WATER-RESOURCES INVESTIGATIONS REPORT 86-4192 

. 

. 

TECHNIQUES FOR 

SIMULATING FLOOD HYDROGRAPHSAND 

ESTIMATING FLOODVOLUMES FOR 

UNGAGED BASINS IN CENTRALTENNESSEE 

,-- 

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U.S. GEOLOGICAL SURVEY 

Prepared in cooperation with the 

TENNESSEE DEPARTMENT OF TRANSPORTATION 
PUBS. SECTION 

FILE COPY 

DO NOT REMOVE 

TECHNIQUES FOR SIMULATING FLOOD HYDROGRAPHS AND ESTIMATING FLOOD 

VOLUMES FOR UNCAGED BASINS IN CENTRAL TENNESSEE 

i 

Clarence H. Robbins 

U.S. GEOLOGICAL SURVEY 

Water-Resources Investigations Report 86-4 192 

Prepared in cooperation with the 

TENNESSEE DEPARTMENT OF TRANSPORTATION 

Nashville, Tennessee 

1986 
UNITED STATES DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary 

GEOLOGICAL SURVEY 

DALLAS L. PECK, DIRECTOR 

i 

For additional information Copies of this report can be 

write to: purchased from: 

District Chief 

U.S. Geological Survey 

A-4 13 Federal Building 

U.S. Courthouse 

Nashville, Tennessee 37203 . * 

Books and Open-File Reports 

U.S. Geological Survey 

Federal Center, Bldg. 41 

Box 25425 

Denver, Colorado 80225 
CONTENTS 

Abstract 1 

Introduction 1 

Inman’s hydrograph simulation method 2 

Testing Inman’s dimensionless hydrograph on central Tennessee streams 3 

Verification of dimensionless hydrograph 6 

Bias 8 

Regionalization of basin lagtime and flood volume 8 

Estimating basin lagtime 8 

Regression analyses 1 2 

Estimating flood volume 17 

Regression analyses 17 

Hydrograph-width relation 2 0 

Application of hydrograph simulation technique 2 3 

Example problem 2 3 

Limitations 2 6 

Conclusions 2 6 

Selected references 2 7 

Symbols, definitions, and units 2 8 

Supplement 2 9 

Regional flood-frequency equations for rural basins in Tennessee 3 0 

Regional flood-frequency equations for urban basins in Tennessee 3 1 

ILLUSTRATIONS 

Figure 1. Map showing location of study area and gaging stations 

used in dimensionless hydrograph analysis 6 

2. Graph showing comparison of Inman and central Tennessee 

dimensionless hydrographs 7 

3-5. Graphs showing comparison of observed and simulated 

hydrographs for: 

3. Richland Creek at Charlotte Avenue at Nashville, 

Tenn. (03431700), for storm of May 3, 1979 9 

4. Wolf River near Byrdstown, Tenn. (03416000), 

for storm of April 3, 1977 10 

5. Duck River near Shelbyville, Tenn. (03598000), 

for storm of March 14, 1973 1 1 

6-9. Graphs showing: 

6. Percent change in rural lagtime and urban lagtime 

resulting from errors in computing channel 

length and percentage of impervious area 18 

7. Percent change in flood volume resulting from 

errors in computing drainage area, lagtime, 

and peak discharge 2 1 

8. Hydrograph-width relation for Inman’s dimensionless 

hydrograph 2 2 

9. Plot of example simulated loo-year flood hydrograph 

for a hypothetical river in central Tennessee 2 6 

IO. Map showing 2-year 24-hour rainfall, in inches 3 2 

111 
TABLES 

Table 1. Time and discharge ratios of Inman’s dimensionless hydrograph 4 

2. Stations and drainage basin characteristics used in 

iagtime regression analyses 1 3 

3. Discharge and hydrograph-width ratios for Inman’s 

dimensionless hydrograph 2 0 

CONVERSION FACTORS 

For readers who may prefer to use the International System of Units (SI) ,rather than 

the inch-pound units used herein, the conversion factors are listed below: 

Multiply EY 

inch (in.) . 2.540 

To obtain 

centimeter (cm) 

foot (ft) 0.3048 meter (m) 

mile (mi) 

square mile (mi2) 

1.609 kilometer (km) 

2.590 square kilometer (km2) 

c&~;~~t per second 0.02832 cubic meter per second 

h3/s) 

iv 
TECHNIQUES FOR SIMULATING FLOOD HYDROCRAPHS AND ESTIMATING 

FLOOD VOLUMES FOR UNCAGED BASINS IN CENTRAL TENNESSEE 

Clarence H. Robbins 

ABSTRACT 

A dimensionless hydrograph developed for a variety of basin conditions in Georgia 

was tested for its applicability to central Tennessee streams by comparing it to a similar 

dimensionless hydrograph developed for central Tennessee streams. Statistical analyses 

were performed by comparing simulated (or computed) hydrographs, derived by application of the Georgia-study dimensionless hydrograph and the central Tennessee dimensionless hydrograph, with 163 observed hydrographs from 38 stations having a wide range of 

drainage area sizes and basin conditions, at 50 and 75 percent of their peak flow widths. 

Test results indicate the two dimensionless hydrographs are essentially the same. Using 

the Georgia-study dimensionless hydrograph, the standard error of estimate was + 21.2 

percent at the 50 percent of peak flow width and + 24.8 percent at the 75 percent 07 peak 

flow width. Study results indicate the Georgia dymensionless hydrograph is applicable to 

central Tennessee streams. 

Equations for rural and urban basin lagtime were derived from multiple-regression 

analyses that relate lagtime to physical basin characteristics. At the 95-percent confidence level, channel length was significant for the rural basin equation and channel length 

and percentage of impervious area were significant for the urban basin equation. A regression equation that related flood volumes to drainage area size, peak discharge, and basin 

lagtime also was developed. The flood-hydrograph and flood-volume techniques are useful 

for estimating a typical (average) flood hydrograph and volume for any specified recurrence interval peak discharge at any ungaged stream site draining areas less than 500 

square miles in central Tennessee. 

INTRODUCTION 

Flood hydrographs and flood volumes are often needed for the design of highway 

drainage structures and embankments or where storage of floodwater or flood prevention 

is part of the design. Additionally, hydrographs may be necessary to estimate the length 

of time of inundation of specific features, for example, roads and bridges. 

A flood of sufficient magnitude for structural design analysis may have occurred 

before systematic streamflow data collection began, or the site of interest is ungaged and 

hydrograph data are not available. Under these conditions, a typical or design hydrograph 

may be simulated using one, or a combination of several, traditional hydrograph estimation 

methods. Each of the traditional methods have inherent characteristics, data requirements, and basin characteristics or coefficients which must be estimated or calculated. 

Most methods rely on the unit hydrograph whereby design hydrographs are computed by 
convolution of the unit hydrograph with rainfall excess. Therefore, rainfall data and 

methods for estimating rainfall excess are necessary for use of any of the unit hydrograph 

methods. 

A need exists for a simple, direct-approach method to estimate the flood hydrograph, 

volume, and width associated with a peak discharge of specific recurrence interval (a design discharge). Recently, a direct-approach method was developed for streams in Georgia 

(Inman, 1986). However, the applicability of this direct-approach method for the Georgia 

streams to other areas was unknown. This report describes the results of a study to determine the applicability of Inman’s method to central Tennessee streams and provides techniques for estimating flood hydrographs (shape, volume, and width) for ungaged basins 

draining areas less than 500 mi2 in central Tennessee. Future studies will determine the 

applicability of Inman’s method to west and east Tennessee streams. This study was conducted in cooperation with the Tennessee Department of Transportation. 

INMAN’S HYDROGRAPH SIMULATION METHOD 

Inman (1986) used 355 actual (observed) streamflow hydrographs from 80 basins and 

harmonic analysis, as described by O’Donnell (1960), to develop unit hydrographs. The 80 

basins represented both urban and rural streamflow characteristics and had drainage areas 

less than 20 mi2. An average unit hydrograph and an average lagtime were computed for 

eat h basin. .These average unit hydrographs were then transformed to unit hydrographs 

having generalized durations of one-fourth, one-third, one-half, and three-fourths lagtime, 

then reduced to dimensionless terms by dividing the time by lagtime and the discharge by 

peak discharge. Representative dimensionless hydrographs developed for each basin were 

combined to generate one typical (average) dimensionless hydrograph for each of the four 

generalized durations. 

Using the four generalized duration dimensionless hydrographs, average basin lagtime, and peak discharge for each observed hydrograph, simulated hydrographs were 

generated for each of the 355 observed hydrography and their widths were compared with 

the widths of the observed hydrographs at 50 and 75 percent of peak flow. Inman (1986) 

concluded that the dimensionless hydrograph based on the one-half lagtime duration provided the best fit of the observed data. At the 50 percent of peak flow width, the standard 

error of estimate was + 3 1.8 percent; and at the 75 percent of peak flow width, the standard error of estimate was + 35.9 percent. 

For verification, the one-half lagtime duration dimensionless hydrograph was applied 

to 138 hydrographs from 37 Georgia stations that were not used in its development. The 

drainage areas of these stations ranged from 20 to 500 mi2. Inman (1986) reported that 

at the 50 percent of peak flow width, the standard error of estimate was 2 39.5 percent; 

and at the 75 percent of peak flow width, the standard error of estimate was + 43.6 

percent. 

Inman (1986) performed a second verification to assess the total or accumulative 

prediction error for large floods through the combined use of the dimensionless hydrograph, estimated lagtimes from regional lagtime equations, and peak discharges from 

regional flood-frequency equations. Inman (1986) reported standard errors of prediction 

of + 5 1.7 and + 57.1 percent at the 50 and 75 percent of peak flow widths, respectively. 

2 
On the basis that Inman’s basic dimensionless hydrograph was developed and tested 

for a variety of conditions (including urban, rural, mountainous, coastal plain, and small 

and large drainage basins), it was theorized that it may be applicable to streams in central 

Tennessee. The time and discharge coordinates of Inman’s (1986) dimensionless hydrograph 

are listed in table 1. 

TESTING INMAN’S DIMENSIONLESS HYDROGRAPH 

ON CENTRAL TENNESSEE STREAMS 

Inman’s dimensionless hydrograph was tested by comparing it to a similar dimensionless hydrograph developed for central Tennessee streams. The test involved several 

phases and is described in detail in this section of the report. 

A total of 245 observed hydrographs from 38 basins across central Tennessee (fig. 1) 

with drainage areas of 0.17 to 481 mi2, representing both rural and urban streamflow 

characteristics, were available for use in the test. However, only 163 observed hydrographs had matching rainfall data and were selected for use in this test. The 38 basins are 

located within hydrologic areas 2 and 3 in Tennessee, as defined by Randolph and Gamble 

(1976). Of the 38 basins, 21 (81 observed hydrographs) are in hydrolo ic area 2, and 17 (82 

observed hydrographs) are in hydrologic area 3. A computer program $ S.E. Ryan, U.S. Geological Survey, written commun., 1986) used by Inman (1986) for development of the dimensionless hydrograph and subsequent statistical analyses also was used for this analysis. 

Unit hydrographs and lagtime were computed from the 163 observed hydrographs, 

and an average unit hydrograph and an average lagtime were computed for each of the 38 

basins. These average unit hydrographs were transformed to unit hydrographs having 

generalized durations of one-fourth, one-third, one-half, and three-fourths lagtime, then 

reduced to dimensionless terms by dividing the time by lagtime and the discharge by peak 

discharge. The 38 dimensionless hydrographs were then separated into their respective 

hydrologic areas (21 in area 2, and 17 in area 3). Representative dimensionless hydrographs 

from eight randomly selected basins in each area were combined to generate one typical 

(average) dimensionless hydrograph for each of the four generalized durations in each 

hydrologic area. 

In each hydrologic area, the four generalized duration dimensionless hydrographs, 

average basin lagtimes, and peak discharges from the 163 observed hydrographs were used 

to generate simulated hydrographs for the corresponding observed hydrographs. The simulated hydrograph widths were compared with the widths of the observed hydrographs at 

50 and 75 percent of peak flow. The dimensionless hydrograph based on the one-half lagtime duration provided the best fit of the observed data in both hydrologic areas. At the 

50 and 75 percent of peak flow widths, standard errors of estirnate were + 21.4 and + 24.4 

percent, respectively, in hydrologic area 2 and + 22.2 and 2 26.0 percent, respectively, in 

hydrologic area 3. 

Inman’s dimensionless hydrograph was then used with the average basin lagtimes, 

and peak discharges from the 163 observed hydrographs to generate simulated hydrographs 

for the observed hydrographs in each hydrologic area. The simulated hydrographs and the 

observed hydrographs were compared at 50 and 75 percent of their peak flow widths. At 

the 50 and 75 percent of peak flow widths, standard errors of estimate were + 20.8 and 

+ 24.1 percent, respectively, in hydrologic area 2 and + 22.1 and + 26.0 percent, respeclively, in hydrologic area 3. Based on the standard errors of estimate, the simulated 
Table 1 .--Time and discharge ratios of Inman's dimensionless hydrograph
Figure 1 .--Location of study area and gaging stations used in dimensionless hydrograph analysis.
hydrographs generated from Inman’s dimensionless hydrograph had a slightly better fit of 

the observed data than the simulated hydrographs generated from the average dimensionless hydrograph for each hydrologic area. 

The eight randomly selected dimensionless hydrographs from each hydrologic area 

were then combined to generate one typical (average) dimensionless hydrograph (based on 

16 dimensionless hydrographs) for each of the four generalized durations for the entire 

area (hydrologic areas 2 and 3). Using the four generalized duration dimensionless hydrographs, average basin lagtimes, and peak discharges from the 163 observed hydrographs, 

simulated hydrographs were generated for the corresponding observed hydrographs and 

their widths were compared with the widths of the observed hydrographs at 50 and 75 

percent of peak flow. As in the preceding test, the dimensionless hydrograph based on the 

one-half lagtime duration provided the best fit of the observed data. At the 50 and 75 percent of peak flow widths, standard errors of estimate were 2 21.7 and + 25.4 percent, 

respectively. 

Finally, Inman’s dimensionless hydrograph, average basin lagtimes, and peak discharges from the 163 observed hydrographs were used to generate simulated hydrographs 

for the corresponding observed hydrographs. The simulated hydrographs and the observed 

hydrographs were compared at 50 and 75 percent of their peak flow widths. Standard 

errors of estimate were 2 21.2 and + 24.8 percent at the 50 and 75 percent of peak flow 

widths, respectively. Based on the standard errors of estimate, the simulated hydrographs 

generated from Inman’s dimensionless hydrograph had a slightly better fit of the observed 

data than the simulated hydrographs generated from the average dirnensioniess hydrograph 

developed f rain Tennessee data. 

Based on the preceding tests, Inman’s dimensionless hydrograph is applicable to 

streams in central Tennessee, and the average dimensionless hydrograph derived from 16 

of the 38 central Tennessee basins is essentially the same as the dimensionless hydrograph 

derived from Georgia basins. Therefore, Inman’s dimensionless hydrograph is the preferred 

one to use for simulating hydrographs in central Tennessee. A comparison of the two 

dimensionless hydrographs is shown in figure 2. At 50 and 75 percent of the ratio Qt/Qp 

widths, the Inman dimensionless hydrograph is 5.4 and 4.8 percent wider, respectively, thall 

the one derived for central Tennessee. The difference between the two dimensionless 

hydrographs is probably due to the number of basins used to develop each dimensionless 

hydrograph. 

Verification Of Dimensionless Hydrograph 

A test was performed to assess the total or accumulative prediction error for large 

floods through the combined use of the Inman dimensionless hydrograph, estimated basin 

lagtimes, and discharges derived from regional flood-frequency equations. Randolph and 

Gamble (1976) provide a technique for estimating the peak discharge of a selected recurrence interval for rural streams in Tennessee, and Robbins (1984) provides a technique for 

estimating the peak discharge of a selected recurrence interval for urban basins draining 

areas less than 25 mi2 in Tennessee. Equations for estimating basin lagtime were developed for central Tennessee streams as part of this study and are presented in a supplement 

at the end of this report. 

This verification test used 36 observed hydrographs (not previously used) from 36 of 

the original 38 basins. These hydrographs were selected from those having the highest 

6 
Figure 2.-- Comparison of Inman and central Tennessee

dimensionless hydrographs.
peaks at each station, and where a station flood-frequency curve was available. The test 

was conducted as follows. The recurrence interval of each observed peak discharge was 

determined from its station-frequency curve. The appropriate regional flood-frequency 

equation, from Randolph and Gamble (1976) or Robbins (19841, was then used to estimate 

the corresponding peak discharge for this recurrence interval. The average recurrence 

interval of the 36 peaks was approximately 25 years. For each station, a basin lagtime 

was estimated from the appropriate regional basin lagtime equation (presented in a later 

section of this report). The estimated peak discharge, the estimated basin lagtime for 

each basin, and Inman’s dimensionless hydrograph were then used to generate simulated 

flood hydrographs. A comparison of the simulated and observed hydrograph widths at 50 

and 75 percent of peak flow showed standard errors of prediction of + 43.9 and + 39.2 

percent, respectively. 

Example comparisons between observed hydrographs and simulated hydrographs 

based on observed peak discharge and measured basin lagtime, and regression discharge 

and regression basin lagtime are shown in figures 3, 4, and 5. A comparison between 

observed and simulated hydrographs for an urban basin in the Nashville, Tenn., area is 

shown in figure 3. Similar comparisons for much larger basins in the rolling hills of central 

Tennessee are shown in figures 4 and 5. All three comparisons show good agreement 

between the observed and simulated hydrographs. Peak discharges of the simulated hydr+ 

i? 

raphs based on regression (estimated) discharge (Q 

LT) may not coincide with the observed peak 

) and regression (estimated) lagtime 

disc R arges because the simulated hydrographs incorporate the error inherent in the regional flood-frequency relations. These 

errors are representative of the total error that might occur at an ungaged site. 

Bias 

Two tests for bias were conducted, one for simulated versus observed hydrograph 

width, and the other for geographical bias. The width-bias test was conducted using the 

residuals (in percent) at 50 and 75 percent of peak flow for the 36 stations in the previous 

verification test. The mean error indicated a small positive error (simulated greater than 

observed) in the hydrograph widths at 50 percent of peak flow and a small negative error 

(simulated less than observed) in the hydrograph widths at 75 percent of peak flow. Additionally, there was a small positive error (regression greater than observed) in the comparison of peak discharge from regional regression equations and peak discharge from station 

frequency curves. The students t-test indicated these errors were not statistically significant at the 0.01 level of significance, and therefore, the simulated hydrograph widths and 

regression discharges are not biased. 

In the geographical-bias test, residual differences in simulated and observed hydrograph widths (in percent) at 50 and 75 percent of peak flow at each station were plotted 

on a map to evaluate the bias of the simulated hydrographs. Although the residual differences in widths varied considerably between some stations, no specific geographic trends 

could be detected. Results of both these tests indicate no bias in the results using the 

outlined procedures and support results reported by Inman (1986). 

REGIONALIZATION OF BASIN LAGTIME AND FLOOD VOLUME 

Estimating Basin Lagtime 

Average basin lagtime is used as the principal time factor in the dimensionless hydrograph. Lagtime (LT) is generally considered to be constant for a basin (as long as basin 

8 
Figure 3.--Comparison of observed and simulated hydrogrophs

for Richland Creek at Charlotte Avenue at Nashville, Tenn.

(03431700), for storm of May 3, 1979 .
Figure 4.-- Comparison of observed and simulated hydrographs

for Wolf River near Byrdstown, Tenn. (03416000), for

storm of April 3, 1977.

10
Figure 5.--Comparison of observed and simulated hydrographs

for Duck River near Shelbyville, Tenn. (03598000), for

storm of March 14, 1973.
conditions remain the same) and is defined as the elapsed time from the centroid of rainfall excess to the centroid of the resultant runoff hydrograph (Stricker and Sauer, 1982). 

The lagtime of a basin is the principal factor in determining the relative shape of a hydrograph from that basin. For example, a long lagtime will produce a broad flat-crested 

hydrograph and a short lagtime will produce a narrow sharp-crested hydrograph. Since 

lagtime is usually not known for a basin, it is often estimated from basin characteristics. 

To provide a method of estimating lagtime for ungaged basins in central Tennessee, 

the 38 average basin lagtimes obtained from the dimensionless hydrograph development 

procedure and 7 basins with measured lagtimes from a rainfall-runoff modeling study by 

Wibben (1976) were related to their basin characteristics. Rural and urban basins were 

analyzed separately because of the effects of urbanization on lagtime. Standard multiple 

linear regression techniques were used to develop equations for estimating rural and urban 

basin lagtimes from five basin characteristics. All five characteristics defined below 

were used in the regression analyses; however, only those characteristics statistically significant at the 9%percent confidence level are included in the final equations. Definitions 

of these five basin characteristics are as follows: 

grainage area (DA) is the contributing drainage area of the basin, in square miles. 

Channel slope (CS) is the slope, in feet per mile, of the main channel determined from the 

difference in elevation at points 10 and 85 percent of the distance along the main 

channel from the discharge site to the drainage-basin divide. 

Channel length (CL) is the distance, in miles, from the discharge site to the drainage-basin 

divide, measured along the main water course. 

CL/a is a ratio, where CL and CS are as previously defined. 

Percentage of impervious area (IA) is the percentage of the contributing drainage area 

that is impervious to infiltration of rainfall. This parameter was measured using the 

grid method on recent aerial photographs. IA can also be measured from topographic 

maps or from population and industrial density reports. 

All of the basins and their characteristics used in the regression analyses are listed in 

table 2. 

Regression Analyses 

Stepwise regression techniques were used with the five basin characteristics of 31 

rural basins and 14 urban basins to derive equations for estimatin basin lagtime during 

the initial regression analyses. Drainage area, channel slope, CL/ t- CS, and percentage of 

impervious area were insignificant for rural lagtime and were deleted from successive 

regression analyses. All of these basin characteristics, with the exception of percentage 

of impervious area, were insignificant for urban lagtime also, and were deleted from 

successive regression analyses. 

The final regression analyses were performed using channel length for rural basin 

lagtime, and channel length and percentage of impervious area for urban basin lagtime. 

These selected variables are easily obtainable and are of practical use in estimating rural 

and urban basin lagtimes in central Tennessee. Drainage area size of the basins used in 

the rural lagtime regression analyses ranged from 0.17 to 481 mi2, and in the urban lagtime regression analyses, from 0.47 to 64 ml ‘2. The following tables summarize the distribution of drainage area size for the basins used. 

12 
Table 2 .--Stations and drainage basin characteristics used in lagtime regression analyses
Table 2 .--Stations and drainage basin characteristics used in lagtime regression analyses--Continued
ns

Channel length ranged from 0.56 to 74.0 miles for the rural lagtime regression analysis and from 0.65 to 17.0 miles for the urban lagtime regression analyses. The following

tables summarize the distribution of channel length for the basins used.

In the urban lagtime analysis, percentage of impervious area ranged from 4.20 to

48.3 percent . The following table summarizes the distribution of impervious area for the

basins used .
A correlation matrix indicated that drainage area, channel slope, and channel length

were highly correlated. In addition, the one-variable equations containing drainage area

or channel slope had higher standard errors of estimate and lower R 2 (coefficient of

determination) values than the equations containing channel length . Therefore, the following equations may be used for estimating rural basin lagtimes and urban basin lagtimes

for ungaged basins in central Tennessee. It should be noted that the standard error of

estimate for the urban basin lagtime equation may be unusually low because of the limited

data base (14 stations) .

where ULT is estimated urban basin lagtime, in hours;

CL is channel length, in miles and

IA is the percentage of the contributing drainage basin occupied by impervious

surface.

The log-linear form of the estimating equations was checked with graphical plots.

Plots of regression residuals versus observed lagtime, residuals versus channel length, and

residuals versus percentage of impervious area were included on each graph. The scatter

of plotting points on each graph appeared to be random with no apparent bias. Therefore,

the form of the estimating equation is assumed to be appropriate.

It should be noted that the urban basin lagtime equation will sometimes predict a

longer lagtime than the rural basin lagtime equation. Conceptually, this situation should

not exist because increasing imperviousness should decrease lagtime. Therefore, when

estimating lagtime for urbanized basins, lagtime should be calculated from both equations,

and the smallest value should be used. Additionally, the estimates of lagtime for urban

basins with channel lengths less than 2 miles may have errors higher than the reported

standard error of estimate.

16
Station residuals were plotted on a map to evaluate geographic bias of estimates

from the rural and urban basin lagtime equations. Although the residuals varied considerably between some stations, no specific geographic trends could be detected. Due to the

limited number of stations available for the rural (31) and urban (14) lagtime regression

analysis,, verification of the regression equatibns was not possible .

A partial analysis of the sensitivity of the rural and urban basin lagtime equations to

channel length (CL) and percentage of impervious area (IA) was performed. Results of

sensitivity of the regression equations are shown graphically in figure 6. For the rural

basin lagtime equation, for example, an error of 40 percent in computing channel length

results in about a 33-percent difference in lagtime. For the urban basin lagtime equation,

an error of 40 percent in computing channel length results in about an 18-percent difference in lagtime, and an error of 60 percent in computing the percentage of impervious

area results in about a 7-percent difference in lagtime.

Estimating Flood Volume

Storage of floodwater or flood prevention may often be part of a particular structure's design . In such cases, it is important to know the volume associated with the design

flood. Therefore, an equation for estimating flood volumes for selected recurrence interval floods on central Tennessee streams was developed . The equation relates flood volumes to drainage area size, flood peak discharge, and basin lagtime. Observed flood

volumes (in inches of runoff) from 245 storms from the 38 basins used in the dimensionless

hydrograph analysis were used in this analysis. Flood volumes were obtained as part of the

unit hydrograph computations discussed earlier. The flood-volume analysis was conducted

using a split-sample technique in which half of the data set (123 storms) was used to

develop the regression equation and the other half (122 storms) was used for verification.

and to determine prediction error.

Regression Analyses

Stepwise regression techniques were used with three basin characteristics to derive

the equation for estimating flood volumes. The three basin characteristics, drainage area

size, flood peak discharge, and basin lagtime, were all statistically significant at the 95-

percent confidence level. Drainage area sizes used in the regression analysis had the same

range and distribution as that used in the rural and urban basin lagtime analysis (0.17 to

481 mi2). Flood peak discharges ranged from 24 to 45,200 ft 3/s. The following table summarizes the distribution of flood peak discharges for the storms used.

Range i n flood

1 7
18

Figure 6.-- Percent ohonqe inrural !

~ enoou!ting

frornerrors inoornputingchannel length ondpercentage o[impervious

area.
i.c:Auuat .a'.iwTrJ." :.xs$`u :r.+ :. ..ACS .t:nuui.±siiLGC.

--Yr .,.u ".nw.. ....r..L:.. ..

Measured basin lagtimes ranged from 0.69 to 41 .6 hours. The following table summarizes the distribution of lagtimes for the basins from which the storms were obtained.

The following equation may be used for estimating flood volumes associated with a

T-year peak discharge for ungaged streams in central Tennessee. Flood volume, for a

given T-year peak discharge, also can be obtained by summing the ordinates of the estimated flood hydrograph.

where V is estimated flood volume, in inches;

DA is drainage area, in square miles;

Q is flood peak discharge, in cubic feet per second; and

LT is basin lagtime, in hours.

A second regression analysis was performed using the other half of the data set.

This regression analysis produced a volume equation that was not significantly different

from the first equation. A third analysis was performed to determine the prediction error

associated with using the volume equation at an ungaged site. This was accomplished by

substituting regression basin lagtime for measured basin lagtime, and regression discharge

associated with the observed recurrence interval flood for observed discharge . The volume equation was then used to predict flood volume for each of the observed storms in

the second half of the data set. Results of this analysis produced an estimate of the standard error of prediction (listed above) which is a measure of how well the regression equation will estimate the dependent variable at sites other than those used to derive the

equation.

The log-linear form of the estimating equation was checked with graphical plots.

Plots of regression residuals versus drainage area, residuals versus flood peak discharge,

and residuals versus basin lagtime were included on each graph. The scatter of plotting

points on each graph appeared to be random with no apparent bias. Therefore, the form

of the estimating equation is assumed to be appropriate .

Station residuals were plotted on a map to evaluate geographic bias of estimates

from the flood-volume equation. Although the residuals varied between stations, no

geographic trends could be detected.
A partial analysis of the sensitivity of the volume equation to drainage area (DA),

flood peak discharge (Qp), and basin lagtime (LT) was performed, and the results are

shown graphically in figure 7. Results of the sensitivity analysis indicate that an error of

40 percent in computing drainage area, for example, results in about a 30-percent difference in flood volume. An error of 20 percent in computing flood peak discharge and an

error of 10 percent in computing basin lagtime results in differences in flood volumes of

21 percent and 11 percent, respectively.

For some hydraulic analyses, it is only necessary to estimate the period of time that

a specific discharge will be exceeded, therefore a complete flood hydrograph is not

needed. In order to estimate this time period, a hydrogaph-width relation was defined by

Inman (1986) for the dimensionless hydrograph in table 1 . Hydrograph-width ratios were

determined by subtracting the value of t/LT on the rising limb of the dimensionless hydrograph from the value of t/LT on the falling limb of the hydrograph at the same discharge

ratio (Qt /Qp) over the full range of the dimensionless hydrograph . The resulting hydrograph-width relations are listed in table 3 and are shown graphically in figure 8. The simulated hydrograph width (W) in hours can be estimated for a specified discharge (Qt ) by

f irst computing the ratio Qt/Qp and then multiplying the corresponding W/LT ratio in

table 3 by the estimated basin lagtime (LT) . The resulting hydrograph width is the period

of time a specified discharge will be exceeded .

- n s npsarw^rn- T,:.~n. / . rnw. ;..

HYDROGRAPH-WIDTH RELATION

Table 3.--Discharge and hydrograph-width ratios for Inman's

dimensionless hydrograph

20
Figure 7.--Percent change in flood volume resulting from

errors in computing drainage area, lagtime, and peak

discharge.
Figure 8.--Hydrograph-width relation for Inman's

dimensionless hydrograph.

22
APPLICATION OF HYDROGRAPH SIMULATION TECHNIQUE 

A step-by-step procedure is described below to assist the user in applying the techniques for simulating flood hydrographs and estimating flood volumes and hydrograph 

widths as presented in this report. In addition, an example is given to demonstrate these 

techniques. The procedure is as follows: 

c 1. 

2. 

3. 

4. 

5. 

6. 

7. 

Determine the drainage area and main-channel length of the basin from the 

best available topographic maps. 

Compute the peak discharge for the desired recurrence-interval flood from the 

applicable flood-frequency report (flood-frequency equations included in the 

supplement). 

Estimate percentage of impervious area if the basin is urbanized. 

Compute the basin lagtime from the appropriate equation (1 or 2, page 16). 

Compute the coordinates of the flood hydrograph by multiplying the value of 

lagtime by the time ratios and the value of peak discharge by the discharge 

ratios (table 1, page 4). 

Compute the volume for the selected recurrence-interval flood using equation 3 

(page 19). 

Compute the period of time a specific discharge will be exceeded using the 

dimensionless hydrograph-width relation (table 3, page 20; or figure 8, page 22). 

: I 

Example Problem 

The following example illustrates the procedure for computing the simulated hydrograph associated with the loo-year discharge estimate in a hypothetical rural basin in 

hydrologic area 3 in central Tennessee. 

1. The drainage area (DA) is determined as 393 mi* and the main-channel length 

is determined to be 60.6 miles. 

2. The peak discharge (Qloo) for the loo-year recurrence-interval flood is 82,500 

f t3/s (Randolph and Gamble, 1976--in supplement). 

3. Using equation 1, rural basin lagtime (RLT) is estimated to be: 

RLT = 0.94 (CL)0.86 

= 0.94 (60.6)0-86 

= 32.1 h. 

23 
4. The coordinates of the simulated flood hydrograph are listed below and are

shown graphically in figure 9 :

24

7,-v,'7-,77':7,

,1.7,.

777, t<? I {u7iry,,f1`

. 77M4~

.

,w~' ,71 .'ii'r. .'imn""W
Figure 9.-- Plot of example simulated 100-year flood hydrograph

for a hypothetical river in central Tennessee.
5. Using equation 3, flood volume (V) is estimated to be: 

V = I.3 x JO-3 (DA)-1.06 (Q )I.05 LT 1.03 

= 1.3 x lo-3 (393)-l-06 $2,500) 5 00 (32.1$03 t, 

= 12.0 in. 

6. If an estimate were needed for the period of time road overflow (beginning at a discharge of 60,000 ftj/s) would occur, compute it as follows: 

ba: from Qt/Qp = 60,000/82,500 8, W/LT = = 0.58 0.73 

figure 

:: rural basin 1 agtime, RLT = 32.1 h, from step 3 

road overflow time = (W/LT)(RLT) 

1 \;.658;(32.1> 

. . 

Limitations 

The techniques for simulating flood hydrographs and estimating flood volumes 

described in this report are limited to streams in hydrologic areas 2 and 3 in central Tennessee. In deriving the rural and urban lagtime equations, basin size ranged from 0.17 to 

481 mi* and from 0.47 to 64.0 mi*, respectively; channel length ranged from 0.56 to 

74.0 miles and from 0.65 to 17.0 miles, respectively; and impervious area in urban basins 

ranged from 4.20 to 48.3 percent. For the flood-volume equation, flood peak discharge 

ranged from 24 to 45,200 ft3/s and basin lagtime ranged from 0.69 to 41.6 hours. Use 

of the hydrograph simulation technique and regression equations should be limited to these 

ranges because the techniques presented have not been tested beyond the indicated ranges 

in values. If values outside these ranges are used, the standard error may be considerably 

higher than for sites where all variables are within the specified ranges. In addition, these 

techniques should not be applied to streams where temporary in-channel storage or overbank detention storage is significant unless suitable estimates of peak discharge and 

lagtime are available which account for these effects. 

CONCLUSIONS 

A dimensionless hydrograph developed for Georgia streams was tested for its 

applicability to central Tennessee streams by comparing it to a similar dimensionless 

hydrograph developed for central Tennessee streams. Test results indicate the two 

dimensionless hydrographs are essentially the same. Therefore, the Georgia dimensionless 

hydrograph can be used to simulate flood hydrographs at ungaged sites for both rural and 

urban streams in central Tennessee. A total of 163 observed flood hydrographs from 38 

basins in central Tennessee were used in the test of the Georgia dimensionless hydrograph 

and an additional 36 flood hydrographs were used for a verification test. 

Multiple-regression techniques were used to develop relations between basin lagtime 

and selected basin characteristics, of which channel length was significant for the rural 

basins, and channel length and percentage of impervious area were significant for urban 

basins. Tests indicated no variable or geographical bias in either the rural or urban 

equations. 

26 
An equation for estimating flood volumes was also developed using multipl+ 

regression techniques. Drainage area, flood peak discharge, and basin lagtime were the 

significant variables in the volume equation. Tests indicated no variable or geographic 

bias in the volume equation. 

A simulated flood hydrograph can be computed by applying lagtime, obtained from 

the appropriate regression equation, and peak discharge of a specific recurrence interval, 

to the dimensionless hydrograph time and discharge ratios in table 1. The coordinates of 

the simulated flood hydrograph are computed by multiplying lagtime by the time ratios 

and peak discharge by the discharge ratios. The volume of the simulated flood hydrograph 

can be estimated from the volume regression equation. 

SELECTED REFERENCES 

Inman, E.J., 1986, Simulation of flood hydrographs for Georgia streams: U.S. Geological 

Survey Water-Resources Investigations Report 86-4004, 48 p. 

O’Donnell, Terrance, 1960, Instantaneous unit hydrograph derivation by harmonic analysis: 

Commission of Surface Waters, Publication 51, International Association of Scientific 

Hydrology, p. 546-557. 

Randolph, W.J., and Gamble, C.R., 1976, Technique for estimating magnitude and frequency of floods in Tennessee: Tennessee Department of Transportation, 52 p. 

Robbins, C.H., 1984, Synthesized flood frequency for small urban streams in Tennessee: 

U.S. Geological Survey Water-Resources Investigations Report 84-4182, 24 p. 

Stricker, V.A., and Sauer, V.B., 1982, Techniques for estimating flood hydrographs for 

ungaged urban watersheds: U.S. Geological Survey Open-File Report 82-365, 24 p. 

U.S. Department of Commerce, 1961, Rainfall frequency atlas of the United States: 

Weather Bureau Technical Paper No. 40, 61 p. 

Wibben, H.C., 1976, Application of the U.S. Geological Survey rainfall-runoff simulation 

model to improve flood-frequency estimates on small Tennessee streams: U.S. Geological Survey Water-Resources Investigations Report 76-120, 53 p. 

, 

27 
SYMBOLS, DEFINITIONS, AND UNITS 

c 

Symbol 

CL 

Definition 

Channel length measured along main water course 

cs Main channel slope 

CL/J.3 Ratio of channel length to the square root 

of channel slope 

DA Contributing drainage area of a basin 

IA 

LT 

Impervious area 

Basin lagtime 

P&P4 2-year 24-hour rainfall amount 

QP 

Qt 

Qt/QP 

Flood peak discharge 

Discharge occuring at time t 

Ratio of discharge occurring at time t 

to flood peak discharge 

Q2,5,10,25,50,100 Rural basin flood-frequency discharge 

for recurrence intervals of 2 through 

loo-years, respectively 

Q(u)2,5, 10,25,50,100 Urban basin flood-frequency discharge 

for recurrence intervals of 2 through 

loo-years, respectively 

R2 Coefficient of determination 

RLT 

t/LT 

Rural basin lagtime 

Ratio of instantaneous time to basin lagtime 

ULT 

V 

W 

W/LT 

Urban basin lagtime 

Flood volume 

Hydrograph width 

Ratio of hydrograph width to basin lagtime 

28 

Unit 

mi 

ft/mi 

mi2 

4 

percent 

h 

in. 

f G/s 

f G/s 

f G/s 

fG/s 

m-m 

h 

-mm 

h 

in. 

h 
SUPPLEMENT 

29 
REGIONAL FLOOD-FREQUENCY EQUATIONS

FOR RURAL BASINS IN TENNESSEE

The following is a list of the rural basin flood-frequency equations from Randolph

and Gamble (1976) for hydrologic areas 2 and 3.

Hydrologic Area 2

Q2 = 199 AO " 744

Q5 = 352 AO . 729

Q10 = 465 AO .723

Q25 = 614 AO " 722

Q50 = 738 AO " 719

Q100 = 867 AO " 718

Hydrologic Area 3

Q2 = 319 AO " 733

Q5 = 512 AO " 725

Q10 = 651 AO-723

Q25 = 836 AO " 720

Q50 = 977 AO " 720

Q100 = 1,125 AO-719

where Q25 is the 25-year recurrence-interval flood, in cubic feet per second; and

A is contributing drainage area, in square miles.

30
ltl?(% ION AL t'l .~)~)1~-i~ltl?~~I113Nc',I-VIIATIONS

I`% S14 1114 OA N 1~A%INs IN I . 111VNIt1410'1%

The following is a list of the urban basin flood-frequency equations frorn Robbins

(1984) which are applicable statewide. The precipitation factor (P2 24) used in each equation can be determined from figure 10.

Where Q(u) is the 25-year recurrence-interval flood, in cubic feet per second;

is contributing drainage area, in square miles;

IA is percentage of the contributing drainage basin occupied by impervious

surface ; and

P2 24 is the 2-year, 24-hour rainfall amount, in inches,
Figure 10.--2-year 24-hour rainfall, in inches

(U.S. Department of Commerce, 1961).

32

*U.S . GOVERNMENT PRINTING OFFICE 1986-631-262/40028