U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2011–5104

Prepared in cooperation with the Harris County Flood Control District and the Texas Department 

of Transportation

A Method for Estimating Peak and Time of Peak Streamflow 

from Excess Rainfall for 10- to 640-Acre Watersheds in the 

Houston, Texas, Metropolitan Area
Cover: Photograph courtesy of Steve Fitzgerald, Harris County Flood Control District, 2005. 
A Method for Estimating Peak and Time of Peak 

Streamflow from Excess Rainfall for 10- to 640-Acre 

Watersheds in the Houston, Texas, Metropolitan Area 

By William H. Asquith, Theodore G. Cleveland, and Meghan C. Roussel 

Prepared in cooperation with the Harris County Flood Control District and the Texas Department of 

Transportation 

Scientific Investigations Report 2011–5104 

U.S. Department of the Interior 

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

KEN SALAZAR, Secretary 

U.S. Geological Survey 

Marcia K. McNutt, Director 

U.S. Geological Survey, Reston, Virginia: 2011 

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U.S. Government. 

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reproduce any copyrighted materials contained within this report. 

Suggested citation:

Asquith, W.H., Cleveland, T.G., and Roussel, M.C., 2011, A method for estimating peak and time of peak streamflow 

from excess rainfall for 10- to 640-acre watersheds in the Houston, Texas, metropolitan area: U.S. Geological Survey 

Scientific Investigations Report 2011–5104, 41 p., at http://pubs.usgs.gov/sir/2011/5104. 
iii 

Contents 

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

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

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

Study Watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
 

Database of Rainfall and Runoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
 

Selected Watershed Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

Analysis of Gamma Unit Hydrographs for the 24 Watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

Computing Gamma Unit Hydrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 

Analysis of Rational Method for the 24 Watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 

Background and Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 

Computational and Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
 

A Method for Estimating Peak and Time of Peak Streamflow from Excess Rainfall for 10- to 640-Acre
 

Watersheds in the Houston, Texas, Metropolitan Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
 

Comparison of Results from Unit Hydrograph and Rational Method Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
 

Nomograph and Example Computations for the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 

Nomograph for the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 

Example Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 

Potential Bias in the Excess Rational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 

On the Relative Influence of Basin-Development Factor on Peak Streamflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 

Influence of Basin-Development Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 

Example Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
 

Appendix 1—Basin-Development Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 

Appendix 2—Selected Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
 

Figures 

1.	 Map showing downstream locations of the 24 selected watersheds in the Houston, Texas, metropolitan area used for gamma unit hydrograph and rational method analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
 

2.	 Example of headers (definitions) and selected data parts of rainfall and runoff data files for storm on
 

January 22, 2006, for BMP Basin 504 on Armand Bayou, Houston, Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 

3.	 Statistical summary of watershed-specific, peak streamflow of gamma unit hydrograph and regression equation for estimation of peak streamflow for applicable watersheds in the Houston, Texas,
 

metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 
iv 

4.	 Statistical summary of watershed-specific, time to peak of gamma unit hydrograph and regression
 

equation for estimation of time to peak for applicable watersheds in the Houston, Texas, metropolitan
 

area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 

5.	 Statistical summary of watershed-specific, gamma unit hydrograph shape parameter and regression equation for estimation of shape parameter for applicable watersheds in the Houston, Texas,
 

metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 

6–9. Graphs showing: 

6.	 Relation between watershed-specific, peak streamflow and fitted values of peak streamflow
 

by regression shown in equation 5 for a gamma unit hydrograph developed for 24 watersheds
 

in the Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 

7.	 Relation between watershed-specific, time to peak and fitted values of time to peak by regression shown in equation 6 for a gamma unit hydrograph developed for 24 watersheds in the
 

Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 

8.	 Dimensionless gamma hydrographs by shape parameter for 24 watersheds in the Houston,
 

Texas, metropolitan area, and for applicable developed and undeveloped, non-Houston
 

watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
 

9.	 Example of a gamma unit hydrograph with a shape factor of 1 from a 0.5-square mile watershed with a basin-development factor of 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
 

10.	 Screenshot of highly specialized, single-purpose software for inversion of rational method for storm
 

on January 22, 2006, for BMP Basin 504 on Armand Bayou, Houston, Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 

11.	 Screenshot of output from highly specialized, single-purpose software showing computational
 

results for storm on January 22, 2006, for BMP Basin 504 on Armand Bayou, Houston, Texas, as
 

shown in figure 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
 

12.	 Statistical summary of watershed-specific, runoff and volumetric runoff coefficients, and runoff
 

coefficient ratios for 24 watersheds in the Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 

13.	 Statistical summary of watershed-specific, time-R and regression equation for time-R estimation for
 

applicable watersheds in the Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 

14–16. Graphs showing: 

14.	 Relation between watershed-specific, time-R and fitted values of time-R by regression
 

shown in equation 19 developed for 24 watersheds in the Houston, Texas, metropolitan area . . . . . . . 19
 

15.	 Relation between time to peak or depth of peak streamflow of the gamma unit hydrograph to
 

drainage area and basin development factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 

16.	 Nomograph of the relation between arithmetic means of peak and time of peak streamflow
 

by gamma unit hydrograph and excess rational method computations for 1 inch of excess
 

rainfall by selected basin-development factor and drainage area for 24 watersheds in the
 

Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 

17.	 Steps for implementing the method for 10- to 640-acre watersheds in the Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 

18.	 Graph showing relation between ratio of watershed-specific (mean) values of runoff coefficient and
 

volumetric runoff coefficient to drainage area for 24 watersheds in the Houston, Texas, metropolitan
 

area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 

Tables 

1.	 Summary of 24 selected watersheds in the Houston, Texas, metropolitan area used for gamma unit
 

hydrograph and rational method analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
 
v 

2.	 Summary of storm counts and watershed-mean parameters of gamma unit hydrograph analysis and 

watershed-mean parameters for rational method analysis of 24 selected watersheds in the Houston, 

Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

3.	 Comparison of values of peak and time of peak streamflow by gamma unit hydrograph and excess 

rational method computations for 1 inch of excess rainfall for limiting developed and undeveloped 

conditions for a 300-acre watershed in the Houston, Texas, metropolitan area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

Conversion Factors
 

Multiply By To obtain 

Length 

inch (in.) 25.4 millimeter (mm) 

foot (ft) .3048 meter (m) 

mile (mi) 1.609 kilometer (km) 

Area 

square mile (mi2) 2.590 square kilometer (km2) 

acre (acre) .001563 square mile (mi2) 

acre (acre) .004047 square kilometer (km2) 

Flow 

cubic foot per second (ft3/s) .02832 cubic meter per second (m3/s) 

Datum 

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

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88). 

A Method for Estimating Peak and Time of Peak 

Streamflow from Excess Rainfall for 10- to 640-Acre 

Watersheds in the Houston, Texas, Metropolitan Area 

By William H. Asquith, Theodore G. Cleveland, and Meghan C. Roussel 

Abstract 

Estimates of peak and time of peak streamflow for 

small watersheds (less than about 640 acres) in a suburban 

to urban, low-slope setting are needed for drainage design 

that is cost-effective and risk-mitigated. During 2007–10, 

the U.S. Geological Survey (USGS), in cooperation with 

the Harris County Flood Control District and the Texas 

Department of Transportation, developed a method to 

estimate peak and time of peak streamflow from excess 

rainfall for 10- to 640-acre watersheds in the Houston, 

Texas, metropolitan area. To develop the method, 24 

watersheds in the study area with drainage areas less than 

about 3.5 square miles (2,240 acres) and with concomitant rainfall and runoff data were selected. The method is 

based on conjunctive analysis of rainfall and runoff data 

in the context of the unit hydrograph method and the rational method. For the unit hydrograph analysis, a gamma 

distribution model of unit hydrograph shape (a gamma 

unit hydrograph) was chosen and parameters estimated 

through matching of modeled peak and time of peak 

streamflow to observed values on a storm-by-storm basis. 

Watershed mean or watershed-specific values of peak and 

time to peak (“time to peak” is a parameter of the gamma 

unit hydrograph and is distinct from “time of peak”) of 

the gamma unit hydrograph were computed. Two regression equations to estimate peak and time to peak of the 

gamma unit hydrograph that are based on watershed characteristics of drainage area and basin-development factor 

(BDF) were developed. For the rational method analysis, 

a lag time (time-R), volumetric runoff coefficient, and 

runoff coefficient were computed on a storm-by-storm 

basis. Watershed-specific values of these three metrics 

were computed. A regression equation to estimate time-R 

based on drainage area and BDF was developed. Overall 

arithmetic means of volumetric runoff coefficient (0.41 

dimensionless) and runoff coefficient (0.25 dimensionless) 

for the 24 watersheds were used to express the rational 

method in terms of excess rainfall (the excess rational 

method). Both the unit hydrograph method and excess 

rational method are shown to provide similar estimates 

of peak and time of peak streamflow. The results from 

the two methods can be combined by using arithmetic 

means. A nomograph is provided that shows the respective relations between the arithmetic-mean peak and time 

of peak streamflow to drainage areas ranging from 10 to 

640 acres. The nomograph also shows the respective relations for selected BDF ranging from undeveloped to fully 

developed conditions. The nomograph represents the peak 

streamflow for 1 inch of excess rainfall based on drainage 

area and BDF; the peak streamflow for design storms 

from the nomograph can be multiplied by the excess rainfall to estimate peak streamflow. Time of peak streamflow 

is readily obtained from the nomograph. Therefore, given 

excess rainfall values derived from watershed-loss models, which are beyond the scope of this report, the nomograph represents a method for estimating peak and time of 

peak streamflow for applicable watersheds in the Houston 

metropolitan area. Lastly, analysis of the relative influence 

of BDF on peak streamflow is provided, and the results 

indicate a 0.04log10 cubic feet per second change of peak 

streamflow per positive unit of change in BDF. This relative change can be used to adjust peak streamflow from 

the method or other hydrologic methods for a given BDF

to other BDF values; example computations are provided. 

Introduction 

Estimation of peak and time of peak streamflow from 

design storms provides for cost-effective, risk-mitigated 

design of drainage structures such as bridges, culverts, 

roadways, and other infrastructure. Relevant guidelines or 

manuals, which provide further context related to infras­
2 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

tructure design, can be found in Texas Department of 

Transportation (2002) or Harris County Flood Control 

District (2004). 

Accordingly, during 2007–10, the U.S. Geological 

Survey (USGS), in cooperation with the Harris County 

Flood Control District (HCFCD) and the Texas Department of Transportation (TxDOT), developed a method for 

estimating peak and time of peak streamflow from excess 

rainfall for 10- to 640-acre watersheds in the Houston, 

Texas, metropolitan area. 

Purpose and Scope 

The primary purpose of this report is to present a 

method for estimating peak and time of peak streamflow from excess rainfall for 10- to 640-acre watersheds 

in the Houston, Texas, metropolitan area that is based 

on watershed characteristics of drainage area and basindevelopment factor. A secondary purpose is to report on 

the conjunctive analysis of rainfall and runoff data in the 

context of the unit hydrograph and the rational method for 

24 watersheds in the Houston metropolitan area. There are 

three major components of this report: 

1.	 A comprehensive summary of the unit hydrograph 

method and statistical results from analysis for the 

24 watersheds is provided in the section titled “Analysis of Gamma Unit Hydrographs for the 24 Watersheds.” The section documents two equations that 

parameterize a gamma unit hydrograph, and each is 

used in the development of the method described in

 component 3. 

2.	 A comprehensive summary of the rational method 

and statistical results from analysis for the 24 watersheds is provided in the section titled “Analysis of 

Rational Method for the 24 Watersheds.” The section 

documents an equation and appropriate runoff coefficient values, which are used in the development of 

the method described in component 3. 

3.	 The method is based on conjunctive analysis of rainfall and runoff data in the context of the unit hydrograph and the rational method for the watersheds and 

is presented in section titled “A Method for Estimating Peak and Time of Peak Streamflow from Excess 

Rainfall for 10- to 640-Acre Watersheds in the Houston, Texas, Metropolitan Area.” The primary result 

of the conjunctive analysis is a nomograph. Example computations involving the nomograph also are 

provided. 

Study Watersheds 

For this investigation, 21 distinct watersheds (based 

on latitude and longitude) were identified as pertinent 

for the Houston metropolitan area. Pertinent watersheds 

were selected on the basis of drainage area as a measure 

of watershed size. These watersheds represent generally the smallest watersheds for which there exist paired 

rainfall and runoff data suitable for conjunctive analysis 

of the unit hydrograph method and the rational method. 

Although the focus of the investigation is the unit hydrograph method and the rational method in the context of 

small watersheds (less than about 640 acres), to support statistical development, watersheds with drainage 

areas less than about 3.5 square miles (2,240 acres) were 

selected. 

Two of the watersheds were identified as having 

considerable changes in extent of land development as 

expressed by a basin-development factor. The period of 

record for the two watersheds was thus segregated (three 

total divisions), resulting in the 24 watersheds that are 

used in this report. The 24 watersheds and ancillary characteristics are listed in table 1 and shown in figure 1. Collectively, these 24 watersheds are assumed to represent the 

generalized hydrologic and hydraulic conditions of many 

small, low-slope watersheds in the Houston metropolitan 

area. 

Database of Rainfall and Runoff 

A database of rainfall and runoff data from about 

1965 through about 2006 for the 24 watersheds was compiled and converted to digital format as needed. The data 

for the 24 watersheds were obtained from various sources 

including Liscum and others (1996) and Liscum (2001) 

and from selected non-USGS rainfall-runoff data (Fred 

Liscum, PBS&J Inc. [now (2011) with HCFCD], written 

commun., 2008; and Steve Johnson, LJA Engineering Inc.; 

written commun., 2008). These collective data used here 

are stored in files available from the Texas Water Science 

Center upon request. 

An example of headers (definitions) and selected data 

parts of two data files for a selected storm for one of the 

study watersheds is shown in figure 2. The definitions 

in the figure are identical to those described in Asquith, 

Thompson, and others (2004, p. 11 and 17). For the rainfall data, the time stamps, which are evenly spaced in the 

example (but not universally in the data), are shown under 

the DATE_TIME field. The cumulative depth of weighted (WTD) 

rainfall among all available rain gages is available under 

the ACCUM_WTD_PRECIP field in units of inches. For the particular example, a single rain gage (PRECIP1) was operated. 
Introduction 3 

Figure 1. Map showing downstream locations of the 24 selected watersheds in the Houston, Texas, metropolitan area used for 

gamma unit hydrograph and rational method analysis. 

The

Woodlands

Houston

§ ̈¦610

§ ̈¦45

§ ̈¦10

§ ̈¦45

§ ̈¦45

£¤90

£¤290

£¤90

£¤59

UV8

UV6

UV288

UV494

UV548

UV225

UV527

UV261

UV8

UV8

HARRIS COUNTY

MONTGOMERYCOUNTY

FORT BEND COUNTY

LIBERTY COUNTY

BRAZORIA

COUNTY

Lake

Houston

Brickhouse Gully

Berry Bayou

Greens

Bayou

Armand

Bayou

Hunting Bayou

9

8 and 19

6 and 7

4

3

2

1

24

23

22

20

18

16

15

14

13

10, 11, and 12

21

5

17

0 57.5

7.5

2.5

0 5 2.5

95°30' 95°15'

30°00'

29°45'

24

EXPLANATION

Downstream location of watershed

and sequence number (table 1)

TEXAS

Map

boundary
4 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Table 1. Summary of 24 selected watersheds in the Houston, Texas, metropolitan area used for gamma unit hydrograph and 

rational method analysis. 

[ no., number; fig., figure; A, drainage area (contributing), L, main-channel length; S, dimensionless main-channel slope; 

BDF, basin-development factor; --, not available; St., Street; Dr., Drive; trib., tributary; BMP, Best-management practice; 

Pkwy., Parkway; Rd., Road; NPDES, National Pollutant Discharge Elimination System ]

Site no. 

used in 

fig. 1 

Station no. and 

calendar year 

range of storms 

Station name Latitude Longitude 

A

(square 

miles) 

L

(miles) 

S

(dimensionless) 

BDF

(dimensionless) 

1 08068438: 1975–87 Swale no. 8 at Woodlands, Tex. 30°08'38" 95°28'09" 0.55 0.74 0.0077 8 

2 08073630: 1979–85 Bettina St. Ditch, Houston, Tex. 29°46'32" 95°32'23" 1.37 .73 .0041 11 

3 08073750: 1967–72 Stoney Brook St. Ditch, Houston, Tex. 29°44'05" 95°30'22" .50 -­ -­ 10 

4 08073800: 1964–72 Bering Ditch at Woodway Dr., Houston, 29°45'22" 95°29'44" 2.77 1.35 .0019 9 

Tex. 

5 08074145: 1980–82 Bingle Rd. storm sewer, Houston, Tex. 29°51'31" 95°29'09" .21 .70 .0011 9 

6 08074200: 1965–75 Brickhouse Gully at Clarblak St., Hous­ 29°49'53" 95°31'42" 2.56 .85 .0030 3 

ton, Tex. 

7 08074200: 1978–84 Brickhouse Gully at Clarblak St., Hous­ 29°49'53" 95°31'42" 2.56 .85 .0030 6 

ton, Tex. 

8 08074400: 1980–82 Lazybrook St. storm sewer, Houston, 29°48'15" 95°26'04" .13 .69 .0047 12 

Tex. 

9 08074910: 1979–83 Hummingbird St. Ditch, Houston, Tex. 29°39'44" 95°29'11" .32 1.45 .0008 9 

10 08075550: 1966–72 Berry Bayou at Gilpin St., Houston, 29°38'32" 95°13'22" 3.26 3.34 .0018 5 

Tex. 

11 08075550: 1974–78 Berry Bayou at Gilpin St., Houston, 29°38'32" 95°13'22" 2.87 3.34 .0018 7 

Tex. 

12 08075550: 1979–84 Berry Bayou at Gilpin St., Houston, 29°38'32" 95°13'22" 2.56 3.34 .0018 7 

Tex. 

13 08075600: 1965–72 Berry Bayou trib. at Globe St., Hous­ 29°39'00" 95°14'18" 1.58 2.27 .0017 5 

ton, Tex. 

14 08075750: 1965–72 Hunting Bayou trib. at Cavalcade St., 29°48'00" 95°20'02" 1.20 1.78 .0013 4 

Houston, Tex. 

15 00000BW8: 2004–06 BMP Beltway 8 mitigation bank near 29°55'54" 95°12'29" .14 -­ -­ 11 

Greens Bayou, Houston, Tex. 

16 0000B504: 2005–06 BMP Basin 504 on Armand Bayou, 29°36'28" 95°06'55" .19 1.03 .0010 9 

Houston, Tex. 

17 0000K542: 2005–06 BMP K542 Eldridge Pkwy. at Louetta 30°00'15" 95°37'23" .18 .81 .0010 12 

Rd., Houston, Tex. 

18 011NS004: 2004–06 NPDES 011NS004 West 11th St., Hous­ 29°47'21" 95°25'58" .36 -­ -­ 9 

ton, Tex. 

19 0LBNS001: 2000–02 NPDES 0LBNS001 Lazybrook storm 29°48'15" 95°26'04" .10 -­ -­ 12 

sewer at White Oak Bayou, Houston, 

Tex. 

20 0TGNS002: 2000–02 NPDES 0TGNS002 Tanglewilde at 29°44'36" 95°32'04" .06 -­ -­ 12 

Houston, Tex. 

21 0WBNS003: 2004–05 NPDES 0WBNS003 Willowbrook Mall 29°57'25" 95°32'17" .13 -­ -­ 12 

near Greens Rd., Houston, Tex. 

22 X8068420: 1985–1986 Ditch C at Wedgewood Lake, Wood­ 30°10'00" 95°29'10" .33 .93 .0041 0 

lands, Tex. 

23 X8068426: 1985–1986 Ditch A at Woodlands Parkway, Wood­ 30°09'00" 95°29'30" .41 1.6 .0030 7 

lands, Tex. 

24 0BFDN000: 1992 Briar Forest Dr. storm sewer at Hous­ 29°45'21" 95°37'59" .07 -­ -­ 4 

ton, Tex. 
Analysis of Gamma Unit Hydrographs for the 24 Watersheds 5 

For other storms or watersheds, two or more rain gages 

might have been operated, and these data also are available within the database. The rainfall used for data processing herein is always provided in the ACCUM_WTD_PRECIP

field. 

For the runoff data, the time stamps, which also are 

evenly spaced in the example but not universally in the 

database, are shown under the DATE_TIME field. The time 

stamps within the separate rainfall and runoff data files 

for the same storm might not have a one-to-one correspondence. The streamflow in cubic feet per second is listed 

under the RUNOFF field. The cumulative volume of runoff in 

inches is provided in the ACCUM_RUNOFF field. 

Selected Watershed Characteristics 

Selected characteristics for the 24 watersheds were 

obtained from various sources including Liscum and 

others (1996) and from colleagues (Duane Barrett, R.G. 

Miller Engineers Inc.; Fred Liscum, PBS&J Inc. [now 

(2011) with HCFCD], and Steve Johnson, LJA Engineering Inc.; written commun., 2008). The characteristics include drainage area (contributing), main-channel 

length, dimensionless main-channel slope, and basindevelopment factor: 

1.	 Values for drainage area (contributing) A for each
 

watershed were obtained and are listed in table 1.
 

2.	 Values for main-channel length L were obtained and 

are listed in table 1. The L is defined as the length 

in stream-course miles of the longest defined channel from the approximate watershed headwaters to 

the outlet. There is considerable ambiguity in estimation of L in the Houston metropolitan area where 

small watersheds can contain interconnected channels, ditches, streets, and storm sewers. As a result, 

some lengths could not be precisely quantified and 

are not listed in table 1. 

3.	 Values for dimensionless main-channel slope S (feet 

per feet) were obtained and are listed in table 1. The 

S is defined as the change in elevation ΔE in feet 

between the two end points of L divided by L in feet: 

S = ΔE/(5,280 × L). Because of the ambiguity in 

estimation of L, some slopes could not be precisely 

quantified and are not listed in table 1. 

4.	 Values for the dimensionless basin-development 

factor BDF were obtained and are listed in table 1. 

BDFs are integers from 0 to 12. There are 13 BDF

categories resulting in 12 discrete changes in BDF, 

from 0 → 1, 1 → 2, and so forth through 11 → 12. 

BDF as a concept and definition for this report is 

described in appendix 1. 

Analysis of Gamma Unit 

Hydrographs for the 24 Watersheds 

Background 

The unit hydrograph method (Dingman, 2002) estimates the runoff hydrograph given an excess rainfall 

hyetograph. Excess rainfall is a volume of rainfall per unit 

area (depth) after watershed losses such as evaporation, 

infiltration, and depression storage are subtracted (Chow 

and others, 1988, p. 135). A unit hydrograph is defined 

as the runoff hydrograph that results from a unit pulse of 

excess rainfall uniformly distributed over a watershed at 

a constant rate for a specific duration (Chow and others, 

1988, p. 213). 

Extensive investigations of the unit hydrograph 

method with Texas data and gamma distributions as the 

form of the unit hydrograph (Haan and others, 1994, 

p. 79) are available in Asquith and others (2005) and 

Asquith and Roussel (2007). In an associated study, Cleveland and others (2006) documented an independent analysis of unit hydrographs for Texas. A gamma distribution 

form of the unit hydrograph is referred to as a gamma unit 

hydrograph. For this report, an analysis of gamma unit 

hydrographs for the 24 watersheds was made. Results of 

the gamma unit hydrograph analysis are listed in table 2 

(columns 2–5). 

Analysis 

In total, 317 data files were considered for the 

24 watersheds. The number of storms (discrete peaks) 

analyzed per watershed and per methods of this report is 

listed in table 2. These numbers were not used as weights 

in statistical computations, such as weight factors in 

weighted-least squares regression, because of the absence 

of numerical similarity in the “No. of storms . . . ” (three 

separate columns) between the USGS monitored watersheds (the first 14 entries in the table) and the remaining 

watersheds. 

For each watershed, unit hydrographs were generated 

from the rainfall and runoff data by using a custom modeling technique developed for Asquith and others (2005); 

Asquith and Roussel (2007). The technique involved an 

analyst-directed approach for 5-minute gamma unit hydrograph estimation. The output consisted of graphics (not 
6 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Table 2. Summary of storm counts and watershed-mean parameters of gamma unit hydrograph analysis and watershed-mean 

parameters for rational method analysis of 24 selected watersheds in the Houston, Texas, metropolitan area. 

[ no., number; GUH, gamma unit hydrograph analysis; qp, peak streamflow; Tp, time to peak; K, gamma hydrograph 

shape; Cv, volumetric runoff coefficient; Cr, runoff coefficient; Tc, time of concentration ]

Station no. and 

calendar year 

range of storms 

No. of 

storms 

for GUH 

qp

(inches 

per 

hour) 

Tp

(hours) 

K

(dimensionless) 

No. of 

storms 

for Cv

No. of 

storms 

for Cr

Tc

(hours) 

Cv

(dimensionless) 

Cr

(dimensionless) 

08068438: 1975–87 11 0.4602 0.9091 1.2529 15 15 1.2625 0.5862 0.3521 

08073630: 1979–85 21 .4935 .8333 1.2157 25 25 1.2952 .5494 .3650 

08073750: 1967–72 12 .5242 .2847 .2439 14 14 .6095 .4866 .3789 

08073800: 1964–72 26 .3450 .5641 .3580 18 15 2.1211 .6398 .4225 

08074145: 1980–82 20 1.6242 .5083 4.4461 35 35 .4809 .5250 .4944 

08074200: 1965–75 18 .1537 2.3610 .9768 19 17 4.4618 .3531 .0968 

08074200: 1978–84 21 .1991 .9048 .3193 32 30 1.8834 .3446 .1761 

08074400: 1980–82 22 2.3764 .1288 .7315 29 29 .2550 .4544 .4757 

08074910: 1979–83 14 .5224 .4702 .5120 23 23 1.1071 .4832 .2647 

08075550: 1966–72 21 .1651 1.8992 .7614 25 24 3.2568 .6089 .2273 

08075550: 1974–78 3 .1282 1.1167 .2301 3 3 2.2776 .5406 .1369 

08075550: 1979–84 12 .1177 1.6597 .3600 15 15 3.2776 .6300 .2079 

08075600: 1965–72 20 .1935 1.2083 .4738 22 20 2.4993 .4783 .1687 

08075750: 1965–72 21 .1856 1.1968 .4375 21 21 1.8116 .4658 .1266 

00000BW8: 2004–06 8 1.0137 .4792 1.6392 10 10 .8649 .2199 .1244 

0000B504: 2005–06 6 1.2564 .4861 2.5042 8 8 .7877 .4485 .4153 

0000K542: 2005–06 6 1.4917 .2778 1.2319 7 7 .5699 .3992 .3273 

011NS004: 2004–06 7 .7308 .3691 .5945 14 14 .7477 .2735 .2206 

0LBNS001: 2000–02 7 1.9788 .3333 2.8948 9 9 .4977 .0334 .0342 

0TGNS002: 2000–02 7 1.2183 .2500 .7256 8 8 .4392 .1880 .1436 

0WBNS003: 2004–05 8 1.1708 .3125 .9906 12 9 .7156 .2204 .2152 

X8068420: 1985–1986 7 .5136 1.7143 5.0355 8 8 1.9466 .2829 .0764 

X8068426: 1985–1986 13 .8785 .8910 4.0126 15 14 1.2142 .5844 .4376 

0BFDN000: 1992 6 1.2202 .1250 .2515 7 7 .4787 .1060 .0689 
# HYETOGRAPH FILE

# INPUT Filename = B504_6.txt; Site=B504_6.

# DATE_TIME=date and time in MM/DD/YYYY@HH:MM:SS

# PRECIP1=preciptation in inches for station 1

# ACCUM_WTD_PRECIP=accumulated weighted precip. in

inches

DATE_TIME HOURS_PASSED PRECIP1 ACCUM_WTD_PRECIP

01/22/2006@14:15:00 0.0000 0.000 0.000

01/22/2006@14:30:00 0.2500 0.010 0.010

01/22/2006@14:45:00 0.5000 0.010 0.020

01/22/2006@15:00:00 0.7500 0.100 0.120

01/22/2006@15:15:00 1.0000 0.030 0.150

01/22/2006@15:30:00 1.2500 0.020 0.170

01/22/2006@15:45:00 1.5000 0.070 0.240

01/22/2006@16:00:00 1.7500 0.130 0.370

01/22/2006@16:15:00 2.0000 0.070 0.440

01/22/2006@16:30:00 2.2500 0.130 0.570

01/22/2006@16:45:00 2.5000 0.060 0.630

01/22/2006@17:00:00 2.7500 0.050 0.680

01/22/2006@17:15:00 3.0000 0.050 0.730

01/22/2006@17:30:00 3.2500 0.010 0.740

01/22/2006@17:45:00 3.5000 0.000 0.740

01/22/2006@18:00:00 3.7500 0.000 0.740

01/22/2006@18:15:00 4.0000 0.000 0.740

01/22/2006@18:30:00 4.2500 0.000 0.740

# HYDROGRAPH FILE

# INPUT Filename = B504_6.txt; Site=B504_6.

# DATE_TIME=date and time in MM/DD/YYYY@HH:MM:SS

# RUNOFF=runoff in cubic feet per second

# ACCUM_RUNOFF=accumulated runoff in inches

DATE_TIME HOURS_PASSED RUNOFF ACCUM_RUNOFF

01/22/2006@14:15:00 0.0000 0.00 0.000000

01/22/2006@14:30:00 0.2500 0.00 0.000000

01/22/2006@14:45:00 0.5000 0.00 0.000000

01/22/2006@15:00:00 0.7500 0.00 0.000000

01/22/2006@15:15:00 1.0000 1.36 0.002786

01/22/2006@15:30:00 1.2500 2.98 0.008883

01/22/2006@15:45:00 1.5000 2.42 0.013844

01/22/2006@16:00:00 1.7500 8.16 0.030560

01/22/2006@16:15:00 2.0000 20.34 0.072231

01/22/2006@16:30:00 2.2500 17.65 0.108399

01/22/2006@16:45:00 2.5000 18.94 0.147199

01/22/2006@17:00:00 2.7500 26.53 0.201552

01/22/2006@17:15:00 3.0000 23.06 0.248801

01/22/2006@17:30:00 3.2500 19.98 0.289730

01/22/2006@17:45:00 3.5000 14.69 0.319839

01/22/2006@18:00:00 3.7500 9.73 0.339783

01/22/2006@18:15:00 4.0000 6.39 0.352868

01/22/2006@18:30:00 4.2500 4.81 0.362714

01/22/2006@18:45:00 4.5000 4.05 0.371009

Analysis of Gamma Unit Hydrographs for the 24 Watersheds 7 

Figure 2. Example of headers (definitions) and selected data parts of rainfall (left) and runoff (right) data files for storm on January 22, 2006, for BMP Basin 504 on Armand Bayou, Houston, Texas. 

shown here) and numerical parameters (statistics of which 

are used here). The numerical parameters are described in 

this section. 

The equation (Asquith and others, 2005; Asquith and 

Roussel, 2007) defining a gamma hydrograph (unit or 

otherwise) is 

 K

q(t) t 1−(t/Tp) = e , (1)

qp Tp

where qp is peak streamflow in inches per hour from the 

watershed, Tp is the time to peak of the gamma distribution in hours, K is a shape parameter that is dependent 

on qp and Tp, and q(t) is streamflow in inches per hour at 

time t in hours. This equation produces the q(t) ordinates 

of a gamma hydrograph. 

The gamma unit hydrograph can attain shapes that 

mimic the general shape of many observed runoff hydrographs (unit or otherwise). Expression and analysis of unit 

hydrographs in terms of qp and Tp are important because 

the magnitude and timing of peak streamflow Qp in cubic 

feet per second are critical for many designs. Two notes 

concerning nomenclature are needed. First, synonymous 

use of lower-case q and upper-case Q for “streamflow” 

with attendant modifiers is made throughout this report; 

the two differ only in context-dependent units. Second, Tp

(time to peak) exists here as a distinct parameter for the 

gamma unit hydrograph. There also exists a time of peak 

that is denoted as T Qp , which represents the real time that 

Qp occurs. 

Because of the importance of Qp estimation in hydrologic engineering practice, the optimal qp and Tp values 

were computed by using techniques described in Asquith 

and others (2005) and Asquith and Roussel (2007) for 

each storm in the database by precisely matching the 

modeled Qp and modeled T Qp to observed values for 291 

individual storm runoff peaks. The count of 291 peaks 

for the unit hydrograph analysis does not match the 317 

data files considered because mathematical solutions for 

the Asquith and others (2005) and Asquith and Roussel 

(2007) technique were not attainable for 26 storm peaks. 

The modeling technique also permitted individual (discrete) analysis of multiple Qp values within a pair (rainfall 

and runoff) of data files if the storm had peaks in streamflow that were substantially separate in time as judged by 

the analyst. 

The relation between streamflow [Q, Q(t), or Qp (a 

peak), in cubic feet per second] and depth runoff [q, q(t), 

or qp (a peak) in inches per hour] is 

Q = 645.33qA, (2) 

where Q is streamflow in cubic feet per second, q is 

streamflow in watershed inches per hour, A is drainage 
8 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

area in square miles, and 645.33 is a unit-conversion factor. A review of unit conversions for equation 2 is provided in appendix 2. 

Although the three parameters qp, Tp, and K are 

shown in equation 1, in practice, K is a function of qp

and Tp, and total runoff volume V. Consequently, any two 

parameters will yield the third because V = 1 (unit volume) for a unit hydrograph. The total runoff volume V of 

a gamma hydrograph (Haan and others, 1994, p. 79) is 

 e(1)

 K V = qp Tp Γ(K) , (3) K

where Γ(K) is the complete gamma function for K. The 

time scale of the unit hydrograph is represented by Tp, but 

Tp does not represent the time base Tb or overall width 

in time of a runoff hydrograph. The complete gamma 

function is expressed as the infinite integral: 

 ∞

u−1 Γ(u) = x e−x dx (4) 

0 

For each watershed, mean values of qp and Tp were 

computed and are referred to as “watershed specific.” 

These watershed-specific values were used to compute, 

through numerical root-solving of equation 3, watershedspecific values of K. 

Summary statistics for each of the three parameters 

for the 24 watersheds were computed and are listed in 

figures 3, 4, and 5 under the SUMMARY STATISTICS heading. 

Lastly, regression equations were developed by using the 

R software (R Development Core Team, 2007) to estimate each parameter from the watershed characteristics of 

drainage area A and basin-development factor BDF. 

The regression equation and ancillary details of the 

analysis for qp are listed in figure 3. The equation is 

= 100.02682×BDF−0.5789log10(A)−0.6575 qp , (5) 

in which the residual standard error is 0.152 log10(inches 

per hour) with an adjusted R-squared of about 0.861. The 

relation between the watershed-specific peak streamflow 

and fitted values from the regression equation is shown in 

figure 6. Plotted next to the data points are corresponding 

values for BDF. 

The regression equation and ancillary details of the 

analysis for Tp are listed in figure 4. The equation is 

= 10−0.03421×BDF+0.3936log10(A)+0.1745 Tp , (6) 

in which the residual standard error is 0.204 log10(hours)

with an adjusted R-squared of about 0.668. The relation 

between watershed-specific, time to peak and fitted values 

of time to peak from the regression equation is made in 

figure 7. Next to the data points are corresponding values 

for BDF. 

The regression equation and ancillary details of the 

analysis for K are listed in figure 5. The equation is considered statistically weak (p-value ≈ 0.08), is not separately typeset, and thus is not suggested for general application. Because the K equation is weak, it is judged that 

the mean (K = 1.3) and median (K = 0.75) values of K for 

the 24 watersheds represent generalized, but acceptable, 

measures of gamma hydrograph shape for the study watersheds. Dimensionless gamma hydrographs for the mean 

value (K = 1.3) and median value (K = 0.75) shape parameter and the shape parameters for developed (K = 5.2) and 

undeveloped (K = 2.9) watersheds in Texas (Asquith and 

Roussel, 2007) are provided in figure 8. 

The dimensionless hydrographs considered by 

Asquith and Roussel (2007) were derived from watersheds 

mostly in central and north central Texas where watershed 

slopes are larger than those in the Houston metropolitan area. As a result, these Asquith and Roussel dimensionless hydrographs (dashed lines in fig. 8) represent 

more “conventional” hydrograph shape than the relatively 

longer tailed (longer recession limb) hydrographs derived 

from the 24 watersheds in the Houston metropolitan area. 

The difference in dimensionless hydrograph shape for the 

Houston metropolitan area likely means that the watersheds have greater storage and depth-driven streamflow 

as opposed to the greater gravitationally driven streamflow of larger sloped watersheds west and northwest of 

the Houston metropolitan area. 

For illustration of the interaction between qp and 

Tp and using K = 1, the relation between qp and Tp by 

equation 3 is straightforward: 1 = qp Tp Γ(1) (e(1)

/1)1 

becomes Tp = 0.3679/qp. 

Comparison of the p-values and adjusted R-squared 

values in conjunction with visual comparison of the plots 

in figures 6 and 7 indicates that the qp and Tp equations 

are reliable for the types of watersheds included in the 

analysis. In practice, K would be determined by iterative 

solution of equation 3 by using qp and Tp estimates from 

equations 5 and 6 to maintain unit volume of the gamma 

hydrograph. 

The authors recognize that the potential qp and Tp

parameter space is not as well populated as statistically 

desired and that the degrees of freedom are barely sufficient to justify two explanatory variables in the regression equations. For example, if figure 7 is used as a guide, 

without a loss of generality the largest watersheds (large 

circles on right of the graph) also tend to have the small­
SUMMARY STATISTICS FOR Tp, IN HOURS

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.1250 0.3281 0.5362 0.8035 1.1370 2.3610

REGRESSION EQUATION
 

Call:
 

lm(formula = log10(Tp) ~ log10(A) + BDF)
 

Residuals:

Min. 1st Qu. Median 3rd Qu. Max.

-0.48622 -0.08179 0.07531 0.12831 0.24908

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 0.17454 0.11866 1.471 0.156140

log10(A) 0.39361 0.08495 4.633 0.000143

BDF -0.03421 0.01520 -2.250 0.035263

--­

Residual standard error: 0.204 on 21 degrees of freedom

Multiple R-Squared: 0.696, Adjusted R-squared: 0.668

F-statistic: 24.09 on 2 and 21 DF, p-value: 3.661e-06

---

SUMMARY STATISTICS FOR qp, IN INCHES PER HOUR

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.1177 0.1977 0.5233 0.7901 1.2190 2.3760

REGRESSION EQUATION
 

Call:
 

lm(formula = log10(qp) ~ log10(A) + BDF)
 

Residuals:

Min. 1st Qu. Median 3rd Qu. Max.

-0.28590 -0.11306 0.01154 0.09148 0.23684

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -0.65746 0.08854 -7.425 2.66e-07

log10(A) -0.57888 0.06339 -9.132 9.27e-09

BDF 0.02682 0.01134 2.364 0.0278

Residual standard error: 0.152 on 21 degrees of freedom

Multiple R-Squared: 0.873, Adjusted R-squared: 0.861

F-statistic: 72.22 on 2 and 21 DF, p-value: 3.869e-10

SUMMARY STATISTICS FOR K, DIMENSIONLESS

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.2301 0.4181 0.7464 1.3420 1.3490 5.0350

REGRESSION EQUATION
 

Call:
 

lm(formula = K ~ log10(A) + BDF)
 

Residuals:

Min. 1st Qu. Median 3rd Qu. Max.

-2.7905 -0.7324 -0.2728 0.2910 2.8134

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 2.26514 0.75702 2.992 0.00694

log10(A) -1.23924 0.54196 -2.287 0.03271

BDF -0.16360 0.09697 -1.687 0.10638

--­

Residual standard error: 1.3 on 21 degrees of freedom
 

Multiple R-Squared: 0.212, Adjusted R-squared: 0.137
 

F-statistic: 2.829 on 2 and 21 DF, p-value: 0.08171
 

ABBREVIATIONS SHOWN ON THESE FIGURES

qp Depth peak streamflow,

in inches per hour

Tp Time to peak streamflow, in hours

K Shape factor, dimensionless

A Drainage area, in square miles

BDF Basin-development factor

Min. Minimum

1st Qu. First quartile

3rd Qu. Third quartile

Max. Maximum

lm() Linear modeling function

Std.Error Standard error

t-value T-statistic

Pr(>|t|) Probability of absolute value

of t-value

DF Degrees of freedom

Analysis of Gamma Unit Hydrographs for the 24 Watersheds 9 

Figure 3. Statistical summary of watershed-specific, peak 

streamflow of gamma unit hydrograph and regression equation for estimation of peak streamflow for applicable watersheds in the Houston, Texas, metropolitan area. 

Figure 5. Statistical summary of watershed-specific, gamma 

unit hydrograph shape parameter and regression equation for 

estimation of shape parameter for applicable watersheds in 

the Houston, Texas, metropolitan area. 

Figure 4. Statistical summary of watershed-specific, time to 

peak of gamma unit hydrograph and regression equation for 

estimation of time to peak for applicable watersheds in the 

Houston, Texas, metropolitan area. 
10 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

5

WATERSHED-SPECIFIC PEAK STREAMFLOW, IN INCHES PER HOUR

FITTED PEAK STREAMFLOW, IN INCHES PER HOUR

0.1 0.2 0.3 0.4 0.5 0.7 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

2

3

8

11

12

9

6

3

6

7 7

8

8

9

11

9

12

9

12

12

12

0

7

4

EXPLANATION

EQUAL VALUE LINE

8

FITTED VALUE FROM REGRESSION EQUATION—Symbol size is in proporation

to drainage area and number indicates basin-development factor.

region of overestimation

region of underestimation

5

Figure 6. Relation between watershed-specific, peak streamflow and fitted values of peak streamflow by regression shown in 

equation 5 for a gamma unit hydrograph developed for 24 watersheds in the Houston, Texas, metropolitan area. 

5

7

WATERSHED-SPECIFIC TIME TO PEAK, IN HOURS

FITTED TIME TO PEAK, IN HOURS

0.05 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 2 3 4 5

0.15

0.2

0.3

0.4

0.5

0.6

0.8

1

2

3

8

11

12

9

5

6 3

7

8

8

9

11

9

12

9

12

12

12

0

7

4

8

FITTED VALUE FROM REGRESSION EQUATION—Symbol size is in proporation

to drainage area and number indicates basin-development factor

EXPLANATION

region of overestimation

region of underestimation EQUAL VALUE LINE

6

Figure 7. Relation between watershed-specific, time to peak and fitted values of time to peak by regression shown in equation 6 for a gamma unit hydrograph developed for 24 watersheds in the Houston, Texas, metropolitan area. 
est BDF values and visa versa. Whether the equations 

for qp and Tp fundamentally lack estimation power for 

largely undeveloped BDF ≤ 3 or moderately developed 

3 ≤ BDF ≤ 6 watersheds with A ≤ 1 square mile cannot be determined because of the general absence of such 

watersheds among the 24 used for this report. 

Computing Gamma Unit Hydrographs 

Computations of the 5-minute gamma unit hydrograph are demonstrated in this section. Suppose that 

1 inch of excess rainfall is uniformly distributed in a 5­

minute interval and across a 0.5-square mile watershed 

with a BDF = 9. What are the estimates of Qp and T Qp

for this watershed? 

An estimate of qp provides a starting point. The qp

for the watershed from equation 5 by substitution is 

= 100.02682×9−0.5789log10(0.5)−0.6575 qp , (7) 

or qp = 0.573 inches per hour of runoff from the watershed. Using a K = 1 (only for these immediate computations), the Tp is Tp = 0.3679/0.573 or Tp = 0.642 

hours (about 40 minutes). Solving for the Qp by using 

equation 2, the gamma unit hydrograph provides UHQp =

645.33 ×0.573 ×0.5 or UHQp = 185 cubic feet per second. 

The gamma unit hydrograph for this watershed is shown 

in figure 9. 

Extending the example, if K = 1 is not assumed, the 

computation of Tp for the watershed is needed, and K will 

have to be computed by numerical methods. The Tp from 

equation 6 by substitution is 

= 10−0.03421×9+0.3936log10(0.5)+0.1745 Tp , (8) 

or Tp = 0.560 hours. The equivalent KT

q

p

p=0.573 

=0.560 value is 

K = 0.79 by iterative-solution to equation 3 for V = 1. 

The value K = 0.79 is less than 1, so the resulting gamma 

unit hydrograph (not shown here) would have a heavier 

tail (longer recession) than the K = 1 hydrograph shown in 

figure 9. 

Lastly, the T Qp of the gamma unit hydrograph for the 

example is simply UHT Qp = Tp because the rainfall has 

a duration of 5 minutes and is equal to that of the gamma 

unit hydrograph; no convolution is required. In general, 

however, UHT Qp requires estimation from convolution of 

the gamma unit hydrograph with the excess rainfall time 

series. Similar time computations to these are described 

in the section titled “Comparison of Unit Hydrograph and 

Rational Method Analysis.” 

Analysis of Rational Method for the 24 Watersheds 11 

Analysis of Rational Method for the 

24 Watersheds 

Background and Mathematical Analysis 

The rational method (Pilgrim and Cordery, 1993; 

Dingman, 2002) is a less complex technique compared to 

the unit hydrograph method to estimate Qp. The method 

originating from Kuichling (1889) is widely used for 

drainage design in small urban watersheds. The estimation of Qp by the rational method (RMQp) is obtained by 

RMQp = 1.008C I A, (9) 

where 1.008 is a unit-conversion factor for indicated units 

of other variables, C is a dimensionless runoff coefficient, 

I, is rainfall intensity in inches per hour, and A is drainage 

area in acres. A review of the unit-conversion factor is 

provided in appendix 2. 

By inspection of equation 9, estimates of RMQp are 

sensitive to values of C. Tables of C for various land use, 

watershed slope, types of surfaces, and rainfall recurrence 

intervals or intensities by various authors for different 

historical contexts are provided in text books (Chow and 

others, 1988, table 15.1.1 on p. 498), intra-agency procedures (Texas Department of Transportation, 2002), and 

jurisdictionally specific guidelines (Freeze and Nichols, 

2010). 

Similar to the gamma unit hydrograph analysis previously described, the rational method can be adapted or 

rephrased for the generalized hydrologic and hydraulic 

conditions represented by the 24 watersheds. To this end, 

the method can be rewritten as 

P(T,F)

RMQp = 1.008C A, (10) T

where the depth of rainfall P(T,F) for a given duration T

and nonexceedance probability F (commonly an expression of recurrence interval) divided by T is equal to I of 

equation 9. Depth-duration relations of rainfall with recurrence interval could be derived from Asquith and Roussel 

(2004) or other sources. Given that P(T,F) and A are 

known quantities in a design context, the rational method 

is not only a function of C but also of T. 

Values for T often are taken to be the “time of concentration,” denoted as Tc, for the watershed. Alternatively, values for T are taken to be a time (perhaps not 

numerically equal to Tc) that is representative of “full 

watershed contribution.” The authors prefer the concept 

of “critical storm duration” over either of the other two. 

Methods to estimate Tc based on empirical equations also 
12 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

RATIO OF TIME (REAL TIME) TO TIME TO PEAK

K = 2.9 FROM DEVELOPED, NON-HOUSTON WATERSHEDS (Asquith and Roussel, 2007)

K = 5.2 FROM UNDEVELOPED, NON-HOUSTON WATERSHEDS (Asquith and Roussel, 2007)

RATIO OF STREAMFLOW TO PEAK STREAMFLOW

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MEAN SHAPE FACTOR K = 1.3 FROM THE 24 WATERSHEDS

MEDIAN K = 0.75 FROM THE 24 WATERSHEDS

EXPLANATION

Figure 8. Dimensionless gamma hydrographs by shape parameter for 24 watersheds in the Houston, Texas, metropolitan area, 

and for applicable developed and undeveloped non-Houston watersheds considered by Asquith and Roussel (2007). 

TIME, IN HOURS

STREAMFLOW, IN CUBIC FEET PER SECOND

0 0.5 1 1.5 2 2.5 3 3.5 4

0

20

40

60

80

100

120

140

160

180

200

Figure 9. Example of a gamma unit hydrograph with a shape factor of 1 from a 0.5-square mile watershed with a basindevelopment factor of 9. 
are available with various lineages or historical contexts 

(Dingman, 2002, table 9–9, and references therein). Roussel and others (2005) provided a review of Tc estimation 

methods pertinent to the needs of TxDOT. Because an 

adaptation of the rational method specific to small lowslope watersheds in the Houston metropolitan area is 

needed, existing equations or techniques for Tc estimation 

were not used in this report. 

If equation 10 reasonably expresses the coupled relation between rainfall intensity and peak streamflow, then 

to maintain proportionality, C cannot be decoupled conceptually from T because the two parameters are interrelated. In a statistical sense, as T becomes too large (is 

overestimated), then C must increase, and the opposite is 

true if T is underestimated. Values for T, therefore, are of 

equal importance to the rational method as values for C. 

The authors note that the method results in one equation 

and two unknowns (C and T). 

A starting point for the analysis of the rational 

method is needed. Schaake and others (1967) observed 

that 

An assumption of the time required for runoff [a 

time of concentration] to flow from the farthest point 

in the drainage area is presently used in design practice. However, there is no known way to determine 

this time of flow, either from measurements in the field 

during storms or from records of rainfall and runoff. 

Except for steady state conditions, which rarely, if 

ever are reached during a thunderstorm, there is no 

good reason to believe that the time of flow from the 

farthest point in a drainage area should necessarily be 

the best rainfall averaging time to use in the [rational 

method.] A study, therefore, was made to find for each 

area the averaging time giving the best correlation 

between average rainfall intensity and peak runoff 

rates. 

Schaake and others (1967) eventually settled on the 

lag time from rainfall centroid to runoff centroid as the 

“best” time. (As discussed in following paragraphs, it 

is effectively this time definition that the authors of this 

report computed and recorded as Tr.) 

Because of the need to lock either C or T down, the 

authors of this report considered 

 Ta = A[square miles] = A[acres]/640, (11) c

as a first-order approximation to Tc or T by association, 

where Ta is in hours and A is in units as shown. Equa­ c

tion 11 can be physically interpreted as follows: if a watershed is square, and thus has si

√

des of equal length, then the 

hypotenuse of a unit area is 2 ≈ 1.4, so the approximation assumes that aggregate travel speed (combination of 

Analysis of Rational Method for the 24 Watersheds 13 

overland flow, storage residence times, and channel flow) 

is about 1.4 miles per hour or about 2 feet per second. 

Although perhaps a fast aggregate velocity for overlandflow conditions for undeveloped, low-slope watersheds in 

the study area, the storm sewers in the Houston metropolitan area generally are engineered for velocities of 3 feet 

per second (Duane Barrett, R.G. Miller Engineers Inc., 

written commun., 2008). Hence, 2 feet per second might 

be considered an appropriate (albeit rough) generalization 

for the method described here. 

Building on equation 11, the rational method for a 

observed storm event that is denoted by i (“storm specific”) becomes 

iP}max a ×A[acres] Tc RMiQp = 1.008 i

Cr × , (12) A[acres]/640 

where RMiQp is the peak streamflow for the storm in 

cubic feet per second, and iCr is a storm-specific runoff 

coefficient (dime

 

nsionless) with subscript r (“root”) as a 

reminder that A[acres]/640 is used to approximate Tc. 

The term iP}max represents the maximum depth of rain- Ta

c

fall in inches for a time window (duration) having a width 

of Ta by equation 11 that is moved or swept through the c

observed rainfall time series. The iP}max effectively deter- Ta

c

mines the maximum rainfall intensity. The units of A are 

explicitly shown as a reminder that a mixed unit system 

for A is involved in mathematical operations in this report. 

Equation 12 can be solved (“inverted”) for iCr

 A[acres]/640 i

Cr = RMiQp × . (13) iP}max ×A[acres] a Tc

Equation 13 is useful because a single runoff coefficient 

can be extracted. Statistical summaries or methods to estimate Cr from processing of multiple storms for a watershed then would provide estimates of C that are congruent 

with the assumption of watershed time given by equation 11. 

A problem still remains. For computation of iCr, 

each observed time series of rainfall and runoff requires 

parsing into discrete epochs for computation of iP}max 

Ta

c

and computation of RMiQp, respectively. This parsing 

requires analyst input for meaningful interpretation. 

To this end, highly specialized, single-purpose software was developed for this report to compute iCr and 

other features of the rational method. A screenshot is 

shown in figure 10, which shows selection of a bounding box (the dashed box was manually superimposed on 

the figure for clarity) on the rainfall time series (top box, 

the “rainfall epoch”) likely producing the observed runoff 
14 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas

hydrograph. The runoff time series also requires parsing into the associated runoff (streamflow) hydrograph

(bottom box, the “runoff epoch”). Extraction of RM

 Qp

from the runoff epoch is straightforward. Computation

of P}

max

T

is algorithmically more complex but readily

c

0

performed by the computer.

The parsing of the rainfall and runoff in figure 10 is

completed by the analyst pressing the “Calculate C” button ( ). The results are shown in figure 11. In

figure 10, there are 24 rainfall pulses of 5-minute width

within the parsed rainfall epoch. Following computational

migration to half the rainfall width, 23 unique durations of

rainfall exist for the example (see the 23 plotted points in

fig. 11). A maximum rainfall intensity and corresponding

C value for each unique duration (5-minute increments for

the example and this study) can be computed by a movable time window. These are referred to as Ci values and

are shown as open circles in figure 11. The Cw value represents the C for the full width of the rainfall epoch, the C

associated with the average intensity of the “entire storm”

as parsed by the analyst. The primary conclusion deriving

from figure 11 is that no unique runoff coefficient exists

without an estimate of watershed time (critical storm duration).

Because no single C exists without watershed time

and because of the need to simplify forthcoming analysis

by having a single representative C by storm, equation 11

was used to provide a first approximation. The filled triangle in figure 11 represents Cr based on P}

max

T

that has

c

0

been “snapped” to the nearest

 

integer multiple

 

of 5 minutes (the time increment of the specific analysis as shown

in fig. 10).

For a given storm, the volumetric runoff coefficient

 Cv can be computed by

 Cv =

 R

, (14)

P

where R is the

 

total runoff in inches for the runoff epoch

and P is the total rainfall in inches for the rainfall epoch.

The Cv is represented in figure 11 by the horizontal

dashed line. In general

A

 Cr = Cv.

useful expression of the rational method for this

investigation is as the “excess rational method,” which is a

solution of the rational method for a unit depth of excess

rainfall. The equation for estimation of Qp by the excess

rational method (ERMQp) is

(E/C ) A[acres]

ERMQp = 1

v

.008Cr

×

, (15)

Tc

where ERMQp is the Qp from the excess rational method

in cubic feet per second, the term E/Cv is the depth of

6

rainfall in inches, E is excess rainfall in inches, and Tc in

hours is explicitly chosen as the critical storm duration.

For a unit depth of excess rainfall (rainfall remaining after

watershed losses) E = 1 inch. The excess rational method

can be expressed as

C ]

ERMQ = 1

r A[acres

p .008 . (16)

Cv Tc

To complete the mathematical analysis of the rational

method and partly inspired by Schaake and others (1967),

a better estimate of time than provided by equation 11 can

be computed after parsing of the rainfall and runoff into

epochs. The Tr can be defined as follows:

Tr — The center of the window of P}

max

T

can be concepc

0

tualized as the centroid of rainfall. Time separation

between this centroid and the time of peak streamflow T

Qp

for the storm is defined as Tr. This time is

known

 

as a lag time (Dingman, 2002, table 9–1, term

TLPC). For this report, Tr also is referred to as time-R.

By further interpretation of the rational method, both

times of rise and recession of a hydrograph are equal to

Tc (Pilgrim and Cordery, 1993, p. 9.15–9.16). The time

of peak Q

ERMT p

(also a “time of rise”) of the excess rational method is equivalent to that for rational method and

therefore is

ERMT

Qp = RMTQp = Tc. (17)

It follows from the definition of Tr and because the

width of rainfall was defined by Tc

0

that the best available

estimate of critical storm duration Tc is given by

Tc

0

p

A[acres]/640

Tc = Tr + = Tr + =

2 2

ERMT

Qp

.

(18)

To reiterate, ERMT

Qp

is the duration from the beginning of rainfall to the time that Qp occurs. (Technically,

this time represents the duration from the beginning of

excess rainfall, but no initial abstraction is formulated in

the rational method.) Therefore, ERMT

Qp can be directly

compared to UHT

Qp

. Equation 18 provides a more reasonable estimate of critical storm duration compared to

equation 11.

Computational and Statistical Analysis

Watershed-specific values for Cv and Cr were computed for each watershed as the arithmetic mean of Cv

and Cr (table 2). Statistical summaries of these Cv and Cr
Analysis of Rational Method for the 24 Watersheds 15
 

Figure 10. Screenshot of highly specialized, single-purpose software for inversion of rational method for storm on January 22, 

2006, for BMP Basin 504 on Armand Bayou, Houston, Texas. 
16 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Figure 11. Screenshot of output from highly specialized, single-purpose software showing computational results for storm on 

January 22, 2006, for BMP Basin 504 on Armand Bayou, Houston, Texas, as shown in figure 10. 

values and the ratio Cr/Cv are provided in figure 12. The 

number of storms varies slightly for the Cv and Cr computations. Because suitable storms had to have less runoff 

than rainfall, values of iCv were available for each storm. 

For a few storms, the width of analyst-selected rainfall 

(the rainfall epoch) was smaller than the Tc = Ta; therefore, c

an estimate for Cr (and Tr) was not available because the 

runoff-producing storm pulse was shorter than Ta in time. c

In total, 394 storms were analyzed and the iCv computed; 

of these, 380 storms provided estimates of iCr. 

An analysis of potential relations between both Cv

and Cr (the watershed-specific values listed in table 2) and 

A and BDF was performed. Consistent with the observation by Chow and others (1988, p. 497) that “C [values 

are] the least precise variable of the rational method,” it 

is concluded from the analysis that reliable estimation of 

these C values by A and BDF is difficult to achieve. There 

are only weak associations of C values with either A or 

BDF. The weak associations are likely related to many 

storm-specific factors including antecedent moisture conditions and deviations from uniform spatial distribution of 

rainfall. 

A useful approximation of Cv and Cr for applicable 

watersheds in the Houston metropolitan area are the mean 

values of the iCv and iCr. These means are reported in the 

statistical summaries shown in figure 12 for the 24 watersheds: Cv = 0.41 and Cr = 0.25. The first and third quartiles of the Cr/Cv ratios shown in figure 12 are about 0.37 

and 0.81, respectively. These ratios can be used to estimate some of the uncertainty associated with the method 

described in this report. The authors explicitly use the 

ratio of respective mean values for Cr and Cv and not the 

mean value for the Cr/Cv ratios in later computations. 

In total, 380 storms were used to estimate iTr. 

Watershed-specific values for Tr were computed as the 

√ mean by watershed of iTr. The term A/2 subsequently 

was added to yield the watershed-specific Tc estimates 

listed in table 2. 

A regression equation was developed to estimate Tr

from watershed characteristics of drainage area A and 

basin-development factor BDF. Because A was used in 

the regression, regression analysis of the Tc values listed 

in table 2 would be inappropriate because of the explicit 

√ inclusion of A/2 in these values. 

The regression equation and ancillary details of the 

analysis for Tr are listed in figure 13. The equation is 

= 10−0.05228×BDF+0.4028log10(A)+0.3926 Tr , (19) 

in which the residual standard error is 0.209 log10(hours)
A Method for Estimating Peak and Time of Peak Streamflow from Excess Rainfall for 10- to 640-Acre Watersheds in . . . 17

with an adjusted R-squared of about 0.718. A comparison

of the watershed-specific, Tr (time-R) and fitted values

from the regression equation is made in figure 14. Next to

the data points are the corresponding values for BDF.

A Method for Estimating Peak and

Time of Peak Streamflow from

Excess Rainfall for 10- to 640-Acre

Watersheds in the Houston, Texas,

Metropolitan Area

A method for estimating peak and time of peak

streamflow from excess rainfall for 10- to 640-acre watersheds in the Houston, Texas, metropolitan area is presented in this section.

Comparison of Results from Unit Hydrograph

and Rational Method Analysis

A comparison of the results from the rainfall and

runoff analysis of the 24 watersheds is informative. Many

of the computations presented in the comparison will be

used for formal definition of the method.

The comparison between (1) the unit hydrograph

analysis using a 5-minute gamma unit hydrograph and (2)

the rational method analysis is made through graphical

depiction of equations 5 (qp), 6 (Tp), 11 (Tc

0

), and 19 (Tr)

and is shown in figure 15. The figure shows the “timing”

and the “peaking” response of the watershed. The figure

helps users visualize the regression equations and facilitates estimation of qp and Tp for applicable watersheds in

the Houston metropolitan area. Independent estimates of

Qp and T

Qp

from the unit hydrograph and rational methods can be obtained with knowledge of

1. The watershed characteristics A and BDF,

2. Representative runoff coefficients (Cr and Cv) for the

excess rational method, and

3. A watershed-loss model that is a conceptual or

numerical model of watershed losses (Chow and

others, 1988, p. 135), which is beyond the scope of

this report.

Hypothetical Watershed—Suppose a watershed

has a drainage area of A = 300 acres (A = 0.469 square

miles). By using undeveloped conditions (as represented

by BDF = 0) and fully developed conditions (as represented by BDF = 12), the T

Qp

(not Tp or Tc) and Qp are

computed from both the unit hydrograph and excess rational methods.

The mean Cv of the 24 watersheds is about Cv = 0.41

(fig. 12), and the mean Cris about Cr = 0.25 (fig. 12).

Combining these runoff coefficients into the excess rational method for 1 inch of excess rainfall for applicable

watersheds in the Houston metropolitan area, the authors

obtain from equation 16 by substitution

0.25 A

ERMQp = 1.008× , (20)

0.41 Tc

or approximately

0.61A

ERMQp = , (21)

Tc

where ERMQp is the Qp in cubic feet per second per 1 inch

of excess rainfall, A is drainage area in acres, and Tc is

estimated by equation 18.

Undeveloped Conditions (BDF = 0)—The Tp of

the gamma unit hydrograph for A = 300 acres is about

Tp = 1.09 hours or about 65 minutes (rounded to nearest

5 minutes) (fig. 15). The qp for the gamma unit hydrograph is about qp = 0.34 inches per hour. The K is computed by inversion of equation 3 for these Tp and qp values and is about K = 1.00. The Tr for the watershed is

about Tr = 1.82 hours or about 110 minutes (rounded

to nearest 5 minutes) for which ( A[acres]/640 )/2 =

0.342 hours or about 20 minutes is added to Tr as per

equation 18 to acquire a Tc for th

p

e watershed (fig. 15).

Therefore, a Tc = 130 minutes (about 2.17 hours) is

used.

The excess rational method for the undeveloped

watershed thus is solved from equation 20:

ERMQp |

BDF=0

0.25 A

= 1.008 (22)

0.41 Tc

0.25 300

= 1.008

0.41 130/60

= 84.5 cubic feet per second .

The division by 60 minutes is made to acquire units of

hours. Because the T

Qp

for the excess rational method is

Tc, the authors estimate that ERMT

Qp

|

BDF=0 or “ERMTQp

given a BDF = 0” is about 130 minutes .

The unit hydrograph method requires convolution of

the 5-minute gamma unit hydrograph GUH(qp=0.34,

Tp = 1.083, K = 1.00) defined by equation 1 with

a unit of excess rainfall uniformly distributed across
ABBREVIATIONS SHOWN ON THIS FIGURE

Min. Minimum

1st Qu. First quartile

3rd Qu. Third quartile

Max. Maximum

# Volumetric runoff coefficient

summary(meansCv) # watershed-specific Cv values

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.03339 0.28060 0.46010 0.41260 0.54280 0.63980

# Runoff coefficient for Rational Method

summary(meansCr) # watershed-specific Cr values

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.03419 0.13430 0.21790 0.24820 0.36850 0.49440

# Ratio of runoff coefficients for Rational Method

summary(meansCr/meansCv)

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.2531 0.3682 0.6549 0.6316 0.8099 1.0470

ABBREVIATIONS SHOWN ON THIS FIGURE

qp Depth peak streamflow,

in inches per hour

Tr Time-R, in hours

K Shape factor, dimensionless

A Drainage area, in square miles

BDF Basin-development factor

Min. Minimum

1st Qu. First quartile

3rd Qu. Third quartile

Max. Maximum

lm() Linear modeling function

Std.Error Standard error

t-value T-statistic

Pr(>|t|) Probability of absolute value

of t-value

DF Degrees of freedom

SUMMARY STATISTICS FOR Tr, DIMENSIONLESS

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.07471 0.35480 0.76710 1.01900 1.32400 3.66200

REGRESSION EQUATION
 

Call:
 

lm(formula = log10(Tr) ~ log10(A) + BDF)
 

Residuals:

Min. 1st Qu. Median 3rd Qu. Max.

-0.53502 -0.09974 0.04105 0.13479 0.36084

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 0.39259 0.11653 3.369 0.002902

log10(A) 0.40275 0.08599 4.684 0.000127

BDF -0.05228 0.01453 -3.597 0.001694

--­

Residual standard error: 0.209 on 21 degrees of freedom

Multiple R-Squared: 0.7428, Adjusted R-squared: 0.718

F-statistic: 30.33 on 2 and 21 DF, p-value: 6.42e-07

18 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Figure 12. Statistical summary of watershed-specific, 

runoff and volumetric runoff coefficients, and runoff coefficient ratios for 24 watersheds in the Houston, Texas, 

metropolitan area. 

Figure 13. Statistical summary of watershed-specific, time-R 

and regression equation for time-R estimation for applicable 

watersheds in the Houston, Texas, metropolitan area. 
A Method for Estimating Peak and Time of Peak Streamflow from Excess Rainfall for 10- to 640-Acre Watersheds in . . . 19

7

5

W

6

ATERSHED-SPECIFIC TIME-R, IN HOURS

FITTED TIME-R, IN HOURS

0.05 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 2 3 4 5

0.15

0.2

0.3

0.4

0.5

0.6

0.8

1

2

3

11

9

12

9

0

7

8

11

10

9

9

3

12

9

5 4

4

12

12

12

11

FITTED VALUE FROM REGRESSION EQUATION—Symbol size is in proporation

to drainage area and number indicates basin-development factor

EXPLANATION

two watersheds

region of overestimation

region of underestimation EQUAL VALUE LINE

7

Figure 14. Relation between watershed-specific, time-R and fitted values of time-R by regression shown in equation 19 developed for 24 watersheds in the Houston, Texas, metropolitan area.

Tc |

BDF=0 = 130 minutes. The convolutions for these

examples were performed in a spreadsheet. Also, because

a 5-minute unit hydrograph is used, for convenience and

the fact that the watershed is small, 5-minute computational steps were used for the convolution process. The

results of the convolution provide a UHQp |

BDF=0 of

about 88.3 cubic feet per second , and a UHT

Qp

|

BDF=0

of about 145 minutes . This timing is on the order of

the Tc |

BDF=0

estimated for the watershed, but logically

is slightly longer because of the nature of the resultant

hydrograph from the convolution—an offset, multiply, and

summation process.

Developed Conditions (BDF = 12)—The Tp of

the gamma unit hydrograph for A = 300 acres is about

Tp = 0.42 hours or about 25 minutes (rounded to nearest 5 minutes) (fig. 15). Also from the figure, the qp for

the gamma unit hydrograph is about qp = 0.73 inches

per hour. The K is computed by inversion of equation 3

for these Tp and qp values and is about K = 0.725. The

Tr for the watershed is about Tr = 0.43 hours or about

25

p

minutes (rounded to nearest 5 minutes) for which

( A[acres]/640 )/2 = 0.342 hours or about 20 minutes is added to Tr as per equation 18 to acquire a Tc for

the watershed (fig. 15). Therefore, a Tc = 45 minutes

(0.75 hours) is used.

The excess rational method (equation 16) for the

developed watershed thus is solved from equation 20:

ERMQp |

BDF=12 C

= 1.008 r A(23)

Cv Tc

0.25 300

= 1.008

0.41 45/60

= 244 cubic feet per second .

Again, because the T

Qp

for the rational method is Tc,

the authors estimate that ERMT

Qp

|

BDF=12 is about

45 minutes .

The unit hydrograph method requires convolution of

the gamma unit hydrograph GUH(qp = 0.73, Tp = 0.417,

K = 0.725) defined in equation 1 with a unit of excess

rainfall uniformly distributed across Tc |

BDF=12 =

45 minutes. The results of the convolution provide a

BDF

UHQp |

=12 of about 202 cubic feet per second , and

a UHT

Qp

|

BDF=12 of about 50 minutes . This timing is on

the order of the Tc |

BDF=12 estimated from figure 15 for

the watershed but logically is slightly longer because of
20 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

DRAINAGE AREA, IN ACRES

TIME, IN HOURS

UNIT HYDROGRAPH PEAK STREAMFLOW, IN INCHES PER HOUR

10 20 30 40 50 60 70 80 100 200 300 400 500 600 800 1,000 1,280

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00 0.15

0.2

0.3

0.5

0.7

1

2

3

5

7

10

12

0Basin Development Factor

The 13 lines (curves) of each 

color represent each of the 

13 basin-development factor 

(BDF) values. The curves are 

stacked top to bottom, 0 to 12 

respectively.

TIME TO PEAK OF 5-MINUTE GAMMA UNIT HYDROGRAPH (T , equation 6)

DEPTH OF PEAK STREAMFLOW OF 5-MINUTE GAMMA UNIT HYDROGRAPH (q , equation 6)

TIME-R FROM ANALYSIS OF RATIONAL METHOD (T , equation 19) 

SQUARE ROOT OF DRAINAGE AREA IN SQUARE MILES (not acres) ( , equation 11)

EXPLANATION

p

p

r

Tc

Figure 15. Relation between time to peak or depth of peak streamflow of the gamma unit hydrograph to drainage area and 

basin development factor. 
A Method for Estimating Peak and Time of Peak Streamflow from Excess Rainfall for 10- to 640-Acre Watersheds in . . . 21

the nature of the resultant hydrograph from the convolution process.

The results for developed and undeveloped watersheds are listed in table 3. The results demonstrate remarkable consistency between the unit hydrograph and rational

methods for the hypothetical watershed and by association

the methods are expected to agree for similar watersheds

less than about 640 acres in the Houston metropolitan

area. As a final step, the arithmetic mean of the two T

Qp

and Qp estimates can be used to establish the best estimate from the analysis; these values are listed in table 3

(columns 6 and 7). This suggestion is the basis for the

nomograph described in the next section.

Nomograph and Example Computations for

the Method

The method is a technique to compute Qp and T

Qp

given excess rainfall for design storms for applicable

watersheds in the Houston metropolitan area. The method

requires graphical lookup from a nomograph of peak

streamflow (cubic feet per second per 1 inch of excess

rainfall) and T

Qp

(minutes) for the watershed. The Qp is

acquired by multiplying the excess rainfall (inches) for

the design storm with the peak streamflow from the nomograph. Given an analyst-selected, watershed-loss model

that is beyond the scope of this report, the excess rainfall

for the design storm for the duration Tc will require an

estimate of Tc.

Nomograph for the Method

To facilitate the method through graphical lookup to

avoid multistep mathematics by the user, a nomograph

is needed. For A values of 10, 20, 40, 80, 160, 320, and

640 acres and BDF values of 0, 3, 6, 9, and 12, the arithmetic means of Qp and T

Qp

from the gamma unit hydrograph and excess rational methods for excess rainfall of

1 inch were estimated by following the computations

described in the previous section. The resultant values

were used as ordinates in the nomograph shown in figure 16. Subsequently, minor smoothing of the lines was

performed.

Example Computations

Example computations that follow show that the

nomograph can be readily used. In this example, the Qp

for a design storm on a 160-acre watershed (0.25 square

miles) that has a BDF = 6 is needed for a design storm

having a recurrence interval of 10 years in the approximate center of the Houston metropolitan area.

First, the nomograph, by graphical lookup, shows

that Qp ≈ 104 cubic feet per second for 1 inch of excess

rainfall and that T

Qp ≈ 60 minutes.

Second, an estimate of the excess rainfall is needed.

The Tc for the watershed is computed by equation 19 with

the addition of ( A[acres]/640)/2 per equation 18. The

estimate of Tris

p

Tr = 10−0.05228×(6)+0.4028log10(0.25)+0.3926

= 0.69 hour, (24)

and the Tc is

Tc = 0.69+

p

160/640

= 0.94 hour . (25)

2

To keep this example brief as possible, 0.94 hour is

rounded to 1 hour, and Asquith and Roussel (2004, fig. 30,

p. 37) is used to estimate the 10-year, 1-hour design storm

for the approximate center of Harris County, Texas. This

depth is about 2.9 inches. This storm has an average

intensity of about 2.9/1 = 2.9 inches per hour. At this

average intensity, a 0.94-hour storm has a depth of about

2.7 inches (0.94×2.9).

Continuing with the example, suppose that a

watershed-loss model for the watershed, which is beyond

the scope of this report, yields an effective rainfall depth

of about 1.4 inches for the 2.7-inch design storm (about

a 50-percent loss for the example). The Qp for the

watershed thus is 1.4×104 ≈ 146 cubic feet per second .

As a means for assessment of uncertainty in Qp estimated from the method, the first and third quartiles of

the Cr and Cv values, which are shown in figure 12, are

useful. These quartiles can be used to compute lower Q

↓

p

and upper Q

↑

p estimates, which might be interpreted as

quartiles of Qp. The computations of lower Q

↓

p are

0.13 146

Q

↓

p = or

0.28 0.61

Q

↓

p = 0.76×146 ≈ 111 cubic feet per second, (26)

and the computations of upper Q

↑

p are

0.37 146

Q

↑

p = or

0.54 0.61

Q

↑

p = 1.12×146 ≈ 164 cubic feet per second, (27)
22 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

UNIT HYDROGRAPH AND EXCESS RATIONAL METHOD APPROACHES

TIME OF PEAK FROM INCEPTION OF EXCESS RAINFALL COMPUTED BY AVERAGING

PEAK STREAMFLOW FOR 1 INCH OF EXCESS RAINFALL COMPUTED BY AVERAGING

UNIT HYDROGRAPH AND EXCESS RATIONAL METHOD APPROACHES

EXPLANATION

DRAINAGE AREA, IN ACRES

PEAK STREAMFLOW, IN CUBIC FEET PER SECOND

TIME OF PEAK, IN MINUTES

10 20 30 40 50 60 80 100 200 300 400 500 640

15

20

30

40

50

60

80

100

200

300

400 0

15

30

45

60

75

90

105

120

135

150

165

180

195

0

3

6

9

12

0

3

6

9

12Basin-development factor

70

Figure 16. Nomograph of the relation between arithmetic means of peak and time of peak streamflow by gamma unit hydrograph and excess rational method computations for 1 inch of excess rainfall by selected basin-development factor and drainage 

area for 24 watersheds in the Houston, Texas, metropolitan area. 
A Method for Estimating Peak and Time of Peak Streamflow from Excess Rainfall for 10- to 640-Acre Watersheds in . . . 23 

Table 3. Comparison of values of peak and time of peak streamflow by gamma unit hydrograph and excess rational method 

computations for 1 inch of excess rainfall for limiting developed and undeveloped conditions for a 300-acre watershed in the 

Houston, Texas, metropolitan area. 

[ BDF, basin-development factor; T Qp , time of peak streamflow; Qp, peak streamflow; ERM, excess rational method; UH, unit hydrograph method; ft3/s, cubic feet per second; ←→, direct (left to right) comparison can be made. The numerical values listed in this table 

are found in the text starting on page 17 and are identifiable by frames or boxes around the value and corresponding units. ]

BDF

(dimensionless) 

ERMT Qp

(minutes) 

UHT Qp

(minutes) 

ERMQp

(ft3/s) 

UHQp

(ft3/s) 

T Qp

(minutes) 

Qp

(ft3/s) 

Developed, BDF = 12 45 ←→ 50 244 ←→ 202 48 223 

Undeveloped, BDF = 0 130 ←→ 145 84.5 ←→ 88.3 138 86.4 

where the value 0.61 (ratio mean Cr to mean Cv) used in 

the denominator is shown in equation 21. The Qp for the 

160-acre watershed that has a BDF = 6 estimated from 

method could be written as 

Qp = 146 ±18 | 35 cubic feet per second ,

where the values 18 and 35 are computed by 18 = 164 −

146 and 35 = 146 − 111. Lastly, the steps of the method 

are shown in figure 17. 

Potential Bias in the Excess Rational 

Method 

The method was developed through conjunctive analysis of the unit hydrograph method and the excess rational 

method. For the excess rational method, as expressed by 

equation 16, a major generalization is made that the ratio 

of Cr to Cv is approximately 0.61 (see equation 20). The 

reliability of the generalization that Cr/Cv ≈ 0.61 is important because the resultant ERMQp is obviously in direct 

proportion with the ratio. 

To explore the reliability, the ratios of the watershedspecific values for Cr and Cv were computed. The relation 

between these ratios and drainage area is shown in figure 18. 

From the figure, it can be concluded that the ratio 

of Cr/Cv tends to be larger than about 0.61 for the 

larger BDF values and smaller for the smaller BDF

values. But exactly how the ratio changes with BDF

and the additional interaction with drainage area is 

uncertain because of the disparity in the distribution 

of low BDF values for very small watersheds. If for 

high BDF (BDF ≈ 10–12) a more appropriate generalization of the Cr/Cv ratio is about 0.9, then the 

ERMQp used for the development of the method are 

a factor of about 0.67 too low (0.61/0.9). If for low 

BDF (BDF ≈ 0–2) a more appropriate generalization of Cr/Cv is about 0.3, then ERMQp used in the creation of the method are a factor of about 2.0 too high 

(0.61/0.3). 

This discussion can be generalized as meaning that 

the excess rational method in equation 20 has the potential 

for considerable overestimation of Qp for undeveloped 

or lightly developed watersheds as expressed by BDF. 

As a result, concerns over the accuracy of the Qp from 

the method naturally arise. A discussion of the relative 

influence of BDF on Qp therefore is needed and provided 

in the next section. 

On the Relative Influence of BasinDevelopment Factor on Peak Streamflow 

Influence of Basin-Development Factor 

There are many hydrologic methods to estimate 

Qp for design storms including empirical equations 

and coupled hydrologic and hydraulic models. The 

method described here is but one that has specific focus 

on excess rainfall and thus is conceptually independent 

from watershed-loss models. A discussion of the relative 

influence of BDF on Qp as demonstrated in this report is 

useful because BDF can be used as an expression of the 

transition of a watershed from undeveloped to developed 

conditions or from a given state of development back to a 

lesser state of development. 

The influence of BDF on Qp can be generalized in 

terms of log10-cycle change per unit change of BDF. 

BDFs are from 0 to 12 (13 categories) and are scored 

representations of the presence of various development 
Implementation of the Method 

1.	 Compute Tr, the center of the window for the maximum rainfall during the rainfall epoch, with equation 19 by using 

area A in square miles and basin-development factor BDF. 

2. Compute Tc with equation 18 by using Tr and A in square miles. 

3.	 Determine design storm depth (the total design rainfall, referred to as Pstorm) for the duration Tc as well as for an 

appropriate annual recurrence interval. 

4.	 Determine excess rainfall Estorm of the design storm depth by using a watershed-loss model; it necessarily follows 

that Estorm < Pstorm . 

5.	 Estimate the peak streamflow (identify as Qfig. 16 

p ) for 1 inch of excess rainfall from figure 16 by using A in acres and 
 

BDF; linearly interpolate to BDF values not shown in the figure.
 

= Qfig. 16 ×Estorm 6. Compute Qp in cubic feet per second for the design storm by Qstorm 

p . p

p 7.	 Estimate the time of peak T Qstorm in minutes for the design storm from figure 16 by using A in acres and BDF. Linearly interpolate to BDF values not shown in the figure. 

24 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Figure 17. Steps for implementing the method for 10- to 640-acre watersheds in the Houston, Texas, metropolitan area. 

DRAINAGE AREA, IN SQUARE MILES

RATIO TO , DIMENSIONLESS

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

11

9

12 9

0

7

8

11

10

9

9

3

6

12

9

5

7

7

5

4

4

12

12

12

8

RATIO OF WATERSHED-SPECIFIC TO . Symbol size is in proportion

to drainage area and number indicates basin-development factor.

EXPLANATION

Cr Cv

Cr Cv

Runoff coefficient is Cr. 

Volumetric runoff coefficient is Cv. 

Figure 18. Graph showing relation between ratio of watershed-specific (mean) values of runoff coefficient and volumetric 

runoff coefficient to drainage area for 24 watersheds in the Houston, Texas, metropolitan area. 
conditions on each third of a watershed. Inspection of 

the Qp lines (solid) in figure 16 on the left axis shows a 

maximum of about 38 and a minimum of about 15, which 

implies a change of 0.031log10 cubic feet per second of 

Qp per unit change in BDF or log10(38) − log10(15) =

0.404 and 0.404/12 = 0.034 for the 12 discrete changes in 

BDF : 0–12. 

In reference to the unit hydrograph method, the 0.034 

value compares favorably with the coefficient of about 

0.027 on BDF in equation 5 and by association the absolute value of the coefficient of about 0.034 on BDF in 

equation 6. (The Qp is inversely dependent on value for 

Tp.) 

In reference to the rational method, the 0.034 value 

also compares favorably with the absolute value of the 

coefficient of about 0.052 on BDF in equation 19. 

After reviewing these various coefficients (about 

0.034, 0.027, 0.034, and 0.052) and with the purpose of 

generalizing the influence of BDF on Qp in terms of a single number that is appropriate for 10- to 640-acre watersheds in the Houston metropolitan area, an ad hoc value of 

0.04log10 cubic feet per second change of Qp per positive 

unit of change in BDF is deemed to be appropriate. 

Example Computations 

Example computations, which demonstrate the generalized influence of BDF on Qp, are informative. For example, assume that a numerical hydrologic and hydraulic 

model of a 200-acre watershed with a BDF = 12 results 

in Qp |

BDF=12= 600 cubic feet per second. Also for this 

hypothetical watershed, rainfall and runoff data only exist 

for BDF = 12 conditions. It is further assumed that the 

hydrologic and hydraulic model model has been reasonably calibrated with these data. 

Can the estimate of Qp |

BDF=0 for this watershed 

be obtained by using the generalized influence of BDF

on Qp? By using the suggested value of 0.04log10 cubic 

feet per second change of Qp per positive unit of change 

in BDF, the computation is made for the base-10 logtransformed estimate of Qp: 

Ψ = log10(Qp) +0.04 ×(ΔBDF) (28) 

where Qp is the peak streamflow in cubic feet per second for the watershed and the respective BDF value and 

ΔBDF is the change in BDF. For a BDF = 12 → 0 the 

ΔBDF = −12 (increments), which yields, in equation 28 

(log10(600) + 0.04 × (−12)), and Ψ = 2.30. When Ψ

is retransformed by 10Ψ, the estimate is Qp |

BDF=0≈

200 cubic feet per second . 

Summary 25 

For a second example computation, a watershed has 

BDF = 9 and an estimated Qp |

BDF=9 = 600 cubic feet 

per second. An estimate of Qp |

BDF=12 for this watershed 

using the generalized influence of BDF on Qp can be 

computed. For a BDF = 9 → 12 the ΔBDF = 3, which 

yields, in equation 28 (log10(600) + 0.04 × (3)), an Ψ =

2.90. When Ψ is retransformed by 10Ψ, the estimate is 

Qp |

BDF=12≈ 791 cubic feet per second . 

Summary 

Estimation of peak and time of peak streamflow from design storms provides for cost-effective, 

risk-mitigated design of drainage structures such 

as bridges, culverts, roadways, and other infrastructure. During 2007–10, the U.S. Geological Survey 

(USGS), in cooperation with the Harris County Flood 

Control District and the Texas Department of Transportation, developed a method to estimate peak and 

time of peak streamflow from excess rainfall for 10­

to 640-acre low-slope watersheds in the Houston, 

Texas, metropolitan area. The method is based on 

conjunctive analysis of rainfall and runoff data in the 

context of the unit hydrograph method and the rational 

method. 

For this investigation, 21 distinct watersheds (based 

on latitude and longitude) were identified as pertinent for 

the Houston metropolitan area. Pertinent watersheds were 

selected on the basis of drainage area as a measure of 

watershed size. These watersheds represent generally the 

smallest watersheds for which there exist paired rainfall 

and runoff data suitable for conjunctive analysis of the 

unit hydrograph method and the rational method. 

The focus of the investigation is the conjunctive 

analysis of the unit hydrograph method and various 

applications of the rational method in the context of 

small watersheds (less than about 640 acres). To support statistical development, watersheds with drainage 

areas less than about 3.5 square miles (2,240 acres) 

were selected. Two of the watersheds were identified 

as having considerable changes in land development 

as expressed by the basin-development factor. The 

period of record for these two watersheds was thus segregated (three total divisions), resulting in the 24 watersheds that are used in this report. Collectively, these 21 

watersheds are assumed to represent the generalized 

hydrologic and hydraulic conditions of many small, 

low-slope watersheds in the Houston metropolitan 

area. 
26 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

For the 24 watersheds, a database of rainfall and 

runoff data for the selected watersheds in the Houston 

metropolitan area was compiled and converted to digital 

format as needed. The data for the 24 watersheds were 

obtained from various sources. Also, selected watershed 

characteristics of drainage area, basin-development factor, and others for the 24 watersheds were obtained from 

various sources. The basin-development factor BDF is 

conceptualized as a measure of runoff transport efficiency 

of a watershed containing interconnections of various 

drainage systems. BDFs are from 0 to 12 point (13 categories) and are scored representations of the presence of 

various development conditions on each third of a watershed. 

For the unit hydrograph analysis, background of the 

unit hydrograph method and discussion of a gamma unit 

hydrograph are provided. A gamma distribution model of 

unit hydrograph shape (a gamma unit hydrograph) was 

chosen and parameters estimated for the storms for the 

24 watersheds through matching of modeled peak and 

time of peak streamflow to observed values on a storm-bystorm basis. Watershed mean or watershed-specific values 

of peak and time to peak (“time to peak” is a parameter of 

the gamma unit hydrograph and is distinct from “time of 

peak”) of the gamma unit hydrograph were recorded. Two 

regression equations to estimate peak and time to peak of 

the gamma unit hydrograph based on watershed characteristics of drainage area and basin-development factor were 

developed. 

For the rational method analysis, background and 

specific mathematical analysis were provided. From analysis of the rational method for the 24 watersheds, a lag 

time (time-R), volumetric runoff coefficient, and runoff 

coefficient were computed on a storm-by-storm basis. 

Watershed-specific values of these three metrics were 

computed. A regression equation to estimate time-R based 

on drainage area and basin-development factor was developed. Overall arithmetic means of volumetric runoff coefficient (0.41 dimensionless) and runoff coefficient (0.25 

dimensionless) for the 24 watersheds were used to express 

the rational method in terms of excess rainfall. 

The peak and time of peak streamflow estimates from 

both the unit hydrograph and rational methods were combined by using the arithmetic mean of the values from 

each method (unit hydrograph and excess rational). Example computations are shown. These computations were 

used to develop the method. 

The method is a technique to compute peak and time 

of peak streamflow given excess rainfall for design storms 

for applicable watersheds in the Houston metropolitan area. The method requires graphical lookup from a 

nomograph of peak streamflow (cubic feet per second 

per 1 inch of excess rainfall) and time of peak streamflow (minutes) for the watershed. The peak streamflow is 

acquired by multiplying the excess rainfall (inches) for 

the design storm with the peak streamflow from the nomograph. Given an analyst-selected, watershed-loss model, 

which is beyond the scope of this report, the excess rainfall for the design storm will require an estimate of time of 

concentration. 

The nomograph shows the respective relations 

between peak and time of peak streamflow to drainage 

areas ranging from 10- to 640-acres. The nomograph 

also shows the respective relations for selected basindevelopment factors, which range from undeveloped to 

fully developed conditions. The nomograph represents 

the peak streamflow for 1 inch of excess rainfall based 

on drainage area and basin-development factor; the peak 

streamflow for design storms from the nomograph can be 

multiplied by the excess rainfall (inches) to estimate peak 

streamflow. Time of peak streamflow is readily obtained 

from the nomograph. Therefore, given excess rainfall 

values derived from watershed-loss models, which are 

beyond the scope of this report, the nomograph represents 

a method for estimating peak and time of peak streamflow 

for applicable watersheds in the Houston metropolitan 

area. Example computations of the method are provided 

for a hypothetical watershed along with an estimate of 

peak streamflow uncertainty associated with the method. 

Lastly, analysis of the relative influence of basindevelopment factor on peak streamflow is provided and 

the results suggest that a 0.04log10 cubic feet per second 

change of peak streamflow per positive unit of change 

in basin-development factor is appropriate. This relative 

change can be used to adjust peak streamflow from alternative hydrologic methods for a given BDF to other basindevelopment factor values; example computations were 

provided. 

References 

Asquith, W.H., and Roussel, M.C., 2004, Atlas of 

depth-duration frequency of precipitation annual 

maxima for Texas: U.S. Geological Survey Scientific Investigations Report 2004–5041, 106 p. 

(Also available at http:// pubs.usgs.gov/ sir/ 2004/

5041/ .) 

Asquith, W.H., and Roussel, M.C., 2007, An 

initial-abstraction, constant-loss model for unit 
Summary 27 

hydrograph modeling for applicable watersheds in Texas: U.S. Geological Survey Scientific Investigation Report 2007–5243, 82 p.
 

(Also available at http:// pubs.usgs.gov/ sir/ 2007/
 

5243/ .)
 

Asquith, W.H., Roussel, M.C., Thompson, D.B., Cleveland, T.G., and Fang, Xing, 2005, Unit hydrograph 

estimation for applicable Texas watersheds: Lubbock, 

Texas Tech Center for Multidisciplinary Research in 

Transportation, Texas Tech University, Texas Department of Transportation Research Report 0–4193–4, 

71 p. 

Asquith, W.H., Thompson, D.B., Cleveland, T.G., and 

Fang, Xing, 2004, Synthesis of rainfall and runoff 

data used for Texas Department of Transportation 

Research Projects 0–4193 and 0–4194: U.S. Geological Survey Open-File Report 2004–1035, 1,050 p. 

(Also available at http:// pubs.usgs.gov/ of/ 2004/

1035/ .) 

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

Applied hydrology: New York, McGraw-Hill, 572 p. 

Cleveland, T.G., He, Xin, Asquith, W.H., Fang, Xing, and 

Thompson, D.B., 2006, Instantaneous unit hydrograph 

selection for rainfall-runoff modeling of small watersheds in north and south central Texas: Journal of Irrigation and Drainage Engineering, v. 132, p. 479–485. 

Dingman, S.L., 2002, Physical hydrology (2nd ed.): 

Upper Saddle River, N.J., Prentice Hall, 646 p. 

Freeze and Nichols, 2010, City of San Angelo stormwater 

design manual: Freeze and Nichols, Fort Worth, Texas, 

adopted by City of San Angelo, Tex. on November 

16, 2010, [variously paged], accessed June 23, 2011, 

at http://www.sanangelotexas.us/vertical/ Sites/

{BD27ED61-E710-4F56-8954-CC319F012B3D}/

uploads/

{1DE248C9-C517-4519-8CF7-04E9EB4645B4}.PDF. 

Haan, C.T., Barfield, B.J., and Hayes, J.C., 1994, Design 

hydrology and sedimentology for small catchments: 

San Diego, Academic Press, 588 p. 

Harris County Flood Control District, 2004, Policy, criteria, and procedure manual—for approval and acceptance of infrastructure: Houston, Tex., Harris County 

Flood Control District, [variously paged], accessed 

June, 23, 2011, at http://www.hcfcd.org/ downloads/

manuals/HCFCD_PCP_Manual_2004.pdf . 

Kuichling, Emil, 1889, The relation between the rainfall 

and the discharge of sewers in populous districts: Transactions, American Society of Civil Engineers, v. 20, 

p. 1–56. 

Liscum, Fred, 2001, Effects of urban development 

on stormwater runoff characteristics for the Houston, Texas, metropolitan area: U.S. Geological Survey Water-Resources Investigations 01–4071, 35 p. 

(Also available at http:// pubs.usgs.gov/wri/

wri01-4071/ .) 

Liscum, Fred, Brown, D.W., and Kasmarek, M.C., 1996, 

Summary of surface-water hydrologic data for the 

Houston metropolitan area, Texas, water years 1964– 

1989: U.S. Geological Survey Open-File Report 96– 

250, 44 p. 

(Also available at http:// pubs.er.usgs.gov/ djvu/

OFR/ 1996/ ofr_96_250.djvu.) 

Pilgrim, D.H., and Cordery, Ian, 1993, Flood runoff, chapter 9, in Maidment, D.R., ed., Handbook of hydrology: 

New York, McGraw-Hill, p. 9.1–9.42. 

R Development Core Team, 2007, R—A language and 

environment for statistical computing: Vienna, Austria, R Foundation for Statistical Computing, ISBN 

3–900051–07–0, http://www.R-project.org. 

Roussel, M.C., Thompson, D.B., Fang, Xing, Cleveland, 

T.G., and Garcia, C.A., 2005, Time-parameter estimation for applicable Texas watersheds: Beaumont, Tex., 

Lamar University, Texas Department of Transportation 

Research Report 0–4696–2, 30 p. 

Schaake, J.C., Geyer, J.C., and Knapp, J.W., 1967, Experimental examination of the rational method: Journal 

of the Hydraulics Division, American Society of Civil 

Engineers, v. 93, no. 6, p. 353–370. 

Texas Department of Transportation, 2002, Hydraulic 

design manual: revised November 2002, Austin, Tex., 

[variously paged], accessed June, 23, 2011, at http://

onlinemanuals.txdot.gov/txdotmanuals/ hyd/index.htm. 

Glossary 

Acronyms 

HCFCD — Harris County Flood Control District (first 

use on page 2). 

TxDOT — Texas Department of Transportation (first use 

on page 2). 

ERM — Excess rational method defined on page 14 and 

by equation 15. 

RM — Rational method defined on page 11 and by equation 9. 

GUH — Gamma unit hydrograph and used in context of 

parameterization of such a unit hydrograph (first use 

on page 17). 

UH — Unit hydrograph and used in context as a modifier 

of other variables defined in this glossary (first use on 

page 11). 

Selected Definitions 

Design Storm — A design storm is a conceptual model 

that can be either a hypothetical or observed storm 

(or combination of both) that is characterized by 

depth, areal extent, duration, and temporal intensities. The design storm is used as an input into one or 

more also conceptual or numerical models of a watershed. The design storm is often chosen to represent 

some type of “hazard” used to illicit hydrologic and 

hydraulic response of similar hazard within the watershed’s interconnected water courses and channels. 

Symbols 

i— Represents “storm specific” (see page 13 and other 

entries in this glossary). 

| — Represents the two concepts of “given” (first use on 

page 17) and “or” (first use on page 23) as mathematical context dictates. For example, Qp | BDF=0 is to 

read “peak streamflow in cubic feet per second given

a basin-development factor of zero.” This is the by 

far the most common context for | used in this report. 

Glossary 29 

For the other example, Qp = 146 ±18 | 35 is to read 

“peak streamflow of 146 cubic feet per second plus 

18 cubic feet per second or minus 35 cubic feet per 

second” (see page 23). 

± — Represents “plus or minus” (see page 23). 

�= — Represents “not equal” as a numerical concept (see 

page 14). 

≈ — Represents “approximate” or “approximately” as a 

numerical concept (see page 8) 

Greek Alphabet 

ΔE — Change in elevation in feet (ΔE) between the two 

end points of the main-channel of a watershed (see 

page 5). 

ΔBDF — Change in BDF (see page 25). For example, 

suppose that a major, albeit unusual, low-impact redevelopment of a watershed causes BDF to go from 

10 to 8 (10 → 8) . Such a watershed has ΔBDF = −2, 

which by the methods described in this report would 

result in a reduction in Qp and lengthening (dilation) 

of T Qp . 

Γ(u) — The complete gamma function is expressed as 

an infinite integral defined in equation 4 and used in 

equation 3. 

Ψ — A quantity defined in equation 28 that represents the 

base-10 transformed estimate of Qp based on ΔBDF. 

Latin Alphabet 

A — Drainage area (contributing) in square miles (or 

acres as indicated in text). A mixed unit system for A

in this report was required based on the recognized 

traditions of the hydrologic engineering community. 

Values for the 24 watersheds are listed in table 1 (see 

page 5). 

BDF — Basin-development factor thoroughly described 

in appendix 1. Values for the 24 watersheds are listed 

in table 1 (see page 5). 
30 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

C — Runoff coefficient of rational method (see equation 9). 

Ci — Runoff coefficient that corresponds to maximum 

rainfall intensity that has been determined for each 

unique duration (5-minute increments in this study) 

from the rainfall epoch (see page 14 and fig. 11). 

Cr — Runoff coefficient from computations involving 

rational method from observed rainfall and runoff for 

storm duration Ta (see page 14). c

iCr — The Cr value for a particular storm (see equations 12 and 13). 

Cv — Volumetric runoff coefficient (total rainfall divided 

by total runoff, dimensionless) (see page 14). 

iCv — Volumetric runoff coefficient for a given storm 

(dimensionless, see equation 14). 

Cw — Runoff coefficient for the full width of the rainfall 

epoch (see page 14 and fig. 11). 

E — Excess rainfall in inches (see equation 15). 

I — Rainfall intensity in inches per hour (see equation 9). 

K — The shape parameter of gamma hydrograph (see 

equation 1). 

L — Main-channel length: The L is defined as the length 

in stream-course miles of the longest defined channel from the approximate watershed headwaters to 

the outlet. Values for the 24 watersheds are listed in 

table 1 (see page 5). 

P — Depth of rainfall in inches (see equation 10). 

iP — Depth of rainfall for a particular storm in inches 

(see equation 14). 

P(T,F) — Depth of rainfall in inches as function of some 

storm duration T and nonexceedance probability F

(see equation 10). 

iP}max 

Ta — Maximum depth of rainfall in inches for a time c

window (duration) having width of Ta (see equa­ c

tions 12 and 13). 

q — Depth of streamflow in watershed inches per hour. 

The relation between Q and q is seen in equation 2. 

q(t) — Depth of streamflow in watershed inches per hour 

at time t of the gamma unit hydrograph of equation 1. 

The relation between Q and q is seen in equation 2. 

qp — Depth of peak streamflow in watershed inches per 

hour of the gamma unit hydrograph of equation 1. 

The estimate of qp for the gamma unit hydrograph is 

provided by equation 5. The relation between Q and 

q is seen in equation 2. 

Q — Streamflow in cubic feet per second. The relation 

between Q and q is seen in equation 2. 

Qp — The peak or maximum instantaneous streamflow in 

cubic feet per second (see page 7). 

Q↓

p — The lower estimate of peak streamflow Qp from the 

method in cubic feet per second (see page 21). 

Q↑

p — The upper estimate of peak streamflow Qp from the 

method in cubic feet per second (see page 21). 

ERMQp — Solution to Qp from the excess rational method 

in cubic feet per second (see equations 15 and 16). 

RMQp — Solution to Qp from the rational method in 

cubic feet per second (see equation 10). 

RMiQp — Peak streamflow Qp for a particular storm from 

the rational method in cubic feet per second (see 

equations 12 and 13). 

UHQp — Solution to Qp from the unit hydrograph method 

in cubic feet per second (see page 11). 

R — Depth of runoff in inches, see iR entry in this glossary. 

iR — Depth of runoff for a particular storm in inches (see 

equation 14). 

S — Dimensionless main-channel slope: The S is defined 

as the change in elevation ΔE in feet between 

the two end points of L divided by L in feet: S =

ΔE/(5,280 × L). Values for the 24 watersheds are 

listed in table 1 (see page 5). 

t — Time metric in hours of equation 1. 

T — Rainfall duration (critical storm duration) in hours 

(see equation 10). 

TLPC — Lag-time of a watershed (see page 14). 
Glossary 31

Tb — Time base of a streamflow hydrograph (see page 8).

Tc —Time of concentration in hours (see equation 18 for

the primary definition of this report).

Tc

0 — First-order approximation to Tc as a critical storm

duration in hours that is equal to p

A[square miles] or

A[acres]/640 (see equation 11).

Tp —

p

The time to qp of a gamma unit hydrograph in

hours of equation 1. Or in other words, the time to

peak of the gamma distribution used to model the

unit hydrograph. The estimate of Tp for the gamma

unit hydrograph is provided by equation 6.

T

Qp — The time of Qp occurrence in hours (see page 7).

p

ERMT

Qp —Solution to T

Q from the excess rational

method in hours (see equation 17).

RMT

Qp —Solution to TQp

from the rational method in

hours, which is same as ERMT

Qp

(see equation 17).

UHT

Qp —Solution to TQp

from the unit hyrograph

method in hours (see page 11).

Tr — The quantity “time-R” defined on page 14, which

represents a time metric between rainfall and resulting time of peak. Equation 19 provides an estimate

from analysis of rainfall and runoff described in this

report.

V —The volume in depth of a runoff hydrograph and for

a unit hydrograph V = 1 inch (see equation 3).

Appendix 1—Basin-Development Factor
 

Basin-Development Factor 

The basin-development factor BDF is conceptualized 

as a measure of runoff transport efficiency of a watershed 

containing interconnections of various drainage systems 

(constructed or otherwise). The BDF is a 0–12 point, categorical variable that is a scored representation of the presence of various development conditions (scored 1 through 

3) on each third of a watershed. Although a categorical 

variable, the BDF is treated as a continuous variable in a 

multilinear regression context herein. Values for BDF are 

defined by dividing the watershed into thirds (see fig. 1.1) 

and evaluating each third with respect to four indices of 

urbanization. 

The BDF is considered by Sauer and others (1983) in 

the context of estimation of urban flooding potential and is 

shown to be a useful predictive variable for that purpose. 

The BDF description that follows is a near verbatim quote 

by Sauer and others (1983, p. 8), who state: 

The most significant index of urbanization that 

resulted from [the 1983] study is a basin-development 

factor (BDF), which provides a measure of the efficiency of the drainage system. This parameter, which 

proved to be highly significant in the regression equations [of the 1983 report], can be easily determined 

from drainage maps and field inspections of the 

drainage basin. The basin is first divided into thirds 

as [shown in figure 1.1]. Then, within each third, four 

aspects of the drainage system are evaluated and each 

assigned a code as follows: 

1. Channel improvements.—If channel 

improvements such as straightening, enlarging, 

deepening, and clearing are prevalent for the 

main drainage channels and principal tributaries 

(those that drain directly into the main channel), 

then a code of 1 is assigned. Any or all of these 

improvements would qualify for a code of 1. To 

be considered prevalent, at least 50 percent of 

the main drainage channels and principal 

tributaries must be improved to some degree 

over natural conditions. If channel 

improvements are not prevalent, then a code of 

zero is assigned. 

2. Channel linings.—If more than 50 percent of the 

length of the main drainage channels and 

principal tributaries has been lined with an 

impervious material, such as concrete, then a 

code of 1 is assigned to this aspect. If less than 

50 percent of these channels is lined, then a code 

of zero is assigned. The presence of channel 

linings would obviously indicate the presence of 

Appendix 1—Basin-Development Factor 35 

channel improvements as well. Therefore, this is 

an added factor and indicates a more highly 

developed drainage system. 

3. Storm drains, or storm sewers.—Storm drains 

are defined as enclosed drainage structures 

(usually pipes), frequently used on the secondary 

tributaries where the drainage is received 

directly from streets or parking lots. Many of 

these drains empty into open channels; however, 

in some basins they empty into channels 

enclosed as box or pipe culverts. When more 

than 50 percent of the secondary tributaries 

within a subarea (third) consists of storm drains, 

then a code of 1 is assigned to this aspect; if less 

than 50 percent of the secondary tributaries 

consists of storm drains, then a code of zero is 

assigned. It should be noted that if 50 percent or 

more of the main drainage channels and 

principal tributaries are enclosed, then the 

aspects of channel improvements and channel 

linings would also be assigned a code of 1. 

4. Curb-and-gutter streets.—If more than 50 

percent of a subarea (third) is urbanized 

(covered by residential, commercial, and/or 

industrial development), and if more than 50 

percent of the streets and highways in the 

subarea are constructed with curbs and gutters, 

then a code of 1 would be assigned to this aspect. 

Otherwise, it would receive a code of zero. 

Drainage from curb-and-gutter streets frequently 

empties into storm drains. 

The above guidelines for determining the various 

drainage-system codes are not intended to be precise measurements. A certain amount of subjectivity 

will necessarily be involved. Field checking should 

be performed to obtain the best estimate. The basindevelopment factor (BDF) is the sum of the assigned 

codes; therefore, with three subareas (thirds) per 

basin, and four drainage aspects to which codes are 

assigned in each subarea, the maximum value for a 

fully developed drainage system would be [BDF = 12]. 

Conversely, if the drainage system were totally undeveloped, then a BDF of zero would result. Such a 

condition does not necessarily mean that the basin is 

unaffected by urbanization. In fact, a basin could be 

partially urbanized, have some impervious area, have 

some improvement of secondary tributaries, and still 

have an assigned BDF of zero. As is discussed later 

in [the 1983 report], such a condition still frequently 

causes peak discharges to increase. 
36 A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-Acre Watersheds in Houston, Texas 

Lower

third

Upper

third

Middle

third

Outlet

Drainage divide

Outlet

Lower

third

Upper

third

Middle

third

Drainage divide

Outlet

Lower

third

Upper

third

Middle

third

Drainage divide

A. Long, narrow basin

B. Fan-shaped basin

C. Short, narrow basin

Figure 1.1. Diagram of typical drainage basin shapes and subdivision into basin thirds. Note that the stream-channel distances 

within any given third of a basin in the examples are approximately equal, but between basin thirds the distances are not equal, 

to compensate for relative basin width of the thirds. 
Appendix 1—Basin-Development Factor 37 

The BDF is a fairly [straightforward] index to estimate 

for an existing urban basin. The 50-percent guideline will usually not be difficult to evaluate because 

many urban areas tend to use the same design criteria, and therefore have similar drainage aspects, 

throughout. Also, the BDF is convenient for projecting 

future development. Obviously, full development and 

maximum urban effects on peaks would occur when 

BDF = 12. Projections of full development or intermediate stages of development can usually be obtained 

from city engineers. 

A basin-development factor was evaluated for each of 

the 269 sites used in [the 1983] study. Approximately 

30 people were involved in making these evaluations, 

using guidelines similar to the ones described in the 

preceding paragraphs but somewhat less explicit. Tests 

have not been made to see how consistently two or 

more people can estimate the BDF for a basin. However, this study indicates that fairly consistent estimates can be made by different people. A relatively 

large group of individuals made the estimates for this 

study and the parameter was statistically very significant in the regression equations. If the results obtained 

by various individuals had not been consistent, it is 

doubtful that the statistical results [reported in the 

1983 study] would be so significant. 

Values of the BDF change with urbanization; therefore, BDF values need to be redefined whenever significant changes take place within a watershed. Further, 

because the current (2010) investigation is focused on 

watersheds in the Houston, Texas, metropolitan area, 

select modifications to the method of BDF computation 

by Sauer and others (1983) were needed (Fred Liscum, 

PBS&J Inc. [now (2011) with HCFCD], written commun., 2008). Modifications to the method were deemed 

important in order to account for conditions commonly 

encountered in the Houston metropolitan area. These modifications consider (1) the presence of enclosed channels 

and (2) the prevalence of roadside ditch drainage in a particular watershed. 

The following discussion presents suggested modifications to the BDF method described by Sauer and others 

(1983). These modifications were used for the 24 watersheds considered in the current (2010) investigation. The 

four “aspects” of Sauer and others (1983) are summarized 

and the modification shown in italic type. 

1.	 Channel Improvements—If channel improvements 

such as straightening, enlarging, deepening, and 

clearing are prevalent for the main drainage channels 

and principal tributaries, then a code of 1 is assigned. 

Otherwise, a code of 0 is assigned unless one of the

following applies:

◦	 If 50 percent or more of the main drainage channel

and/or principal tributaries are enclosed (for

example by storm sewers), then a code of 1 is

assigned.

◦	 For areas with roadside ditch drainage that satisfy

the 50 percent or more criteria, a code of 0.5 is

assigned in lieu of the normal value of 1.

2.	 Channel Linings—If more than 50 percent of the 

length of the main drainage channel and principal 

tributaries has been lined with an impervious material such as concrete, a code of 1 is assigned. Otherwise, a code of 0 is assigned unless one of the following applies:

◦	 If 50 percent or more of the main drainage channel

and/or principal tributaries are enclosed (for

example by storm sewers), then a code of 1 is

assigned.

◦	 For areas with roadside ditch drainage that satisfy

the 50 percent or more criteria, a code of 0.5 is

assigned in lieu of the normal value of 1.

3.	 Storm Sewers—If more than 50 percent of the main 

channel and secondary tributaries are enclosed as 

storm sewers, a code of 1 is assigned. Otherwise, a 

code of 0 is assigned. 

◦	 In the absence of channels (conventional openchannels), the main trunk of the storm sewer system

is treated as a “lined” channel for the purposes of

scoring the Channel Linings aspect.

◦	 In the absence of channels (conventional openchannels), the main trunk of the storm sewer system

is scored as Channel Improvements.

4.	 Curb-and-Gutter Streets—If more than 50 percent 

of a third is urbanized, and if more than 50 percent of 

the streets and roads within the area are constructed 

with curbs and gutters, a code of 1 is assigned. Otherwise, a code of 0 is assigned. 

Reference 

Sauer, V.B., Thomas, W.O., Stricker, V.A., Wilson, K.V., 1983, Flood characteristics of urban 

watersheds in the United States: U.S. Geological Survey Water-Supply Paper 2207, 63 p. 

(Also available at http:// pubs.er.usgs.gov/ djvu/

WSP/wsp_2207.djvu.) 

Appendix 2—Selected Unit Conversions
 

 

 

Appendix 2—Selected Unit Conversions 41 

The relation between q in inches per hour and Q in cubic feet per second needed for equation 2 is readily obtained by 

dimensional analysis as follows: 

feet3 inches miles2 1 hour 1 foot 5,2802 feet2 

Q = q ×A × × × , second hour 1 3,600 seconds 12 inches 1 mile2

 

645.33 

which upon simplification and dropping of units becomes Q = 645.33 ×q×A. 

The unit-conversion factor 1.008 for the rational method in equation 9 is readily obtained by dimensional analysis as 

follows: 

Q

feet3 

second = κ dimensionless ×I

inches 

hour 

×A

acre 

1 

×

1 hour 

3,600 seconds 

 

×

1 foot 

12 inches 

×

208.72 feet2 

acre 

, 

1.008 

which upon simplification becomes κ = 1.008. 

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Asquith, Cleveland, and Roussel— A Method for Estimating Peak and Time of Peak Streamflow for 10- to 640-AcreWatersheds in Houston, Texas—SIR 2011–5104 

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