U.S. DEPARTMENT OF THE INTERIOR

U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 00–4184

Equations for Estimating Clark Unit-Hydrograph 

Parameters for Small Rural Watersheds in Illinois

In cooperation with the Illinois Department of Natural Resources, Office of Water Resources
Equations for Estimating Clark Unit-Hydrograph 

Parameters for Small Rural Watersheds in Illinois

By Timothy D. Straub, Charles S. Melching, and Kyle E. Kocher

Urbana, Illinois

2000

In cooperation with the Illinois Department of Natural Resources, Office of Water Resources

Water-Resources Investigations Report 00–4184

U.S. DEPARTMENT OF THE INTERIOR

U.S. GEOLOGICAL SURVEY
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

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

not constitute endorsement by the U.S. Geological Survey.

For additional information write to:

District Chief

U.S. Geological Survey

221 North Broadway Avenue

Urbana, IL 61801

Copies of this report can be purchased from:

U.S. Geological Survey

Branch of Information Services

Box 25286

Denver, CO 80225-0286
Contents III

CONTENTS

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

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

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

Small Rural Watersheds for Which Rainfall and Runoff Data are Available .............................................................. 2

Acknowledgments ....................................................................................................................................................... 4

Synthetic Unit-Hydrograph Method...................................................................................................................................... 6

Clark Unit-Hydrograph Method .................................................................................................................................. 6

Previous Clark Unit-Hydrograph Parameter Relations for Watersheds in Illinois ...................................................... 7

Determining and Evaluating Clark Unit-Hydrograph Parameters......................................................................................... 9

Storm Selection............................................................................................................................................................ 9

Hydrograph-Parameter Determination ........................................................................................................................ 10

Equation Development ................................................................................................................................................ 12

Results Based on Methods Similar to the Graf and others (1982b) Study ........................................................ 12

Results Based on Average Values of Time of Concentration and Storage Coefficient for Each Watershed...... 12

Results Using Time of Concentration and Storage Coefficient for All Storms Independently ......................... 13

Comparison of Calibrated Hydrograph-Parameter Values with Results Obtained Using the 

Graf and others (1982b) Method............................................................................................................................. 13

Equation Verification ................................................................................................................................................... 15

Testing with Lake County Data ................................................................................................................................... 18

Application Limits for the Estimation Equations ........................................................................................................ 19

Application Example ................................................................................................................................................... 22

Summary and Conclusions .................................................................................................................................................... 23

References Cited.................................................................................................................................................................... 24

FIGURES

1–2. Maps showing:

1. Location of stations where rainfall and discharge data were collected and used in the development 

and verification of equations for estimating time of concentration and storage coefficient for the 

Clark unit hydrograph for small watersheds in Illinois...................................................................................... 5

2. Regional values of R/(TC + R) determined by Graf and others (1982b) for Illinois........................................... 8

3–4. Diagrams showing:

3. Time of concentration for storms on 41 watersheds in Illinois for the Clark unit-hydrograph method 

measured (average) and computed as a function of watershed main-channel length and slope........................ 14

4. Storage coefficient for storms on 41 watersheds in Illinois for the Clark unit-hydrograph method 

measured (average) and computed as a function of watershed main-channel length and slope........................ 15

5–7. Measured direct-runoff hydrograph and computed direct-runoff hydrograph simulated with the Clark 

unit-hydrograph method in the U.S. Army Corps of Engineers Flood Hydrograph Package HEC-1 with 

the time of concentration and storage coefficient determined from the estimation equations for Illinois 

developed in this study for the storms of 

5. April 21, 1973, on South Branch Kishwaukee River......................................................................................... 21

6. August 17, 1959, on Andy Creek Tributary ....................................................................................................... 21

7. April 21, 1973, on West Bureau Creek Tributary .............................................................................................. 22

TABLES

1. Characteristics of small rural watersheds in Illinois and the number of storms used for developing and 

verifying equations for the estimation of Clark unit-hydrograph parameters......................................................... 3
Contents IV

2. Average values of measures of calibration quality for the calibrated Clark unit-hydrograph method for all 

storms on selected small rural watersheds in Illinois utilized to develop and verify the equations for 

estimation of time of concentration and storage coefficient................................................................................... 11

3. Coefficient of determination and standard error for logarithmic data in the equations for estimating 

time of concentration for the Clark unit hydrograph .............................................................................................. 13

4. Coefficient of determination and standard error for logarithmic data in the equation for estimating the 

watershed storage coefficient for the Clark unit hydrograph.................................................................................. 13

5. Time of concentration for the Clark unit-hydrograph method estimated with equations developed in this 

study and with the method of Graf and others (1982b) compared to the average values determined from 

calibration for all storms used in equation development on selected small rural watersheds in Illinois................ 16

6. Storage coefficient estimated with equations developed in this study and with the method of Graf and 

others (1982b) compared to the average values determined from calibration for all storms used in 

equation development on selected small rural watersheds in Illinois..................................................................... 17

7. Time of concentration for the Clark unit-hydrograph method estimated with equations developed in this 

study and with the method of Graf and others (1982b) compared to the average values determined from 

calibration for all storms used for verification of the developed equation on selected small rural 

watersheds in Illinois .............................................................................................................................................. 18

8. Storage coefficient estimated with an equation developed in this study and with the method of Graf 

and others (1982b) compared to the average values determined from calibration for all storms used for 

verification of the developed equation on selected small rural watersheds in Illinois............................................ 19

9. Percentage error in estimated peak discharge and time-to-peak discharge for the verification storms 

on selected small rural watersheds in Illinois, simulated with the Clark unit-hydrograph method utilizing 

estimated values of time of concentration and watershed-storage coefficient developed in this study.................. 20

10. Characteristics of watersheds in Lake County, Ill., selected for testing the equations developed to 

estimate time of concentration and storage coefficient for small rural watersheds in Illinois................................ 22

11. Average percentage error in the estimated peak discharge and time-to-peak discharge for the storms on 

selected watersheds in Lake County, Ill., simulated with the Clark unit-hydrograph method utilizing 

estimated values of time of concentration and storage coefficient computed with equations developed 

in this study for small rural watersheds in Illinois.................................................................................................. 23

12. Parameters for the Clark unit-hydrograph method determined from calibration of the U.S. Army Corps 

of Engineers (1990) Flood Hydrograph Package HEC-1 and measures of calibration quality for data 

from small rural watersheds in Illinois, for storms utilized to develop and verify the equations for 

estimation of Clark unit-hydrograph parameters.................................................................................................... 27

CONVERSION FACTORS

Multiply By To obtain

Length

inch (in.) 25.4 millimeter

mile (mi) 1.609 kilometer

Area

square mile (mi2) 2.590 square kilometer 

Flow rate

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

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer
Introduction 1

Equations for Estimating Clark Unit-Hydrograph

Parameters for Small Rural Watersheds in Illinois

By Timothy D. Straub, Charles S. Melching, and Kyle E. Kocher

Abstract

Equations for estimating the time of concentration (TC) and storage coefficient (R) of the Clark 

unit-hydrograph method were developed for small 

rural watersheds [0.02–2.3 square miles (mi2)] in 

Illinois. The equations will provide State and local 

engineers and planners with more accurate methods to estimate the TC and R for use in simulating 

discharge hydrographs on small rural watersheds 

when designing stormwater-management facilities 

and other hydraulic structures, determining floodplain boundaries, and assessing the safety of structures in rivers.

The rainfall and runoff data from gaged 

small rural watersheds (0.02–2.3 mi2) with insignificant amounts of impervious land cover in Illinois were used to develop the equations. Equations 

were developed on the basis of data for 121 storms 

that occurred in 39 watersheds. Data for 29 storms 

in 18 watersheds were used to verify the equations.

TC and R were determined by calibrating 

available rainfall and runoff data, using the 

U.S. Army Corps of Engineers Flood Hydrograph 

Package HEC-1. The mathematical relations 

between watershed and storm characteristics, 

and TC and R were determined by multiple-linear 

regression of the logarithms of the values. Mainchannel length and slope were identified as important watershed characteristics for estimating TC

and R. The estimation equations had coefficients 

of determination of 0.73 and 0.64 for the logarithms of TC and R, respectively. When storm 

characteristics were added in the regression of 

hydrograph parameters utilizing length and slope, 

only minimal increases to the coefficient of determination resulted. Thus, storm characteristics were 

not considered further in development of the equations.

Simulation of the measured discharge 

hydrographs for the verification storms utilizing TC

and R obtained from the estimation equations 

yielded good results. The error in peak discharge 

for 21 of the 29 verification storms was less than 25 

percent, and the error in time-to-peak discharge for 

18 of the 29 verification storms also was less than 

25 percent. Therefore, applying the estimation 

equations to determine TC and R for design-storm 

simulation may result in reliable design hydrographs, as long as the physical characteristics of 

the watersheds under consideration are within the 

range of those characteristics for the watersheds 

in this study [area: 0.02–2.3 mi2, main-channel 

length: 0.17–3.4 miles, main-channel slope: 

10.5–229 feet per mile, and insignificant percentage of impervious cover].

[A compact disk containing the rainfall and 

runoff data, HEC-1 input files and digital format of 

this report is included with the report.]

INTRODUCTION

Designing stormwater-management facilities 

and other hydraulic structures (such as culverts and 

bridge waterways), determining flood-plain boundaries, and assessing the safety of structures in rivers 

typically involve applying a design hydrograph. These 

design hydrographs are computed on the basis of 

design storms of a specified probability of occurrence 

determined from standard references, such as the 
2 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

U.S. Weather Bureau Technical Paper Number 40 

(Hershfield, 1961) or the Illinois State Water Survey 

Bulletin 70 (Huff and Angel, 1989). Abstractions from 

rainfall resulting from interception, depression storage, 

and infiltration then are determined on the basis of 

available data from the literature and considering 

the effects of the soil type, land cover/land use, and 

antecedent-moisture conditions. Typically, the Soil 

Conservation Service (SCS, now known as the Natural 

Resources Conservation Service) (1985) curve-number 

method is applied to determine the abstractions. By 

subtracting the abstractions from the design rainfall, the 

precipitation excess, which approximately equals the 

direct runoff (effective precipitation) resulting from the 

design storm, is obtained. By utilizing a synthetic unit 

hydrograph, the precipitation excess then is transformed into a simulated discharge hydrograph at the 

outlet of the watershed. Areas larger than 1 mi2

 often 

are subdivided into a number of subwatersheds, and the 

runoff hydrographs from each subwatershed is routed 

to the watershed outlet with hydrologic- or hydraulicrouting methods.

This procedure for determining design hydrographs described above is utilized by the Illinois 

Department of Natural Resources, Office of Water 

Resources (IDNR-OWR) for many different waterresources-management issues. When determining 

design hydrographs, the IDNR-OWR typically divides 

watersheds into subwatersheds less than 1 mi2

 in area. 

Synthetic hydrographs then are developed for each subwatershed, utilizing the Clark (1945) unit-hydrograph 

method as implemented in the U.S. Army Corps of 

Engineers (1990) Flood Hydrograph Package (HEC-1). 

In the Clark method, the time of concentration (TC) and 

storage coefficient (R) for a watershed must be specified. Therefore, values of TC and R must be estimated 

for each subwatershed. Equations have been developed 

by Graf and others (1982a,b) and by Melching and 

Marquardt (1996) that relate TC and R to watershed 

characteristics for watersheds in Illinois. Most data 

used to derive these equations were collected for 

watersheds with drainage areas greater than 10 mi2

. 

Therefore, the U.S. Geological Survey (USGS), in 

cooperation with the IDNR-OWR, began a study in 

1998 to develop a new set of equations to estimate TC

and R derived from data for small rural watersheds 

(0.02–2.3 mi2

). These estimated values then could be 

used to apply the HEC-1 model with the Clark unithydrograph method in hydrologic design and analysis 

in Illinois.

Purpose and Scope

This report describes the results of the study to 

develop improved equations for estimating TC and R for 

small rural watersheds (0.02–2.3 mi2) in Illinois. The 

new equations will provide State and local engineers 

and planners with more accurate methods to estimate 

the TC and R typically used to estimate design hydrographs. These more accurate estimates of TC and R

should result in more reliable design hydrographs 

relative to the current practice for water-resourcesmanagement activities, including designing 

stormwater-management facilities and other 

hydraulic structures, determining flood-plain 

boundaries, and assessing the safety of structures 

in rivers.

Selection and analysis of storms for use in calibrating and verifying the estimation equations for the 

hydrograph parameters is described. The established 

data base of rainfall and runoff data from gaged small 

rural watersheds in Illinois with insignificant amounts 

of impervious land cover is listed in table 1. For all 

storms in the data base, the direct-runoff depth is 

greater than 0.4 in. Watershed areas used in the storm 

analysis range from 0.02 to 2.3 mi2

. TC and R for each 

storm were derived by calibrating rainfall and runoff 

data using the HEC-1 model. Multiple-linear regression 

techniques were used to develop mathematical relations 

that express TC and R as functions of watershed characteristics for 39 small rural watersheds in Illinois. To verify and test the accuracy of the developed equations, 

two methods were used. For the verification storms, 

verifying the equations involved comparing TC and R

values derived by calibration using HEC-1 to TC and R

values estimated with the equations developed in this 

study. Verification also included comparing hydrographs computed on the basis of the estimated values of 

TC and R to measured hydrographs for the verification 

storms.

Small Rural Watersheds for Which Rainfall and 

Runoff Data are Available

For the purpose of flood-frequency analysis on 

small streams in Illinois, a network of streamflow and 

rainfall gages on watersheds less than 10 mi2

 was established throughout the State (Curtis, 1977). The USGS 

operated these gages from 1956 to 1975. Typically, 

continuous records of rainfall and streamflow were 

collected for a period of 2-4 years, which was long 
Introduction 3

Table 1. Characteristics of small rural watersheds in Illinois and the number of storms used for developing and verifying equations for the 

estimation of Clark unit-hydrograph parameters

[mi2, square miles; mi, miles; ft/mi, feet per mile; ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture; na, not applicable]

Station

number Watershed

Number of

storms for

equation

development

Number of

storms for

verification

Drainage

area

(mi2)

Main channel Forest

area

(percent)

Length

(mi)

Slope

(ft/mi)

03336100 Big Four Ditch Tributary near Paxton 1 0 1.05 2.16 21.0 3.72

03338100 Salt Fork Tributary near Catlin 2 1 2.20 3.40 15.8 .00

03338800 North Fork Vermilion River Tributary

near Danville

2 0 1.42 2.21 33.2 1.61

03341900 Raccoon Creek Tributary near Annapolis 3 2 .04 .303 52.8 .00

03344250 Embarras River Tributary near Greenup 1 1 .08 .38 10.5 .00

03380300 Dums Creek Tributary near Iuka 6 0 .08 .403 98.7 29.4

03380450 White Feather Creek near Marlow 5 2 .43 1.11 87.7 17.5

03381600 Little Wabash River Tributary near New Haven 2 0 .16 .62 89.8 43.0

03382025 Little Saline Creek Tributary near Goreville 4 1 .52 1.13 75.5 33.0

03385500 Lake Glendale Inlet near Dixon Springs 1 0 1.05 1.86 145 72.0

03612200 Q Ditch Tributary near Choat 2 0 .27 .80 141 40.0

05438850 Middle Branch of South Branch Kishwaukee River

near Malta

1 1 1.67 2.60 28.7 .00

05439550 South Branch Kishwaukee River Tributary

near Irene

0 2 1.71 2.22 53.8 6.00

05440900 Leaf River Tributary near Forreston 1 0 .15 .814 144 .00

05448050 Sand Creek near Milan 1 0 .22 .758 67.1 .00

05469750 Ellison Creek Tributary near Roseville 4 0 .26 1.67 28.8 .00

05495200 Little Creek near Breckenridge 8 1 1.45 1.82 34.5 5.27

05496900 Homan Creek Tributary near Quincy 1 0 .50 1.29 106 4.00

05502120 Kiser Creek Tributary near Barry 7 3 .78 1.20 78.7 15.0

05551800 Fox River Tributary Number Two near Fox 2 0 .45 1.02 87.1 7.00

05554600 Mud Creek Tributary near Odell 7 0 .16 .79 60.7 .00

05557100 West Bureau Creek Tributary near Wyanet 0 1 .33 1.65 97.2 .00

05558050 Coffee Creek Tributary near Florid 1 1 .03 .303 229 .00

05558075 Coffee Creek Tributary near Hennepin 3 1 .22 .852 139 2.00

05572100 Wildcat Creek Tributary near Monticello 3 0 .10 .330 18.0 .00

05577700 Sangamon River Tributary at Andrew 5 0 1.50 1.36 40.1 1.13

05586200 Illinois River Tributary at Florence 3 0 .49 1.11 132 23.8

05586500 Hurricane Creek near Roodhouse 3 0 2.30 3.30 24.3 5.16

05586850 Bear Creek Tributary near Reeders 3 0 .02 .17 63.4 .00

05587850 Cahokia Creek Tributary near Carpenter 2 0 .45 .918 42.5 12.7

05592700 Hurricane Creek Tributary near Witt 5 3 .14 .44 27.1 .00

05594200 Williams Creek near Cordes 3 0 1.90 2.88 17.2 2.12

05596100 Andy Creek Tributary at Valier 3 1 1.03 1.78 39.0 10.0

05599640 Green Creek Tributary near Jonesboro 3 0 .43 1.19 112 74.6

ISWSFS01 ISWS Field Site 1 near Highland 2 0 .073 .47 26.4 .00

ISWSFS06 ISWS Field Site 6 near Highland 3 1 .096 .48 24.5 .00

USDAIA1M USDA Watershed IA1 near Monticello 1 2 .048 .33 54.4 .00

USDAIAMO USDA Watershed IA near Monticello 2 3 .128 .49 33.5 .00

USDAIBMO USDA Watershed IB near Monticello 6 2 .071 .53 31.9 .00

USDAW1ED USDA Watershed W-1 near Edwardsville 4 0 .043 .21 72.3 .00

USDAW4ED USDA Watershed W-4 near Edwardsville 5 0 .453 .81 196 .00

Total 121 29 na na na na
4 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

enough to measure six or more storms that could be 

used for model calibration. The continuous-record 

gages then were moved to another location and a creststage gage was used to measure annual peak flows at 

the given site.

Rainfall and streamflow data from two stations 

near Edwardsville, Ill., which were operated by the 

U.S. Department of Agriculture (USDA) from 1938 to 

1955, were used in this study. The USDA and University of Illinois at Urbana-Champaign operated rainfall 

and streamflow gages on small watersheds at Allerton 

Farms and Park near Monticello, Ill., from 1949 to 

1983. Data from three USDA Monticello gages are 

used in this study. Monitoring of the USDA watersheds 

at Allerton Farms and Park near Monticello, Ill., was 

supported by USDA Hatch project funds. The Illinois 

State Water Survey (ISWS) operated rainfall and 

streamflow gages on small watersheds near Highland 

Silver Lake, Ill., from 1981 to 1984. Data from two 

ISWS gages were used in this study.

Data collected from 150 storms with directrunoff depths greater than 0.4 in. occurring in 41 watersheds were used in the study (fig. 1, table 1). Useful 

storm rainfall and runoff data are available for only 

30 watersheds with areas less than 1 mi2

 in Illinois, 

which are of interest with respect to current IDNROWR practice. To broaden the range of storms, watershed conditions, locations within the State, and data 

from slightly larger watersheds (up to 2.3 mi2

) were 

included in the analysis. Inclusion of data from slightly 

larger watersheds should increase the general applicability of the developed equations for estimating TC and 

R on watersheds less than 1 mi2 without biasing the 

equations.

In the hydrologic data base, watershed areas 

range from 0.02 to 2.3 mi2

 and have insignificant 

impervious areas (only one or two country roads and 

an occasional rooftop from a house or shed). Most of 

the watersheds also have insignificant forest cover 

(table 1). The main-channel length of streams within 

the watersheds ranged from 0.17 to 3.4 mi (table 1). 

The main-channel length is measured along the main 

channel from the watershed outlet to the watershed 

divide. Main-channel slope ranged from 10.5 to 

229 ft/mi. For all USGS, ISWS, and USDA Monticello 

sites, the main-channel slope was determined from 

elevations at points 10 and 85 percent of the distance 

along the main channel from the watershed outlet to 

the watershed divide. The slope and length were determined from USGS topographic maps (if an adequate 

number of contours were within the watershed) or from 

a field survey (for the smaller watersheds with an inadequate number of contours). Slopes for the two USDA 

watersheds near Edwardsville were determined from a 

report describing the length, area, and percent slope 

(U.S. Department of Agriculture, 1957). The slope 

values were given in the form “63% of the watershed is 

in 0–1.5% class; 21% in 1.5–4%; 9% in 4–7%; 7% in 

7–12%.” For these two watersheds, 10 percent of the 

gentlest sloping area and 15 percent of the steepest 

sloping area were removed and a weighted average was 

computed from the remaining values to determine the 

slope.

Data from 150 storms that occurred in 41 watersheds were used in the calibration and verification of 

the equations (table 1). Equations were developed using 

data for 121 storms that occurred in 39 watersheds 

(table 1). In the verification, data for 29 storms that 

occurred in 18 watersheds were used (table 1). The 

storms used in the verification generally were double 

peaked and too complicated to use in equation development. Two of the watersheds, from which verification 

data were available, were not used in the equation 

development. Additional testing was done using 

74 storms that occurred in 9 watersheds from a 

study done in Lake County, Illinois (Melching and 

Marquardt, 1996). To test the limitations of the equations, these watersheds were chosen because the physical characteristics of these watersheds were not within 

the limitations of the types of watersheds used in the 

equation development and verification. The watersheds 

that were used violated one or more limitation: larger 

than 2.3 mi2

, a significant amount of impervious cover, 

or a main-channel slope smaller than 10.5 ft/mi.

Acknowledgments

Timothy B. Kosiek, USGS employee from 1997 

to 1999 in Urbana, Ill., greatly contributed to retrieving, 

processing, and tabulating the data. Data from Allerton 

Farms and Park watersheds were provided by J.K. 

Mitchell, Department of Agricultural Engineering, 

University of Illinois, Urbana-Champaign. Vernon 

Knapp provided data collected at watersheds monitored 

by the ISWS. USDA and ISWS data were useful in 

compiling a large data base of storms with a wide 

spatial distribution of watersheds.
Introduction 5

Figure 1. Location of stations where rainfall and discharge data were collected and used in the 

development and verification of equations for estimating time of concentration and storage 

coefficient for the Clark unit hydrograph for small watersheds in Illinois.
6 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

SYNTHETIC UNIT-HYDROGRAPH METHOD

Synthetic unit-hydrograph methods are utilized 

to describe the entire unit hydrograph for a gaged 

watershed with only a few hydrograph parameters. 

Needed hydrograph parameters vary among the 

different synthetic unit-hydrograph methods. These 

hydrograph parameters can be related to the characteristics of the watersheds and storms from which the 

parameters were determined. This method can be 

applied to ungaged watersheds with geomorphology, 

soils, land cover/land use, and climate similar to the 

gaged watersheds. Many synthetic unit-hydrograph 

methods have been proposed in the hydrologic literature. In this report, only the Clark (1945) unithydrograph method is considered because this method 

commonly is applied for hydrologic design and analysis in Illinois.

Clark Unit-Hydrograph Method

The processes of translation and attenuation 

dominate the movement of flow through a watershed. 

Translation is the movement of flow downgradient 

through the watershed in response to gravity. Attenuation results from the frictional forces and channelstorage effects that resist the flow. Clark (1945) noted 

that the translation of flow throughout the watershed 

could be described by a time-area curve, which 

expresses the curve of the fraction of watershed area 

contributing runoff to the watershed outlet as a function 

of time since the start of effective precipitation. Effective precipitation is that precipitation that is neither 

retained on the land surface nor infiltrated into the soil 

(Chow and others 1988, p. 135). The time-area curve 

is bounded in time by the watershed TC. Thus, TC is a 

hydrograph parameter of the Clark unit-hydrograph 

method. Attenuation of flow can be represented with a 

simple, linear reservoir for which storage is related to 

outflow as

S = RO, (1)

where

S is the watershed storage,

R is the watershed-storage coefficient, and

O is the outflow from the watershed.

Therefore, Clark (1945) proposed that a synthetic unit hydrograph could be obtained by routing 

1 in. of direct runoff to the channel in proportion to the 

time-area curve and routing the runoff entering the 

channel through a linear reservoir.

Numerous researchers have found that determining the time-area curve for the watershed was not 

needed to obtain a reasonable unit hydrograph. For 

example, Turner and Burdoin (1941) and O’Kelly 

(1955) found that reasonable unit hydrographs were 

obtained when simple geometric shapes were substituted for the actual time-area curve. Experience with 

the Clark unit-hydrograph method at the U.S. Army 

Corps of Engineers, Hydrologic Engineering Center, 

indicates that a detailed time-area curve usually is 

not necessary for accurate synthetic unit-hydrograph 

estimation (Ford and others, 1980). In most instances, 

the dimensionless time-area curve included in HEC-1 

(U.S. Army Corps of Engineers, 1990) is satisfactory 

for obtaining a reliable synthetic unit hydrograph.

In Illinois, HEC-1 (U.S. Army Corps of Engineers, 1990) typically is utilized to compute the Clark 

unit hydrograph. TC and R are the hydrograph parameters required for HEC-1 computation of the Clark unit 

hydrograph. The TC for the Clark unit hydrograph is 

slightly different than the typical definition applied in 

stormwater management, such as that in the Rational 

method (Kuichling, 1889). In the typical definition, 

the time of concentration (tc) is the traveltime for the 

first drop of effective precipitation at the hydraulically 

most distant point in the watershed to reach the watershed outlet. In the Clark unit-hydrograph method, TC

 is the time from the end of effective precipitation 

to the inflection point of the recession limb of the 

runoff hydrograph. The inflection point on the runoff 

hydrograph corresponds to the time when overland flow 

to the channel network ceases and beyond that time the 

measured runoff results from drainage of channel storage. Therefore, Clark’s TC is the traveltime required for 

the last drop of effective precipitation at the hydraulically most distant point in the watershed to reach the 

channel network. From a linear-system theory and 

the conceptual model of pure translatory flow, the two 

definitions of time of concentration are equivalent. The 

subtle differences, however, between the definition of 

time of concentration in the Rational method and in 

the Clark unit-hydrograph method imply the time of 

concentration estimation equations commonly applied 

in the Rational method may not be appropriate for 

application to the Clark unit-hydrograph method. In 

most applications of HEC-1, TC is determined from 

values calibrated with measured rainfall and runoff data 

either directly, by scaling from hydrologically similar 
Synthetic Unit-Hydrograph Method 7

watersheds, or from equations, such as those developed 

in this study.

Previous Clark Unit-Hydrograph Parameter 

Relations for Watersheds in Illinois

Methods and equations have been developed previously that relate TC and R to watershed characteristics 

for watersheds in Illinois (Graf and others, 1982a,b; 

Melching and Marquardt, 1996). Most data used to 

derive these equations were collected for watersheds 

with areas larger than 10 mi2

.

TC and R values for the study by Graf and others 

(1982a,b) were determined for 98 watersheds in Illinois 

ranging in size from 0.45 to 362 mi2

 by calibration 

of HEC-1 for rainfall and runoff data for six to eight 

storms per watershed. Multiple-regression analysis 

was applied to determine relations among (TC+R), 

R/(TC+R), and watershed characteristics. These combined parameters were utilized to reduce the effects 

of correlation between TC and R. The relation among 

(TC+R), main-channel length, and main-channel slope 

was determined as

(TC+R) = 35.2 L0.39S -0.78, (2)

where

L is the stream length measured along the main

channel from the watershed outlet to the

watershed divide, in mi, and

S is the main-channel slope determined from 

elevations at points that represent 10 and 

85 percent of the distance along the channel

from the watershed outlet to the watershed

divide, in ft/mi.

Regional values of R/(TC+R) were determined 

for various areas of the State (fig. 2). The hypothesis 

was that these regional values account, in part, for 

aspects of watershed geomorphology and land cover/

land use not considered in the analysis, such as impervious and wetland areas. Scattergrams of the estimated 

and measured TC and R values showed no clear separation of the results for the 19 urban watersheds studied 

relative to the results for all other watersheds.

TC and R values for the Melching and Marquardt 

(1996) study were determined from 66 storms with 

effective-precipitation depths greater than 0.4 in. on 

9 watersheds (areas between 0.06 and 37 mi2

, mainchannel length: 0.33–16.6 mi, main-channel slope: 

3.13–55.3 ft/mi, and percentage of impervious cover: 

7.32–40.6). Data from 11 storms on 8 of these watersheds were utilized to verify (test) the Melching and 

Marquardt (1996) estimation equations. The peak 

discharge for 8 of the 11 storms was estimated within 

25 percent and the time-to-peak discharge for 10 of the 

11 storms was estimated within 20 percent. Separate 

sets of equations were developed with watershed area 

and main-channel length as the starting parameters. 

Percentage of impervious cover, main-channel slope, 

and depth of effective precipitation also were identified 

as important watershed and storm characteristics for 

estimation of TC and (or) R. The equations for estimating TC and R, in hours, as a function of watershed and 

storm characteristics with watershed area as the primary watershed characteristic are

TC= 39.1A0.577(I+1)-1.146D0.781 and (3)

R=123 A0.390(I+1)-0.722S -0.303, (4)

where

I is the percentage of impervious cover;

A is the watershed area, in mi2

; and

D is the effective precipitation depth, in inches.

The equations for estimating TC and R, in hours, as a 

function of watershed and storm characteristics with 

main-channel length (L, in mi) as the primary watershed characteristic are

 TC =87.5 L0.868(I+1)-1.563D0.780 and (5)

R=81.1 L0.759(I+1)-0.994. (6)

The differences in TC and R estimated with the 

area-based equations and length-based equations may 

be substantial; however, the differences in the final 

computed hydrographs may be small. Melching and 

Marquardt (1996, p. 26) recommended that the design 

hydrographs obtained using the area-based and lengthbased equations should be compared and the most 

reasonable hydrograph applied.

When applying the results of Graf and others 

(1982b) and Melching and Marquardt (1996) to estimate TC and R for small rural watersheds (less than 

1 mi2

) in Illinois, three general problems occur. These 

problems are discussed in detail below.
8 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Figure 2. Regional values of R/(TC + R) determined by Graf and others (1982b) for Illinois.
Determining and Evaluating Clark Unit-Hydrograph Parameters 9

The first problem is that most data were obtained 

from watersheds larger than 1 mi2

. In the study by Graf 

and others (1982a,b), only 8 of the 98 watersheds used 

were within selected parameters for this study. Also, 

seven of the nine watersheds analyzed by Melching and 

Marquardt (1996) were larger than 1 mi2

. The substantial amount of data from larger watersheds may appreciably affect the reliability of estimating TC and R for 

small watersheds.

The second problem is that the locations of 

the rain gages, which were used in the study by Graf 

and others (1982a,b) to determine the watershed-average storm rainfall and the temporal rainfall distribution, 

commonly were 5 to 25 mi outside of the watershed, 

where runoff data were available. Therefore, 

uncertainties in the temporal distribution of the 

effective precipitation could substantially affect 

the reliability of the TC and R values determined 

from calibration in the HEC-1 modeling.

The third problem is that both Graf and others 

(1982a,b) and Melching and Marquardt (1996) used 

watersheds with appreciable amounts of impervious 

area. Also, the substantial amount of wetland areas 

coupled with extremely small watershed slopes (six 

of the nine watersheds with slopes less than 10 ft/mi) 

could affect the TC and R values in the Melching and 

Marquardt (1996) study that focused on data from Lake 

County, Illinois. The 41 watersheds used in this study 

were primarily small agricultural areas with insignificant amounts of impervious and wetland land cover.

To assess the utility of the method of Graf and 

others (1982b), TC and R are estimated for each watershed studied using that method and the equations developed in this study. The estimated values then are 

compared with each other and to the values obtained 

from calibrating the HEC-1 model. Because impervious area is an important factor in the Melching and 

Marquardt (1996) equations for determining TC and R, 

the methods of their study were not assessed on runoff 

data for each watershed in this study. The data used by 

Melching and Marquardt (1996), however, were used 

to test and determine limitations of the new statewide 

equations developed herein.

DETERMINING AND EVALUATING CLARK 

UNIT-HYDROGRAPH PARAMETERS

Selected storms were calibrated with HEC-1 to 

obtain optimal TC and R values for the Clark unit 

hydrograph. TC and R values determined in this study 

were evaluated by comparing the fit of the calibrated 

hydrographs to the observed hydrographs and by 

comparing TC and R values derived during this study 

to values from previous studies.

Storm Selection

Storms for determining parameters for 

synthetic unit hydrographs should be selected to 

conform as closely as possible to the definition of a 

unit hydrograph. A unit hydrograph is the dischargetime graph (hydrograph) of a unit volume of direct 

runoff resulting from a spatially uniformly distributed 

effective precipitation (approximately equal to precipitation excess if interflow is small) with a uniform intensity over a given duration. Viessman and others (1989, 

p.186) recommend that, ideally, the storms utilized to 

determine unit hydrographs should include the following characteristics:

• a simple-storm structure, resulting in well defined 

hydrographs with distinct peaks;

• uniform rainfall distribution throughout the period of 

effective precipitation; and

• uniform spatial distribution (of rainfall) over the 

entire watershed.

Calibrating HEC-1 (U.S. Army Corps of Engineers, 

1990) reduces the importance of the second characteristic because the multiple periods of effective precipitation are adequately deconvoluted in the calibration 

process if the direct-runoff hydrograph is well defined 

with a distinct peak (characteristic 1). Further, Viessman and others (1989, p. 186) recommend the direct 

runoff for the selected storm should range from 0.5 

to 1.75 in. The design storms to be simulated with 

the synthetic unit hydrographs typically will result in 

direct-runoff values in this range. Further, Laurenson 

and Mein (1985, p. 87) stated that small storms, resulting in less than about 0.4 in. of runoff, often are more 

difficult to fit than large storms because of extreme 

areal variability of runoff, partial-area runoff, and large 

differences in the time distribution of effective precipitation resulting from small errors in the applied model. 

Therefore, storms that resulted in at least 0.4 in. of 

direct runoff were selected for analysis in this study. 

Hydrographs affected by snowmelt were not considered.

Base flow was not a part of the total-runoff 

hydrographs for the majority of the storms selected. 

Some storms, however, did include base flow, and the 

base flow had to be separated (subtracted) from the 
10 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

total-runoff hydrograph to obtain the direct-runoff 

hydrograph. For the majority of the storms for which 

the total-runoff hydrographs included base flow, the 

base flow was estimated by extending the trend in 

flow throughout the entire hydrograph prior to the start 

of the storm. This method was used because the trend 

in flow before and after the storm was approximately 

equal for most of the storms. For those storms that had 

an unusually large amount of time elapse before the 

flow returned to its prestorm trend, base flow was 

estimated by extending the trend in flow prior to the 

start of the storm to the time-of-peak discharge. After 

the time-of-peak discharge, the base flow was assumed 

to increase linearly to the time when the total-runoff 

hydrograph consisted of only base flow. This time was 

defined as the point on a semilogarithmic plot of the 

total-runoff hydrograph (with discharge on the logarithmic scale) at which the recession limb is approximately 

linear (as described in Chow, 1964, p. 10-14).

Storms may be distributed in time such that welldefined rises in the hydrograph with distinct peaks 

result, but a second rise begins in the latter part of the 

recession curve of the first rise. In this case, rises in the 

hydrograph must be separated so that the direct-runoff 

hydrographs from each storm may be evaluated. Storms 

were separated on the basis of a standard recession 

curve. The standard recession curve was developed 

on the basis of an average recession for the storms 

that were not affected by additional rainfall during the 

recession period on the given watershed. Typically, the 

agreement among these recession curves was close. In 

storm separation, the standard recession curve was 

matched to the recession curve of the first rise and 

utilized to extend the normal recession under the 

second rise. In some cases, the second rise began at 

discharges above those utilized in the standard recession curve, and the direct-runoff hydrographs resulting 

from the two storms could not be reliably separated. For 

storms that the hydrograph could not be separated, the 

data were used for equation verification.

Hydrograph-Parameter Determination

The TC and R needed for the Clark (1945) unithydrograph method were determined by calibrating the 

HEC-1 model (U.S. Army Corps of Engineers, 1990) 

for hyetographs from rain gages at or very near (within 

the watershed) the streamflow-gaging station and 

direct-runoff hydrographs. A total of 121 storms that 

occurred in 39 watersheds were utilized to develop the 

TC and R equations. To verify the TC and R estimation 

equations, 29 storms that occurred in 18 watersheds 

were utilized. Optimal values of the initial-loss and 

continuing-loss rate also were determined in the HEC1 calibration, primarily to match the effective-precipitation depths (that is, the direct runoff), and were not 

used further in the development of the estimation equations. The calibration quality was assessed on the basis 

of the coefficient of model-fit efficiency (Nash and Sutcliffe, 1970) as

, (7)

where

EFF is the coefficient of model-fit efficiency,

Qmi is the measured direct runoff at time i,

Qm is the average measured direct runoff for 

the storm,

Qsi is the simulated direct runoff at time i, and

n is the number of simulated hydrograph 

ordinates.

Multiple starting points (for initial-loss and continuingloss rates, TC, and R) were utilized, as necessary, in 

the nonlinear optimization applied in HEC-1 to 

ensure a close match between the measured and 

simulated direct-runoff hydrographs. The percentage 

error between the measured and simulated directrunoff peak discharges was computed as a measure of 

the reliability when applying the Clark unit-hydrograph 

method. The TC and R values, the model-fit efficiency, 

and the percentage error in the simulated direct-runoff 

peak discharge for the storms utilized to develop and 

verify the hydrograph-parameter estimation equations 

are listed in table 12 (at the back of this report). The 

average coefficient of model-fit efficiency and percentage error in simulation of direct-runoff peak discharge 

for each watershed are listed in table 2. Model-fit 

efficiency coefficients greater than 0.9, generally, indicate a close match between measured and simulated 

direct-runoff hydrographs. The model-fit efficiency for 

6 of the 121 storms utilized to develop the hydrographparameter estimation equations and for 5 of the 29 

storms utilized to verify equations was less than 0.9 

(table 12 at the back of the report).

EFF

( ) Qmi – Qm 2

i = 1

n

∑ Qmi Qsi ( ) – 2

i = 1

n

– ∑

( ) Qmi – Qm 2

i = 1

n

∑

= -----------------------------------------------------------------------------------------
Determining and Evaluating Clark Unit-Hydrograph Parameters 11

Table 2. Average values of measures of calibration quality for the calibrated Clark unit-hydrograph

method for all storms on selected small rural watersheds in Illinois utilized to develop and verify the

equations for estimation of time of concentration and storage coefficient

[Negative percent indicates the simulated peak discharge was less than the measured peak discharge; 

ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Number

of storms

Model-fit

efficiency

Error in

simulated

peak discharge

(percent)

Big Four Ditch Tributary 1 0.985 3.70

Salt Fork Tributary 3 .981 -.95

North Fork Vermilion River Tributary 2 .963 -2.00

Raccoon Creek Tributary 5 .968 .75

Embarras River Tributary 2 .958 -1.31

Dums Creek Tributary 6 .975 8.00

White Feather Creek 7 .956 1.16

Little Wabash River Tributary 2 .991 2.26

Little Saline Creek Tributary 5 .911 -12.35

Lake Glendale Inlet 1 .915 -7.40

Q Ditch Tributary 2 .981 -7.20

Middle Branch of South Branch Kishwaukee River 2 .941 -5.77

South Branch Kishwaukee River Tributary 2 .982 2.70

Leaf River Tributary 1 .968 -2.60

Sand Creek 1 .973 6.40

Ellison Creek Tributary 4 .976 .83

Little Creek 9 .957 -2.74

Homan Creek Tributary 1 .982 -.40

Kiser Creek Tributary 10 .944 -.21

Fox River Tributary Number Two 2 .967 6.51

Mud Creek Tributary 7 .962 -5.95

West Bureau Creek Tributary 1 .958 -10.80

Coffee Creek Tributary near Florid 2 .947 -3.26

Coffee Creek Tributary near Hennepin 4 .867 .05

Wildcat Creek Tributary 3 .923 1.07

Sangamon River Tributary 5 .954 -4.44

Illinois River Tributary 3 .974 -2.67

Hurricane Creek 3 .994 -.75

Bear Creek Tributary 3 .929 -.07

Cahokia Creek Tributary 2 .976 1.83

Hurricane Creek Tributary 8 .961 2.68

Williams Creek 3 .989 -.22

Andy Creek Tributary 4 .973 2.68

Green Creek Tributary 3 .960 -.93

ISWS Field Site 1 2 .942 2.33

ISWS Field Site 6 4 .968 -.25

USDA Watershed IA1 3 .933 2.50

USDA Watershed IA 5 .882 -1.21

USDA Watershed IB 8 .943 -1.44

USDA Watershed W-1 4 .963 2.07

USDA Watershed W-4 5 .952 -4.47
12 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Equation Development

For small rural watersheds in Illinois, three 

methods were used to develop new equations for estimating TC and R. Similar to the Graf and others (1982b) 

study, a multiple-linear regression analysis was used 

to determine mathematical relations among watershed 

characteristics and (TC+R), and an attempt was made 

to determine regional values of R/(TC+R). The second 

method involved using multiple-linear regression analysis to determine mathematical relations among watershed characteristics and average values of TC and R for 

each watershed. No storm characteristics or seasonal 

effects were analyzed in the second method. The third 

method involved using multiple-linear regression analysis to determine mathematical relations among watershed, storm, and seasonal characteristics and values of 

TC and R for each storm. Overall, the second method 

yielded the best equations, as described in the following 

sections.

Results Based on Methods Similar to the Graf and others 

(1982b) Study

In the first method, equations for estimating 

(TC+R) were developed utilizing multiple-linear 

regression to relate the logarithm of the average (TC+R) 

for each watershed to logarithms of watershed area and 

main-channel length and slope. The multiple-linear 

regression of logarithms resulted in an estimation 

equation

, (8)

where

hpi is hydrograph parameter i

[in this case (TC+ R)],

Wj are watershed characteristics j,

bj are exponents corresponding to 

watershed characteristics j, and

a is a coefficient.

Watershed characteristics were added one at a time 

to the regression model (eq. 8), and characteristics 

were retained in the regression model only if the 

corresponding exponents were statistically significant 

(the corresponding 95-percent confidence interval for 

the parameter did not include zero) and the sign of 

the exponent was correct from a physical viewpoint. 

For example, hydrograph-timing parameters should 

increase with increasing area and main-channel length 

hpi aW b1

1 Wb2 = 2

and decrease with increasing main-channel slope. From 

the regression, an equation involving the length and 

slope was determined to yield the highest coefficient of 

determination (R2

=0.74).

Next, the average [R/(TC+R)] values for each 

watershed were plotted on a map of the State of Illinois. 

Contours were drawn to try to determine regional 

trends in the values, but all such attempts were unsuccessful. Before abandoning the method similar to that 

used by Graf and others (1982b) because of the inability to find regional trends in average [R/(TC+R)], the 

logarithm of these values were regressed against watershed characteristics similar to the previous regression, 

using values of average (TC+R). All combinations of 

the regression yielded very poor coefficient of determination (R2

) values (highest equaling 0.38). With no reliable method of determining [R/(TC+R)], this method 

was abandoned.

Interestingly, the coefficient and exponents for 

the Graf and others (1982b) equation for (TC+R) were 

not within the 95-percent confidence intervals when 

compared to the equation developed for (TC+R) for this 

study, except for a slight overlap in the length exponent. 

The upper and lower confidence bounds for the length 

exponent were 0.650 and 0.339, respectively, for the 

(TC+R) equation developed in this study.

Results Based on Average Values of TC and R for Each 

Watershed

The second method, determined to be the overall 

best method in this study, utilized multiple-linear 

regression analysis to relate the logarithms of the average TC and average R for each watershed to logarithms 

of watershed area and main-channel length and slope. 

Equations for the TC and R estimations (in hours) that 

yield the highest R2

 values included main-channel 

length and slope and are

TC = 1.54 L 0.875 S -0.181 and (9)

R = 16.4 L 0.342 S -0.790, (10)

where

L is the stream length measured along the main

channel from the watershed outlet to the

watershed divide, in mi, and

S is the main-channel slope determined from 

elevations at points that represent 10 and 

85 percent of the distance along the channel
Determining and Evaluating Clark Unit-Hydrograph Parameters 13

from the watershed outlet to the watershed

divide, in ft/mi.

The coefficient of determination and standard error 

for the logarithmic data resulting from the stepwise 

multiple-linear regression for estimating TC and R

are listed in tables 3 and 4, respectively.

Equation 9 explains 73 percent of the variance in 

the logarithms of TC. Equation 10 explains 64 percent 

of the variance in the logarithms of R. The TC and R values estimated using equations 9 and 10, respectively, 

and the values determined through calibration are 

shown in scattergrams in figures 3 and 4, respectively. 

The TC and R values estimated using equations developed in this study, the Graf and others (1982b) method, 

and the values determined through calibration in this 

study are listed in tables 5 and 6, respectively.

Results Using TC and R for All Storms Independently

The third method utilized multiple-linear 

regression analysis to relate the logarithm of the TC

and R for each storm to logarithms of watershed area, 

main-channel length and slope, storm duration and 

intensity, and Julian day (as a measure of time in the 

growing season). The multiple-linear regression in 

logarithms resulted in estimation equations of the form

, (11)

where

hpi is hydrograph parameter I,

Wj are watershed characteristics j,

bj are exponents corresponding to watershed 

characteristics j,

Sk are storm characteristics k,

ck are exponents corresponding to storm 

characteristics k,

J denotes the Julian day,

d is an exponent corresponding to the 

Julian day, and

a is a coefficient.

Nonlinear equations, such as equation 11, among 

hydrograph parameters, and watershed and storm 

characteristics have been determined theoretically 

from the kinematic wave approximation (Ragan and 

Duru, 1972), experimentally in the laboratory (Shen, 

1974), and empirically from field data (Snyder, 1938; 

Rao and others, 1972; and others).

Regressions involving length and slope resulted 

in the highest R2

 values when considering all possible 

watershed parameters regressed against TC and R. Only 

minimal increases in R2

 resulted from adding storm 

characteristics to the regression of length and slope, and 

no increase in R2

 resulted when adding Julian day to the 

regression. The extra effort needed to obtain storm 

characteristics did not warrant the slight increase in the 

accuracy of estimating TC and R. The distribution of 

Julian day was limited primarily to the summer months. 

This limitation possibly has caused the Julian day to 

have an insignificant effect on the regression.

Comparison of Calibrated Hydrograph-Parameter 

Values with Results Obtained Using the Graf and 

others (1982b) Method

When applying the method of Graf and others 

(1982b) to estimate TC, the mean square error (in real 

space) is 0.432 when comparing computed TC and 

measured TC for all 150 storms on the 41 watersheds. 

When estimating TC using equation 9, the mean square 

error (in real space) is 0.292 when comparing computed 

and measured TC for all 150 storms on the 41 watersheds.

When applying the method of Graf and others 

(1982b) to estimate R, the mean square error (in real 

space) is 0.341 when comparing computed R and 

hpi aWb1

1 Wb22 ...S c11 S c22 ...J d = ...

Table 3. Coefficient of determination 

and standard error for logarithmic data 

in the equations for estimating time of 

concentration for the Clark unit 

hydrograph

[R2, coefficient of determination]

Parameter R2 Standard

error

Slope 0.10 0.8096

Area .67 .4920

Length .70 .4673

Length and slope .73 .4523

Table 4. Coefficient of determination 

and standard error for logarithmic data 

in the equation for estimating the 

watershed storage coefficient for the 

Clark unit hydrograph

[R2, coefficient of determination]

Parameter R2 Standard

error

Area 0.17 0.8188

Length .18 .8143

Slope .55 .6036

Slope and length .64 .5501
14 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Figure 3. Time of concentration for storms on 41 watersheds in Illinois for the Clark unithydrograph method measured (average) and computed as a function of watershed mainchannel length and slope.

measured R for all 150 storms on the 41 watersheds. 

When estimating R using equation 10, the mean square 

error (in real space) is 0.333 when comparing computed 

R and measured R for all 150 storms on the 41 watersheds.

When applying the method of Graf and others 

(1982b) to estimate TC+R, the mean square error (in 

real space) is 0.777 when comparing computed TC+R

and measured TC+R for all 150 storms on the 

41 watersheds. When estimating TC+R using equations 

9 and 10, the mean square error (in real space) is 0.636 

when comparing computed and measured TC+R for all 

150 storms on the 41 watersheds.

When comparing the results utilizing equation 9 

to estimate TC with the results utilizing the Graf and 

others (1982b) method to estimate TC for the same data 

set, the Graf and others results are not as favorable 

because the mean square error is larger when using the 

Graf and others method than the mean square error 

when using equation 9. Further, equation 9 provides a 

better TC estimate for 93 of the 150 storms compared to 

the estimates obtained with the Graf and others (1982b) 

method.

When the results of utilizing equation 10 to estimate R are compared to the results of the Graf and 

others (1982b) method to estimate R for the same data 

set, the Graf and others results are similar to the results 

utilizing equation 10 because the mean square error is 

nearly identical for the two cases. These results also are 

reflected in the computations for estimating R. The Graf 

and others (1982b) method provided a better R estimate 

for 77 storms, and equation 10 provided a better R

estimate for 73 storms. The Graf and others (1982b) 

method is more cumbersome than using equations to 
Determining and Evaluating Clark Unit-Hydrograph Parameters 15

Figure 4. Storage coefficient for storms on 41 watersheds in Illinois for the Clark unit-hydrograph method 

measured (average) and computed as a function of watershed main channel-length and slope.

solve directly for TC and R. Therefore, although the 

results based on the Graf and others (1982b) method 

are similar for the computation of R, equation 10 is 

much easier to use.

When the results of utilizing equations 9 and 10 

to estimate TC+R were compared to the results of the 

Graf and others (1982b) method to estimate TC+R for 

the same data set, the Graf results are not as favorable 

because the mean square error is larger than the mean 

square error when using equations 9 and 10. Further, 

equation 9 provides a better estimate of TC+R for 85 of 

the 150 storms compared to the estimates derived by 

the Graf and others (1982b) method.

The better performance of equations 9 and 10 

in estimating TC and TC+R (compared to the Graf and 

others (1982b) method) was expected because 80 percent of the storms considered were used developing 

equations 9 and 10. The primary purpose of this comparison, however, was to assess the accuracy of the Graf 

and others (1982b) method for small watersheds (less 

than 2.3 mi2

). The good performance of the Graf and 

others (1982b) method in estimating R indicates this 

method may yield reasonable results for some small 

rural watersheds. The poor performance of the Graf and 

others (1982b) method in estimating TC and TC+R, 

however, indicates generally this method will not 

yield reliable results.

Equation Verification

Data for 29 storms in 18 watersheds were used to 

verify equations 9 and 10 (table 1). Two of the watersheds for which verification data were available were 

not used in the equation development. The storms used 

in the verification generally were double peaked and 

too complicated to use in equation development. The 

verification results using the complicated storms should 
16 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Table 5. Time of concentration for the Clark unit-hydrograph method estimated with equations developed in this study 

and with the method of Graf and others (1982b) compared to the average values determined from calibration for all 

storms used in equation development on selected small rural watersheds in Illinois

[TC, time of concentration; ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Number

of storms

TC estimated

from

equations

developed

in this study

(hours)

TC estimated

from

equations

developed

in Graf and

others study

(hours)

Mean TC

determined

from storm

calibration

in this study

(hours)

Big Four Ditch Tributary 1 1.741 2.210 2.205

Salt Fork Tributary 2 2.727 3.952 2.651

North Fork Vermilion River Tributary 2 1.635 1.872 1.390

Raccoon Creek Tributary 3 .264 .801 .350

Embarras River Tributary 1 .432 2.312 .330

Dums Creek Tributary 6 .303 .412 .244

White Feather Creek 5 .751 .671 .855

Little Wabash River Tributary 2 .449 .613 .637

Little Saline Creek Tributary 4 .784 .760 1.735

Lake Glendale Inlet 1 1.077 .554 1.138

Q Ditch Tributary 2 .517 .340 .905

Middle Branch of South Branch Kishwaukee River 1 1.935 1.862 1.760

Leaf River Tributary 1 .523 .336 .537

Sand Creek 1 .564 .594 .475

Ellison Creek Tributary 4 1.313 1.564 .586

Little Creek 8 1.370 1.405 1.115

Homan Creek Tributary 1 .827 .616 .469

Kiser Creek Tributary 7 .820 .753 .556

Fox River Tributary Number Two 2 .698 .544 .741

Mud Creek Tributary 7 .596 .653 1.172

Coffee Creek Tributary near Florid 1 .203 .160 .258

Coffee Creek Tributary near Hennepin 3 .548 .352 .757

Wildcat Creek Tributary 3 .346 1.436 .356

Sangamon River Tributary 5 1.033 1.337 1.887

Illinois River Tributary 3 .697 .489 .373

Hurricane Creek 3 2.457 2.794 1.975

Bear Creek Tributary 3 .154 .416 .157

Cahokia Creek Tributary 2 .725 1.097 .948

Hurricane Creek Tributary 5 .413 1.170 .771

Williams Creek 3 2.322 3.475 2.368

Andy Creek Tributary 3 1.314 1.518 2.091

Green Creek Tributary 3 .763 .570 .508

ISWS Field Site 1 2 .440 1.225 .763

ISWS Field Site 6 3 .454 1.307 .355

USDA Watershed IA1 1 .283 .607 .088

USDA Watershed IA 2 .437 1.034 .239

USDA Watershed IB 6 .472 1.107 .740

USDA Watershed W-1 4 .181 .507 .124

USDA Watershed W-4 5 .493 .398 .373
Determining and Evaluating Clark Unit-Hydrograph Parameters 17

Table 6. Storage coefficient estimated with equations developed in this study and with the method of Graf and others (1982b) 

compared to the average values determined from calibration for all storms used in equation development on selected small 

rural watersheds in Illinois

 [R, storage coefficient; ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Number of

storms

R estimated

from equations

developed in

this study

(hours)

R estimated

from equations

developed in

Graf and

others study

(hours)

Mean R

determined

from storm

calibration

in this study

(hours)

Big Four Ditch Tributary 1 1.926 2.210 2.035

Salt Fork Tributary 2 2.816 2.635 2.715

North Fork Vermilion River Tributary 2 1.352 1.248 .920

Raccoon Creek Tributary 3 .475 .200 .197

Embarras River Tributary 1 1.838 1.541 1.320

Dums Creek Tributary 6 .319 .275 .398

White Feather Creek 5 .496 .448 .511

Little Wabash River Tributary 2 .399 .263 .334

Little Saline Creek Tributary 4 .562 .506 .606

Lake Glendale Inlet 1 .398 .369 1.572

Q Ditch Tributary 2 .305 .340 .426

Middle Branch of South Branch Kishwaukee River 1 1.603 1.862 3.130

Leaf River Tributary 1 .301 .336 .343

Sand Creek 1 .538 .594 .605

Ellison Creek Tributary 4 1.374 1.564 1.009

Little Creek 8 1.227 1.405 .579

Homan Creek Tributary 1 .449 .410 .191

Kiser Creek Tributary 7 .555 .502 .629

Fox River Tributary Number Two 2 .484 .544 .674

Mud Creek Tributary 7 .590 .653 1.071

Coffee Creek Tributary near Florid 1 .149 .160 .042

Coffee Creek Tributary near Hennepin 3 .315 .352 .219

Wildcat Creek Tributary 3 1.144 .957 .804

Sangamon River Tributary 5 .986 .891 1.033

Illinois River Tributary 3 .359 .326 .147

Hurricane Creek 3 1.984 1.863 2.105

Bear Creek Tributary 3 .337 .277 .729

Cahokia Creek Tributary 2 .824 .731 .803

Hurricane Creek Tributary 5 .914 .780 .605

Williams Creek 3 2.488 2.317 1.786

Andy Creek Tributary 3 1.105 1.012 .879

Green Creek Tributary 3 .419 .380 .362

ISWS Field Site 1 2 .954 .816 2.382

ISWS Field Site 6 3 1.019 .872 1.505

USDA Watershed IA1 1 .478 .405 .792

USDA Watershed IA 2 .802 .689 1.366

USDA Watershed IB 6 .856 .738 .843

USDA Watershed W-1 4 .327 .338 .282

USDA Watershed W-4 5 .236 .265 .393
18 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

be of similar quality to the results for a simple (single 

peaked) storm because, once derived, a unit hydrograph 

should be applicable to any type of storm through the 

principle of linear superposition. Therefore, storms not 

suitable for deriving a unit hydrograph may be applicable for testing the unit hydrograph.

The verification storms were analyzed through 

HEC-1 calibration to determine hydrograph characteristics, TC, and R in the same manner as the 121 storms 

utilized to develop the estimation equations. The TC, R, 

model-fit efficiency, and percentage error in the peak 

discharge from the HEC-1 calibration of the Clark unithydrograph method for the verification storms are listed 

in table 12 (in the back of this report). Equations 9 and 

10 were used to estimate TC and R values for the verification storms. The computed and average measured TC

and R values for the verification storms are presented in 

figures 3 and 4, respectively, and tables 7 and 8, respectively.

The percentage errors in the estimated peak 

discharge and time-to-peak discharge for the verification storms simulated with the Clark unit-hydrograph 

method utilizing TC and R estimated with equations 9 

and 10, respectively, are listed in table 9. For 21 of the 

29 verification storms, the error in the peak discharge

 is less than 25 percent. For 18 of the 29 verification 

storms, the error in the time-to-peak discharge is less 

than 25 percent.

Selected computed and measured hydrographs 

are shown in figures 5-7. The selected graphs show a 

representative sample of the 29 verification storms. 

Approximately one-third of the verification storms 

resulted in good agreement (less than 15 percent 

error in peak discharge and time-to-peak discharge) 

with measured values (fig. 5). Approximately another 

one-third of the verification storms resulted in fair 

agreement (between 15 and 35 percent error in peak 

discharge and time-to-peak discharge) with measured 

values (fig. 6). The final one-third of the verification 

storms resulted in poor agreement (greater than 

35 percent error in peak discharge or time-to-peak 

discharge) with the measured values (fig. 7).

Testing with Lake County Data

TC and R values were estimated using equations 

9 and 10, respectively, for the nine watersheds analyzed 

Table 7. Time of concentration for the Clark unit-hydrograph method estimated with equations developed in this study and 

with the method of Graf and others (1982b) compared to the average values determined from calibration for all storms used 

for verification of the developed equation on selected small rural watersheds in Illinois

[TC, time of concentration; ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Number

of storms

TC estimated

from equations

developed in

this study

(hours)

TC estimated

from equations

developed in

Graf and others

study

(hours)

Mean TC

determined

from storm

calibration

in this study

(hours)

Salt Fork Tributary 1 2.734 3.952 4.019

Raccoon Creek Tributary 2 .265 .801 .300

Embarras River Tributary 1 .432 2.312 .197

White Feather Creek 2 .754 .671 .816

Little Saline Creek Tributary 1 .787 .760 1.849

Middle Branch of South Branch Kishwaukee River 1 1.942 1.862 .413

South Branch Kishwaukee River Tributary 2 1.510 1.073 1.725

Little Creek 1 1.375 1.405 1.040

Kiser Creek Tributary 3 .818 .753 .468

West Bureau Creek Tributary 1 1.047 .603 .263

Coffee Creek Tributary near Florid 1 .202 .160 .310

Coffee Creek Tributary near Hennepin 1 .550 .352 .927

Hurricane Creek Tributary 3 .415 1.170 .864

Andy Creek Tributary 1 1.319 1.518 2.691

ISWS Field Site 6 1 .455 1.307 .321

USDA Watershed IA1 2 .284 .607 .775

USDA Watershed IA 3 .438 1.034 .495

USDA Watershed IB 2 .474 1.107 2.484
Determining and Evaluating Clark Unit-Hydrograph Parameters 19

Table 8. Storage coefficient estimated with an equation developed in this study and with the method of Graf and others (1982b) 

compared to the average values determined from calibration for all storms used for verification of the developed equation on 

selected small rural watersheds in Illinois

[R, storage coefficient; ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Number of

storms

R estimated

from equations

developed in

this study

(hours)

R estimated

from equations

developed in

Graf and others

study

(hours)

Mean R

determined

from storm

calibration

in this study

(hours)

Salt Fork Tributary 1 2.815 2.635 2.911

Raccoon Creek Tributary 2 .475 .200 .341

Embarras River Tributary 1 1.837 1.541 1.114

White Feather Creek 2 .496 .448 .685

Little Saline Creek Tributary 1 .562 .506 .521

Middle Branch of South Branch Kishwaukee River 1 1.603 1.862 4.747

South Branch Kishwaukee River Tributary 2 .925 1.073 1.201

Little Creek 1 1.228 1.405 .960

Kiser Creek Tributary 3 .555 .502 1.062

West Bureau Creek Tributary 1 .524 .602 .467

Coffee Creek Tributary near Florid 1 .149 .160 .160

Coffee Creek Tributary near Hennepin 1 .314 .352 .293

Hurricane Creek Tributary 3 .914 .780 .943

Andy Creek Tributary 1 1.105 1.012 .759

ISWS Field Site 6 1 1.019 .872 1.369

USDA Watershed IA1 2 .478 .405 1.621

USDA Watershed IA 3 .802 .689 .708

USDA Watershed IB 2 .856 .738 1.737

in the Lake County, Ill., study (Melching and 

Marquardt, 1996). Watershed characteristics of 

these watersheds are presented in table 10. The 

average percentage errors in the estimated peak 

discharge and time-to-peak discharge for the Lake 

County storms simulated with the Clark unithydrograph method utilizing TC and R estimated 

with equations 9 and 10, respectively, are listed in 

table 11. The average error in peak discharge is greater 

than 100 percent for five of the nine watersheds, and 

four of these five watersheds have a negative percentage 

of error for the time-to-peak discharge. The Green Lake 

Ditch watershed had different results, with a negative 

average percentage error in peak discharge and a large 

positive error in the time-to-peak discharge. Further 

discussion of the results of testing the equations with 

Lake County data is included in the following section.

Application Limits for the Estimation Equations

Unaccounted for storage depressions (for example, wetlands, undersized culverts causing ponding 

upstream from the culvert, and (or) extremely flat 

slopes) could explain the extremely high overestimation of the peak discharge and the underestimation of 

the time-to-peak discharge in the verification and testing of equations. This result seems to be the case on 

many of the Lake County watersheds (Green Lake 

Ditch is the only exception) where wetlands and flat 

slopes are scattered throughout the watershed. Among 

the watersheds considered in this study, the flattest 

slope is 10.5 ft/mi. In the Lake County data set, six of 

the nine watersheds have a slope of less than 10 ft/mi. 

The effects of extremely flat slopes and large depression storage in eight Lake County watersheds seem 

to overshadow any effects of impervious area on the 

measured hydrograph shape. In general, equation 10 

seems to underestimate storage in watersheds in Lake 

County.

The testing results for Green Lake Ditch 

(table 11) show the effects of simulating discharge for 

an urban watershed using TC and R values computed 

with equations developed for small rural watersheds. 

The results show a late and undersimulated peak of the 

computed hydrograph compared to the peak of the 

measured hydrograph of the urban (Green Lake Ditch) 

watershed.
20 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Table 9. Percentage error in estimated peak discharge and time-to-peak discharge for the verification storms 

on selected small rural watersheds in Illinois, simulated with the Clark unit-hydrograph method utilizing estimated 

values of time of concentration and watershed-storage coefficient developed in this study

[Negative percent indicates the simulated peak discharge was less than or occurred before the measured peak discharge; 

ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]

Watershed Date

Error in peak

discharge

(percent)

Error in

 time-to-peak

discharge

(percent)

Salt Fork Tributary August 4, 1968 11.0 -14.6

Raccoon Creek Tributary June 27, 1957 -22.4 8.00

Raccoon Creek Tributary June 20, 1959 8.80 .00

Embarras River Tributary June 13, 1958 -31.3 8.50

White Feather Creek May 18, 1959 24.8 -17.6

White Feather Creek May 8, 1961 14.0 -2.83

Little Saline Creek Tributary February 20, 1971 12.3 -60.8

Middle Branch of South Branch Kishwaukee River June 14, 1972 58.4 30.2

South Branch Kishwaukee River Tributary September 13, 1972 9.86 -15.8

South Branch Kishwaukee River Tributary April 21, 1973 12.0 -3.85

Little Creek December 31, 1964 -13.9 11.6

Kiser Creek Tributary September 13, 1961 -13.3 50.0

Kiser Creek Tributary October 11, 1973 31.5 3.78

Kiser Creek Tributary April 21, 1974 14.1 -18.8

West Bureau Creek Tributary April 21, 1973 -34.0 42.0

Coffee Creek Tributary near Florid July 18, 1969 1.47 .00

Coffee Creek Tributary near Hennepin September 23, 1970 17.8 -26.4

Hurricane Creek Tributary June 13, 1958 12.7 -40.0

Hurricane Creek Tributary June 27, 1958 22.1 -20.0

Hurricane Creek Tributary August 7, 1958 40.9 -27.3

Andy Creek Tributary August 17, 1959 -2.21 7.76

ISWS Field Site 6 July 10, 1982 19.6 90.2

USDA Watershed IA1 July 2, 1982 123 -15.9

USDA Watershed IA1 May 30, 1982 76.9 -40.6

USDA Watershed IA July 23, 1973 6.69 17.0

USDA Watershed IA June 22, 1974 -5.68 27.2

USDA Watershed IA May 30, 1982 -6.05 .00

USDA Watershed IB February 4, 1971 23.1 -18.5

USDA Watershed IB April 19, 1972 202 -58.3

The equations developed during this study for 

estimating TC and R for the Clark unit-hydrograph 

method work well on watersheds that are within the 

characteristic limitations used in the equation development. Most data used in equation development was 

obtained for watersheds less than 0.5 mi2

 in area (26 of 

39 watersheds). Therefore, these equations probably 

are most accurate for watersheds less than 0.5 mi2

 in 

area, and the equation’s accuracy probably decreases as 

the upper limit of the area is approached. The equations 

do not estimate TC and R well for watersheds that contain wetlands, undersized culverts that cause ponding 

upstream from the culvert, or very flat slopes (less than 

10 ft/mi). The use of TC and R values computed with 

equations 9 and 10 tend to undersimulate peak discharge and oversimulate time-to-peak discharge for 

highly impervious watersheds that do not have any 

items mentioned in the previous sentence.
Determining and Evaluating Clark Unit-Hydrograph Parameters 21

Figure 6. Measured direct-runoff hydrograph and computed direct-runoff hydrograph simulated with the Clark unithydrograph method in the U.S. Army Corps of Engineers Flood Hydrograph Package HEC-1 with the time of 

concentration and storage coefficient determined from the estimation equations for Illinois developed in this study for 

the storm of August 17, 1959, on Andy Creek Tributary.

Figure 5. Measured direct-runoff hydrograph and computed direct-runoff hydrograph simulated with the Clark unithydrograph method in the U.S. Army Corps of Engineers Flood Hydrograph Package HEC-1 with the time of 

concentration and storage coefficient determined from the estimation equations for Illinois developed in this study for 

the storm of April 21, 1973, on South Branch Kishwaukee River.
22 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Figure 7. Measured direct-runoff hydrograph and computed direct-runoff hydrograph simulated with the Clark unithydrograph method in the U.S. Army Corps of Engineeers Flood Hydrograph Package HEC-1 with the time of 

concentration and storage coefficient determined from the estimation equations for Illinois developed in this study for 

the storm of April 21, 1973, on West Bureau Creek Tributary.

Application Example

The Sangamon River Tributary near Andrew, Ill., 

watershed is 1.36 mi long with a slope of 40.1 ft/mi. 

The TC and R values for the watershed can be estimated 

with equations 9 and 10, respectively, as

, and

.

TC 1.54 1.36 ( )0.875( ) 40.1 –0.181 = = 1.03 hours

R 16.4 1.36 ( )0.342( ) 40.1 –0.790 = = 0.986 hours

Table 10. Characteristics of watersheds in Lake County, Ill., selected for testing the equations developed to estimate 

 time of concentration and storage coefficient for small rural watersheds in Illinois

[mi2, square miles; mi, miles; ft/mi, feet per mile]

Watershed

Drainage

area

(mi2)

Length

(mi)

Slope

(ft/mi)

Impervious

area

(percent)

Forest area

(percent)

Wetland area

(percent)

Bull Creek 6.3 6.4 3.13 13.9 7.48 6.80

Terre Faire Ditch .077 .33 55.3 27.7 .00 2.00

Indian Creek 35.7 11.6 13.6 15.8 3.48 4.22

Green Lake Ditch .06 .6 14.0 40.6 .00 .00

North Branch Chicago River 19.7 13.5 3.24 21.3 32.5 .77

Skokie River at Lake Forrest 13.0 10.8 5.58 29.4 24.0 .15

Skokie River near Highland Park 21.1 16.6 5.29 34.4 30.1 .24

Squaw Creek 17.2 7.8 4.79 7.32 3.73 7.32

Flint Creek 37.0 12.9 7.99 8.83 8.97 5.09
Summary and Conclusions 23

These TC and R values then could be input to HEC-1 

(U.S. Army Corps of Engineers, 1990) along with the 

design hyetograph.

SUMMARY AND CONCLUSIONS

Equations for estimating the time of concentration (TC) and storage coefficient (R) for use with the 

Clark unit-hydrograph method were developed for 

small rural watersheds [0.02–2.3 square miles (mi2

)] in 

Illinois. The equations provide State and local engineers and planners with more accurate methods to estimate design hydrographs relative to current practice for 

water-resources-management activities, including 

designing stormwater-management facilities and other 

hydraulic structures, determining flood-plain boundaries, and assessing the safety of structures in rivers.

The established hydrologic data base contains 

rainfall and runoff data from gaged small rural watersheds (0.02–2.3 mi2

) in Illinois with insignificant 

amounts of impervious land cover. Equations were 

developed using 121 storms with effective rainfall 

depths greater than 0.4 inches (in.) on 39 watersheds. 

In the verification, 29 storms with effective rainfall 

depths greater than 0.4 in. on 18 watersheds were used. 

Two of the watersheds used in verification were not 

used for equation development. The limitations of the 

developed equations were tested for 74 storms with 

effective rainfall depths greater than 0.4 in. on 9 larger 

and (or) urbanized watersheds in Lake County, Illinois.

The Clark unit-hydrograph parameters (TC and 

R) were determined by calibration, using the U.S. Army 

Corps of Engineers Flood Hydrograph Package HEC-1. 

Mathematical relations among watershed and storm 

characteristics and seasonal effects, and TC and R were 

determined by multiple-linear regressions of the logarithms of the values. Main-channel length and slope 

were identified as important characteristics for estimating TC and R. The estimation equations had coefficients 

of determination (R2

) of 0.73 and 0.64 for logarithms of 

TC and R, respectively. When adding storm characteristics, only minimal increases to the R2

 resulted. No 

increase in R2

 resulted when adding Julian day (as a 

measure of seasonal effects) to the regression with 

length and slope; therefore, equations utilizing storm 

characteristics and seasonal effects were not developed. 

Attempts to estimate TC and R using methods similar to 

those used in an earlier study were abandoned because 

a reliable way to determine R/(TC+R) was not found.

For the verification storms, simulation of the 

measured hydrographs utilizing TC and R obtained 

from the estimation equations (utilizing main-channel 

length and slope) yielded good results. The error in 

peak discharge for 21 of the 29 storms was less than 

25 percent, and the error in time-to-peak discharge for 

18 of the 29 storms was less than 25 percent. Application of the estimation equations to determine TC and R

for design-storm simulation may result in reliable 

design-discharge hydrographs, as long as the physical 

characteristics of the watersheds under consideration 

are within the range of the physical characteristics for 

the watersheds used in this study [area: 0.02–2.3 mi2

, 

main-channel length: 0.17 –3.4 miles, main-channel 

Table 11. Average percentage error in the estimated peak discharge and time-to-peak discharge for the storms on 

selected watersheds in Lake County, Ill., simulated with the Clark unit-hydrograph method utilizing estimated values 

of time of concentration and storage coefficient computed with equations developed in this study for small rural 

watersheds in Illinois

[TC, time of concentration; R, storage coefficient; negative percent indicates the simulated peak discharge was less than or occurred 

before the measured peak discharge]

Watershed Number of

storms

Computed TC

(hours)

Computed R

(hours)

Error peak

discharge

(percent)

Error

time-to-peak

discharge

(percent)

Bull Creek 11 6.305 12.485 123 16.0

Terre Faire Ditch 4 .285 .471 123 -17.0

Indian Creek 11 8.120 4.789 222 -27.9

Green Lake Ditch 2 .614 1.710 -27.2 64.6

North Branch Chicago River 10 11.995 15.659 34.6 18.9

Skokie River at Lake Forrest 9 8.958 9.447 69.1 28.1

Skokie River near Highland Park 6 13.146 11.405 46.3 21.6

Squaw Creek 10 6.937 9.542 300 -47.5

Flint Creek 11 9.801 7.557 438 -51.0
24 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

slope: 10.5–229 feet per mile, and insignificant percentage of impervious cover].

REFERENCES CITED

Clark, C.O., 1945, Storage and the unit hydrograph: Transactions: American Society of Civil Engineers, vol. 110, 

p. 1419-1488.

Chow, V.-T., 1964, Runoff, section 14 in Chow, V-T., ed., 

Handbook of applied hydrology: New York, McGrawHill Book Company, variable pagination.

Chow, V.T., Maidment, D.R., and Mays, L.W., Jr., 1988, 

Applied hydrology: New York, McGraw-Hill Book 

Company, 572 p.

Curtis, G.W., 1977, Frequency analysis of Illinois floods 

using observed and synthetic streamflow records: 

U.S. Geological Survey Water-Resources Investigations 

Report 77-104, 32 p.

Ford, D.T., Morris, E.C., and Feldman, A.D., 1980, Corps of 

Engineers’ experience with automatic calibration precipitation-runoff model in Haimes, Y., and Kindler, J., 

eds., Water and related land resource systems: 

New York, Pergamon Press, p. 467–476.

Graf, J.B., Garklavs, G., and Oberg, K.A., 1982a, Time of 

concentration and storage coefficient values for Illinois 

streams: U.S. Geological Survey Water-Resources 

Investigations Report 82-13, 35 p.

———1982b, A technique for estimating time of concentration and storage coefficient values for Illinois streams: 

U.S. Geological Survey Water-Resources Investigations 

Report 82-22, 10 p.

Hershfield, D.M., 1961, Rainfall frequency atlas of the 

United States for durations from 30 minutes to 24 hours 

and return periods from 1 to 100 years: Washington, 

D.C., U.S. Department of Commerce, Weather Bureau 

Technical Paper 40, 115 p.

Huff, F.A., and Angel, J.R., 1989, Frequency distributions 

and hydroclimatic characteristics of heavy rainstorms in 

Illinois: Champaign, Ill., Illinois State Water Survey 

Bulletin 70, 177 p.

Kuichling, E., 1889, The relation between the rainfall and the 

discharge of sewers in populous districts: Transactions: 

American Society of Civil Engineers, vol. 20, p. 1–60.

Laurenson, E.M., and Mein, R.G., 1985, RORB-version 3 

runoff routing program user manual: Monash University, Clayton, Victoria, Australia, Department of Civil 

Engineering, 155 p.

Melching, C.S., and Marquardt, J.S., 1996, Equations for 

estimating synthetic unit-hydrograph parameter values 

for small watersheds in Lake County, Illinois: U.S. Geological Survey Open-File Report 96–474, 49 p.

Nash, J.E., and Sutcliffe, J.V., 1970, River flow forecasting 

through conceptual models, Part 1-A discussion of principles: Journal of Hydrology, v. 10, p. 282–290.

O’Kelly, J.J., 1955, The employment of unit hydrographs to 

determine the flows of Irish arterial drainage channels: 

Proceedings, Institution of Civil Engineers (Ireland), 

vol. 4, no.3, p. 365–412.

Ragan, R.M., and Duru, J.O., 1972, Kinematic wave nomograph for times of concentration: Journal of Hydraulics 

Division, American Society of Civil Engineers, vol. 98, 

no. HY10, p. 1765–1771.

Rao, A.R., Delleur, J.W., and Sarma, B.S.P., 1972, Conceptual hydrologic models for urbanizing basins: Journal of 

the Hydraulics Division, American Society of Civil 

Engineers, vol. 98, no. HY7, p. 1205–1220.

Shen, Y.-Y., 1974, Experimental investigation of the effects 

of rainstorm and watershed characteristics on surface 

runoff: Urbana, Ill., Ph.D. Thesis, Department of 

Civil Engineering, University of Illinois at UrbanaChampaign, 197 p.

Soil Conservation Service, 1985, National engineering 

handbook, section 4. hydrology: Washington, D.C., 

U.S. Department of Agriculture, variable pagination.

Snyder, F.F., 1938, Synthetic unit-graphs: Transactions, 

American Geophysical Union, vol. 19, p. 447–454.

Turner, H.M., and Burdoin, A.J., 1941, The flood 

hydrograph: Journal of the Boston Society of Civil 

Engineers, vol. 28, no. 3, p. 232-281.

U.S. Army Corps of Engineers, 1990, HEC-1 flood 

hydrograph package, user’s manual: Davis, Calif., 

U.S. Army Corps of Engineers Hydrologic Engineering 

Center, variable pagination.

U.S. Department of Agriculture, Agricultural Research 

Service, 1957, Monthly precipitation and runoff for 

small agricultural watersheds in the United States: 

Washington, D.C.

Viessman, W., Jr., Lewis, G.L., and Knapp, J.W., 1989, Introduction of hydrology, third edition: New York, Harper 

and Row Publishers, 780 p.
TABLE 12

Table 12. 27

Watershed Date TC R Model-fit

efficiency

Error in peak

discharge

(percent)

Big Four Ditch Tributary May 26, 1956 2.205 2.035 0.985 3.7

Salt Fork Tributary August 4, 19681 4.019 2.911 .990 -1.6

April 19, 1970 2.345 3.375 .974 -6.2

June 22, 1974 2.956 2.054 .980 4.9

North Fork Vermillion River Tributary July 4, 1956 1.224 .886 .957 -5.7

July 16, 1956 1.556 .954 .968 1.7

Raccoon Creek Tributary June 27, 19571 .299 .191 .956 3.8

June 28, 1957 .369 .182 .958 1.3

July 10, 1958 .449 .242 .993 2.7

July 11, 1958 .232 .168 .973 .8

June 22, 19591 .300 .490 .958 -4.8

Embarras River Tributary June 13, 19581 .197 1.114 .927 -3.4

June 23, 1960 .330 1.320 .988 .8

Dums Creek Tributary July 3, 1958 .200 .490 .986 9.6

May 18, 1959 .274 .916 .988 1.6

June 12, 1960 .202 .228 .994 7.2

June 28, 1960 .155 .265 .979 6.7

June 30, 1960 .342 .248 .957 15

July 1, 1960 .292 .239 .945 7.9

White Feather Creek December 19, 1957 .542 .748 .941 -2.7

July 30, 1958 .756 .294 .950 1.4

May 18, 19591 .951 .689 .928 -.7

May 6, 1961 .991 .779 .943 -.2

May 7, 1961 1.058 .372 .976 1.1

May 8, 1961 .929 .361 .978 4.6

May 8, 19611 .680 .680 .978 4.6

Little Wabash Tributary March 31, 1968 .509 .451 .996 2.5

April 3, 1968 .764 .216 .985 2.0

Little Saline Creek Tributary April 19, 1970 1.733 .518 .967 -13

May 1, 1970 1.363 .767 .940 -8.9

May 10, 1970 2.495 .475 .921 -13

February 20, 19711 1.849 .521 .831 -12

December 30, 1971 1.347 .663 .897 -15

Lake Glendale Inlet March 29, 1960 1.138 1.572 .915 -7.4

Q Ditch Tributary August 6, 1959 .904 .406 .968 -15

August 17, 1959 .905 .446 .993 .1

Middle Branch South Branch Kishwaukee River June 14, 19721 .413 4.747 .950 -5.6

August 25, 1972 1.760 3.130 .932 -5.9

South Branch Kishwaukee River Tributary September 13, 19721 1.887 1.483 .977 -5.5

April 21, 19731 1.562 .918 .987 10.9

Leaf River Tributary June 7, 1969 .537 .343 .968 -2.6

Sand Creek June 7, 1967 .475 .605 .973 6.4

Ellison Creek Tributary April 8, 1965 .714 1.386 .963 2.21

June 10, 1967 .534 .836 .986 -1.2

June 21, 1967 .452 1.288 .987 4.5

July 29, 1967 .644 .527 .966 -2.2

Little Creek April 19, 1964 1.150 .470 .963 -7.7

Table 12. Parameters for the Clark unit-hydrograph method determined from calibration of the U.S. Army Corps of Engineers (1990) Flood 

Hydrograph Package HEC-1 and measures of calibration quality for data from small rural watersheds in Illinois, for storms utilized to develop 

and verify the equations for estimation of Clark unit-hydrograph parameters

[TC, time of concentration; R, storage coefficient; negative percent indicates that the simulated peak discharge was less than the measured peak discharge; 

ISWS, Illinois State Water Survey; USDA, U.S. Department of Agriculture]
28 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

Little Creek December 31, 19641 1.040 0.960 0.947 -5.6

July 13, 1965 .802 .558 .898 -.85

April 21, 1973 1.218 .522 .958 -4

April 30, 1973 .891 .569 .985 -.5

May 27, 1973 1.188 .612 .951 -2.4

June 18, 1973 1.151 .620 .975 1.3

July 29, 1973 1.359 .732 .997 2.95

May 19, 1974 1.163 .547 .935 -7.9

Homan Creek Tributary July 4, 1962 .469 .191 .982 -.4

Kiser Creek Tributary August 10, 1961 .657 .323 .982 2.92

September 13, 19611 .607 .213 .972 -2.7

May 10, 1962 .322 .548 .873 -4.3

June 9, 1962 .780 .350 .902 -7

July 2, 1962 .623 .307 .996 1.95

August 5, 1962 .620 .790 .959 -6.3

July 28, 1973 .360 1.440 .973 -.01

October 11, 19731 .275 2.225 .921 11.5

April 21, 19741 .521 .749 .928 3.2

May 29, 1974 .527 .644 .938 -1.4

Fox River Tributary Number Two August 25, 1972 .897 .573 .979 8.2

May 15, 1974 .585 .775 .954 4.82

Mud Creek Tributary May 14, 1970 1.409 .901 .960 -3.6

June 20, 1970 1.336 1.184 .984 -8.3

July 17, 1972 1.062 1.298 .947 -3

July 18, 1972 1.476 .944 .925 -8.1

August 6, 1972 .448 1.212 .994 1.07

August 25, 1972 1.505 1.135 .983 -5.8

June 16, 1973 .967 .823 .942 -13.9

West Bureau Creek Tributary April 21, 19731 .263 .467 .958 -10.8

Coffee Creek Tributary near Florid July 18, 19691 .310 .160 .988 -1.5

July 17, 1972 .258 .042 .906 -5.01

Coffee Creek Tributary near Hennepin July 30, 1970 .866 .274 .825 -11.8

September 23-24, 19701 .927 .293 .771 9.6

August 23, 1972 .774 .126 .969 1.5

May 16, 1974 .632 .258 .904 .9

Wildcat Creek Tributary June 10, 1958 .216 .724 .957 6

July 11, 1958 .558 .802 .923 1.7

June 23, 1960 .295 .885 .888 -4.5

Sangamon River Tributary April 19, 1964 1.813 .977 .972 -8.6

April 20, 1964 1.761 .949 .950 -6.3

April 27, 1964 2.284 1.026 .950 -1.0

June 1, 1965 2.218 1.142 .916 -7.0

August 30, 1965 1.361 1.069 .981 .7

Illinois River Tributary September 3, 1961 .506 .134 .968 -9.8

July 2, 1962 .264 .136 .983 1.2

July 4, 1962 .348 .172 .971 .6

Hurricane Creek August 9, 1961 1.554 3.016 .998 .04

June 3, 1962 2.035 1.535 .994 .01

June 8, 1962 2.337 1.763 .990 -2.3

Watershed Date TC R Model-fit

efficiency

Error in peak

discharge

(percent)

Table 12. Parameters for the Clark unit-hydrograph method determined from calibration of the U.S. Army Corps of Engineers (1990) Flood 

Hydrograph Package HEC-1 and measures of calibration quality for data from small rural watersheds in Illinois, for storms utilized to develop 

and verify the equations for estimation of Clark unit-hydrograph parameters —Continued
Table 12. 29

Bear Creek Tributary July 7, 1962 0.202 0.678 0.941 3.7

July 13, 1962 .105 .645 .921 2.8

October 2, 1962 .165 .865 .925 -6.7

Cahokia Creek Tributary May 8, 1961 1.340 .660 .988 -3.4

August 10, 1961 .555 .945 .963 7.1

Hurricane Creek Tributary April 5, 1958 .750 .780 .985 1.8

June 13, 19581 .735 .935 .880 2.6

June 27, 1958 1.336 .784 .950 -3.2

June 27, 19581 .920 .920 .979 7.7

August 7, 19581 .936 .974 .951 6.1

May 27, 1959 .572 .398 .961 -.2

August 6, 1959 .631 .559 .995 1.4

August 6, 1959 .567 .503 .989 5.2

Williams Creek March 5, 1961 2.888 1.422 .994 1.6

May 6, 1961 2.330 2.330 .992 -.1

May 8, 1961 1.885 1.605 .982 -2.1

Andy Creek Tributary June 12, 1958 1.878 1.012 .957 4.7

July 13, 1958 2.081 .770 .971 .6

July 22, 1958 2.314 .856 .984 5.0

August 17, 19591 2.691 .759 .980 .4

Green Creek Tributary March 16, 1963 .318 .422 .965 3.4

July 15, 1966 .897 .253 .960 -13

August 19, 1966 .310 .410 .955 6.7

ISWS Field Site 1 April 2, 1982 .304 3.076 .944 .1

April 3, 1984 1.222 1.688 .954 2.9

ISWS Field Site 6 April 2, 1982 .396 1.404 .987 .5

May 28, 1982 .318 .953 .959 .0

July 10, 19821 .321 1.369 .936 4.1

April 3, 1984 .351 2.159 .988 -5.6

USDA Watershed IA1 May 30, 19821 .937 1.193 .917 -2.6

July 2, 19821 .612 2.048 .946 6.8

June 28, 1983 .088 .792 .936 3.3

USDA Watershed IA July 23, 19731 1.076 .554 .661 -5.8

June 22, 19741 .087 .883 .893 -2.9

February 16, 1976 .249 1.531 .979 -6.8

August 7, 1977 .229 1.201 .968 6.3

May 30, 19821 .323 .687 .907 3.1

USDA Watershed IB April 20, 1964 1.165 .745 .974 4.9

December 21, 1967 .924 .726 .962 -11

June 15, 1970 .270 .260 .890 -1.0

February 4, 19711 2.645 .395 .964 -3.6

April 19, 19721 2.322 3.078 .937 5.0

June 22, 1974 .137 .623 .909 1.0

February 16, 1976 1.311 1.669 .932 -4.4

August 7, 1977 .635 1.035 .974 -2.2

USDA Watershed W-1 May 27, 1938 .085 .145 .959 3.0

March 31, 1952 .157 .423 .953 -3.6

March 31, 1952 .162 .288 .965 2.3

July 2, 1952 .090 .270 .974 6.6

Watershed Date TC R Model-fit

efficiency

Error in peak

discharge

(percent)

Table 12. Parameters for the Clark unit-hydrograph method determined from calibration of the U.S. Army Corps of Engineers (1990) Flood 

Hydrograph Package HEC-1 and measures of calibration quality for data from small rural watersheds in Illinois, for storms utilized to develop 

and verify the equations for estimation of Clark unit-hydrograph parameters —Continued
30 Equations for Estimating Clark Unit-Hydrograph Parameters for Small Rural Watersheds in Illinois

USDA Watershed W-4 May 27, 1938 0.345 0.345 0.965 -8.4

June 21, 1942 .322 .348 .972 -1.5

March 31, 1952 .441 .719 .914 -8.8

March 31, 1952 .515 .265 .929 -7.5

July 2, 1952 .244 .286 .981 3.9

1Storm utilized to verify the relations for estimating the Clark unit-hydrograph parameters.

Watershed Date TC R Model-fit

efficiency

Error in peak

discharge

(percent)

Table 12. Parameters for the Clark unit-hydrograph method determined from calibration of the U.S. Army Corps of Engineers (1990) Flood 

Hydrograph Package HEC-1 and measures of calibration quality for data from small rural watersheds in Illinois, for storms utilized to develop 

and verify the equations for estimation of Clark unit-hydrograph parameters —Continued
U.S. Geological Survey Water-Resources Investigations Report 00–4184

Straub and others—EQUATIONS FOR ESTIMATING CLARK UNIT-HYDROGRAPH PARAMETERS FOR SMALL RURAL WATERSHEDS IN ILLINOIS—

U.S. GEOLOGICAL SURVEY

221 NORTH BROADWAY AVE.

URBANA, ILLINOIS 61801