U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2008–5011

Prepared in cooperation with the Federal Emergency Management Agency

Climate Simulation and Flood Risk Analysis for 2008–40 

for Devils Lake, North Dakota
Cover photograph: Landsat image of Devils Lake, North Dakota, 1997. Data available from U.S. Geological Survey 

Center for Earth Resources Observation and Science (EROS), Sioux Falls, South Dakota.
Climate Simulation and Flood Risk Analysis 

for 2008–40 for Devils Lake, 

North Dakota

By Aldo V. Vecchia

Prepared in cooperation with the Federal Emergency Management Agency

Scientific Investigations Report 2008–5011

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2008

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Vecchia, A.V., 2008, Climate simulation and flood risk analysis for 2008–40 for Devils Lake, North Dakota: 

U.S. Geological Survey Scientific Investigations Report 2008–5011, 28 p.
iii

Contents

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

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

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

Input Data for Development of Stochastic Simulation Model................................................................6

Statistical Properties of Model Input Data.......................................................................................9

Long-Term Climate Variability for Devils Lake................................................................................10

Stochastic Simulation of Climate and Lake Levels for Devils Lake.....................................................12

Flood Risk Analysis for Devils Lake...........................................................................................................16

Summary........................................................................................................................................................21

References Cited..........................................................................................................................................22

Appendix........................................................................................................................................................23

Estimated Monthly Precipitation, Evaporation, and Inflow for Devils Lake for 

2000–2006................................................................................................................................24

Climate Transition Model...................................................................................................................25

Time Series Model for Generating Precipitation, Evaporation, and Inflow ..............................27

Assumptions Used for Emergency Outlet.......................................................................................28

Figures

1–2. Maps showing:

1. Location of the Devils Lake Basin in North Dakota .......................................................3

2. Locations of U.S. Geological Survey gaging stations used for 

Devils Lake flood risk analysis...........................................................................................4

3–13. Graphs showing:

3. Recorded elevations of Devils Lake for 1868–2007........................................................5

4. Measured monthly volume changes for Devils Lake for 2000–2006 

and fitted volume changes from the water mass-balance model...............................8

5. Measured monthly volume changes for Stump Lake for 2000–2006 

and fitted volume changes from the water mass-balance model...............................8

6. Estimated annual precipitation for Devils Lake for 1950–2006...................................10

7. Estimated annual evaporation for Devils Lake for 1950–2006....................................11

8. Estimated annual inflow for Devils Lake for 1950–2006...............................................11

9. Five-year moving average of annual precipitation for 1,500 years 

generated from the climate transition model with 30-year average 

duration of the wet periods and 120-year average duration of the 

dry periods ..........................................................................................................................13

10. Historical and generated annual maximum lake levels for Devils Lake 

for 1950–2040, assuming the current wet period lasts until 2010 ..............................14

11. Historical and generated annual maximum lake levels for Devils Lake 

for 1950–2040, assuming the current wet period lasts until 2038 ..............................15

12. Cumulative exceedance elevations for Devils Lake for 2008–40, 

computed by using 10,000 traces from the Devils Lake stochastic 

simulation model................................................................................................................17
iv

13. Flood elevations for Devils Lake for 2008–40 for annualized risks of 

0.5, 1, and 2 percent, computed by using 10,000 traces from the 

Devils Lake stochastic simulation model.......................................................................19

A1. Five-year moving average of annual precipitation for 1,500 years 

generated from the climate transition model with 24-year average 

duration of the wet periods and 78-year average duration of the 

dry periods ..........................................................................................................................26

Tables

1. Selected U.S. Geological Survey gaging stations in the Devils Lake Basin.......................7

2. Monthly multipliers applied to precipitation and potential evapotranspiration data from the Devils Lake National Weather Service 

station to obtain estimated precipitation and evaporation for the 

Devils Lake water-balance model..............................................................................................7

3. Cumulative flood elevations for Devils Lake for 2008–40 for cumulative 

exceedance probabilities of 1, 5, 10, 20, and 50 percent .....................................................18

4. Flood elevations for Devils Lake for 2008–40 for annualized risks of 

0.5, 1, and 2 percent....................................................................................................................20

A1. Recurrence intervals and average durations of wet and dry periods...............................25

A2. Statistical properties of generated annual precipitation, evaporation, and 

inflow data for Devils Lake for representative climate periods 1950–79, 

1980–99, and 1980–2006. ............................................................................................................27

A3. Assumptions regarding North Dakota Devils Lake Emergency Outlet..............................28

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 4,047 square meter (m2)

square mile (mi2) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3) 0.02832 cubic meter (m3) 

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

All lake levels and elevations in this report are referenced to the National Geodetic Vertical 

Datum (NGVD) of 1929.
Climate Simulation and Flood Risk Analysis for 2008–40 

for Devils Lake, North Dakota

By A.V. Vecchia

Therefore, without adjusting for wind or ice, a residence near 

Devils Lake at elevation 1,454.6 feet has the same chance of 

being flooded sometime during the next 10 years as a residence at the edge of the 100-year flood plain along a river. 

Adjusting for the effects of wind or ice, which will increase 

the flood elevations for many locations near the lakes, was not 

within the scope of this study.

Introduction

The Devils Lake Basin is a 3,810-square-mile subbasin of 

the Red River of the North (Red River) Basin (fig. 1). About 

3,320 square miles of the basin is tributary to Devils Lake 

and the remainder is tributary to Stump Lake. At an elevation 

of 1,446.5 feet, Devils Lake begins to spill into Stump Lake, 

and at an elevation of about 1,459 feet, the combined Devils 

Lake and Stump Lake system begins to spill from Stump Lake, 

through Tolna Coulee, to the Sheyenne River (fig. 2).

Lake levels of Devils Lake were recorded sporadically 

from 1867 to 1890, and in 1901 the U.S. Geological Survey 

(USGS) established a gaging station on Devils Lake. From 

1867 through 2007, the lake level fluctuated between a minimum of 1,400.9 feet (nearly dry) in 1940 and a maximum of 

1,449.2 feet in May 2006 (fig. 3). On October 1, 1992, the 

combined volume of Devils Lake and Stump Lake was about 

590,000 acre-feet and the combined surface area was about 

49,000 acres. Water began spilling from Devils Lake to Stump 

Lake in May 1999, and by September 2007 Devils Lake and 

Stump Lake became essentially one continuous water body 

with an elevation of 1,447.1 feet, a combined volume of about 

2.9 million acre-feet, and a combined surface area of about 

140,000 acres. Therefore, from 1992 to 2007 the combined 

volume increased by about 2.3 million acre-feet and the combined area increased by about 91,000 acres.

The rising water of Devils Lake and Stump Lake has 

destroyed hundreds of homes and businesses and inundated 

thousands of acres of productive farmland. Since 1993, the 

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

more than $450 million in flood-mitigation efforts, including 

more than $59 million for raising levees to protect the city 

of Devils Lake and surrounding areas and $178 million for 

Abstract

Devils Lake and Stump Lake in northeastern North 

Dakota receive surface runoff from a 3,810-square-mile 

drainage basin, and evaporation provides the only major water 

loss unless the lakes are above their natural spill elevation to 

the Sheyenne River. In September 2007, flow from Devils 

Lake to Stump Lake had filled Stump Lake and the two lakes 

consisted of essentially one water body with an elevation of 

1,447.1 feet, about 3 feet below the existing base flood elevation (1,450 feet) and about 12 feet below the natural outlet 

elevation to the Sheyenne River (1,459 feet).

Devils Lake could continue to rise, causing extensive 

additional flood damages in the basin and, in the event of an 

uncontrolled natural spill, downstream in the Red River of 

the North Basin. This report describes the results of a study 

conducted by the U.S. Geological Survey, in cooperation with 

the Federal Emergency Management Agency, to evaluate 

future flood risk for Devils Lake and provide information for 

developing updated flood-insurance rate maps and planning 

flood-mitigation activities such as raising levees or roads.

In about 1980, a large, abrupt, and highly significant 

increase in precipitation occurred in the Devils Lake Basin 

and elsewhere in the Northern Great Plains, and wetter-thannormal conditions have persisted through the present (2007). 

Although future precipitation is impossible to predict, paleoclimatic evidence and recent research on climate dynamics indicate the current wet conditions are not likely to end 

anytime soon. For example, there is about a 72-percent chance 

wet conditions will last at least 10 more years and about a 

37-percent chance wet conditions will last at least 30 more 

years.

A stochastic simulation model for Devils Lake and Stump 

Lake developed in a previous study was updated and used to 

generate 10,000 potential future realizations, or traces, of precipitation, evaporation, inflow, and lake levels given existing 

conditions on September 30, 2007, and randomly generated 

future duration of the current wet period. On the basis of the 

simulations, and assuming ice-free conditions and calm wind, 

the Devils Lake flood elevation for an annualized flood risk of 

1 percent (analogous to a “100-year” riverine flood) was estimated to be 1,454.6 feet for a 10-year time horizon (2008–17). 
2 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

raising or improving roads and highways (North Dakota State 

Water Commission, written commun., 2002; North Dakota 

Department of Transportation, written commun., 2005).

If Devils Lake and Stump Lake continue to rise, an 

uncontrolled natural spill to the Sheyenne River could occur. 

According to Murphy and others (1997), Stump Lake (and 

hence Devils Lake) spilled to the Sheyenne River at least 

twice in the past 4,000 years. If such a spill occurred in future 

years and the Tolna Coulee outlet was not modified to control erosion, the outlet could erode quickly, causing extensive 

downstream flooding, channel erosion, and water quality 

degradation (U.S. Army Corps of Engineers, 2003).

 In 1993, Devils Lake rose above its highest level for 

more than 100 years, prompting a study by the North Dakota 

State Water Commission (NDSWC), in cooperation with the 

USGS, to evaluate the lake level used for establishing flood 

insurance rate maps (FIRMs) for Devils Lake and surrounding 

areas. In 1996, the results of that study were published (Wiche 

and Vecchia, 1996). On the basis of the best information available at that time, the long-term 1-percent exceedance elevation 

(the lake level that was exceeded an average of 1 percent of 

the time since the end of glaciation, about 10,000 years ago) 

was estimated to be approximately 1,445 feet. That elevation 

was about 8 feet higher than the lake level in 1996, and about 

10 feet higher than the previously established flood elevation 

(1,435 feet). However, Devils Lake continued to rise and by 

1999 the lake level exceeded 1,445 feet, prompting the Federal 

Emergency Management Agency (FEMA) to raise the base 

flood elevation for most of the areas surrounding Devils Lake 

to the current elevation of 1,450 feet (Federal Emergency 

Management Agency, 2000). In 2000, Congress directed the 

U.S. Army Corps of Engineers (COE) to complete preconstruction engineering and design and an associated Environmental Impact Statement for an emergency outlet from Devils 

Lake.

As part of the emergency-outlet planning process, the 

USGS developed a Devils Lake stochastic simulation model 

for the COE (Vecchia, 2002). Updated information through 

1999 and more detailed hydrologic and climatologic analyses than what were available in earlier reports indicated that 

long-term hydroclimatic variability in the Devils Lake Basin 

and elsewhere in the Northern Great Plains was much greater 

than previously thought. The long-term geologic record of 

Devils Lake levels (Murphy and others, 1997) was found to 

be consistent with a two-state climate model in which climatic conditions randomly shifted between “normal” periods, 

similar to conditions that occurred during 1950–79, and “wet” 

periods, similar to conditions that occurred during 1980–99. 

The average duration of the normal periods was estimated to 

be 120 years and the average duration of the wet periods was 

estimated to be 20 years. There was estimated to be only a 

small chance (about 1 in 50) of Devils Lake rising above the 

spill elevation to Stump Lake during extended normal periods 

(such as occurred from before European settlement in the mid1800’s through most of the 1900’s). Conversely, there was estimated to be more than a 1 in 2 chance of Devils Lake spilling 

to Stump Lake and more than a 1 in 20 chance of Devils Lake 

spilling to the Sheyenne River sometime during a 20-year wet 

climatic period. Because strong evidence suggests that a wet 

period began in about 1980 and continues through the present (2007), it seems imperative that flood risk for Devils Lake 

needs to be re-examined. Given that the region is in the midst 

of a prolonged wet period, Stump Lake is full, and the current 

lake level (1,447.1 feet) is less than 3 feet below the existing base flood elevation, a study was initiated in cooperation 

with FEMA to provide the most up-to-date and scientifically 

defensible estimates of future flood risk for Devils Lake and 

Stump Lake.

Purpose and Scope

The purpose of this report is to describe climate simulation and flood risk analysis for 2008–2040 for Devils Lake, 

North Dakota. In the report, previously published estimates of 

monthly precipitation, evaporation, and inflow data for Devils 

Lake for 1950–99 are extended to include data for 2000–2006, 

the data are used to recalibrate the Devils Lake stochastic 

simulation model, and new estimates of future flood risk for 

Devils Lake and Stump Lake are provided. Recent research on 

long-term regional climate variability will be used to provide 

a sound scientific basis for the climatic assumptions used to 

generate potential future lake levels and estimate future flood 

risk. The report can be used by FEMA, COE, and other agencies for developing updated flood-insurance rate maps and 

planning flood-mitigation activities such as levee raises, road 

raises, or other infrastructure improvements.

The flood elevations described in this report are for the 

USGS Devils Lake gage on Creel Bay (site 10, fig. 2). Flood 

elevations for some areas, such as areas surrounding Pelican 

Lake, Lakes Alice and Irvine, Dry Lake, or other upstream 

lakes that are currently connected to Devils Lake, may need 

to be increased to account for flow restrictions. Such adjustments are not part of the scope of this report. Also, the flood 

elevations described in this report are for calm conditions and 

open water, and thus do not include adjustments for the effects 

of wind or ice. Finally, the effects of recent (2007) or pending 

construction to improve roads acting as water barriers were not 

incorporated into this report. The reduction in available storage resulting from that construction is estimated to result in an 

increase of 0.0 to 0.12 foot in future water-surface elevations 

(Rick Hauck, U.S. Army Corps of Engineers, written commun., 2007).
0 40 80 120 MILES

0 40 80 120 KILOMETERS Base from U.S. Geological Survey

1:2,000,000, 1972

103°

101° 99° 97°

95°

93°

91°

49°

47°

45°

43°

43°

45°

47°

49°

51° 

91° 

95° 93°

97°

99° 101° 103°

51°

Lake

of the

Woods

 Upper

Red

Lake Devils

 Lake

Mille

Lacs

Lake

Sakakawea

Lake

Oahe

Lower

Red

Lake

Lake

River

Souris

Red

River of

the North

Minnesota

River

River

Mississippi

Missouri

River

River

Missouri

James

River

River

Assiniboine

Big

Sioux

River

 EXPLANATION

Red River of the North Basin

(excluding Assiniboine Basin)

Devils Lake Basin

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

Purpose and Scope 3
Channel A

St. Joe Coulee

East

Stump

Lake

Lake

Laretta

Lake Mary

McHugh

Rose Lake

Slough

Swan

Lake

Tolna

Coulee

West

Stump

 Lake

Sixmile Bay

Creel Bay

Big Coulee

Little

Coulee

Pelican Lake

Coulee

Comstock

Peterson

Coulee

DEVILS

SHEYENNE

RIVER

LAKE

 East Devils Lake

Bay

East

Main Bay

West

Bay

Mission

Bay

Fort

Totten

Bay

BlackTiger Bay

Ziebach

Pass

LakeIrvine

Ibsen Lake

LakeAlice

Chain Lake

Mikes Lake

Coulee

Starkweather

Webster Coulee

Morrison Lake

Cavanaugh Lake

Sweetwater Lake

 DryLake

Mauvais

Mauvais

Coulee

Coulee

Tributary

Tributary #3

Edmore

Edmore

Calio

Coulee

Coulee

Coulee

FortTotten

Devils Lake

9

11

10

8

6 7

5 3

4

2 1

Woods Rutten Road

281

281

57

20

2

2

19

20

Levee

 EXPLANATION

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

SOUTH DAKOTA

MONTANA

MINNESOTA

SASKATCHEWAN MANITOBA

RED RIVER OF THE NORTH

MISSOURI

RIVER

River

River

Sheyenne

James

River

Souris

101° 103°

101° 103° 97°

49°

FARGO

BISMARCK

Devils Lake

Grand

Forks

Minot

Index Map

StudyArea

Levee

0 5 10 MILES0 5 10 KILOMETERS

2

State

Outlet

47°

99°

99° 49°

47°

Grahams Island Rd

Figure 2. Locations of U.S. Geological Survey gaging stations used for Devils Lake flood risk analysis.

4 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

1,400

1,405

1,410

1,415

1,420

1,425

1,430

1,435

1,440

1,445

1,450

1,455

1,460

1,465

ELEVATION, IN FEET ABOVE NGVD OF 1929

Overflow to Stump Lake (1,446.5 feet)

Overflow to Sheyenne River (1,459 feet)

Infrequent measurements

Lake-gage measurements

YEAR

Figure 3. Recorded elevations of Devils Lake for 1868–2007.

Purpose and Scope 5
(1)

(2)

6 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

Input Data for Development of 

Stochastic Simulation Model

Developing new modeling techniques and gathering 

extensive data not previously available were beyond the scope 

of this study. Therefore, the simulation model described by 

Vecchia (2002) was modified and used in this study to provide 

stochastic simulations of potential future water levels for 

Devils Lake and Stump Lake. Inputs to the simulation model 

include monthly estimates of precipitation, evaporation, and 

inflow for Devils Lake and Stump Lake.

Monthly water-balance equations for Devils Lake and 

Stump Lake for 2000–2007 take the following form:

VDL(t) = VDL(t–1) + (1/12) [P(t) – E(t)] ADL(t–1) 

+ QDL(t) – QDL>SL(t) + eDL(t)

VSL(t) = VSL(t–1) + (1/12) [P(t) – E(t)] ASL(t–1) 

+ QSL(t) + QDL>SL(t) + eSL(t),

where

VDL(t) is the total volume of Devils Lake at the end 

of month t, in acre-feet;

VSL(t) is the total volume of Stump Lake at the end 

of month t, in acre-feet;

ADL(t) is the surface area of Devils Lake at the end of 

month t, in acres;

ASL(t) is the surface area of Stump Lake at the end of 

month t, in acres;

QDL(t) is total inflow to Devils Lake for month t, in 

acre-feet;

QSL(t) is local inflow to Stump Lake for month t, in 

acre-feet;

QDL>SL(t) is flow from Devils Lake to Stump Lake for 

month t, in acre-feet;

P(t) is precipitation on the lake surface for 

month t, in inches;

E(t) is evaporation from the lake surface for 

month t, in inches;

eDL(t) is the water-balance error for Devils Lake for 

month t, in acre-feet; and

eSL(t) is the water-balance error for Stump Lake for 

month t, in acre-feet.

None of the terms in the water-balance equations 1 and 2 

were known with certainty, so minimum variance, unbiased 

estimates (estimates for which the errors have a small variance 

and a mean of zero) were computed as follows. Monthly flows 

from Devils Lake to Stump Lake, QDL>SL(t), were estimated 

by using streamflow for USGS gaging station 05056636 

(Devils Lake outlet to Stump Lake near Lakota, N. Dak.; 

site 11, fig. 2; table 1). Monthly inflows for Devils Lake for 

1950–99 were computed using USGS streamflow gaging 

stations 05056400 (Big Coulee near Churchs Ferry, N. Dak.), 

05056270 (Big Coulee below Churchs Ferry, N. Dak.), 

05056340 (Little Coulee near Leeds, N. Dak.), and 05056410 

(Channel A near Penn, N. Dak.; sites 6–9, fig. 2; table 1). 

Most of the surface inflow to Devils Lake during 1950–99 

could be computed by using data from those stations. By 1999, 

however, backwater from Devils Lake had reached Lakes 

Alice and Irvine and the control structure on Channel A, and 

the streamgages on Big Coulee and Channel A had been discontinued. Therefore, for 2000–2006, several upstream lakes 

(Lakes Alice and Irvine, Chain Lake, Mikes Lake, and Dry 

Lake) were considered part of Devils Lake and the inflows 

needed to be estimated by using a different method. Monthly 

inflows to Devils Lake for 2000–2006 thus were expressed by 

using the equation

QDL(t) = M[f Q*(t) + (1–f) Q*(t–1)], (3)

where

Q*(t) is total flow for month t for USGS gages 

05056060 (Mauvais Coulee Tributary no. 3 

near Cando, N. Dak.), 05056100 (Mauvais 

Coulee near Cando, N. Dak.), 05056200 

(Edmore Coulee near Edmore, N. Dak.), 

05056215 (Edmore Coulee Tributary 

near Webster, N. Dak.), 05056239 

(Starkweather Coulee near Webster, 

N. Dak.), and 05056340 (Little Coulee 

near Leeds, N. Dak.; sites 1, 2, 3, 4, 5, and 

7, fig. 2; table 1);

M>0 is a multiplier to be estimated as described 

later; and

f (0≤f≤1) is a fraction to be estimated as described later.

Use of equation 3 was motivated by several considerations. A weighted sum of the current and previous month’s 

streamflows for the upstream stations was used because a time 

delay was expected to occur between the time flows passed 

the upstream stations and the time those flows became part of 

Devils Lake. Therefore, it was reasonable to assume that part 

of the current month’s streamflow, namely (1–f)Q*(t), does not 

reach the lake until the following month. The multiplier, M, 

was expected to be greater than one because there is a considerable fraction of the total Devils Lake drainage area that is 

not tributary to one of the upstream stations. For example, the 

Calio, St. Joe, and Comstock Coulee drainage areas (fig. 2) 

and other small drainage areas adjacent to Devils Lake were 

not gaged. An intercept term was not included in equation 3 

because previous studies indicated ground-water inflow or outflow for Devils Lake was negligible (Wiche and Pusc, 1994; 

Vecchia, 2002).

Precipitation and evaporation (P(t) and E(t), equations 1 

and 2) were estimated by multiplying monthly precipitation 

and potential evapotranspiration for the National Weather 

Service station at Devils Lake (High Plains Regional Climate 

Center, 2006) by the monthly coefficients from table 2. 

Because precipitation data from the same station used by 
Table 1. Selected U.S. Geological Survey gaging stations in the Devils Lake Basin

 [USGS, U.S. Geological Survey]

Map number 

(fig. 2)

USGS gaging 

station number Station name

1 05056060 Mauvais Coulee Tributary no. 3 near Cando, N. Dak.

2 05056100 Mauvais Coulee near Cando, N. Dak.

3 05056200 Edmore Coulee near Edmore, N. Dak.

4 05056215 Edmore Coulee Tributary near Webster, N. Dak.

5 05056239 Starkweather Coulee near Webster, N. Dak.

6 05056270 Big Coulee below Churchs Ferry, N. Dak.

7 05056340 Little Coulee near Leeds, N. Dak.

8 05056400 Big Coulee near Churchs Ferry, N. Dak.

9 05056410 Channel A near Penn, N. Dak.

10 05056500 Devils Lake near Devils Lake, N. Dak.

11 05056636 Devils Lake outlet to Stump Lake near Lakota, N. Dak.

Table 2. Monthly multipliers applied to precipitation and potential evapotranspiration data 

from the Devils Lake National Weather Service station to obtain estimated precipitation and 

evaporation for the Devils Lake water-balance model.

Month Multiplier for precipitation Multiplier for evaporation

January 1.172 0

February 1.322 0

March 1.346 0

April 1.222 .262

May 1.062 .498

June 1.000 .724

July 1.070 .669

August 1.174 .726

September 1.191 .851

October 1.103 .701

November 1.010 .543

December 1.029 0

Input Data for Development of Stochastic Simulation Model 7

Vecchia (2002) were used in this report, the monthly coefficients for precipitation are the same as the coefficients given 

in table 2 of that report. However, pan evaporation data used to 

compute earlier estimates of E(t) were no longer available for 

2000–2006. Therefore, the coefficient for E(t) for a particular 

month was computed so that the average of the estimated lake 

evaporation for that month for 2000–2006 was equal to the 

average lake evaporation for that month for 1980–99 (as estimated using the previous methods). Because 1980–2006 was a 

homogeneous climatic period as shown later, lake evaporation 

for 2000–2006 was assumed to be similar to lake evaporation 

for 1980–99.

The values used for M and f in equation 3 were selected 

to minimize the sum of the squared errors [the eDL(t)’s in equation 1], subject to the constraint that the average error equals 

zero. The resulting estimates were M=1.57 and f=0.87, so 

equation 3 can be written

QDL(t) = 1.57 [0.87 Q*(t) + 0.13 Q*(t–1)]. (4)

The fitted monthly volume changes for Devils Lake 

during 2000–2006 based on estimates from equation 1, 

VDL(t) – VDL(t–1) – eDL(t), are plotted versus the measured volume changes, VDL(t) – VDL(t–1), in figure 4. The water-balance 
-100

-50

0

50

100

150

200

-100 -50 0 50 100 150 200

FITTED VOLUME CHANGE, IN THOUSAND ACRE-FEET

MEASURED VOLUME CHANGE, IN THOUSAND ACRE-FEET

Line: measured volume change

equals fitted volume change 

Figure 4. Measured monthly volume changes for Devils Lake 

for 2000–2006 and fitted volume changes from the water massbalance model.

-10

0

10

20

30

40

-10 0 10 20 30 40

FITTED VOLUME CHANGE, IN THOUSAND ACRE-FEET

MEASURED VOLUME CHANGE, IN THOUSAND ACRE-FEET

Line: measured volume change

equals fitted volume change 

Figure 5. Measured monthly volume changes for Stump Lake 

for 2000–2006 and fitted volume changes from the water massbalance model.

8 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

model provided a good fit to the measured monthly volume 

changes for 2000-2006. There was no systematic deviation of 

the measured volume changes from the fitted volume changes, 

and the magnitude of the errors was reasonable. Considering 

the fact that Devils Lake covers about 120,000 acres, a 1-inch 

error in P(t)–E(t) translates into a 10,000-acre-foot error in the 

calculated water balance.

As a cross-check to see if equation 4 was reasonable, the 

same equation was used to compute estimated inflow to Devils 

Lake for 1994–97 for comparison with the total flow for Big 

Coulee near Churches Ferry and Channel A near Penn (fig. 2) 

for that time period. The period 1994–97 was used because all 

the gaging stations for the upstream tributaries and the downstream tributaries were operating during that time. Average 

annual inflow for 1994–97 computed by using equation 4 was 

285,000 acre-feet per year, and average annual flow for Big 

Coulee and Channel A for 1994–97 was 267,000 acre-feet per 

year. The difference (18,000 acre-feet per year) is consistent 

with net evaporation loss that would be expected for that time 

from the upstream chain of lakes. The upstream lakes were 

full during 1994–97 and their total surface area was about 

24,000 acres. Therefore, 18,000 acre-feet per year corresponds 

to about 9 inches of net evaporation per year.

Local inflow to Stump Lake could not be estimated 

directly because of the lack of gaging stations in the Stump 

Lake subbasin. Therefore, local inflows to Stump Lake were 

estimated from Devils Lake inflows as described by Vecchia 

(2002):

QSL(t) = 200 + 0.046QDL(t). (5)

The intercept represents an estimated ground-water inflow 

of 200 acre-feet per month, which is added to 4.6 percent of 

Devils Lake inflow to obtain estimated Stump Lake inflow. 

Comparison of the fitted and measured volume changes for 

Stump Lake for 2000–2006 (fig. 5) indicated a reasonable fit. 

Although there were some relatively large errors considering 

the size of Stump Lake (about 12,000 acres), the errors were 

small in relation to the combined volume of Devils Lake and 

Stump Lake. Therefore, potential errors in computing Stump 

Lake inflows were expected to have a negligible effect on the 

combined water-balance for Devils Lake and Stump Lake.

Estimated monthly precipitation, evaporation, and inflow 

values for Devils Lake for 2000–2006, obtained as described 

previously, are provided in the appendix. Annual inputs for 

2000–2006, obtained by summing the estimated monthly 

inputs, are shown in figures 6–8 along with previously computed annual inputs for 1950–99. The next two sections of the 

report discuss statistical properties of the model input data and 

some important implications for flood-risk analysis for Devils 

Lake and Stump Lake.
Input Data for Development of Stochastic Simulation Model 9

Statistical Properties of Model Input Data

Devils Lake and Stump Lake receive surface runoff from 

a large basin, and evaporation is the only major water-loss 

mechanism unless water levels are above the spill elevation 

to the Sheyenne River. Therefore, water in the lakes at the 

present time (2007) depends on past inflow, precipitation, and 

evaporation all the way back to the last time Devils Lake was 

essentially empty (1940). Thus, long-term climatic conditions 

in the basin have a much larger effect on lake levels than yearto-year changes in precipitation or evaporation. By contrast, 

a riverine flood commonly depends on short-term precipitation events, such as the summer rains of 1993 or the winter 

blizzards of 1996–97, as the primary sources of flooding. For 

example, the highest recorded annual inflow to Devils Lake 

occurred during 1997 and consisted of about 540,000 acrefeet (fig. 8). During that year, record floods occurred on many 

rivers and streams throughout the Red River Basin, including 

the Devils Lake Basin. However, the 1997 inflow comprised 

only about 16 percent of the total inflow for 1993–2006 (about 

3.5 million acre-feet). On the basis of equation 5, an estimated 

193,000 acre-feet of local inflow to Stump Lake occurred 

during 1993–2006. When added to Devils Lake inflow, the 

total inflow to Devils Lake and Stump Lake during 1993–2006 

was about 3.7 million acre-feet. As mentioned previously, the 

combined volume of Devils Lake and Stump Lake increased 

by about 2.3 million acre-feet from 1992 through the present (2007). The difference of 1.4 million acre-feet (3.7–2.3) 

was the amount of net evaporative loss from the lakes during 

1993–2006.

The previous discussion indicates the importance of 

understanding climatic and hydrologic variability in the 

Devils Lake Basin before analyzing flood risk. As indicated by figure 6, precipitation for 1950–79 averaged about 

18.3 inches per year, compared to about 22.4 inches per year 

for 1980–2006. The increase of 4.1 inches per year is highly 

significant from a statistical standpoint: a two-sample t-test 

(Devore, 1987) for equality of the two means yielded a p-value 

of 0.0006. In other words, there is only about a 6 in 10,000 

chance of obtaining such a large difference between the two 

means if there is no change in precipitation patterns. Generally, a p-value less than 0.01 indicates a highly significant 

difference. Thus, there is strong evidence for an abrupt change 

in Devils Lake precipitation in about 1980. This change is 

in agreement with a regional pattern of change detected by 

Garbrecht and Rossel (2002), in which an abrupt trend toward 

higher precipitation occurred in parts of Minnesota, North 

Dakota, and South Dakota during 1980–99. Furthermore, 

Garbrecht and Rossel (2002) analyzed precipitation patterns 

from 1895–1999 and found no significant changes in precipitation before 1980. Judging by figure 6, the increase in 

precipitation occurred primarily in July-December when there 

tended to be a much higher frequency of wet conditions during 

1980–2006 than during 1950–79. In fact, 15 of the 18 wettest 

July-December seasons during 1950–2006 occurred during 

1980–2006.

Lake evaporation also increased from 1950–79 to 

1980–2006, but by a much smaller amount than precipitation 

(fig. 7). Evaporation during 1950–79 averaged 29.6 inches 

per year, compared to 30.9 inches per year during 1980–2006. 

The increase is not statistically significant (the p-value for a 

two-sample t-test was 0.07) and was due primarily to a string 

of very high evaporation years during the extreme drought 

of 1988–90. During those years, lake evaporation averaged 

37.1 inches and precipitation averaged 17.9 inches, so net 

evaporation averaged more than 19 inches per year.

Unlike precipitation, lake inflows did not increase 

until 1993 (fig. 8). Inflows during 1950–92 averaged only 

33,800 acre-feet per year, compared to 244,200 acre-feet per 

year for 1993–2006. [Note that the annual inflow estimates 

given in this report may differ from estimates obtained using 

alternate water-balance equations. For example, Wiche and 

others (2000) obtained an estimated average annual inflow 

of 60,100 acre-feet for 1950–92, compared with 33,800 acrefeet in this report. Small changes in estimated precipitation 

or evaporation can lead to substantial changes in estimated 

annual inflow for a large lake such as Devils Lake.] The long 

lag between the onset of wetter conditions in about 1980 and 

the more than sevenfold increase in inflow during 1993–2006 

can be attributed to the unusual hydrologic conditions of the 

Devils Lake Basin (Vecchia, 2002). Much of the increase in 

precipitation during 1980–87 probably went toward filling soil 

moisture deficits, the upstream chain of lakes, and the thousands of smaller lakes and wetlands in the upper basin; thus 

little of the precipitation reached Devils Lake as runoff. Then, 

the extreme drought of 1988–90 delayed the onset of higher 

flows until the summer flood of 1993, which filled most of 

the lakes and wetlands in the upper basin. Once most of the 

available storage in the upper basin was filled, inflow to Devils 

Lake increased dramatically. Runoff for the 3,320-square-mile 

Devils Lake Basin averaged only about 0.2 inch per year during 1950–92, compared to about 1.4 inches per year during 

1993–2006.

Although a definitive cause for the change in precipitation in 1980 is not known, it is likely related to sea-surface 

temperature anomalies and atmospheric-pressure anomalies 

in the Pacific and Atlantic Oceans (Wiche and others, 2000), 

including the El Niño/Southern Oscillation (ENSO), the 

Pacific Decadal Oscillation (PDO), and the Atlantic Multidecadal Oscillation (AMO). Recent discussions of the effects of 

these anomalies on North American climate can be found in 

Enfield and Cid-Serrano (2006); Schwierz and others (2006); 

Shabbar (2006); and McCabe and others (2007). Unfortunately, the complex nature of the interactions between the 

various anomalies and the geographic location of the Devils 

Lake Basin seems, at least at present, to make future long-term 

projections of precipitation in the Devils Lake Basin of limited 

use.
0

4

8

12

16

20

24

28

32

36

1950 1960 1970 1980 1990 2000

January-June

July-December

PRECIPITATION, IN INCHES

Mean (1950–79)

Mean (1980–2006)

YEAR

Figure 6. Estimated annual precipitation for Devils Lake for 1950–2006.

10 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

Long-Term Climate Variability for Devils Lake

Although future precipitation may be unpredictable, there 

is mounting evidence that long-term climate variation follows 

some interesting probabilistic structure that can be exploited. 

In particular, long-term climate reconstructions based on tree 

rings or lake sediments (Vance and others, 1992; Shapley and 

others, 2005) indicate that climate in the Northern Great Plains 

has consisted of relatively long dry periods interspersed with 

wet periods, that the transitions from dry to wet or wet to dry 

are relatively abrupt (that is, little time elapses between the 

dry and wet states), and that the durations of the individual 

dry or wet periods are highly variable and seemingly random. 

Climate modeling research also indicates that coupled global 

ocean-temperature, atmospheric-circulation, and terrestrialfeedback mechanisms can lead to multiple “equilibrium” 

climate states with abrupt transitions between states (Demaree 

and Nicolis, 1990; Yonetani and Gordon, 2001; Wang, 2004). 

Furthermore, physical considerations support the existence 

of either one or two primary equilibrium states because more 

than two states tend to result in rapid transitions between states 

and continuation in any one state for an extended period of 

time is improbable (Demaree and Nicolis, 1990). The highenergy state (the wet state) generally has shorter duration, on 

average, and more variability from year to year than the lowenergy state (the dry state).

The previous discussion points to a rather interesting 

possibility with regard to Devils Lake precipitation (fig. 6). 

Climatic conditions in the Devils Lake Basin in the distant 

past and the near future may consist of two equilibrium 

climate states: a dry state similar to 1950–79 and a wet state 

similar to 1980–2006. Existence of any intermediate states, or 

more extreme dry or wet states, is unlikely. Further evidence 

of the plausibility of this assumption was given by Vecchia 

(2002), who showed that the geologic history of lake-level 

fluctuations for Devils Lake for the past 5,000 years (Bluemle, 

1996) is consistent with a two-state climate model with random transitions between states. He estimated that the average 

duration of the dry states (or “normal” states, similar to conditions in 1950–79) was 120 years and the average duration of 

the wet states (similar to conditions in 1980–99) was 20 years. 

The average duration of the wet states is a key parameter for 

evaluating future flood risk for Devils Lake. Therefore, that 

parameter was updated for this study using the longer period 

(1980–2006) to represent the wet state and using a more 

detailed analysis based on the model of Demaree and Nicolis 

(1990).

The new estimation results, described in the appendix, 

indicate that an average duration of 30 years for the wet states 

and 120 years for the dry states best matched the historical evidence from long-term climate reconstructions in the 

region. The average recurrence interval of the wet periods 
January-June

July-December

Mean (1993–2006)

0

50

100

150

200

250

300

350

400

450

500

550

1950 1960 1970 1980 1990 2000

INFLOW, IN THOUSAND ACRE-FEET

Mean (1950–92)

YEAR

Figure 8. Estimated annual inflow for Devils Lake for 1950–2006.

0

4

8

12

16

20

24

28

32

36

40

44

1950 1960 1970 1980 1990 2000

January-June

July-December

EVAPORATION, IN INCHES

Mean (1950–79)

Mean (1980–2006)

YEAR

Figure 7. Estimated annual evaporation for Devils Lake for 1950–2006.

Input Data for Development of Stochastic Simulation Model 11
12 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

(the time from the start of one wet period to the start of 

the next wet period) was 150 years, which is in agreement 

with Shapley and others (2005), who estimated an average 

recurrence interval of 140–160 years by using tree-ring and 

lake-sediment records. An example of a simulated precipitation record with a length of 1,500 years (a small segment of a 

much longer record of 50,000 years; see appendix) is shown 

in figure 9. There were 10 wet periods (shaded in gray) during 

the 1,500-year simulation period, so the average recurrence 

interval for this example was equal to the expected long-term 

average of 150 years. The average duration of the wet periods 

for this example was 32 years and the average duration of the 

dry periods was 145 years, both of which are somewhat higher 

than the long-term averages of 30 years and 120 years. However, there was a large range in the durations of the individual 

intervals—the wet periods ranged from 3 to 80 years in 

duration and the dry periods ranged from 15 to 367 years in 

duration.

The simulated precipitation data have a peculiar property 

commonly observed in nature—the number of years remaining 

until a particular wet (or dry) period ends has the same probability distribution no matter how long the wet (or dry) period 

has already lasted (Demaree and Nicolis, 1990). Therefore, the 

27-year duration (1980-2006) of the current wet period for the 

Devils Lake Basin has no influence on the remaining length 

of the period. In particular, the probability distribution of the 

number of years remaining in the current wet period is

Prob[Y>k] = exp(–k/30), (6)

where Prob[.] denotes the probability of the event in brackets, Y is the remaining length of the current wet period, in 

years, and k is any non-negative integer. For example, there 

is a 72-percent chance the current wet cycle will last at least 

another 10 years (Prob[Y>10]), a 37-percent chance it will last 

at least another 30 years, and a 14-percent chance it will last 

at least another 60 years. Therefore, it is not likely the current 

wet cycle will end any time soon, and there is better than a 

1-in-3 chance it will last at least another 30 years.

The stochastic simulations were generated by using 

initial conditions existing on September 30, 2007. The only 

variables required to initialize the model were the starting 

lake levels for Devils Lake and Stump Lake (1,447.1 feet), the 

estimated amount of Devils Lake inflow for September 2007 

(2,315 acre-feet), the starting sulfate concentrations for each 

of the bays of Devils Lake (see appendix), and the current 

climate state (wet). A time series model was used to generate future sequences of monthly precipitation and evaporation 

for Devils Lake that had the same statistical properties (such 

as means, variances, serial correlations) as the historical data 

for either 1950–79 or 1980–2006, depending on whether the 

climate state was dry (1950–79) or wet (1980–2006). Then, 

the generated precipitation and evaporation data were used to 

generate monthly inflows, and the combined monthly input 

data were used in a water and sulfate mass-balance model to 

simulate monthly lake levels and sulfate concentrations for 

Devils Lake and Stump Lake, streamflow and sulfate concentrations for the Sheyenne River receiving waters, and outlet 

discharges for the emergency outlet.

Because of the current (2007) high water levels, the 

remaining duration of the current wet period is by far the most 

important consideration for evaluating future risk. To demonstrate, two sets of simulation runs with 1,000 traces each 

were generated by using a fixed duration for the wet period of 

2 more years for the first set and 30 more years for the second 

set. Each trace was based on randomly generated possible 

future time series of precipitation, evaporation, and inflow that 

were consistent with the assumed duration of the wet period. 

The recorded annual maximum lake levels for 1950–2007, 

along with examples of 10 (out of the 1,000) future lake level 

traces for 2008–40 are shown in figures 10 and 11. The 5th, 

50th, and 95th percentiles of the generated lake levels for 

each year, computed from all 1,000 simulated traces, also are 

shown. For the simulations with the wet period lasting 2 more 

years (fig. 10), most (90 percent) of the simulated lake levels 

in 2010 were between about 1,444 and 1,452 feet. After the 

wet period ended, however, lake levels steadily declined and 

in 2040, most of the traces were between about 1,420 and 

1,440 feet. For the simulations with the wet period lasting 

30 more years (fig. 11), the simulated lake levels were highly 

variable and the median lake level declined only slightly from 

2008 through 2040. In 2040, most of the traces were between 

about 1,434 and 1,456 feet. Thus, if wet conditions continue, 

lake levels could well increase more than 10 feet during the 

next 10 years, or lake levels could well decline several feet 

during the next 10 years before rising again to 1,456 feet or 

higher during subsequent decades.

Because it was impossible to predict exactly how long the 

current wet cycle will continue, the risk analysis results in the 

next section are based on a weighted average over all possible 

future climate conditions. A total of 10,000 simulated traces 

were generated from the stochastic model, in which each trace 

represents one possible future realization based on randomly 

generated climate states and randomly generated time series of 

precipitation, evaporation, and inflow that were consistent with 

Stochastic Simulation of Climate and 

Lake Levels for Devils Lake

The simulation model developed by Vecchia (2002) was 

modified as described in the appendix to generate potential 

future realizations, or traces, of monthly Devils Lake precipitation, evaporation, and inflow and resulting lake levels and 

sulfate concentrations for Devils Lake and Stump Lake for 

2008–40. Sulfate concentrations were required along with lake 

levels because the North Dakota Devils Lake Emergency Outlet (North Dakota State Water Commission, 2007) is designed 

to control the amount of water discharged from Devils Lake in 

accordance with downstream sulfate standards in the Sheyenne 

River receiving waters. The assumptions used for operation of 

the emergency outlet are described in the appendix.
15

16

17

18

19

20

21

22

23

24

25

0 200 400 600 800 1000 1200 1400

SIMULATION YEAR

PRECIPITATION, IN INCHES

Average annual precipitation for 1950–79

Distinct wet periods

EXPLANATION

Transition between wet and dry states

Average annual precipitation for 1980–2006

Figure 9. Five-year moving average of annual precipitation for 1,500 years generated from the climate transition model with 30-year 

average duration of the wet periods (similar to 1980–2006) and 120-year average duration of the dry periods (similar to 1950–79).

Stochastic Simulation of Climate and Lake Levels for Devils Lake 13

the generated climate states. For each trace and each simulation year, the climate state was selected at random by using 

the climate transition model with 30 years average duration 

of the wet periods and 120 years average duration of the dry 

periods. On the basis of the previous results from equation 6, 

72 percent (7,200) of the traces have the wet cycle lasting at 

least another 10 years, 37 percent (3,700) of the traces have 

the wet cycle lasting at least another 30 years, and 14 percent 

(1,400) of the traces have the wet cycle lasting at least another 

60 years.
1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

1,410

1,420

1,430

1,440

1,450

1,460

LAKE LEVEL, IN FEET ABOVE NGVD OF 1929

Recorded lake level

Simulated lake level

95th percentile

Median

5th percentile

YEAR

Figure 10. Historical and generated annual maximum lake levels for Devils Lake for 1950–2040, assuming the current wet period lasts 

until 2010.

14 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
Figure 11. Historical and generated annual maximum lake levels for Devils Lake for 1950–2040, assuming the current wet period lasts 

until 2038.

1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

1,410

1,420

1,430

1,440

1,450

1,460

LAKE LEVEL, IN FEET ABOVE NGVD OF 1929

Recorded lake level

Simulated lake level

95th percentile

Median

5th percentile

YEAR

Stochastic Simulation of Climate and Lake Levels for Devils Lake 15
16 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

Flood Risk Analysis for Devils Lake

 For riverine floods, a key parameter for flood-insurance 

rate maps is the so-called 100-year flood elevation, which 

is the water level that is exceeded once every 100 years, on 

average. It is generally assumed that the annual maximum 

water level obtained at a particular river cross-section has the 

same probability distribution every year and that the maximum 

water level for a particular year is statistically independent of 

all past or future annual maximum water levels. Under these 

assumptions, the average recurrence interval of a flood (the 

average length of time between any two successive floods) is 

the reciprocal of the annual exceedance probability,

RI(h) = 1/P(h), (7)

where

h is a specified elevation, in feet;

RI(h) is the recurrence interval, in years, associated 

with elevation h; and

P(h) is the probability of exceeding elevation h in 

any given year.

The 100-year flood elevation, h100 , is the value of h for which 

RI(h100 )=100 and P(h100 )=0.01. The actual amount of time 

until the next 100-year flood (starting this year, for example) is 

a random variable with a mean of 100 years and a probability 

distribution given by

Prob[X>k] = (0.99)k, (8) 

where

X is the time, in years, until the next 100-year 

flood; and

k=0,1,2,... is any non-negative integer.

Equation 8 holds because the event [X>k] is equivalent to the 

event that no 100-year flood occurs in each of the next k years, 

which with the previous assumptions equals (0.99)k. The event 

[X>k] also is equivalent to the event that the maximum elevation obtained during the next k years is less than the 100-year 

flood elevation, therefore

Prob[Hk < h100] = (0.99)k = (1–1/100)k, (9)

where

Hk is the maximum elevation obtained during the 

next k years.

Equation 9 can be expressed in a more general form,

Prob[Hk < h100/AEP] = (1–AEP/100)k, (10)

where

AEP is the annual exceedance probability, in 

percent.

A key observation regarding equation 10 is that, for riverine 

floods, h100/AEP is assumed to not depend on the number of 

future years, k, or on the starting year. However, for Devils 

Lake (or any closed lake), the probability distribution of the 

maximum lake level for future years is highly dependent 

on the starting lake level. Therefore, analogous with equation 10, the flood elevation for Devils Lake for a given annual 

exceedance probability was defined in this report as follows:

 Prob[Hk(2008) < hk,100/AEP(2008)] = (1–AEP/100)k, (11) 

where

Hk(2008) is the maximum level of Devils Lake during 

2008 to 2008+k–1;

hk,100/AEP(2008) is the flood elevation for 2008 to 2008+k–1; 

and

AEP is the annualized exceedance probability, in 

percent.

Equation 11 was developed for use in determining flood-insurance rate maps (FIRMs) for Devils Lake that are analogous 

to FIRMs for rivers, which are based on a specified annual 

exceedance probability, such as 1 percent. For other purposes, 

such as planning road or levee raises, cumulative flood elevations are also useful. Cumulative flood elevations for Devils 

Lake are defined as follows:

Prob[Hk(2008) < hk*(2008)] = 1–CEP/100, (12)

where

hk*(2008) is the cumulative flood elevation for 2008 to 

2008+k–1; and

CEP is the cumulative exceedance probability, in 

percent.

The cumulative exceedance probability is the probability of 

exceeding a given elevation sometime during the next k years, 

and the annual exceedance probability is the average probability of exceeding a given elevation in any one of the next 

k years. Therefore, cumulative flood elevations generally are 

higher than corresponding flood elevations with AEP=CEP.

Cumulative flood elevations for Devils Lake for 2008–40 

(k=1 to 33), computed by using 10,000 lake level traces 

described previously, indicate there is a relatively high risk 

of much higher lake levels occurring in future years (fig. 12 

and table 3). For example, there is about a 1-percent chance 

of exceeding 1,459.9 feet (0.9 foot above the spill elevation 

to the Sheyenne River), about a 5-percent chance of exceeding 1,455.7 feet, and about a 10-percent chance of exceeding 1,453.8 feet sometime between 2008 and 2015. In the 

nearer future, there is about a 1-percent chance of reaching 

1,456.2 feet, about a 5-percent chance of reaching 1,452.8 feet, 

and about a 10-percent chance of reaching 1,451.2 feet sometime between 2008 and 2010. In the more distant future, there 

is about a 1-percent chance of reaching 1,461.8 feet, about a 

5-percent chance of reaching 1,459.3 feet, and about a 
Cumulative exceedance probability

1-percent

5-percent

10-percent

20-percent

50-percent

2005 2010 2015 2020 2025 2030 2035 2040 2045

1,444

1,446

1,448

1,450

1,452

1,454

1,456

1,458

1,460

1,462

1,464

LAKE LEVEL, IN FEET ABOVE NGVD OF 1929

YEAR

Figure 12. Cumulative exceedance elevations for Devils Lake for 2008–40, computed by using 10,000 traces from the Devils Lake 

stochastic simulation model (exceedance elevations are for calm conditions and open water).

Flood Risk Analysis for Devils Lake 17

10-percent chance of reaching 1,457.3 feet sometime between 

2008 and 2040. Although the risk of greater lake levels is 

relatively high, the lake is by no means certain to rise more 

than another foot above the historical record level (1,449.2 feet 

in 2006). For example, there is about a 50-percent chance the 

lake will not exceed 1,450 feet any time between 2008 and 

2040.

Flood elevations for 2008–40 corresponding with a 

0.5-percent, 1-percent, and 2-percent annualized exceedance 

probability (equation 11) also were computed using the same 

10,000 lake level traces (fig. 13 and table 4). For a 1-percent 

annualized exceedance probability, the flood elevation 

increases from about 1,452.3 feet in 2008 to about 1,454.6 feet 

in 2017 before declining. Thus, for 2008, a residence near 

Devils Lake at elevation 1,452.3 feet has about the same 

chance of being flooded as a residence on the edge of the 

100-year flood plain for, say, the Red River at Fargo, N. Dak. 

However, when considering the next 10 years (2008–17), a 

residence near Devils Lake at elevation 1,454.6 feet has about 

the same chance of being flooded as a residence on the edge 

of the 100-year flood plain for the Red River at Fargo, N. Dak. 

The Devils Lake residence at 1,454.6 feet would be less likely 

to flood early in the 10-year period and more likely to flood 

later in the 10-year period than the residence on the Red River 

at Fargo, but the average risk over the 10-year period would 

be the same. Considering a 0.5-percent annualized flood risk 

(equivalent to a “200-year” riverine flood) and a 10-year time 

horizon (2008–17), the Devils Lake flood elevation would be 

1,456.5 feet; for a 2-percent annualized flood risk (equivalent 

to a “50-year” riverine flood) and a 10-year time horizon, it 

would be 1,452.5 feet. Note that these flood elevations for 

Devils Lake do not include the effects of ice or wind, which 

may increase the flood elevations for certain locations around 

the lake. The basic flood-elevation maps produced by FEMA 

for locations around the lake will include adjustments for the 

effects of wind and ice, and thus the published flood elevations 

for many locations may be higher than the elevations reported 

here.
Table 3. Cumulative flood elevations for Devils Lake for 2008–40 for cumulative exceedance probabilities of 1, 5, 

10, 20, and 50 percent.

[Flood elevations are for calm conditions and open water]

Year Cumulative exceedance probability, in percent

1 5 10 20 50

2008 1,452.3 1,450.2 1,449.4 1,448.6 1,447.7

2009 1,454.4 1,451.6 1,450.4 1,449.2 1,447.9

2010 1,456.2 1,452.8 1,451.2 1,449.8 1,448.0

2011 1,457.3 1,453.5 1,451.9 1,450.2 1,448.2

2012 1,458.2 1,454.2 1,452.5 1,450.7 1,448.3

2013 1,459.0 1,454.8 1,453.0 1,451.0 1,448.4

2014 1,459.5 1,455.3 1,453.4 1,451.3 1,448.6

2015 1,459.9 1,455.7 1,453.8 1,451.6 1,448.7

2016 1,460.2 1,456.1 1,454.1 1,451.9 1,448.8

2017 1,460.4 1,456.4 1,454.4 1,452.1 1,448.9

2018 1,460.6 1,456.7 1,454.7 1,452.4 1,449.0

2019 1,460.7 1,457.0 1,454.9 1,452.6 1,449.0

2020 1,460.8 1,457.2 1,455.1 1,452.7 1,449.1

2021 1,461.0 1,457.4 1,455.3 1,452.9 1,449.2

2022 1,461.0 1,457.6 1,455.4 1,453.0 1,449.3

2023 1,461.1 1,457.8 1,455.6 1,453.2 1,449.4

2024 1,461.2 1,457.9 1,455.7 1,453.3 1,449.4

2025 1,461.2 1,458.1 1,455.9 1,453.4 1,449.5

2026 1,461.3 1,458.2 1,456.0 1,453.6 1,449.6

2027 1,461.4 1,458.4 1,456.2 1,453.7 1,449.6

2028 1,461.4 1,458.5 1,456.3 1,453.8 1,449.6

2029 1,461.5 1,458.6 1,456.4 1,453.8 1,449.7

2030 1,461.5 1,458.6 1,456.4 1,453.9 1,449.7

2031 1,461.6 1,458.7 1,456.5 1,454.0 1,449.8

2032 1,461.6 1,458.8 1,456.6 1,454.1 1,449.8

2033 1,461.6 1,458.9 1,456.7 1,454.2 1,449.8

2034 1,461.7 1,459.0 1,456.8 1,454.2 1,449.9

2035 1,461.7 1,459.1 1,456.8 1,454.3 1,449.9

2036 1,461.7 1,459.2 1,456.9 1,454.4 1,449.9

2037 1,461.7 1,459.2 1,457.0 1,454.4 1,450.0

2038 1,461.8 1,459.2 1,457.1 1,454.5 1,450.0

2039 1,461.8 1,459.3 1,457.2 1,454.5 1,450.0

2040 1,461.8 1,459.3 1,457.3 1,454.6 1,450.0

18 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
2005 2010 2015 2020 2025 2030 2035 2040 2045

1,448

1,449

1,450

1,451

1,452

1,453

1,454

1,455

1,456

1,457

1,458

LAKE LEVEL, IN FEET ABOVE NGVD OF 1929

YEAR

0.5-percent annualized risk

1-percent annualized risk

2-percent annualized risk

Figure 13. Flood elevations for Devils Lake for 2008–40 for annualized risks of 0.5, 1, and 2 percent, computed by using 

10,000 traces from the Devils Lake stochastic simulation model (flood elevations are for calm conditions and open water).

Flood Risk Analysis for Devils Lake 19
Table 4. Flood elevations for Devils Lake for 2008–40 for annualized risks of 0.5, 1, and 2 percent. 

[Flood elevations are for calm conditions and open water]

Year Annualized risk, in percent

0.5 1 2

2008 1,453.3 1,452.3 1,451.3

2009 1,454.5 1,453.2 1,452.0

2010 1,455.3 1,453.9 1,452.4

2011 1,455.7 1,454.2 1,452.6

2012 1,455.9 1,454.4 1,452.7

2013 1,456.1 1,454.5 1,452.7

2014 1,456.3 1,454.6 1,452.6

2015 1,456.4 1,454.6 1,452.6

2016 1,456.4 1,454.6 1,452.6

2017 1,456.5 1,454.6 1,452.5

2018 1,456.5 1,454.6 1,452.4

2019 1,456.5 1,454.5 1,452.3

2020 1,456.5 1,454.5 1,452.2

2021 1,456.5 1,454.4 1,452.1

2022 1,456.4 1,454.4 1,452.0

2023 1,456.4 1,454.3 1,452.0

2024 1,456.4 1,454.2 1,451.8

2025 1,456.4 1,454.2 1,451.7

2026 1,456.4 1,454.1 1,451.6

2027 1,456.3 1,454.1 1,451.5

2028 1,456.3 1,454.0 1,451.4

2029 1,456.2 1,453.9 1,451.3

2030 1,456.2 1,453.8 1,451.2

2031 1,456.1 1,453.8 1,451.1

2032 1,456.0 1,453.7 1,451.0

2033 1,456.0 1,453.6 1,450.9

2034 1,455.9 1,453.5 1,450.8

2035 1,455.9 1,453.4 1,450.7

2036 1,455.8 1,453.4 1,450.6

2037 1,455.8 1,453.3 1,450.5

2038 1,455.7 1,453.2 1,450.4

2039 1,455.7 1,453.1 1,450.3

2040 1,455.6 1,453.0 1,450.2

20 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
Summary 21

Summary

Devils Lake and Stump Lake in northeastern North 

Dakota receive surface runoff from a 3,810-square-mile 

drainage basin, and evaporation provides the only major water 

loss unless the lakes are above their natural spill elevation 

to the Sheyenne River. From 1992 to 2007, above-normal 

precipitation and runoff caused Devils Lake and Stump Lake 

to increase in combined volume by about 2.3 million acre-feet 

and increase in combined area by about 91,000 acres. The 

rising water has destroyed hundreds of homes and businesses, 

inundated thousands of acres of productive farmland, and cost 

more than $450 million in flood-mitigation efforts such as 

raising levees and roads. On September 30, 2007, flow from 

Devils Lake to Stump Lake had filled Stump Lake and the two 

lakes consisted of essentially one water body with an elevation of 1,447.1 feet, about 3 feet below the existing base flood 

elevation (1,450 feet) and about 12 feet below the natural 

outlet elevation (1,459 feet). The lakes could continue to rise 

above the natural outlet, causing extensive additional flood 

damages in the basin and, in the event of an uncontrolled natural spill, downstream in the Red River of the North Basin. This 

report describes the results of a study conducted by the U.S. 

Geological Survey, in cooperation with the Federal Emergency 

Management Agency, to evaluate future flood risk for Devils 

Lake and provide information for developing updated floodinsurance rate maps and planning flood-mitigation activities 

such as raising levees or roads.

Unlike riverine floods, which primarily depend on shortterm precipitation or snowmelt events, Devils Lake flooding 

primarily depends on long-term climatic conditions in the 

basin. Therefore, understanding long-term climate variability 

is important for evaluating future flood risk for Devils Lake. In 

about 1980, a large, abrupt, and highly significant increase in 

precipitation occurred in the Devils Lake Basin and elsewhere 

in the Northern Great Plains, and wetter-than-normal conditions have persisted through the present (2007). Although 

future precipitation is impossible to predict, paleoclimatic data 

and recent research on climate dynamics indicate that climatic 

conditions in the Devils Lake Basin in the distant past and the 

near future may consist of two equilibrium states: a dry state 

similar to 1950–79 and a wet state similar to 1980–2006. Existence of any intermediate states, or more extreme dry or wet 

states, is unlikely. The average duration of the wet states was 

estimated to be 30 years and the average duration of the dry 

states was estimated to be 120 years. However, the number of 

years left in the current wet period is highly random and it is 

not likely the current wet period will end anytime soon—there 

is about a 72-percent chance it will last at least 10 more years 

and about a 37-percent chance it will last at least 30 more 

years.

A stochastic simulation model for Devils Lake developed in a previous study was updated and used to generate 

10,000 potential future realizations, or traces, of Devils Lake 

precipitation, evaporation, inflow, and lake levels given existing conditions on September 30, 2007, and randomly generated future duration of the current wet period. The generated 

traces were used to compute cumulative flood elevations for 

2008–40 by computing the elevations that have a fixed probability of being exceeded sometime between now and a given 

future year. For example, there is about a 1-percent chance of 

Devils Lake exceeding 1,459.9 feet (0.9 foot above the natural 

spill elevation), a 5-percent chance of exceeding 1,455.7 feet, 

and a 10-percent chance of exceeding 1,453.8 feet sometime 

between 2008 and 2015. Although the risk of much higher 

lake levels in future years is relatively high, there is also about 

a 50-percent chance the lake will not exceed 1,450 feet (less 

than 1 foot above the historical record level of 1,449.2 feet set 

in 2006) anytime during 2008–40.

Flood elevations used for flood-insurance rate maps generally assume a fixed probability, such as 1 percent, of exceeding the flood elevation in a single year. Unlike riverine floods, 

in which the flood elevation is the same each year, the flood 

elevation for Devils Lake depends on which year is specified. 

For example, there is about a 1-percent chance of Devils Lake 

exceeding 1,452.3 feet in 2008 and about a 1-percent chance 

of exceeding 1,456 feet in 2010. Therefore, flood elevations 

were computed for Devils Lake for which the “annualized” 

flood risk, defined as the average annual probability of exceeding the flood elevation for a certain future time horizon, was 

fixed. The Devils Lake flood elevation for an annualized 

flood risk of 1 percent (analogous to a “100-year” riverine 

flood) was estimated to be 1,454.6 feet for a 10-year time 

horizon (2008–17). Therefore, a residence near Devils Lake 

at elevation 1,454.6 feet has the same chance of being flooded 

sometime during the next 10 years as a residence at the edge 

of the 100-year flood plain along a river. The Devils Lake 

residence would be less likely to flood early in the 10-year 

period and more likely to flood later in the 10-year period 

than the residence along the river, but the average risk over 

the next 10 years would be the same. The flood elevations in 

this report are for the USGS gage on Creel Bay of Devils Lake 

and were not adjusted for the effects of wind or ice. The basic 

flood-elevation maps produced by FEMA for locations around 

the lake will include adjustments for the effects of wind and 

ice, and thus the published flood elevations for many locations 

may be higher than the elevations reported here.
22 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota

References Cited

Bluemle, J.P., 1996, North Dakota Geological Survey newsletter: v. 23, no. 1, p. 1–2.

Demaree, G.R., and Nicolis, C., 1990, Onset of Sahelian 

drought viewed as a fluctuation-induced transition: Quarterly Journal of the Royal Meteorological Society, v. 116, 

p. 221–238.

Devore, J.L., 1987, Probability and Statistics for Engineering 

and the Sciences: Brooks/Cole, Monterey, California, 672 p.

Enfield, D.B., and Cid-Serrano, L., 2006, Projecting the risk of 

future climate shifts: International Journal of Climatology, 

v. 26, p. 885–895.

Federal Emergency Management Agency, 2000, Code of Federal Regulations, Title 44, Part 67.

Garbrecht, J.D., and Rossel, F.E., 2002, Decade-scale precipitation increase in the Great Plains at the end of the 20th 

century: Journal of Hydrologic Engineering, v. 7, Issue 1, 

p. 64–75.

High Plains Regional Climate Center, 2006, Climate data, 

accessed June 15, 2007 at http://www.hprcc.unl.edu/

McCabe, G.J, Betancourt, J.L,, and Hidalgo, H.G., 2007, 

Associations of decadal to multidecadal sea-surface temperature variability with upper Colorado River Flow: Journal 

of the American Water Resources Association, v. 43, no. 1, 

p. 183–192.

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

Jerusalem and Tolna outlets in the Devils Lake Basin, North 

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

North Dakota State Water Commission, 2007, Devils 

Lake Emergency Outlet Information, accessed June 15, 

2007 at http://www.swc.state.nd.us/4dlink9/4dcgi/

GetSubCategoryRecord/ Devils%20Lake%20Flooding/

Outlet

Schuh, W.M., and Hove, M.H., 2006, Sources and processes 

affecting dissolved sulfate concentrations in the Upper 

Sheyenne River: North Dakota State Water Commission 

Report, 52 p.

Schwierz, C., Appenzeller, C., Davies, H.C., Liniger, M.A., 

Muller, W., Stocker, T.F., and Yoshimori, M., 2006, Challenges posed by and approaches to the study of seasonal-todecadal climate variability: Climatic Change, 

v. 79, p. 31–63.

Shabbar, A., 2006, The impact of El Niño-Southern Oscillation on the Canadian climate: Advances in Geosciences, 

v. 6, p. 149–153.

Shapley, M.D., Johnson, W.C., Engstrom, D.R., and 

Osterkamp, W.R., 2005, Late-Holocene flooding and 

drought in the Northern Great Plains, USA, reconstructed 

from tree rings, lake sediments, and ancient shorelines: The 

Holocene, v. 15, p. 29–41.

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

North Dakota Integrated Planning Report and Environmental Impact Statement: St. Paul District, COE, various 

pagination.

Vance, R.E., Mathewes, R.W., and Clague, J.J, 1992, 

7,000-year record of lake-level change on the northern 

Great Plains: A high-resolution proxy of past climate: 

Geology, v. 20, p. 879–882.

Vecchia, A.V., 2002, Simulation of a proposed emergency 

outlet from Devils Lake, North Dakota: U.S. Geological 

Survey Water-Resources Investigations Report 02–4042, 

129 p.

Wang, G., 2004, A conceptual modeling study on biosphereatmosphere interactions and its implications for physically based climate modeling: Journal of Climate, v. 17, 

p. 2572–2583.

Wiche, G.J., and Pusc, S.W., 1994, Hydrology of the Devils 

Lake area, North Dakota: North Dakota State Water Commission, Water Resources Investigation 22, 24 p.

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.

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

Fey, J.T., 2000, Climatology, hydrology, and simulation 

of an emergency outlet, Devils Lake Basin, North Dakota: 

U.S. Geological Survey Water-Resources Investigations 

Report 00–4174, 16 p.

Yonetani, T., and Gordon, H.B., 2001, Abrupt changes as 

indicators of decadal climate variability: Climate Dynamics, 

v. 17, p. 249–258.
Appendix
Estimated Monthly Precipitation, Evaporation, and Inflow for Devils Lake for 

2000–2006.

Month

Year Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

Precipitation, in inches

2000 0.25 1.02 0.65 0.51 2.07 2.91 6.10 5.10 3.13 1.84 2.62 0.60

2001 .23 .78 .35 2.03 2.20 5.40 5.31 1.58 .64 2.06 .21 .42

2002 .23 .09 .49 .70 1.03 4.27 2.33 5.78 2.55 .67 .18 .69

2003 .63 .13 .96 .89 4.05 3.80 3.62 2.46 1.42 .66 .65 .73

2004 1.44 .48 4.78 .43 6.04 1.12 1.97 4.02 3.38 1.25 .15 .48

2005 .69 .11 .53 .95 3.83 8.32 3.11 2.95 .31 1.51 1.56 1.06

2006 .30 .63 1.43 1.73 1.23 2.06 1.08 3.33 .94 1.26 .07 .81

Evaporation, in inches

2000 0 0 0 1.36 4.24 4.89 4.60 5.97 4.80 3.19 0.48 0

2001 0 0 0 1.02 4.15 4.75 4.88 6.39 4.62 2.73 1.67 0

2002 0 0 0 .89 4.46 5.77 5.45 5.37 5.00 1.96 1.15 0

2003 0 0 0 1.40 3.70 5.23 5.49 6.82 5.06 3.85 .68 0

2004 0 0 0 1.35 3.48 5.22 5.12 4.71 4.01 2.85 2.00 0

2005 0 0 0 1.57 3.51 4.64 5.42 6.05 5.26 2.86 .97 0

2006 0 0 0 1.29 4.23 5.90 7.37 6.89 5.01 2.93 1.60 0

Inflow, in thousand acre-feet

2000 0 0 1.8 1.9 1.0 2.0 11.1 4.6 3.8 0.5 0 0

2001 0 0 0 151.6 50.2 23.4 16.2 13.9 2.2 .1 0 0

2002 0 0 0 .5 .9 42.4 17.2 3.3 3.4 .5 0 0

2003 0 0 6.9 37.3 14.3 6.3 2.6 1.8 1.2 .1 0 0

2004 0 0 5.7 177.5 55.4 56.2 17.9 5.9 3.0 .3 0 0

2005 0 0 .9 50.3 14.7 18.5 99.2 25.5 4.2 .4 0 0

2006 0 0 2.7 172.1 49.3 11.6 1.8 .8 .1 .6 1.8 1.7

24 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
Climate Transition Model

The climate transition model was derived by using the annual precipitation data (fig. 6) and the techniques of Demaree and 

Nicolis (1990). They show that the following stochastic nonlinear differential equation can be used to model long-term climate 

fluctuations:

 d

dt

X t ( )   = −[ ( X t)] + −   [ ( X t)]   [ ( X t)]+  3 2 +σF(t), (A1)

where X(t*) (with t*=ρt) is a climatic variable for time t*, in decimal years, F(t) is a Gaussian white noise time series with mean 

zero and variance one, and ρ, λ, μ, ν, and σ are model parameters. They also show that under general conditions on the parameters, X(t*) has two equilibrium states, X(t*) ≈ M1

+S1 Y1(t*) and X(t*) ≈ M2+S2 Y2(t*), where Mi and Si are the mean and standard 

deviation of the ith state and Yi(t*) is a stochastic process with a mean of zero and a standard deviation of one (the properties of 

Yi(t*) are not important for purposes of this report). They express the parameters of the model (A1) in terms of the means and 

standard deviations of the two equilibrium states. In particular,

λ=M1+M2+B; μ=B(M1+M2)+M1M2; and ν=BM1M2 , (A2)

where

B= (M1S1

2

+M2S2

2

) / (S1

2

+S2

2

), (A3)

and

σ2=2S2

2 [M22

–M1M2–B(M2–M1)]. (A4)

The remaining parameter (ρ) needs to be estimated through simulation analysis as described later in this section.

The climatic time series selected for the climate transition model (X) was derived from the annual precipitation time series 

for Devils Lake (fig. 6),

X(t*) = P5 (t*)–20.3, (A5)

where

P5 (t*) is average annual precipitation for Devils Lake for the 5 years up to and including time t*.

The 5-year moving average was used to smooth out some of the noise in the data that was not expected to represent regional 

climate variability. The constant (20.3) is the average annual precipitation for 1950–2006 and was subtracted strictly for numerical purposes. Values obtained by using 1950–79 to represent the first state and 1980–2006 to represent the second state were 

M1= –1.9, M2=2.1, S1=0.30, and S2=0.58. Using these values in the previous formulas (A1–A4) gives

B =1.2; λ =1.32; μ = –4.08; ν = –5.14; and σ2 = 2.41.

The remaining parameter (ρ) determines the rate at which X(t*) cycles between the two equilibrium states. It was estimated by 

substituting values into model A1 (with the other parameters fixed), numerically simulating a long (50,000-year) precipitation 

time series, and selecting the value such that the average recurrence interval of the wet periods was 150 years. A recurrence 

interval of 150 years was selected to agree with Shapley and others (2005), who estimated an average recurrence interval of 

140–160 years for the wet periods on the basis of tree-ring and lake-sediment data. The average recurrence intervals for selected 

values of ρ are given in the following table.

Table A1. Recurrence intervals and average durations of wet and dry periods.

Value for ρ Average recurrence interval for 

wet periods, in years

Average duration of 

wet periods, in years

Average duration of 

dry periods, in years

0.35 169 32 137

.40 146 29 117

.45 102 24 78

Appendix—Climate Transition Model 25
15

16

17

18

19

20

21

22

23

24

25

0 200 400 600 800 1000 1200 1400

SIMULATION YEAR

PRECIPITATION, IN INCHES

Average annual precipitation for 1950–79

Distinct wet periods

EXPLANATION

Transition between wet and dry states

Average annual precipitation for 1980–2006

Figure A1. Five-year moving average of annual precipitation for 1,500 years generated from the climate transition model with 24-year 

average duration of the wet periods and 78-year average duration of the dry periods.

On the basis of table A1, ρ =0.40 was selected, which yields an average recurrence interval of 146 years for the wet 

periods, an average duration of 29 years for the wet periods, and an average duration of 117 years for the dry periods. Part of the 

50,000-year generated time series for the selected value of ρ is shown in figure 9. For comparison, to demonstrate the sensitivity 

to the value of ρ, a segment of the generated data with ρ=0.45 is shown in figure A1.

The middle line in figure A1 (and fig. 9) is the transition point between the wet and dry periods and is given by B+20.3=21.5 

inches. A wet period was assumed to start when the time series first crossed the transition line, and the subsequent dry period 

was assumed to start when the time series was below the transition line for 10 years in a row.

26 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
Time Series Model for Generating Precipitation, Evaporation, and Inflow

Minor modifications to the model developed and verified by Vecchia (2002, p. 74–82) were needed to generate input data 

for this study. The model depends on whether the climate state is wet or dry. The same period (1950–79) used in the previous 

study to represent the dry period was used here, so no modifications were needed. However, 1980–2006 was used in this study 

to represent the wet period whereas 1980–99 was used previously. The only other modification was with regard to the climate 

transition model, which used average durations of 30 years and 120 years for the wet and dry periods, respectively, for this study 

compared with 20 years and 120 years in the previous study.

Some statistical properties of the generated annual precipitation, evaporation, and inflow data for the representative climate 

periods (obtained by aggregating the monthly data) are given in table A2. For precipitation, use of the 1980–2006 climate period 

resulted in small decreases of 0.1 to 0.2 inch in the mean and all of the percentiles compared to the 1980–99 climate period. For 

evaporation, the mean and median of the data from use of the 1980–2006 and 1980–99 climate periods were the same. However, 

the spread of the data was slightly smaller using the 1980–2006 period: the 1st and 99th percentiles were 24.1 and 38.2 inches, 

respectively, for 1980–2006 compared to 23.9 and 38.8 inches for 1980–99. For inflows, the mean was somewhat smaller for 

1980–2006 compared to 1980–99 (about 108 compared to about 116 thousand acre-feet), primarily owing to the small decrease 

in precipitation. The largest differences occurred for high inflow percentiles, in which the percentiles for 1980–2006 were 

considerably smaller than those for 1980–99. The 99th percentile for 1980–2006 was about 860 thousand acre-feet, compared 

to about 932 thousand acre-feet for 1980–99. The decrease in the upper tail of the inflow distribution was due to a combination 

of the increase in the lower tail of the evaporation distribution and the decrease in the upper tail of the precipitation distribution. A somewhat smaller chance of having an extremely cool, wet summer, such as occurred in 1993, resulted from the use of 

1980–2006 as the representative climate period.

Table A2. Statistical properties of generated annual precipitation, evaporation, and inflow data for Devils Lake for 

representative climate periods 1950–79, 1980–99, and 1980–2006.

Climatic period Mean

Percentile

1 10 50 90 99

Precipitation, in inches

1950–1979 18.4 10.0 13.3 18.2 24.6 30.9

1980–1999 22.5 13.1 16.8 22.1 28.9 35.7

1980–2006 22.4 13.0 16.5 22.0 28.7 35.5

Evaporation, in inches

1950–1979 28.5 23.4 25.6 28.5 31.4 34.2

1980–1999 30.9 23.9 26.9 30.8 35.0 38.8

1980–2006 30.9 24.1 27.0 30.8 34.8 38.2

Inflow, in thousand acre–feet

1950–1979 24.9 0.0 0.2 4.5 65.4 273.3

1980–1999 115.7 .1 1.9 44.5 310.5 931.7

1980–2006 108.2 .1 1.5 39.2 294.2 859.5

Appendix—Time Series Model for Generating Precipitation, Evaporation, and Inflow 27
Assumptions Used for Emergency Outlet

Besides the differences between the simulations from Vecchia (2002) and this report that were described in the previous 

section, the only other difference was with regard to the assumptions for the Devils Lake outlet. In the previous report, the 

outlet simulations were for a proposed outlet from Devils Lake that was being considered by the COE. That outlet has not been 

constructed at this time (2007). However, the State of North Dakota did construct an emergency outlet from West Bay of Devils 

Lake (fig. 2) that became operational in 2005 (North Dakota State Water Commission, Devils Lake online information, accessed 

September 1, 2007). The outlet operational details effective on October 1, 2007, were used for all simulations described in this 

report (table A3).

Table A3. Assumptions regarding North Dakota Devils Lake Emergency Outlet.

Window of operation: April 1 through November 30

Pump capacity: 100 cubic feet per second

Downstream channel capacity: 600 cubic feet per second (combined outlet and ambient Sheyenne River flow cannot exceed 600 cubic 

feet per second)

Downstream sulfate constraint: If ambient Sheyenne River concentration is less than 260 milligrams per liter, combined outlet and 

ambient Sheyenne River flow cannot exceed 300 milligrams per liter. If ambient concentration is 

between 260 and 390 milligrams per liter, combined outlet and ambient flow cannot exceed 

1.15 times the ambient concentration. If ambient concentration is greater than 390 milligrams 

per liter, combined outlet and ambient flow cannot exceed 450 milligrams per liter.

Starting sulfate concentrations: Concentrations for Devils Lake bays on September 30, 2007, were computed by using concentrations measured by North Dakota Department of Health as close to that date as possible (Mike Ell, 

North Dakota Department of Health, written commun., 2007): Pelican Lake (490 mg/L); West Bay 

(620 mg/L); Main Bay (640 mg/L); East Bay (880 mg/L); East Devils Lake (1,140 mg/L).

Ambient Sheyenne River sulfate: Generated concentrations 2.2 times the concentrations generated using the algorithm from Vecchia 

(2002)

Recent measured sulfate concentrations from the Sheyenne River near the outlet insertion were much higher than assumed in 

previous simulations (Schuh and Hove, 2006). Therefore, the concentrations generated by using the algorithm from Vecchia 

(2002) were multiplied by 2.2, which was the average ratio of the measured concentrations at the insertion point for 2007 to 

those generated by using the previous algorithm.

The amount of water discharged from the outlet was highly variable from trace to trace in the climate simulations. As 

expected, the traces for which the lake levels increased substantially during the next few years (beginning in 2008) generally 

were the traces with the most outlet discharges. Ten percent (1,000) of the traces had at least 25,800 acre-feet of total outlet 

discharge during 2008–10 and at least 72,200 acre-feet of total discharge during 2008–17. One percent (100) of the traces had 

at least 45,900 acre-feet of total outlet discharge during 2008–10 and at least 143,000 acre-feet of discharge during 2008–17. 

Although analyzing the effect of the outlet on flood elevations was not part of the scope of this study, the traces were run both 

with and without the outlet. The cumulative flood elevations without the outlet generally were 0.1 to 0.3 foot higher than the 

values shown in table 3. The largest differences (about 0.3 foot) were for the 5- and 10-percent cumulative exceedance probabilities, intermediate differences (about 0.2 foot) for the 1- and 20-percent cumulative exceedance probabilities, and the smallest 

differences (about 0.1 foot) for the 50-percent cumulative exceedance probability.

28 Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota
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821 E. Interstate Ave.

Bismarck, ND 58503

(406) 457–5900

Or visit the North Dakota Water Science Center Web site at:

http://nd.water.usgs.gov
Vecchia, A.V.—Climate Simulation and Flood Risk Analysis for 2008–40 for Devils Lake, North Dakota—Scientific Investigations Report 2008–5011

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