U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2018–5155

Prepared in cooperation with the North Dakota State Water Commission

Stochastic Model for Simulating Souris River Basin 

Regulated Streamflow Upstream from Minot, North Dakota
Cover: Rafferty Dam located in Saskatchewan, Canada. Back cover: Lake Darling located near 

Minot, North Dakota (top photograph); Alameda Dam (renamed Grant Devine Dam in 2017) located in 

Saskatchewan, Canada (lower photograph). Photographs by Joel Galloway, U.S. Geological Survey, 

June 11, 2011.
Stochastic Model for Simulating Souris 

River Basin Regulated Streamflow 

Upstream from Minot, North Dakota

By Kelsey A. Kolars, Aldo V. Vecchia, and Joel M. Galloway

Prepared in cooperation with the North Dakota State Water Commission

Scientific Investigations Report 2018–5155

U.S. Department of the Interior

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

DAVID BERNHARDT, Acting Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2019

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Suggested citation:

Kolars, K.A., Vecchia, A.V., and Galloway, J.M., 2019, Stochastic model for simulating Souris River Basin regulated 

streamflow upstream from Minot, North Dakota: U.S. Geological Survey Scientific Investigations Report 2018–5155, 

24 p., https://doi.org/10.3133/sir20185155.

ISS328-0328 (online)
iii

Acknowledgments

The authors thank Beth Faber of the U.S. Army Corps of Engineers for her helpful technical 

review. We also thank Chanel Mueller and Mitch Weier of the U.S. Corps of Engineers and Tim 

Fay, Laura Ackerman, and Chris Korkowski of the North Dakota State Water Commission for 

providing valuable technical guidance.

v

Contents

Acknowledgments........................................................................................................................................iii

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

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

Purpose and Scope ..............................................................................................................................3

Review of Stochastic Natural (Unregulated) Streamflow Model.................................................3

Stochastic Regulated Streamflow Model..................................................................................................4

Methods for Developing the Regulated Reservoir Storage/Streamflow Routing Model .........4

Reservoir Water-Balance Approach........................................................................................5

Reservoir Operation ....................................................................................................................7

Streamflow Routing...................................................................................................................10

Regulated Reservoir Storage/Streamflow Routing Model Results ...................................12

Methods for Developing the Stochastic Regulated Streamflow Model....................................12

Stochastic Simulation Results..........................................................................................................20

Summary........................................................................................................................................................22

References Cited..........................................................................................................................................23

Figures

1. Map showing Souris River Basin showing locations of major reservoirs and

streamgages used for developing the stochastic model for simulating regulated

streamflow in the Souris River upstream from Minot, North Dakota...................................2

2. Schematic of reservoir water balance accounting for reservoir inflows,

evaporation, seepage, and reservoir releases........................................................................5

3. Graph showing comparison of Rafferty Reservoir gains and losses in 2001, a flood

year .................................................................................................................................................6

4. Graph showing comparison of Rafferty Reservoir gains and losses in 2004, a

nonflood year.................................................................................................................................6

5. Graph showing comparison of measured and simulated Rafferty Reservoir volumes

through use of a water-balance approach with the inclusion of a seepage rate .............7

6. Schematic of reservoir operation given four scenarios for the current reservoir

level in relation to the target drawdown level and ideal reservoir level.............................9

7. Schematic of regulated reservoir storage/streamflow routing model and the order

in which streamflows are determined and routed down to Minot, North Dakota...........11

8. Graph showing comparison of measured and simulated reservoir volumes and

releases for Rafferty Reservoir from 1997 through 2011 ......................................................13

9. Graph showing comparison of measured and simulated reservoir volumes and

releases for Alameda Reservoir from 1999 through 2011 ....................................................13

10. Graph showing comparison of measured and simulated reservoir volumes and

releases for Boundary Reservoir from 1992 through 2011...................................................14

11. Graph showing comparison of measured and simulated reservoir volumes and

releases for Lake Darling from 1992 through 2011 ................................................................14
vi

12. Graph showing 10-day mean measured and simulated streamflow at Souris River 

above Minot, North Dakota, 1992–2011...................................................................................16

13. Graph showing annual measured and simulated streamflow volumes at Souris

River above Minot, North Dakota, 1993–2011 ........................................................................16

14. Graphs showing simulated 10-day mean streamflow for 100-year simulation period

for the Souris River above Minot, North Dakota ...................................................................17

15. Graphs showing simulated 10-day mean streamflow for 100-year simulation period

for the Souris River above Minot, North Dakota ...................................................................18

16. Graph showing difference between recorded annual peak streamflow and

recorded annual maximum 10-day mean streamflow, as a percent of annual

maximum 10-day mean streamflow, for the streamgages Souris River above Minot,

North Dakota, and Souris River near Sherwood, North Dakota.........................................20

17. Graph showing estimated flood frequency curves for annual maximum 10-day

mean streamflow for the Souris River above Minot, North Dakota, for natural and

regulated conditions and three climatic conditions .............................................................21

18. Graph showing estimated flood frequency curves for annual instantaneous peak

streamflow for the Souris River above Minot, North Dakota, for regulated

conditions and three future climatic conditions....................................................................21

Tables

1. Monthly evaporation rates for April–October for Rafferty, Alameda, and Boundary

Reservoirs and Lake Darling.......................................................................................................5

2. Seepage rates for Rafferty, Alameda, and Boundary Reservoirs and Lake Darling

for a 10-day time step...................................................................................................................6

3. Reservoir and channel capacities, as defined in Annex A, used within the stochastic 

regulated streamflow model.......................................................................................................8

4. Lags applied to 10-day streamflow routing model................................................................10

5. Correlation coefficients and root mean-square error between measured and

simulated 10-day mean reservoir volumes for Rafferty, Alameda, and Boundary

Reservoirs and Lake Darling.....................................................................................................15

6. Correlation coefficients and root mean-square error between measured and

simulated 10-day mean reservoir releases for Rafferty, Alameda, and Boundary

Reservoirs and Lake Darling.....................................................................................................15

7. Estimated peak streamflow magnitudes for the Souris River above Minot, North

Dakota, for annual exceedance probabilities of 10, 5, 2, 1, 0.5, and 0.2 percent and

three climatic conditions...........................................................................................................22
vii

Conversion Factors

U.S. customary units to International System of Units

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

Area

square mile (mi2) 2.590 square kilometer (km2)

Volume

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

acre-foot (acre-ft) 0.001233 cubic hectometer (hm3) 

Flow rate

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

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

 °F = (1.8 × °C) + 32.

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

 °C = (°F – 32) / 1.8.

Datum

Vertical coordinate information for Canadian reservoirs is referenced to the Geodetic Survey 

of Canada Datum and for the U.S. reservoir it is referenced to the National Geodetic Vertical 

Datum of 1929 (NGVD 1929).
viii

Abbreviations

AM10DF annual maximum 10-day mean streamflow 

AM10DF

p flood magnitude for annual maximum 10-day mean streamflow corresponding 

 to exceedance probability p 

AM10D

p

 annual maximum 10-day mean streamflow corresponding to exceedance 

 probability p

APF annual peak streamflow

APF

p

 flood magnitude for annual peak streamflow corresponding to exceedance 

 probability p

FSL full supply level

IRL ideal reservoir level

IRV ideal reservoir volume

p exceedance probability

PD percent difference between the annual peak streamflow and the annual 

 maximum 10-day mean streamflow

r correlation coefficient

RMSE root mean square error

TDL target drawdown levels

TF target flow

USACE U.S. Army Corps of Engineers

USGS U.S. Geological Survey

W weight used for the weighted least-squares regression
Stochastic Model for Simulating Souris River Basin 

Regulated Streamflow Upstream from Minot, 

North Dakota

By Kelsey A. Kolars, Aldo V. Vecchia, and Joel M. Galloway

Abstract

The Souris River Basin is a 24,000 square-mile basin in 

the Provinces of Saskatchewan and Manitoba in Canada, and 

the State of North Dakota in the United States. Above-average 

snowpack during the winter of 2010–11, along with record-setting rains in May and June of 2011, led to record flooding that 

caused extensive damage to Minot, North Dakota, and numerous smaller communities in Saskatchewan, Manitoba, and 

North Dakota. As a result, the International Souris River Board 

created the Souris River Flood Task Force to evaluate potential reservoir operation changes and flood control measures 

to manage future floods and droughts. Part of this evaluation 

involved identifying a need for a stochastic streamflow model 

to estimate the likelihood of future flooding or drought.

A stochastic natural (unregulated) streamflow simulation 

model described in a previous report was built upon in this 

report to include the effects of regulation of four reservoirs 

(Rafferty, Alameda, and Boundary Reservoirs and Lake Darling) and their operation guidelines. First, a regulated reservoir 

storage/streamflow routing model was developed and calibrated from when all four reservoirs were in operation until the 

end of the reconstructed natural streamflow dataset provided by 

the U.S. Army Corps of Engineers (1992–2011). The regulated 

reservoir storage/streamflow routing model then was combined 

with the stochastic natural (unregulated) streamflow model to 

provide a stochastic regulated streamflow simulation model for 

the Souris River Basin upstream from Minot, North Dakota.

The stochastic regulated streamflow simulation model 

was used to estimate regulated flood frequency curves, which 

are useful for feasibility and design of critical structures such 

as levees or bridges. Three potential future climatic conditions were considered in this analysis: condition A (wet 

equilibrium), representing wet (similar to 1970–2017) climatic 

conditions; condition B (transition), representing transition 

from wet to dry (similar to 1912–69) climatic conditions; 

and condition C (dry equilibrium), representing dry climatic 

conditions. Comparison of the estimated flood frequency 

curves for regulated flow among the three climatic conditions 

indicated large differences in flood magnitudes for the more 

extreme (1-percent or less) annual exceedance probabilities. 

The estimated 0.2-percent annual exceedance probability 

flood magnitude for the Souris River upstream from Minot, N. 

Dak., was 29,300 cubic feet per second for condition A (wet 

equilibrium), compared to 14,800 cubic feet per second for 

condition C (dry equilibrium). For comparison, the recorded 

peak streamflow for 2011 for the Souris River upstream from 

Minot, N. Dak., was 26,900 cubic feet per second. Although 

it is not possible to predict how long the current (1970–2017) 

wet climatic conditions may persist, flood risk for at least the 

next 25 years, or until about 2040, may be represented best by 

climatic condition A.

Introduction

The Souris River Basin is a 24,000 square-mile basin 

in the Provinces of Saskatchewan and Manitoba in Canada 

and the State of North Dakota in the United States (fig. 1). 

Above-average snowpack during the winter of 2010–11, along 

with record-setting rains in May and June of 2011, resulted in 

historically unprecedented flooding in the Souris River Basin 

(Vining and others, 2013). The 2011 flood caused extensive 

damage to Minot, N. Dak., and numerous smaller communities in Saskatchewan, Manitoba, and North Dakota. The severe 

flooding prompted the International Souris River Board to create a Souris River Flood Task Force, which prepared a plan of 

study for evaluating potential reservoir operation changes and 

flood control measures to manage future floods and droughts 

(International Souris River Board, 2013). The task force plan 

indicated a need for developing stochastic streamflow simulation methods to estimate the likelihood of future floods or 

droughts that may be extreme on the basis of the available 

historical record but may not be so extreme in a much longer 

historical context. In response to this need, the U.S. Geological Survey (USGS), in cooperation with the North Dakota 

State Water Commission, developed a stochastic model for 

simulating precipitation, evapotranspiration, and natural 

(unregulated) streamflow in the Souris River Basin (Kolars 

and others, 2016).
2 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota Des Lacs NWR Upper Souris NWR J. Clark Salyer NWR Moose Mountain Reservoir 4015344 Estevan A 4012400 Minot Experiment Station GHCND USC00325993 Boundary Reservoir diversion canal near Estevan (05NB038)

Deep River

Willow

River

Wintering

Assiniboine

Pipestone Creek

Moose

Souris River

Moose Jaw River

Long Creek

Mountain

Creek

Plum Creek

Lake Sakakawea

Creek

River

SASKATCHEWAN

MANITOBA

CANADA

UNITED STATES

MONTANA

NORTH DAKOTA

99° 100° 101° 102° 103° 104° 105°

50°49°48°

Lake Winnipeg

Assiniboine River Basin

Red River of the North Basin

Souris  River  Basin

Devils Lake Basin

ALAMEDA(05ND012)

LAKE DARLING(05115500)

BOUNDARY(05NB012)

RAFFERTY

(05NB032)

0 30 15 45 60 MILES

0 50 25 75 100 KILOMETERS

Des Lacs 

River

05NB036

05113600

05NA003

05ND004

05114000

05116000

05117500

05116500

05NB001

EXPLANATION

National Wildlife Refuge (NWR)Reservoir/dam and identifierStreamgage and identifierMeteorological station and identifier

ALAMEDA

(05ND012)

05114000

4012400

Estevan A

Oxbow

Estevan

Alameda

Noonan Sherwood

Westhope

Wawanesa

Souris

Willow City

Bantry

Minot

Karlsruhe

Verendrye

Melita

Upham

Foxholm

Figure 1. Souris River Basin showing locations of major reservoirs and streamgages used for developing the stochastic model for simulating regulated streamflow in the Souris River upstream from Minot, North Dakota.
Introduction 3

Stochastic modeling takes into account the frequency of 

past climatic events and simulates a series of future realizations of climatic inputs with the same probability distribution 

as the historical record but with climatic events happening in a 

variety of orders within each realization. These future realizations of the climatic inputs represent all potential outcomes 

including multiyear wet or dry periods that are more extreme 

than any wet or dry periods in the historical period of record. 

The task force also identified the need to develop a 

stochastic model incorporating regulated streamflow for the 

Souris River Basin upstream from Minot, N. Dak. The flow 

upstream from Minot, N. Dak., is affected by regulation from 

Rafferty, Alameda, and Boundary Reservoirs in Saskatchewan 

and Lake Darling in North Dakota (fig. 1). The reservoirs 

currently (2017) are operated according to an international 

agreement between Canada and the United States (International Joint Commission, 1989). To address this need, the 

USGS, in cooperation with the North Dakota State Water 

Commission, built upon the stochastic natural (unregulated) 

streamflow simulation model described in Kolars and others 

(2016) by incorporating a reservoir storage/streamflow routing 

model to simulate regulated streamflow for the Souris River 

Basin upstream from Minot, N. Dak. The stochastic model for 

simulating regulated streamflow presented in this report uses 

the simulated climatic inputs from the unregulated model and 

accounts for the accumulation (or depletion) of available water 

storage in the soil profile and the resulting runoff to nearby 

streams given intermittent periods of wet and dry. Incorporating regulation into a stochastic model, specifically regulation 

through a series of reservoirs, allows for the accumulation or 

depletion of water stored within the reservoirs. Rules governing the storage and release of water from the reservoirs affect 

streamflow and corresponding regulated flood frequency relations. Standard statistical methods for flood frequency analysis 

are based on an observed streamflow record and do not 

necessarily take into account regulation or the accumulation 

of water, either in the soil profile or through reservoirs, and 

their effect on streamflow. The stochastic model for simulating regulated streamflow in the Souris River Basin described 

in this report accounts for cumulative effects; however, the 

model was developed using a relatively short historical record 

of observations available to validate model predictions and 

assumptions. Model assumptions used for the regulated 

streamflow routing model, described in this report, as well as 

model assumptions used for the natural streamflow simulation 

model, described in Kolars and others (2016), need to be carefully considered.

The statistical software R (version 3.4.0; R Core Development Team, 2015) was used to develop the stochastic 

natural streamflow simulation model described in Kolars and 

others (2016). The natural streamflow model, which simulates 

streamflow for an approximately 10-day (3 values per month) 

time step, was modified as described in this report to incorporate reservoir operation guidelines described in Annex A of 

the International Joint Commission Agreement (International 

Joint Commission, 1989; hereafter referred to as “Annex A”). 

The natural (unregulated) streamflow values generated from 

the first report were fed as inputs to the stochastic regulated 

streamflow model to generate regulated 10-day mean streamflow values. The generated 10-day mean streamflow values 

were used to compute estimated flood frequency distributions 

for annual maximum 10-day mean streamflow, and a weighted 

least-squares regression model was developed to estimate 

flood frequency distributions for annual maximum instantaneous flow given the estimated flood frequency distributions 

for annual maximum 10-day mean streamflow.

Overall, the stochastic model for simulating regulated 

streamflow in the Souris River Basin can be used to estimate 

regulated flood frequency curves and provide useful information for feasibility and design of critical structures such as 

levees or bridges. The model also can be used to evaluate the 

effects of proposed changes to the international agreement 

between Canada and the United States to optimize future 

reservoir operation for the joint objectives of hydropower 

generation, water delivery, and flood protection.

Purpose and Scope

The purpose of this report is to describe the stochastic 

model for simulating regulated streamflow in the Souris River 

Basin upstream from Minot, N. Dak., and present results 

of model simulations of future regulated streamflow for the 

Souris River upstream from Minot, N. Dak., and the flood 

frequency estimates to support levee feasibility and design 

studies being completed by the U.S. Army Corps of Engineers (USACE), the City of Minot, and the North Dakota 

State Water Commission. The stochastic regulated streamflow 

model was developed by first developing a regulated reservoir 

storage/streamflow routing model, and then combining that 

with the stochastic natural streamflow model described in 

Kolars and others (2016). The simulated regulated streamflow 

data from the stochastic model were used to evaluate future 

flood risk. Flood risk is evaluated assuming current (1992–

2017) reservoir infrastructure and operation is maintained in 

future years; however, the model can be modified easily for 

future use to evaluate the effects of proposed changes to the 

current regulation.

Review of Stochastic Natural (Unregulated) 

Streamflow Model

The stochastic model for simulating regulated streamflow described in this report builds upon the stochastic model 

for simulating natural (unregulated) streamflow described in 

a previous report (Kolars and others, 2016). Therefore, it is 

useful to review some key concepts from the previous report 

before describing the methods and results for the stochastic 

regulated streamflow model.

The stochastic natural streamflow model (Kolars and 

others, 2016) consisted of three components: (1) a stochastic 

climate model for simulating monthly time series of spatially 
4 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

distributed precipitation, temperature, and potential evapotranspiration for the Souris River Basin; (2) a water-balance 

model for estimating monthly runoff (in millimeters per 

square kilometer) for specified subbasins based on monthly 

precipitation, snow accumulation and melt, soil-moisture 

storage, and evapotranspiration; and (3) a temporal disaggregation and streamflow-routing model to simulate approximate 

10-day (3 values per month) streamflow values for specified streamgage locations. The streamflow values consist of 

mean streamflow (in cubic feet per second) for the beginning 

(days 1–10), middle (days 11–20), and end (days 21 through 

the end of the month) of each month, and are hereafter referred 

to as “10-day mean streamflow.”

The stochastic climate model was developed using 

recorded monthly precipitation, temperature, and potential 

evapotranspiration data for 1912–2011 from Canadian and 

United States meteorological stations in and surrounding the 

Souris River Basin (Government of Canada, 2014; Menne and 

others, 2014; Kolars and others, 2016), along with tree-ring 

chronologies dating back to the early 1700s (National Oceanic and Atmospheric Administration, National Climatic Data 

Center, 2014; Ryberg and others, 2016, Kolars and others, 

2016). A key conclusion of the climate analysis was that longterm (multidecadal to century-scale) climatic variability in the 

Souris River Basin is consistent with alternating wet and dry 

climate states of highly variable and unpredictable durations. 

One such transition from a dry to wet state likely occurred in 

the Souris River Basin in about 1970, as evidenced by a highly 

significant step-like increase in March–October precipitation around that time (Kolars and others, 2016). Thus, for the 

Souris River Basin, the period from 1970 to present (2017) is 

characterized by generally wetter conditions (and higher flood 

risk) compared to prior decades dating back to at least the 

early 20th century. The alternating wet and dry climate states 

are thought to be a manifestation of natural, long-term climatic 

persistence and are not thought to be associated with climate 

change or global warming. The implications of this long-term 

climatic persistence for evaluating future flood risk for the 

Souris River Basin are described later in this report in the 

“Methods for Developing the Stochastic Regulated Streamflow Model” section.

The water-balance model for estimating monthly runoff 

and the temporal disaggregation/streamflow routing model for 

estimating natural 10-day mean streamflow assumed no reservoir regulation. The models were calibrated and verified using 

reconstructed natural streamflow data for 1946–2011 provided 

by the U.S. Army Corps of Engineers (2013), as described in 

Kolars and others (2016). The water-balance model and the 

temporal disaggregation part of the disaggregation/streamflow routing model described in that report are identical to 

the methods used for this report describing the stochastic 

regulated streamflow model. The only modifications required 

for developing the stochastic regulated streamflow simulation model were with respect to the streamflow routing part 

of the model. These modifications are described in the section 

“Methods for Developing the Regulated Reservoir Storage/

Streamflow Routing Model.”

Stochastic Regulated Streamflow 

Model

The stochastic regulated streamflow model was developed by first developing a regulated reservoir storage/streamflow routing model, which is described in the following 

section “Methods for Developing the Regulated Reservoir 

Storage/Streamflow Routing Model,” and then combining 

that with the stochastic natural streamflow model described 

in Kolars and others (2016). The combining of the regulated 

reservoir storage/streamflow routing model and the previous 

stochastic natural streamflow model is described later in the 

report in the section “Methods for Developing the Stochastic Regulated Streamflow Model.” The simulated regulated 

streamflow data from the stochastic model were used to evaluate future flood risk as described later in the report in the section “Stochastic Simulation Results.” Flood risk is evaluated 

assuming current (1992–2017) reservoir infrastructure and 

operation is maintained in future years; however, the model 

can be modified easily for future use to evaluate the effects of 

potential changes to the current regulation.

Methods for Developing the Regulated 

Reservoir Storage/Streamflow Routing Model

The regulated reservoir storage/streamflow routing model 

builds upon the stochastic natural streamflow model developed previously in Kolars and others (2016) by including the 

effects of four reservoirs (Rafferty, Alameda, and Boundary 

Reservoirs and Lake Darling) and their operation according to 

guidelines described in Annex A (International Joint Commission, 1989). The regulated reservoir storage/streamflow 

routing model was calibrated for the period between 1992 and 

2011, the period between construction of the last reservoir 

(1992) and the end of the natural streamflow dataset (2011) 

provided by the U.S. Army Corps of Engineers (2013). Development of the regulated reservoir storage/streamflow routing 

model consisted of three main parts. First, the regulated reservoir storage/streamflow routing model required an understanding of the reservoir water balance to determine seepage rates 

and effectively capture reservoir storage changes. Second, 

operation of the reservoirs was simulated considering current and forecasted inflows, reservoir storage changes, and 

guidelines provided in Annex A. Third, reservoir releases and 

intervening runoff from downstream subbasins were routed 

down to Minot, N. Dak. (Souris River above Minot, N. Dak., 

USGS streamgage 05117500; fig. 1; U.S. Geological Survey, 

2017), using the regulated reservoir storage/streamflow routing model.
Stochastic Regulated Streamflow Model 5

Reservoir Water-Balance Approach

A reservoir water-balance approach was used to estimate 

reservoir storage changes in response to reservoir inflow and 

outflow (releases from the reservoir plus evaporation), along 

with estimated losses due to seepage (fig. 2). If all inflows and 

outflows (including evaporation) from a reservoir are known, 

and the reservoir still experiences a loss of storage during the 

10-day period, then that loss is assumed to be attributed to 

seepage. The USACE estimated daily natural inflows to the 

Canadian reservoirs as a residual through the use of a waterbalance approach by taking known reservoir outflows and storage relations and backcalculating reservoir inflows (U.S. Army 

Corps of Engineers, 2013). More details on how inflows were 

estimated are provided in U.S. Army Corps of Engineers 

(2013). For the current study, a water-balance approach similar 

to the one used by the U.S. Army Corps of Engineers (2013) 

was used to backcalculate seepage rates with the same evaporation, reservoir releases, and storage relations being used in 

the USACE water balance and the reservoir water balance in 

this report, except the USACE considers a daily time step and 

the reservoir water balance in this report considers a 10-day 

time step. Additionally, the U.S. Army Corps of Engineers 

(2013) water balance used reservoir water levels that had 

been smoothed with an n-day mean to compensate for wind 

wave effects. The reservoir water-balance approach described 

in this report simply used the mean daily recorded reservoir 

level at the start of the 10-day time step. Specifically, for Lake 

Darling, the recorded reservoir levels were sporadic, so reconstructed reservoir levels provided by the U.S. Army Corps 

of Engineers (2013) were used. A schematic of the reservoir 

water balance is given in figure 2.

Evaporation rates for Rafferty and Boundary Reservoirs 

were measured at Estevan, Saskatchewan (meteorological 

station Estevan A 4012400; fig. 1), and for Alameda Reservoir, they were measured at Moose Mountain Reservoir 

Evaporation

Reservoir 

releases

Seepage

Inflows

Reservoir storage

Figure 2. Reservoir water balance accounting for reservoir 

inflows, evaporation, seepage, and reservoir releases.

meteorological station 4015344 (fig. 1); both were provided by the Saskatchewan Water Security Agency (table 1) 

(U.S. Army Corps of Engineers, 2013). For Lake Darling, the 

evaporation rates were provided by the National Oceanic and 

Atmospheric Administration’s (NOAA) National Climatic 

Data Center and measured at the Minot Experiment Station 

GHCND: USC00325993 (fig. 1) near Minot, N. Dak. (table 1) 

(U.S. Army Corps of Engineers, 2013).

The storage relations (elevation capacity and elevation area) for the Canadian reservoirs were provided 

through Annex A (International Joint Commission, 1989, 

plates 7A–8B). For Lake Darling, a modified storage relation 

that was provided by the USACE Lake Darling Water Control 

Manual and presented in the USACE Regional and Reconstructed Hydrology of the Souris River Report for use with a 

HEC–ResSIM model (Lake Darling Water Control Manual, 

2012; U.S. Army Corps of Engineers, 2013) was used by the 

USACE. These storage relations helped to determine reservoir 

volume and corresponding surface area, which was used to 

determine evaporation volumes, from reservoir elevations. 

Reservoir elevations/levels were provided through Environment and Climate Change Canada water-level gages for Rafferty Reservoir near Estevan (05NB032), Alameda Reservoir 

near Alameda (05ND012), and Boundary Reservoir near Estevan (05NB012) (Government of Canada, 2017), and through 

the USGS water-level gage Lake Darling near Foxholm, N. 

Dak. (streamgage 05115500; U.S. Geological Survey, 2017) 

(fig. 1). Reservoir releases were measured by Environment 

and Climate Change Canada streamgages (Government of 

Canada, 2017) on the Souris River below Rafferty Reservoir 

(streamgage 05NB036), Moose Mountain Creek near Oxbow 

(streamgage 05ND004), Boundary Reservoir diversion canal 

near Estevan (05NB038), and Long Creek near Estevan 

below Boundary Reservoir (streamgage 05NB001), and by 

the USGS streamgage on the Souris River near Foxholm, 

N. Dak. (streamgage 05116000; U.S. Geological Survey, 

Table 1. Monthly evaporation rates for April–October for Rafferty, 

Alameda, and Boundary Reservoirs and Lake Darling.

Month

Evaporation, in inches

Rafferty and 

Boundary Alameda Lake Darling

April 2.30 2.40 2.40

May 5.64 4.39 4.68

June 6.39 5.40 4.45

July 7.37 6.90 5.26

August 7.18 5.53 5.11

September 4.91 3.49 3.61

October 2.44 1.47 2.92
6 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

2017) (fig. 1). Inflows to the reservoirs were provided by the 

U.S. Army Corps of Engineers (2013). Specifically, for Rafferty and Boundary Reservoirs, the inflow volumes accounted 

for the operation of Boundary canal and diverted all historical 

gaged streamflow through the canal into Rafferty Reservoir 

(U.S. Army Corps of Engineers, 2013). As a result, during 

calibration, the reservoir water balance for Rafferty Reservoir 

includes additional inflows from Boundary canal (Boundary 

Reservoir diversion canal near Estevan 05NB038; fig. 1) and 

the reservoir water balance for Boundary Reservoir includes 

additional outflows to Boundary canal. Inclusion of Boundary 

canal was necessary to reproduce historical inflows and outflows from the reservoirs for calibration of seepage rates and 

validation of the reservoir water balance. Operation guidelines 

for Boundary canal have been left largely to the discretion 

of the operator with mention in Annex A that the canal “may 

be used for flood control provided that storage is available in 

Rafferty Reservoir in excess of the amount required to meet 

United States flood control requirements in that year, by the 

amount of volume to be diverted” (International Joint Commission, p. A–22, 1989). For the purpose of this study, operation of Boundary canal was left out of the stochastic regulated 

streamflow model but was included during calibration of the 

model to historical streamflows.

The reservoir water-balance model was calibrated by 

adjusting the seepage rate. Including a seepage rate in the 

reservoir water-balance model helped compensate for drift 

and prevented the model from overpredicting actual reservoir 

volume. Visual inspection (ensuring simulated and measured 

reservoir volumes were similar) and a high correlation coefficient (greater than 0.9) for the relation between simulated and 

measured reservoir volumes were considered when determining the seepage rate. The resulting seepage rates, estimated as 

a percentage of reservoir volume at the start of a 10-day time 

step, for each reservoir are listed in table 2. The estimated 

seepage represents the total seepage (in depth equivalent) during the 10-day time step. Seepage rates were relatively small 

compared to reservoir inflow and evaporation volumes (figs. 3 

and 4).

Table 2. Seepage rates for Rafferty, Alameda, and Boundary 

Reservoirs and Lake Darling for a 10-day time step.

Reservoir Seepage rate1 Correlation 

coefficient 

Rafferty 0.175 0.997

Alameda 0.080 0.999

Boundary 0.225 0.926

Lake Darling 0.080 0.988

1

Seepage rates are presented as a percentage of the reservoir volume at the 

start of the 10-day time step and represent the total seepage (in depth equivalent) over the 10-day time step.

ence between the U.S. Army Corps of Engineers (2013) waterbalance approach and the reservoir water balance used in this 

report relates to the USACE use of a daily time step compared 

to the 10-day time step used in this report. Hence, the resulting 

reservoir volumes/levels were expected to be similar to historical values because most components of the water-balance 

approach for both models were the same (fig. 5).

In addition to the inclusion of seepage, the main differVolume, in acre−feet

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

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

Reservoir inflow

Evaporation

Seepage

EXPLANATION

2001

Figure 3. Comparison of Rafferty Reservoir gains (reservoir 

inflows) and losses (evaporation, seepage) in 2001, a flood year.

0

2,000

4,000

6,000

8,000

10,000

Volume, in acre−feet

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

Reservoir inflow

Evaporation

Seepage

EXPLANATION

2004

Figure 4. Comparison of Rafferty Reservoir gains (reservoir 

inflows) and losses (evaporation, seepage) in 2004, a nonflood 

year.
Stochastic Regulated Streamflow Model 7

Reservoir volume, in acre−feet

1992 1993 1995 1997 1994 1996 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

Full supply level

Maximum allowable flood level

Year

Measured

Simulated—With seepage

Simulated—Without seepage

EXPLANATION

Figure 5. Comparison of 

measured and simulated Rafferty 

Reservoir volumes (1992–2011) 

through use of a water-balance 

approach with the inclusion of a 

seepage rate.

Reservoir Operation

The reservoir water balance was combined with guidelines from Annex A to determine reservoir operation and 

releases. Annex A describes the operating plan for Rafferty, 

Alameda, and Boundary Reservoirs and Lake Darling and 

forms part of the agreement between the Government of 

Canada and the Government of the United States for water 

supply and flood control in the Souris River Basin (International Joint Commission, 1989). The primary objectives of 

the operating plan are to “(1) provide 1-percent (100-year) 

flood protection at Minot, N. Dak.; (2) provide flood protection to urban and rural areas downstream from Rafferty Dam, 

Alameda Dam, and Lake Darling Dam; and (3) ensure, to the 

extent possible, that the existing benefits from the supply of 

water in the Souris River Basin and the supply of water to 

the Souris Basin project are not compromised” (International 

Joint Commission, 1989). The operating plan was designed 

to accommodate floods similar to those that had been experienced in 1969, 1974, 1975, 1976, 1979, and 1982 (International Joint Commission, 1989). The operating plan provides 

detailed guidelines on managing the four reservoirs during 

flood and nonflood years and modifies these guidelines based 

on the time of year. Much detail is offered in Annex A, but for 

purposes of this report, a simplification of Annex A guidelines is used to replicate past operation of the reservoirs. The 

guidelines in Annex A that have been used in this report and 

the simplification of some guidelines are further described in 

the remainder of this section.

The first step in converting the natural streamflow routing model provided in the previous study (Kolars and others, 

2016) to a model that included regulation from four reservoirs 

was determining how much to release from the Canadian reservoirs. Determining the amount to release from each reservoir 

first depended on the current volume/level of the reservoir and 

forecasted 30-day inflow volumes. Given the USACE natural 

streamflow dataset covered the period between 1992 and 2011, 

the forecasted 30-day inflows already were known and are 

similar to forecasting 30-day flows with 100-percent accuracy. It is assumed that historical forecasts would have been 

reasonably close to 30-day inflows derived based on observed 

data. Annex A provides target drawdown levels (TDLs) 

for each of the reservoirs depending on forecasted 30- and 

90-percent, 90-day inflows (International Joint Commission, 

1989, Annex A, plates 1–4). These TDLs increase or decrease 

depending on the forecasted 30- or 90-percent, 90-day inflow 

volumes. A large forecasted inflow volume would push the 

TDL lower to create more storage in the reservoir, whereas a 

small forecasted inflow volume would push the TDL higher 

because not nearly as much storage would be needed in the 

reservoir. The TDLs for Canadian reservoirs are based on 

forecasted (90-percent chance of exceedance) 90-day inflow 

volumes to the reservoirs, whereas TDLs for Lake Darling 

are based on forecasted 30-day inflow volumes (International Joint Commission, 1989). For this report, the TDLs for 

the Canadian reservoirs were determined using forecasted 

30-day inflow volumes, instead of the 90-percent, 90-day 

inflow volumes, because historical forecasts of the 90 percent, 

90-day flow are unavailable, making it difficult to replicate 

past operation of the reservoirs. Furthermore, actual inflow 

could be quite different from forecasted values when looking 

at a large time extent like 90 days. Additionally, during large 

flood events, much of the flow volume happens within 1 or 

2 months (usually during the spring); hence, 30-day inflows 
8 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

were considered to resemble operating decisions based on 

forecasted flows more closely than 90-day inflows.

In addition to identifying the TDLs for the reservoirs, 

an ideal reservoir level (IRL) was set such that the reservoir 

either would release or contain flows to maintain the reservoir 

level at the IRL. Unlike the TDL, the IRL is static throughout 

the year and does not change depending on forecasted inflow 

volumes. When there is no threat of a flood, the IRL is still the 

objective. If there is a series of dry years, the reservoir volume 

will fall below the IRL and, without sufficient inflows to keep 

the volume at the IRL, the reservoir will remain below the IRL 

and provide more storage for the following years (more storage for potential flood years). For Rafferty and Alameda Reservoirs, this IRL was set to the “normal level prior to spring 

runoff” defined in Annex A (International Joint Commission, 

1989) and provided in table 3. An exception was made for 

Boundary Reservoir, which does not have a specified “normal 

level prior to spring runoff.” For Boundary Reservoir, the IRL 

was set 0.33 foot (ft) below the full supply level (1,840 ft) at 

1,839.67 ft. The TDL (determined by looking at forecasted 

30-day inflows) and the IRL were used to determine releases 

from the Canadian reservoirs. If the current reservoir level 

(RL) was less than or equal to (≤; below) the IRL and the TDL, 

then there would be no releases from the reservoirs (fig. 6A; 

eq. 1):

Qout=0, if RL≤IRL and RL≤TDL (1)

where

RL is the current reservoir level, in feet;

IRL is the ideal reservoir level, in feet;

TDL is the target drawdown level, in feet; and

Qout is the reservoir release, in acre-feet, for the 

current 10-day period.

If the current reservoir level was below the IRL but greater 

than (>; above) the TDL, then the difference between the current reservoir volume and the TDL volume would be released 

(fig. 6B; eq. 2):

Qout=RV–TDV, if RL≤IRL and RL>TDL (2)

where

RV is the current reservoir volume, in acre-feet; 

and

TDV is the target drawdown volume, in acre-feet.

For this last scenario, an exception was made for Boundary 

Reservoir where releases were not just the difference between 

RV and TDV, but rather releases were the minimum of either 

RV–TDV or one-half of the reservoir inflows. This exception 

was made for Boundary Reservoir to reduce releases because 

the reservoir also serves as a source of cooling water for the 

adjacent coal-fired Boundary Dam Power Station.

If the current reservoir level is above the IRL (even 

when it is below the TDL) and there is enough storage in 

the reservoir to hold the anticipated 30-day inflow volume 

(that is, the IRL is greater than or equal to [≥] the forecasted 

Table 3. Reservoir and channel capacities, as defined in Annex A (International Joint Commission, 1989), used within the stochastic 

regulated streamflow model. 

[--, no data; ft3/s, cubic foot per second; acre-ft, acre-foot]

Threshold

Reservoir

Lake Darling Rafferty Alameda Boundary

Ideal reservoir level (IRL), in feet1 2

1,802.82 21,840.55 1,839.67 31,597.00

Normal level before spring runoff, in feet 1,802.82 1,840.55 -- --

Full supply level, in feet 1,806.10 1,843.83 1,840.00 1,597.00

Maximum allowable flood level, in feet 1,817.59 1,860.24 1,840.00 1,601.00

Downstream reservoir channel capacity, in ft3

/s 500 4

2,295.5

 1,800 900 2,500 (March–May) 

500 (other) 

Maximum recorded 10-day release volume, in acre-ft 69,0651 46,580 85,729 515,701.92

1

IRL; static throughout entire year.

2

For Rafferty and Alameda Reservoirs, the IRL is equivalent to the normal level before spring runoff value provided in Annex A.

3

For Lake Darling, the IRL is equivalent to the full supply level provided in Annex A.

4

In the model, the capacity of Rafferty Reservoir’s low-flow outlet was used in place of the downstream reservoir channel capacity.
Stochastic Regulated Streamflow Model 9

No outflow

Outflow

Maximum allowable

flood level 

Reservoir

volume/level 

A B C D

Outflow

Outflow=minimum

(channel capacity, 

reservoir volume–IRL volume) 

30-day inflow

30-day inflow

Outflow

Outflow=minimum

(maximum record,

reservoir volume–IRL volume) 

Target drawdown level

(TDL)

Outflow=reservoir volume–TDL volume

Ideal reservoir level 

(IRL)

Figure 6. Reservoir operation given four scenarios for the current reservoir level in relation to the target drawdown level (TDL) 

and ideal reservoir level (IRL). A, current reservoir level is less than both the TDL and IRL and there is no outflow from the reservoir; 

B, current reservoir level is greater than the TDL but less than the IRL and outflow is the difference between the current reservoir 

volume and the TDL volume; C, current reservoir level is greater than the IRL, the forecasted 30-day inflow volume does not exceed 

reservoir storage, and outflow is the minimum of either the channel capacity or the difference between the current reservoir volume 

and the IRL volume; and D, current reservoir level is greater than the IRL, the forecasted 30-day inflow volume does exceed reservoir 

storage, and outflow is the minimum of either the maximum recorded outflow or the difference between current reservoir volume and 

the IRL volume. Note that in this figure, the TDL is depicted as lying below the IRL, but the TDL is not a constant value and could rest 

either above or below the IRL depending on forecasted inflows.

30-day inflow volume), then reservoir releases are calculated to be the minimum of either the channel capacity (just 

downstream from the reservoir) or the difference between the 

current reservoir volume and the ideal reservoir volume (IRV). 

For Lake Darling, the channel capacity changes depending 

on time of year and is further described later in this section. 

Additionally, it is important to note that when the current 

reservoir level is greater than the IRL, the target level is no 

longer the TDL but instead the IRL because a target level of 

TDL, in this scenario, resulted in too much release from the 

reservoir during the 10-day period. Setting the target level to 

IRL helped reduce releases and lower the reservoir level at a 

slower rate. For Rafferty Reservoir, using channel capacity as 

a threshold resulted in simulated values that were much lower 

than measured values, and, for this reason, the capacity of the 

low-flow outlet, 2,295.5 cubic feet per second (ft3/s, table 3) 

or about 45,530 acre-feet (acre-ft) for the 10-day period, was 

used instead of channel capacity (fig. 6C; eq. 3):

Qout=minimum(CCmax, RV–IRV), 

if RL >IRL and RS ≥IV30d

where

RS is the current available reservoir storage 

(maximum allowable flood volume–current 

reservoir volume), in acre-feet;

IV30d is the forecasted 30-day inflow volume, in 

acre-feet;

CCmax is the channel capacity, in acre-feet; and

IRV is the ideal reservoir volume, in acre-feet.

Lastly, when the current reservoir level is above the IRL and 

there is not enough available storage to hold the anticipated 

30-day inflow volume (in other words, the maximum allowable flood level described in table 3 will be exceeded), then the 

releases are calculated to be the minimum of either the historical maximum measured outflow volume during 10 days or the 

difference between the current volume and the IRV (fig. 6D; 

eq. 4). For this last scenario, the reservoir releases are no 

(3)
10 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

longer limited by downstream channel capacities and instead 

are limited only by the historical maximum flow. This rule 

holds even when the current reservoir level is below the TDL:

Qout=minimum(Q10day , RV–IRV), 

if RL>IRL and RS<IV30d

(4)

where

Q10day is the 10-day maximum measured outflow 

from the reservoir, in acre-feet.

A description of each of these critical thresholds for each 

of the Canadian reservoirs is given in table 3 and shown in 

figure 6.

Similar to the releases from Canadian reservoirs, releases 

from Lake Darling are based on an IRL and the use of TDL 

(International Joint Commission, 1989, Annex A, plate A–4); 

however, release guidelines for Lake Darling change according to time of year (International Joint Commission, 1989). 

If the reservoir level is below the IRL but above the TDL, 

then during the months of March, April, and May, reservoir 

releases are capped at channel capacity (2,500 ft3/s, or about 

23,500 acre-ft for a 10-day period); for the other months, 

reservoir releases are capped at 500 ft3/s. However, once the 

reservoir level is above the IRL, then the channel capacity is 

considered to be 2,500 ft3/s, no matter the month. For Lake 

Darling, the IRL was set at the full supply level (FSL) of 

1,597 ft given the elevation/capacity curves for this reservoir 

included values above FSL, unlike Boundary Reservoir, which 

only provided elevation/capacity values up to the FSL in 

Annex A. Additionally, the TDL for Lake Darling is determined through looking at uncontrolled 30-day streamflow 

volumes at the Souris River near Sherwood, N. Dak. (USGS 

streamgage 05114000, U.S. Geological Survey, 2017; fig. 1; 

hereafter referred to as “Sherwood Crossing”), and the use of 

Annex A (International Joint Commission, 1989; Annex A, 

plate A–4). Uncontrolled volume at Sherwood Crossing is the 

volume of runoff that cannot be controlled by the available 

flood control storage (International Joint Commission, 1989).

Streamflow Routing

After determining the releases from the Canadian reservoirs, based on forecasted 30-day inflow volumes and the IRL, 

releases were routed down to Sherwood Crossing using the 

same 10-day streamflow routing model presented in Kolars 

and others (2016) with the lags represented by the fractions 

of current and previous 10-day streamflows and streamflow 

loss to evaporation and aquifer recharge shown in table 4. 

After flows are routed to Sherwood Crossing, routed flow 

volume and the target flow (TF) volume are compared. The 

TF volume was determined using forecasted natural 30-day 

streamflow volumes at Sherwood Crossing and Annex A 

(International Joint Commission, 1989; Annex A, plate A–5). 

The forecasted natural 30-day streamflow volume is the flow 

that would have happened if the reservoirs were not in place; 

these natural streamflow volumes were produced from the 

natural streamflow routing model in Kolars and others (2016). 

If the current routed streamflow volume (considering regulation) is greater than the TF volume, then the excess (that above 

the TF volume) is stored back in the reservoirs by reducing 

releases. When placing the excess volume back in the reservoirs, the reservoir with the largest available storage is filled 

first. During a flood, the maximum amount retained in the 

reservoir is equivalent to either the quantity of water required 

to fill the reservoir to the top of its maximum allowable flood 

level or the amount of water that had been released from the 

reservoir—whichever is less. If there is still excess flow to 

be retained, then the reservoir with the next largest available 

Table 4. Lags applied to 10-day streamflow routing model (from Kolars and others, 2016). 

[NA, not available because of lack of reconstructed streamflow data] 

Streamgage 

identifier 

(fig. 1)

Streamgage name State or 

Province

Fraction of 

current 

10-day 

streamflow

Fraction of 

previous 

10-day 

streamflow

Streamflow 

loss, 

in percent

05ND004 Moose Mountain Creek near Oxbow Saskatchewan 1.0 0 0

05NA003 Long Creek at western crossing of international boundary Saskatchewan 1.0 0 0

05113600 Long Creek near Noonan North Dakota 0.80 0.20 5

05NB001 Long Creek near Estevan Saskatchewan NA NA NA

05NB036 Souris River below Rafferty Reservoir Saskatchewan 0.80 0.20 5

05114000 Souris River near Sherwood North Dakota 0.90 0.10 0

05116000 Souris River near Foxholm North Dakota 1.0 0 0

05116500 Des Lacs River at Foxholm North Dakota 1.0 0 0

05117500 Souris River above Minot North Dakota 0.90 0.10 0
Stochastic Regulated Streamflow Model 11

storage is filled in a similar manner as the first. After most, if 

not all, excess flow at Sherwood Crossing has been retained 

in the Canadian reservoirs, the updated releases from the 

reservoirs are routed again using the same streamflow routing model in Kolars and others (2016). If there is still excess 

after the last reservoir has been filled to its maximum allowable flood level, then the excess is allowed to flow through at 

Sherwood Crossing.

The updated flow volumes at Sherwood Crossing then are 

routed, using the streamflow routing model from Kolars and 

others (2016), and fed into Lake Darling as inflows. Lake Darling outflows (determined using the TDL and IRL as described 

in the “Reservoir Operation” section) then are routed down to 

Minot, N. Dak., using the same streamflow routing model as 

in Kolars and others (2016) but using Lake Darling outflows 

in place of natural streamflows at Souris River near Foxholm, 

N. Dak. (USGS streamgage 05116000; fig. 1). After streamflows have been routed down to Minot, N. Dak., streamflow 

volumes in excess of the channel capacity (5,000 ft3/s, or 

about 98,000 acre-ft for a 10-day period) are reduced by taking the excess volume and storing it back in Lake Darling. 

If Lake Darling is at its maximum allowable flood level, the 

excess is stored in the Canadian reservoirs following the same 

format as storing excess flows at Sherwood Crossing. If there 

still is excess flow after all the reservoirs have been filled, 

then the excess is allowed to flow through Minot in excess of 

5,000 ft3/s. A schematic of the streamflow routing model is 

presented in figure 7, where (1) releases from the Canadian 

reservoirs are determined; (2) reservoir releases and intervening flows (according to the natural 10-day flow routing model 

described in Kolars and others [2016]) are routed down to 

Sherwood Crossing; (3) control point at Sherwood Crossing 

where routed flows from step 2 are compared with target flows 

provided in Annex A (International Joint Commission, 1989, 

plate A–5) and adjusted according to the description given 

in the “Streamflow Routing” section of this report; (4) flows 

from Sherwood Crossing are routed to Lake Darling; (5) based 

on routed flows into Lake Darling and the TDL, Lake Darling 

Reservoir releases are determined; (6) Lake Darling releases 

are routed to Minot, N. Dak.; and lastly (7) routed flows to 

Minot, N. Dak., are compared with the channel capacity of 

5,000 ft3/s and adjusted according to the description given 

previously in the “Streamflow Routing” section of this report.

Alameda

Rafferty

Boundary

Sherwood Crossing

Lake Darling

Minot, North Dakota

Canada

United States

Reservoir

Control point

Gaged tributary

Number represents the order 

in which reservoir releases 

and streamflows are 

determined and routed

1

1

1

1

5

2

3

4

6

7

EXPLANATION

Figure 7. Regulated reservoir 

storage/streamflow routing 

model and the order in which 

streamflows are determined 

and routed down to Minot, 

North Dakota (Souris River 

above Minot, North Dakota; 

U.S. Geological Survey 

streamgage 05117500).
12 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

Regulated Reservoir Storage/Streamflow 

Routing Model Results

A comparison of measured and simulated 10-day reservoir volumes for Rafferty, Alameda, and Boundary Reservoirs and Lake Darling is shown in figures 8, 9, 10, and 11, 

respectively. There was generally good agreement between 

the measured and simulated data for the four reservoirs. The 

correlation coefficients (r) between measured and simulated 

reservoir volumes (table 5) ranged from r=0.64 for Alameda 

Reservoir to r=0.96 for Boundary Reservoir, and the root 

mean square error (RMSE) ranged from 4,412 acre-ft for 

Boundary Reservoir to 33,902 acre-ft for Rafferty Reservoir. 

For reservoir releases (table 6), the correlation coefficients 

ranged from r=0.66 for Alameda Reservoir to r=0.99 for 

Boundary Reservoir, and the RMSE ranged from 3,525 acreft for Alameda Reservoir to 23,640 acre-ft for Lake Darling. 

The larger RMSE of the reservoir releases for Lake Darling 

probably result from larger differences between actual and 

forecasted 30-day inflow volumes for Lake Darling compared 

to the other reservoirs. For Lake Darling, Annex A (International Joint Commission, 1989) uses 30-day uncontrolled 

flows at Sherwood, N. Dak., instead of inflows at the reservoir structure. Given that measured and simulated reservoir 

levels at Lake Darling remained below the IRL for most of 

the calibration period (1992–2011), the amount released 

from the reservoir depended on the TDL, which depends on 

streamflows at the Sherwood crossing. Scenarios A and B in 

figure 6 represent the reservoir operation situation for Lake 

Darling during most of the calibration period. The lack of fit, 

with respect to replicating reservoir volumes, was somewhat 

expected given operation of Lake Darling depends on many 

factors, including wildlife management, flood protection, 

agriculture, and water supply. When comparing measured and 

simulated 10-day releases from the Canadian reservoirs, simulated values seemed to follow the same pattern as measured 

values, and periods of high and low flow were observed in the 

measured and simulated record, indicating that during high 

flow periods, using TDL partially compensated for the errors 

between measured and simulated reservoir volumes (figs. 8, 9, 

10, and 11).

After streamflow was routed through the regulated 

reservoir storage/streamflow routing model and excess flows 

were placed back in the reservoirs, simulated 10-day streamflows at Minot, N. Dak., were observed to visually follow the 

same pattern as historical flows (fig. 12) and had a correlation 

coefficient of 0.91 and RMSE of 1,184 ft3/s from 1992 through 

2011. Similarly, simulated annual streamflow volumes at 

Minot, N. Dak., followed the same pattern as historical gaged 

(measured) volumes with overpredictions during the flood 

years of 1999, 2009, and 2011 and underpredictions during the 

flood years of 1995, 1996, 2005, and 2010 (fig. 13).

Methods for Developing the Stochastic 

Regulated Streamflow Model

The regulated reservoir storage/flow routing model 

described earlier in this report was combined with the stochastic natural streamflow model described in Kolars and others 

(2016) to provide a stochastic model for simulating regulated 

streamflow for the upper Souris River Basin. The simulated 

regulated streamflow data from the stochastic model were used 

to evaluate future flood risk. The methods used to simulate 

regulated streamflow and evaluate flood risk are described in 

this section, and flood risk results for the Souris River above 

Minot, N. Dak. (USGS streamgage 05117500) (fig. 1) are 

provided in the “Stochastic Simulation Results” section.

In the previous study, Kolars and others (2016) stochastically generated time series of potential future 10-day 

mean inflow volumes to the upstream reservoirs (Boundary, Alameda, and Rafferty), and incremental flow volumes for subbasins downstream from the reservoirs and 

upstream from the Souris River above Minot, N. Dak. (USGS 

streamgage 05117500) (fig. 1), were used as inputs to a natural 

(unregulated) streamflow routing model to generate time series 

of natural 10-day mean streamflow for streamgages at the terminus of each subbasin. The same generated reservoir inflow 

volumes and downstream incremental flow volumes from 

Kolars and others (2016) were used as inputs to the reservoir 

storage/flow routing model described in the “Methods for 

Developing the Regulated Reservoir Storage/streamflow Routing Model” section of this report to generate time series of 

regulated 10-day mean streamflow for the same streamgages. 

These generated time series are referred to as “traces.” There 

were 100 traces generated for each streamgage, each 100 years 

long. As in the previous report, starting conditions (initial 

reservoir storage volumes and soil moisture storage conditions) for each trace were generated at random by discarding 

values for a 10-year initialization period at the beginning of 

each trace. Each trace assumes 50 years of wet (similar to 

1970–2017) climatic conditions followed by 50 years of dry 

(similar to 1912–69) climatic conditions; thus, there are a total 

of 10,000 simulation years (5,000 for the wet period and 5,000 

for the dry period). Note that although each trace is generated assuming a 50-year wet period followed by a 50-year dry 

period, no assumption is made regarding when the current wet 

state (beginning in 1970) actually will end. Furthermore, using 

a 100-year simulation period is not related to the expected 

design life of potential flood control projects being considered. 

The lengths of the wet and dry periods simply were chosen to 

represent long enough periods to provide an ample number of 

simulation years for estimating flood magnitudes corresponding to small annual exceedance probabilities (for example, 

1 percent or less).

An example of the natural (unregulated) and regulated 

10-day mean flows for the Souris River above Minot, N. Dak. 

(USGS streamgage 05117500), for one of the traces (trace 

number 23) is shown in figure 14. The maximum recorded 

10-day mean streamflow for this streamgage for the historical 
Stochastic Regulated Streamflow Model 13

Volume, in acre−feet

1997 1998 1999 2000 2001 2002 2004 2003 2005 2006 2007 2008 2009 2010 2011 2012

0

100,000

200,000

300,000

400,000

500,000

Ideal reservoir level

Maximum allowable flood level

Low flow outlet maximum release

Year

Measured reservoir volume

Simulated reservoir volume

Measured reservoir release volume

Simulated reservoir release volume

EXPLANATION

Figure 8. Comparison of measured and simulated reservoir volumes and releases (outflows) for Rafferty Reservoir from 1997 (when 

the reservoir first filled to full supply level) through 2011. Volume, in acre−feet

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

0

50,000

100,000

150,000

200,000

Ideal reservoir level

Maximum allowable flood level

Channel capacity

Year

Measured reservoir volume

Simulated reservoir volume

Measured reservoir release volume

Simulated reservoir release volume

EXPLANATION

Figure 9. Comparison of measured and simulated reservoir volumes and releases (outflows) for Alameda Reservoir from 1999 (when 

the reservoir first filled to full supply level) through 2011.
14 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota Volume, in acre−feet

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

0

20,000

40,000

60,000

80,000

100,000

Ideal reservoir level

Full supply level

Channel capacity (reservoir outlet plus diversion canal)

Year

Measured reservoir volume

Simulated reservoir volume

Measured reservoir release volume

from both the reservoir outlet and

diversion canel

Simulated reservoir release volume

from both the reservoir outlet and

diversion canel

EXPLANATION

Figure 10. Comparison of measured and simulated reservoir volumes and releases (outflows) for Boundary Reservoir from 1992 

through 2011. Volume, in acre−feet

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

Ideal reservoir volume

Maximum allowable flood volume

Channel capacity

Year

Measured reservoir volume

Simulated reservoir volume

Measured reservoir release volume 

Simulated reservoir release volume 

EXPLANATION

Figure 11. Comparison of measured and simulated reservoir volumes and releases (outflows) for Lake Darling from 1992 through 2011.
Stochastic Regulated Streamflow Model 15

Table 5. Correlation coefficients and root mean square error 

between measured and simulated 10-day mean reservoir volumes 

for Rafferty, Alameda, and Boundary Reservoirs and Lake Darling.

[RMSE, root mean square error]

Reservoir

Maximum 

allowable 

flood level, 

in acre-feet

Correlation 

coefficient 

RMSE, 

in acrefeet

Rafferty (1997–2011) 513,000 0.87 33,902

Alameda (1999–2011) 153,710 0.64 8,783

Boundary (1992–2011) 49,800 0.96 4,412

Lake Darling (1992–2011) 158,600 0.77 16,445

Table 6. Correlation coefficients and root mean square error 

between measured and simulated 10-day mean reservoir releases 

for Rafferty, Alameda, and Boundary Reservoirs and Lake Darling.

[RMSE, root mean square error]

Reservoir

Channel 

capacity, in 

acre-feet

Correlation 

coefficient 

RMSE, 

in acrefeet

Rafferty (1997–2011) 9,917

1

45,531

0.88 12,487

Alameda (1999–2011) 35,703 0.66 3,525

Boundary (1992–2011) 17,851 0.99 5,782

Lake Darling (1992–2011) 49,587 0.90 23,640

1

In the model, the capacity of Rafferty Reservoir’s low-flow outlet was used 

in place of the downstream reservoir channel capacity.

period of record (1904–2017) was in 2011 and was about 

18,000 ft3/s (U.S. Geological Survey, 2017). For this particular 

trace, there were 3 years during the wet period and 1 year during the dry period when the simulated natural flow exceeded 

the 2011 record. Regulation generally resulted in a large 

reduction in the annual maximum flow compared to unregulated flow except for a single year during the wet period for 

which the regulated maximum flow was about twice as high 

as the 2011 record. An example of another trace (number 66) 

is shown in figure 15. For this trace, there were 3 years during the wet period for which unregulated flow exceeded the 

2011 record compared to no exceedances for the dry period. 

When looking at regulated flows, there was 1 year during 

the wet period that was slightly higher than the 2011 record. 

Note that because extremely high 30-day inflow volumes to 

the upstream reservoirs in the spring of 2011 caused reservoir 

outflows to be at maximum allowable levels, the maximum 

unregulated flow for 2011 would have been similar to the 

recorded, historical maximum regulated flow during 2011. 

However, as indicated by trace 66, there can be simulation 

years when the maximum unregulated flows exceeded the 

2011 value, but the maximum regulated flow was reduced 

substantially compared to the unregulated flow because the 

30-day inflow volumes and corresponding reservoir releases 

for one or more of the upstream reservoirs were less than the 

2011 values.

The annual maximum 10-day mean streamflow was computed for each of the 10,000 simulation years and for unregulated and regulated conditions. The resulting annual maxima 

were used to estimate flood magnitudes associated with specified annual exceedance probabilities, which is called a flood 

frequency analysis. A flood frequency analysis was completed 

assuming three conditions:

• Condition A, wet equilibrium.—Flood frequency 

estimates based on the first 50 years of each trace 

(5,000 simulation-years). This condition represents 

flood risk for the remaining duration of the current 

(1970–2017) wet period.

• Condition B, transition.—Flood frequency estimates 

based on years 26–75 of each trace (5,000 simulation 

years). This condition represents flood risk during the 

transition period between the end of the current wet 

period and start of the subsequent dry period.

• Condition C, dry equilibrium.—Flood frequency estimates based on years 51–100 of each trace (5,000 simulation years). This condition represents flood risk 

during a prolonged period of dry conditions (similar to 

1912–69).

It is not possible to predict when the current (1970–2017) 

wet climatic conditions may revert back to drier conditions similar to 1912–69. The flood frequency curves for the 

three conditions are meant to bracket the range of reasonable 

possibilities. Based on climate modeling and paleolake-level 

information for the Devils Lake Basin (just east of the Souris 

Basin in northeastern North Dakota), Vecchia (2011) determined that, starting from any given initial year during a wet 

state (such as 2016), there was about a 70-percent chance of 

the wet climatic conditions continuing at least 10 more years 

(at least until 2026) and about a 40-percent chance of the wet 

climate state continuing at least 30 more years (at least until 

2046). Furthermore, wet soil moisture conditions and a tendency for high runoff in the basin may persist for as much as 

10 years after the transition back to drier climatic conditions. 

Therefore, mean annual flood risk during the next 25 years 

(2016–40) may be represented best by climatic condition A, 

which represents equilibrium conditions for the wet (similar 

to 1970–2017) climate state. The curve for climatic condition B represents mean annual flood risk for the next 50 years 

(2016–65) under a hypothetical scenario in which the wet 

conditions persist for an additional 25 years (2016–40) before 

reverting back to dry conditions (similar to 1912–69) for the 

next 25 years (2041–65). However, the curve for climatic 

condition B would tend to underestimate flood risk during the 

earlier years (2016–40) and overestimate flood risk during the 
16 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota 10−day mean streamflow, in cubic feet per second

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

0

5,000

10,000

15,000

Channel capacity

Year

Measured

Simulated

EXPLANATION

Figure 12. 10-day mean measured and simulated streamflow at Souris River above Minot, North Dakota (U.S. Geological 

Survey streamgage 05117500), 1992–2011.

Year

Flow volume, in acre−feet

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

0

500,000

1,000,000

1,500,000

2,000,000

Measured

Simulated

EXPLANATION

Figure 13. Annual measured and simulated streamflow volumes at Souris River above Minot, North Dakota (U.S. Geological 

Survey streamgage 05117500), 1993–2011.
Stochastic Regulated Streamflow Model 17

40,000

35,000

30,000

25,000

20,000

15,000

10,000

5,000

0

40,000

35,000

30,000

25,000

20,000

15,000

10,000

5,000

0

0 10 20 30 40 50 60 70 80 90 100

Simulation year

A. Natural (unregulated) streamflow

B. Regulated streamflow

Trace 23

Natural 10-day mean streamflow,

in cubic feet per second Regulated 10-day mean streamflow, in cubic feet per second

Wet period Dry period

Wet period Dry period

Historical maximum (2011)

Historical maximum (2011)

Figure 14. Simulated 10-day mean streamflow for 100-year simulation period (50 years of wet climate state followed by 50 years 

of dry climate state) for the Souris River above Minot, North Dakota (U.S. Geological Survey streamgage 05117500) (stochastic 

trace 23). A, natural (unregulated) flow; and B, regulated flow.
18 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

40,000

35,000

30,000

25,000

20,000

15,000

10,000

5,000

0

40,000

35,000

30,000

25,000

20,000

15,000

10,000

5,000

0

0 10 20 30 40 50 60 70 80 90 100

Simulation year

A. Natural (unregulated) streamflow

B. Regulated streamflow

Trace 66

Natural 10-day mean streamflow,

in cubic feet per second Regulated 10-day mean streamflow, in cubic feet per second

Wet period Dry period

Wet period Dry period

Historical maximum (2011)

Historical maximum (2011)

Figure 15. Simulated 10-day mean streamflow for 100-year simulation period (50 years of wet climate state followed by 50 years 

of dry climate state) for the Souris River above Minot, North Dakota (U.S. Geological Survey streamgage 05117500) (stochastic 

trace 66). A, natural (unregulated) flow; and B, regulated flow.
Stochastic Regulated Streamflow Model 19

later years (2041–65) and thus may not represent the actual 

flood risk very well in either period. The curve for climatic 

condition C represents mean annual flood risk during the next 

50 years (2016–65) under a hypothetical scenario in which 

drier climatic conditions return immediately (2016) and persist 

for at least 50 years, and thus would even more severely 

underestimate flood risk during the earlier years (2016–40) 

compared to climatic condition B.

Flood frequency estimates for design of levees and 

other critical infrastructure are based on the annual maximum 

instantaneous flow, or annual peak streamflow, whereas the 

simulations described in the previous paragraph are based on 

the annual maximum 10-day mean streamflow. Flood frequency values for a generic year are defined as follows:

Prob[APF>APFp]=Prob[AM10DF>AM10DFp]=p/100, (5)

where

APF is annual peak streamflow, in cubic feet per 

second;

AM10DF is annual maximum 10-day mean streamflow, 

in cubic feet per second;

Prob[.] denotes probability;

p is a specified exceedance probability, in 

percent;

APFp is the flood magnitude for annual peak 

streamflow corresponding to exceedance 

probability p, in cubic feet per second; and

AM10Dp is flood magnitude for annual maximum 

10-day mean streamflow corresponding to 

exceedance probability p, in cubic feet per 

second.

The annual peak streamflow (APF) is always greater than 

annual maximum 10-day mean streamflow (AM10DF); consequently, the flood magnitude for annual peak streamflow 

corresponding to exceedance probability p (APFp ) will be 

greater than the flood magnitude for annual maximum 10-day 

mean streamflow corresponding to exceedance probability 

p (AM10Dp ) for all values of p. An empirical method was 

therefore used to increase estimates of AM10Dp

 to “equivalent” estimates for APFp

. The empirical method is based on 

the percent difference between recorded values of APF and 

AM10DF:

PD=100(APF–AM10DF)/AM10DF, (6)

where

PD is the percent difference between APF and 

AM10DF.

The relations between AM10DF and the percent difference between the annual peak streamflow and the annual 

maximum 10-day mean streamflow (PD) for the Souris 

River above Minot, N. Dak. (USGS streamgage 05117500, 

fig. 1) and the Souris River near Sherwood, N. Dak. (USGS 

streamgage 05114000, fig. 1) are shown in figure 16. Only 

points for which AM10DF was greater than or equal to 

1,000 ft3/s were used in this analysis. The historical period of 

record is 1937–2016 for Sherwood Crossing and 1904–2016 

for the USGS streamgage Souris River above Minot, N. Dak. 

(U.S. Geological Survey, 2017). The lines show the fitted 

geometric mean (an estimate of the median PD as a function 

of AM10DF) for each streamgage. The equations for each line 

were determined by fitting a weighted simple linear regression 

model (Helsel and Hirsch, 2002) for logarithm-transformed 

PD given logarithm-transformed AM10DF for the respective 

streamgages. The equation for the Souris River above Minot, 

N. Dak., (USGS streamgage 05117500) is given by

FV[log(PD)]=0.82+0.56 {log(AM10DF)–3}; 

AM10DF≥1,000 ft3/s (7)

and the equation for the Souris River near Sherwood, N. Dak., 

(USGS streamgage 05114000) is given by 

FV[log(PD)]=1.24+0.45 {log(AM10DF)–3}; 

AM10DF≥1,000 ft3/s (8)

where

FV[log(PD)] is the fitted value of logarithm-transformed 

percent difference; and

 log( ) is the base-10 logarithm.

Weighted least-squares regression (Helsel and Hirsch, 2002) 

was used to estimate the coefficients in equations 7 and 8 

because the variance of the regression residuals decreased as 

AMD10F increased. The weights were the same for both equations and were given by 

W=1+[log(AM10DF)–3]2 (9)

where 

W is the weight used for the weighted leastsquares regression.

For example, the weight for AM10DF=10,000 ft3

/s (W=2) is 

twice as large as the weight for AM10DF=1,000 ft3

/s (W=1). 

Although there was a lot of scatter in the points, there was 

a highly significant (p-value less than 0.01) increase in the 

geometric mean for both streamgages. Based on the good 

agreement with recorded flows for years with AM10DF>2,000 

ft3

/s (regardless of regulated conditions), the curves provided 

a reasonable approximation to PD for AM10DF>2,000 ft3

/s. 

Because of the similarity between the curves for the two 

streamgages and the apparent temporal consistency of the 

relations despite differing regulated conditions during the 

historical period, it was assumed that the relation between 

PD and AM10DF is not affected by regulated conditions for 

AM10DF>2,000 ft3

/s. 

Estimated flood frequency curves for the USGS 

streamgage Souris River above Minot, N. Dak., for specified exceedance probabilities were computed as follows. The 

estimated flood magnitude for AM10DF (AM10DFp

 in eq. 5) 

is equal to the 100(1−p/100)th percentile of the simulated 

annual maxima for all 5,000 simulation years (for each of the 
20 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

0

20

40

60

80

100

120

Percent difference between annual peak streamflow

and annual maximum 10−day mean streamflow

 1,000 2,000 5,000 10,000 20,000

Annual maximum 10−day mean streamflow, in cubic feet per second

Sherwood, 1937–2016

Minot, 1937–2016

Minot, 1904–36

Fitted geometric mean

Fitted geometric mean

EXPLANATION

Figure 16. Difference between recorded annual peak streamflow and recorded annual maximum 10-day mean streamflow, 

as a percent of annual maximum 10-day mean streamflow, for the streamgages Souris River above Minot, North Dakota (U.S. 

Geological Survey streamgage 05117500), and Souris River near Sherwood, North Dakota (U.S. Geological Survey streamgage 

05114000). [Period of record 1937–2016 for Sherwood streamgage and 1904–2016 for Minot streamgage; points for which annual 

maximum 10-day mean streamflow is less than 1,000 cubic feet per second are not shown.]

climatic conditions A–C). The estimated flood magnitudes for 

AM10DF for regulated conditions were adjusted using equation 7 to obtain estimated flood magnitudes for APF, namely

APFp=AM10DFp (1+(0.01)10[0.82+0.56 {log(AM10DFp)–3}]) 

AM10DFp≥1,000 ft3/s. (10)

The flood frequency analysis results for all three conditions 

are described in the following section.

Stochastic Simulation Results

Estimated flood frequency curves for both unregulated and regulated annual maximum 10-day mean 

streamflow for the Souris River above Minot, N. Dak. 

(USGS streamgage 05117500), and exceedance probabilities less than 10 percent are shown in figure 17. For all three 

potential future climatic conditions, there were large reductions in the curves for regulated streamflow compared to 

unregulated streamflow. The largest reductions were for 

moderate (5-percent to 1-percent) exceedance probabilities; 

for example, for the 3-percent exceedance probability and 

condition A (wet equilibrium), the unregulated flood magnitude was about 16,000 ft3

/s, compared to about 6,000 ft3/s for 

the regulated flood magnitude, a reduction of about 60 percent. 

The reductions were less (in relative terms) for the extreme 

(less than 1-percent) exceedance probabilities. For example, 

for the 0.2-percent exceedance probability and condition A, 

the unregulated flood magnitude was about 38,000 ft3/s compared to about 21,000 ft3

/s for the regulated flood magnitude, a 

reduction of about 45 percent. A comparison among the regulated curves for the three future climatic conditions indicates 

that the regulated curves were most sensitive to the future 

climatic conditions for the more extreme exceedance probabilities. For example, for the 1-percent exceedance probability, 

the regulated flood magnitude for condition A (wet equilibrium) was about 12,500 ft3

/s compared to about 7,000 ft3/s for 

condition C (dry equilibrium), a reduction of about 44 percent.

The estimated regulated frequency curves for annual 

maximum 10-day flows shown in figure 17 were adjusted 

using equation 10 to obtain estimated regulated frequency 

curves for annual peak streamflow. The resulting curves are 

shown in figure 18. For comparison purposes, the 11 largest 

recorded annual peak streamflows for the Souris River above 

Minot, N. Dak. (USGS streamgage 05117500; U.S. Geological 

Survey, 2017), for 1904–2016 are plotted as well. The plotting 

positions for the recorded peaks use the empirical (Weibull) 

exceedance probabilities p=100 K/(n+1), where K denotes the 

Kth largest recorded peak flow and n=113 is the record length 

(Helsel and Hirsch, 2002). Note that the recorded annual peak 
Stochastic Regulated Streamflow Model 21

 2,000

 5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

Annual maximum 10−day mean streamflow, in cubic feet per second 

10 9 8 7 6 5 4 3 2 1 0.8 0.6 0.5 0.4 0.3 0.2

Annual exceedance probability, in percent

Unregulated Regulated

Scenario A

Scenario B

Scenario C

EXPLANATION

Figure 17. Estimated flood frequency curves for annual maximum 10-day mean streamflow for the Souris River above Minot, North 

Dakota (U.S. Geological Survey streamgage 05117500), for natural (unregulated) and regulated conditions and three climatic conditions. 

Scenario A, (wet equilibrium) representing wet (similar to 1970–2017) climatic conditions; scenario B, (transition) representing transition 

from wet to dry (similar to 1913–69) climatic conditions; and scenario C, (dry equilibrium) representing dry climatic conditions.

 2,000

 5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

Annual peak streamflow, in cubic feet per second

10 9 8 7 6 5 4 3 2 1 0.8 0.6 0.5 0.4 0.3 0.2

Annual exceedance probability, in percent

1923

1974

2013

1927

1916

1975 1979

1969

1976

1904

2011 Regulated

Scenario A

Scenario B

Scenario C

EXPLANATION

Figure 18. Estimated flood frequency curves for annual instantaneous peak streamflow for the Souris River above Minot, North 

Dakota (U.S. Geological Survey streamgage 05117500), for regulated conditions and three future climatic conditions. Scenario A, (wet 

equilibrium) representing wet (similar to 1970–2017) climatic conditions; scenario B, (transition) representing transition from wet to dry 

(similar to 1913–69) climatic conditions; and scenario C, (dry equilibrium) representing dry climatic conditions. [The 11 largest recorded 

peak flows for 1904–2017, regardless of regulated conditions, are plotted using empirical (Weibull) probability plotting positions. Note 

that the regulated frequency curves assume future reservoir operation is the same as current (1991–2017) operation.]
22 Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota

streamflows are shown regardless of past regulated conditions 

and have not been adjusted to represent current (1991–2017) 

regulation, whereas the estimated regulated curves are assuming current regulated conditions. For example, although the 

recorded annual peak streamflow for 1904 was the second 

highest peak of record, if the current reservoirs were in place 

during 1904, that point would be shifted down (and possibly 

to the left, depending on where it ranked among the other peak 

streamflows). The peak streamflow for 1976 also would be 

shifted down (and possibly to the left) had Rafferty and Alameda Dams been in place at that time. Only 2 of the 11 highest peak streamflows (2011 and 2013) happened after all the 

reservoirs were in place. The 2011 peak streamflow plots well 

above even the condition A curve, indicating that it was an 

extreme flood (exceedance probability well below 1 percent) 

even in context of the recent wet conditions.

As described previously in the “Review of Stochastic 

Natural (Unregulated) Streamflow Model” section of this 

report, there is believed to have been a transition in about 

1970 from a period of drier climatic conditions to wetter climatic conditions, resulting in a higher risk of extreme floods, 

similar to the 2011 flood, since 1970 compared to earlier years 

(1912–69) (Kolars and others, 2016; Nustad and others, 2016). 

Depending on how long the wetter conditions may persist, 

flood risk for the next 25 years (through about 2040), and perhaps longer, may be represented best by scenario A. If a transition to the drier climate state happens in about 25 years, then 

mean annual flood risk for the next 50 years (that is, through 

about 2065) may be more indicative of scenario B. However, 

the curve for scenario B will underestimate flood risk for the 

earlier years (until the transition happens) and overestimate 

flood risk for the later years and may not represent the actual 

flood risk very well in either period. The curve for scenario C 

likely will underestimate the true flood risk regardless of when 

the transition may happen.

Estimated peak flow magnitudes for selected exceedance probabilities and all three future conditions are given in 

table 7. For condition A (wet equilibrium), which probably 

best represents flood risk for at least the next 25 years (through 

about 2040), a 10-percent-per-year chance of regulated 

streamflow exceeding 6,000 ft3/s is estimated for the Minot, 

N. Dak., streamgage, a 1-percent-per-year chance of regulated streamflow exceeding 15,700 ft3

/s is estimated, and a 

0.2-percent-per-year chance of regulated streamflow exceeding 29,300 ft3

/s is estimated. The recorded peak streamflow in 

2011 was 26,900 ft3/s, which is similar to the 0.2-percent-peryear chance estimated peak streamflow magnitude for condition A (29,300 ft3

/s) and nearly twice as large as the 0.2-percent-per-year chance estimated peak streamflow magnitude 

for condition C (14,800 ft3/s) (fig. 18; table 7). The estimated 

1-percent annual exceedance probability flood magnitude was 

8,800 ft3/s for condition C (dry equilibrium).

Summary

The Souris River Basin is a 24,000-square-mile basin 

in the Provinces of Saskatchewan and Manitoba in Canada 

and the State of North Dakota in the United States. Aboveaverage snowpack during the winter of 2010–11, along with 

record-setting rains in May and June of 2011, led to record 

flooding that caused extensive damage to Minot, N. Dak., and 

numerous smaller communities in Saskatchewan, Manitoba, 

and N. Dak. As a result, the International Souris River Board 

created the Souris River Flood Task Force to evaluate potential reservoir operation changes and flood control measures 

to manage future floods and droughts. Part of this evaluation 

involved identifying a need for a stochastic model to simulate 

streamflow in the Souris River Basin upstream from Minot, 

Table 7. Estimated peak streamflow magnitudes for the Souris River above Minot, North Dakota (U.S. 

Geological Survey streamgage 05117500), for annual exceedance probabilities of 10, 5, 2, 1, 0.5, and 0.2 

percent and three climatic conditions.

[A, (wet equilibrium) representing wet (similar to 1970–2015) climatic conditions; B, (transition) representing transition from wet to dry (similar to 1913–69) climatic conditions; C, (dry equilibrium) representing dry climatic conditions; 

estimates assume current (1991–2017) reservoir operation]

Annual 

exceedance 

probability, 

in percent

Estimated peak streamflow magnitude, 

in cubic feet per second

Condition A 

(wet equilibrium)

Condition B 

(transition)

Condition C 

(dry equilibrium)

10 5,800 to 6,000 4,500 to 4,400 3,000 to 2,800

5 7,400 to 8,000 6,500 to 7,100 5,600 to 5,700

2 11,000 to 11,300 8,700 to 9,100 7,500 to 7,800

1 15,600 to 15,700 10,900 to 11,100 8,700 to 8,800

0.5 20,800 to 21,000 15,100 to 15,200 10,500 to 10,600

0.2 29,100 to 29,300 23,500 to 23,000 14,700 to 14,800
References Cited 23

North Dakota, to estimate the likelihood of future flooding or 

drought; this need was partly addressed through developing a 

stochastic model to simulate natural (unregulated) streamflow 

in the Souris River Basin as described in a previous report.

The stochastic natural streamflow model was built upon 

through this report to include the effects of regulation from 

four reservoirs (Rafferty, Alameda, and Boundary Reservoirs 

and Lake Darling) which are operated according to their 

operational guidelines. A regulated reservoir storage/streamflow routing model was first developed and calibrated for the 

period when all four reservoirs were in operation until the end 

of the reconstructed natural streamflow dataset provided by 

the U.S. Army Corps of Engineers (1992–2011). Measured 

and simulated reservoir volumes produced through the regulated reservoir storage/streamflow routing model compared 

well with correlation coefficients of 0.87 (Rafferty Reservoir), 

0.64 (Alameda Reservoir), 0.96 (Boundary Reservoir), and 

0.77 (Lake Darling). Additionally, measured and simulated 

10-day releases (outflows) from all four reservoirs had similar 

peaks and low points, as did 10-day mean streamflows at the 

streamgage Souris River above Minot, N. Dak. The regulated reservoir storage/streamflow routing model then was 

combined with the previously published stochastic natural 

streamflow model to provide a stochastic model for simulating 

regulated streamflow for the upper Souris River Basin.

The stochastic regulated streamflow model can be used to 

estimate regulated flood frequency curves and provide useful 

information for feasibility and design of critical structures 

such as levees or bridges. There was a transition in about 1970 

from a period of drier (similar to 1912–69 climatic conditions) 

to a period of wetter (similar to 1970–2017) climatic conditions in the Souris River Basin. Therefore, three potential 

future climatic conditions were considered in this analysis: 

condition A (wet equilibrium), representing wet (similar to 

1970–2017) climatic conditions; condition B (transition), 

representing transition from wet to dry (similar to 1912–69) 

climatic conditions; and condition C (dry equilibrium), 

representing dry climatic conditions. For all three climatic 

conditions, there were large reductions in the flood frequency 

curves for regulated flow compared to unregulated, and the 

largest reduction was for moderate (5-percent to 1-percent) 

exceedance probabilities. The reductions were less for the 

extreme (less than 1-percent) exceedance probabilities. A 

comparison among regulated conditions for the three different 

climatic conditions indicated the regulated flood frequency 

curves were highly sensitive to the climatic conditions for the 

more extreme exceedance probabilities (less than 1 percent). 

The estimated 1-percent annual exceedance probability flood 

magnitude for the Souris River upstream from Minot, N. 

Dak., was 15,700 cubic feet per second (ft3/s) for condition A 

(wet equilibrium), compared to 8,800 ft3/s for condition C 

(dry equilibrium). The estimated 0.2-percent annual exceedance probability flood magnitude for the same streamgage 

was 29,300 ft3/s for condition A (wet equilibrium), compared 

to 14,800 for condition C (dry equilibrium). For comparison, 

the recorded peak flow for 2011 for the Souris River upstream 

from Minot, N. Dak., was 26,900 ft3/s. 

Although it is not possible to predict how long the current (1970–2017) wet climatic conditions may persist, flood 

risk for the near future (at least the next 25 years, or until 

about 2040)) may be represented best by climatic condition A, 

which represents equilibrium conditions for the wet (similar to 

1970–2017) climatic conditions. The curve for climatic condition B, which represents a mixture of wet and dry (similar to 

1911–69) climatic conditions, may underestimate flood risk 

for the next two decades (until the transition back to the drier 

climatic conditions happens) and overestimate flood risk for 

the later years and may not represent the actual flood risk 

very well in either period. The curve for climatic condition C, 

which represents equilibrium conditions for the dry climatic 

state, likely will underestimate the true flood risk during the 

next 25 years and perhaps longer depending on when the transition back to the dry climatic conditions may occur.

References Cited

Government of Canada, 2014, Adjusted and homogenized 

Canadian climate data: Government of Canada web page, 

accessed July 15, 2015, at https://ec.gc.ca/dccha-ahccd/

default.asp?lang=En&n=B1F8423A-1.

Government of Canada, 2017, Real-time hydrometric data: 

Government of Canada web page, accessed December 16, 

2017, at https://wateroffice.ec.gc.ca/mainmenu/real_time_

data_index_e.html.

Helsel, D.R., and Hirsch, R.M., 2002, Statistical methods in 

water resources: U.S. Geological Survey Techniques of 

Water-Resources Investigations, book 4, chap. A3, 522 p.

International Joint Commission, 1989, Boundary waters—

Agreement between the Government of Canada and the 

Government of the United States of America for water 

supply and flood control in the Souris River Basin (with 

annexes and Canada/Saskatchewan agreement): Ottawa, 

Queen’s Printer for Canada, 36 p.

International Souris River Board, 2013, Plan of study—For 

the review of the operating plan contained in Annex A of 

the 1989 international agreement between the Government 

of Canada and the Government of the United States: Souris 

River Basin Task Force Report, 197 p.

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For more information about this publication, contact:

Director, USGS Dakota Water Science Center

821 East Interstate Avenue

Bismarck, ND 58503

701–250–7400

For additional information, visit: https://www.usgs.gov/centers/dakotawater

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Kolars and others—Stochastic Model for Simulating Souris River Basin Regulated Streamflow Upstream from Minot, North Dakota—SIR 2018–5155

ISSN 2328-0328 (online)

https://doi.org/10.3133/sir20185155