U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2015–5170

Prepared in cooperation with the Minnesota Pollution Control Agency

Methods for Estimating Flow-Duration Curve and LowFlow Frequency Statistics for Ungaged Locations on Small 

Streams in Minnesota
Cover. Redwood River near Redwood Falls, Minnesota. Photograph by the U.S. Geological Survey.
Methods for Estimating Flow-Duration 

Curve and Low-Flow Frequency Statistics 

for Ungaged Locations on Small Streams in 

Minnesota

By Jeffrey R. Ziegeweid, David L. Lorenz, Chris A. Sanocki, and Christiana R. 

Czuba

Prepared in cooperation with the Minnesota Pollution Control Agency

Scientific Investigations Report 2015–5170

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015

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Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials 

as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner.

Suggested citation:

Ziegeweid, J.R., Lorenz, D.L., Sanocki, C.A., and Czuba, C.R., 2015, Methods for estimating flow-duration curve and 

low-flow frequency statistics for ungaged locations on small streams in Minnesota: U.S. Geological Survey Scientific 

Investigations Report 2015–5170, 23 p., http://dx.doi.org/10.3133/sir20155170.

ISSN 2328-0328 (online)
iii

Contents

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

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

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

Description of Study Area ...................................................................................................................2

Methods for Data Development ..................................................................................................................3

Trend Analyses......................................................................................................................................3

Flow-Duration Curves...........................................................................................................................5

Low-Flow Frequency............................................................................................................................6

Basin Characteristics...........................................................................................................................6

Regression Analysis.............................................................................................................................8

Regional Regression Analysis .....................................................................................................................9

Flow-Duration Curve Regression Equations ....................................................................................9

Low-Flow Frequency Regression Equations....................................................................................9

Drainage-Area Ratio Calculations ............................................................................................................13

Limitations of Regression Equations ........................................................................................................16

Constructing Consistent Flow-Duration Curves .....................................................................................16

StreamStats ..................................................................................................................................................18

Summary........................................................................................................................................................19

Acknowledgments.......................................................................................................................................20

References Cited..........................................................................................................................................20

Appendix 1. Detailed Streamgage Information Used to Conduct Analyses...................................23

Figures

1. Map showing hydrologic regions in Minnesota and streamgages in study area .............4

2. Graph showing flow-duration curve developed using 13 exceedance-probability

quantiles for the South Fork Crow River near Mayer, Minnesota, for water years

1941–79 ...........................................................................................................................................5

3. Graphs showing relation between observed 7-day annual low-flow values and

values estimated using log-Pearson type III, three-parameter lognormal, and the

log-generalized extreme value distributions for computing a 7-day annual

low-flow frequency graph for the South Fork Crow River near Mayer, Minnesota ..........7

4. Graph showing relation between drainage-area ratio and the observed 0.5 flowduration curve exceedance probabilities for the 25 pairs of streamgages listed in 

table 4 ...........................................................................................................................................15

5. Graph showing example of how to construct a consistent flow-duration curve 

from regional regression estimates.........................................................................................18
iv

Tables

1. Basin characteristics tested as explanatory variables in the development of

low-flow regression equations...................................................................................................8

2. Regression equations for estimating selected flow-duration curve exceedance

probabilities for unregulated streams in Minnesota ............................................................10

3. Regression equations for estimating selected low-flow frequency statistics for

unregulated streams in Minnesota..........................................................................................12

4. Paired streamgages selected to develop within-basin equations that use

drainage-area ratios to estimate streamflow statistics at ungaged locations on

streams with a streamgage in another location....................................................................14

5. Standard errors of selected flow-duration curve and low-flow frequency 

statistics based on statewide drainage area-ratio equations developed in 

this study ......................................................................................................................................15

6. Maximum, minimum, and mean values of the basin characteristics used in the

regional regression analysis of each region in this study ...................................................17

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

cubic foot per second (ft3

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

Hydraulic conductivity

foot per day (ft/d) 0.3048 meter per day (m/d)

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

Water year is the 12-month period of October 1 through September 30 and is designated by the 

calendar year in which it ends.

Climate year is the 12-month period of April 1 through March 31 and is designated by the 

calendar year in which it ends.
v

Datums

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Altitude, as used in this report, refers to distance above the vertical datum.

Abbreviations

BSLDEM10M mean basin slope, in feet per foot

CROPSNLCD01 percent area of cultivated land

DAR drainage-area ratio

DRNAREA drainage area, in square miles

FDC flow-duration curve

FOREST percent area of forested land

GIS geographic information system

LAKEAREA percent area of lakes

LFF low-flow frequency

NRCS Natural Resources Conservation Service

PFLATLOW percent mean basin low-lying flatland

PFLATUP percent mean basin upland flatland

PMPE mean basin precipitation minus potential evapotranspiration, in millimeters

PPCC probability plot correlation coefficient

SEE standard error of the estimate

SOILA percent streamgage drainage basin area consisting of hydrologic soil group A

SOILB percent streamgage drainage basin area consisting of hydrologic soil group B

SOILC percent streamgage drainage basin area consisting of hydrologic soil group C

SOILD percent streamgage drainage basin area consisting of hydrologic soil group D

SSURGCLAY percent mean basin clay

SSURGO NRCS Soil Survey Geographic database

SSURGOKSAT mean basin hydraulic conductivity of saturated soils, in micrometers per 

second
vi

SSURGOM percent mean basin organic matter content

SSURGSAND percent mean basin sand

SSURGSILT percent mean basin silt 

STORAGE percent streamgage drainage basin area consisting of lakes and wetlands

USGS U.S. Geological Survey
Methods for Estimating Flow-Duration Curve and LowFlow Frequency Statistics for Ungaged Locations on Small 

Streams in Minnesota

By Jeffrey R. Ziegeweid, David L. Lorenz, Chris A. Sanocki, and Christiana R. Czuba

Abstract

Knowledge of the magnitude and frequency of low flows 

in streams, which are flows in a stream during prolonged dry 

weather, is fundamental for water-supply planning and design; 

waste-load allocation; reservoir storage design; and maintenance of water quality and quantity for irrigation, recreation, 

and wildlife conservation. This report presents the results of a 

statewide study for which regional regression equations were 

developed for estimating 13 flow-duration curve statistics and 

10 low-flow frequency statistics at ungaged stream locations in 

Minnesota. The 13 flow-duration curve statistics estimated by 

regression equations include the 0.0001, 0.001, 0.02, 0.05, 0.1, 

0.25, 0.50, 0.75, 0.9, 0.95, 0.99, 0.999, and 0.9999 exceedance-probability quantiles. The low-flow frequency statistics 

include annual and seasonal (spring, summer, fall, winter) 

7-day mean low flows, seasonal 30-day mean low flows, and 

summer 122-day mean low flows for a recurrence interval of 

10 years. Estimates of the 13 flow-duration curve statistics 

and the 10 low-flow frequency statistics are provided for 196 

U.S. Geological Survey continuous-record streamgages using 

streamflow data collected through September 30, 2012. 

The study area includes 196 streamgages located within 

Minnesota and 50 miles beyond the State’s borders in North 

Dakota, South Dakota, Iowa, and Wisconsin. The study area 

was divided into five regions that were developed in a previous study using the concept of hydrologic landscape units. 

Geographic information system software was used to calculate 18 characteristics investigated as potential explanatory 

variables in regression analyses for each streamgage drainage 

basin. Trend analyses indicated statistically significant trends 

in summer 7-day low flows that were not related to precipitation patterns for 19 streamgages. For 16 of these streamgages, 

the streamflow record was subset using structural change 

analysis to identify the most recent period of record without a 

significant trend. The three remaining streamgages with significant trends were excluded from the final analysis because 

the effective period of record without a significant trend was 

less than 10 years.

Because several streams in this study have zero flow as 

their minimum reported flow, weighted left-censored regression was used to analyze the flow data in an unbiased manner, 

with weights based on the number of years of record. A total 

of 115 regression equations were developed in this study to 

calculate flow-duration curve and low-flow frequency statistics 

for ungaged locations in the study area. In addition, data from 

25 pairs of streamgages were used to develop drainage-area 

ratio equations that can be used to estimate streamflow statistics at ungaged locations on streams that have a streamgage 

in another location. Streamflow statistics estimated using 

regional regression and drainage-area ratio equations were 

compared among regions. For regions A, D, and E, drainagearea ratio equations were more accurate than regional regression equations for flows, but regional regression equations 

were more accurate for high flows. For region F, regional 

regression equations were consistently more accurate than 

drainage-area ratio equations. For region BC, the pattern in 

accuracies of regional regression and drainage-area ratio equations between low flows and high flows was not consistent. 

Equations developed in this study apply only to stream 

locations where flows are not substantially affected by regulation, diversion, or urbanization. All equations presented in 

this study will be incorporated into StreamStats, a web-based 

geographic information system tool developed by the U.S. 

Geological Survey. StreamStats allows users to obtain streamflow statistics, basin characteristics, and other information for 

user-selected locations on streams through an interactive map.

Introduction

Low flow can be defined as “the flow of water in a 

stream during prolonged dry weather” (U.S. Environmental 

Protection Agency, 2013). Knowledge of the magnitude and 

frequency of low flows for streams is fundamental for watersupply planning and design; waste-load allocation; reservoir 

storage design; and maintenance of water quality and quantity 

for irrigation, recreation, and wildlife conservation (Eash and 

Barnes, 2012). Because low-flow frequency (LFF) statistics 
2 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

indicate the probable availability of water in streams during 

times when conflicts between water supply and demand are 

most prevalent, LFF statistics can be used by Federal, State, 

tribal, and local agencies for water-quality regulatory activities, water-supply planning, and water management. For 

example, LFF statistics can be used as thresholds when setting 

wastewater-treatment plant effluent limits and allowable pollutant loads to meet water-quality regulations. In addition, 

LFF statistics can be used by commercial, industrial, and 

hydroelectric facilities to determine availability of water for 

water supply, waste discharge, and power generation. Finally, 

LFF statistics can be used in ecological research because lowflow conditions can disturb ecosystems and create biological 

responses that include changes in habitat, reduced populations 

of aquatic species, and shifts in the relative distribution of species (Miller and Golladay, 1996).

In 2014, 1,649 stream reaches in Minnesota were designated as impaired (Minnesota Pollution Control Agency, 

2014) and are scheduled to have pollutant loads analyzed and 

maximum loading rates established by total maximum daily 

load (TMDL) assessments (U.S. Environmental Protection 

Agency, 2012). Reliable estimates of expected streamflow are 

needed for specific periods of the year when determining the 

maximum allowable load of a pollutant in a stream. Estimates 

of expected streamflow are especially important for low-flow 

periods when agencies need to determine waste-load allocations (WLAs) for National Pollution Discharge Elimination 

System (NPDES; U.S. Environmental Protection Agency, 

2012) discharge permits for municipalities, industries, and 

other entities with facilities that release treated wastewater into 

a stream. A WLA is the loading capacity or maximum quantity of a pollutant each point-source discharger is allowed to 

release into a particular stream (U.S. Environmental Protection 

Agency, 2012), and WLAs are used to establish water-qualitybased limits for point-source discharges.

The U.S. Geological Survey (USGS) operates a network 

of streamgages in Minnesota and throughout the Nation that 

are used to compute streamflow data for a variety of purposes. 

Flow-duration curve (FDC) and LFF statistics are calculated 

using streamflow data collected at these locations; however, 

streamgages are not available for every stream, and therefore, 

methods are needed for estimating FDC and LFF statistics 

at ungaged stream locations. The USGS initiated a statewide 

study in 2012 in cooperation with the Minnesota Pollution 

Control Agency (MPCA) to estimate FDC and LFF statistics 

for ungaged stream locations in Minnesota using streamflow 

data collected through September 30, 2012. Major components of the study included (1) computing 13 selected FDC 

statistics and 10 selected LFF statistics at 196 continuousrecord streamgages within Minnesota and adjacent States, (2) 

measuring 18 basin characteristics for each streamgage, (3) 

developing 115 regional regression equations to estimate 23 

selected statistics for ungaged stream locations based on basin 

characteristics, and (4) developing drainage-area ratio equations to provide estimates of low flow at ungaged locations 

on gaged streams in Minnesota. The regression equations 

developed for this study will be incorporated into StreamStats, 

a web-based geographic information system (GIS) tool developed by the USGS (Ries and others, 2008).

This study builds upon the work presented in several previous USGS reports (Warne, 1978; Arntson and Lorenz, 1987; 

Winterstein and others, 2007; Lorenz and others, 2010; Eash 

and Barnes, 2012). Warne (1978) presented a map in atlas 

form to demonstrate variability in LFF data for streamgages 

in Minnesota. Arntson and Lorenz (1987) and Winterstein and 

others (2007) compiled LFF characteristics for continuousrecord streamgages in Minnesota. Lorenz and others (2010) 

used data from streamgages in Minnesota and adjacent States 

to develop techniques for estimating the magnitude and 

frequency of peak flows on small, ungaged streams in Minnesota, and the results of their study were incorporated into 

StreamStats. Finally, Eash and Barnes (2012) used data from 

continuous-record streamgages in Iowa and adjacent States to 

estimate selected LFF statistics and harmonic mean flows for 

ungaged stream locations in Iowa.

Purpose and Scope

This report presents methods used to develop regional 

regression equations for use in estimating FDC and LFF 

statistics at ungaged locations on unregulated streams in Minnesota. The regression equations relate selected FDC and LFF 

statistics to physical and hydrologic soil characteristics of 

gaged drainage basins for five defined hydrologic regions. The 

regression equations can be used to estimate 13 selected FDC 

statistics and 10 selected LFF statistics for ungaged locations 

on unregulated streams in Minnesota. The 13 FDC statistics 

estimated by regression equations include the 0.0001, 0.001, 

0.02, 0.05, 0.1, 0.25, 0.50, 0.75, 0.9, 0.95, 0.99, 0.999, and 

0.9999 exceedance-probability quantiles. The LFF statistics 

include annual and seasonal (spring, summer, fall, winter) 

7-day mean low flows, seasonal 30-day mean low flows, and 

summer 122-day mean low flows for a recurrence interval of 

10 years. 

The FDC and LFF statistics were computed using 

streamflow data collected through September 30, 2012. Names 

of LFF statistics used in this report were selected to maintain consistency with names used within StreamStats (http://

water.usgs.gov/osw/streamstats/StatisticsDefinitions.html). 

All statistics reported in this study were computed for each 

streamgage using the longest, most recent period of record 

without a significant trend in low flow related to factors other 

than precipitation patterns. The accuracy and limitations of the 

regression equations and the methodology used to develop the 

equations are described in the report.

Description of Study Area

The study area (fig. 1) includes the State of Minnesota 

and a 50-mile (mi) buffer around Minnesota in the neighboring 

States of Iowa, North Dakota, South Dakota, and Wisconsin. 
Methods for Data Development 3

Canadian portions of the basins were not included because 

many of the characteristics were not available in the same 

format as those for U.S. portions of the basins. The study area 

is divided into five hydrologic regions (fig. 1) developed using 

a combination of statistical analyses from previous studies 

(Jacques and Lorenz, 1987; Lorenz and others, 1997; Lorenz 

and others, 2010) and the concept of hydrologic landscape 

units (Winter, 2001; Wolock and others, 2004). Region BC 

represents the combined regions B and C from Lorenz and 

others (2010) because not enough streamgages with low-flow 

data were available in region C to develop regional regression 

equations. Analyses of residuals from regressions confirmed 

that regions B and C could be combined without affecting the 

ability of regional regression equations to accurately estimate 

FDC and LFF statistics. Streamgages included in this study 

were assigned a map number, and information associated with 

each map number (streamgage) can be found in table 1–1 of 

the appendix.

Differences among the five hydrologic regions are 

addressed using the hydrologic landscape unit information 

presented in Lorenz and others (2010). Region A has the most 

heterogeneous landscapes, with generally low slopes that 

become more moderately sloped near the boundary. Most of 

region BC is dominated by sandy soils and low to moderate 

slopes, but the northeastern part of region BC along the north 

shore of Lake Superior and the northeastern Canadian border 

generally is high in slope. Low slopes near the center of region 

D change to moderate slopes and less sandy soils around the 

regional boundaries. Region E is similar to region D, with 

moderate slopes and low sand content in the soils; however, 

there are distinct differences in the drainage patterns between 

regions D and E (Lorenz and others, 2010). Finally, high slope 

areas in region F change to moderately sloped areas along the 

western regional boundary.

Methods for Data Development

Data used in this report were collected for 196 active and 

inactive continuous-record streamgages located in Minnesota 

or within 50 miles of Minnesota in the neighboring States of 

Iowa, North Dakota, South Dakota, and Wisconsin (fig. 1; 

table 1‒1). Selected streamgages met the following criteria: 

(1) at least 10 complete water years of daily mean streamflows 

(a water year is the 12-month period from October 1 through 

September 30 and is designated by the calendar year in which 

it ends); (2) streamflows unaffected by regulation, diversion, 

or urbanization; and (3) drainage areas less than 3,000 square 

miles. For 5 of the 196 streamgages, additional streamflow 

records available from the Minnesota Department of Natural Resources (MDNR; Minnesota Department of Natural 

Resources, 2013) were appended to the USGS streamflow 

data. Streamflow data from MDNR streamgages 01063003 

(April 1, 2008, through September 30, 2012), 01092001 

(September 1, 2008, through September 30, 2012), 70018001 

(September 1, 2007, through September 30, 2012), 41043001 

(July 1, 2009, through September 30, 2012), and 40016001 

(April 1, 2008, through September 30, 2012) were appended 

to streamflow data from USGS streamgages 04012500, 

04014500, 05095000, 05374900, and 05376800, respectively. 

Streamflow data from USGS streamgages in neighboring 

States were used to improve the representativeness of selected 

FDC and LFF statistics and to provide better estimates of the 

error associated with the regression equations for ungaged 

locations near the Minnesota border. Nationwide, USGS 

streamflow data are collected using consistent methods and 

instrumentation specifications to ensure data are comparable 

(Rantz and others, 1982). Daily mean streamflow data collected through water year 2012 (September 30, 2012) were 

retrieved for the 196 streamgages from the USGS National 

Water Information System (U.S. Geological Survey, 2013) and 

used in computing selected FDC and LFF statistics. All statistical analyses presented in this report were performed using R 

version 3.1.3 (Venables and others, 2010).

Trend Analyses

Trends in data can introduce bias into the calculation 

of FDC and LFF statistics because the statistical analysis is 

based on the assumption that statistical parameters associated 

with the data do not change over time (National Institute of 

Standards and Technology, 2013). Such parameters include the 

percentiles of flow, which are used in FDCs for the distribution of low flows and for determining the 10 selected LFF 

statistics. For this study, trends were evaluated based on 

the summer (June through September) 7-day low flows. An 

initial assessment of trends using the Mann-Kendall trend test 

(Helsel and Hirsch, 2002) determined that 69 streamgages had 

significant trends at a significance level of 0.05 (p-values less 

than 0.05); however, precipitation patterns have a considerable 

influence on streamflow. 

To further evaluate those trends in streamflow, a linear 

regression model was used to relate the summer 7-day low 

flow, minimum monthly summer precipitation (June through 

September); the total antecedent precipitation in the year 

(January through September); the total precipitation from the 

previous calendar year; and the year of the observation (to 

indicate the time trend). Mean monthly precipitation data were 

available by climate division for the period of January 1895 

to December 2012 from the National Climatic Data Center 

(National Oceanic and Atmospheric Administration, 2013). 

After accounting for precipitation effects on streamflow, 50 

streamgages had significant trends at a significance level of 

0.05. To reduce the statistical chance of making a type I error 

(identifying a trend when it is not present in all of the trend 

analyses), p-values from the linear regression analyses were 

adjusted for a false discovery rate (Benjamini and Hochberg, 

1995); 19 streamgages had significant trends at a false discovery level of 0.05.
4 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

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0 5025

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100 MILES

100 KILOMETERS U.S. Geological Survey streamgage 

with map number corresponding 

to detailed information 

(table 1–1)— Streamgages located 

outside Minnesota were assigned 

to hydrologic regions shown in 

table 1–1

Hydrologic region

EXPLANATION

A

BC

D

E

F

Base from U.S. Geological Survey and Minnesota 

Department of Natural Resources, various scales

Universal Transverse Mercator projection, zone 15

Horizontal coordinate information is referenced 

to the North American Datum of 1983 

157

CANADA

MINNESOTA

WISCONSIN

IOWA

NORTH

DAKOTA

SOUTH

DAKOTA

Lake of the Woods

Upper

Red Lake

Lower

Red Lake

LAKE

SUPERIOR

Mille

 Lacs

UNITED

STATES

Figure 1. Hydrologic regions in Minnesota and streamgages in study area.
Methods for Data Development 5

The 19 streamgages with significant trends not related to 

precipitation patterns could indicate changes within the basin 

due to human activities, such as changes in cropping patterns, 

drainage, wetland restoration, and other factors. Such temporal 

changes would affect the relation between the basin characteristic and the FDC and LFF statistics. For example, previous 

studies attributed increases in streamflow in many rivers to 

increasing precipitation beginning in the 1940s (Houghton and 

others, 1996; Griggs and Noguer, 2002); however, Zhang and 

Schilling (2006) demonstrated that increases to streamflow 

and base flow in the Mississippi River basin actually resulted 

from reduced evapotranspiration and increased groundwater 

recharge associated with conversion of land from perennial 

vegetation to seasonal row crops. For 16 of the 19 streamgages, 

the streamflow record was subset using structural change 

analysis (Zeileis and others, 2002), which identified changes 

in the relation between the summer low flows and precipitation patterns, to define the most recent period of record of at 

least 10 years without a significant trend at a significance level 

of 0.05 (table 1‒1). Three streamgages were excluded from 

the final analysis because their effective period of record was 

less than 10 years; the three excluded streamgages were not 

included in the 196 streamgages listed in table 1–1. 

Flow-Duration Curves

For a given streamgage, the FDC was computed from 

daily mean streamflows from complete water years for the 

entire streamflow period of record (Searcy, 1959), except for 

the 16 streamgages with significant trends identified in the 

streamflow data not attributable to precipitation patterns and 

for which a subset streamflow record was used, as described 

previously in the “Trend Analyses” section. The FDCs were 

computed by sorting daily-value data and assigning exceedance probabilities to each value by means of the plotting position formula: 

pi=(i−a)/(n+1−2a), (1)

where 

pi is the non-exceedance probability,

i is the rank (1 to n, smallest to largest),

n is the number of values, and

a is a constant, taken here as 0.4 (Helsel and 

Hirsch, 2002). 

The 0.0001, 0.001, 0.02, 0.05, 0.1, 0.25, 0.50, 0.75, 0.9, 0.95, 

0.99, 0.999, and 0.9999 exceedance-probability quantiles were 

obtained from the daily streamflow data. The computed FDC 

quantiles are provided in table 1‒1 of the appendix, along with 

information on the period of record used to compute the quantiles. The FDC for USGS streamgage 05279000 (map number 

78; fig. 1), the South Fork Crow River near Mayer, Minnesota, 

is shown in figure 2 as an example. Figure 2 also illustrates 

that flow values of zero prevent the FDC from reaching 

100 percent exceedance.

Streamflow, in cubic feet per second

Percentage of time streamflow is exceeded

0.1

1

10

100

1,000

10,000

100,000

0 10 20 30 40 50 60 70 80 90 100

Figure 2. Flow-duration 

curve developed using 13 

exceedance-probability 

quantiles for the South Fork 

Crow River near Mayer, 

Minnesota, (U.S. Geological 

Survey streamgage 

05279000) for water years 

1941–79.
6 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Low-Flow Frequency

A frequency analysis was conducted to estimate 10 

selected low-flow statistics at the 196 streamgages used in the 

study. The LFF statistics typically consist of mean low flows 

for a specified number of days within a season or climate year 

for a specified recurrence interval. A climate year is defined 

as the 12-month period from April 1 through March 31 and 

is designated by the calendar year in which the period ends. 

Recurrence interval, in years, is the reciprocal of the annual or 

seasonal non-exceedance probability. The MPCA uses a recurrence interval of 10 years for regulatory protocols (Minnesota 

Pollution Control Agency, 2008), which was used for all LFF 

analyses in this study. 

Climate years were used to divide annual and seasonal 

low-flow data because the start of the climate year (April 1) 

generally corresponds with high flow in the spring, so lowflow conditions do not span across multiple years. Low-flow 

frequencies were estimated for the annual (climate year) 7-day 

mean low flows, the seasonal (June through September, October through November, December through March, and April 

through May) 7- and 30-day mean low flows, and the summer 

seasonal (June through September) 122-day mean low flows. 

More simply, 7-day mean low flow with a recurrence interval of 10 years is the lowest 7-day mean flow that occurs (on 

average) once every 10 years (U.S. Environmental Protection 

Agency, 2013). 

For a given streamgage, the occurrence of each lowflow value was computed over the climate year or season 

by identifying consecutive days with the lowest 7-, 30-, or 

122-day mean streamflows. If there were fewer than six 

nonzero values, then an empirical fit was used to fit the data. 

Otherwise, the magnitude and frequency for each of the LFF 

statistics were computed by fitting each nonzero value to one 

of three distributions: log-Pearson type III, three-parameter 

lognormal, or log-generalized extreme value (Vogel and Kroll, 

1989; Helsel and Hirsch, 2002). The distribution used to determine LFF statistics at each streamgage is listed in table 1‒1 

of the appendix. The log-Pearson type III distribution was 

used unless the data fit the distribution poorly. Fit of the data 

to the log-Pearson type III distribution was assessed by using 

the probability plot correlation coefficient (PPCC) from the 

observed nonzero data and the predicted values (Helsel and 

Hirsch, 2002); the closer the linear correlation coefficient was 

to 1.0, the better the data fit the distribution. 

In this study, the log-Pearson type III distribution 

automatically was used when computed linear correlation 

coefficients were greater than 0.985. Otherwise, PPCCs were 

computed for all three distributions (log-Pearson type III, 

three-parameter lognormal, log-generalized extreme value), 

and the distribution that resulted in the highest PPCC was 

selected. In addition, a conditional probability adjustment for 

zero-flow values (appendix 5 in U.S. Interagency Advisory 

Committee on Water Data, 1982) was used for streamgages 

with one or more annual 7-, 30-, or 122-day low-flow values 

of zero. The 7-day annual LFF graph for USGS streamgage 

05279000 (map number 78; fig. 1), South Fork Crow River 

near Mayer, Minn., is shown in figure 3 to provide a comparison of the three distributions used to compute LFF statistics 

in this study. The log-Pearson type III analysis was used for 

the 7-day annual LFF analysis of the 33 nonzero values for 

streamgage 05279000, and a conditional probability adjustment was included in the analysis for the 5 zero-flow years in 

the annual 7-day low-flow record.

Basin Characteristics

Low-flow characteristics of streams are related to the 

physical, geologic, and climatic properties of drainage basins 

(Smakhtin, 2001). In most basins, drainage area is a statistically significant variable in explaining low-flow variability 

(Funkhouser and others, 2008; Kroll and others, 2004). Basin 

characteristics investigated in this study as potential explanatory variables in regression analyses were selected on the basis 

of their theoretical relation to low flows, results of previous 

studies in similar hydrologic areas, and the ability to quantify 

the basin characteristics using GIS technology. The use of GIS 

enables the automation of the basin-characteristic calculations and solution of the regional regression equations using 

StreamStats (http://water.usgs.gov/osw/streamstats/index.

html).

Using GIS technology, 18 basin characteristics (table 1) 

were calculated for each of the 196 streamgage basins in 

this study and investigated as potential explanatory variables 

in regression analyses with 23 selected low-flow statistics as response variables (table 1‒1). Basin characteristics 

generally can be divided into three categories: (1) physical 

measurements, (2) land cover, and (3) soils. Physical measurement characteristics include drainage area, in square 

miles (DRNAREA); mean basin slope, in feet per foot 

(BSLDEM10M); percent streamgage drainage basin area 

consisting of lakes and wetlands (STORAGE); precipitation 

minus potential evapotranspiration, in millimeters (PMPE); 

and percent mean basin low-lying flatland (PFLATLOW) 

and upland flatland (PFLATUP). Land-cover characteristics 

include percent areas of lakes (LAKEAREA), forested land 

(FOREST), and cultivated land (CROPSNLCD01). Soil characteristics include percent mean basin organic matter content 

(SSURGOM), sand (SSURGSAND), silt (SSURGSILT), clay 

(SSURGCLAY); mean basin hydraulic conductivity of saturated soils, in micrometers per second (SSURGOKSAT); and 

percent streamgage drainage basin areas consisting of hydrologic soil groups A (SOILA), B (SOILB), C (SOILC), and D 

(SOILD). Sources of data were the National Elevation Dataset 

(http://ned.usgs.gov/; Gesch and others, 2009), National 

Wetlands Inventory (http://www.fws.gov/wetlands/Data/

Data-Download.html; U.S. Fish and Wildlife Service, 2013), 

hydrologic landscape units (David M. Wolock, U.S. Geological Survey, unpub.data, 2003), National Land Cover Dataset 

(http://www.mrlc.gov/nlcd2006.php; Homer and others, 2012), 
Methods for Data Development 7

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1,000

1:1 line

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1,000

1:1 line

0.01

0.1

1

10

100

0.01

1:1 line

0.1 1 10 100 1,000

7-day annual low flow, in cubic feet per second

Estimated 7-day low-flow values using log-Pearson type III, in cubic feet per second

7-day annual low flow, in cubic feet per second

Estimated 7-day low-flow values using 3-parameter log-normal, in cubic feet per second

7-day annual low flow, in cubic feet per second

Estimated 7-day low-flow values using log-generalized extreme value, in cubic feet per second

Figure 3. Relation between observed 7-day annual low-flow values and values estimated using log-Pearson type III, threeparameter lognormal, and the log-generalized extreme value distributions for computing a 7-day annual low-flow frequency graph 

for the South Fork Crow River near Mayer, Minnesota (U.S. Geological Survey streamgage 05279000).
8 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Table 1. Basin characteristics tested as explanatory variables in the development of low-flow regression equations.

[StreamStats and NHDPlus are web-based tools used to measure areas of watersheds. Basin mean is the mean of all raster values in the basin area. Percent drainage area is the area of drainage basin covered by feature, in percent. DRNAREA, drainage area; BSLDEM10M, mean basin slope; STORAGE, area of lakes 

and wetlands; PMPE, precipitation minus potential evapotranspiration; PFLATLOW, low-lying flatland; PFLATUP, upland flatland; LAKEAREA, area of lakes; 

FOREST, area of forested land; CROPSNLCD01, area of cultivated land; SSURGOM, organic matter; SSURGSAND, sand; SSURGSILT, silt; SSURGCLAY, 

clay; SSURGOKSAT, hydraulic conductivity of saturated soils; SOILA, soil in hydrologic group A; SOILB, soil in hydrologic group B; SOILC, soil in hydrologic group C; SOILD, soil in hydrologic group D]

StreamStats 

name

Units Method Source data

Physical measurements

DRNAREA square miles StreamStats

NHDPlus

http://water.usgs.gov/osw/streamstats/index.html

http://www.horizon-systems.com/NHDPlus/NHDPlusV1_data.php

BSLDEM10M feet per foot Basin mean http://ned.usgs.gov/

STORAGE percent Percent drainage area http://www.fws.gov/wetlands/Data/Data-Download.html

PMPE millimeters Basin mean David M. Wolock, U.S. Geological Survey, unpub.data, 2003

PFLATLOW percent Basin mean David M. Wolock, U.S. Geological Survey, unpub.data, 2003

PFLATUP percent Basin mean David M. Wolock, U.S. Geological Survey, unpub.data, 2003

Land cover

LAKEAREA percent Percent drainage area http://www.fws.gov/wetlands/Data/Data-Download.html

FOREST percent Percent drainage area http://www.mrlc.gov/nlcd2006.php

CROPSNLCD01 percent Percent drainage area http://www.mrlc.gov/nlcd2006.php

Soils

SSURGOM percent Basin mean http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SSURGSAND percent Basin mean http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SSURGSILT percent Basin mean http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SSURGCLAY percent Basin mean http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SSURGOKSAT micrometers per second Basin mean http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SOILA percent Percent drainage area http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SOILB percent Percent drainage area http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SOILC percent Percent drainage area http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

SOILD percent Percent drainage area http://www.nrcs.usda.gov/wps/portal/nrcs/main/soils/survey/

Soil Survey Geographic Database (http://www.nrcs.usda.gov/

wps/portal/nrcs/main/soils/survey/; Soil Survey Staff, 2012), 

StreamStats (http://water.usgs.gov/osw/streamstats/index.html; 

Ries and others, 2008) and the National Hydrography Dataset Plus (http://www.horizon-systems.com/NHDPlus/NHDPlusV1_data.php; Bondelid and others, 2006). Data for the 18 

basin characteristics will be incorporated into StreamStats for 

use in calculating FDC and LFF statistics at ungaged locations 

on small streams in Minnesota.

Regression Analysis

Because some streams in this study have values of zero 

for selected FDC or LFF statistics, special regression techniques are required to analyze these statistics in an unbiased 

manner. Estimates of zero flow computed from observed 

streamflow commonly are treated as left-censored data (Kroll 

and Stedinger, 1996; Kroll and Vogel, 2002), and the use of 

multiple-linear regression is not recommended for censored 

data (Helsel and Hirsch, 2002). For this study, weighted 

left-censored regression was used for all regression analyses, 

with weights based on the number of years of record (Lorenz, 

2014). When no data are censored, weighted left-censored 

regression provides the same results as weighted least-squares 

regression (Helsel and Hirsch, 2002). Because of the uncertainty in measuring low flows and estimating FDC and LFF 

statistics less than 0.1 cubic foot per second (ft3

/s), the censoring threshold used to develop the left-censored regression 

equations was set at 0.1 ft3

/s.

Regional regression equations were developed for the 

five regions shown in figure 1, and residuals from regression 

analyses were analyzed to ensure that the regional divisions were supported by the hydrologic data collected at the 

streamgages within the study area. Residuals were required to 

be approximately normally distributed, homoscedastic (having 

constant variance throughout the range of fitted values), and 
Regional Regression Analysis 9

show no regional pattern. Region BC (fig. 1) represents the 

combined regions B and C from Lorenz and others (2010) 

because the number of streamgages in region C is too small 

to develop regional regression equations. A consistent set of 

explanatory variables was used in all of the regression equations for each region to make consistent estimations at the 

selected FDC exceedance probabilities or LFF statistics. A set 

of consistent explanatory variables that had logical coefficients 

and were statistically significant at the 0.05 level was selected 

for the regression equations for all FDC exceedance probabilities or LFF statistics within that region. This approach guarantees consistency for uncensored regressions but does not 

guarantee consistency when censored data are included. In this 

report, “inconsistency” refers to situations when an estimated 

low flow for a smaller number of consecutive days is greater 

than an estimated low flow for a larger number of consecutive 

days. For example, an estimated 7-day low flow may be larger 

than the estimated 30-day low flow. 

Regional Regression Analysis

In a regional regression study, dividing a large study area 

into subregions that are relatively homogeneous in terms of 

hydrology typically helps to reduce error in the regression 

equations. Hydrologic regions were developed for peak flows 

in Minnesota by Lorenz and others (2010) using the concept of 

hydrologic landscape units (Winter, 2001; Wolock and others, 

2004) and not peak-flow residuals. For this study, the regions 

of Lorenz and others (2010) were used as the initial hydrologic 

regions, but residual analyses based on the regressions were 

used to verify that the hydrologic regions form reasonably 

homogenous FDC and LFF regression equations. As previously mentioned, regions B and C were combined. 

Flow-Duration Curve Regression Equations

The weighted left-censored regression method (Helsel 

and Hirsch, 2002) was used to develop regression equations 

for each exceedance probability for each region in figure 1. 

One streamgage (05412100, Roberts Creek above Saint Olaf, 

Iowa) was excluded from the analysis for development of 

regression equations in region F because it was an extreme 

outlier, but regression results for the gage are provided for 

completeness (table 1–1). The regression equations and corresponding standard errors of the estimate (SEEs) are listed in 

table 2. The number of explanatory variables was minimized 

in regression equations to maximize residual degrees of freedom and minimize multicollinearity, which is the condition 

where at least one explanatory variable is closely related to 

one or more other explanatory variables (Helsel and Hirsch, 

2002). 

Nine of the 18 examined basin characteristics were used 

as explanatory variables in the development of FDC regional 

regression equations (table 2). Explanatory variables used in 

regression equations for region A were DRNAREA, SSURGOM, and BSLDEM10M. Explanatory variables used in 

regression equations for region BC were DRNAREA, STORAGE, and SOILC. Explanatory variables used in regression 

equations for region D were DRNAREA, PMPE, and PFLATLOW. Explanatory variables used in regression equations for 

region E were DRNAREA and CROPSNLCD01. Finally, 

explanatory variables used in regression equations for region F 

were DRNAREA, FOREST, and PFLATLOW. For each 

region, selected explanatory variables minimized SEEs and 

produced consistent FDCs. Generally, SEEs (expressed as 

percentage-based values) were largest for the largest exceedance probabilities and generally increased with increased 

censored data. 

Low-Flow Frequency Regression Equations

The weighted left-censored regression method was used 

to develop regression equations for each LFF statistic for each 

region shown in figure 1. One streamgage (05412100, Roberts 

Creek above Saint Olaf, Iowa; map number 159; fig. 1) was 

excluded from the analysis for development of regression 

equations in region F because it was an extreme outlier, but 

regression results for the gage are provided for completeness 

(table 1–1). The regression equations and corresponding SEEs 

are listed in table 3. The number of explanatory variables 

were minimized in regression equations to maximize residual 

degrees of freedom and minimize multicollinearity (Helsel and 

Hirsch, 2002). 

Eight of the 18 examined basin characteristics were used 

as explanatory variables in the development of LFF regional 

regression equations (table 3). Explanatory variables used in 

regression equations for region A were DRNAREA, SSURGOM, and BSLDEM10M. Explanatory variables used in 

regression equations for region BC were DRNAREA, STORAGE, and SOILC. Explanatory variables used in regression 

equations for region D were DRNAREA, PMPE, and PFLATLOW. Explanatory variables used in regression equations for 

region E were DRNAREA and CROPSNLCD01. Finally, 

explanatory variables used in regression equations for region F 

were DRNAREA, CROPSNLCD01, and BSLDEM10M. For 

each region, explanatory variables were selected to minimize 

SEEs, produce consistent LFF statistics, and minimize multicollinearity (Helsel and Hirsch, 2002). Ranges of SEEs varied 

by region, with regions BC and F having smaller SEEs compared to regions A, D, and E. Among all regions, SEEs ranged 

from 23 to 665 percent. Fewer censored values occurred in 

regions BC and F, and SEEs generally increased as the number 

of censored values increased among regions. 

The number of censored values in each analysis varied 

by region. Annually, across regions A, BC, D, and E, most 

of the censored values occurred in the winter (December 

through March) and the summer (June through September) 

seasonal regression analyses. Region F had the fewest number 

of censored flow values; many of the streams in region F 
10 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Table 2. Regression equations for estimating selected flow-duration curve exceedance probabilities for unregulated streams in 

Minnesota.

[SEE, standard error of the estimate; number censored, number of streamflow values less than or equal to 0.1 cubic foot per second; D0_01, streamflow with 

an exceedance probability of 0.0001; DRNAREA, drainage area, in square miles; SSURGOM, percentage organic matter estimated using the Soil Survey 

Geographic Database; BSLDEM10M, mean basin slope computed from a 10-meter digital elevation model; D0_1, streamflow with an exceedance probability 

of 0.001; D2, streamflow with an exceedance probability of 0.02; D5, streamflow with an exceedance probability of 0.05; D10, streamflow with an exceedance 

probability of 0.10; D25, streamflow with an exceedance probability of 0.25; D50, streamflow with an exceedance probability of 0.50; D75, streamflow with an 

exceedance probability of 0.75; D90, streamflow with an exceedance probability of 0.90; D95, streamflow with an exceedance probability of 0.95; D99, streamflow with an exceedance probability of 0.99; D99_9, streamflow with an exceedance probability of 0.999; D99_99, streamflow with an exceedance probability 

of 0.9999; STORAGE, the percentage of area of water storage in lakes, ponds, reservoirs, and wetlands; SOILC, percentage of area of hydrologic soil type C; 

PMPE, precipitation minus potential evapotranspiration, in millimeters; PFLATLOW, percentage of low-lying flatland; CROPSNLCD01, percentage of area of 

cultivated land; FOREST, percentage of area covered by forest]

Regression equation SEE (percent) Number censored

Region A (35 streamgages)

D0_01 = 118.4 (DRNAREA)0.6475 (SSURGOM + 1)-0.08101 (BSLDEM10M)-0.2437 51 0

D0_1 = 47.16 (DRNAREA)0.7086 (SSURGOM + 1)-0.08814 (BSLDEM10M)-0.1047 30 0

D2 = 2.881 (DRNAREA)0.8819 (SSURGOM + 1)0.2393 (BSLDEM10M)-0.08488 32 0

D5 = 0.5466 (DRNAREA)0.9751 (SSURGOM + 1)0.4639 (BSLDEM10M)0.04280 45 0

D10 = 0.1105 (DRNAREA)1.047 (SSURGOM + 1)0.7049 (BSLDEM10M)0.2260 58 0

D25 = 0.006626 (DRNAREA)1.180 (SSURGOM + 1)1.040 (BSLDEM10M)0.6323 71 2

D50 = 0.0007381 (DRNAREA)1.203 (SSURGOM + 1)1.279 (BSLDEM10M)1.234 79 6

D75 = (6.929 x 10-5) (DRNAREA)1.299 (SSURGOM + 1)1.553 (BSLDEM10M)1.716 95 9

D90 = (2.892 x 10-6) (DRNAREA)1.436 (SSURGOM + 1)2.146 (BSLDEM10M)2.305 150 13

D95 = (6.658 x 10-7) (DRNAREA)1.435 (SSURGOM + 1)2.470 (BSLDEM10M)2.765 238 17

D99 = (1.256 x 10-7) (DRNAREA)1.363 (SSURGOM + 1)2.557 (BSLDEM10M)3.735 604 22

D99_9 = (2.835 x 10-8) (DRNAREA)1.599 (SSURGOM + 1)0.9545 (BSLDEM10M)4.882 675 28

D99_99 = (4.183 x 10-7) (DRNAREA)1.143 (SSURGOM + 1)1.187 (BSLDEM10M)4.296 420 28

Region BC (49 streamgages)

D0_01 = 37.88 (DRNAREA)0.7700 (STORAGE + 1)-0.1406 (SOILC + 1)0.2069 65 0

D0_1 = 16.58 (DRNAREA)0.7986 (STORAGE + 1)-0.06526 (SOILC + 1)0.2098 50 0

D2 = 4.068 (DRNAREA)0.8549 (STORAGE + 1)0.02530 (SOILC + 1)0.1813 39 0

D5 = 2.458 (DRNAREA)0.8785 (STORAGE + 1)0.06252 (SOILC + 1)0.1305 35 0

D10 = 1.227 (DRNAREA)0.9333 (STORAGE + 1)0.03388 (SOILC + 1)0.1166 27 0

D25 = 1.673 (DRNAREA)0.9455 (STORAGE + 1)-0.1367 (SOILC+ 1)-0.02305 30 0

D50 = 2.436 (DRNAREA)1.001 (STORAGE + 1)-0.4477 (SOILC + 1)-0.1631 38 0

D75 = 3.903 (DRNAREA)1.021 (STORAGE + 1)-0.6862 (SOILC + 1)-0.2812 54 0

D90 = 4.759 (DRNAREA)1.061 (STORAGE + 1)-0.8323 (SOILC + 1)-0.4041 68 1

D95 = 5.226 (DRNAREA)1.113 (STORAGE + 1)-0.9446 (SOILC + 1)-0.5080 80 3

D99 = 7.486 (DRNAREA)1.147 (STORAGE + 1)-1.133 (SOILC + 1)-0.6666 112 7

D99_9 = 8.96 (DRNAREA)1.231 (STORAGE + 1)-1.303 (SOILC + 1)-0.9244 217 8

D99_99 = 9.861 (DRNAREA)1.218 (STORAGE + 1)-1.271 (SOILC+ 1)-1.037 261 8

Region D (47 streamgages)

D0_01 = 513.1 (DRNAREA)0.7935 (1.003)PMPE (PFLATLOW + 1)-0.7232 43 0

D0_1 = 210.9 (DRNAREA)0.8329 (1.003)PMPE (PFLATLOW + 1)-0.7058 38 0

D2 = 18.45 (DRNAREA)0.9443 (1.005)PMPE (PFLATLOW + 1)-0.5820 35 0

D5 = 6.698 (DRNAREA)0.9972 (1.006)PMPE (PFLATLOW + 1)-0.5603 41 0
Regional Regression Analysis 11

Table 2. Regression equations for estimating selected flow-duration curve exceedance probabilities for unregulated streams in 

Minnesota.—Continued

Regression equation SEE (percent) Number censored

Region D (47 streamgages)—Continued

D10 = 3.279 (DRNAREA)1.042 (1.007)PMPE (PFLATLOW + 1)-0.6111 51 0

D25 = 2.171 (DRNAREA)1.109 (1.009)PMPE (PFLATLOW + 1)-0.9597 91 1

D50 = 1.650 (DRNAREA)1.215 (1.011)PMPE (PFLATLOW + 1)-1.500 108 3

D75 = 0.5481 (DRNAREA)1.404 (1.013)PMPE (PFLATLOW + 1)-1.955 157 9

D90 = 0.4237 (DRNAREA)1.387 (1.014)PMPE (PFLATLOW + 1)-2.135 182 13

D95 = 0.6143 (DRNAREA)1.260 (1.016)PMPE (PFLATLOW + 1)-2.216 255 14

D99 = 0.09255 (DRNAREA)1.215 (1.021)PMPE (PFLATLOW + 1)-1.983 497 19

D99_9 = 0.02581 (DRNAREA)1.578 (1.023)PMPE (PFLATLOW + 1)-2.750 1,248 25

D99_99 = 0.04036 (DRNAREA)1.512 (1.023)PMPE (PFLATLOW + 1)-2.831 1,301 26

Region E (20 streamgages)

D0_01 = 2.034 (DRNAREA)0.6332 (CROPSNLCD01 + 1)0.9775 56 0

D0_1 = 0.4470 (DRNAREA)0.6875 (CROPSNLCD01 + 1)1.113 40 0

D2 = 0.03186 (DRNAREA)0.9674 (CROPSNLCD01 + 1)0.9967 39 0

D5 = 0.01140 (DRNAREA)1.069 (CROPSNLCD01 + 1)0.9321 53 0

D10 = 0.002035 (DRNAREA)1.146 (CROPSNLCD01 + 1)1.090 65 0

D25 = (3.855 x 10-5) (DRNAREA)1.226 (CROPSNLCD01+ 1)1.659 84 0

D50 = (1.941 x 10-7) (DRNAREA)1.340 (CROPSNLCD01 + 1)2.449 107 0

D75 = (1.078 x 10-10) (DRNAREA)1.557 (CROPSNLCD01 + 1)3.578 187 3

D90 = (1.707 x 10-19) (DRNAREA)1.635 (CROPSNLCD01 + 1)7.858 104 6

D95 = (1.070 x 10-22) (DRNAREA)1.665 (CROPSNLCD01 + 1)9.343 102 8

D99 = (6.127 x 10-30) (DRNAREA)1.999 (CROPSNLCD01 + 1)12.33 128 11

D99_9 = (1.368 x 10-33) (DRNAREA)2.364 (CROPSNLCD01 + 1)13.35 366 14

D99_99 = (3.975 x 10-38) (DRNAREA)3.312 (CROPSNLCD01 + 1)14.12 747 15

Region F (44 of 45 streamgages)

D0_01 = 255.5 (DRNAREA)0.9331 (FOREST + 1)-0.5002 (PFLATLOW + 1)-0.3321 31 0

D0_1 = 68.29 (DRNAREA)0.9256 (FOREST + 1)-0.4113 (PFLATLOW + 1)-0.05605 28 0

D2 = 4.081 (DRNAREA)1.102 (FOREST + 1)-0.2792 (PFLATLOW + 1)-0.1492 33 0

D5 = 1.313 (DRNAREA)1.122 (FOREST + 1)-0.1566 (PFLATLOW + 1)-0.08057 34 0

D10 = 0.7140 (DRNAREA)1.106 (FOREST + 1)-0.06831 (PFLATLOW + 1)-0.0492 33 0

D25 = 0.2575 (DRNAREA)1.073 (FOREST + 1)0.1421 (PFLATLOW + 1)0.004680 31 0

D50 = 0.09218 (DRNAREA)1.036 (FOREST + 1)0.3885 (PFLATLOW + 1)0.06342 30 0

D75 = 0.03823 (DRNAREA)1.018 (FOREST + 1)0.5786 (PFLATLOW + 1)0.1189 30 0

D90 = 0.01424 (DRNAREA)1.018 (FOREST + 1)0.7888 (PFLATLOW + 1)0.1919 27 0

D95 = 0.008009 (DRNAREA)1.030 (FOREST + 1)0.8961 (PFLATLOW + 1)0.2253 28 0

D99 = 0.004000 (DRNAREA)1.040 (FOREST + 1)1.022 (PFLATLOW + 1)0.2531 32 1

D99_9 = 0.002113 (DRNAREA)1.028 (FOREST + 1)1.151 (PFLATLOW + 1)0.2955 42 1

D99_99 = 0.002002 (DRNAREA)0.9861 (FOREST + 1)1.152 (PFLATLOW + 1)0.3661 55 1
12 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Table 3. Regression equations for estimating selected low-flow frequency statistics for unregulated streams in Minnesota.

[SEE, standard error of the estimate; number censored, number of streamflow values less than or equal to 0.1 cubic feet per second; M7D10Y, annual 7-day 

low flow for a recurrence interval of 10 years; DRNAREA, drainage area, in square miles; SSURGOM, percentage organic matter estimated using the Soil 

Survey Geographic Database; BSLDEM10M, mean basin slope computed from a 10-meter digital elevation model; M7D10Y_FAL, fall (October and November) 7-day low flow for a recurrence interval of 10 years; M30D10Y1011, fall 30-day low flow for a recurrence interval of 10 years; M7D10Y1203, winter 

(December through March) 7-day low flow for a recurrence interval of 10 years; M30D10Y123, winter 30-day low flow for a recurrence interval of 10 years; 

M7D10Y0405, spring (April and May) 7-day low flow for a recurrence interval of 10 years; M30D10Y45, spring 30-day low flow for a recurrence interval of 

10 years; M7D10Y0609, summer (June through September) 7-day low flow for a recurrence interval of 10 years; M30D10Y69, summer 30-day low flow for a 

recurrence interval of 10 years; M122D10Y69, summer 122-day low flow for a recurrence interval of 10 years; STORAGE, percentage of area of water storage 

in lakes, ponds, reservoirs, and wetlands; SOILC, percentage of area of hydrologic soil type C; PMPE, precipitation minus potential evapotranspiration, in millimeters; PFLATLOW, percentage of low-lying flatland; CROPSNLCD01, percentage of area of cultivated land]

Regression equation SEE (percent) Number censored

Region A (35 streamgages)

M7D10Y = (1.897 x 10-8) (DRNAREA)1.684 (SSURGOM + 1)2.546 (BSLDEM10M)3.664 665 22

M7D10Y_FAL = (8.463 x 10-6) (DRNAREA)1.222 (SSURGOM + 1)2.065 (BSLDEM10M)2.472 147 15

M30D10Y1011 = (1.596 x 10-5) (DRNAREA)1.275 (SSURGOM + 1)1.93 (BSLDEM10M)2.157 109 13

M7D10Y1203 = (3.166 x 10-7) (DRNAREA)1.449 (SSURGOM + 1)2.536 (BSLDEM10M)3.004 245 19

M30D10Y123 = (3.968 x 10-7) (DRNAREA)1.457 (SSURGOM +1)2.508 (BSLDEM10M)2.917 229 18

M7D10Y0405 = (5.596 x 10-5) (DRNAREA)1.33 (SSURGOM + 1)1.592 (BSLDEM10M)1.776 125 10

M30D10Y45 = (1.536 x 10-3) (DRNAREA)1.162 (SSURGOM + 1)1.259 (BSLDEM10M)0.9878 80 5

M7D10Y0609 = (2.390 x 10-7) (DRNAREA)1.461 (SSURGOM + 1)2.372 (BSLDEM10M)3.188 336 19

M30D10Y69 = (2.907 x 10-6) (DRNAREA)1.399 (SSURGOM + 1)1.828 (BSLDEM10M)2.485 171 14

M122D10Y69 = (3.847 x 10-4) (DRNAREA)1.200 (SSURGOM + 1)1.409 (BSLDEM10M)1.187 88 6

Region BC (49 streamgages)

M7D10Y = (5.491) (DRNAREA)1.152 (STORAGE + 1)-1.106 (SOILC + 1)-0.5761 95 5

M7D10Y_FAL = (4.823) (DRNAREA)1.066 (STORAGE + 1)-0.7602 (SOILC + 1)-0.5446 80 2

M30D10Y1011 = (4.979) (DRNAREA)1.027 (STORAGE + 1)-0.6926 (SOILC + 1)-0.4697 71 1

M7D10Y1203 = (5.295) (DRNAREA)1.13 (STORAGE + 1)-1.018 (SOILC + 1)-0.5363 93 5

M30D10Y123 = (6.096) (DRNAREA)1.101 (STORAGE + 1)-1.001 (SOILC + 1)-0.508 88 4

M7D10Y0405 = (3.741) (DRNAREA)1.031 (STORAGE + 1)-0.6304 (SOILC + 1)-0.3486 57 0

M30D10Y45 = (1.763) (DRNAREA)0.9214 (STORAGE + 1)-0.1277 (SOILC + 1)-0.1699 63 0

M7D10Y0609 = (4.391) (DRNAREA)1.118 (STORAGE + 1)-0.9835 (SOILC + 1)-0.5133 86 5

M30D10Y69 = (4.493) (DRNAREA)1.093 (STORAGE + 1)-0.9214 (SOILC + 1)-0.4692 80 4

M122D10Y69 = (2.648) (DRNAREA)0.8943 (STORAGE + 1)-0.4049 (SOILC + 1)-0.1264 62 0

Region D (47 streamgages)

M7D10Y = (2.161 x 10-3) (DRNAREA)1.212 (1.022)PMPE (0.901315)PFLATLOW 498 19

M7D10Y_FAL = (1.263 x 10-2) (DRNAREA)1.155 (1.016)PMPE (0.9029)PFLATLOW 242 13

M30D10Y1011 = (1.401 x 10-2) (DRNAREA)1.193 (1.016)PMPE (0.9026)PFLATLOW 267 13

M7D10Y1203 = (7.681 x 10-3) (DRNAREA)1.142 (1.017)PMPE (0.8970)PFLATLOW 329 17

M30D10Y123 = (1.077 x 10-2) (DRNAREA)1.14 (1.017)PMPE (0.8953)PFLATLOW 289 15

M7D10Y0405 = (6.765 x 10-2) (DRNAREA)1.177 (1.011)PMPE (0.9107)PFLATLOW 101 5

M30D10Y45 = (2.117 x 10-1) (DRNAREA)1.087 (1.011)PMPE (0.9125)PFLATLOW 98 4

M7D10Y0609 = (7.028 x 10-3) (DRNAREA)1.162 (1.020)PMPE (0.8948)PFLATLOW 402 16

M30D10Y69 = (7.587 x 10-3) (DRNAREA)1.237 (1.018)PMPE (0.8989)PFLATLOW 246 14

M122D10Y69 = (6.237 x 10-2) (DRNAREA)1.132 (1.014)PMPE (0.9238)PFLATLOW 93 4
Drainage-Area Ratio Calculations 13

Table 3. Regression equations for estimating selected low-flow frequency statistics for unregulated streams in Minnesota.—

Continued

Regression equation SEE (percent) Number censored

Region E (20 streamgages)

M7D10Y = (3.866 x 10-25) (DRNAREA)1.973 (CROPSNLCD01 + 1)9.971 115 11

M7D10Y_FAL= (2.036 x 10-21) (DRNAREA)1.681 (CROPSNLCD01 + 1)8.717 110 7

M30D10Y1011 = (1.040 x 10-17) (DRNAREA)1.631 (CROPSNLCD01 + 1)6.940 120 6

M7D10Y1203 = (3.350 x 10-28) (DRNAREA)1.978 (CROPSNLCD01 + 1)11.54 92 10

M30D10Y123 = (1.039 x 10-27) (DRNAREA)1.961 (CROPSNLCD01 + 1)11.38 89 10

M7D10Y0405 = (4.178 x 10-10) (DRNAREA)1.616 (CROPSNLCD01 + 1)3.149 285 5

M30D10Y45 = (8.254 x 10-6) (DRNAREA)1.403 (CROPSNLCD01 + 1)1.346 158 1

M7D10Y0609 = (4.648 x 10-23) (DRNAREA)1.612 (CROPSNLCD01 + 1)9.548 172 8

M30D10Y69 = (1.859 x 10-22) (DRNAREA)1.592 (CROPSNLCD01 + 1)9.401 122 7

M122D10Y69 = (1.844 x 10-10) (DRNAREA)1.487 (CROPSNLCD01 + 1)3.619 182 2

Region F (44 of 45 streamgages)

M7D10Y = (0.2560) (DRNAREA)1.084 (0.9663)CROPSNLCD01 (BSLDEM10M)0.4765 36 1

M7D10Y_FAL = (0.3163) (DRNAREA)1.069 (0.9707)CROPSNLCD01 (BSLDEM10M)0.4283 33 0

M30D10Y1011 = (0.3469) (DRNAREA)1.068 (0.9712)CROPSNLCD01 (BSLDEM10M)0.4129 30 0

M7D10Y1203 = (0.2940) (DRNAREA)1.083 (0.9660)CROPSNLCD01 (BSLDEM10M)0.4536 34 1

M30D10Y123 = (0.3243) (DRNAREA)1.084 (0.9670)CROPSNLCD01 (BSLDEM10M)0.4410 32 1

M7D10Y0405 = (0.4147) (DRNAREA)1.079 (0.9751)CROPSNLCD01 (BSLDEM10M)0.2612 30 0

M30D10Y45 = (0.6311) (DRNAREA)1.082 (0.9750)CROPSNLCD01 (BSLDEM10M)0.1574 30 0

M7D10Y0609 = (0.3130) (DRNAREA)1.077 (.9710)CROPSNLCD01 (BSLDEM10M)0.3580 37 0

M30D10Y69 = (0.3060) (DRNAREA)1.061 (0.9742)CROPSNLCD01 (BSLDEM10M)0.3691 31 0

M122D10Y69 = (0.4153) (DRNAREA)1.065 (0.9798)CROPSNLCD01 (BSLDEM10M)0.2417 23 0

are spring-fed as a result of the underlying karst topography 

(Ruhl, 1989), and the springs likely sustain streamflows during low-flow periods and reduce ice thicknesses in the winter 

time. Also, region F generally has higher-gradient streams than 

regions A, D, and E (Gesch and others, 2009), which corresponds to higher BSLDEM10M values (table 1‒1). Region BC 

is characterized by high-gradient streams in the northeastern 

part of the region (Gesch and others, 2009), lake-fed streams 

in the central and northern parts of the region, and spring-fed 

streams in the northwestern part of the region. In contrast, 

regions A, D, and E generally consist of low-gradient streams 

(Gesch and others, 2009) that develop thick ice in the winter. 

Ice formation shrinks the effective depth of stream channels and reduces winter streamflows (Melcher and Walker, 

1992). Summer flows in regions A, D, and E generally are low 

because potential evapotranspiration exceeds precipitation in 

most years, reducing recharge from groundwater (Delin and 

others, 2007; Lorenz and Delin, 2007). This hypothesis is further supported by small or negative PMPE values for regions 

A, D, and E (table 1‒1). 

Drainage-Area Ratio Calculations

For some ungaged locations on streams that have a 

streamgage at another location, a more accurate estimation 

of FDC or LFF statistics can be made based on ratios of the 

drainage areas between ungaged and gaged locations than 

from regional regression equations, provided landscapes did 

not change drastically within examined watersheds. For all 

FDC and LFF statistics computed in this study, the direct 

application of the drainage-area ratio (DAR) provides a 

good estimate of the value at the ungaged location for DARs 

ranging from 0.25 to 4. Of the 196 streamgages used in this 

statewide analysis, there were 25 pairs of streamgages from 

the same stream or tributary that were used to develop a DAR 

equation (table 4). The selected streamgages produced DARs 

greater than 0.25 and less than 1 (table 4). 

For the 25 selected pairs of streamgages, the relation 

between DARs and ratios of corresponding values for FDC 

and LFF statistics was compared for selected FDC exceedance probabilities (0.02, 0.10, 0.50, 0.90, and 0.99) and LFF 
14 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Table 4. Paired streamgages selected to develop within-basin equations that use drainage-area ratios to estimate streamflow 

statistics at ungaged locations on streams with a streamgage in another location.

[Basin name, the river basin containing the paired streamgages; hydrologic region, the hydrologic region of the basin; upstream, the farthest upstream of the 

paired streamgages; downstream, the farthest downstream of the paired streamgages; DAR, drainage-area ratio calculated by dividing the drainage area at the 

upstream streamgage by the drainage area at the downstream streamgage]

Basin name Hydrologic region 

(fig. 1) Upstream Downstream DAR

Mustinka River D 05047500 05049000 0.2536

Maple River A 05059700 05060000 0.5737

Buffalo River A 05061000 05062000 0.3403

Buffalo River A 05061500 05062000 0.4673

Clearwater River A 05078000 05078500 0.4078

Roseau River A 05104500 05107500 0.4246

Roseau River A 05107500 05112000 0.7140

Crow River D 05279000 05280000 0.4363

Redwood River D 05315000 05316500 0.4150

St. Croix River BC 05333500 05336000 0.5219

Eau Galle River F 05369945 05370500 0.5234

Zumbro River F 05374000 05374500 0.8590

Zumbro River F 05374500 05374900 0.9417

Whitewater River F 05376000 05376800 0.3701

Whitewater River F 05376800 05377500 0.9416

Root River F 05384000 05385000 0.4927

Root River F 05385000 05386000 0.8096

Upper Iowa River F 05387500 05388250 0.6655

Kickapoo River F 05407470 05408000 0.4407

Kickapoo River F 05410000 05410490 0.8970

Turkey River F 05411850 05412020 0.7079

Wapsipinicon River D 05420560 05420680 0.2784

Des Moines River D 05476000 05476500 0.8859

Skunk Creek E 06481480 06481500 0.4226

Little Sioux River D 06605600 06605850 0.8642

statistics (annual and seasonal 7-day mean low flows for a 

recurrence interval of 10 years; table 5). The relation was 

assessed visually and statistically using nonlinear least squares 

(Bates and Watts, 1988). None of the nonlinear least-squares 

results indicated that the coefficient for the DAR was significantly different from 1, resulting in the general DAR equation:

Statu = Statg * (DRNAREAu/DRNAREAg) (2)

where

Statu is the selected FDC or LFF statistic for the 

ungaged stream location,

Statg is the selected FDC or LFF statistic for the 

gaged stream location,

DRNAREAu is the drainage area of the ungaged stream 

location, and

DRNAREAg is the drainage area of the gaged stream 

location.

Figure 4 shows the relation between the DARs and the 

observed 0.5 FDC exceedance-probability ratios for the 25 

pairs of streamgages as an example, as well as a line that represents a 1:1 relation between DAR and 0.5 FDC exceedanceprobability ratio. The SEEs for selected FDC and LFF statistics calculated using DAR equations for DARs ranging from 

0.25 to 1 are reported in table 5. The relation between DARs 

and 0.5 exceedance-probability ratios for DARs ranging from 

1 to 4 can be obtained by plotting the reciprocals of DARs and 

reciprocals of the associated 0.5 exceedance-probability ratios 

presented in figure 4.
Drainage-Area Ratio Calculations 15

Table 5. Standard errors of selected flow-duration curve and 

low-flow frequency statistics based on statewide drainage arearatio equations developed in this study.

[SEE, standard error of the estimate; D2, streamflow with an exceedance 

probability of 0.02; D10, streamflow with an exceedance probability of 0.10; 

D50, streamflow with an exceedance probability of 0.50; D90, streamflow 

with an exceedance probability of 0.90; D99, streamflow with an exceedance 

probability of 0.99; M7D10Y, annual 7-day low flow for a recurrence interval 

of 10 years; M7D10Y_FAL, fall (October and November) 7-day low flow for 

a recurrence interval of 10 years; M7D10Y1203, winter (December through 

March) 7-day low flow for a recurrence interval of 10 years; M7D10Y0405, 

spring (April and May) 7-day low flow for a recurrence interval of 10 years; 

M7D10Y0609, summer (June through September) 7-day low flow for a recurrence interval of 10 years]

Statistic SEE

D2 62

D10 63

D50 65

D90 69

D99 113

M7D10Y 79

M7D10Y_FAL 72

M7D10Y1203 73

M7D10Y0405 66

M7D10Y0609 79

The SEEs of the selected flow statistics calculated using 

DAR equations (table 5) were compared to SEEs of the same 

flow statistics calculated using the regional regression equations (tables 2 and 3). For regions A, D, and E, comparisons 

of the SEEs (tables 2 and 5) indicate that regional regression equations may be more accurate than DAR equations 

for higher flows with exceedance probabilities less than or 

equal to 0.10. However, DAR equations seem to be more 

accurate than regional regression equations for lower flows 

with exceedance probabilities of greater than or equal to 0.50 

(tables 2 and 5). Furthermore, in regions A, D, and E, SEEs 

consistently were lower using DAR equations for annual and 

seasonal 7-day LFF statistics (tables 3 and 5). 

Comparisons of SEEs for region BC were less consistent 

than comparisons for regions A, D, and E. For FDC statistics, 

SEEs were lower for regional regression equations than for 

DAR equations for exceedance probabilities of 0.02, 0.10, 

and 0.50. Standard errors for exceedance probabilities of 0.90 

and 0.99 were within 1 percent but still slightly lower using 

regional regression equations (tables 2 and 5). In contrast, 

SEEs for annual and seasonal 7-day LFF statistics generally 

were lower using DAR equations than using regional regression equations (tables 3 and 5), with the exception of the 

spring (April and May) 7-day low flow. For the spring 7-day 

low flow, the SEE using the DAR equation was 66 percent 

(table 5), but the SEE using the regional regression equation 

was 57 percent (table 3).

For streamgages in region F, SEEs of FDC and LFF statistics indicate that regional regression equations consistently 

were more accurate than DAR equations. The observed differences between region F and other regions in the study may 

be related to the karst topography of region F (Ruhl, 1989). In 

karst topography, low flows could be affected substantially by 

gains from large springs and by losses from sinkholes (Eash 

and Barnes, 2012).

0

0.5

1.0

1.5

2.0

Observed 0.5 exceedence-probability ratio

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Drainage-area ratio

1:1 line

Figure 4. Relation between drainage-area ratio and the observed 0.5 flow-duration curve exceedance probabilities for the 25 pairs 

of streamgages listed in table 4.
16 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

Limitations of Regression Equations

Several limitations warrant consideration when using 

the regression equations presented in this report. The regional 

regression equations apply only to stream locations in Minnesota where flows are not substantially affected by regulation, 

diversion, or urbanization. The applicability and accuracy of 

the regional equations depend on whether the basin characteristics calculated for an ungaged stream location are within the 

range of the values for variables used to develop the regression 

equations. The acceptable range of basin characteristic values 

used to develop each regional regression equation are tabulated as maximum and minimum values in table 6. The mean 

is included to represent the most typical values for the range of 

the data; for PMPE, the mean is the arithmetic mean, and for 

all other basin characteristics, the mean represents geometric 

mean (table 6). Where the minimum value is 0, the geometric 

mean is adjusted by adding 1 to each value, computing the 

geometric mean of the adjusted values and subtracting 1 to 

produce the adjusted geometric mean. The applicability of the 

regional equations is unknown when any characteristic value 

calculated for an ungaged location is outside the acceptable 

range. In addition, basin-characteristic calculations at ungaged 

locations should be computed using the same GIS datasets and 

calculation methods used in this study. The USGS StreamStats 

Web-based GIS tool (http://water.usgs.gov/osw/streamstats/

index.html; Ries and others, 2008) includes the same GIS data 

layers and calculation methods as used to develop the regression equations in this study.

The regression equations presented in this report should 

be used with caution for ungaged stream locations with basincharacteristic values approaching the minimum or maximum 

values (table 6) because of the potential for inconsistencies in 

the estimates. Although the occurrence of inconsistencies were 

minimized by using the same set of explanatory variables for 

each regional set of LFF equations, inconsistencies still may 

occur. Inconsistencies may occur because regional regression 

equations were developed separately and have variable estimation intervals depending on the size and variability of the datasets used to develop regression equations. If inconsistencies 

are obtained for an ungaged stream location, a comparison of 

all LFF estimates for the location and a check of streamgage 

data or other published data may help determine which LFF 

statistic is inconsistent (Eash and Barnes, 2012). 

Several factors affect the estimation accuracy of the presented regression equations. Estimation accuracy depends on 

the sample size, the accuracy of each recorded streamflow, and 

how well the chosen distribution fits the actual distribution of 

the data (Lorenz and others, 2010). The accuracies of regression estimates are affected by errors in explanatory variables, 

and systematic errors in the computation of the response variable can bias estimates. Although streamgages used to develop 

the regression were weighted based on the number of years of 

record, the period of record also can bias estimates of response 

variables because of patterns in precipitation, flow conditions, 

other weather-related factors, and land use changes. Finally, 

estimating flow statistics at a location immediately downstream from a lake or wetland also could affect the accuracy of 

estimates made using the presented regression equations. 

The DAR equations to estimate streamflow statistics at 

ungaged locations on streams that have a streamgage were 

developed using paired streamgage data from all regions 

presented in this study (fig. 1; table 4). However, each 

region was not equally represented in the DAR equations. 

Regions BC and E each were represented only by one pair 

of streamgages. Regions A and D each were represented 

by 6 pairs of streamgages, and region F was represented by 

11 pairs of streamgages. Region E is the smallest region in 

Minnesota, and region BC contains a small number of suitable streamgages because of a lack of streamgages along the 

high-gradient north shore of Lake Superior, extensive connected wetlands in the northern part of the region, and rivers 

affected by regulation or diversion in the southwestern part of 

the region. 

The regression equations presented in this report also 

should be used with caution in areas where flows are affected 

by substantial gains as a result of large springs or substantial 

losses as a result of sinkholes common to karst topography. 

Region F, in southeastern Minnesota (fig. 1), contains karst 

areas where flows may vary considerably because of gaining or losing stream reaches (Eash and Barnes, 2012). User 

judgement may be required to decide if an ungaged location 

in a karst area may be affected by substantial gains or losses 

in flow, and regression estimates should be compared against 

streamgage data or other published data. If the equations are 

used at ungaged locations on regulated streams, or on streams 

affected by water-supply and agricultural withdrawals, then 

the estimates will need to be adjusted by the amount of regulation or withdrawal to estimate the actual flow conditions if that 

is of interest.

Special attention must be given to censored values and 

the number of significant figures used. Because of the uncertainty in measuring and estimating flows less than 0.1 ft3

/s, the 

censoring threshold used to develop the left-censored regression equations was set at 0.1 ft3

/s. Thus, any regression estimates that are 0.1 ft3

/s or less should be reported as less than 

0.1 ft3

/s. Because the precision of response- and explanatoryvariable data used to develop the equations commonly was 

limited to three significant figures, selected-statistic streamflows estimated from the regression equations also should be 

limited to three significant figures.

Constructing Consistent Flow-Duration 

Curves

Flow-duration curves are constructed from point estimates of streamflow for 13 exceedance probabilities using the 

regression equations developed for this report. The regression 

equations were developed to produce reasonably consistent 

FDCs, which means the magnitude of the predicted value 
Constructing Consistent Flow-Duration Curves 17

Table 6. Maximum, minimum, and mean values of the basin characteristics used in the regional regression analysis of each region in this study.[DRNAREA, drainage area; BSLDEM10M, mean basin slope; STORAGE, area of lakes and wetlands; PMPE, precipitation minus potential evapotranspiration; PFLATLOW, low-lying flatland; PFLATUP, upland flatland; LAKEAREA, area of lakes; FOREST, area of forested land; CROPSNLCD01, area of cultivated land; SSURGOM, organic matter; SSURGSAND, sand; SSURGSILT, silt; SSURGCLAY, clay; SSURGOKSAT, hydraulic conductivity of saturated soils; SOILA, soil in hydrologic group A; SOILB, soil in hydrologic group B; SOILC, soil in hydrologic group C; SOILD, soil in hydrologic group D]

Statistic

DRNAREA

BSLDEM10M

STORAGE 1

PMPE

PFLATLOW

PFLATUP

LAKEAREA

FOREST

CROPSNLCD01

SSURGOM

SSURGSAND

SSURGSILT

SSURGCLAY

SSURGOKSAT

SOILA

SOILB

SOILC

SOILD

Region A

Maximum 1,440 5.99 95.3 38.7 89.0 62.7 6.02 44.0 92.4 52.2 67.4 77.2 31.2 66.7 98.9 68.9 66.9 7.95

Mean 221 1.55 12.4 -45.7 36.0 29.1 0.46 5.62 46.9 2.29 42.7 30.9 16.6 26.3 21.3 24.9 22.1 1.29

Minimum 11.2 0.46 0.52 -123 5.99 4.00 0 0.18 1.34 0.29 13.8 15.8 7.16 8.61 2.01 0.25 0.49 0

Region BC

Maximum 2,960 10.7 48.0 308 40.8 28.4 21.4 76.5 74.5 19.8 76.4 41.0 36.5 72.5 75.1 63.0 100 43.5

Mean 177 4.15 22.0 184 16.6 5.49 2.91 40.7 4.93 5.86 47.5 21.7 9.33 31.5 16.8 14.8 28.7 3.07

Minimum 2.98 1.61 2.44 43.7 1.44 0.49 0 8.41 0 0 15.6 5.30 3.65 8.34 0 0 2.61 0

Region D

Maximum 2,640 9.38 17.4 203 56.9 40.7 6.40 7.76 90.0 3.05 60.4 46.5 30.6 48.9 32.2 82.8 77.1 17.0

Mean 327 2.56 4.60 42.0 26.3 17.7 1.13 1.43 72.3 0.81 36.7 34.4 23.6 16.4 2.78 44.7 38.9 0.93

Minimum 7.65 1.18 0.09 -132 4.88 1.01 0 0.22 3.46 0.32 20.8 22.8 13.9 5.17 0.10 18.6 11.1 0

Region E

Maximum 2,400 4.76 10.5 34.4 24.3 23.8 3.81 1.27 89.5 1.33 47.1 64.3 28.9 114 2.79 97.0 99.0 10.8

Mean 237 2.96 1.62 -18.8 13.2 11.0 0.42 0.51 72.2 0.58 25.0 43.0 24.5 23.9 0 33.1 31.3 1.06

Minimum 49.0 2.08 0.02 -82.7 2.61 5.76 0 0.30 41.1 0.34 6.87 30.0 20.7 6.78 0 0.43 2.21 0

Region F

Maximum 1,540 21.1 5.80 222 15.2 12.9 0.24 48.9 89.0 1.43 74.0 63.0 28.5 60.6 42.3 99.1 78.5 28.8

Mean 159 9.78 1.05 188 2.46 2.48 0.03 18.2 60.4 0.52 28.6 38.2 18.7 19.2 2.08 54.3 18.1 1.72

Minimum 2.28 3.41 0.01 129 0 0 0 0.69 20.8 0.29 4.67 15.4 7.37 4.87 0 8.76 0.50 0

1For PMPE, the mean is the arithmetic mean, but for all other basin characteristics, the mean represents the geometric mean.
18 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

decreases with increasing exceedance probability. However, 

the regression equations for the largest exceedance probabilities are less stable than the equations for the smaller exceedance probabilities because regression equations for the largest 

FDC exceedance probabilities were developed using censored 

data. Decreased stability resulting from the use of censored 

data can result in inconsistent FDCs.

Inconsistencies can be removed from FDCs by forcing 

succeeding magnitudes of increasing exceedance probability 

to be less than or equal to the previous value. The process of 

removing inconsistencies in regression-equation estimates 

of FDC exceedance probabilities is illustrated in figure 5. In 

the example (fig. 5), the predicted flow for the 0.999 exceedance probability (0.143 ft3

/s) is greater than the predicted 

value for the 0.99 exceedance probability (0.110 ft3

/s), but the 

predicted flow for the 0.9999 exceedance probability (0.061 

ft3

/s) is less than the predicted flow for the 0.99 exceedance 

probability (0.110 ft3

/s). Log-q-normal interpolation is used 

to compute a consistent flow value for the 0.999 exceedance 

probability. First, the logarithms of the flows for the 0.99 and 

0.9999 exceedance probabilities are computed. Second, the 

value for the logarithm of the flow for the 0.999 exceedance 

probability is linearly interpolated along the normal quantiles 

for the 0.99, 0.999, and 0.9999 exceedance probabilities. The 

normal quantiles of the 0.99, 0.999, and 0.9999 exceedance 

probabilities in this example are 2.326, 3.090, and 3.719, 

respectively. Back-transformation of the interpolated logarithm results in an estimated flow of 0.079 ft3

/s for the 0.999 

exceedance probability. However, if the flow for the 0.9999 

exceedance probability also had been larger than the flow for 

the 0.99 exceedance probability, then the flows for the 0.999 

and 0.9999 exceedance probabilities would have been set to 

the flow for the 0.99 exceedance probability (0.110 ft3/s).

StreamStats

StreamStats is a USGS Web-based GIS tool (http://water.

usgs.gov/osw/streamstats/index.html; Ries and others, 2008) 

that allows users to obtain streamflow statistics, basin characteristics, and other information for user-selected locations on 

streams. Users can select stream locations of interest from an 

interactive map and can obtain information for these locations. If a user selects the location of a USGS streamgage, 

the user will receive previously published information for the 

streamgage from a database. If a stream location is selected 

where no data are available (an ungaged location), a GIS program will estimate information for the location. The GIS program determines the boundary of the drainage basin upstream 

from the stream location, calculates the basin characteristics of 

the drainage basin, and solves the appropriate regression equations to estimate streamflow statistics for that location. The 

results are presented in a table and a map showing the basinboundary outline. The estimates are applicable for stream 

locations not substantially affected by regulation, diversions, 

or urbanization. In the past, it could take an experienced person more than a day to estimate this information at an ungaged 

location. StreamStats reduces the effort to only a few minutes.

EXPLANATION

Original flow-duration curve

Consistent flow-duration curve

0.01

0.1

1

10

100

1,000

10,000

Streamflow, in cubic feet per second

0.01 0.1 1 5 10 25 50 75 90 95 99 99.9 99.99

Percentage of time streamflow is exceeded

Figure 5. Example of 

how to construct a 

consistent flow-duration 

curve from regional 

regression estimates.
Summary 19

StreamStats makes the process of computing streamflow 

statistics for ungaged locations much faster, more accurate, 

and more consistent than previously used manual methods. Examples of streamflow statistics that can be provided 

by StreamStats include the 1-percent flood probability, the 

median annual flow, and the mean 7-day, 10-year low flow. 

Examples of basin characteristics include the drainage area, 

stream slope, mean annual precipitation, and percent of area 

underlain by hydrologic soil groups. Basin characteristics provided by StreamStats are the physical, geologic, and climatic 

properties that have been statistically related to the movement 

of water through a drainage basin to a stream location.

Use of the regression equations developed in this study 

to estimate FDC and LFF statistics will be facilitated through 

incorporation into the USGS StreamStats Web-based GIS tool 

(http://water.usgs.gov/osw/streamstats/index.html). Streamflow statistics can be estimated at any location along a stream 

and can assist with the following: (1) water-resources planning, management, and permitting; (2) design and permitting of facilities such as wastewater-treatment plants and 

water-supply reservoirs; and (3) design of structures such as 

roads, bridges, culverts, dams, locks, and levees. In addition, 

planners, regulators, and resource managers often need to 

know the physical, geologic, and climatic properties (basin 

characteristics) of the drainage basins upstream from locations 

of interest to help them understand the processes that control 

water availability and water quality at these locations.

Summary

Knowledge of the magnitude and frequency of low flows 

in streams, which are flows in a stream during prolonged dry 

weather, is fundamental for water-supply planning and design; 

waste-load allocation; reservoir storage design; and maintenance of water quality and quantity for irrigation, recreation, 

and wildlife conservation. The U.S. Geological Survey initiated a statewide study in 2012 in cooperation with the Minnesota Pollution Control Agency to estimate selected flowduration curve and low-flow frequency statistics for stream 

locations in Minnesota.

Regional regression equations were developed to estimate 

13 flow-duration curve statistics and 10 low-flow frequency 

statistics at ungaged stream locations in Minnesota. The 13 

flow-duration curve statistics estimated by regression equations include the 0.0001, 0.001, 0.02, 0.05, 0.1, 0.25, 0.50, 

0.75, 0.9, 0.95, 0.99, 0.999, and 0.9999 exceedance-probability quantiles. The low-flow frequency statistics include annual 

and seasonal (spring, summer, fall, winter) 7-day mean low 

flows, seasonal 30-day mean low flows, and summer 122-day 

mean low flows for a recurrence interval of 10 years. Estimates of the 13 flow-duration curve statistics and the 10 lowflow frequency statistics are provided for 196 U.S. Geological 

Survey continuous-record streamgages using streamflow data 

collected through September 30, 2012. 

The study area includes 196 streamgages located within 

Minnesota and 50 miles beyond the Minnesota border in the 

states of North Dakota, South Dakota, Iowa, and Wisconsin. The study area was divided into five regions that were 

developed in a previous study using the concept of hydrologic 

landscape units. Geographic information system software was 

used to calculate 18 selected basin characteristics for each 

streamgage, which were investigated as potential explanatory 

variables in regression analyses. Trend analyses indicated 

statistically significant trends in summer 7-day low flows that 

were not related to precipitation patterns for 19 streamgages. 

For 16 of these streamgages, the streamflow record was subset 

using structural change analysis to identify the most recent 

period of record without a significant trend. The three remaining streamgages with significant trends were excluded from 

the final analysis because the effective period of record without a significant trend was less than 10 years.

Because several streamgages in this study have minimum 

reported flows of zero, weighted left-censored regression was 

used to analyze the flow data in an unbiased manner, with 

weights based on the number of years of record. Flow-duration 

curve and low-flow frequency estimates calculated from censored regression equations that are 0.1 cubic foot per second 

or less should be reported as less than 0.1 cubic foot per second to maintain a consistent prediction-streamflow-reporting 

limit for Minnesota. Geographic information system software 

is required to calculate the basin characteristics included as 

explanatory variables in the regression equations. A total of 

115 regression equations were developed in this study to estimate 23 selected flow-duration curve and low-flow frequency 

statistics for ungaged locations in the study area. 

In addition to regional regression equations, data from 25 

pairs of streamgages were used to develop drainage-area ratio 

equations that can be used to estimate streamflow statistics 

at ungaged locations on streams or tributaries that have a 

streamgage in another location. Standard errors of selected 

streamflow statistics determined using drainage-area ratio and 

regional regression equations were compared among regions. 

For regions A, D, and E, drainage-area ratio equations were 

more accurate than regional regression equations for flows, 

but regional regression equations were more accurate for 

high flows. For region F, regional regression equations were 

consistently more accurate than drainage-area ratio equations. 

For region BC, there was not a consistent pattern in accuracies of regional regression and drainage-area ratio equations 

between low flows and high flows. All regional regression and 

drainage-area ratio equations presented in this study will be 

incorporated into StreamStats, a web-based geographic information systems tool developed by the U.S. Geological Survey. 

StreamStats allows users to obtain streamflow statistics, basin 

characteristics, and other information for user-selected locations on streams through an interactive map.

Several limitations warrant consideration when applying the results presented in this study. The regression equations developed in this study apply only to stream locations 

in Minnesota where flows are not substantially affected by 
20 Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams

regulation, diversion, or urbanization. Furthermore, the regression equations developed in this study are not intended for use 

at ungaged stream locations in which the basin characteristics 

are outside the range of those used to develop the equations. 

Inconsistencies in estimates may result for flow-duration curve 

and low-flow frequency equation estimates if basin-characteristic values approach the minimum or maximum values of the 

range. The regression equations should be used with caution 

in areas where low flows are affected by substantial gains 

as a result of large springs or substantial losses as a result of 

sinkholes common to karst topography. If the equations are 

used at ungaged locations on regulated streams, or on streams 

affected by water-supply and agricultural withdrawals, then 

the estimates will need to be adjusted if actual flow conditions 

are of interest.

Acknowledgments

David Bender and Kristen O’Connor of the U.S. Geological Survey (USGS) South Dakota Water Science Center 

developed the initial regional regression equations. Erich 

Kessler and Sarah Elliott of the USGS Minnesota Water Science Center helped conduct statistical analyses. Scott Bennett, 

a contract employee at the USGS Minnesota Water Science 

Center, created the study area map and compiled the basin 

characteristics. David Eash of the USGS Iowa Water Science 

Center and James Fallon of the USGS Minnesota Water Science Center provided technical reviews.

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Appendix 1. Detailed Streamgage Information Used to Conduct Analyses 23

Appendix 1. Detailed Streamgage Information Used to

Conduct Analyses

Table 1‒1 presents a summary of streamgages used in the analyses of this report. The 

map numbers presented in table 1‒1 correspond to the map numbers listed for the streamgages 

presented in figure 1. Table 1‒1 is presented as a Microsoft Excel® spreadsheet (http://dx.doi.

org/10.3133/sir20155170).
Publishing support provided by: 

Rolla Publishing Service Center

For more information concerning this publication, contact:

Director, USGS Minnesota Water Science Center

2280 Woodale Drive

Mounds View, Minnesota 55112

(763) 783–3100

Or visit the Minnesota Water Science Center Web site at: 

http://mn.water.usgs.gov/

Ziegeweid and others—Methods for Estimating Flow-Duration Curve and Low-Flow Frequency Statistics for Ungaged Locations on Small Streams—SIR 2015–5170

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20155170