0.067

(.025)

-0.015

(-.016)

0.001

(.014)

Nitrite plus nitrate as nitrogen, dissolved

7 Big Coulee near Churchs Ferry, North Dakota -0.115

(-.018)

0.662

(.951)

-0.030

(-.025)

0.100

(.235)
Figure A6. Estimated variogram for flow-adjusted dissolved sulfate concentrations for the Big 

Coulee near Churchs Ferry, North Dakota, station for season 1 (March through May).

0

0.4

0.8

1.2

1.6

VARIOGRAM

TIME LAG, IN WEEKS

0 4 8 12 16 20

Line with variogram equal to 1

Estimated variogram

Figure A7. Estimated variogram for flow-adjusted dissolved sulfate concentrations for the Big 

Coulee near Churchs Ferry, North Dakota, station for season 2 (June through November).

0

0.4

0.8

1.2

1.6

VARIOGRAM

TIME LAG, IN WEEKS

0 4 8 12 16 20

Line with variogram equal to 1

Estimated variogram

Model Robustness 63
Figure A8. Estimated variogram for flow-adjusted total phosphorus concentrations for the 

Channel A near Penn, North Dakota, station for season 1 (March through May).

Figure A9. Estimated variogram for flow-adjusted total phosphorus concentrations for the 

Channel A near Penn, North Dakota, station for season 2 (June through November).

0

0.4

0.8

1.2

1.6

VARIOGRAM

TIME LAG, IN WEEKS

0 4 8 12 16 20

Line with variogram equal to 1

Estimated variogram

0

0.4

0.8

1.2

1.6

VARIOGRAM

TIME LAG, IN WEEKS

0 4 8 12 16 20

Line with variogram equal to 1

Estimated variogram

64  Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota
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/
Ryberg and Vecchia—Water-Quality Trend Analysis and Sampling Design, Devils Lake Basin, North Dakota—Scientific Investigations Report 2006-5238

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