Emerson, Vecchia, and Dahl–Evaluation of Drainage-Area Ratio Method Used to Estimate Streamflow for the

Red River of the North Basin, North Dakota and Minnesota–Scientific Investigations Report 2005–5017

In cooperation with the Bureau of Reclamation

Evaluation of Drainage-Area Ratio Method Used to Estimate 

Streamflow for the Red River of the North Basin, North 

Dakota and Minnesota

Scientific Investigations Report 2005–5017

U.S. Department of the Interior

U.S. Geological Survey Printed on recycled paper
Evaluation of Drainage-Area Ratio Method 

Used to Estimate Streamflow for the Red River of the North Basin, North Dakota and MinnesotaBy Douglas G. Emerson, Aldo V. Vecchia, and Ann L. Dahl
U.S. Department of the Interior

Gale A. Norton, SecretaryU.S. Geological SurveyCharles G. Groat, DirectorU.S. Geological Survey, Reston, Virginia: 2005For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal CenterDenver, CO 80225For more information about the USGS and its products:Telephone: 1-888-ASK-USGSWorld Wide Web: http://www

usgs

gov/
 iii

ContentsAbstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Description of Red River of the North Basin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Drainage-basin characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Evaluation of drainage-area ratio method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figures

1. Map showing locations of streamflow-gaging stations used in study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22. Isohyetal map of mean annual winter/spring (January, February, March, April, May,

November, and December) precipitation for 1971 through 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

3. Isohyetal map of mean annual summer (June, July, August, September, and

October) precipitation for 1971 through 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

4. Graph showing recorded spring streamflow and fitted spring streamflow computed

from the regression model with the logarithm of drainage area and the logarithmof winter/spring precipitation as explanatory variables

9
iv 

Conversion Factors and Datum

Multiply By To obtain

Length

inch 25.4 millimeterfoot 0.3048 metermile 1.609 kilometer

Area

square mile 2.590 square kilometer

Flow rate

cubic foot per second 0.02832 cubic meter per second

Hydraulic gradient

foot per mile 0.1894 meter per kilometer

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88)
Evaluation of Drainage-Area Ratio Method Used to

Estimate Streamflow for the Red River of the NorthBasin, North Dakota and MinnesotaBy Douglas G. Emerson, Aldo V. Vecchia, and Ann L. DahlAbstract

The drainage-area ratio method commonly is used to estimate streamflow for sites where no streamflow data were collected. To evaluate the validity of the drainage-area ratio 

method and to determine if an improved method could be developed to estimate streamflow, a multiple-regression technique was used to determine if drainage area, main channel slope, and precipitation were significant variables for estimating streamflow in the Red River of the North Basin. A separate regression analysis was performed for streamflow for each of three seasons--winter

spring

and summer

Drainage area and summer 

of variance extension type 1 (MOVE.1), can be used to extend the streamflow record (Hirsch, 1982). 

Hirsch (1979) evaluated the drainage-area ratio method, 

regional regression equations, linear regression equations, and 

log-log regression equations to reconstruct streamflow records and noted that “...log regressions appear superior to the linear. The regional statistics method is distinctly superior to the drainage area ratio method, but comparison between the regional statistics and the log regression method is ambiguous.” Hirsch 

(1979) also noted that the drainage-area ratio method works relatively well if streams have similar flow characteristics. How EXPLANATION

Red River of the North Basin boundaryDevils Lake Basin boundaryDevils Lake BasinStation and map number (given in table 1)

Red

River

of

the

North

M A N I T O B A

99 ̊ 97 ̊

95 ̊

95 ̊

49 ̊

47 ̊ 47 ̊

49 ̊

99 ̊

Base from U.S. Geological Survey

1:2,000,000, 1972

1

2 3

4

5

6

1

7

12

11

10

13

18

17

15

14

16

9

8

24 27

23

26 25

19

2220

21

Grand

Forks

Thief River Falls

Red Lake Falls

Oklee

Plummer

Argyle

Malung

Caribou

Lake Bronson

Minto

Fordville

Grafton

Akra

Neche

FARGO

Twin

Valley

MOORHEAD

Dilworth Hawley

Hendrum

Climax

Crookston

Enderlin

Valley City

Dazey

Harvey

Warwick

Cooperstown

Amenia

Hillsboro

Abercrombie

Upper

Red

Lake

Devils

 Lake

Sheyenne

River

River

River

River

River

Pelican

Tail

River

River

R.

R.

R.

R.

Tongue

Pembina

R.

River

Rivers

S. Br.

N. Br. Roseau

River

Clearwater

Lost

R.

R.

Two

River

R.

R.

Middle

Tamarac

Thief

Red

Lake

Snake

Forest

Turtle

Park

Rice

Marsh River

R.

Sand

Hill

Wild

Buffalo

R.

R.

R.

Elm

Goose

Maple

Rush

Stump

 Lake

Lake

Ashtabula

Beaver

Baldhill

Cr.

Cr.

Cart

Cr.

Lower

Red

Lake

RED RIVER OF THENORTH BASIN

NORTH DAKOTA

SOUTH DAKOTA

MINNESOTA

State Ditch #51

2 Evaluation of Drainage-Area Ratio Method Used to Estimate Streamflow for the Red River of the North Basin
Drainage-Basin Characteristics 3

Drainage-Basin Characteristics

Topographic and meteorologic characteristics of a drainage basin affect streamflow from the basin. The characteristics 

that were used in this study to evaluate the drainage-area ratio method were drainage area, main channel slope, and precipitation. If available, the noncontributing drainage area was excluded and only the contributing drainage area was used. The main channel slope that was used was the slope, as defined by Benson (1962), that is between points that are 85 and 10 percent 

upstream from the site of interest. The drainage area and the main channel slope for the streamflow-gaging stations used in this study were obtained from the U.S. Geological Survey’s basin characteristics file, Williams-Sether (1992), or Lorenz and others (1997). The precipitation data used in this study were obtained from the U.S. Department of Commerce, National Climatic Data Center (2002). Data for 1971 through 2000 were 

used to compute mean annual winter/spring precipitation and mean annual summer precipitation for each of 90 sites in or near the Red River of the North Basin. Winter/spring precipitation was defined as the average precipitation for January, February, March, April, May, November, and December. Those months were chosen to represent the months for which snowfall accumulation and precipitation would affect spring streamflow. S

i

it

ti

d

fi

d 

th

i

it

ti

where

Y

 ̃

ij is the estimated streamflow, in cubic feet per 

second, for month i and year j for the site of interest;

Ay is the drainage area, in square miles, for the 

site of interest;

Ax is the drainage area, in square miles, for the 

streamflow-gaging station; and

Xij is the streamflow, in cubic feet per second, for 

month i and year j for the streamflowgaging station.

For the drainage-area ratio method (eq. 1), the assumption 

is made that the exponent of 

Ay

Ax----- ⎝ ⎠ ⎛ ⎞ is 1. To test this assumption, the drainage areas for the 27 streamflow-gaging stations 

were used in equation 1 to estimate streamflow. To simplify the evaluation and to determine if the estimated streamflow was an unbiased estimate of the actual streamflow, the average streamflow for various seasons for the period of record (1971-2000) was used rather than the streamflow for individual months. Thus, estimated streamflow for a particular season and period of record is given by
 EXPLANATION

Red River of the North Basin boundaryDevils Lake Basin boundaryDevils Lake BasinLine of equal mean annual winter/springprecipitation

Interval 0

25 inch

99 ̊ 97 ̊

95 ̊

95 ̊

49 ̊

47 ̊ 47 ̊

49 ̊

99 ̊

Red

River

of

the

North

Upper

Red

Lake

Devils

 Lake

Sheyenne

River

Lower

Red

Lake

Base from U.S. Geological Survey

1:2,000,000, 1972

M A N I T O B A

6

4.25

4.25

5

25

5.25

4.75

3.75

3.5

4.5 4

5

6

5.75

5

4

4

4

4

5.25

Grand

Forks

FARGO

4 Evaluation of Drainage-Area Ratio Method Used to Estimate Streamflow for the Red River of the North Basin
 EXPLANATION

Red River of the North Basin boundaryDevils Lake Basin boundaryDevils Lake BasinLine of equal mean annual summerprecipitation

Interval 0

5 inch

99 ̊ 97 ̊

95 ̊

95 ̊

49 ̊

47 ̊ 47 ̊

49 ̊

99 ̊

Red

River

of

the

North

Upper

Red

Lake

Devils

 Lake

Sheyenne

River

Lower

Red

Lake

Base from U.S. Geological Survey

1:2,000,000, 1972

M A N I T O B A

12.5

13.5

20.5

21.5

14.5

15.5

18.5

19.5

18.5

16.5

17.5

19.5

15.5

14.5 14.5

13.5

15.5

13

14

14

15

13

15 15 19

18

19

16 17

16

19

Grand

Forks

FARGO

Evaluation of Drainage-Area Ratio Method 5
Map

number

(figure 1)

Streamflowgaging

station

number

Streamflow-gaging

station name

Drainage

area

(square

miles)

Main

channel

slope

(feet

per

mile)

weighted

mean

winter/

spring

precipitation

(inches)

Area

weighted

mean

summer

precipitation

(inches)

Annual

streamflow

(cubic

feet

per

second)

Win

stream

(cub

fee

pe

seco

1 05053000 Wild Rice River near

Abercrombie, N. Dak.

1,490 2.1 5.12 15.29 129.43 11

2 05054500 Sheyenne River above

Harvey, N. Dak.

154 3.0 4.04 12.72 14.72 3

3 05056000 Sheyenne River near

Warwick, N. Dak.

760 1.6 4.08 13.46 79.82 12

4 05057000 Sheyenne River near

Cooperstown, N. Dak.

1,270 .8 4.17 13.93 155.27 26

5 05057200 Baldhill Creek near Dazey,

N. Dak.

351 3.0 4.41 14.88 30.48 3

6 05059700 Maple River near Enderlin,

N. Dak.

796 3.0 4.63 15.07 65.69 6

7 05060500 Rush River at Amenia,

N. Dak.

116 3.5 4.63 14.69 15.71 1

8 05061000 Buffalo River near Hawley,

Minn.

325 6.2 5.44 17.71 92.78 37

9 05062000 Buffalo River near Dilworth,

Minn.

975 2.6 5.26 16.85 188.01 52

10 05062500 Wild Rice River at Twin

Valley, Minn.

934 4.2 5.33 18.33 245.55 90

11 05064000 Wild Rice River at Hendrum,

Minn.

1,560 4.4 5.27 17.82 382.60 111

12 05066500 Goose River at Hillsboro,

N. Dak.

1,093 4.1 4.68 15.56 128.21 15
13 05069000 Sand Hill River at Climax,

Minn.

420 4.8 4.81 16.90 95.82 26

14 05076000 Thief River near Thief

River Falls, Minn.

985 1.9 4.41 16.50 226.78 40

15 05078000 Clearwater River at

Plummer, Minn.

555 3.4 4.60 18.12 177.80 73

16 05078230 Lost River at Oklee, Minn. 254 4.9 4.58 18.06 73.71 19

17 05078500 Clearwater River at Red

Lake Falls, Minn.

1,380 5.3 4.59 17.94 370.52 127

18 05079000 Red Lake River at

Crookston, Minn.

5,270 2.2 4.62 17.89 1,446.46 805

19 05084000 Forest River near Fordville,

N. Dak.

336 11.0 4.08 14.86 43.96 9

20 05085000 Forest River at Minto,

N. Dak.

620 10.0 4.15 14.91 55.38 6

21 05087500 Middle River at Argyle,

Minn.

255 4.3 4.27 14.74 48.57 4

22 05090000 Park River at Grafton,

N. Dak.

695 6.5 3.91 14.77 68.02 4

23 05094000 South Branch Two Rivers

at Lake Bronson, Minn.

422 4.0 4.36 14.61 100.80 8

24 05100000 Pembina River at Neche,

N. Dak.

3,410 4.0 3.78 14.31 320.60 27

25 05101000 Tongue River at Akra,

N. Dak.

160 21.7 3.68 14.63 20.64 5

Map

number

(figure 1)

Streamflowgaging

station

number

Streamflow-gaging

station name

Drainage

area

(square

miles)

Main

channel

slope

(feet

per

mile)

Areaweighted

mean

winter/

spring

precipitation

(inches)

Areaweighted

mean

summer

precipitation

(inches)

Annual

streamflow

(cubic

feet

per

second)

Win

stream

(cub

fee

pe

seco
26 05104500 Roseau River below South

Fork near Malung, Minn.

424 3.3 4.56 15.98 127.35 22

27 05112000 Roseau River below State

Ditch 51 near Caribou, Minn.

1,420 1.3 4.52 15.73 332.23 64

Map

number

(figure 1)

Streamflowgaging

station

number

Streamflow-gaging

station name

Drainage

area

(square

miles)

Main

channel

slope

(feet

per

mile)

weighted

mean

winter/

spring

precipitation

(inches)

Area

weighted

mean

summer

precipitation

(inches)

Annual

streamflow

(cubic

feet

per

second)

Win

stream

(cub

fee

pe

seco
Evaluation of Drainage-Area Ratio Method 9

uted, random variables that have the same distribution as ε in equation 4. This limit exists provided the distribution of ε satisfies certain constraints.

From equation 5, it follows that

E

YM

*

XM

--------- ⎝ ⎠ ⎛ ⎞ Ay Ax ----- ⎝ ⎠ ⎛ ⎞ α Py Px ----- ⎝ ⎠ ⎛ ⎞ β Sy Sx ---- ⎝ ⎠ ⎛ ⎞ γE 10ηy ηx – = ( ) (6)

where

E denotes the expected value.

Therefore, equation 3, multiplied by a bias-correction factor, 

E 10ηy ηx – ( ) , is an unbiased estimate of YM* (given XM ).

Assuming the 27 streamflow-gaging stations could be 

treated as a homogeneous group--that is, assuming equation 5 

holds for all 27 stations--the equation was fitted to the streamflow data using ordinary least-squares regression. A separate 

regression analysis was performed for each of three seasons--winter (January, February, November, and December), spring (March, April, and May), and summer (June, July, August, September

and October)

Drainage area

main channel slope

win

where

Q

ˆ

is the estimated spring streamflow, in cubic 

feet per second, for the site of interest.

The coefficients given in equation 7 for both drainage area and winter/spring precipitation were highly significant (p-values of less than 0.01). The remaining variables were not significant when drainage area and winter/spring precipitation were included in the regression equation. The recorded spring streamflow values and the fitted spring streamflow values computed from the regression model are shown in figure 4

. The 

coefficient of determination was 89 percent. However, the 

recorded streamflow values for the North Dakota stations tended to be less than the fitted streamflow values, and the recorded streamflow values for the Minnesota stations tended to be greater than the fitted streamflow values.

The results of the regression analyses for the winter and 

summer streamflows were similar to the results of the regression analysis for the spring streamflow. Drainage area was the 

most significant variable for both the winter and summer streamflows, but summer precipitation also was a significant variable. The regression equations generally overestimated streamflows for the North Dakota stations and underestimated streamflows for the Minnesota stations.
C C1 C2 = + I MN ( )

10 Evaluation of Drainage-Area Ratio Method Used to Estimate Streamflow for the Red River of the North Basin

(8)

where

C1 and C2 are regression coefficients, and

I MN ( ) is equal to zero for the North Dakota stations 

and 1 for the Minnesota stations.

The indicator variables for the coefficient for logA were given by

α α1 α2 = + I MN ( ) (9)

where

α1 and α2 are regression coefficients.

Results from the revised regression model indicate C2 was positive and highly significant for all three seasons and α2 was not significantly different from zero for all three seasons. Furthermore, main channel slope, winter/spring precipitation, and summer precipitation were not significant variables and drainage area was the only significant variable. The revised regression equations--that is, the equations that include the indicator 

variables--are given in table 2

Table 2. Regression equations used to estimate log-transformed seasonal streamflow for the Red River of the North Basin, NorthDakota and Minnesota.[R2, coefficient of determination; log, base-10 logarithm; Q , average seasonal streamflow, in cubic feet per second, for 1971 through 2000; C , intercept; A , drainage area, in square miles; the number in parentheses (0.10) is the standard error of the coefficient]

Season Regression equation R2 (percent)

Winter (January, February, November, and December) logQ = C + 0.85logA

( ) 0.10

C = –1.43 for North Dakota stations

1

C = –0.67 for Minnesota stations1

, and the fitted regression lines for 

90
0

123401234

WINTERSPRING

OW, AS BASE-10 LOGARITHM OF CUBIC FEET PER SECOND

South Branch Two Rivers atLake Bronson, Minnesota

Middle River at Argyle, Minnesota

Evaluation of Drainage-Area Ratio Method 11
12 Evaluation of Drainage-Area Ratio Method Used to Estimate Streamflow for the Red River of the North Basin

all three seasons. Therefore, streamflow at a Minnesota station should not be used with equation 3 to estimate streamflow at a 

North Dakota station or vice versa. No significant differences 

occurred between the coefficient for log

A for the North Dakota 

stations and the coefficient for log

A for the Minnesota stations.

As shown in figure 5, the regression equations provided a 

good fit to streamflow data for the spring and summer seasons. For the winter season, however, the streamflow data for two Minnesota stations (South Branch Two Rivers at Lake Bronson, Minn., and Middle River at Argyle, Minn.--sites 21 and 23, 

respectively; fig. 1) were outliers when the stations were 

included in the Minnesota group of stations. Therefore, those two stations were included in the North Dakota group of stations for the winter season. If the stations had been included in 

the Minnesota group of stations, a significant difference would have occurred between the slopes of the regression lines for the two groups of stations, and the slope of the regression line for the Minnesota group of stations would have been greater than one. Winter streamflow for all stations was predominantly ground-water discharge, and the assumption could be made that ground-water discharge would increase less, in relation to an increased drainage area, than precipitation or snowmelt runoff. The bias-correction factor, 

E 10ηy ηx – ( ) , given in equation 6 

was estimated using the nonparametric method described by 

Summary

Many water-management programs require streamflow 

data for sites where no data were collected or for streamflowgaging stations for periods during which the gage was not operated. Methods used to estimate streamflow for sites where no 

streamflow data were collected include the drainage-area ratio method. To evaluate the validity of the drainage-area ratio method and to determine if an improved method could be developed to estimate streamflow, a multiple regression technique was used to determine if drainage area, main channel slope, and precipitation were significant variables for estimating streamflow. Monthly streamflows for 27 streamflow-gaging stations in the Red River of the North Basin and precipitation data for 90 sites were used in the study. The period of record for each station and each site was 1971 through 2000. The precipitation data were used to compute mean annual winter/spring precipitation and mean annual summer precipitation for each of the 90 sites. Winter/spring precipitation was defined as the average precipitation for January, February, March, April, May, November, and December. Summer precipitation was defined as the average precipitation for June, July, August, September, and October.
References 13

ReferencesBenson, M.A., 1962, Factors influencing the occurrence of 

floods in a humid region of diverse terrain: U.S. Geological Survey Water-Supply Paper 1580-B, 64 p.

Burr, M.J., Teller, R.W., and Neitzert, K.M., 2003, Water 

resources data, South Dakota, Water year 2002: U.S. Geological Survey Water-Data Report SD-02-1, 515 p.

Duan, N., 1983, Smearing estimate--A nonparametric retransformation method: Journal of the American Statistical Association, v. 78, p. 605-610.Emerson, D.G., and Dressler, V.M., 2002, Historic and unregulated monthly streamflow for selected sites in the Red River 

of the North Basin in North Dakota, Minnesota, and South Dakota, 1931-99: U.S. Geological Survey Water-Resources Investigations Report 02-4095, 271 p.

Guenthner, R.S., Weigel, J.F., and Emerson, D.G., 1990, Gaged 

and estimated monthly streamflow during 1931-84 for selected sites in the Red River of the North Basin in North Dakota and Minnesota: U.S. Geological Survey WaterResources Investigations Report 90-4167, 230 p.

Ghupta, V.K., and Waymire, E.C., 1990, Multiscaling properties of spatial rainfall and river flow distributions: Journal of 

Geophysical Research, v. 95, p. 1999-2009.

Harkness

R

E

Lundgren

R

F

Norbeck

S

W

Robinson

Lorenz, D.L., Carlson, G.H., and Sanocki, C.A., 1997, Techniques for estimating peak flow on small streams in Minnesota: U.S. Geological Survey Water-Resources Investigations Report 97-4249, 42 p.Mitton, G.B., Guttormson, K.G., Stratton, G.W., and Wakeman, E.S., 2003, Water resources data, Minnesota, Water 

year 2002: U.S. Geological Survey Water-Data Report MN02-1, 333 p.

Stoner, J.D., Lorenz, D.L., Wiche, G.J., and Goldstein, R.M., 

1993, Red River of the North Basin, Minnesota, North Dakota, and South Dakota: American Water Resources Association Monograph Series No. 19 and Water Resources Bulletin, v. 29, no. 4, p. 575-614.

Troutman, B.M., and Vecchia, A.V., 1999, Estimation of Renyi 

exponents in random cascades: Bernoulli, v. 5, no. 2, p. 191-207.

U.S. Department of Commerce, National Climatic Data Center, 

2002, Monthly station normals of temperature, precipitation, and heating and cooling degree days 1971-2000 (Minnesota and North Dakota): Asheville, North Carolina, Climatography of the United States, No. 81.

U.S. Geological Survey, 2003, National elevation dataset: 

accessed July 15, 2003, at http://gisdata.usgs.gov/NED/.