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U.S. Department of the Interior 

U.S. Geological Survey

Supplement to Water-Resources Investigations 

Report 99-4122, Lake-Level Frequency Analysis for 

the Waubay Lakes Chain, Northeastern South Dakota

By Colin A. Niehus, Aldo V. Vecchia, and Ryan F. Thompson

Water-Resources Investigations Report 99-4251

Prepared in cooperation with the Federal Emergency Management Agency
U.S. Department of the Interior

Bruce Babbitt, Secretary

U.S. Geological Survey

Charles G. Groat Director

The use of firm, trade, and brand names in this report is for identification purposes 

only and does not constitute endorsement by the U.S. Geological Survey.

Rapid City, South Dakota: 1999

For additional information write to:

District Chief 

U.S. Geological Survey 

1608 Mt. View Road 

Rapid City, SD 57702

Copies of this report can be purchased from:

U.S. Geological Survey 

Information Services 

Box 25286 

Denver, CO 80225-0286
CONTENTS

Abstract..................................................................^ 1

Introduction........................................................................................................_ 2

Changes to Water Mass-Balance and Stochastic Time-Series Models.................................................................................. 4

Lake-Level Frequency Analysis ............................................................................................................................................ 4

Unconditional Frequency Analysis ............................................................................................................................ 7

Conditional Frequency Analysis.................................................................................................................................. 11

Climatic Scenarios....................................................................................................................................................... 19

Summary.........................................................^ 22

Selected References......................................................................................................................................................^ 24

ILLUSTRATIONS

1. Map showing location of the study area................................................................................................................... 3

2. Schematic showing lake connections for Waubay Lakes Chain with the farm crossings removed at the

outlet between Rush and Bitter Lakes...................................................................................................................... 5

3. Map showing outlet between Rush and Bitter Lakes ............................................................................................... 6

4. Graph showing rating curve for the outlet between Rush and Bitter Lakes with the farm crossings

removed at the outlet between Rush and Bitter Lakes ............................................................................................. 7

5. Graph showing generated 20-year maximum lake levels for Waubay Lake using an unconditional

frequency analysis with the farm crossings removed at the outlet between Rush and Bitter Lakes ........................ 8

6. Graph showing historic (water years 1963-99) and generated (water years 2000-49) annual 

maximum lake levels for Waubay Lake with the farm crossings removed at the outlet between 

Rush and Bitter Lakes .............................................................................................................................................. 11

7-16. Graphs showing upper and lower probability bounds computed from 10,000 50-year traces from 

the water mass-balance model, with each trace starting October 1, 1999, and with the farm 

crossings removed at the outlet between Rush and Bitter Lakes for:

7. Bitter Lake..................................................................._ 14

8. Blue Dog Lake................................................................................................................................................. 14

9. Enemy Swim Lake .......................................................................................................................................... 15

10. Hfflebrands Lake ............................ 15

11. MinnewastaLake............................................................................................................................................. 16

12. Pickerel Lake................................................................................................................................................... 16

13. RushLake.............................................................................................................................................^ 17

14. Spring Lake ..................................................................................................................................................... 17

15. SwanPond..................................................................^ 18

16. Waubay Lake..................................................................^ 18

17. Graph showing simulated lake levels of Bitter Lake for selected 16-year climatic scenarios, 

starting with October 1, 1999, lake level and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes ................................................................................................................................ 21

18. Graph showing simulated lake levels of Waubay Lake for selected 16-year climatic scenarios, 

starting with October 1,1999, lake level and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes ................................................................................................................................ 21

19. Graph showing probability bounds and selected 16-year scenario results for Waubay Lake, 

starting with October 1, 1999, lake level and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes ................................................................................................................................ 22

Contents III
TABLES

1. Probability of high lake levels using an unconditional frequency analysis for the Waubay Lakes

Chain with the farm crossings removed at the outlet between Rush and Bitter Lakes........................................... 9

2. Probability of low lake levels using an unconditional frequency analysis for the Waubay Lakes

Chain with the farm crossings removed at the outlet between Rush and Bitter Lakes........................................... 10

3. Probability of Waubay Lake exceeding selected lake levels in each of the next 50 years, using 

October 1,1999, starting lake levels and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes .............................................................................................................................. 13

PLATES

[Plates are in pocket]

1. Probabilities of lake extents, based on October 1, 1999, starting lake levels and with the farm crossings 

removed at the outlet between Rush and Bitter Lakes

2. Lake extents in year 2015 for two climatic scenarios and for the 1-percent exceedance probability 

with the farm crossings removed at the outlet between Rush and Bitter Lakes

CONVERSION FACTORS AND VERTICAL DATUM

Multiply By To obtain

cubic foot (ft3 )

cubic foot (ft3 )

cubic foot per second (ft3/s)

foot (ft)

inch (in.)

inch (in.)

28.32

0.02832

0.02832

0.3048

2.54

25.4

cubic decimeter

cubic meter

cubic meter per second

meter

centimeter

millimeter

Sea level: In this report, "sea level" refers to the National Geodetic Vertical Datum of 1929 (NGVD 

of 1929) a geodetic datum derived from a general adjustment of the first-order level nets of both 

the United States and Canada, formerly called Sea Level Datum of 1929.

Water year: In U.S. Geological Survey reports, water year is the 12-month period, October 1 

through September 30. The water year is designated by the calendar year in which it ends; thus, the 

water year ending September 30, 1999, is called the "1999 water year."

IV Contents
Supplement to Water-Resources Investigations 

Report 99-4122, Lake-Level Frequency Analysis for 

the Waubay Lakes Chain, Northeastern South Dakota

By Colin A. Niehus, Aldo V. Vecchia, and Ryan R Thompson

ABSTRACT

A supplement to a U.S. Geological Survey 

report was prepared to present lake-level 

frequency results for the Waubay Lakes Chain, 

northeastern South Dakota, incorporating a modified outlet between Rush and Bitter Lakes. The 

modification involved removal of two farm 

crossings from the drainage way between Rush 

and Bitter Lakes. The farm crossings were 

removed during July-August 1999 as a result of a 

court ruling. Water mass-balance and stochastic 

time-series models were revised to incorporate the 

modified outlet and also to change the starting lake 

levels to October 1,1999.

The outlet elevation between the upstream 

lakes (Hillebrands Lake, Rush Lake, Spring Lake, 

Swan Pond, and Waubay Lake) and Bitter Lake 

was changed from an elevation of 1,803.6 feet to 

1,798.69 feet to reflect the modified outlet 

between Rush and Bitter Lakes. Two 24-inch 

culverts through an east-west township road 

embankment now control outflow from Rush Lake 

to Bitter Lake. At higher lake levels, outflow from 

Rush Lake to Bitter Lake is controlled by a 

42-inch culvert through a railroad embankment.

Both an unconditional and a conditional 

lake-level frequency analysis were done. An 

unconditional frequency analysis estimates the 

frequencies of high or low lake levels for a long 

time period. A conditional frequency analysis 

estimates the frequencies of hypothetical future 

lake levels for a relatively short time period.

Because studies that relate to flood-risk-assessment or flood-mitigation alternatives usually are 

pertinent only for relatively short time periods into 

the future, conditional frequency analysis usually 

is more appropriate than unconditional frequency 

analysis.

The unconditional frequency analysis was 

performed for Bitter, Enemy Swim, Pickerel, and 

Waubay Lakes using 10,000 years of generated 

lake levels with the farm crossings removed at the 

outlet between Rush and Bitter Lakes. Waubay 

Lake was above an elevation of 1,811.0 ft, which 

is approximately equal to the closed-basin spill 

elevation to the Big Sioux River, 1 percent of the 

time (100 out of 10,000 years). However, only 

nine separate spill events with an average duration 

of 11 years occurred during the 10,000-year simulation period, giving an average return period of 

1,111 years. Bitter Lake exceeded an elevation of 

1,814.0 ft, which is high enough to result in large 

flows to the Big Sioux River, an average of once 

every 2,500 years and stayed above this elevation 

an average of 5 years before subsiding. The 

exceedance-level results for Blue Dog Lake, 

Hillebrands Lake, Minnewasta Lake, Rush Lake, 

Spring Lake, and Swan Pond are nearly identical 

to those for Waubay Lake.

The unconditional frequency analysis also 

shows that the recent high levels of the Waubay 

Lakes Chain occur relatively infrequently. For 

example, the peak lake level of Waubay Lake was 

about an elevation of 1,802.1 ft during 1998 and 

1999; however, based on model simulations,

Abstract 1
Waubay Lake only reaches this elevation an 

average of once every 154 years and stays above 

this elevation an average of 15 years before subsiding. Enemy Swim Lake reaches an elevation of 

1,856.1 ft, which was exceeded in 1997, an average of once every 500 years and stays above this 

elevation an average of 1 year before subsiding. 

Pickerel Lake reaches an elevation of 1,846.7 ft, 

which was equalled in 1997, an average of once 

every 500 years and stays above this elevation an 

average of 1 year before subsiding.

The conditional frequency analysis was performed for the 10 major lakes of the Waubay 

Lakes Chain using 10,000 simulated 50-year lakelevel traces, each starting on October 1,1999, with 

the farm crossings removed at the outlet between 

Rush and Bitter Lakes. The generated data can be 

used to estimate the probability of virtually any 

event that involves future lake levels.

The conditional frequency analysis for 

Bitter Lake, Blue Dog Lake, Hillebrands Lake, 

Minnewasta Lake, Rush Lake, Spring Lake, Swan 

Pond, and Waubay Lake indicates that the chance 

the lakes will continue to rise to the spill elevation 

with the Big Sioux River within the next 10 to 

15 years is greater than 1 percent. Conversely, 

there is about a 1-percent chance that Hillebrands 

Lake, Spring Lake, Swan Pond, and Waubay Lake 

will return to levels comparable to historic 

1960-92 levels in the next 25 years and that Bitter 

Lake will return to 1960-92 levels in the next 

50 years. However, there is about a 10-percent 

chance that Blue Dog and Rush Lakes could return 

to historic 1960-92 levels within 10 years.

Because Enemy Swim and Pickerel Lakes 

spill frequently and never join with the downstream lakes, the conditional frequency analysis 

for these lakes shows that both lakes are unlikely 

to increase significantly over current levels, and 

both lakes could return to historic 1960-92 levels 

much sooner than the other lakes under a reversal 

of the current wet conditions.

The climatic scenarios analysis also was 

revised to account for the removal of the farm 

crossings at the outlet between Rush and Bitter 

Lakes and using October 1, 1999, starting conditions. The wet-wet scenario, representing two

1991-98 climatic segments, results in Bitter Lake, 

after 16 years, rising 21.4 ft above the starting lake 

level and Waubay Lake rising 9.8 ft. The dry-dry 

scenario, representing two consecutive 1969-76 

climatic segments, results in Bitter Lake, after 

16 years, declining 3.9 ft and Waubay Lake 

declining 14.7 ft.

The conditional frequency analysis performed for Waubay Lake with October 1,1999, 

starting lake levels was compared to the scenario 

results. The five selected scenarios seemed to represent essentially the full range of hypothetical 

future lake levels. The lake levels after 16 years 

range from about the lower 1-percent probability 

bound for the dry-dry scenario to about the upper 

0.2-percent probability bound for the wet-wet 

scenario.

INTRODUCTION

A lake-level frequency analysis for the Waubay 

Lakes Chain in northeastern South Dakota was recently 

conducted by the U.S. Geological Survey (USGS) in 

cooperation with the Federal Emergency Management 

Agency (FEMA). Results are presented in USGS 

Water-Resources Investigations Report 99-4122 

(WRIR 99-4122), entitled "Lake-Level Frequency 

Analysis for the Waubay Lakes Chain, Northeastern 

South Dakota" (Niehus and others, 1999).

The purpose of the study was to obtain information on high lake levels, possible causes of the high lake 

levels, and probabilities of hypothetical future lake 

levels of 10 lakes in the Waubay Lakes Chain (fig. 1). 

Those lakes are Bitter Lake, Blue Dog Lake, Enemy 

Swim Lake, Hillebrands Lake, Minnewasta Lake, 

Pickerel Lake, Rush Lake, Spring Lake, Swan Pond, 

and Waubay Lake.

The outlet between Rush and Bitter Lakes was 

modified during July-August 1999 as a result of a 

July 6,1999, ruling by the Fifth Judicial Circuit Court 

that specified removal of two farm crossings in the 

area. Consequently, lake-level frequency results 

needed to be revised. The purpose of this report is to 

present the revised results. A description of the study 

area, and of a water mass-balance model and a 

stochastic time-series model that were developed for 

the lake-level frequency analysis, are presented in 

WRIR 99-4122.

2 Supplement to Water-Resources Investigations Report 99-4122
97°30' 97°22'30" 97°15' 97°07'30"

Study 

SOUTH ^ Area

DAKOTA

45°30'

45°22'30'

45°15'

45°07'30"

X

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Waubay National 

Wildlife Refuge 

\Headquarters

EXPLANATION

|" ~"| WAUBAY LAKES CHAIN 

* ' BASIN

A U.S. GEOLOGICAL SURVEY 

06479215 STREAMFLOW GAGING 

STATION-Number is 

station identification

 * *~ POSSIBLE DIRECTIONS 

OF LAKE OVERFLOW

*.

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nan \ >nd L'ampbell

r^ I I' \ r-f 1 V^tOHUh Danling Slouehg ' K

I Minneu'f.yf Lake

> --' Snum \munu\ 

Lake

Blue Dog 

lake

Webster

 .. Antelope 

$5) f Lake

yr^

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GRANT CO

Y

Solomon 

Slough

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0

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CODINGTON CO

Florence

LrtAr

06479215

Cmtmiwood "str. 

LaAt

.S»// 

LoAf

Base from U.S. Geological Survey digital data, 1:100,000, Webster, 1985; Sisseton, 1979

Lakes from U.S. Geological Survey digital bathymetric data, 1999, and South Dakota

Department of Game, Fish, and Parks, 1963-96

Study area from 10-meter digital elevation model data, 1998

Universal Transverse Mercator porjection zone 14

North American Horizontal Datum 1927

Figure 1 . Location of the study area.

012345 MILES I i I i i i

01234 5 KILOMETERS

Introduction
CHANGES TO WATER MASS-BALANCE 

AND STOCHASTIC TIME-SERIES 

MODELS

Changes to the water mass-balance and 

stochastic time-series models involved incorporating 

the modified outlet between Rush and Bitter Lakes and 

also using October 1,1999, starting lake levels instead 

of October 1, 1998, starting lake levels. All other 

information concerning the models is described in 

WRIR 99-4122.

The elevations of the outlets between lakes in the 

Waubay Lakes Chain are shown in figure 2. The outlet 

elevation between the upstream lakes (Hillebrands 

Lake, Rush Lake, Spring Lake, Swan Pond, and 

Waubay Lake) and Bitter Lake was changed from an 

elevation of 1,803.6 ft to an elevation of 1,798.69 ft to 

reflect the modified outlet between Rush and Bitter 

Lakes. Prior to July 1999, the outflow from Rush Lake 

through Little Rush Lake to Bitter Lake was by overtopping of two farm crossings (shown in figure 3). The 

Fifth Judicial Circuit Court ruled on July 6, 1999, that 

the two farm crossings be removed by August 13, 

1999. Outflow from Rush Lake to Bitter Lake is now 

controlled by two 24-in. culverts through an east-west 

township road embankment and at higher lake levels by 

a 42-in. culvert through the Burlington Northern 

Railroad embankment as shown in figure 3. As a result 

of the subsequent removal of the two farm crossings 

during July-August 1999, the new spill elevation from 

Rush Lake to Bitter Lake is 1,798.69 ft. The water 

mass-balance model was revised to incorporate this 

change.

Empirical flow-rating equations could not be 

developed for Rush Lake to Bitter Lake and for Bitter 

Lake to the Big Sioux River because historic data are 

not available for spills. Therefore, the spill calculations 

were based on flow ratings supplied by the U.S. Army 

Corps of Engineers (USAGE) (Doug Clemetson, 

written commun., 1999). A continuous curve (relating 

water-surface elevation to flow) was used to approximate the outlet rating curves for flow from Rush Lake 

to Bitter Lake. If Rush Lake is above an elevation of 

1,811.5 ft, the railroad embankment (fig. 3) is overtopped and flow is assumed to be unimpeded; if Rush 

Lake is below 1,798.69 ft (the spill elevation with 

Bitter Lake), no flow is allowed. Although an assumption is made that Bitter Lake will never spill into Rush 

Lake, if that situation does occur the water is allowed 

to flow from Bitter Lake to Rush Lake until the levels

of the two lakes are equal or until the level of Bitter 

Lake is below the spill elevation.

The revised USAGE flow ratings between Rush 

Lake and Bitter Lake are shown in figure 4. Three 

different rating curves are shown, depending on what 

controls the flow between Rush and Bitter Lakes (the 

east-west township road, the north-south county road, 

or the Burlington Northern Railroad). The rating used 

in the water mass-balance model is shown in red in 

figure 4. The assumption was made that the east-west 

township road is the control until the lake level reaches 

an elevation of 1,807.5 ft, when the railroad embankment becomes the control. An assumption was made 

that the north-south county road never is the control. It 

was assumed that the east-west township road will be 

maintained so that it is not breached, although some 

piping is occurring at the present time (September 

1999) around the two 24-in. culverts through the road 

embankment.

The water mass-balance and stochastic timeseries models are available at the USGS Subdistrict 

Office in Huron, S. Dak. These models simulate semiannual lake volumes of the 10 major lakes of the 

Waubay Lakes Chain using precipitation, evaporation, 

total inflow to the system, and starting lake levels as 

input. Lake levels that existed on September 13,1999, 

were used to represent October 1,1999, starting 

conditions.

LAKE-LEVEL FREQUENCY ANALYSIS

The frequencies of extreme high or low lake 

levels for the Waubay Lakes Chain were estimated 

using the stochastic time-series model as described in 

WRIR 99-4122 to generate inputs for the water massbalance model. Both an unconditional and a conditional lake-level frequency analysis were done. An 

unconditional frequency analysis estimates the frequencies of high or low lake levels for a long time 

period during which the lake levels cycle many times 

between wet and dry periods. The starting conditions 

at the beginning of the simulation period have no effect 

on the analysis because of the long simulation period. 

An unconditional frequency analysis is used to evaluate 

the frequency of an individual event, such as the historic lake-level rise for water years 1993-99, in the context of a much longer record. A conditional frequency 

analysis estimates the frequencies of hypothetical 

future lake levels for a relatively short time period, such 

as 50 years. Because of the short time period, the

4 Supplement to Water-Resources Investigations Report 99-4122
Elevation 

1,853.6 feet

Minnewasta 

Lake

Elevation 

1,844.9 feet

Bitter Lake

Elevations 

1,811.1 feet or 

1,812.1 feet

Approximate 

elevation 

1,808 feet

Big Sioux 

River

Hillebrands Lake, Rush Lake, Spring Lake,

Swan Pond, and Waubay Lake 

(joined above elevation 1,797.6 feet)

NOTE: Elevations are measured in feet above sea level. Above figure has been modified from

SDDENR's schematic located on the Internet at www.state.sd.us/denr/DES/WaterRights/wr_flood_schematic.htm

Figure 2. Lake connections for the Waubay Lakes Chain with the farm crossings removed at the outlet between Rush 

and Bitter Lakes.

Lake-Level Frequency Analysis 5
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6 Supplement to Water-Resources Investigations Report 99-4122
1,814

20 40 60 80 100 120 140 

DISCHARGE, IN CUBIC FEET PER SECOND

160 180 200

EXPLANATION

OUTLET RATING CURVE USED IN REPORT--East-west township road (figure 3) is control

 OUTLET RATING CURVE IF NORTH-SOUTH COUNTY ROAD (FIGURE 3) IS CONTROL

_______ OUTLET RATING CURVE IF BURLINGTON NORTHERN RAILROAD (FIGURE 3) IS CONTROL

 DISCHARGE MEASUREMENTS BY U.S. GEOLOGICAL SURVEY

Figure 4. Rating curve for the outlet between Rush and Bitter Lakes with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

starting conditions at the beginning of the simulation 

period have an effect on the frequencies of the lake 

levels. Because studies that relate to flood-riskassessment or flood-mitigation alternatives usually are 

pertinent only for relatively short time periods into the 

future, a conditional frequency analysis usually is more 

appropriate than an unconditional frequency analysis.

Unconditional Frequency Analysis

An unconditional frequency analysis was performed for Bitter, Enemy Swim, Pickerel, and Waubay 

Lakes using 10,000 years of generated data with the 

two farm crossings removed at the outlet between Rush 

and Bitter Lakes. The generated data for Waubay Lake 

are shown in figure 5. To smooth some of the highfrequency variation in the data, the maximum lake 

levels for successive 20-year intervals are shown in the 

figure. Also shown is the (unconditional) I-percent 

exceedance level, which is defined as the lake level

above which 1 percent of the annual maximum lake 

levels lie. For Waubay Lake, the 1 -percent exceedance 

level is at an elevation of 1,811.0 ft, which is approximately equal to the closed-basin spill elevation to the 

Big Sioux River. However, only nine separate spill 

events occurred during the 10,000-year simulation 

period, giving an average return period of 1,111 years. 

If the lake levels were independent from year to year 

(as usually is the case when considering annual stream 

discharges), the recurrence interval for the 1-percent 

exceedance level would be 1 divided by the exceedance 

probability, or 100 years. However, because of the 

high serial persistence in the lake levels, the return 

period corresponding to the 1-percent exceedance level 

is much longer than 100 years. The average duration of 

each event is the number of years the lake level is above 

an elevation of 1,811.0 ft (100) divided by the number 

of occurrences (9). Thus, the lake level is above an 

elevation of 1,811.0 ft an average of once every 

1,111 years but stays above that level an average of 

11 years per event.

Lake-Level Frequency Analysis
1,820p

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000

NUMBER OF YEARS FROM PRESENT

Figure 5. Generated 20-year maximum lake levels for Waubay Lake using an unconditional frequency analysis with the 

farm crossings removed at the outlet between Rush and Bitter Lakes.

The exceedance levels, average return periods, 

and average durations for selected exceedance probabilities for Bitter, Enemy Swim, Pickerel, and Waubay 

Lakes are shown in table 1. Bitter Lake exceeded an 

elevation of 1,814.0 ft, which is high enough to result 

in large flows to the Big Sioux River, an average of 

once every 2,500 years and stayed above this elevation 

an average of 5 years before subsiding. The exceedance levels for Hillebrands Lake, Rush Lake, Spring 

Lake, and Swan Pond are nearly identical to those for 

Waubay Lake because they are joined with Waubay 

Lake above an elevation of 1,797.6 ft (fig. 2). The 

exceedance-level results for Waubay Lake also are 

assumed to be identical to Blue Dog and Minnewasta 

Lakes because their outlet elevation with Waubay Lake 

is substantially lower than most of the exceedancelevel elevations shown in table 1.

On September 13, 1999, Bitter Lake was at an 

elevation of 1,792.05 ft, Enemy Swim Lake was at an 

elevation of 1,853.58 ft, Pickerel Lake was at an elevation of 1,845.54 ft, and Waubay Lake was at an elevation of 1,803.75 ft. With the exception of Bitter Lake,

these values are all near the 10-percent exceedance 

probability (table 1). However, the average return 

periods are much higher than 10 years (204 years for 

Bitter Lake, 50 years for Enemy Swim Lake, 30 years 

for Pickerel Lake, and 154 years for Waubay Lake), 

and the average durations are longer than 1 year 

(20 years for Bitter Lake, 5 years for Enemy Swim 

Lake, 3 years for Pickerel Lake, and 15 years for 

Waubay Lake). Generally, the smaller upstream lakes, 

such as Pickerel and Enemy Swim Lakes, are more sensitive to short-term climatic fluctuations than the larger 

downstream lakes, such as Bitter and Waubay Lakes. 

Therefore, Enemy Swim and Pickerel Lakes generally 

reach levels similar to those in 1999 more often than 

Bitter and Waubay Lakes and stay at those levels for 

shorter time periods.

High lake-level frequency results for Waubay 

Lake with and without the farm crossings at the outlet 

between Rush Lake and Bitter Lake (table 1 of this 

report and table 14 of WRIR 99-4122) were compared. 

The exceedance levels corresponding to the 1-, 0.5-, 

and 0.2-percent exceedance probabilities are very

8 Supplement to Water-Resources Investigations Report 99-4122
Table 1 . Probability of high lake levels using an unconditional frequency analysis for the Waubay Lakes Chain with the farm 

crossings removed at the outlet between Rush and Bitter Lakes

Lake

Bitter

Enemy Swim

Pickerel

Waubay

Exceedance 

probability1

10 (1 in 10)

4 (1 in 25)

2 (1 in 50)

1 (1 in 100)

0.5(1 in 200)

0.2 (1 in 500)

10 (1 in 10)

4(1 in 25)

2 (1 in 50)

1 (1 in 100)

0.5 (1 in 200)

0.2(1 in 500)

10 (1 in 10)

4(1 in 25)

2(1 in 50)

1 (1 in 100)

0.5 (1 in 200)

0.2 (1 in 500)

10 (1 in 10)

4(1 in 25)

2 (1 in 50)

1 (1 in 100)

0.5(1 in 200)

0.2 (1 in 500)

Exceedance level 

(feet above 

sea level)

1,799,7

1,805.0

1,807.7

1,810.0

1,812.3

1,814.0

1,854.0

1,854.5

1,854.9

1,855.3

1,855.7

1,856.1

1,845.6

1,845.9

1,846.1

1,846.4

1,846.6

1,846.7

1,802.1

1,806.3

1,808.9

1,811.0

1,812.3

1,814.1

Average return 

period2 

(years)

204

385

769

1,111

1,667

2,500

50

71

116

185

312

500

30

55

83

155

244

500

154

345

625

1,111

1,667

2,500

Average 

duration3 

(years)

20

15

15

11

8

5

5

3

2

2

2

1

3

2

2

2

1

1

15

14

13

11

8

5

Percent of time the lake is above a given level. 

"Average time between events when the lake is above a given level. 

3Average time per event that the lake is above a given level.

similar (within 0.5 ft). Because the lake levels corresponding to these exceedance probabilities are substantially above elevation 1,807.5 ft, above which the 

Burlington Northern Railroad is the control (fig. 4), it 

was expected that the results wouldn't change greatly. 

However, the exceedance levels for the 10- and 

4-percent exceedance probabilities decreased about 

2 ft without the farm crossings because more water is 

removed from Waubay Lake.

High lake-level frequency results for Bitter Lake 

with and without the farm crossings at the outlet 

between Rush Lake and Bitter Lake also were compared. The exceedance levels are very similar, with the 

exception of the 10-percent exceedance probability, 

which increased about 3 ft.

Even though the extreme high exceedance levels 

for both Bitter and Waubay Lakes have changed only 

slightly as the result of removing the two farm crossings, the average return periods and durations have 

increased significantly. For example, the average 

return period for the 1-percent exceedance probability 

for Waubay Lake increased from 769 years to 1,111 

years, and the average duration increased from 8 years 

to 11 years. With the farm crossings removed, Waubay 

Lake reaches elevation 1,811.0 ft less often and stays 

there for longer periods than with the farm crossings in 

place.

The unconditional frequency analysis shows that 

the recent high levels of the Waubay Lakes Chain occur 

relatively infrequently. For example, the peak lake 

level of Waubay Lake was about an elevation of

Lake-Level Frequency Analysis
1,802.1 ft in 1998 and 1999; however, based on model 

simulations, Waubay Lake only reaches this elevation 

an average of once every 154 years and stays above this 

elevation an average of 15 years before subsiding. 

Enemy Swim Lake reaches an elevation of 1,856.1 ft, 

which was exceeded in 1997, an average of once every 

500 years and stays above this elevation an average of 

1 year before subsiding. Pickerel Lake reaches an elevation of 1,846.7 ft, which was equalled in 1997, an 

average of once every 500 years and stays above this 

elevation an average of 1 year before subsiding. The 

lake levels for the 1-percent exceedance probabilities in 

table 1 are about 18 ft higher for Bitter Lake, 1.7 ft 

higher for Enemy Swim Lake, 0.9 ft higher for Pickerel 

Lake, and 7.3 ft higher for Waubay Lake than the levels 

of those lakes that existed on September 13,1999. 

However, the average return periods are 1,111 years for 

Bitter Lake, 185 years for Enemy Swim Lake, 

155 years for Pickerel Lake, and 1,111 years for 

Waubay Lake. For example, if in the next several years 

Bitter Lake continues to rise to an elevation of 

1,810.0 ft, that would be an event that happens only 

once in an average of 1,111 years. If Bitter Lake does 

rise to an elevation of 1,810.0 ft, it could stay above 

that level for 11 years or longer before beginning to 

subside.

The average return periods for Enemy Swim and 

Pickerel Lakes for the 0.2-percent exceedance probabilities in table 1 are 500 years, indicating the extreme 

high lake levels of the smaller upstream lakes probably 

are caused by 1-year climatic extremes. However, the 

average return periods for Bitter and Waubay Lakes for 

the 0.2-percent exceedance probabilities are 

2,500 years, indicating the extreme high lake levels of 

the larger downstream lakes are caused by extreme 

climatic conditions lasting several years.

Results from the unconditional frequency 

analysis for low lake levels for Bitter, Enemy Swim, 

Pickerel, and Waubay Lakes are given in table 2. The 

annual maximum lake levels generated using the 

stochastic time-series model were reduced by 2 ft as 

described in WRIR 99-4122 because, in dry years, lake 

levels usually are about 2 ft lower at the end of the 

summer than peak lake levels following spring runoff.

Low lake-level frequency results for Waubay and 

Bitter Lakes with and without the farm crossings at the 

outlet between Rush Lake and Bitter Lake were compared. The exceedance levels for Waubay and Bitter 

Lakes corresponding to all the exceedance probabilities 

are slightly higher (0.1 to 1.1 ft) without the farm crossings. Because the higher starting lake levels have no 

effect on the results from an unconditional frequency

Table 2. Probability of low lake levels using an unconditional frequency analysis for the Waubay Lakes Chain with the farm 

crossings removed at the outlet between Rush and Bitter Lakes

Lake

Bitter

Enemy Swim

Pickerel

Waubay

Non-exceedance 

probability1

10 (1 in 10)

4 (1 in 25)

2 (1 in 50)

1 (1 in 100)

10 (1 in 10)

4 (1 in 25)

2(1 in 50)

1 (1 in 100)

10 (1 in 10)

4 (1 in 25)

2(1 in 50)

1 (1 in 100)

10 (1 in 10)

4 (1 in 25)

2 (1 in 50)

1 (1 in 100)

Exceedance level 

(feet above 

sea level)

1,778.1

1,776.5

1,775.4

1,774.3

1,846.9

1,845.7

1,844.8

1,844.1

1,842.9

1,842.5

1,842.2

1,841.9

1,781.4

1,779.3

1,778.1

1,777.2

Average return period2 

(years)

161

294

435

833

70

182

294

476

27

40

67

125

175

256

385

625

Average 

duration3 

(years)

16

12

9

8

7

7

6

5

3

2

1

1

18

10

8

6

Percent of time the lake is below a given level. 

"Average time between events when the lake is below a given level. 

3Average time per event that the lake is below a given level.

10 Supplement to Water-Resources Investigations Report 99-4122
analysis, the increase is attributed entirely to the 

removal of the farm crossings at the Rush-Bitter outlet. 

Bitter and Waubay Lakes have less variability when the 

lakes are above an elevation of 1,798.69 ft because 

water can flow both ways between them. Therefore, 

both lakes likely will spend less time below elevation 

1,798.69 ft with the farm crossings removed, which 

results in the lower exceedance levels being increased 

for the corresponding exceedance probability.

The frequencies for extreme low lake levels are 

difficult to accurately determine because small changes 

in lake volumes result in large changes in lake levels. 

For example, as described in WRIR 99-4122, the 

volume of Bitter Lake at an elevation of 1,775 ft differs 

only slightly from the volume at an elevation of 

1,770 ft (essentially dry). Likewise, the volume of 

Waubay Lake at an elevation of 1,780 ft differs only 

slightly from the volume at an elevation of 1,775 ft 

(essentially dry). Therefore, the very small exceedance 

probabilities (0.5 and 0.2 percent) are not given in 

table 2. However, as indicated in table 2 and in WRIR 

99-4122, the lake levels of Enemy Swim, Pickerel, and 

Waubay Lakes in the 1930's and 1940's were very low 

compared to the long-term lake levels. For example, 

for 12 years in the 1930's and early 1940's, Enemy

Swim Lake was below an elevation of 1,845 ft, which 

corresponds to a return period of about 300 years. 

Similarly, for 4 years in the 1930's, Pickerel Lake was 

below an elevation of 1,842 ft, which corresponds to a 

return period of about 125 years. For several years in 

the 1930 's, Waubay Lake was below an elevation of 

1,780 ft, which corresponds to a return period of about 

256 years.

Conditional Frequency Analysis

A conditional frequency analysis was performed 

for the 10 major lakes of the Waubay Lakes Chain 

using 10,000 simulated 50-year lake-level traces, each 

starting on October 1, 1999, with the two farm crossings removed at the outlet between Rush Lake and 

Bitter Lake. The generated data can be used to estimate 

the probability of virtually any event that involves 

future lake levels.

The historic lake-level data for water years 

1963-99 for Waubay Lake are shown in figure 6 along 

with five traces of generated hypothetical future lake 

levels. Also shown are the upper and lower 1-percent 

probability bounds for future (water years 2000-49)

LLJ

LU

w

LJJ

O

LJJ

1,820

1,815

1,810

1,805

1,800

1,795

1,790

1,785

1,780

 - - Historic data

 Generated trace 1 data

 Generated trace 2 data

 Generated trace 3 data

 Generated trace 4 data

 - Generated trace 5 data

upper 1-percent probability bound

lower 1-percent probability bound

"l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II II I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I LLLLL

1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

Figure 6. Historic (water years 1963-99) and generated (water years 2000-49) annual maximum lake levels for Waubay 

Lake with the farm crossings removed at the outlet between Rush and Bitter Lakes.

Lake-Level Frequency Analysis 11
lake levels. The frequency results for Waubay Lake 

(fig. 6) indicate the upper I-percent probability bound 

rises quickly to a constant level at an elevation of 

1,811 ft at the end of water year 2011, which corresponds to the unconditional 1-percent exceedance level 

discussed previously (table 1). Therefore, the chance 

that Waubay Lake will continue to rise to the spill elevation with the Big Sioux River (elevation 1,811.1 ft) 

within the next 10 to 15 years is greater than 1 percent. 

The lower 1 -percent probability bound falls slowly 

during the next 50 years. It reaches an elevation of 

1,785 ft in 2024, indicating that there is about a 

1-percent chance Waubay Lake will return to levels 

comparable to historic 1960-92 levels (about elevation 

1,785.0 ft) in the next 25 years. Also, Waubay Lake is 

unlikely to return to levels comparable to those during 

the extreme drought of the 1930's (about elevation 

1,775 ft) for much longer than 50 years. Table 3 shows 

the probability of Waubay Lake exceeding lake levels 

between elevations 1,785 to 1,815 ft during the next 

50 years.

A comparison was done between the frequency 

results for Waubay Lake with (modified outlet) and 

without (unmodified outlet) the farm crossings at the 

outlet between Rush and Bitter Lakes. The upper 0.2- 

and 1 -percent probability bounds with the modified 

outlet rise slightly slower than the upper 0.2- and 

1 -percent probability bounds with the unmodified 

outlet. However, by the end of water year 2010, the 

bounds are nearly identical because, at high lake levels, 

Waubay Lake will be nearly the same level as Bitter 

Lake, and the farm crossings will have little impact on 

the lake levels in the Waubay Lakes Chain. The 0.2- 

and 1-percent probability bounds with the modified 

outlet do not rise as quickly as the unmodified outlet 

bounds because more water is being moved from 

Waubay Lake to Bitter Lake. The lower 0.2- and 

1-percent probability bounds with the modified outlet 

are very close to the lower 0.2- and 1-percent probability bounds with the farm crossings in place at the 

outlet. The upper and lower 10-percent probability 

bounds are affected the most by the modified outlet. 

The upper 10-percent probability bound is about 2 ft 

below the upper 10-percent probability bound with the 

unmodified outlet, and the lower 10-percent probability bound is about 1 ft below the upper 10-percent 

probability bound with the unmodified outlet. The 

removal of the farm crossings results in less water in 

Waubay Lake and causes the lower probability bounds.

The 0.2-, 1-, and 10-percent upper and lower 

probability bounds for the 10 major lakes of the 

Waubay Lakes Chain are shown in figures 7 through 

16. The 1-percent probability bounds shown in 

figure 6 for Waubay Lake are the same as those shown 

in figure 16.

The frequency results for Bitter Lake (fig. 7) 

indicate the upper 10-percent probability bound 

increases about 10 ft during water years 1999-2010 and 

then slowly decreases, and the upper 1-percent probability bound increases about 19 ft during water years 

1999-2015 and then becomes nearly constant at about 

elevation 1,810 ft. The chance that the level of Bitter 

Lake will continue to rise to the spill elevation with the 

Big Sioux River within the next 15 to 20 years is 

greater than 1 percent. Conversely, the lower 

10-percent probability bound decreases about 9 ft 

during water years 1999-2025 and then slowly 

decreases to an elevation of about 1,780.5 ft by the end 

of water year 2049. Furthermore, the lower I-percent 

probability bound decreases about 11 ft during water 

years 1999-2025 and then slowly decreases to an elevation of about 1,777.5 ft by the end of water year 2049. 

Therefore, there is about a 1-percent chance that Bitter 

Lake will return to levels comparable to historic 

1960-92 levels (about elevation 1,775 ft) in the next 

50 years.

A comparison was done between the frequency 

results for Bitter Lake with and without the farm crossings at the outlet between Rush and Bitter Lakes. The 

upper 0.2-percent probability bounds are nearly identical for both results. The upper 1-percent probability 

bound rises slightly quicker than the upper 1-percent 

probability bound with the unmodified outlet, but by 

the end of water year 2010, the bounds are nearly identical. This is because, at high lake levels, Bitter Lake 

will be at nearly the same lake level as the other lakes 

in the Waubay Lakes Chain (with the exception of 

Pickerel and Enemy Swim Lakes) and the farm crossings will have little effect on the lake levels. Until 

about the end of water year 2010, the lower 0.2- and 

1 -percent probability bounds initially are about 4 to 5 ft 

above the lower 0.2- and I-percent probability bounds 

with the unmodified outlet. From water years 2011 -48, 

this difference slowly decreases until, by the end of 

water year 2048, the difference in the probability 

bounds is less than 1 ft. This is caused by the higher 

initial lake level in Bitter Lake plus the inflow from 

Rush Lake to Bitter Lake, but as the years pass and 

Bitter Lake separates from the other lakes of the

12 Supplement to Water-Resources Investigations Report 99-4122
Table 3. Probability of Waubay Lake exceeding selected lake levels in each of the next 50 years, using October 1,1999, 

starting lake levels and with the farm crossings removed at the outlet between Rush and Bitter Lakes

Water yea

2000

j 2001

2002

: 2003

2004

2005

2006

I ' 2007

2008

2009

2010

in 2011

2012

2013

2014

2015

2016

\\\ 2017

2018

2019

2020

''-. 2021

2022

2023 

2024

2025

2026

2027

2028

2029

2030

t 2031

2032

f=; 2033

2034

I 2035

2036

\ 2037

2038

}^ 2039

2040

f . 2041

2042

: 2043

2044

,' 2045

2046

\ 2047

2048

; 2049

1,785

100

100

100

100

100

100

100

100

100

100

100

, ? «i, ;.;10Q, : _

100

100

100

100

100

J|if ftf ifjjgfrtil

100

100

100

100'

100

99

98

98

97

97

' 9*6"

95

95

94

---;;93,,

93

92

92

91

91

AC. 7*J * 1 » nW-iB

90

89

88

88

88

-"''' ' ; 8T"'

87

87

86

86

Exceedance probability (in percent) for lake level (in feet above sea level)

1,790

100

too

100

100:

100

100

100

100

100

100

100

100 ,, 3i? .

100

100

99

98

97

i!iiua$Ui.:a.iij;.; * "92 *

91

88

86

85

-. -.= - -83 ...-.,,. 

81

79

78

..<--,- 76

75

' "' 74

73

72

71

,. -^.-.JQ,,,..,.^,

69

69

68

67

66

fir r 66 = " - -

65

65

64

63

63

! - ?z?i!i fe3 i; iif ' T : -

62

62

62

62

1,795

100

100

100

100

100

100

100

99 - - -

97

94

88

.,,82.^..,

76?^ - -

67

63

60

-:i-A8. 56 U :unij ~^~

54

53

51

50

48

47

45

45 -, t ....

45

44:""

44

43

42

,.;f2,-... TT;: ,

41

40

40

40

40

:4° rttiii:

39

39

38

38

38

at?5^ri«v^7

37

131

31

37

1,800

100

100:

99

;85

62

46

38

34:

31

29

28

.27

26

" :24'' "

24

23

22

22

22

21:

21

;20

20

20

20

19

19

18

18

18

18

18

17

17

17

16

16

16

16

16

16

15

15

15

15

15

15

15

15

'&.:....

1,805

0

2;5

2.5

32

4.6

54

6.0

64

6.6

6.8

7.0

7.1

6.9

7.0

7.1

7-0

6.8

6,8

6.7

6.6

6.5

62

6.1

6.1

5.8

5J

5.6

'54

5.4

;5.2: : ' : '

5.0

'5.0!

4.8

'4,9

5.0

4.9

4.8

4.7^

4.7

4.6

4.7

4J

4.8

4,5

4.4

44

4.2

4J

4.1

4,1

1,810

0

0

0

;.02

.1

.3

.6

.8

1.1

12

1.5

1,6

1.7

L7

1.7

1.7

1.7

1:7

1.7

1;.7

1.6

14

1.6

1.6 

1.5

1.6

1.5

1.5

1.5

 :-., j^

1.5

1.5

1.4

14

1.3

1.3

1.4

14

1.3

14

14

13

1.2

1.2

1.1

1,1

1.2

U

1.1

1-2

1,815

0

0

0

0

0

^0,

.01

;.03

.04

,04

.03

,02;

.01

m

.02

.04

.03

.05

.03

.03

.03

.02

.04

.02

;04

.05

.01

.03

.04

.04

,03

.03

.05

.04

,02

0.04

:03

.09

.05 ,

.06

;07

.07

.07

.03

.01

.03

.04

.01

.02

Lake-Level Frequency Analysis 13
1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

upper 0.2 percent

upper 1 percent

upper 10 percent

lower 10 percent 

lower 1 percent

lower 0.2 percent

LLJ

1999 2004 2009 2014 2019 2024 2029

WATER YEAR

2034 2039 2044 2049

Figure 7. Upper and lower probability bounds for Bitter Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1,1999, and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes.

LLJ 

LLJ

LLJ 

CO 

111

O

CO

LLJ

LLJ

_l

LLJ

1,820

1,815

1,810

1,805

1,800

1,795

1,790

upper 0.2 percent ~

upper 1 percent

lower 1 percent ~

upper 10 percent 

lower 10 percent

lower 0.2 percent

1999 2004 2009 2014 2019 2024 2029 

WATER YEAR

2034 2039 2044 2049

Figure 8. Upper and lower probability bounds for Blue Dog Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1 , 1999, and with the farm crossings removed at the outlet 

between Rush and Bitter Lakes.

14 Supplement to Water-Resources Investigations Report 99-4122
1,858

LLJ

1999 2004 2009 2014 2019 2024 2029 2034 2039 2044 2049

WATER YEAR

Figure 9. Upper and lower probability bounds for Enemy Swim Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

1,820

lower 10 percent

 - 

lower 1 percent

 _

lower 0.2 percent

1,780

1999 2004 2009 2014 2019 2024 2029

WATER YEAR

2034 2039 2044 2049

Figure 10. Upper and lower probability bounds for Hillebrands Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1,1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

Lake-Level Frequency Analysis 15
1,820

1999 2004 2009 2014 2019 2024 2029 

WATER YEAR

2034 2039 2044 2049

Figure 11. Upper and lower probability bounds for Minnewasta Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

LJJ 

UJ

O 

CO

UJ 

UJ 

LJJ

1,848

1,847

1,846

1,845

1,844

1,843

upper 0.2 percent

upper 1 percent

upper 10 percent

lower 10 percent

lower 1 percent

lower 0.2 percent

"I 842 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 

1999 2004 2009 2014 2019 2024 2029 2034 2039 2044 2049

WATER YEAR

Figure 12. Upper and lower probability bounds for Pickerel Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

16 Supplement to Water-Resources Investigations Report 99-4122
1,820

i 790 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i

1999 2004 2009 2014 2019 2024 2029 2034 2039 2044 2049

WATER YEAR

Figure 13. Upper and lower probability bounds for Rush Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1,1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

1,820

lower 10 percent 

lower 1 percent

1,775

1999 2004 2009 2014 2019 2024 2029 

WATER YEAR

2034 2039 2044 2049

Figure 14. Upper and lower probability bounds for Spring Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

Lake-Level Frequency Analysis 17
1999 2004 2009 2014 2019 2024 2029

WATER YEAR

2034 2039 2044 2049

Figure 15. Upper and lower probability bounds for Swan Pond computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

1,820

1999 2004 2009 2014 2019 2024 2029

WATER YEAR

2034 2039 2044 2049

Figure 16. Upper and lower probability bounds for Waubay Lake computed from 10,000 50-year traces from the water 

mass-balance model, with each trace starting October 1, 1999, and with the farm crossings removed at the outlet between 

Rush and Bitter Lakes.

18 Supplement to Water-Resources Investigations Report 99-4122
Waubay Lakes Chain, the effect of the farm crossings 

is diminished. The upper 10-percent probability bound 

is affected the most by the modified outlet. The 

removal of the farm crossings causes the upper 

10-percent bound to rise much quicker and stay about 

3 ft above the upper 10-percent bound with the unmodified outlet for water years 1999-2048. The removal of 

the farm crossings causes more inflow to Bitter Lake 

from the other lakes of the Waubay Lakes Chain. The 

lower 10-percent bounds are affected much less and 

actually converge within 1 ft at the end of water year 

2048.

The frequency results for Hillebrands Lake 

(fig. 10), Rush Lake (fig. 13), Spring Lake (fig. 14), 

Swan Pond (fig. 15), and Waubay Lake (fig. 16) indicate the upper 10-, 1-, and 0.2-percent probability 

bounds are about the same for all five lakes because the 

lakes are joined with Waubay Lake above an elevation 

of 1,797.6 ft. The frequency results for Blue Dog 

(fig. 8) and Minnewasta (fig. 11) Lakes also are similar 

to Waubay Lake because they are joined with Waubay 

Lake above an elevation of 1,800.2 ft. The upper 

10-percent probability bounds are constant until the 

end of water year 2010 then slowly decrease, and the 

upper 1-percent probability bounds increase about 8 ft 

from the current level and reach an elevation of 

1,811.5 ft in water year 2015. These increases are 

smaller than the increases for Bitter Lake, partly 

because of water spilling from Rush Lake to Bitter 

Lake as the level of Rush Lake continues to increase. 

The upper 0.2-percent probability bounds for Bitter 

and Rush Lakes become about equal after year 2010 

because enough water spills from the upstream lakes to 

fill Bitter Lake to the extreme high levels of Rush Lake. 

As with Waubay Lake, the conditional frequency 

analysis for Blue Dog Lake, Hillebrands Lake, 

Minnewasta Lake, Rush Lake, Spring Lake, and Swan 

Pond indicates that the chance the lakes will continue 

to rise to the spill elevation with the Big Sioux River 

within the next 10 to 15 years is greater than 1 percent.

The lower probability bounds for Hillebrands 

Lake, Minnewasta Lake, Spring Lake, Swan Pond, and 

Waubay Lake are similar, with each bound decreasing 

slowly during the entire 50-year simulation period. 

There is about a 1-percent chance that Hillebrands 

Lake, Spring Lake, Swan Pond, and Waubay Lake will 

decrease to historic 1960-92 levels in the next 25 years. 

However, the lower probability bounds for Blue Dog 

and Rush Lakes become nearly constant by year 2010 

and beyond and, indicate that there is about a 

10-percent chance Blue Dog and Rush Lakes could

return to historic 1960-92 levels within 10 years. Blue 

Dog and Rush Lakes are much shallower than the other 

lakes, and both lakes spill regularly under normal 

climatic conditions. Therefore, levels of both lakes 

respond quickly to short-term climatic reversals.

The frequency results for Enemy Swim Lake 

(fig. 9) and Pickerel Lake (fig. 12) are totally 

unaffected by the removal of the farm crossings at the 

outlet between Rush and Bitter Lakes. Because Enemy 

Swim and Pickerel Lakes spill frequently and never 

join with the downstream lakes, the upper probability 

bounds for both lakes remain nearly constant for the 

entire simulation period. Therefore, Enemy Swim and 

Pickerel Lakes are unlikely to increase significantly 

over current levels, and both lakes could return to historic 1960-92 levels much sooner than the other lakes 

under a reversal of the current wet conditions. The 

lower probability bounds for Enemy Swim Lake 

decrease more slowly than those for Pickerel Lake. 

The bounds for Enemy Swim Lake become nearly constant in about year 2030, and the bounds for Pickerel 

Lake become nearly constant in about 2005.

The lake extents for the Waubay Lakes Chain 

corresponding to the 0.2-, 1 -, 2-, 4-, 10-, and 50-percent 

probabilities (conditional analysis) based on 

October 1,1999, starting lake levels and with the farm 

crossings removed at the outlet between Rush and 

Bitter Lakes are shown on plate 1. Lake extents shown 

on plate 1 do not reflect areas that may be protected by 

dikes constructed near the city of Waubay during 1998. 

Exceedance elevations for different probabilities may 

be similar and, therefore, their respective inundated 

areas may be nearly indistinguishable at the scale 

shown (1:100,000).

Climatic Scenarios

The climatic scenarios analysis summarized in 

WRIR 99-4122 also was revised to account for the 

removal of the farm crossings at the outlet between 

Rush and Bitter Lakes. The beginning lake levels used 

in the revised climatic scenarios also were changed to 

represent October 1,1999, starting conditions.

As previously described in WRIR 99-4122, each 

of five 8-year descriptive net lake evaporation classifications was paired to form 25 possible 16-year 

sequences. Of these 25, five were selected to represent 

the range of resulting lake levels. The five scenarios 

include wet-wet, wet-dry, average-average, dry-wet, 

and dry-dry net lake evaporation.

Lake-Level Frequency Analysis 19
The climatic-scenario revised results for Bitter 

(fig. 17) and Waubay (fig. 18) Lakes due to the changes 

in the outlet elevation between Rush and Bitter Lakes 

were used to represent all the lakes of the Waubay 

Lakes Chain. Bitter Lake is most affected by the modified outlet because, with the farm crossings removed, 

it receives more water from the upstream lakes. 

Waubay Lake is representative of the upstream lakes 

when they are connected at high levels. Figures 17 and 

18 correspond to figures 41 and 42 in WRIR 99-4122. 

Both lakes have higher lake levels at the beginning of 

the scenarios than in WRIR 99-4122, with Bitter Lake 

being 1.1 ft higher, and Waubay Lake being 1.7 ft 

higher. Because the outlet between Rush and Bitter 

Lakes was at an elevation of 1,803.6 ft, and the culvert 

invert now joining the two lakes is at an elevation of 

1,798.69 ft, the lakes begin to interact with one another 

at an elevation 4.91 ft lower than previously. Thus, 

Rush Lake will begin to spill to Bitter Lake at a lower 

elevation, causing earlier rises in Bitter Lake and 

delayed rises in Waubay Lake (and the other lakes connected to it) compared to the previous climatic scenarios with the farm crossings in place. For both lakes, 

lake levels at the end of the wet-wet and wet-dry 

scenarios are very similar (within 1 ft) to the levels 

presented in WRIR 99-4122. With the farm crossings 

removed, Bitter Lake rises 21.4 ft and Waubay Lake 

rises 9.8 ft by the end of a 16-year wet-wet scenario. 

Bitter and Waubay Lakes, after 16 years, would combine into one lake that has an elevation of about 

1,813.5 ft, which is 2.4 ft above the level at which 

Bitter Lake would begin to flow out of the closed basin. 

Even though both lakes started the simulation with 

higher lake levels than in WRIR 99-4122, they do not 

end the simulation with appreciably higher levels 

because the outlet to the Big Sioux River effectively 

sets the upper limit of lake levels. The maximum 

annual water-level rise in Bitter Lake would be 4.0 ft, 

and the maximum annual water-level rise in Waubay 

Lake would be 1.8 ft, occurring in year 2000 and 2004, 

respectively (corresponding to 1991 and 1995 climatic 

conditions). In the wet-dry scenario, Bitter Lake rises 

9.5 ft and Waubay Lake declines 1.9 ft by the end of the 

16-year period. In the dry-wet and average-average 

scenarios with the farm crossings removed, Bitter Lake 

rises 11.2 ft and 6.4 ft, respectively, while Waubay 

Lake rises 0.90 ft and declines 4.6 ft, respectively, by 

the end of the 16-year period. In the dry-dry scenario 

with the farm crossings removed, Bitter Lake declines

3.9 ft and Waubay Lake declines 14.7 ft by the end of 

the 16-year period. The maximum annual water-level 

decline in Bitter and Waubay Lakes would be 2.6 ft 

occurring in year 2007 (corresponding to 1976 climatic 

conditions).

Differences in the ending lake levels for the three 

scenarios that start with either average or dry conditions are markedly different than the scenario results 

with the farm crossings in place. Bitter Lake is higher 

and Waubay Lake is lower at the end of these three 

scenarios. Under the average-average scenario, Bitter 

Lake ends 5.6 ft higher and Waubay Lake ends 3.2 ft 

lower than the corresponding scenarios presented in 

WRIR 99-4122. Under the dry-wet scenario, Bitter 

Lake ends 3.5 ft higher and Waubay Lake ends 2.2 ft 

lower, while under the dry-dry scenario, Bitter Lake 

ends 3.8 ft higher and Waubay Lake ends 2.6 ft lower 

than the corresponding scenarios presented in WRIR 

99-4122.

The lake extents for the Waubay Lakes Chain at 

the end of the dry-dry and wet-wet climatic scenarios, 

as well as the October 1,1999, and upper 1-percent 

probability lake levels (conditional analysis), are 

shown on plate 2. The lake extents for the wet-wet 

climatic scenario are only slightly different than those 

presented in WRIR 99-4122. As on plate 1, the lake 

extents do not reflect areas that may be protected by 

dikes constructed near the city of Waubay during 1998.

Results from the lake-level frequency analysis 

were used to determine the probability of the five 

selected scenarios (wet-wet, wet-dry, average-average, 

dry-wet, and dry-dry). The conditional frequency 

analysis was performed for Waubay Lake using 10,000 

simulated 16-year lake-level traces generated using the 

water mass-balance and stochastic time-series models 

and October 1,1999, starting lake levels. The 0.2-, 1-, 

and 10-percent upper and lower probability bounds and 

the five 16-year traces generated for the selected scenarios for Waubay Lake are shown in figure 19. The 

five selected scenarios seem to represent essentially the 

full range of future lake levels. The lake levels after 

16 years range from about the lower 1-percent probability bound for the dry-dry scenario, which corresponds to very dry conditions, to about the upper 

0.2-percent probability bound for the wet-wet scenario, 

which corresponds to very wet conditions. The 

average-average scenario is near the 50-percent 

probability bound after 16 years.

20 Supplement to Water-Resources Investigations Report 99-4122
1815

LU 1810 

LLJ

LLJ

w 

LU

o

CD

LU 

LU

LU

LU

_l

LU

1805

1800

1795

1790

1785

Spill elevation to Big Sioux River

Wet + wet 

Wet + dry 

Average + average 

Dry + wet 

Dry + dry

1991-98 + 1991-£ 

1991-98 + 1969-76 

1985-92 + 1985-92 

1969-76 + 1991-98 

1969-76 + 1969-76

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

WATER YEAR

Figure 17. Simulated lake levels of Bitter Lake for selected 16-year climatic scenarios, starting with October 1, 1999, 

lake level and with the farm crossings removed at the outlet between Rush and Bitter Lakes.

1815

LU 1810 

LU

LU

O

CO

<

ILU 

LU

1805

1800

T 1795

LU

LU

_l

LU

1790

1785

CLIMATIC 

SCENARIO

Wet + wet

Wet + dry

Average + average

Dry + wet

Dry + dry

1991-98 + 1991-98 

1991-98 + 1969-76 

1985-92 + 1985-92 

1969-76 + 1991-98 

1969-76 + 1969-76

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

WATER YEAR

2012 2013 2014 2015

Figure 18. Simulated lake levels of Waubay Lake for selected 16-year climatic scenarios, starting with October 1, 1999, 

lake level and with the farm crossings removed at the outlet between Rush and Bitter Lakes.

Lake-Level Frequency Analysis 21
1,820

upper 10 percent

CLIMATIC 

SCENARIO

Wet + wet 

Wet + dry 

Average + average 

Dry + wet 

Dry + dry

1991-98 + 1991-98 

1991-98 + 1969-76 

1985-92+1985-92 

1969-76 + 1991-98 

1969-76+1969-76

1,785

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

WATER YEAR

Figure 19. Probability bounds and selected 16-year scenario results for Waubay Lake, starting with October 1, 1999, 

lake level and with the farm crossings removed at the outlet between Rush and Bitter Lakes.

SUMMARY

A supplement to a USGS report, WRIR 99-4122, 

entitled "Lake-Level Frequency Analysis for the 

Waubay Lakes Chain, Northeastern South Dakota," 

was prepared to present lake-level frequency results for 

the Waubay Lakes Chain incorporating a modified 

outlet between Rush and Bitter Lakes. The modification involved removal of two farm crossings from the 

drainage way between Rush and Bitter Lakes during 

July-August 1999 as a result of a court ruling. Water 

mass-balance and stochastic time-series models were 

revised to incorporate the modified outlet and also to 

change the starting lake levels to October 1, 1999.

The outlet elevation between the upstream lakes 

(Hillebrands Lake, Rush Lake, Spring Lake, Swan 

Pond, and Waubay Lake) and Bitter Lake was changed 

from an elevation of 1,803.6 ft in WRIR 99-4122 to an 

elevation of 1,798.69 ft to reflect the modified outlet 

between Rush and Bitter Lakes. Prior to July 1999, the 

outflow from Rush Lake through Little Rush Lake to 

Bitter Lake was by overtopping of two farm crossings. 

The Fifth Judicial Circuit Court ruled on July 6, 1999, 

that the two farm crossings be removed. Two 24-in. 

culverts through an east-west township road embankment now control outflow from Rush Lake to Bitter 

Lake. At higher lake levels, outflow from Rush Lake to 

Bitter Lake is controlled by a 42-in. culvert through the 

Burlington Northern Railroad embankment.

The U.S. Army Corps of Engineers supplied flow 

ratings between Rush and Bitter Lakes used in the 

water mass-balance model. The ratings are based on 

the assumption that the east-west township road is the 

control until the lake level reaches an elevation of 

1,807.5 ft, when the railroad embankment becomes the 

control. It was assumed that the east-west township 

road will be maintained so that it is not breached.

Both an unconditional and a conditional lakelevel frequency analysis were done. An unconditional 

frequency analysis estimates the frequencies of high or 

low lake levels for a long time period during which the 

lake levels cycle many times between wet and dry 

periods. A conditional frequency analysis estimates 

the frequencies of hypothetical future lake levels for a 

relatively short time period, such as 50 years. Because 

studies that relate to flood-risk-assessment or floodmitigation alternatives usually are pertinent only for 

relatively short time periods into the future, conditional 

frequency analysis usually is more appropriate than 

unconditional frequency analysis.

22 Supplement to Water-Resources Investigations Report 99-4122
The unconditional frequency analysis was performed for Bitter, Enemy Swim, Pickerel, and Waubay 

Lakes using 10,000 years of generated lake levels with 

the farm crossings removed at the outlet between Rush 

and Bitter Lakes. The 1-percent exceedance level for 

Waubay Lake is at an elevation of 1,811.0 ft, which is 

approximately equal to the closed-basin spill elevation 

to the Big Sioux River. However, only nine separate 

spill events with an average duration of 11 years 

occurred during the 10,000-year simulation period, 

giving an average return period of 1,111 years. Therefore, because of the high serial persistence in the lake 

levels, the return period corresponding to the 1-percent 

exceedance level for Waubay Lake is much longer than 

100 years. Bitter Lake exceeded an elevation of 

1,814.0 ft, which is high enough to result in large flows 

to the Big Sioux River, an average of once every 2,500 

years and stayed above this elevation an average of 

5 years before subsiding. The exceedance-level results 

for Blue Dog Lake, Hillebrands Lake, Minnewasta 

Lake, Rush Lake, Spring Lake, and Swan Pond are 

nearly identical to those for Waubay Lake.

The unconditional frequency analysis also shows 

that the recent high levels of the Waubay Lakes Chain 

occur relatively infrequently. For example, the peak 

lake level of Waubay Lake was about an elevation of 

1,802.1 ft in 1998 and 1999; however, based on model 

simulations, Waubay Lake only reaches this elevation 

an average of once every 154 years and stays above this 

elevation an average of 15 years before subsiding. 

Enemy Swim Lake reaches an elevation of 1,856.1 ft, 

which was exceeded in 1997, an average of once every 

500 years and stays above this elevation an average of 

1 year before subsiding. Pickerel Lake reaches an elevation of 1,846.7 ft, which was equalled in 1997, an 

average of once every 500 years and stays above this 

elevation an average of 1 year before subsiding.

The average return periods for Enemy Swim and 

Pickerel Lakes for the 0.2-percent exceedance probabilities were 500 years, indicating that the extreme high 

lake levels of the smaller upstream lakes are caused by 

1-year climatic extremes. However, the average return 

periods for Bitter and Waubay Lakes for the 

0.2-percent exceedance probabilities are 2,500 years, 

indicating that the extreme high lake levels of the larger 

downstream lakes probably are caused by extreme 

climatic conditions lasting several years.

The frequencies for extreme low lake levels are 

difficult to accurately determine because small changes 

in lake volumes result in large changes in lake levels.

The lake levels of Enemy Swim, Pickerel, and Waubay 

Lakes in the 1930's and 1940's were very low 

compared to the long-term lake levels.

The conditional frequency analysis was performed for the 10 major lakes of the Waubay Lakes 

Chain using 10,000 simulated 50-year lake-level 

traces, each starting on October 1, 1999, with the farm 

crossings removed at the outlet between Rush and 

Bitter Lakes. The generated data can be used to estimate the probability of virtually any event that involves 

future lake levels.

The upper 1-percent probability bound (conditional analysis) for Waubay Lake rises quickly to a 

nearly constant level at an elevation of about 1,811 ft, 

which corresponds to the unconditional 1-percent 

exceedance level. Therefore, the chance that Waubay 

Lake will continue to rise to the spill elevation with the 

Big Sioux River within the next 10 to 15 years is 

greater than 1 percent. The lower 1-percent probability 

bound (conditional analysis) for Waubay Lake falls 

slowly during the next 50 years. The lower 1-percent 

probability bound is at elevation 1,785 ft in 2024, indicating that there is about a 1-percent chance Waubay 

Lake will return to levels comparable to historic 

1960-92 levels in the next 25 years. Also, Waubay 

Lake is unlikely to return to levels comparable to those 

during the extreme drought of the 1930's for much 

longer than 50 years.

The upper 10-percent probability bound (conditional analysis) for Bitter Lake with the farm crossings 

removed at the Rush-Bitter outlet increases about 10 ft 

during water years 1999-2010 and then slowly 

decreases, and the upper 1-percent probability bound 

increases about 19 ft during water years 1999-2015 and 

then becomes nearly constant at an elevation of about 

1,810 ft. The chance that the level of Bitter Lake will 

continue to rise to the spill elevation with the Big Sioux 

River within the next 15 to 20 years is greater than 

1 percent. Conversely, the lower 10-percent probability bound for Bitter Lake decreases about 9 ft during 

water years 1999-2025 and then slowly decreases to 

about an elevation of 1,780.5 ft by the end of water year 

2049. Furthermore, the lower 1-percent probability 

bound decreases about 11 ft by the end of water year 

2025 and then slowly decreases to about an elevation of 

1,777.5 ft by the end of water year 2049. Therefore, 

there is about a 1-percent chance that Bitter Lake will 

return to levels comparable to historic 1960-92 levels 

in the next 50 years.

Summary 23
The conditional frequency results for Blue Dog 

Lake, Hillebrands Lake, Minnewasta Lake, Rush Lake, 

Spring Lake, Swan Pond, and Waubay Lake with the 

farm crossings removed at the Rush-Bitter outlet indicate the upper 10-, 1-, and 0.2-percent probability 

bounds are about the same for all seven lakes because 

the lakes are joined at these lake levels. The upper 

10-percent probability bounds are constant until water 

year 2010, then slowly decrease, and the upper 

1-percent probability bounds increase about 8 ft from 

the current level and reach an elevation of 1,811.5 ft in 

2015. The upper 0.2-percent probability bounds for 

Bitter and Rush Lakes become about equal after 2010 

because enough water spills from Rush Lake to fill 

Bitter Lake to the extreme high levels of Rush Lake. 

As with Waubay Lake, the conditional frequency 

analysis for Blue Dog Lake, Hillebrands Lake, 

Minnewasta Lake, Rush Lake, Spring Lake, and Swan 

Pond indicates that the chance the lakes will continue 

to rise to the spill elevation with the Big Sioux River 

within the next 10 to 15 years is greater than 1 percent.

The lower probability bounds (conditional 

analysis) for Hillebrands Lake, Minnewasta Lake, 

Spring Lake, Swan Pond, and Waubay Lake are similar, with each bound decreasing slowly during the 

entire 50-year simulation. There is about a 1-percent 

chance that Hillebrands Lake, Spring Lake, Swan 

Pond, and Waubay Lake will decrease to 1960-92 

levels in the next 25 years. However, the lower probability bounds for Blue Dog and Rush Lakes become 

nearly constant in year 2010 and beyond and indicate 

that there is about a 10-percent chance that Blue Dog 

and Rush Lakes could return to historic 1960-92 levels 

within 10 years.

The frequency results for Enemy Swim and 

Pickerel Lakes are totally unaffected by the removal of 

the farm crossings at the outlet between Rush and 

Bitter Lakes. Because Enemy Swim and Pickerel 

Lakes spill frequently and never join with the downstream lakes, the upper probability bounds (conditional 

analysis) for both lakes remain nearly constant for the 

entire simulation period. Therefore, levels of both 

lakes are unlikely to increase significantly over current

levels, and both lakes could return to historic 1960-92 

levels much sooner than the other lakes under a reversal 

of the current wet conditions.

The climatic scenarios analysis summarized in 

WRIR 99-4122 also was revised to account for the 

removal of the farm crossings at the outlet between 

Rush and Bitter Lakes. The starting lake levels used in 

the revised climatic scenarios represent October 1, 

1999, levels.

The wet-wet scenario, representing two 1991-98 

climatic segments, results in Bitter Lake, after 

16 years, rising 21.4 ft above the starting lake level and 

Waubay Lake rising 9.8 ft. When an 8-year wet segment (1991-98) is combined with an 8-year dry segment (1969-76), Bitter Lake, after 16 years, rises 9.5 ft 

above the starting lake level, and Waubay Lake declines

I.9 ft. Under the dry-wet scenario, Bitter Lake rises

II.2 ft and Waubay Lake rises 0.90 ft, while under the 

dry-dry scenario, Bitter Lake declines 3.9 ft and 

Waubay Lake declines 14.7 ft.

The conditional frequency analysis performed 

for Waubay Lake with October 1, 1999, starting lake 

levels was compared to the scenario results. The five 

selected scenarios seemed to represent essentially the 

full range of hypothetical future lake levels. The lake 

levels after 16 years range from about the lower 

1-percent probability bound for the dry-dry scenario to 

about the upper 0.2-percent probability bound for the 

wet-wet scenario. The average-average scenario is 

near the 50-percent probability bound after 16 years.

SELECTED REFERENCES

Niehus, C.A., Vecchia, A.V., and Thompson, R.F., 1999, 

Lake-level frequency analysis for the Waubay Lakes 

Chain, northeastern South Dakota: U.S. Geological 

Survey Water-Resources Investigations Report 

99-4122, 166 p., 2 pis.

U.S. Department of Commerce, 1950-99, Monthly climatological data. South Dakota: Asheville, North Carolina.

Wiche, G.J., and Vecchia, A.V., 1996, Lake-level frequency 

analysis for Devils Lake, North Dakota: U.S. 

Geological Survey Water-Supply Paper 2469, 57 p.

24 Supplement to Water-Resources Investigations Report 99-4122

* U.S. GOVERNMENT PRINTING OFFICE: 1999 573-085 / 50001 Region No. 8