Development of Guidance for Runoff Coefficient Selection and Modified Rational 
Unit Hydrograph Method for Hydrologic Design 
 
by 
 
Nirajan Dhakal 
 
 
 
 
A dissertation submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Doctor of Philosophy  
 
Auburn, Alabama 
May 7, 2012 
 
 
 
 
Keywords: Rational Method, Runoff Coefficient, Unit Hydrograph, 
Modified Rational Unit Hydrograph 
 
 
Copyright 2012 by Nirajan Dhakal 
 
 
Approved by 
 
Xing Fang, Chair, Associate Professor of Civil Engineering 
 T. Prabhakar Clement, Professor of Civil Engineering 
Jose G. Vasconcelos, Assistant Professor of Civil Engineering 
Luke J. Marzen, Professor of Geography 
 
 
 
 
 
 
ii 
 
Abstract 
 
 
The rational method is the most widely used method by hydraulic and drainage 
engineers to estimate design discharges. The runoff coefficient (C) is a key parameter for 
the rational method. Literature-based C values (C
lit
) are listed for different land-use/land-
cover conditions in various design manuals and textbooks, but C
lit
 appear not to be 
derived from any observed data. In this study, C
lit
 values were derived for 90 watersheds 
in Texas from two sets of land-cover data for 1992 and 2001. C values were also 
estimated using observed rainfall and runoff data for more than 1,600 events in the study 
watersheds using two different approaches (1) the volumetric approach (C
v
) (2) the rate-
based approach (C
rate
).When compared with the C
v 
values, about 80 percent of C
lit
 values 
were greater than C
v
 values. This result might indicate that literature-based C 
overestimate peak discharge for drainage design when used with the rational method. 
Similarly, when compared with the C
rate
 values, about 75 percent of C
lit
 values were 
greater than C
rate 
values, however, for developed watersheds with more impervious cover, 
C
lit
 values were greater than C
rate 
values. Rate-based C were also developed as function of 
return period for 36 undeveloped watersheds in Texas using peak discharge frequency 
from previously published regional regression equations and rainfall intensity frequency 
for return periods of 2, 5, 10, 25, 50, and 100 years. The C values of this study increased 
with return period more rapidly than the increase suggested in prior literature.
 
 iii
To use the rational method for hydraulic structures involving storage, the 
modified rational method (MRM) was developed. The hydrograph developed using the 
MRM can be considered application of a special unit hydrograph (UH) that is termed the 
modified rational unit hydrograph (MRUH) in this study. Being a UH, the MRUH can be 
applied to nonuniform rainfall distributions and for watersheds with drainage areas 
greater than typically used for the rational method (a few hundred acres). The MRUH 
was applied to 90 watersheds in Texas using 1,600 rainfall-runoff events. The MRUH 
performed as well as other three UH methods (Gamma, Clark-HEC-1, and NRCS) when 
the same rainfall loss model was used. 
 
iv 
 
Acknowledgments 
 
 
I would like to express sincere gratitude to my academic advisor, Dr. Xing Fang, 
for his valuable guidance, support, encouragement and dynamism throughout the work. 
He is an excellent professional researcher as well a very good human being and he will 
always remain as an inspiration to me throughout my life. 
I would also like to thank Professors T. Prabhakar Clement, Jose G. Vasconcelos 
and Luke J. Marzen for serving as my committee members, and for their valuable time to 
review my proposal, dissertation and give suggestions that have helped to improve the 
research. 
The research fund has been supported by the Texas Department of Transportation 
(TxDOT). I am grateful to Dr. Theodore G. Cleveland of the Texas Tech University, Dr. 
David B. Thompson of R. O. Anderson Engineering, Inc., and Dr. William H. Asquith of 
the USGS for their technical guidance in the research. 
Finally, I would like to express my deepest gratitude to my family and friends for 
their support and encouragement throughout the work. 
 v
Table of Contents 
 
 
Abstract ............................................................................................................................... ii 
Acknowledgments.............................................................................................................. iv  
List of Tables ................................................................................................................... viii 
List of Figures ......................................................................................................................x  
List of Abbreviations ....................................................................................................... xiv 
Chapter 1.  Introduction 
 1.1  Background .......................................................................................................1 
 1.2  Research Objectives ..........................................................................................5 
 1.3  Study Area and Rainfall-Runoff Database .......................................................6 
 1.4  Organization of Dissertation .............................................................................7 
 1.5  References .........................................................................................................9 
Chapter 2.  Estimation of Volumetric Runoff Coefficients for Texas Watersheds 
Using Land-Use and Rainfall-Runoff Data 
 2.1  Abstract ...........................................................................................................12 
 2.2  Introduction .....................................................................................................13 
 2.3  Watershed Studied and Rainfall-Runoff Database .........................................15 
 2.4  Estimation of Runoff Coefficients Using LULC Data ...................................18 
 2.5  Estimation of the Back-Computed Volumetric Runoff Coefficients 
(C
vbc
) Using Observed Rainfall-Runoff Data .................................................22 
 2.6  Estimation of Volumetric Runoff Coefficients from the Rank-Ordered 
Pairs of Observed Rainfall and Runoff Depths ..............................................28
vi 
 
 2.7  Discussion .......................................................................................................32
 2.8  Summary .........................................................................................................34 
 2.9  Acknowledgements .........................................................................................37 
 2.10  Notation.........................................................................................................37 
 2.11  References .....................................................................................................39 
Chapter 3.  Rate-based Estimation of the Runoff Coefficients for Selected 
Watersheds in Texas 
 3.1  Abstract ...........................................................................................................54 
 3.2  Introduction .....................................................................................................54 
 3.3  Study Area and Rainfall-Runoff Database .....................................................58 
 3.4  Runoff Coefficients Estimated from Event Rainfall-Runoff Data .................61 
 3.5  Runoff Coefficients from the Frequency-Matching Approach .......................65 
 3.6  Summary .........................................................................................................72 
 3.7  Acknowledgements .........................................................................................74 
 3.8  Notation...........................................................................................................74 
 3.9  References .......................................................................................................76 
Chapter 4. Return Period Adjustments for Runoff Coefficients Based on Analysis 
in Texas Watersheds   
 4.1  Abstract ...........................................................................................................89 
 4.2  Introduction .....................................................................................................89 
 4.3  Study Watersheds............................................................................................92 
 4.4  Runoff Coefficients for Different Return Periods ..........................................94 
 4.5  Discussion .......................................................................................................96 
 4.6  Summary .........................................................................................................99 
 4.7  Acknowledgements .......................................................................................100 
 4.8  Notation.........................................................................................................101 
 vii
 4.9  References .....................................................................................................102 
Chapter 5.  Modified Rational Unit Hydrograph Method and Applications in Texas 
Watersheds 
 5.1  Abstract .........................................................................................................110 
 5.2  Introduction ...................................................................................................110 
 5.3  Applications of MRUH in Texas Watersheds ..............................................117 
5.4  Estimated Runoff Hydrographs from Different Unit Hydrograph 
Methods ........................................................................................................125 
 5.5  Sensitivity of the MRUH to unit hydrograph duration .................................128 
 5.6  Discussion and Summary ..............................................................................129 
 5.7  Notation.........................................................................................................132 
 5.8  Appendix A: Statistical Measures to Evaluate Model Performance ............133 
 5.9  References .....................................................................................................135 
Chapter 6.  Conclusions and Recommendations 
 6.1 Conclusions ....................................................................................................153 
 6.2 Recommendations ..........................................................................................158 
 6.3 References ......................................................................................................160 
 viii
List of Tables 
 
 
Table 2.1 Runoff Coefficients Selected for Various Land-Cover Classes from 
NLCD 2001 ............................................................................................................42 
Table 2.2 Statistical Summary of Average C
lit
 Using NLCD 1992 and Watershed-
Minimum, -Average and -Maximum C
lit
 Using NLCD 2001................................43 
Table 2.3 Statistical Summary of Watershed-average C
lit
 
Using NLCD 2001 for 
Developed and Undeveloped Watersheds .............................................................43 
Table 2.4 Statistical Summary of C
vbc
 
and R
v
 from NURP (USEPA 1983) .......................44 
Table 2.5 Statistical Summary of Watershed-average C
vbc
 for Developed and 
Undeveloped Watersheds .......................................................................................44 
Table 2.6 Statistical Summary of C
vr
 and Absolute Difference (ABS) of C
vr
 with 
Watershed-average C
lit
 for 83 Texas Watersheds ..................................................44 
Table 2.7 Statistical Summary of C
vr
 for Developed and Undeveloped Watersheds 
and Absolute Difference (ABS) of C
vr
 with Watershed-average C
vbc
 and C
lit
 
for Developed Watersheds .....................................................................................45 
Table 3.1 Statistical Summary of C
rate
 Calculated from Observed Rainfall-Runoff 
Event Data ..............................................................................................................80 
Table 3.2 Statistical Summary of Watershed-Median, Watershed-Mean and 
Standard Deviation of C
rate
 for 80 Texas Watersheds ...........................................80 
Table 3.3 Statistical Summary of Watershed-Median C
rate
 for Developed and 
Undeveloped Watersheds .......................................................................................80 
Table 3.4 Statistical Summary of C
r
, and Differences among C
r
, C
rate
 and C
lit
 for 80 
Texas watersheds ...................................................................................................81 
Table 3.5 Statistical Summary of C
r
 and C
lit
 for Developed and Undeveloped 
Watersheds .............................................................................................................81 
Table 4.1 Statistical Summary of C(T) for Select Return Periods (T) for Texas 
Watersheds ...........................................................................................................105
ix 
 
Table 4.2 Statistical Summary of the Frequency Factors C(T)/C(10) for Texas 
Watersheds ...........................................................................................................105 
Table 4.3 Frequency Factor or Multiplier for Literature-based Rational Runoff 
Coefficient C from Different Sources ..................................................................105 
Table 5.1 Quantitative Measures of the Success of the Peak Discharges Modeled 
Using MRUH with Back- Computed Volumetric Runoff Coefficient (C
vbc
) 
and Time of Concentration Estimated Using Four Equations .............................139 
Table 5.2 Quantitative Measures of the Success of the Time to Peak Modeled 
Using MRUH with Back-Computed Volumetric Runoff Coefficient (C
vbc
) 
and Time of Concentration Estimated Using Four Equations .............................139 
Table 5.3 Quantitative Measures of the Success of the Peak Discharges and Time to 
Peak Modeled Using MRUH with Time of Concentration Estimated Using 
Kirpich Equation and Runoff Coefficients Estimated Using Two Different 
Methods (C
vbc
 and C
lit
) .........................................................................................140 
Table 5.4 Quantitative Measures of the Success of the Peak Discharges Modeled 
Using Four Unit Hydrograph Models for 1,600 Rainfall-Runoff Events in 
90 Texas Watersheds ...........................................................................................140 
Table 5.5 Sensitivity of Peak Discharges Modeled using MRUH on Unit 
Hydrograph Duration for the Rainfall Event on 07/28/1973 for the 
Watershed Associated with the USGS Streamflow-gaging Station 08178600 
Salado Creek, San Antonio, Texas ......................................................................141 
Table 5.6 Sensitivity of Peak Discharges Modeled using MRUH on Unit 
Hydrograph Duration for 1,600 rainfall events in 90 Texas Watersheds ............141 
 x
List of Figures 
 
 
Fig 2.1. Map showing the U.S. Geological Survey streamflow-gaging stations 
(dots) associated with the watershed locations in Texas  ......................................46 
Fig 2.2. Cumulative distributions of C
lit
 obtained using NLCD 1992 and NLCD 2001 
(top) and C
lit
 using NLCD 2001 for developed and undeveloped watersheds 
(bottom)..................................................................................................................47 
Fig 2.3. Cumulative distributions of C
vbc
 for watershed-average, watershed-median, 
and all rainfall-runoff events (top) and watershed-average C
vbc
 for 
developed and undeveloped watersheds (bottom) .................................................48?
Fig 2.4. Volumetric runoff coefficients from different studies versus percent 
impervious area including regression lines (top) and runoff coefficients with 
one standard deviations for 45 Texas watersheds and 60 NURP watersheds 
(bottom)..................................................................................................................49 
Fig 2.5. The rank-ordered pairs of observed runoff and rainfall depths (top) and 
runoff coefficients derived from the rank-ordered pairs of observed runoff 
and rainfall depths versus total rainfall depths (bottom). All data presented 
are for the USGS gage station 08042650 (North Trinity Basin in Texas) .............50 
Fig 2.6. Cumulative distributions of C
vr
 and C
lit 
(top) and C
vr
 for developed and 
undeveloped watersheds (bottom) .........................................................................51 
Fig 2.7. Runoff coefficients C
vr
, C
vbc
 (watershed-average), and runoff coefficients 
R
v
 from 60 NURP watersheds versus percent impervious area including 
lines for the regression equation (2.6) and the regression equation (2.7) ..............52 
Fig 2.8. Runoff coefficients C
lit
 (watershed-average), C
vbc
 (watershed-average), C
vr
, 
and R
v
 plotted against watershed area (km
2
) ..........................................................52 
Fig 2.9. Runoff coefficients C
vbc
 (watershed-average) and C
vr
 plotted against C
lit
 
(watershed-average) for 83 Texas watersheds .......................................................53 
Fig 3.1. Map showing U.S. Geological Survey streamflow-gaging stations 
representing 80 developed and undeveloped watersheds in Texas. .......................82
 
 xi
Fig 3.2. Average rainfall intensity I and runoff coefficient C
rate
 as a function of t
av
 
for two storm events: (A) on 09/22/1969 in Dry Branch in Fort Worth, 
Texas (U.S. Geological Survey [USGS] streamflow-gaging station 
08048550 Dry Branch at Blandin Street, Fort Worth Texas), and (B) on 
04/25/1970 in Honey Creek near Dallas, Texas. (USGS streamflow-gaging 
station 08058000 Honey Creek subwatershed number 12 near McKinney, 
Texas). ....................................................................................................................83 
Fig 3.3. Cumulative distributions of runoff coefficient (C
rate
 values): (A) for 
rainfall-runoff events when the time to peak (T
p
) was less than the time of 
concentration (T
c
); for rainfall-runoff events when the time to peak (T
p
) was 
greater than or equal to the time of concentration (T
c
); watershed-average 
(mean); and watershed-median; and (B) watershed-median C
rate
 for 
developed and undeveloped Texas watersheds. .....................................................84 
Fig 3.4. (A) Runoff coefficient (C
r
) values derived from the rank-ordered pairs of 
observed peak discharge and maximum rainfall intensity during each storm 
event in mm/hr at USGS streamflow-gaging station 08042650 North Creek 
Surface Water Station 28A near Jermyn, Texas, and (B) the rank-ordered 
pairs of the observed peak discharge and the average rainfall intensity for 
the same station. .....................................................................................................85 
Fig 3.5. Cumulative distributions of: (A) C
r
, watershed-median C
rate
 and C
lit
, and 
(B) distributions of C
r
 and C
lit
 for developed and undeveloped watersheds. ........86 
Fig 3.6. Runoff coefficients versus the percentage of impervious area, IMP. ...................87 
Fig 3.7. Modeled peak discharges (Q
p
) from rational equation (3.1) using C
r
 and 
watershed-median C
rate
 for 1,500 rainfall-runoff events in 80 Texas 
watersheds against observed peak discharges. .......................................................88 
Fig 3.8. Rate-based runoff coefficients C
rate
 for all events, watershed-mean C
rate
, 
and C
r
 versus drainage area (km
2
) in 80 Texas watersheds. ..................................88 
Fig 4.1. Map showing the locations of U.S. Geological Survey streamflow-gaging 
stations in Texas associated with the 36 undeveloped watersheds considered 
for this study (two stations are very close and overlapped each other). ..............106 
Fig 4.2. Estimation of C(T) versus T for three undeveloped Texas watersheds. .............107 
Fig 4.3. Box plot for the distribution of runoff coefficients for different return 
periods from Texas watersheds, from Hotchkiss and Provaznik (1995) and 
from Young et al. (2009). ....................................................................................108 
Fig 4.4. Frequency factors for different return periods from Texas watersheds, 
Young et al. (2009), French et al. (1974) and FHWA (Jens 1979) and Gupta 
(1989). ..................................................................................................................109 
 xii
Fig 5.1. The modified rational hydrographs for three different cases: (A) Duration 
of rainfall (D) is equal to time of concentration (T
c
), (B) Duration of rainfall 
is greater than T
c
, and (C) Duration of rainfall is less than T
c
. ............................142 
Fig 5.2. The modified rational unit hydrographs (MRUH) developed for: (A) two 
lab settings from Yu and McNown (1964) and (B) for the watershed 
associated with USGS streamflow-gaging station 08157000 Waller Creek, 
Austin, Texas. ......................................................................................................143 
Fig 5.3. Incremental rainfall hyetograph and observed and modeled runoff 
hydrographs using the MRUH for the two lab tests on concrete surfaces: 
(A) 152.4 m
 
?
 
0.3 m and (B) 76.8 m
 
?
 
0.3 m reported by Yu and McNown 
(1964). ..................................................................................................................144 
Fig 5.4. Map showing the U.S. Geological Survey streamflow-gaging stations 
(dots) associated with the watershed locations in Texas. ....................................145 
Fig 5.5. Incremental rainfall hyetograph and observed and modeled runoff 
hydrographs using the MRUH with T
c
 estimated by four empirical 
equations for the event on 07/08/1973 for the watershed associated with the 
USGS streamflow-gaging station 08157000 Waller Creek, Austin, Texas. ........146 
Fig 5.6. Modeled versus observed peak discharges for 1,600 rainfall-runoff events 
in 90 Texas watersheds. Modeled results were developed from MRUH 
using event C
vbc
 and T
c
 estimated using four different methods: (A) 
Haktanir-Sezen equation, (B) Johnstone-Cross equation, (C) Williams 
equation and (D) Kirpich equation.   ...................................................................147 
Fig 5.7. Modeled versus observed time to peak for 1,600 rainfall-runoff events in 
90 Texas watersheds. Modeled results were developed from MRUH using 
event C
vbc
 and T
c
 estimated using four different methods: (A) Haktanir-
Sezen equation, (B) Johnstone-Cross equation, (C) Williams equation and 
(D) Kirpich equation.  ..........................................................................................148 
Fig 5.8. Modeled versus observed peak discharges developed from MRUH using 
C
vbc
 (triangles) and C
lit
 (circles) with time of concentration estimated using 
Kirpich equation for 90 Texas watersheds. ..........................................................149 
Fig 5.9. (A) Modified rational, Gamma, Clark-HEC-1, and NRCS unit hydrographs 
developed for the watershed associated with USGS streamflow-gaging 
Station 08048520 Sycamore, Fort Worth, Texas; and (B) Rainfall 
hyetograph, observed and modeled runoff hydrographs using the four 
different unit hydrographs for the rainfall event on 07/28/1973 for the same 
watershed. ............................................................................................................150 
Fig 5.10. Modeled versus observed peak discharges using: (A) MRUH, (B) Gamma 
UH, (C) Clark-HEC-1 UH, and (4) NRCS UH for 1,600 rainfall-runoff 
events in 90 Texas watersheds. ............................................................................151 
 xiii
Fig 5.11. Observed and modeled runoff hydrographs using MRUH with six unit 
hydrograph durations (5, 10, 20, 30, 40 and 50 minutes) for the rainfall 
event on 07/28/1973 for the watershed associated with the USGS 
streamflow-gaging station 08178600 Salado Creek, San Antonio, Texas. ..........152 
 
 
 xiv
List of Abbreviations 
 
 
ASCE American Society of Civil Engineers 
DEM Digital Elevation Model 
GIS Geographic Information System 
IDF Intensity-Duration-Frequency  
IUH Instantaneous Unit Hydrograph 
LULC Land-use/land-cover 
MRM Modified Rational Method 
MRUH Modified Rational Unit Hydrograph Method 
NLCD National Land Cover Dataset 
NRCS Natural Resources Conservation Service 
NURP National Urban Runoff Program 
RHM Rational Hydrograph Method 
SBUH Santa Barbara Unit Hydrograph  
TxDOT Texas Department of Transportation 
USEPA United States Environmental Protection Agency  
USGS U.S. Geological Survey 
WPCF Water Pollution Control Federation
 1
Chapter1. Introduction 
 
 
1.1 Background 
Rational method and Runoff coefficient 
Early storm water or catchment runoff estimation throughout the world was based 
on designer?s experience and judgment. Current practice is that the watershed that is to be 
drained by a proposed storm sewer system will be generally divided into one or more 
sub-catchments or sub-watersheds that are of reasonable size and are approximately 
homogeneous in nature. These urban watersheds may include residential, commercial or 
industrial areas, but usually have larger proportions of pavement and the streets and roads 
which are the principal surface drainage conveyance, have short time of concentration, 
and have well-defined flow paths, typically through gutters, ditches and medians of 
streets and roads. Each year, billions of dollars are spent on new construction of drainage 
structures. For the safety, design of hydraulic structures is done based on the peak 
discharge (Q
p
) as the design flow. Therefore, Q
p
 is the major hydrological parameter 
required for the hydraulic design purpose. Additional parameters such as volume of 
runoff and time of peak flow are required in some cases such as for the design of 
facilities that use storage such as detention and retention basins. 
 The rational method is the most widely used method by hydraulic and drainage 
engineers to estimate peak design discharges, which are used to size a variety of drainage 
2 
 
structures for small urban (developed) and rural (undeveloped) watersheds (Viessman 
and Lewis 2003). The rational method was developed in the United States by Emil 
Kuichling (1889) and introduced to Great Britain by Lloyd-Davies (1906). The peak 
discharge (Q
p
 in m
3
/s in SI units or ft
3
/s in English units) for the method is computed 
using: 
CIAmQ
p 0
?  (1.1) 
where C is the runoff coefficient (dimensionless), I is the average rainfall intensity 
(mm/hr or in./hr) for a storm with a duration equal to a critical period of time (typically 
assumed to be the time of concentration), A is the drainage area (hectares or acres), and 
m
o
 is a dimensional correction factor (1/360 = 0.00278 in SI units, 1.008 in English 
units). 
 C is the variable of the rational method least amenable to precise determination, 
and estimation of C calls for judgment on the part of the engineer (ASCE and WPCF 
1960; TxDOT 2002). C can vary substantially depending on watershed conditions. 
Therefore, research to document appropriate values of C is needed. Typical C values, 
representing the integrated effects of many watershed conditions (C
lit
), are listed for 
different land-use/land-cover (LULC) conditions in various design manuals and 
textbooks (Chow et al. 1988; Viessman and Lewis 2003). Values of C
lit
 published by the 
joint committee were obtained from a survey, which received ?71 returns of an extensive 
questionnaire submitted to 380 public and private organizations throughout the United 
States.? The results represented decades of professional practice experience using the 
rational method to determine runoff volumes in storm-sewer design applications (ASCE 
and WPCF 1960). No justification based on observed rainfall and runoff data for the 
 3
selected C
lit
 values was provided in the ASCE and WPCF (1960) manual. In short, we 
conclude that C
lit
 values appeared heuristically determined, and therefore comparison of 
C values derived from observed rainfall and runoff data to the C
lit
 values is important. 
In this study, we focus on applying different methods to estimate C for 90 
watersheds in Texas using observed rainfall and runoff data for 1,600 events. C
lit
 values 
were derived from two sets of LULC data for 1992 and 2001. Volumetric runoff 
coefficients (C
v
) were estimated by event totals of observed rainfall and runoff depths 
from more than 1,600 events observed in the watersheds. C
v
 values were also estimated 
using rank-ordered pairs of rainfall and runoff depths (frequency matching). 
It is important to stress that rational method is the rate-based method (eq. 1.1). 
Current runoff coefficients given in textbooks and design manuals are neither volumetric 
nor rate-based because they were not derived from observed data but are used for the 
rate-based rational method. In this study, the rate-based runoff coefficients, C
rate
, were 
estimated for each of 1,600 rainfall-runoff events and the time window used to determine 
average rainfall intensity was time of concentration computed using the Kirpich method. 
Subsequently, the frequency-matching approach was used to extract a representative 
runoff coefficient, C
r
, for each watershed. C values developed from both the volumetric 
and rate-based approaches are compared independently with C
lit
 values. 
A substantial criticism of the rational method arises because observed C values 
vary from storm to storm (Schaake et al. 1967; Pilgrim and Cordery 1993). The C has 
been considered a function of return period by various researchers (Jens 1979; Pilgrim 
and Cordery 1993; Hotchkiss and Provaznik 1995; Titmarsh et al. 1995; Young et al. 
2009). Using watershed parameters, such as drainage area, slope, and channel length and 
 4
regression equations of discharge and rainfall intensity at different return periods T of 2-, 
5-, 10-, 25-, 50-, and 100-years, the C(T) were estimated in 36 undeveloped watersheds in 
Texas. Subsequently, frequency factors C
f
(T) = C(T)/C(10) were computed. Results of 
C(T) and frequency factors C
f
(T) were analyzed and compared with previous studies. 
 
Modified Rational Method (MRM) 
Incorporation of detention basins to mitigate effects of urbanization on peak flows 
required design methodologies to include the volume of runoff as well as the peak 
discharge (Rossmiller 1980). To use the rational method for hydraulic structures 
involving storage, the modified rational method (MRM) was developed (Poertner 1974). 
While the original rational method is meant to produce only the peak design discharge, 
the MRM produces a runoff hydrograph and the runoff volume of the entire watershed. 
The MRM, which has found widespread use in the engineering practice since 1970s, is 
used to size detention/retention facilities for a specified recurrence interval and 
concurrent release rate. The MRM is based on the same assumptions as the conventional 
rational method, that is, the rainfall is uniform in space over the drainage area being 
considered and the rainfall intensity is uniform throughout the duration of the storm 
(Rossmiller 1980).  
The MRM was revisited and reevaluated in this study. The hydrograph developed 
from application of the MRM is a special case of the unit hydrograph method and will be 
termed the modified rational unit hydrograph (MRUH) in this study. Being a unit 
hydrograph, the MRUH can be applied to nonuniform rainfall distributions. Furthermore, 
the MRUH can be used on watersheds with drainage areas in excess of the typical limit 
 5
for application of the rational method or the modified rational method (a few hundred 
acres). The MRUH was applied to 90 watersheds in Texas using 1,600 rainfall-runoff 
events. The Gamma UH, Clark-HEC-1 UH, and NRCS dimensionless UH were also used 
to predict peak discharges of all events in the database. 
 
1.2 Research Objectives 
This research work is a part of TxDOT Project 0-6070 ?Use of the Rational and 
the Modified Rational Methods for TxDOT Hydraulic Design?. The principal objective 
of the project is to evaluate appropriate conditions for the use of the rational method and 
modified rational methods for designs on small watersheds, evaluate and refine, if 
necessary, current tabulated values of the runoff coefficient and construct guidelines for 
TxDOT analysts for the selection of appropriate parameter values for Texas conditions. 
The specific objectives are:  
1. Estimation of areally-weighted literature-based runoff coefficients (C
lit
) for the study 
watersheds using land-use data. 
2. Estimation of volumetric runoff coefficients (C
v
) for the study watersheds using 
rainfall-runoff data. 
3. Estimation of the rate based runoff coefficients (C
rate
) for the study watersheds using 
rainfall-runoff data and comparison of C
lit
 with both the C
v 
and C
rate
. 
4. Estimation of the runoff coefficients for different return periods and compare the 
current frequency multiplier C(T)/C(10) in the literature with our results. 
 6
5. Evaluate the applicability of the modified rational unit hydrograph method (MRUH) if 
blindly applied to watersheds of sizes greater than originally intended with either the 
rational method or the modified rational method (that is, a few hundred acres).  
6. Study the effects of runoff coefficient and the timing parameters on predictions of 
runoff hydrographs using MRUH. 
 
 
1.3 Study Area and Rainfall-Runoff Database 
Watershed data from a larger dataset accumulated by researchers from the U.S. 
Geological Survey (USGS) Texas Water Science Center, Texas Tech University, 
University of Houston, and Lamar University (Asquith et al. 2004) and previously used 
in a series of research projects funded by the Texas Department of Transportation 
(TxDOT) were used for this study. The data were collected as a part of USGS small-
watershed projects and urban watershed studies during 1959?1986 (Asquith et al. 2004). 
The original data, available in the form of 220 printed USGS data reports, were 
transcribed to digital format manually (Asquith et al. 2004). Incidentally, these data also 
are used by Cleveland et al. (2006), Asquith and Roussel (2007), Fang et al. (2007, 
2008), and Dhakal et al. (2012). 
The dataset comprises 90 USGS streamflow-gaging stations in Texas, each 
representing a different watershed (Fang et al. 2007, 2008). There are 29, 21, 7, 13 
watersheds in Austin, Dallas, Fort Worth, and San Antonio areas, respectively, and 
remaining 20 watersheds are small rural watersheds in Texas. The drainage area of study 
watersheds ranged from approximately 0.8?440.3 km
2
 (0.3?170 mi
2
), with median and 
mean values of 17.0 km
2
 (6.6 mi
2
) and 41.1 km
2
 (15.9 mi
2
), respectively.  There are 33, 
 7
57, and 80 study watersheds with drainage areas less than 13 km
2
 (5 mi
2
), 26 km
2
 (10 
mi
2
), and 65 km
2
 (25 mi
2
), respectively.  The stream slope of study watersheds ranged 
from approximately 0.0022?0.0196, with median and mean values of 0.0075 and 0.081, 
respectively.  The percentage of impervious area (IMP) of study watersheds ranged from 
approximately 0.0?74.0, with median and mean values of 18.0 and 28.4, respectively. 
The rainfall-runoff dataset comprised about 1,600 rainfall-runoff events. The 
number of events available for each watershed varied; for some watersheds only a few 
events were available whereas for some others as many as 50 events were available 
(Cleveland et al. 2006). Values of rainfall depths for 1,600 events ranged from 3.56 mm 
(0.14 in.) to 489.20 mm (19.26 in.), with median and mean values of 57.15 mm (2.25 in.) 
and 66.29 mm (2.61 in.), respectively. Values of maximum rainfall intensities calculated 
using time of concentration for 1,600 events ranged from 0.01 mm/min (0.03 in./hr) to 
2.54 mm/min (6.01 in./hr), with median and mean values of 0.25 mm/min (0.58 in./hr) 
and 0.30 mm/min (0.72 in./hr), respectively. 
 
1.4 Organization of Dissertation 
 
This dissertation is organized into six chapters. Chapter two to five are organized 
in journal paper format prepared for ASCE journal publication. Parts of results of the 
study were presented in two conference papers: 
1. Dhakal, N., Fang, X., Cleveland, T. G., Thompson, D. B., and Marzen, L. J. (2010). 
"Estimation of rational runoff coefficients for Texas watersheds." Proceeding (CD-ROM) 
for 2010 World Environmental and Water Resources Congress, Providence, Rhode 
Island. 
 8
 2. Nirajan Dhakal, Xing Fang, Theodore G. Cleveland, and David B. Thompson, 2011. 
?Revisiting Modified Rational Method?.  Proceeding (CD-ROM) for 2011 World 
Environmental and Water Resources Congress, Palm Springs, CA, May 22-26, 2011. 
Chapter two deals with the estimation of the volumetric runoff coefficients (C
v
) 
for the study watersheds using both land-use and rainfall-runoff data. Two regression 
equations of C
v
 versus percent impervious area were developed and combined into a 
single equation which can be used to rapidly estimate C
v
 values for similar Texas 
watersheds. The work of this chapter has been published in the ASCE Journal of 
Irrigation and Drainage Engineering (Nirajan Dhakal, Xing Fang, Theodore G. 
Cleveland, David B. Thompson, William H. Asquith, and Luke J. Marzen, 2012 
(January). ?Estimation of Volumetric Runoff Coefficients for Texas Watersheds Using 
Land-Use and Rainfall-Runoff Data.? ASCE Journal of Irrigation and Drainage 
Engineering, 138(1):43-54, DOI=10.1061/(ASCE)IR.1943-4774.0000368). 
Chapter three deals with the estimation of the rate based runoff coefficients for 
the study watersheds from the rainfall-runoff data. An equation applicable to many Texas 
watersheds is proposed to estimate C as a function of impervious area. The work of this 
chapter has been revised and resubmitted for publication in the ASCE Journal of 
Hydrologic Engineering (Nirajan Dhakal, Xing Fang, William H. Asquith, Theodore G. 
Cleveland, and David B. Thompson. ?Rate-based Estimation of the Runoff Coefficients 
for Selected Watersheds in Texas?. ASCE Journal of Hydrologic Engineering). 
Chapter four deals with the estimation of the runoff coefficients based on the 
return period. The work of this chapter has been submitted for review and publication in 
the ASCE Journal of Irrigation and Drainage Engineering (Nirajan Dhakal, Xing Fang, 
 9
William H. Asquith, Theodore G. Cleveland, and David B. Thompson. ?Return Period 
Adjustments for Runoff Coefficients Based on Analysis in Texas Watersheds?. ASCE 
Journal of Irrigation and Drainage Engineering. In Review). 
Chapter five deals with the development and application of the Modified Rational 
Unit Hydrograph Method (MRUH) for the study watersheds. The Gamma UH, Clark-
HEC-1 UH, and NRCS dimensionless UH were also used to predict peak discharges of 
all events in the database.  The work of this chapter has been submitted for review and 
publication in the ASCE Journal of Hydrologic Engineering (Nirajan Dhakal, Xing Fang, 
David B. Thompson, and Theodore G. Cleveland. ?Modified Rational Unit Hydrograph 
Method and Applications in Texas Watersheds?. ASCE Journal of Hydrologic 
Engineering. In Review). 
Chapter six summarizes the conclusion of the study and provides some 
recommendations for the future study in this area. 
 
 
1.5 References 
 
ASCE, and WPCF. (1960). Design and construction of sanitary and storm sewers, 
American Society of Civil Engineers (ASCE) and Water Pollution Control Federation 
(WPCF). 
Asquith, W.H., Thompson, D.B., Cleveland, T.G., and Fang, X. (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, Austin, 
Texas. 
Asquith, W.H., and Roussel, M.C. (2007). ?An initial-abstraction, constant-loss model 
for unit hydrograph modeling for applicable watersheds in Texas.? U.S. Geological 
Survey Scientific Investigations Report 2007?5243, 82 p. 
[http://pubs.usgs.gov/sir/2007/5243]. 
Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). Applied hydrology, McGraw 
Hill, New York. 
 10
Cleveland, T.G., He, X., Asquith, W.H., Fang, X., and Thompson, D.B. (2006). 
?Instantaneous unit hydrograph evaluation for rainfall-runoff modeling of small 
watersheds in north and south central Texas.? Journal of Irrigation and Drainage 
Engineering, 132(5), pp. 479?485. 
Dhakal, N., Fang, X., Cleveland, T.G., Thompson, D.B., Asquith, W.H., and Marzen, L.J. 
(2012). ?Estimation of volumetric runoff coefficients for Texas watersheds using 
land-use and rainfall-runoff data.? Journal of Irrigation and Drainage Engineering, 
138(1):43-54. 
Fang, X., Thompson, D.B., Cleveland, T.G., and Pradhan, P. (2007). ?Variations of time 
of concentration estimates using NRCS velocity method.? Journal of Irrigation and 
Drainage Engineering, 133(4), pp. 314?322. 
Fang, X., Thompson, D.B., Cleveland, T.G., Pradhan, P., and Malla, R. (2008). ?Time of 
concentration estimated using watershed parameters determined by automated and 
manual methods.? Journal of Irrigation and Drainage Engineering, 134(2), pp. 202?
211. 
Hotchkiss, R. H., and Provaznik, M. K. (1995). ?Observations on the rational method C 
value.? Watershed management: planning for the 21st century; proceedings of the 
symposium sponsored by the Watershed  Management Committee of the Water 
Resources, New York. 
Jens, S. W. (1979). Design of Urban Highway Drainage, Federal Highway 
Administration (FHWA), Washington DC. 
Kuichling, E. (1889). "The relation between the rainfall and the discharge of sewers in 
populous areas." Transactions, American Society of Civil Engineers 20, 1?56. 
Lloyd-Davies, D. E. (1906). "The elimination of storm water from sewerage systems." 
Minutes of Proceedings, Institution of Civil Engineers, Great Britain, 164, 41. 
Poertner, H. G. (1974). "Practices in detention of urban stormwater runoff: An 
investigation of concepts, techniques, applications, costs, problems, legislation, legal 
aspects and opinions." American Public Works Association, Chicago, IL. 
Pilgrim, D. H., and Cordery, I. (1993). ?Flood runoff.? In Handbook of Hydrology, D. R. 
Maidment, ed., McGraw-Hill, New York, 9.1?9.42. 
Rossmiller, R. L. (1980). "Rational formula revisited." International Symposium on 
Urban Storm Runoff, University of Kentucky, Lexington, KY., 1-12. 
Roussel, M.C., Thompson, D.B., Fang, D.X., Cleveland, T.G., and Garcia, A.C. (2005). 
?Timing parameter estimation for applicable Texas watersheds.? 0?4696?2, Texas 
Department of Transportation, Austin, Texas. 
 11
Schaake, J. C., Geyer, J. C., and Knapp, J. W. (1967). ?Experimental examination of the 
rational method.? Journal of the Hydraulics Division, ASCE, 93(6), pp. 353?370. 
Titmarsh, G. W., Cordery, I., and Pilgrim, D. H. (1995). ?Calibration Procedures for 
Rational and USSCS Design Flood Methods.? Journal of Hydraulic Engineering, 
121(1), pp. 61?70. 
Viessman, W., and Lewis, G. L. (2003). Introduction to hydrology, 5th Ed., Pearson 
Education, Upper Saddle River, N.J., 612. 
Young, C. B., McEnroe, B. M., and Rome, A. C. (2009). ?Empirical determination of 
rational method runoff coefficients.? Journal of Hydrologic Engineering, 14, 1283 p. 
 
 
 
 12
Chapter 2. Estimation of Volumetric Runoff Coefficients for Texas Watersheds 
Using Land-Use and Rainfall-Runoff Data 
 
 
2.1 Abstract 
 The rational method for peak discharge (Q
p
) estimation was introduced in the 
1880s. Although the rational method is considered simplistic, it remains an effective 
method for estimating peak discharge for small watersheds. The runoff coefficient (C) is 
a key parameter for the rational method and there are various ways to estimate C. 
Literature-based C values (C
lit
) are listed for different land-use/land-cover (LULC) 
conditions in various design manuals and textbooks. However, these C
lit
 values were 
developed without much basis on observed rainfall and runoff data. C
lit
 values were 
derived for 90 watersheds in Texas from two sets of LULC data for 1992 and 2001; C
lit
 
values derived from the 1992 and 2001 LULC datasets were essentially the same. Also 
for this study, volumetric runoff coefficients (C
v
) were estimated by event totals of 
observed rainfall and runoff depths from more than 1,600 events observed in the 
watersheds. Watershed-median and watershed-average C
v
 values were computed and both 
are consistent with the data from the National Urban Runoff Program. C
v
 values also 
were estimated using rank-ordered pairs of rainfall and runoff depths (frequency 
matching). As anticipated, C values derived by all three methods (literature-based, event 
totals, and frequency matching) consistently have larger values for developed watersheds 
than undeveloped watersheds. Two regression equations of C
v
 versus percent impervious 
13 
 
area were developed and combined into a single equation which can be used to rapidly 
estimate C
v
 values for similar Texas watersheds. 
 
2.2 Introduction 
Estimation of peak discharge and runoff values for use in designing certain 
hydraulic structures (e.g., crossroad culverts, drainage ditches, urban storm drainage 
systems, and highway bridge crossings) are important and challenging aspects of 
engineering hydrology (Viessman and Lewis 2003). Various methods are available to 
estimate peak discharges and runoff volumes from urban watersheds (Chow et al. 1988). 
The rational method is the most widely used method by hydraulic and drainage engineers 
to estimate design discharges, which are used to size a variety of drainage structures for 
small urban (developed) and rural (undeveloped) watersheds (Viessman and Lewis 
2003). The rational method was developed in the United States by Emil Kuichling (1889) 
and introduced to Great Britain by Lloyd-Davies (1906). The peak discharge (Q
p
 in m
3
/s 
in SI units or ft
3
/s in English units) for the method is computed using: 
,AICmQ
op
?
 
(2.1) 
where C is the runoff coefficient (dimensionless), I is the average rainfall intensity 
(mm/hr or in./hr) for a storm with a duration equal to a critical period of time (typically 
assumed to be the time of concentration), A is the drainage area (hectares or acres), and 
m
o
 is a dimensional correction factor (1/360 = 0.00278 in SI units, 1.008 in English 
units). 
 The precise definition and subsequent interpretation of C varies. The C of a 
watershed can be defined either as the ratio of total depth of runoff to total depth of 
 14
rainfall or as the ratio of peak rate of runoff to rainfall intensity for the time of 
concentration (Wanielista and Yousef 1993). Kuichling (1889) analyzed observed rainfall 
and discharge data for developed urban watersheds in Rochester, NY, and computed the 
percentage of the rainfall discharged during the period of greatest flow as Q
p
/(IA), which 
is equal to runoff coefficient C from equation (2.1). Kuichling concluded that the 
percentage of the rainfall discharged for any given watershed studied is nearly equal to 
the percentage of impervious surface within the watershed, and this is the original 
meaning of C introduced by Kuichling (1889). Using Kuichling?s definition, C = 0 for a 
strictly pervious surface and C = 1 for a strictly impervious surface. 
 C is the variable of the rational method least amenable to precise determination, 
and estimation of C calls for judgment on the part of the engineer (ASCE and WPCF 
1960; TxDOT 2002). Typical C values, representing the integrated effects of many 
watershed conditions, are listed for different land-use/land-cover (LULC) conditions in 
various design manuals and textbooks (Chow et al. 1988; Viessman and Lewis 2003). 
The source of these published C values (literature-based C, called C
lit
 in this paper) 
derives from the 1960 sanitary and storm sewer design manual produced by a joint 
committee of the American Society of Civil Engineers (ASCE) and the Water Pollution 
Control Federation (WPCF). Values of C
lit
 published by the joint committee were 
obtained from a survey, which received ?71 returns of an extensive questionnaire 
submitted to 380 public and private organizations throughout the United States.? The 
results represented decades of professional practice experience using the rational method 
to determine runoff volumes in storm-sewer design applications (ASCE and WPCF 
1960). No justification based on observed rainfall and runoff data for the selected C
lit
 
 15
values was provided in the ASCE and WPCF (1960) manual. However, analysis of 
observed rainfall and runoff data was presented by Kuichling (1889). 
 In this paper, three methods were implemented to estimate C for 90 watersheds in 
Texas (Figure 2.1). The first method used LULC information for a watershed and 
published C
lit
 values for various land uses to derive a watershed-composite C
lit
. The 
second method estimated the volumetric runoff coefficient (C
v
) values by the ratio of 
total runoff depth to total rainfall depth for individual storm events. About 1,600 rainfall-
runoff events measured from 90 Texas watersheds were analyzed to determine event, 
watershed-median, and watershed-average C
v
 values. C
v
 determined from storm events is 
called the back-computed volumetric runoff coefficient (C
vbc
) in this paper. The third 
method computed probabilistic C
v
 values from the rank-ordered pairs of observed rainfall 
and runoff depths of a watershed and extracted a representative C
v
 for the watershed from 
the plot of C
v
 versus rainfall depths. C
v
 determined from the rank-ordered data is called 
the rank-ordered volumetric runoff coefficient (C
vr
) in this paper. The third method is 
similar to the procedure used by Schaake et al. (1967). C values estimated by the three 
different methods were analyzed and compared. Regression equations of C
vbc
 and C
vr
 
versus percent impervious area are presented. 
 
2.3 Watersheds Studied and Rainfall-Runoff Database 
Watershed data taken from a larger dataset (Asquith et al. 2004) accumulated by 
researchers from the U.S. Geological Survey (USGS) Texas Water Science Center, Texas 
Tech University, University of Houston, and Lamar University, and previously used in a 
series of research projects funded by the Texas Department of Transportation (TxDOT) 
 16
were used for this study. The dataset comprises 90 USGS streamflow-gaging stations in 
Texas, each representing a different watershed (Fang et al. 2007, 2008). Location and 
distribution of the stations in Texas are shown in Figure 2.1. There are 29, 21, 7, 13 
watersheds in Austin, Dallas, Fort Worth, and San Antonio areas, respectively, and 
remaining 20 watersheds are small rural watersheds in Texas (Figure 2.1). The drainage 
area of study watersheds ranged from approximately 0.8?440.3 km
2
 (0.3?170 mi
2
), with 
median and mean values of 17.0 km
2
 (6.6 mi
2
) and 41.1 km
2
 (15.9 mi
2
), respectively.  
There are 33, 57, and 80 study watersheds with drainage areas less than 13 km
2
 (5 mi
2
), 
26 km
2
 (10 mi
2
), and 65 km
2
 (25 mi
2
), respectively.  The stream slope of study 
watersheds ranged from approximately 0.0022?0.0196, with median and mean values of 
0.0075 and 0.081, respectively.  The percentage of impervious area (IMP) of study 
watersheds ranged from approximately 0.0?74.0, with median and mean values of 18.0 
and 28.4, respectively. 
Many would argue that the application of the rational method is not appropriate 
for the range of watershed areas presented in this study.  For example, watershed 
drainage area is a criteria used to select a hydrologic method (Chow et al. 1988) to 
compute peak discharge according to TxDOT guidelines for drainage design. The 
TxDOT guidelines recommend the use of the rational method for watersheds with 
drainage areas less than 0.8 km
2
 (200 acres) (TxDOT 2002). However, French et al. 
(1974) estimated values of the runoff coefficient in New South Wales, Australia, for 37 
rural watersheds, ranging in size up to 250 km
2
 (96 mi
2
).  Young et al. (2009) determined 
runoff coefficients for 72 rural watersheds in Kansas with drainage areas up to 78 km
2
 
 17
(30 mi
2
).  ASCE and WPCF (1960) made the following statements when the rational 
method was introduced for design and construction of sanitary and storm sewers: 
?Although the basic principles of the rational method are applicable to large drainage 
areas, reported practice generally limits its use to urban areas of less than 5 sq miles.  
Development of data for application of hydrograph methods is usually warranted on 
larger areas? (ASCE and WPCF 1960, p. 32). 
Chow et al. (1988) and Viessman and Lewis (2003) do not specify an area limit 
for application of the rational method.  Pilgrim and Cordery (1993) stated that the rational 
method is one of the three methods widely used to estimate peak flows for small to 
medium sized basins.  ?It is not possible to define precisely what is meant by ?small? and 
?medium? sized, but upper limits of 25 km
2
 (10 mi
2
) and 550 km
2
 (200 mi
2
), respectively, 
can be considered as general guides? (Pilgrim and Cordery 1993, p. 9.14).  Results of this 
study will further indicate that there is no demonstrable trend in runoff coefficient with 
drainage area. 
The rainfall-runoff dataset comprised about 1,600 rainfall-runoff events recorded 
during 1959?1986. The number of events available for each watershed varied; for some 
watersheds only a few events were available whereas for some others as many as 50 
events were available (Cleveland et al. 2006). Values of rainfall depths for 1,600 events 
ranged from 3.56 mm (0.14 in.) to 489.20 mm (19.26 in.), with median and mean values 
of 57.15 mm (2.25 in.) and 66.29 mm (2.61 in.), respectively. Values of maximum 
rainfall intensities calculated using time of concentration for 1,600 events ranged from 
0.01 mm/min (0.03 in./hr) to 2.54 mm/min (6.01 in./hr), with median and mean values of 
0.25 mm/min (0.58 in./hr) and 0.30 mm/min (0.72 in./hr), respectively. 
 18
 A geospatial database was developed from another TxDOT project (Roussel et al. 
2005), containing watershed boundaries for the 90 watersheds, delineated using a 30-
meter digital elevation model (DEM). The geospatial database contains watershed 
drainage area, longitude and latitude of the USGS streamflow-gaging station, which was 
treated as the outlet of the watershed, and 42 watershed characteristics (e.g., main 
channel length, channel slope, basin width, etc.) of each individual watershed (Roussel et 
al. 2005).  Each of the 90 watersheds was classified as either developed (urbanized) or 
undeveloped (Roussel et al. 2005, Cleveland et al. 2008).  Forty-four developed 
watersheds are located in four metropolitan areas in Texas (Austin, Dallas, Fort Worth, 
and St. Antonio) and used for USGS urban studies from 1959 to 1986.  Thirty-six 
undeveloped watersheds include 20 small rural watersheds and 16 watersheds in 
suburban of the four metropolitan areas.  The classification scheme of developed and 
undeveloped watersheds parallels and accommodates the disparate discussion and 
conceptualization in more than 220 USGS reports that provided the original data for the 
rainfall and runoff database (Asquith et al. 2004).  Although this binary classification 
seems arbitrary, it was purposeful and reflected the uncertainty in watershed development 
condition at the time the rainfall-runoff data were collected (Asquith and Russell 2007). 
This binary classification was successfully used to develop regression equations to 
estimate the shape parameter and the time to peak for regional Gamma unit hydrographs 
for Texas watersheds (Asquith et al. 2006). 
 
2.4 Estimation of Runoff Coefficients Using LULC Data 
 19
 C is strongly dependent on land use, and to a lesser extent, on watershed slope 
(Schaake et al. 1967; ASCE 1992). For watersheds with multiple land-use classes, a 
composite (area-weighted average) runoff coefficient, C
lit
, can be estimated using: 
 
,
1
1
?
?
?
?
?
n
i
i
n
i
ii
lit
A
AC
C  (2.2) 
where, i = i
th
 sub-area with particular land-use type, n = total number of land-use classes 
in the watershed, C
i
 = literature-based runoff coefficient for i
th
 land-use class, and A
i
 = 
sub-area size for i
th 
land-use class in the watershed (TxDOT 2002). In this study, 
watershed-composite C
lit
 values were derived for the 90 watersheds in Texas using LULC 
information and published C
lit
 values from various literatures. A geographic information 
system (GIS) was used for sub-areal extraction of different LULC classes within a 
particular watershed (ESRI 2004). The 1992 and 2001 National Land Cover Data 
(NLCD) for Texas were obtained from the USGS website http://seamless.usgs.gov/ 
(accessed on May 30, 2008). 
 Each watershed has different LULC classes distributed within its boundary. Of 16-
LULC classes from the NLCD 2001 data, 15 were used for the 90 watersheds studied; 
definitions of NLCD LULC classes are available at 
http://www.epa.gov/mrlc/definitions.html (accessed on May 30, 2008). Runoff 
coefficients were assigned for the 12-NLCD, 2001-LULC classes or mixed classes as 
listed in Table 2.1, which are based on 15-NLCD 2001-LULC classes; the table includes 
sources and references for the selected C values. From all sources considered, C values 
typically were not available for most of 15-NLCD LULC classes, but similar land-use 
types from literature were identified to match NLCD LULC (Table 2.1). A C value of 1 
 20
was assigned to open water, woody wetlands and emergent herbaceous wetlands and is 
not shown in Table 2.1. For the other LULC classes a range of C values were available 
from the mentioned sources under similar LULC types, and the average values (listed in 
column 3 of Table 2.1) were taken as literature C values for the study before a sensitivity 
analysis of C
lit
 on selected C values for different LULC classes from literature was 
conducted. 
 Using NLCD 2001 data and standard published mean C values (Table 2.1), 
composite runoff coefficients, C
lit
, for the 90 Texas watersheds were developed using 
equation (2.2) (Table 2.2). Values of C
lit
 ranged from 0.29 to 0.63, with median and mean 
values of 0.50 and 0.47, respectively (Table 2.2). Estimates of C
lit
 for a given watershed 
may differ, depending on the experience and judgment used in assigning them to LULC 
classes and estimating areas for land-use classes. For example, Harle (2002) determined 
C
lit
 for a subset of 36 watersheds from the 90 Texas watersheds using standard C
lit
 tables 
published by TxDOT (2002). The average absolute difference between Harle?s estimate 
of C
lit
 and that presented in this study was 0.06 and the maximum absolute difference was 
0.13. 
 The NLCD 1992 data were used to examine the potential for temporal differences 
in composite C
lit
 estimates. When NLCD 1992 data were used, 18 out of 21 LULC 
classes for the 90 watersheds were used. This difference (from the 15-LULC classes 
determined using NLCD 2001) occurred because there were more land use (LU) codes 
for some land-cover classes in NLCD 1992. Summary statistics of the composite runoff 
coefficients, C
lit
,
 
obtained using NLCD 1992 and 2001 data are listed in Table 2.2. The 
average values of the runoff coefficients for 90 watersheds derived from two LULC 
 21
datasets are the same (0.47, see Table 2.2). The median absolute difference of C
lit
 derived 
from the two LULC datasets is 0.03 with a minimum difference of 0.00 and a maximum 
difference of 0.14. The differences between C
lit
 obtained using two LULC datasets are 
plotted in Figure 2.2 (top). The authors conclude that there is no substantial difference 
between C
lit
 values derived from the 1992 and 2001 LULC datasets because the paired t-
test gives p-value (Ayyub and McCuen 2003) of 0.88, much larger than the level of 
significance 0.05 for the level of confidence 95%. 
 A sensitivity analysis was conducted to examine effect of selected C values for 
different LULC classes from literature on watershed composite runoff coefficient C
lit
.  
The minimum and maximum C values for each LULC class from literature (Table 2.1) 
were used to derive watershed-minimum and watershed-maximum C
lit
 values for each 
watershed, respectively.  The NLCD 2001 data was used for the sensitivity analysis. The 
cumulative distributions of watershed-minimum, -average, and -maximum C
lit
 values 
obtained using NLCD 2001 are shown in Figure 2.2 (top) and summary statistics of these 
C
lit
 values are listed in Table 2.2.  Values of watershed-minimum C
lit
 ranged from 0.13 to 
0.60, with median and mean values of 0.41 and 0.38, respectively (Table 2.2).  Values of 
watershed-maximum C
lit
 ranged from 0.38 to 0.68, with median and mean values of 0.58 
and 0.55, respectively (Table 2.2).  The differences of watershed-maximum and 
watershed-minimum C
lit
 values for 90 Texas watersheds ranged from 0.04 to 0.34, with 
median and mean differences of 0.14 and 0.17, respectively.  ASCE and WPCF (1960) 
and design manuals (e.g., TxDOT 2002) and textbooks (e.g., Viessman and Lewis 2003) 
give a range of C values (not a single value) for different land use types, and the range of 
published C values for the same land use is from 0.04 to 0.3, same variations of C
lit
 for 90 
 22
Texas watersheds.  This indicates uncertainty and variation of peak discharge estimation 
using the rational method. 
The amount of developed land in a watershed is a key factor governing the runoff 
from the watershed. To study the relation of the composite runoff coefficients to the 
development factor of the watersheds, statistical summaries of C
lit
 (Table 2.3) from 
NLCD 2001 data were obtained separately for the 90 watersheds, which were classified 
as developed or undeveloped (Roussel et al. 2005). The corresponding cumulative 
frequency distributions are shown in Figure 2.2 (bottom). The median value of C
lit
 
(watershed-average) for undeveloped watersheds is 0.37 and for the developed 
watersheds is 0.54. The average values of C
lit
 for undeveloped and developed watersheds 
are 0.39 and 0.54, respectively (Table 2.3). C
lit
 values of the developed watersheds are 
distinctly greater than those from undeveloped watersheds (p-value < 0.0001 from the 
pooled t-test), as shown in Figure 2.2 (bottom); the combination of LULC data and 
published C
lit
 values provides representative estimates of C
lit
 to reflect land-use 
development in a watershed. 
 
2.5 Estimation of the Back-Computed Volumetric Runoff Coefficients (C
vbc
) Using 
Observed Rainfall-Runoff Data 
The concept of a rainfall-runoff event volumetric runoff coefficient (C
v
) in 
hydrology dates to the beginning of the 20
th
 century. An example is Sherman (1932), who 
used the percent of rainfall when he introduced the unit-hydrograph method. C
v
 is defined 
as the portion of rainfall that becomes runoff during an event (Merz et al. 2006). 
Estimates of C
v
 from an individual event are usually determined by three steps: (1) 
 23
separation into single events, (2) separation of observed streamflow into base flow and 
direct runoff, and (3) estimation of event C
v
 as the ratio of direct flow or runoff volume to 
event rainfall volume (Merz et al. 2006). C
v
 is based on the integrated response of the 
watershed, that is, the transformation of rainfall volume to runoff volume.  
 French et al. (1974) evaluated C
v
 for several rural catchments in New South 
Wales; Calomino et al. (1997) computed C
v
 for 66 events for a urban watershed (91.5% 
impervious area and total drainage area of 0.019 km
2 
(1.89 ha)). For the urbanized 
watershed studied by Calomino et al. (1997), event C
v
 ranged from 0.31 to 0.88, and C
v
 
was strongly correlated to the total rainfall depth (P): C
v
 = 0.57 P
0.042
 (R
2 
= 0.96) 
(Calomino et al. 1997). The Water Planning Division of the United States Environmental 
Protection Agency (USEPA) operated the National Urban Runoff Program (NURP). This 
program had 20 projects throughout the United States to study pollutants from 76 urban 
watersheds, with drainage area ranging from 0.004 to 115 km
2
 (USEPA 1983). NURP 
researchers collected rainfall and runoff data from these watersheds, with the number of 
events ranging from 5 to 121. A runoff coefficient, R
v
 (USEPA 1983), defined as the ratio 
of runoff volume to rainfall volume, was determined for each of the NURP-monitored 
storm events. The median value of the runoff coefficients, the coefficient of variation, 
and the percent impervious area were reported for all watersheds used in the study 
(USEPA 1983). 
In this study, estimates of the volumetric runoff coefficient are called the back-
computed volumetric runoff coefficient (C
vbc
)
,
 and an individual-event C
vbc
 was obtained 
for kth storm event by the ratio of the total runoff depth, R
k
 (mm or in.), to the total 
rainfall depth, P
k
 (mm or in.), by: 
 24
 
k
kk
vbc
Peventtheforrainalltotal
Rrunoffeventtotal
C
,
,
?   (2.3) 
The study database comprised 1,600 rainfall-runoff events with observed rainfall and 
runoff data collected from 90 watersheds in Texas (Fang et al. 2007). Therefore, 1,600 
event runoff coefficients C
vbc
 were obtained using equation (2.3). Event C
vbc
 ranged from 
near 0.0 to 1.0, covering the range of possible values. The cumulative distributions of 
C
vbc
 are presented in Figure 2.3 and summary statistics are listed in Table 2.4. For the 90 
study watersheds in Texas, no substantial relation between rainfall depth and C
vbc
 was 
detected (Pearson?s correlation coefficient r = 0.2 at the 0.1 percent level of significance 
because of p-value less than 0.0001). For example, for 19 events with total rainfall depth 
less than 12.7 mm (0.5 in.), computed C
vbc
 ranged from 0.050 to 0.844. For 253 events 
with total rainfall depth between 76.2 mm (3 in.) and 101.6 mm (4 in.), the computed C
vbc
 
ranged from 0.006 to 0.982. Based on review of Figure 2.3, C
vbc
 is less than 0.1 for 13 
percent of events. Furthermore, C
vbc
 exceeds 0.9 for one percent of events. The regression 
relation between the runoff coefficient C
vbc
 and the total runoff depth (R) was C
vbc
 = 
0.374 R
0.699
. The regression explained about 76 percent of the variance between runoff 
depth and runoff coefficient and the regression coefficients were statistically significant 
at the 0.1 percent level of significance (p-value less than 0.0001). 
 C
vbc
 values calculated for all events in the same watershed varied from one event to 
another, e.g., depending on antecedent moisture condition before a rainfall event.  
Statistical parameters of the range of C
vbc
 values as the difference between maximum and 
minimum C
vbc
 values calculated for all events in the same watershed are given in Table 
2.4.  The maximum and average values of the range of event C
vbc
 in the same watershed 
is 0.97 and 0.52 (Table 2.4) for 1,600 rainfall-runoff events in 90 Texas watersheds, 
 25
respectively.  This finding is supported by previous studies by French et al. (1974) and 
USEPA (1983).  Variations of event C
vbc
 in the same watershed determined from 
observed rainfall and runoff data are much larger than ranges of published C values for 
the same land use type. 
 Watershed-average and median values of C
vbc
 were calculated from C
vbc
 values for 
all rainfall-runoff events observed in the same watershed and developed for 83 of the 90-
watershed dataset in Texas. Of the 90 watersheds in Texas, 7 were excluded; less than 4 
rainfall-runoff events were available for analysis in each of these 7 watersheds. 
Computed watershed-average C
vbc
 ranged from about 0.1 to 0.67 and from about 0.06 to 
0.76 for the watershed-median C
vbc
 (Table 2.4). These values are similar to values of C
lit
 
estimated from LULC data, which ranged from 0.29 to 0.68 (Figure 2.2 and Table 2.2). 
About 80% of the C
vbc
-median and -average values were less than 0.5. Watershed-median 
R
v
 ranged from 0.02 to 0.93 for 76 watersheds studied in the National Urban Runoff 
Program (USEPA 1983).  The average values of the watershed-average and the 
watershed-median C
vbc
 are approximately the same, 0.33 and 0.31, respectively (Table 
2.4). As shown in Figure 2.3, the cumulative frequency distributions of the watershed-
average and watershed-median of C
vbc
 values are similar and the maximum absolute 
difference of watershed-average and median C
vbc
 is less than 0.10. 
 The developed and undeveloped watershed classifications (Roussel et al. 2005) 
were used to sort the watershed-average C
vbc
 values for additional statistical analysis. The 
results are listed in Table 2.5 and cumulative distributions of C
vbc
 are shown in Figure 
2.3. The cumulative distributions are distinctly different: developed watersheds have 
greater C
vbc
 (watershed-average) in comparison to undeveloped watersheds (p-value < 
 26
0.0001 from the pooled t-test). The median values of the watershed-average C
vbc
 for 
undeveloped and developed watersheds are 0.19 and 0.37, respectively (Table 2.5). 
 For this study, the percentage of impervious area (IMP) was computed using 1992 
NLCD. Of the 90 study watersheds, 45 have percent impervious area greater than 15 
percent. The watershed-median runoff coefficients C
vbc
 and R
v
 versus percent impervious 
area for the 45 developed watersheds in Texas and the 60 watersheds from NURP are 
shown in Figure 2.4 (top). For 76 watersheds among those studied in NURP (USEPA 
1983), two separate graphs of watershed-median runoff coefficient versus percent 
impervious area were developed and reported by USEPA (1983): one graph is for the 60 
watersheds and another is for 16 watersheds. 
?The separate grouping is based on the fact that the relationship for these sites (16 
watersheds) is internally consistent and significantly different (much lower) than the bulk 
of the project results? (USEPA 1983 p. 6-60). 
 Polynomial regression lines were fit to the 60 NURP watershed data and to the 
combined watershed data for the combined group of 60 NURP and 45 Texas watersheds 
(watershed-median C
vbc
). The regression lines are displayed in Figure 2.4 (top). 
Coefficients of determination R
2
 for the two datasets are 0.79 and 0.57, respectively. 
 The regression equation obtained from the combined 60 NURP watershed data and 
the 45 Texas watershed data (watershed-median C
vbc
) is 
 ,036.0289.1275.2843.1
23
???? IMPIMPIMPC
v
 (2.4) 
where, C
v
 = volumetric runoff coefficient and IMP = percent impervious area expressed 
as a fraction (50% = 0.5) of the watershed area.  Urbanization alters the land surface and 
increases IMP.  Although other watershed parameters, e.g., basin development factor 
 27
(Sauer et al 1983), can be used to quantify the degree of urbanization, IMP was used in 
this study to correlate it to C
v
 because Kuichling (1889) concluded that runoff coefficient 
for any given watershed he studied is nearly equal to the percentage of impervious 
surface within the watershed. 
 For comparison, Urbonas et al. (1989) used watershed-median runoff coefficients 
from the group of 60 NURP watersheds and several runoff coefficients developed for 
watersheds in the Denver area to develop a polynomial regression equation between 
runoff coefficient and percent impervious area. The Urbonas et al. equation (not repeated 
here) currently (2010) is used by the Denver Urban Drainage and Flood Control District 
in its Drainage Criteria Manual (from http://www.udfcd.org/downloads/down_ 
critmanual.htm, accessed on January 10, 2010) to determine C for hydrologic soil group 
(HSG) Types C and D. The curve for the Urbonas et al. equation also is shown in Figure 
2.4 (top). Although the regression parameters differ between the three regressions shown, 
the three regression curves are similar and have a maximum absolute difference of C
v
 less 
than 0.1. 
 Values of watershed-median C
vbc
 for the 45 Texas watersheds are generally 
consistent with those from the 60 NURP watershed data (Figure 2.4). Standard deviations 
from the watershed-average C
vbc 
were calculated for Texas watersheds and shown in 
Figure 2.4 (bottom) as solid circles with thick error bars. Standard deviations and 
coefficients of variation from watershed-averages C
vbc 
ranged from 0.04 to 0.30 (Table 
2.4) and 0.15 to 1.34 for 83 Texas watersheds, respectively. 
 For the NURP data, watershed-median values and the coefficients of variation 
were reported (USEPA 1983), but the watershed-average runoff coefficients (R
v
) were 
 28
not reported. In order to examine the variation of runoff coefficient for NURP data, 
reported watershed-median values were used as watershed-averages to estimate the 
standard deviations from reported coefficient of variations, and the statistical distribution 
parameters of estimated standard deviations for NURP watersheds is listed in Table 2.4.  
Estimated maximum standard deviation from NURP watershed data is greater than 1.0, 
which is impossible if R
v
 ranged from 0.0 to 1.0, possibly because watershed-median R
v
 
were used. That NURP watershed has watershed-median R
v
 of 0.17 and the coefficient of 
variation of 6.64, which is much larger than 1.34, the maximum coefficient of variation 
for 83 Texas watersheds.  There are 15 NURP watersheds having estimated standard 
deviations greater than 0.3, the maximum standard deviation for 83 Texas watersheds 
(Table 2.4).  The median standard deviations are approximately equal for both datasets 
(Table 2.4). The NURP data (median R
v
 for the 60 watersheds) plus and minus estimated 
one standard deviation are shown in Figure 2.4 (bottom) as open squares with wide error 
bars.  Standard deviations from watershed-average C
vbc
 for the 45 Texas watersheds are 
consistently less than those from the 60 NURP watersheds (Figure 2.4). The NURP data 
cover a greater range of percent impervious area for watersheds (Figure 2.4) that are 
useful to develop the regression equation (2.4), which make the regression applicable to a 
wider range of watersheds. 
 
2.6 Estimation of Volumetric Runoff Coefficients from the Rank-Ordered Pairs of 
Observed Rainfall and Runoff Depths 
 
 29
Schaake et al. (1967) examined the rational method using observed rainfall and 
runoff data collected from 20 gaged urban watersheds in Baltimore, MD. The size of 
watershed drainage area used by Schaake et al. was 0.6 km
2
 (150 acres) or smaller. 
Schaake et al. (1967) used a frequency-matching approach to prepare their data for 
analysis. The frequency-matching approach was independently sorting observed rainfall 
intensity (average intensity over the watershed lag time) and peak runoff rate before 
computing the runoff coefficient using the rational method. That is, the rainfall intensity 
and peak runoff rate are paired on the rank order and not the event order. 
Schaake et al. (1967) concluded that the frequency of occurrence of the computed 
design peak runoff rate is the same as the frequency of the rainfall intensity selected by 
the designer. Schaake et al. (1967) developed a regression equation to relate rate-based C 
(determine from peak discharge and rainfall intensity) to the imperviousness of the 
watershed and the main channel slope. Hjelmfelt (1980) and Hawkins (1993) used a 
similar frequency matching procedure as Schaake et al. (1967), except they used rank-
ordered rainfall and runoff depths for computing actual curve numbers from historical 
rainfall-runoff events. 
For each of the 90 Texas watersheds in this study, the total rainfall depth and the 
total runoff depth were ranked independently from greatest to least. As an example, the 
rank-ordered pairs of total rainfall depth (mm) and total runoff (mm) for 13 events at 
USGS gage station 08042650 (North Trinity Basin, TX) are presented in Figure 2.5 (top 
panel). The volumetric runoff coefficient C
v
 was computed from the rank-ordered pairs of 
total runoff and rainfall depths using: 
 
,
j
j
vrj
P
R
C ?
 
(2.5) 
 30
where, C
vrj
 is the C
v
 corresponding to the total runoff depth R
j
 and the total rainfall depth 
P
j
 of the j
th
 order of rainfall-runoff pairs (subscript ?r? for C
vrj
 stands for rank-ordered). 
A plot of runoff coefficient (C
vrj
) versus the total rainfall depth was prepared for 
each watershed. For example, the plot for USGS gage station 08042650 is presented in 
Figure 2.5 (bottom panel). For most of the study watersheds, C
vrj
 increases until acquiring 
an approximate constant value. This constant value was considered as representative C
vr
 
for the watershed; for example, watershed representative runoff coefficient C
vr
 = 0.17 for 
North Trinity Basin watershed associated with USGS gage station 08042650 (Figure 2.5).  
In addition to applying Hawkins?s procedure (Hawkins 1993), i.e., asymptotic 
determination of C
vr
 from C
vrj
 versus rainfall depth, watershed C
vr
 can be estimated from 
the slope of the regression line obtained from the plots of the rank-ordered total runoff 
depth versus the rank-ordered total rainfall depth as shown in the top panel of Figure 2.5. 
For example, for USGS station 08042650, the regression equation developed from rank-
ordered runoff and rainfall data is ?Total-runoff (mm) = 0.167 ? Total-rainfall-depth 
(mm)?. Therefore, the watershed representative C
vr
 is 0.167 and equal to the slope of the 
regression equation.  For most of the study watersheds, C
vr
 obtained from both 
procedures have approximately the same value.  Statistical distribution parameters of C
vr
 
for 83 Texas watersheds (7 out of 90 watersheds were excluded because of the too few 
rainfall-event data available) are listed in Table 2.6. The mean and median values of C
vr
 
are the same and equal to 0.40.  The values of C
vr
 range from 0.10 to 0.78. The 
cumulative distributions of the runoff coefficients C
vr
 and C
lit
 are shown in Figure 2.6 
(top). The median value of the absolute differences |C
vr
 -C
lit
| is 0.14 (Table 2.6). 
 31
Examining the cumulative frequency distributions of C
lit
 and C
vr
 for the 90 study 
watersheds (Figure 2.6 top), about 80 percent of the C
lit 
values exceed the C
vr
 values. 
 Watershed-representative runoff coefficients C
vr
 were grouped into two categories: 
those from developed and those from undeveloped watersheds (Roussel et al. 2005). The 
statistical summary of C
vr
 for the two groups is listed in Table 2.7 and the corresponding 
cumulative frequency distributions are presented in Figure 2.6 (bottom). The median 
runoff coefficient C
vr
 from undeveloped watersheds is 0.24, and the median value from 
developed watersheds is 0.48. Based on this observation, C
vr
 derived from rank-ordered 
rainfall-runoff data reflects effects of watershed development, specifically the increase of 
percent impervious area (Figure 2.7). A statistical summary of the absolute differences 
|C
vr
 - C
vbc
|
 
and |C
vr
 - C
lit
| for the 45 developed watersheds in Texas is listed in Table 2.7. 
Small average and median values of absolute differences |C
vr
 - C
vbc
| indicate that C
vr
 is 
similar to C
vbc
 because both were derived from observed rainfall-runoff data. Average 
and median values of absolute differences |C
vr
 - C
lit
| are greater than those of |C
vr
 - C
vbc
| 
(Table 2.7), and indicate that C
vr
 derived from rainfall-runoff data differs from C
lit
 
derived from land-use data and published runoff coefficients (see Figures 2.2 and 2.6). 
 The watershed representative C
vr
 and watershed-median C
vbc
 from 45 developed 
Texas watersheds and runoff coefficient R
v
 from 60 NURP watersheds were plotted 
against percent impervious area (Figure 2.7). Results from these three datasets are 
consistent ? overall increasing volumetric runoff coefficient with the increase of percent 
impervious area or degree of development. A polynomial regression line was fitted to 
combined data from 60 NURP watersheds and watershed representative C
vr
 for the 45 
 32
Texas developed watersheds (Figure 2.7). R
2
 for the regression equation (2.6) = 0.57, 
which is the same as R
2
 for equation (2.4). 
 
043.0315.1940.1469.1
23
???? IMPIMPIMPC
v
 (2.6) 
The regression equations (2.4) and (2.6) were combined by averaging their coefficients to 
get a single equation for general application in Texas watersheds similar to 45 developed 
Texas watersheds. 
 
04.030.111.266.1
23
???? IMPIMPIMPC
v
 (2.7)
 
Equation (2.7) can be used to estimate C
v
 for developed (urban) watersheds based on 
impervious cover.  Equation (2.7) is plotted on Figure 2.7, which also includes a curve 
for Equation (2.6) and data points of C
vbc
, C
vr
, and R
v
 values versus percent impervious 
area (IMP).  Figure 2.7 and Equation (2.7) indicate that C
v
 is not equal to 1.0 when IMP = 
100%. This is because R
v
 estimated in the NURP study is for watersheds greater than 
0.004 km
2
 (1 acre) and C
v
 estimated in this study for watersheds greater than 0.8 km
2
 
(200 acres); therefore, Equation (2.7) does not apply to very small 100% impervious 
catchment such as a small parking lot. Further study is needed to correlate runoff 
coefficients for undeveloped watersheds (Figure 2.6) to soil types and other watersheds 
characteristics. 
 
2.7 Discussion 
Volumetric runoff coefficients, watershed-average C
vbc
 and C
vr
 for 83 Texas 
watersheds, watershed-average C
lit
 for 90 Texas watersheds, and watershed-median R
v
 
for NURP 60 watersheds (USEPA 1983), were plotted against drainage area A in km
2
 
(Figure 2.8).  Pearson?s correlation coefficients between C
vbc
, C
vr
, C
lit
, R
v
 and A (km
2
) are 
-0.20, -0.12, -0.27, and -0.26 with p-values of 0.060, 0.256, 0.009, and 0.044, 
 33
respectively.  Therefore, at the 90% confidence level, C
vbc
, C
lit
, and R
v
 has no substantial 
relation with area (Figure 2.8).  C
vr
 has no substantial relation with area only at the 70% 
confidence level.  Above statistical analyses between volumetric C values and drainage 
area indicated that there is no demonstrable relation in volumetric C with drainage area as 
Young et al. (2009) reported. This finding supports the conclusion by ASCE and WPCF 
(1960), Pilgrim and Cordery (1993), and Young et al. (2009) that the rational method 
may be applied to much larger drainage areas than typically assumed in some design 
manuals, as long as the watershed is unregulated (Young et al. 2009).  The authors do not 
advocate specifying specific limit that should be imposed on drainage area for application 
of the rational method.  It is the duty and responsibility of the end-user of the rational 
method to apply appropriate engineering judgment and experience in developing designs. 
The authors explicitly are not advocating application of the rational method for 
larger watersheds because the steady-state assumption of the rational method for design 
purposes is questionable. However, extensive data analysis (Asquith 2010) suggests that 
inherent relations between runoff coefficient and drainage area are insubstantial if time of 
concentration of a watershed is reasonably estimated for determining rainfall intensity.  
The authors support the recommendation of ASCE and WPCF in 1960 ?Development of 
data for application of hydrograph methods is usually warranted on larger areas? (ASCE 
and WPCF 1960, p. 32). 
The authors explicitly recognize that volumetric runoff coefficients might not 
have direct applicability in use of the rational method for engineering design purposes.  
Therefore, the authors did not apply the rational method and use volumetric runoff 
coefficients (C
vbc
 and C
vr
) to predict peak discharges for 1,600 events and compare 
 34
predicted and observed peak discharges.  To predict peak flows using volumetric runoff 
coefficients is inconsistent with the assertion that rate-based values for the runoff 
coefficient be used. In a subsequent paper, the authors determined rate-based rational 
runoff coefficients for these 90 Texas watersheds, and applied the rational method with 
rate-based runoff coefficients to predict peak discharges and compared predicted and 
observed peak discharges. 
 Volumetric runoff coefficients estimated from observed rainfall and runoff data 
for 83 Texas watersheds were plotted against literature-based C
lit
 (watershed-average) 
determined for the same watersheds (Figure 2.9).  Regression equations between 
watershed-average C
vbc
, C
vr
 and C
lit
 were developed and shown in Figure 2.9, with 
Pearson?s correlation coefficients r = 0.36 and 0.26 at the 95% confidence level (p-values 
of 0.0007 and 0.01), respectively.  Therefore, regression analyses indicated volumetric 
runoff coefficients determined from rainfall and runoff data are weakly correlated to 
literature-based C
lit
. 
 
2.8 Summary 
Volumetric runoff coefficients were estimated for 90 Texas watersheds using 
three different methods. The first method is estimation of literature-based runoff 
coefficients (C
lit
) using published values and GIS analysis of LULC classes to construct 
areally weighted values over a watershed. C
lit
 was obtained independently from 1992 and 
2001 NLCD and using minimum, average, and maximum published C values for 
different LULC. No substantial difference in the results of watershed-average C
lit
 was 
obtained using the 1992 or 2001 versions of the land-use data. For the study watersheds, 
 35
watershed-average C
lit
 ranged from 0.29 to 0.68 with median and average values about 
0.5.  The differences of watershed-maximum and watershed-minimum C
lit
 values for 90 
Texas watersheds ranged from 0.04 to 0.34, with median and mean differences of 0.14 
and 0.17, respectively. When C
lit
 (watershed-average) is grouped into developed and 
undeveloped watersheds, the range of C
lit
 for developed watersheds was between 0.37 
and 0.63, with a median value of 0.54. The median value of C
lit
 for developed watersheds 
exceeds that for undeveloped watersheds. This result stems from the fact that published 
runoff coefficients, even though they were not developed from observed rainfall-runoff 
measurements and instead resulted from a survey on engineering practices in 1950s, 
reflect the physical meanings of the original runoff coefficients introduced by Kuichling 
in 1889 ? the runoff coefficient is related to the percent impervious area within the 
watershed. Therefore, published runoff coefficients remain useful for engineering design 
of drainage systems. 
The second method is based on use of back-computed volumetric runoff 
coefficients (C
vbc
) from observed rainfall-runoff measurements of more than 1,600 events 
by the ratio of total runoff depth to total rainfall depth for individual storm event. Event 
volumetric runoff coefficients cover all possible values from 0.0 to 1.0 with 10% of all 
values less than 0.08 and 10% of all values greater than 0.63 (Figure 2.3).  The maximum 
and average values of the range of event C
vbc
 in the same watershed is 0.97 and 0.52 
(Table 2.4) for 1,600 rainfall-runoff events in 90 Texas watersheds, respectively. 
Watershed-average and watershed-median values of C
vbc
 and estimates of standard 
deviations were extracted. The distributions of the watershed-average and watershed-
median C
vbc
 are similar. Watershed-median values of C
vbc
 ranged from 0.06 to 0.76 with 
 36
an average of 0.31. Watershed-median values of C
vbc
 for 45 developed watersheds in 
Texas with percent imperviousness greater than 15% are consistent with median values of 
runoff coefficient R
v
 reported for 60 NURP watersheds by the USEPA. 
The third method involved the computation of runoff coefficients by the 
frequency matching procedures of observed total rainfall-runoff depths from a watershed. 
A single watershed-specific value of the runoff coefficient (C
vr
) was developed from the 
plot of rank-ordered runoff coefficients versus rainfall depths. The values of C
vr
 ranged 
from 0.10 to 0.78 with the median value 0.40. The C
vr 
values for the developed 
watersheds are consistently higher than those for the undeveloped watersheds. The 
distribution of C
vr
 is different from that of C
lit
 with about 80 percent of C
lit
 value greater 
than C
vr
 value. This result might indicate that literature-based runoff coefficients 
overestimate peak discharge for drainage design when used with the rational method. 
Runoff coefficients derived from observed rainfall and runoff data in 90 Texas 
watersheds in this study are volumetric based (ratio of total runoff and rainfall depth) and 
are useful in transforming rainfall depth to runoff depth such as is done in the curve 
number method (SCS 1963) and for watershed rainfall-runoff modeling, e.g., the 
fractional loss model (McCuen 1998, p. 493). Current runoff coefficients given in 
textbooks and design manuals are neither volumetric nor rate-based (determined from 
peak discharge and rainfall intensity) because they were not derived from observed data 
but are used for the rate-based rational method. 
Two regression equations (2.4) and (2.6) were developed using watershed-median 
C
vbc
 and watershed-representative C
vr
 data combined with median runoff coefficients R
v
 
from 60 NURP watersheds.  Coefficient of determination R
2
 for both equations are 0.57 
 37
and these equations were combined into a single equation (2.7) which can be used to 
estimate volumetric runoff coefficients for developed urban watersheds that are similar to 
the 45 developed watersheds in Texas.  The published limits on drainage area for 
application of the rational method seem to be arbitrary.  Results from this study supports 
the conclusion by ASCE and WPCF (1960), Pilgrim and Cordery (1993), and Young et 
al. (2009) that the rational method may be applied to much larger drainage areas than 
typically assumed in some design manuals. 
 
2.9 Acknowledgments 
The authors thank TxDOT project director Mr. Chuck Stead, P.E., and project 
monitoring advisor members for their guidance and assistance. They also express their 
thanks to technical reviewers Meghan Roussel and Glenn Harwell from Texas Water 
Science Center in Austin, and Fort Worth, respectively, and to three anonymous 
reviewers; the comments and suggestions greatly improved the paper.  This study was 
partially supported by TxDOT Research Projects 0?6070, 0?4696, 0?4193, and 0?4194. 
 
2.10 Notation 
The following symbols are used in this paper: 
A = watershed drainage area in hectares or acres; 
A
i
 = sub area for ith land cover classes in the watershed; 
C = runoff coefficient; 
C
i
 = literature-based runoff coefficient for ith land-cover class (Table 2.1); 
 38
C
v
 = volumetric runoff coefficient, portion of rainfall that becomes runoff, determined 
from regression equations; 
C
vbc
 
= watershed average or median back-computed volumetric runoff coefficient; 
C
k
vbc
 
= back-computed volumetric runoff coefficient for the kth event; 
C
lit
 = literature?based runoff coefficient developed from landuse data; 
C
vrj 
= runoff coefficient estimated from the ratio of jth rank-ordered runoff and rainfall 
data pair; 
C
vr 
= watershed representative runoff coefficient estimated from the distribution of ratios 
of rank-ordered runoff and rainfall; 
I = average rainfall intensity (mm/hr or in./hr) with the duration equal to time of 
concentration; 
IMP = percent of impervious area expressed as a fraction (50% = 0.5) for a watershed 
area; 
m
o
 = the dimensional correction factor (1.008 in English units, 1/360 = 0.00278 in SI 
units); 
no = total number of land cover classes in a watershed (Equation 2.2); 
P
j
 = total rainfall depth of the j
th
 order of rank runoff data series; 
P
k
 = total rainfall depth of the k
th
 event; 
Q
p
 = peak discharge or runoff rate in m
3
/s or ft
3
/s; 
R
j
 = total runoff depth of the j
th
 order of rank rainfall data series; 
R
k
 = total runoff depth of the k
th
 event; 
R
v 
= runoff coefficient as the ratio of runoff volume to rainfall volume determined by 
USEPA for the NURP data. 
 39
 
2.11 References 
ASCE, and WPCF. (1960). Design and construction of sanitary and storm sewers, 
American Society of Civil Engineers (ASCE) and Water Pollution Control Federation 
(WPCF). 
ASCE. (1992). "Hydrology and introduction to water quality." Design and construction 
of urban stormwater management systems, American Society of Civil Engineers, 
New York, 63-97. 
Asquith, W. H., Thompson, D. B., Cleveland, T. G., and Fang, X. (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, Austin, 
Texas. 
Asquith, W. H., Thompson, D. B., Cleveland, T. G., and Fang, X. (2006). "Unit 
hydrograph estimation for applicable Texas watersheds." Texas Tech University, 
Center for Multidisciplinary Research in Transportation, Austin, Texas. 
Asquith, W. H., and Roussel, M. C. (2007). "An initial-abstraction, constant-loss model 
for unit hydrograph modeling for applicable watersheds in Texas." U.S. Geological 
Survey Scientific Investigations Report 2007-5243, 82 p. [ 
http://pubs.usgs.gov/sir/2007/5243 ]. 
Ayyub, B. M., and McCuen, R. H. (2003). Probability, statistics, and reliability for 
engineers and scientists, 2nd Ed., Chapman & Hall/CRC, Boca Raton, 640 p. 
Calomino, F., Veltri, P., Prio, P., and Niemczynowicz, J. (1997). "Probabilistic analysis 
of runoff simulations in a small urban catchment." Water Science and Technology, 
36(8-9), 51-56. 
Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). Applied hydrology, McGraw 
Hill, New York. 
Cleveland, T. G., He, X., Asquith, W. H., Fang, X., and Thompson, D. B. (2006). 
"Instantaneous unit hydrograph evaluation for rainfall-runoff modeling of small 
watersheds in north and south central Texas." Journal of Irrigation and Drainage 
Engineering, 132(5), 479-485. 
Cleveland, T. G., Thompson, D. B., Fang, X., and He, X. (2008). "Synthesis of unit 
hydrographs from a digital elevation model." Journal of Irrigation and Drainage 
Engineering, 134(2), 212-221. 
 40
Dunne, T., and Leopold, L. B. (1978). Water in Environmental Planning, San Francisco, 
California: W. H. Freeman and Company. 
ESRI. (2004). "Using ArcGIS spatial analyst." Environmental Systems Research 
Institute, Inc. (ESRI), Redlands, California. 
Fang, X., Thompson, D. B., Cleveland, T. G., and Pradhan, P. (2007). "Variations of time 
of concentration estimates using NRCS velocity method." Journal of Irrigation and 
Drainage Engineering, 133(4), 314-322. 
Fang, X., Thompson, D. B., Cleveland, T. G., Pradhan, P., and Malla, R. (2008). "Time 
of concentration estimated using watershed parameters determined by automated and 
manual methods." Journal of Irrigation and Drainage Engineering, 134(2), 202-211. 
French, R., Pilgrim, D. H., and Laurenson, E. M. (1974). "Experimental examination of 
the rational method for small rural catchments." Civil Engineering Transactions, The 
Institution of Engineers, Australia, CE16(2), 95-102. 
Harle, H. K. (2002). "Identification of appropriate size limitations for hydrologic 
modeling for the state of Texas," Master Thesis, Department of Civil and 
Environmental Engineering, Texas Tech University, Lubbock, Texas. 
Hawkins, R. H. (1993). "Asymptotic determination of runoff curve numbers from data." 
Journal of Irrigation and Drainage Engineering, 119(2), 334-345. 
Hjelmfelt, A. T. (1980). "Empirical investigation of curve number technique." Journal of 
the Hydraulics Division, 106(9), 1471-1476. 
Institute of Hydrology (1976). "Water balance of the headwater catchments of the Wye 
and Severen 1970-1975." Report No. 33, Institute of Hydrology, UK. 
Kuichling, E. (1889). "The relation between the rainfall and the discharge of sewers in 
populous areas." Transactions, American Society of Civil Engineers 20, 1?56. 
Law, F. (1956). "The effect of afforestation upon the yield of water catchment areas." 
Technical report, British Association for Advancement of Science, Sheffield, 
England. 
Lloyd-Davies, D. E. (1906). "The elimination of storm water from sewerage systems." 
Minutes of Proceedings, Institution of Civil Engineers, Great Britain, 164, 41. 
Merz, R., Bl?schl, G., and Parajka, J. (2006). "Spatio-temporal variability of event runoff 
coefficients." Journal of Hydrology, 331(3-4), 591-604. 
 41
Mulholland, P. J., Wilson, G. V., and Jardine, P. M. (1990). "Hydrogeochemical response 
of a forested watershed to storms: Effects of preferential flow along shallow and deep 
pathways." Water Resources Research 26(12), 3021-3036.  
Roussel, M. C., Thompson, D. B., Fang, D. X., Cleveland, T. G., and Garcia, A. C. 
(2005). "Timing parameter estimation for applicable Texas watersheds." 0-4696-2, 
Texas Department of Transportation, Austin, Texas. 
Sauer, V. B., Thomas, W. O., Stricker, V. A., and Wilson, K. V. (1983). "Flood 
characteristics of urban watersheds in the United States." U.S. Geological Survey 
Water-Supply Paper 2207, U.S. Government Printing Office, Washington D.C. 
Schaake, J. C., Geyer, J. C., and Knapp, J. W. (1967). "Experimental examination of the 
rational method." Journal of the Hydraulics Division, ASCE, 93(6), 353?370. 
Schwab, G. O., and Frevert, R. K. (1993). Elementary Soil and Water Engineering, 
Krieger Publishing Company. 
SCS. (1963). "Section 4, Hydrology." National Engineering Handbook, The Soil 
Conservation Service (SCS), U.S. Dept. of Agriculture Washington, D.C. 
Sherman, L. K. (1932). "Stream-flow from rainfall by the unit-graph method." Eng. 
News-Rec., 108, 501-505. 
TxDOT. (2002). Hydraulic design manual, The bridge division of the Texas Department 
of Transportation (TxDOT), 125 East 11th street, Austin, Texas 78701-2483. 
Urbonas, B., Guo, C. Y. J., and Tucker, L. S. (1989). "Optimization of stormwater quality 
capture volume." Proceeding of an Engineering Foundation Conference "Urban 
Stormwater Quality Enhancement - Source Control, Retrofitting, and Combined 
Sewer Technology", Devos Platz, Switzerland, ASCE. 
USEPA. (1983). "Results of the nationwide urban runoff program, Volume I - Final 
Report." National Technical Information Service (NTIS) Accession Number: PB84-
185552, Water Planning Division, United States Environmental Protection Agency 
(USEPA), Washington D.C. 
Viessman, W., and Lewis, G. L. (2003). Introduction to hydrology, 5th Ed., Pearson 
Education, Upper Saddle River, N.J., 612. 
Wanielista, M. P., and Yousef, Y. A. (1993). Stormwater Management, John Wiley & 
Sons, Inc., New York. 
Young, C. B., McEnroe, B. M., and Rome, A. C. (2009). "Empirical determination of 
rational method runoff coefficients." Journal of Hydrologic Engineering, 14, 1283. 
 42
Table 2.1 Runoff Coefficients (C) Selected for Various Land-Cover Classes from NLCD 
2001. 
 
NLCD  
classification 
NLCD classification description    C Land use or description in the source 
 
21 
 
Developed, Open Space 
 
0.4
(1),(2)
 
 
Residential: single family areas (0.3-0.5) 
 
22 Developed , Low Intensity 0.55
(3)
 50 % of area impervious (0.55) 
23 Developed,  Medium Intensity 0.65
(3)
 70% of area impervious (0.65) 
24 Developed , High  Intensity 0.83
(2)
 Business: downtown areas (0.7-0.95) 
31 Barren Land 0.3
(2),(7)
 Sand or sandy loam soil, 0-5% (0.15-0.25); 
black or loessial soil, 0-5% (0.18-0.3); 
heavy clay soils; shallow soils over 
bedrock: pasture (0.45) 
 
41 Deciduous  Forest 0.52
(4)
 Deciduous forest (Tennessee) (0.52) 
42 Evergreen Forest 0.48
(5),(6)
 Forest (UK) (0.28-0.68); Forest (Germany) 
(0.33-0.59) 
 
43 Mixed Forest 0.48
(5),(6)
 Forest (UK) (0.28-0.68); Forest (Germany) 
(0.33-0.59) 
 
52 Shrub/Scrub 0.3
(7)
 Woodland, sandy and gravel  soils (0.1); 
loam soils (0.3); heavy clay soils (0.4); 
shallow  soil on rock (0.4) 
 
71 Grassland/Herbaceous 0.22
(3)
 Pasture, grazing HSG A (0.1); HSG B 
(0.2); HSG C (0.25); HSG D (0.3) 
 
81 Pasture/Hay 0.35
(7)
 Pasture, sandy and gravel  soils (0.15); 
loam soils (0.35); heavy clay soils (0.45); 
shallow  soil on rock (0.45) 
 
82 Cultivated Crops 0.4
(7)
 Cultivated, sandy and gravel  soils (0.2); 
loam soils (0.4); heavy clay soils (0.5); 
shallow  soil on rock (0.5) 
Sources: - 
(1)
 ASCE (1992), 
(2) 
TxDOT (2002), 
 (3) 
Schwab and Frevert (1993),
 (4) 
Mulholland et al. (1990) ,  
    
(5) 
Law (1956), 
(6) 
Hydrology (1976), 
(7) 
Dunne and Leopold (1978)  
Note Numbers in parenthesis are ranges for runoff coefficients given in the source (literature). 
 HSG = hydrologic soil group 
  
 43
Table 2.2 Statistical Summary of Average C
lit
 Using NLCD 1992 and Watershed-
Minimum, -Average and -Maximum C
lit
 Using NLCD 2001. 
Statistical 
distribution 
parameters 
Watershed-average 
1
 C
lit 
using NLCD 
1992 
(1) 
C
lit
using NLCD 2001 
Absolute 
difference 
(1) ? (3) 
Watershed-
minimum 
1
 
(2) 
Watershed-
average 
(3) 
Watershed-
maximum
 1
 
(4) 
Minimum 0.32 0.13
0.29 0.38
0.00 
Maximum 0.68 0.60 
0.63 0.68 
0.14 
25% Quartile  0.40 0.24 
0.38 0.48
0.02 
Median  0.47 0.41 
0.50 0.58 
0.03 
75% Quartile 0.52 0.50 
0.55 0.60
0.05 
Average 0.47 0.38
0.47 0.55 
0.04 
Standard deviation 0.09 0.14 
0.10 0.09
0.03 
1
 Watershed-average, -minimum, and ?maximum C
lit
 values were derived using mean, minimum, and 
maximum C values for each LULC from literature (Table 2.1), respectively. 
 
 
 
 
Table 2.3 Statistical Summary of Watershed-average C
lit
 
Using NLCD 2001 for 
Developed and Undeveloped Watersheds. 
  Undeveloped Developed 
Minimum 0.29 0.37
Maximum 59 0.63
25% Quartile 0.33 0.52 
Median 0.37 0.54
75% Quartile 0.43 0.58 
Average  0.39 0.54 
Standard deviation 0.07 0.06 
 
 
 44
 
Table 2.4?Statistical Summary of C
vbc
 
and R
v
 from NURP (USEPA 1983). 
  
C
vbc
 
All events 
Range of 
C
vbc
  1
 Watershed- 
Median C
vbc
 
Watershed- 
average C
vbc
 
Standard 
deviation 
C
vbc
 
Standard 
deviation R
v
  
2
 
Minimum 0.00 
 
0.02 0.06 0.10 0.04 0.02 
Maximum 0.99 
 
0.97 0.76 0.67 0.30 1.13 
25% Quartile 0.17 
 
0.37 0.17 0.20 0.12 0.10 
Median 0.29 
 
0.53 0.30 0.31 0.16 0.16 
75% Quartile 0.47 
 
0.66 0.42 0.42 0.19 0.28 
Average  0.33 
 
0.52 0.31 0.33 0.16 0.21 
Standard 
deviation 0.21 
 
0.22 0.17 0.15 0.05 0.18 
1
 Range of C
vbc
 is difference between maximum and minimum C
vbc
 values calculated for all events in the 
same watershed.  
2
 Standard deviations for R
v
 (USEPA 1983) were estimated from median values and 
coefficients of variation of R
v
 for 60 NURP watersheds. 
 
 
Table 2.5 Statistical Summary of Watershed-average C
vbc
 for Developed and 
Undeveloped Watersheds. 
  Undeveloped Developed 
Minimum 0.10 0.17
Maximum 0.56 0.67
25% Quartile 0.15 0.30 
Median 0.19 0.37
75% Quartile 0.36 0.48 
Average  0.24 0.39 
Standard deviation 0.12 0.13 
?
?
?
Table 2.6 Statistical Summary of C
vr
 and Absolute Difference (ABS) of C
vr
 with 
Watershed-average C
lit
 for 83 Texas Watersheds. 
  C
vr
 ABS (C
vr 
- C
lit
) 
Minimum 0.10 0.01 
Maximum 0.78 0.40
25% Quartile 0.24 0.07 
Median 0.40 0.14
75% Quartile 0.52 0.24 
Average 0.40 0.16
Standard deviation 0.18 0.11 
?
 45
Table 2.7 Statistical Summary of C
vr
 for Developed and Undeveloped Watersheds and 
Absolute Difference (ABS) of C
vr
 with Watershed-average C
vbc
 and C
lit
 for Developed 
Watersheds.?
   Undeveloped Developed ABS (C
vr
 - C
vbc
) ABS (C
vr 
- C
lit
) 
Minimum 0.10 0.20 0.00 0.01 
Maximum 0.70 0.74 0.45 0.38
25% Quartile 0.18 0.34 0.02 0.06 
Median 0.24 0.48 0.04 0.12
75% Quartile 0.44 0.60 0.12 0.18 
Average  0.31 0.46 0.08 0.14 
Standard deviation 0.18 0.15 0.10 0.10 
?
 46
 
Fig. 2.1 Map showing the U.S. Geological Survey streamflow-gaging stations (dots) 
associated with the watershed locations in Texas. 
 47
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Composite Runoff Coefficient,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Undeveloped
Developed
1992 (watershed-average)
2001 ( watershed-minimum)
2001 (watershed-average)
2001 (watershed-maximum)
C
lit
 
Fig. 2.2 Cumulative distributions of C
lit
 obtained using NLCD 1992 and NLCD 2001 
(top) and C
lit
 using NLCD 2001 for developed and undeveloped watersheds (bottom). 
 48
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Watershed-average,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Runoff Coefficient,
Undeveloped
Developed
Watershed-average
Wat ershed-median
All rainfall-runoff events
C
vbc
C
vbc
 
Fig 2.3. Cumulative distributions of C
vbc
 for watershed-average, watershed-median, and 
all rainfall-runoff events (top) and watershed-average C
vbc
 for developed and 
undeveloped watersheds (bottom). 
 
 49
-0.8
-0.4
0.0
0.4
0.8
1.2
R
u
n
o
f
f
 
c
o
e
f
f
i
c
i
e
n
t
0.0
0.2
0.4
0.6
0.8
1.0
R
u
n
o
f
f
 
c
o
e
f
f
i
c
i
e
n
t
0 10 20 30 40 50 60 70 80 90 100
Percent impervious
0 10 20 30 40 50 60 70 80 90 100
Texas (45 watersheds) NURP (60 watersheds)
Texas (45 watersheds)
NURP (60 watersheds)
Denver (HSG C & D)
NURP
NURP + Texas
 
Fig. 2.4 Volumetric runoff coefficients from different studies versus percent impervious 
area including regression lines (top) and runoff coefficients with one standard deviations 
for 45 Texas watersheds and 60 NURP watersheds (bottom). 
 50
0.00
0.05
0.10
0.15
0.20
0.25
C
v
r
j
0.0
0.2
0.4
0.6
0.8
1.0
1.2
T
o
t
a
l
 
R
u
n
o
f
f
 
(
i
n
c
h
e
s
)
0 1 2 3 4 5
Total Rainfall (inches)
0 1 2 3 4 5
C
vr
Runoff = 0.167 x Rainfall
R
2
= 0.81
 
Fig. 2.5 The rank-ordered pairs of observed runoff and rainfall depths (top) and runoff 
coefficients derived from the rank-ordered pairs of observed runoff and rainfall depths 
versus total rainfall depths (bottom). All data presented are for the USGS gage station 
08042650 (North Trinity Basin in Texas). 
 51
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
D
i
s
t
r
i
b
u
t
i
o
n
 
(
%
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Runoff Coefficient,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Runoff Coefficient
Undeveloped
Developed
C
vr
C
lit
 using NLCD 2001
C
vr
 
Fig. 2.6 Cumulative distributions of C
vr
 and C
lit 
(top) and C
vr
 for developed and 
undeveloped watersheds (bottom). 
 52
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
u
n
o
f
f
 
c
o
e
f
i
c
i
e
n
t
0 10 20 30 40 50 60 70 80 90 100
Percent impervious
C
vr
C
vbc
R
v 
(NURP) Equation (2.6) Equation (2.7)
 
Fig. 2.7 Runoff coefficients C
vr
, C
vbc
 (watershed-average), and runoff coefficients R
v
 
from 60 NURP watersheds versus percent impervious area including lines for the 
regression equation (2.6) and the regression equation (2.7). 
 
 
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
u
n
o
f
f
 
c
o
e
f
f
i
c
i
e
n
t
0.01 0.1 1 10 100
Watershed area (mi
 2
)
C
vbc
C
vr
C
lit
R
v
(NURP)
 
Fig. 2.8 Runoff coefficients C
lit
 (watershed-average), C
vbc
 (watershed-average), C
vr
, and 
R
v
 plotted against watershed area (km
2
). 
 53
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
u
n
o
f
f
 
c
o
e
f
f
i
c
i
e
n
t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(average)
Watershed-average C C
vr
1:1 line
C
lit
C
vbc
 = 0.53 C
lit 
+ 0.075  (R
2
 = 0.13)
C
vr
 = 0.49 C
lit 
+ 0.161  (R
2
 = 0.07)
vbc
 
Fig. 2.9 Runoff coefficients C
vbc
 (watershed-average) and C
vr
 plotted against C
lit
 
(watershed-average) for 83 Texas watersheds. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 54
Chapter 3. Rate-based Estimation of the Runoff Coefficients for Selected 
Watersheds in Texas 
 
 
3.1 Abstract 
 The runoff coefficient, C, of the rational method is an expression of rate 
proportionality between rainfall intensity and peak discharge. Values of C were derived 
for 80 developed and undeveloped watersheds in Texas using two distinct methods. First, 
the rate-based runoff coefficients, C
rate
, were estimated for each of 1,500 rainfall-runoff 
events. Second, the frequency-matching approach was used to extract a runoff 
coefficient, C
r
, for each watershed. Using the 80 Texas watersheds, comparison of the 
two methods shows that about 75 percent of literature-based runoff coefficients are 
greater than C
r
 and the watershed-median C
rate
, but for developed watersheds with more 
impervious cover, literature-based runoff coefficients are less than C
r
 and C
rate
. An 
equation applicable to many Texas watersheds is proposed to estimate C as a function of 
impervious area. 
 
3.2 Introduction 
The search for a reliable method for estimation of peak discharges for small and 
ungaged undeveloped (rural) watersheds has led to various engineering-design methods 
(French et al. 1974). These various methods often are applicable for developed (suburban 
 55
to urban) watersheds (Chow et al. 1988). The rational method likely is the most often 
applied method used by hydraulic and drainage engineers to estimate design discharges 
for small watersheds. These design discharges are used to size a variety of drainage 
structures for small undeveloped and developed watersheds throughout the United States 
(Viessman and Lewis 2003). 
The rational method (Kuichling 1889) computes the peak discharge, Q
p
, (in m
3
/s 
in SI units or ft
3
/s in English units) by: 
CIAmQp
0
?  (3.1) 
where C is the runoff coefficient (dimensionless), I is the rainfall intensity (mm/hr or 
in/hr) over a critical period of storm time (typically taken as the time of concentration, 
T
c
), A is the drainage area (hectares or acres), and m
o
 is the dimensional correction factor 
(1/360 = 0.00278 in SI units, 1.008 in English units). Steady-state conditions are needed 
for the application of the rational method (French et. al 1974). From inspection of the 
equation, it is evident that C is an expression of rate proportionality between rainfall 
intensity and peak discharge (flow rate).  The theoretical range of values for C is between 
0 and 1.  The typical ?whole watershed? C values, that is, C values representing the 
integrated effects of various surfaces in the watershed and other watershed properties, are 
listed for different general land-use conditions in various design manuals and textbooks. 
Examples of textbooks that include tables of C values are Chow et al. (1988) and 
Viessman and Lewis (2003). Published C values, C
lit
, were sourced from American 
Society of Civil Engineers (ASCE) and the Water Pollution Control Federation (WPCF) 
in 1960 (ASCE and WPCF 1960). The C
lit
 values were obtained from a response survey, 
which received ?71 returns of an extensive questionnaire submitted to 380 public and 
 56
private organizations throughout the United States.? No justification based on analyses of 
observed rainfall intensity and peak discharge data for the C
lit
 values is apparent in ASCE 
and WPCF (1960)?a few analyses of observed relations between rainfall intensity and 
peak discharge were considered by Kuichling (1889). In short, the authors of this paper 
conclude that C
lit
 values appear to be heuristically determined, and therefore a 
comparison of rate-based C values derived from observed rainfall and runoff data to the 
C
lit
 values is made in this study. 
Estimation of reliable values of C presents a substantial difficulty in the rational 
method and a major source of uncertainty in many small watershed projects (Pilgrim and 
Cordery 1993). Furthermore, the concept of ?runoff coefficient? for a watershed is a term 
fraught with ambiguities. The volumetric runoff coefficient, C
v
, is the ratio of total runoff 
to rainfall (Merz et al. 2006; Dhakal et al. 2012). Because C as an expression of rate 
proportionality (equation 3.1), such a coefficient is termed a rate-based runoff coefficient, 
C
rate
. It is important to stress that although the C
lit
 found in ASCE and WPCF (1960) are 
intended for use in the rate-based rational method, C
lit
 appear not to be derived from any 
observed data. 
When observed rainfall and runoff data are available, the C
rate
 is computed through the 
rational method (Pilgrim and Cordery 1993) as: 
AtIm
Q
C
av
p
rate
)(
0
?  ,        (3.2)  
where I(t
av
) is the average rainfall intensity over a t
av
 time period, which is the rainfall 
intensity averaging time period (a critical time period) rather than the entire rainfall event 
duration (Kuichling 1889; Schaake et al. 1967). The t
av
 should be a period of time for a 
 57
storm that contributes runoff that produces the observed Q
p
. Kuichling (1889) argues 
against using the entire rainfall event duration, t
w
, to obtain average rainfall intensity, I
w
, 
because I
w
 is in general not appropriate, which results from rainfall durations of real 
storms often being greater than the characteristic time of small watersheds considered by 
Kuichling. Thus, a lasting contribution of Kuichling (1889) was the introduction of the 
concept of the time of concentration, T
c
, of a watershed. This time also is termed a 
?critical storm duration? because this uses an average rainfall intensity that produces 
reliable peak discharge estimates. The T
c
 is influenced in part by drainage area, which is a 
major criterion used to assess the applicability of the rational method (Chow et al. 1988). 
In particular, TxDOT (2002) recommends the use of the rational method for watersheds 
with very small drainage areas of < 0.8 km
2
 (200 acres). 
Other investigators have reported C values derived from analysis of observed rainfall and 
runoff data for various watersheds throughout the world. Schaake et al. (1967) examined 
the rational method using experimental rainfall and runoff data collected from 20 small 
urban watersheds of < 0.6 km
2
 (150 acres) in Baltimore, MD. Those authors used 
watershed lag time to compute average rainfall intensity and used a frequency-matching 
approach. 
Hotchkiss and Provaznik (1995) estimated C
rate
 for 24 rural watersheds in south-
central Nebraska using event-paired and frequency-matched data. Young et al. (2009) 
estimated C
rate
 for 72 rural watersheds in Kansas with drainage areas of < 78 km
2
 (30 
mi
2
) for different return periods. The peak discharge for each return period was estimated 
using annual peak frequency analysis of the gaged peak discharges and rainfall intensity 
obtained from rainfall intensity-duration-frequency tables (Young et al. 2009). 
 58
In this study, two methods were used to estimate C
rate
 for 80 selected watersheds 
in Texas. Both methods rely on analysis of observed rainfall and runoff data. First, C
rate
 
was estimated using equation (3.2); the I(t
av
) was computed as the maximum intensity for 
a moving time window of duration T
c
 before and up to the time to peak, T
p
. The T
c
 was 
derived for the study watersheds using the Kerby-Kirpich approach (Roussel et al. 2005; 
Fang et al. 2008). A total of about 1,500 rainfall-runoff events from 80 Texas watersheds 
were analyzed to determine event-specific, watershed-median, and watershed-mean C
rate
 
values. Second, the frequency-matching approach (Schaake et al., 1967) was used to 
derive a representative C referred to as C
r
 for each of the 80 watersheds. The study also 
compares C values from the two different methods and those published in the literature. 
Finally, an equation of C as a function of the percentage of impervious area is proposed 
for the 80 watersheds. 
 
3.3 Study Area and Rainfall-Runoff Database 
Watershed data from a larger dataset accumulated by researchers from the U.S. 
Geological Survey (USGS) Texas Water Science Center, Texas Tech University, 
University of Houston, and Lamar University (Asquith et al. 2004) are used for this 
study. Ten watersheds out of about 90 represented by USGS streamflow-gaging stations 
in the source database (Asquith et al. 2004) are not used in this study because less than 
four rainfall and runoff events were recorded for each of these 10 watersheds. The 
locations of 80 USGS streamflow-gaging stations representing 80 watersheds in Texas 
are shown in Figure 3.1. Incidentally, these data also are used by Asquith and Roussel 
(2007), Cleveland et al. (2006), Fang et al. (2007, 2008), and Dhakal et al. (2012). The 
 59
rainfall-runoff dataset consists of about 1,500 rainfall-runoff events which occurred 
between 1959 and 1986. The number of events available for each watershed varied from 
4 to 50 events with median and mean values of 16 and 19 events, respectively. Values of 
rainfall depths for about 1,500 events ranged from 3.56 mm (0.14 in.) to 489.20 mm 
(19.26 in.), with median and mean values of 57.66 mm (2.27 in.) and 66.8 mm (2.63 in.), 
respectively. 
The drainage area of study watersheds range from approximately 0.2?320 km
2
 
(0.1?123.6 mi
2
); the median and mean values are 17.0 km
2
 (6.6 mi
2
) and 37.3 km
2
 (14.4 
mi
2
), respectively.  The stream slope of study watersheds range from approximately 
0.0022?0.0196 dimensionless; the median and mean values are 0.0076 and 0.0081, 
respectively.  The percentage of impervious area (IMP) of study watersheds range from 
approximately 0 to 73%; the median and mean values are 18.0 and 28.2, respectively. 
There has been discussion in the literature concerning the size of watersheds for which 
the application of the rational method is appropriate. For application of the rational 
formula, Kuichling (1889, pages 40?41) stated: ?For large areas, on the other hand, a 
more elaborate analysis becomes necessary in order to find under what condition the 
absolute maximum discharge will occur, although the method of procedure above 
indicated will remain the same.? Kuichling (1889) did not suggest a specific large area 
limit. ASCE and WPCF (1960, p. 32), made the following statement when the rational 
method was introduced for design and construction of sanitary and storm sewers: 
?Although the basic principles of the rational method are applicable to large drainage 
areas, reported practice generally limits its use to urban areas of less than 5 square miles.?  
Pilgrim and Cordery (1993, p. 9.14) explained that the rational method is one of the three 
 60
methods widely used to estimate peak flows for small to medium sized basins, and wrote 
?it is not possible to define precisely what is meant by ?small? and ?medium? sized, but 
upper limits of 25 km
2
 (10 mi
2
) and 550 km
2
 (200 mi
2
), respectively, can be considered as 
general guides.? Young et al. (2009) stated that the rational method might be applied to 
much larger drainage areas than typically assumed in some design manuals, as long as the 
watershed is unregulated. Results of this study will further indicate that there is no 
demonstrable relation between runoff coefficient and drainage area. 
For any watershed (regardless of its size), some of the attributes necessary to 
apply the Kuichling method are the time of concentration (T
c
), main channel length (L
c
), 
and channel slope (S
c
). For each of the 80 Texas watersheds, a geospatial database was 
developed by Roussel et al. (2005) containing L
c
 and S
c
 for each watershed, along with 
drainage area, basin width, longitude, latitude, and 39 other watershed characteristics. For 
this paper, L
c 
and S
c
 are used to estimate time of concentration T
c
 by Kirpich (1940) for 
channel flow plus travel time for overland flow using Kerby (1959). A combination of 
the methods of Kirpich (1940) and Kerby (1959) is discussed by Roussel et al. (2005) and 
Fang et al. (2008). The Kirpich equation (1940) was developed from the Soil 
Conservation Services (SCS) data for rural watersheds with drainage areas less than 0.45 
km
2
 and is presented below: 
,978.3
385.077.0 ?
?
ccc
SLT  (3.3) 
where, L
c
 is the channel length in km and S
c
 is the channel slope in m/m. Fang et al. 
(2007, 2008) demonstrated that, for watersheds with relatively large drainage areas (more 
than 50 km
2
), the Kirpich equation provides as reliable an estimate of T
c
 as the other 
empirical equations developed for large watersheds and the SCS velocity method 
 61
(Viessman and Lewis 2003). The T
c
 estimated using the Kirpich equation reasonably 
approximate the average T
c
 estimated from observed rainfall and runoff data (Fang et al. 
2007). The T
c
 for the study watersheds ranged from 1.1 hours to 16.7 hours with median 
and mean values of 2.8 hours and 3.8 hours respectively. 
 Each of the 80 Texas watersheds was previously classified as either developed or 
undeveloped (Roussel et al. 2005, Cleveland et al. 2008). The classification scheme of 
developed and undeveloped watersheds is consistent with the characterization of 
watersheds in more than 220 USGS reports of Texas data from which the original data 
for the rainfall and runoff database were obtained (Asquith et al. 2004). Although this 
binary classification seems arbitrary, it does take into account the uncertainty in 
watershed development conditions for the time period of available data (Asquith and 
Roussel 2007). This binary classification was used by Asquith et al. (2006) in a 
regionalization study of unit hydrographs for the Texas watersheds (Asquith et al., 2004). 
Using the binary classification scheme for the 80 Texas watersheds, there are 44 
developed watersheds in four metropolitan areas in Texas (Austin, Dallas, Fort Worth, 
and San Antonio) and 36 undeveloped watersheds. The 36 undeveloped watersheds 
consist of 16 watersheds near these four cities and 20 rural watersheds. 
 
3. 4 Runoff Coefficients Estimated from Event Rainfall-Runoff Data 
 
Rate-based C Derived for Individual Rainfall-Runoff Events 
 62
For this study, the intensity I in equation (3.2) is the maximum rainfall intensity 
before the time to peak, T
p
, of a runoff hydrograph and is calculated as the maximum 
intensity found by a moving time window of duration t
av
 through the 5-minute interval 
rainfall hyetograph for the storm event. For data processing, only the largest Q
p
 for each 
storm event (in the case of multiple peaks in the overall hydrograph) was used. 
The computation of C
rate
 is illustrated by example. In Figure 3.2, the I and C
rate
 values are 
shown as a function of t
av
 for two storm events gaged by the USGS:  one on 09/22/1969 
at USGS streamflow-gaging station 08048550 Dry Branch at Blandin Street, Fort Worth, 
Texas (hereinafter Dry Branch) and the second on 04/25/1970 at 08058000 Honey Creek 
near McKinney, Texas (hereinafter Honey Creek). As shown in Figure 3.2, as t
av
 
increases I decreases and C
rate
 increases. For example, for the storm event at Dry Branch, 
as t
av
 increases from 5 minutes to 3.5 hours, I decreases from about 119 mm/hr (4.7 in/hr) 
to about 16.3 mm/hr (0.64 in/hr) and C
rate 
increases from 0.04 to 0.30. For the Dry 
Branch and Honey Creek watersheds, estimated T
c
 is 1.8 hours and 1.5 hours, 
respectively (Figure 3.2).  These T
c
 values are derived from the Kerby-Kirpich method 
(Roussel et al. 2005; Fang et al. 2008); the corresponding C
rate
 values for the Dry Branch 
and Honey Creek watersheds are 0.23 and 1.70, respectively (Figure 3.2). Following the 
analysis leading to Figure 3.2, one C
rate
 value was determined using I corresponding to a 
moving time window T
c
 for each of about 1,500 events from the 80 Texas watersheds. 
The occurrence of C
rate
 > 1 is related to unknown errors in T
c
 used to calculate I (see 
Figure 3.2), rainfall characteristics, fundamental measurement errors of rainfall and 
runoff data, or other unusual hydrologic factors. Several studies (French et. al 1974; 
Pilgrim and Cordery 1993; Young et al. 2009) have shown that values of C
rate
 greater 
 63
than 1 are possible when rate-based C was determined from observed peak flow rate and 
computed rainfall intensity over a critical period of storm time. If the time of 
concentration is exactly correct for the watershed, and if the rainfall were spatially and 
temporally homogeneous and isolated (no preceding rainfall), then C
rate
 would have to be 
less than or equal to 1, but the rainfall normally varies in time and in space. Averaging 
temporal and spatial variability of rainfall leads to lower rainfall intensities and 
consequently lower predicted peak discharge values.  Thus, when using an average 
rainfall as a predictor of Q
p
, the C value will necessarily be higher than if the rainfall 
were truly uniform in space and time. 
Most of the C
v
 values derived from about 1,500 rainfall events in Texas 
watersheds (Dhakal et al. 2012) are between 0 and 1.  Rate-based runoff coefficient C
rate
 
and volumetric-based runoff coefficient C
v
 are defined differently and were determined 
using different approaches from observed rainfall and runoff data. The use of rate-based 
runoff coefficients is appropriate if one wants to determine peak discharge using the 
rational method, and volumetric runoff coefficient can be used to estimate fractional 
rainfall loss using the constant fraction method (McCuen 1998) or for hydrologic 
modeling and runoff volume design purposes for a stormwater quality control basin 
(USEPA 1983; Guo and Urbonas 1996; Mays 2004). 
Frequency distributions of C
rate
 values computed for about 1,500 events from the 
80 Texas watersheds are shown in Figure 3.3, and summary statistics are listed in Table 
3.1. Recalling that T
c
 was computed using the Kerby-Kirpich approach, for events where 
T
p
 < T
c
, the mean value of C
rate
 is 0.31. In contrast, for events where T
p 
? T
c
, the mean 
value of C
rate
 is 0.50 (Table 3.1). From inspection of the frequency distributions of C
rate
 
 64
shown in Figure 3.3, estimates of C
rate
 are significantly greater (Welch-Satterthwaite t-
test, p-value < 0.0001) for storm events when T
p
 ? T
c
 than those from events when T
p
 < 
T
c
. Therefore, values of C
rate
 are dependent on the duration of rainfall event, which 
supports the idea proposed by Kuichling (1889). Kuichling?s idea is that as T
c
 is reached, 
discharge for a watershed becomes a maximum (a peak) because the entire area is 
contributing runoff to the outlet. For cases considered in this research, the maximum 
value of Crate sometimes exceeded 1 (up to 4.48 when T
p
 ? T
c
). For 124 of about 1,500 
events, the calculated C
rate
 is greater than 1. 
 
Watershed Mean and Median Runoff Coefficients 
Watershed mean and median values of C
rate
 
for the 80 Texas watersheds were 
calculated for all observed storms in the same watershed (regardless of whether T
p
 was 
less than or greater than T
c
). Computed watershed means of C
rate
 range from 0.07 to 1.79, 
and the watershed medians of C
rate
 range from 0.07 to 1.73 (Table 3.2). The average 
values of the individual watershed mean and median C
rate
 are 0.44 and 0.40, respectively 
(Table 3.2). Standard deviations from the watershed-means C
rate
 range from 0.03 to 0.87. 
Frequency distributions of the watershed mean and median of C
rate
 are shown in Figure 
3.3; these distributional locations (means) are similar at a significance level of 0.01 
(paired t-test, p-value = 0.04). 
The amount of developed land in a watershed influences various runoff 
characteristics of a watershed. To study the relation between C
rate
 and the binary 
watershed development classification, statistical summaries of watershed median C
rate
 
were computed (Table 3.3). The watershed median C
rate
 for developed watersheds range 
 65
from 0.17 to 1.73, and for undeveloped watersheds, the watershed median C
rate
 ranged 
from 0.07 to 0.73. Of the developed watersheds, only two had a watershed-median C
rate
 > 
1. The corresponding frequency distributions of watershed-median C
rate
 for developed 
and undeveloped watersheds are shown in Figure 3.3B. The watershed-median value of 
C
rate
 for developed watersheds is 0.40 and for undeveloped watersheds watershed-median 
value of C
rate
 is 0.20 (Table 3.3). The C
rate
 values of the developed watersheds are 
significantly larger than those from the undeveloped watersheds (Figure 3.3) as 
anticipated (Welch-Satterthwaite t-test, p-value <0.0001). 
 
3.5 Runoff Coefficients from the Frequency-Matching Approach 
The frequency-matching approach assumes return periods of rainfall and runoff 
events are the same (Hawkins 1993). Specifically, the T-year storm produces the T-year 
peak discharge. An alternative viewpoint is that the frequency-matching approach forces 
the largest rainfall intensity to produce the largest peak discharge within a given dataset. 
The authors observed that this assumption is implicit in circumstances of practical 
application of the rational method. Many design engineers assume that the T-year storm 
produces the T-year discharge. Although not a physical requirement, this assumption 
generally is appropriate in small watersheds. 
The maximum rainfall intensities and the observed peak discharges were 
independently ranked from largest to smallest for each of the 80 Texas watersheds. The 
frequency-matched C was computed from the rank-ordered pairs of the observed peak 
discharge and the maximum rainfall intensity for each storm event using: 
 66
AIm
Q
C
j
pj
rj
0
?  ,         (3.4)  
where C
rj
 is the runoff coefficient corresponding to the maximum rainfall intensity I
j
, the 
observed peak discharge Q
pj
 of the j
th
 rank-order of I
max
-Q
p
 data pairs, and drainage area 
A. A plot of runoff coefficients, C
rj
, versus the maximum rainfall intensity was prepared 
for each watershed. For most of the watersheds, the C
rj
 increases until acquiring an 
approximate constant value as judged by an analyst. This constant value is referred to as 
C
r
. For example, the plot for USGS streamflow-gaging station 08042650 North Creek 
Surface Water Station 28A near Jermyn, Texas (hereinafter North Creek near Jermyn) is 
presented in Figure 3.4A; C
r
 = 0.20 for this watershed. 
The C
r
 also can be estimated from the slope of the regression line obtained from the plots 
of the rank-ordered Q
pj
/(0.00278*A) or Q
pj*
 values versus the rank-ordered I
j
. For 
example, the regression equation for North Creek near Jermyn is Q
pj*
 (m
3
/s/ha) = 0.19 * I
j
 
(mm/hr) [Figure 3.4B]. The slope of this (and other similar equations) also is 
representative of C
r
. Using the slope of the line, C
r
 = 0.19 for the North Creek near 
Jermyn. For most of the 80 Texas watersheds, C
r
 values obtained using analyst judgment 
or the regression slope method have approximately the same value, and C
r
 values ranged 
from 0.10 to 1.2 (one outlying C
r
 value of 1.2 was the only C
r
 value > 1). The mean and 
medians for C
r
 were 0.42 and 0.37, respectively. A statistical summary of C
r
 is listed in 
Table 3.4. 
 
Comparison C
r
 to Watershed Median C
rate
 and Literature-based C
lit 
 67
For the 80 Texas watersheds, distributions of C
r
 and watershed median C
rate
 
follow the same shape (Figure 3.5A) and are not statistically different at the 0.05 
significance level (paired t-test, p-value = 0.27). The difference between C
r
 and 
watershed median C
rate
 for each watershed was calculated, and a statistical summary of 
the differences is listed in Table 3.4. The median value of C
r
 minus C
rate
 differences is 
0.03 (Table 3.4). The minimum and maximum differences are -0.53 and 0.52 (Table 3.4), 
respectively, and quartiles of the differences between C
r
 and watershed-median C
rate
 are 
considered acceptably small (less than 0.06). About 74% of C
r
 and watershed-median 
C
rate
 values differ less than ?0.1 (Table 3.4). 
The frequency distributions of literature-based C
lit
 from land-use data (Dhakal et 
al. 2012) for developed and undeveloped watersheds are shown in Figures 3.5A and 
3.5B. The differences between C
r
 and C
lit
 or C
rate
 and C
lit
 are larger than the differences 
between C
r
 and C
rate
. When the runoff coefficient is less than 0.55, C
lit
 is greater than C
r
, 
otherwise C
lit
 is smaller than C
r 
(Figure 3.5A). About 75% of C
lit
 values are greater than 
C
r
 (Figure 3.5A). For typical applications of the rational method in urban (developed) 
watersheds, using the typically smaller C
lit
 value for the watershed would underestimate 
Q
p
 for design purposes. The difference between watershed-median C
r
 and C
lit
 or C
rate
 and 
C
lit
 for each watershed was calculated, and statistical summary of the differences is listed 
in Table 3.4. The median (50th percentile) of C
r
 minus C
lit
 and C
rate
 minus C
lit
 are -0.11 
and -0.14, respectively, compared to the smaller mean differences between C
r
 and C
lit
 and 
C
rate
 and C
lit
 (-0.06 or -0.07, respectively) (Table 3.4). 
 
C
r
 and C
lit
 for Developed and Undeveloped Watersheds  
 68
C
r
 and C
lit
 were grouped into two categories of developed and undeveloped 
watersheds (Roussel et al. 2005). Statistical summaries of C
r
 and C
lit
 for developed and 
undeveloped watersheds are listed in the Table 3.5; C
r
 and C
lit
 frequency distributions are 
shown in Figure 3.5B. The median value of C
r
 for undeveloped watersheds is 0.26 and 
the median value for the developed watershed is 0.45.  These median values are similar to 
those for watershed-median C
rate
 (Table 3.3).  The median and mean values of C
lit
 are 
larger than those of C
r
 for both developed and undeveloped watersheds (Table 3.5).  
About 68 and 78% of C
lit
 are larger than C
r
 for developed and undeveloped watersheds, 
respectively (Figure 3.5). 
 
C
r
 in relation to impervious area 
For this study, the percentage of impervious area for each watershed was 
computed using 1992 National Land Cover Data for Texas (Vogelmann and others, 
2001). C
r
 for 45 Texas watersheds with watershed imperviousness (IMP) greater than 
10% are plotted in Figure 3.6. Schaake et al. (1967) developed the regression equation 
SIMPC 05.065.014.0 ???
 (referred to herein as the ?Schaake et al. equation?) for 
urban drainage areas in Baltimore, MD to relate C
r
 (for a return period of 5 years) to the 
relative imperviousness of the drainage area and channel slope of the watershed. C
r
 was 
calculated using the Schaake et.al equation for the 45 Texas watersheds with watershed 
imperviousness greater than 10%.  For comparison purposes with C
r
 values for 45 Texas 
watersheds with watershed imperviousness (IMP) greater than 10%, C
r
 values calculated 
using the Schaake et. al equation also are plotted on Figure 3.6, along with C values 
extracted from Jens (1979), and equation 3.5 from Asquith (2011). 
 69
The results of these three studies [this study, Schaake et al. (1967), and Jens 
(1979)] are consistent?the value of C increases with increasing IMP. Asquith (2011) 
proposes a single equation to estimate C for Texas watersheds as a function of IMP. The 
equation is 
15.085.0 ?? IMPC  (3.5) 
The equation was used to estimate the runoff coefficient C* (?C-star?) for the unified 
rational method (URAT) developed for a TxDOT research project summarized in 
Cleveland et al. (2011). Equation 3.5 is plotted in Figure 3.6 and is consistent with the 
general pattern of the data. 
 Several studies (Jens 1979; Pilgrim and Cordery 1993; Hotchkiss and Provaznik 
1995; Titmarsh et al. 1995; Young et al. 2009) have demonstrated that C is highly 
dependent on the return period T.  In this study, rate-based runoff coefficients were not 
derived for any return period because the observed data do not include all events that 
would constitute the complete annual series needed for the frequency analysis. Return-
period based C(T) values were computed by the authors using regional regression 
equations for Q
p
 and I for the 36 undeveloped Texas watersheds in the database and 
presented as a separate paper (Dhakal et al. 2011). 
 
Correlation between C and Watershed Area
 
In order to evaluate the C values for the 80 Texas watersheds, C
r
 and watershed-
median C
rate
 were used to estimate the peak discharge rates (Q
p
) for each of about 1,500 
rainfall-runoff events using the rational equation (3.1). The observed versus the modeled 
 70
Q
p
 are shown in Figure 3.7. The peak relative error (QB) between the observed and the 
modeled peak discharges was estimated to analyze the model results (Cleveland et al. 
2006): 
,
i
ii
O
OP
QB
?
?  (3.6) 
where, P
i
 are the modeled peak discharge values, O
i
 are the observed peak discharge 
values. Cleveland et al. (2006) suggested the following range of the QB for the 
acceptance of model performance: 
25.025.0 ??? QB  (3.7) 
Median QB values derived using C
r
 and watershed-median C
rate
 are 0.11 and 0.00, 
respectively. Similarly, use of C
r
 resulted in about 56% and use of watershed-median 
C
rate
 resulted in about 59% of storms with QB less than ?50%. About 87 percent of the 
modeled Q
p
 values from both cases are within about half of a log cycle from the equal 
value line (Figure 3.7). The differences between the observed and modeled Q
p
 are 
generally within about a third of a log cycle, which is an uncertainty similar to that 
reported for regional regression equations of peak discharge in Texas by Asquith and 
Roussel (2009). 
The observation that about 87% of the modeled Q
p
 values from both cases are 
within about half of a log cycle from the equal value line supports the conclusion by 
ASCE and WPCF (1960), Pilgrim and Cordery (1993), and Young et al. (2009) that the 
rational method may be applied to much larger drainage areas than typically indicated 
(assumed) in some design manuals, as long as streamflows in the watershed are 
unregulated (Young et al. 2009).  ASCE and WPCF (1960) state ?Although the basic 
 71
principles of the rational method are applicable to large drainage areas, reported practice 
generally limits its use to urban areas of less than 5 square miles. Development of data 
for application of hydrograph methods is usually warranted on larger areas.?  Kuichling 
(1889, pages 40?41) made a similar statement for application of the rational formula to 
large watershed areas. 
C
rate
 for all events, watershed-mean C
rate
, and C
r
 versus watershed area (km
2
) are 
displayed on Figure 3.8.  Of note on Figure 3.8 is that the runoff coefficients are subject 
to substantial variability. That is, based on visual examination, there appears to be no 
relation between watershed drainage area and runoff coefficient. To apply a quantitative 
test, Pearson correlation coefficients between watershed-mean C
rate 
and C
r
 and watershed 
area are -0.27 and -0.26 with p-values of 0.012 and 0.018, respectively.  Therefore, at the 
95% confidence level (n = 80 observations), Pearson correlation coefficients between 
C
rate 
and C
r
 and watershed area are statistically significant but correlations are weak 
(determination coefficient r
2
 ? 0.07) and C exhibits high variability (Fig. 3.8).  Only 
about 7% of the variance is described by the correlation. Although statistically 
significant, the contribution of the correlation to description of the variability of C
rate
 and 
C
r
 is not useful in an engineering context. 
Of the 80 study watersheds, drainage area exceeds 40 km
2
 (15 mile
2
) for 17.  The 
choice of 40 km
2
 is completely arbitrary for the purposes of examining the runoff 
coefficient for relatively large watersheds. For this group of largest watersheds, average 
values of watershed-mean C
rate
 and C
r
 are 0.27 and 0.29, respectively. Standard 
deviations of watershed-mean C
rate
 and C
r
 are 0.11 and 0.12, respectively. In comparison, 
literature-based C
lit
 from land-use data (Dhakal et al. 2012) for these watersheds ranged 
 72
from 0.30 to 0.55 with average value of 0.42 and standard deviation of 0.10.  By 
inference, literature-based C
lit
 might be too large (Fig. 3.5), so estimates derived from 
application of C
lit
 to relatively large watersheds might lead to overly conservative 
estimates of discharge. Therefore, literature-based C values, e.g., published by ASCE and 
WPCF (1960) and current textbooks and design manuals, should not be used for 
watersheds with large drainage area. 
Although published values for C are not appropriate for relatively large 
watersheds, the rational method can be applied if reasonable estimates of runoff 
coefficient can be derived.  One source would be observations of runoff coefficient from 
hydrologically similar watersheds. Another would be derivation from observations of 
rainfall and runoff from the watershed of interest.  The published limits [5 square miles, 
ASCE and WPCF (1960); 200 acres, TxDOT (2002)] on the maximum drainage area for 
application of the rational method seem to be arbitrary. 
The authors do not advocate any specific limits that should be imposed on 
drainage area for application of the rational method. Therefore, it remains the 
responsibility of the end-user to apply appropriate engineering judgment when applying 
the rational method and the assumptions associated with the method, such as steady-state 
conditions. 
 
3.6 Summary 
The runoff coefficient, C, of the rational method is an expression of rate 
proportionality between rainfall intensity and peak discharge.  Two methods were used to 
estimate C. Both methods used about 1,500 observed rainfall and runoff events data from 
 73
80 Texas watersheds to derive C. For the first method, the rate-based runoff coefficient, 
C
rate
, was estimated for each rainfall-runoff event by the ratio of event peak discharge in 
a time series to the corresponding largest average rainfall intensity, I, in the same time 
series, averaged over the time window length. Time of concentration, T
c
, was used as the 
time window length to estimate I. The T
c
 values estimated using the Kerby-Kirpich 
method were used for the 80 watersheds studied. The rate-based C is dependent on 
rainfall intensity averaging time t
av
 used for the study, because based on equation (3.2), 
estimates of the runoff coefficient based on observed data cannot be decoupled from the 
selection of the time-response characteristic. Watershed-mean and watershed-median 
values of C
rate
 were derived. The distributions of the watershed-mean and watershed-
median C
rate
 are similar. Lastly, the C
rate
 values for the developed watersheds are 
consistently higher than those for undeveloped watersheds. For the second method, the 
frequency-matching approach, similar to the procedure used by Schaake et al. (1967), 
was used to sort peak discharges and average rainfall intensities independently and then 
to compute the rate-based C from the rational formula. A constant runoff coefficient C
r
 
for the watershed was derived from the plot of the rate-based C versus I. The C
r
 values 
for the developed watersheds are consistently greater than those for the undeveloped 
watersheds; about 74% of C
r
 and watershed-median C
rate
 differ less than ?0.1 (Table 3.4). 
The values of C
r
 and C
rate
 were compared with the literature based runoff coefficients 
(C
lit
) developed from land-use data for these study watersheds (Dhakal et al. 2012). 
About 75% of C
lit
 values are greater than C
r
 (Figure 3.5). For typical applications of the 
rational method in developed (urban) watersheds, watershed C
lit
 is less than C
r
 (Figure 
3.5); using smaller C
lit
 would underestimate Q
p
 for design. An equation was proposed to 
 74
estimate rate-based C as a function of the percentage of impervious area (IMP) for Texas 
watersheds, and prediction from the equation is consistent with the results from Schaake 
et al. (1967) and Jens (1979). 
 
3.7 Acknowledgments 
The authors thank TxDOT project director Mr. Chuck Stead, P.E., and project 
monitoring advisor members for their guidance and assistance. They also express thanks 
to technical reviewers Jennifer Murphy and Nancy A. Barth from USGS Tennessee and 
California Water Science Centers, respectively, and to three anonymous reviewers; the 
comments and suggestions greatly improved the paper.  This study was partially 
supported by TxDOT Research Projects 0?6070, 0?4696, 0?4193, and 0?4194. 
 
3.8 Notation 
The following symbols are used in this paper: 
A = drainage area in hectares or acres; 
C
lit 
= literature?based runoff coefficient developed from landuse data; 
C
rate 
= rate-based runoff coefficient; 
C
r
 
= runoff coefficients from the frequency matching approach; 
C
rj
 = runoff coefficient estimated from the ratio of jth rank-ordered peak discharge and 
the maximum rainfall intensity data pairs; 
C
v
 
= volumetric runoff coefficient; 
 75
C* = runoff coefficients as a function of percentage of impervious area from equation 
(3.5); 
I = average rainfall intensity (mm/hr or in. /hr) with the duration equal to time of 
concentration; 
I
j
 = the maximum rainfall intensity
 
of the j
th
 order; 
IMP = percentage of impervious area expressed as a decimal (50% = 0.5) for a watershed 
area; 
I
w
 = average rainfall intensity from the entire rainfall event duration; 
j = j
th
 term in the sequence of ordered peak discharge and the maximum rainfall intensity 
data pairs; 
L
c
 = channel length in km; 
m
o
 = the dimensional correction factor (1.008 in English units, 1/360 = 0.00278 in SI 
units); 
O
i
 = observed peak discharge for computing QB; 
P
i
 = modeled peak discharge for computing QB; 
QB = peak relative error between the observed and simulated peak discharges; 
Q
p
 = peak discharge in m
3
/s or ft
3
/s; 
Q
pj
 = peak discharge of the j
th
 rank-order of maximum rainfall intensity and peak 
discharge data pairs in cubic meters per second; 
Q
pj*
 = Q
pj 
divided by 0.0028 times the drainage area in cubic meters per second per 
hectare; 
 76
S = channel slope (Schaake et al. (1967)); 
S
c
 = channel slope in m/m; 
t
av
 = rainfall intensity averaging time period; 
T
c
 = time of concentration; 
T
p
 = time to peak; 
t
w
 = rainfall event duration; 
 
3.9 References: 
ASCE and WPCF. (1960). ?Design and construction of sanitary and storm sewers.? 
American Society of Civil Engineers (ASCE) and Water Pollution Control Federation 
(WPCF). 
Asquith, W.H., Thompson, D.B., Cleveland, T.G., and Fang, X. (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, Austin, 
Texas. 
Asquith, W.H., Thompson, D.B., Cleveland, T.G., and Fang, X. (2006). ?Unit 
hydrograph estimation for applicable Texas watersheds.? Texas Tech University, 
Center for Multidisciplinary Research in Transportation, Austin, Texas. 
Asquith, W.H., and Roussel, M.C. (2007). ?An initial-abstraction, constant-loss model 
for unit hydrograph modeling for applicable watersheds in Texas.? U.S. Geological 
Survey Scientific Investigations Report 2007?5243, 82 p. 
[http://pubs.usgs.gov/sir/2007/5243]. 
Asquith, W.H., and Roussel, M.C. (2009). "Regression equations for estimation of annual 
peak-streamflow frequency for undeveloped watersheds in Texas using an L-
moment-based, PRESS-Minimized, residual-adjusted approach." U.S. Geological 
Survey Scientific Investigations Report 2009?5087, 48 p. 
Asquith, W.H. (2011). ?A proposed unified rational method for Texas? in Cleveland, 
T.G., Thompson, D.B., Fang, X., (2011), ?Use of the Rational and Modified Rational 
Methods for TxDOT Hydraulic Design.? Research Report 0?6070?1, section 3, pp. 
18?58, Texas Tech University, Lubbock, TX. 
 77
Chow, V.T., Maidment, D.R., and Mays, L.W. (1988). Applied hydrology, McGraw Hill 
New York. 
Cleveland, T.G., He, X., Asquith, W.H., Fang, X., and Thompson, D.B. (2006). 
?Instantaneous unit hydrograph evaluation for rainfall-runoff modeling of small 
watersheds in north and south central Texas.? Journal of Irrigation and Drainage 
Engineering, 132(5), pp. 479?485. 
Cleveland, T.G., Thompson, D.B., Fang, X., and He, X. (2008). ?Synthesis of unit 
hydrographs from a digital elevation model.? Journal of Irrigation and Drainage 
Engineering, 134(2), pp. 212?221. 
Cleveland, T.G., Thompson, D.B., and Fang, X. (2011). ?Use of the Rational and 
Modified Rational Methods for TxDOT Hydraulic Design.? Research Report 0?
6070?1, 143 p., Texas Tech University, Lubbock, TX. 
Chow, V.T., Maidment, D.R., and Mays, L.W. (1988). Applied hydrology, McGraw Hill, 
New York. 
Dhakal, N., Fang, X., Cleveland, T.G., Thompson, D.B., Asquith, W.H., and Marzen, L.J. 
(2012). ?Estimation of volumetric runoff coefficients for Texas watersheds using 
land-use and rainfall-runoff data.? Journal of Irrigation and Drainage Engineering, 
138(1):43-54. 
Dhakal, N., Fang, X., Asquith, W.H., Cleveland, T.G., and Thompson, D.B., (2011). 
?Return period adjustments for runoff coefficients based on analysis in Texas 
watersheds.? Journal of Irrigation and Drainage Engineering (submitted).  
Fang, X., Thompson, D.B., Cleveland, T.G., and Pradhan, P. (2007). ?Variations of time 
of concentration estimates using NRCS velocity method.? Journal of Irrigation and 
Drainage Engineering, 133(4), pp. 314?322. 
Fang, X., Thompson, D.B., Cleveland, T.G., Pradhan, P., and Malla, R. (2008). ?Time of 
concentration estimated using watershed parameters determined by automated and 
manual methods.? Journal of Irrigation and Drainage Engineering, 134(2), pp. 202?
211. 
French, R., Pilgrim, D.H., and Laurenson, B.E. (1974). ?Experimental examination of the 
rational method for small rural catchments.? Civil Engineering Transactions, The 
Institution of Engineers, Australia CE16(2), pp. 95?102. 
Guo, J.C.Y., and Urbonas, B. (1996). "Maximized detention volume determined by 
runoff capture ratio." Journal of Water Resources Planning and Management, 122(1), 
33-39. 
 78
Hawkins, R.H. (1993). ?Asymptotic determination of runoff curve numbers from data.? 
Journal of Irrigation and Drainage Engineering, 119(2), pp. 334?345. 
Hotchkiss, R.H., and Provaznik, M.K. (1995). ?Observations on the rational method C 
value.? Watershed management; planning for the 21st century; proceedings of the 
symposium sponsored by the Watershed  Management Committee of the Water 
Resources, New York. 
Jens, S.W. (1979). Design of urban highway drainage, Federal Highway Administration 
(FHWA), Washington DC. 
Kerby, W.S. (1959). ?Time of concentration for overland flow.? Civil Engineering, 29(3), 
174 p. 
Kirpich, Z.P. (1940). ?Time of concentration of small agricultural watersheds.? Civil 
Engineering, 10(6), 362 p. 
Kuichling, E. (1889). ?The relation between the rainfall and the discharge of sewers in 
populous areas.? Transactions, American Society of Civil Engineers 20, pp. 1?56. 
Mays, L.W. (2004). Urban stormwater management tools, McGraw-Hill, New York. 
McCuen, R.H. (1998). Hydrologic analysis and design, Prentice-Hall, Inc., Upper Saddle 
River, N.J. 
Merz, R., Bl?schl, G., and Parajka, J. (2006). ?Spatio-temporal variability of event runoff 
coefficients.? Journal of Hydrology, 331(3?4), pp. 591?604. 
Pilgrim, D.H., and Cordery, I. (1993). ?Flood runoff.? Handbook of hydrology, D. R. 
Maidment, ed., McGraw-Hill, New York, 9.1?9.42. 
Roussel, M.C., Thompson, D.B., Fang, D.X., Cleveland, T.G., and Garcia, A.C. (2005). 
?Timing parameter estimation for applicable Texas watersheds.? 0?4696?2, Texas 
Department of Transportation, Austin, Texas. 
Schaake, J.C., Geyer, J.C., and Knapp, J.W. (1967). ?Experimental examination of the 
rational method.? Journal of the Hydraulics Division, ASCE, 93(6), pp. 353?370. 
Titmarsh, G.W., Cordery, I., and Pilgrim, D.H. (1995). ?Calibration procedures for 
rational and USSCS design flood methods.? Journal of Hydraulic Engineering, 
121(1), pp. 61?70. 
TxDOT. (2002). Hydraulic design manual, The Bridge Division of the Texas Department 
of Transportation (TxDOT), 125 East 11th street, Austin, Texas 78701. 
 79
USEPA. (1983). "Results of the nationwide urban runoff program, Volume I - Final 
Report." National Technical Information Service (NTIS) Accession Number: PB84-
185552, Water Planning Division, United States Environmental Protection Agency 
(USEPA), Washington D.C. 
Viessman, W., and Lewis, G.L. (2003). Introduction to hydrology, 5th Ed., Pearson 
Education, Upper Saddle, N.J., 612 p. 
Vogelmann, J.E., S.M. Howard, L. Yang, C. R. Larson, B. K. Wylie, and J. N. Van Driel. 
(2001). ?Completion of the 1990s National Land Cover Data Set for the conterminous 
United States.? Photogrammetric Engineering and Remote Sensing, 67, pp. 650-662. 
Young, C.B., McEnroe, B.M., and Rome, A.C. (2009). ?Empirical determination of 
rational method runoff coefficients.? Journal of Hydrologic Engineering, 14, 1283 p. 
 
 80
Table 3.1 Statistical Summary of C
rate
 Calculated from Observed Rainfall-Runoff Event 
Data. 
 T
p
 ? T
c
 1 
T
p
 < T
c
 2 
All events 
Minimum 0.01 0.01
0.01 
Maximum 4.48 2.68
4.48 
25% Quartile 0.21 0.13 
0.17 
Median 0.38 0.24
0.32 
75% Quartile 0.68 0.40 
0.56 
Mean 0.50 0.31
0.43 
Standard deviation 0.42 0.27 
0.39 
Note: 
1
 for 952 events, and 
2
 for 548 events. 
 
 
 
Table 3.2 Statistical Summary of Watershed-Median, Watershed-Mean and Standard 
deviation Values of C
rate
 for 80 Texas Watersheds. 
  Watershed-Median C
rate
 Watershed-Mean C
rate
 Standard deviation 
Minimum 0.07 0.07 0.03 
Maximum 1.73 1.79 0.87 
25 % Quartile 0.20 0.27 0.16 
Median 0.31 0.36 0.22
75 % Quartile 0.55 0.56 0.37 
Mean 0.40 0.44 0.27
Standard deviation 0.29 0.27 0.15 
 
 
Table 3.3 Statistical Summary of Watershed-Median C
rate
 for Developed and 
Undeveloped Watersheds. 
  Undeveloped
 1 
Developed
 2 
Minimum 0.07 0.17 
Maximum 73 1.73
25%Quartile 0.13 0.30
Median 0.20 0.40
75%Quartile 0.28 0.71 
Mean 0.24 0.53
Standard deviation 0.16 0.31 
Note: 
1
 for 36 undeveloped watersheds, and 
2
 for 44 developed watersheds. 
 
 81
Table 3.4 Statistical Summary of C
r
, and Differences among C
r
, C
rate
 and C
lit
 for 80 
Texas watersheds. 
  C
r 
C
r
 - C
rate 
1 
C
r
 - C
lit
C
rate
 
1
 - C
lit 
Minimum 0.09 -0.53 -0.44 -0.46 
Maximum 1.20 0.52 0.66 1.19 
25 % Quartile 0.25 -0.02 -0.17 -0.24 
Median 0.37 0.03 -0.11 -0.14 
75 % Quartile 0.54 0.06 0.02 -0.01 
Mean 0.42 0.02 -0.06 -0.07 
Standard deviation 0.23 0.14 0.21 0.27 
 1
 is watershed-median C
rate
. 
 
Table 3.5 Statistical Summary of C
r
 and C
lit
 for Developed and Undeveloped Watersheds. 
 
C
r  
Undeveloped 
watersheds 
C
r  
Developed 
watersheds 
C
lit  
Undeveloped 
watersheds 
C
lit 
 Developed 
watersheds 
Minimum 0.09 0.18 0.29 0.37 
Maximum 0.69 1.20 0.59 0.63
25%Quartile 0.20 0.34 0.33 0.52 
Median 0.26 0.45 0.37 0.54
75%Quartile 0.42 0.62 0.44 0.58 
Mean 0.32 0.50 0.39 0.54
Standard deviation 0.17 0.23 0.08 0.06 
 
 82
 
 
Fig. 3.1 Map showing U.S. Geological Survey streamflow-gaging stations representing 
80 developed and undeveloped watersheds in Texas. 
 83
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
C
r
a
t
e
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
 
C
r
a
t
e
0
20
40
60
80
100
120
I
(
m
m
/
h
r
)
20
40
60
80
100
120
I
(
m
m
/
h
r
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
t
av 
(hr)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
C
rate
I(mm/hr)
U.S. Geological Survey streamflow-gaging station 
08058000 Honey Creek subwatershed number 12 
near McKinney, Texas 
(Area = 3.11 km    )
U.S. Geological Survey streamflow-gaging station 
08048550 Dry Branch at Blandin Street, Fort Worth, Texas
(Area = 2.87 km    ) 
T
c 
= 1.8 hr., C
rate
 = 0.23
T
c 
= 1.5 hr., C
rate
 = 1.7
(A)
(B)
2
2
 
 
Fig 3.2 Average rainfall intensity I and runoff coefficient C
rate
 as a function of t
av
 for two 
storm events: (A) on 09/22/1969 in Dry Branch in Fort Worth, Texas (U.S. Geological 
Survey [USGS] streamflow-gaging station 08048550 Dry Branch at Blandin Street, Fort 
Worth Texas), and (B) on 04/25/1970 in Honey Creek near Dallas, Texas. (USGS 
streamflow-gaging station 08058000 Honey Creek subwatershed number 12 near 
McKinney, Texas). 
 84
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
P
e
r
c
e
n
t
a
g
e
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
P
e
r
c
e
n
t
a
g
e
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Watershed-median,
0.01 0.1 1
Runoff Coefficient,
Undeveloped (36 Texas watersheds)
Developed (44 Texas watersheds)
All 80 Texas watersheds
T  < T
T  >= T
watershed-mean
watershed-median
C
rate
C
rate
(A)
(B)
p
c
p c
 
 
Fig. 3.3 Cumulative distributions of runoff coefficient (C
rate
 values): (A) for rainfall-
runoff events when the time to peak (T
p
) was less than the time of concentration (T
c
); for 
rainfall-runoff events when the time to peak (T
p
) was greater than or equal to the time of 
concentration (T
c
); watershed-average (mean); and watershed-median; and (B) 
watershed-median C
rate
 for developed and undeveloped Texas watersheds. 
 85
0
2
4
6
8
10
12
Q
p
j
*
 
(
m
3
/
s
/
h
a
)
0.00
0.05
0.10
0.15
0.20
0.25
C
r
j
0 10 20 30 40 50
I
j
  (mm/hr)
0 10 20 30 40 50
C
r
Q
pj* 
= 0.19 x I
j
R
2
 = 0.88
(A)
(B)
 
 
Fig. 3.4 (A) Runoff coefficient (C
r
) values derived from the rank-ordered pairs of 
observed peak discharge and maximum rainfall intensity during each storm event in 
mm/hr at USGS streamflow-gaging station 08042650 North Creek Surface Water Station 
28A near Jermyn, Texas, and (B) the rank-ordered pairs of the observed peak discharge 
and the average rainfall intensity for the same station. 
 86
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
P
e
r
c
e
n
t
a
g
e
0
10
20
30
40
50
60
70
80
90
100
C
u
m
u
l
a
t
i
v
e
 
P
e
r
c
e
n
t
a
g
e
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Runoff Coefficient
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
C
r 
 for 36 undeveloped Texas watersheds
C
r 
 for 44 developed Texas watersheds
C
lit
 for undeveloped watersheds
C
lit
 for developed watersheds
C 
r
 for 80 Texas watersheds
watershed-median C         for 80 Texas watersheds
C
lit 
 for undeveloped and developed watersheds
(A)
(B)
rate
 
 
Fig. 3.5 Cumulative distributions of: (A) C
r
, watershed-median C
rate
 and C
lit
, and (B) 
distributions of C
r
 and C
lit
 for developed and undeveloped watersheds. 
 87
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R
u
n
o
f
f
 
C
o
e
f
f
i
c
i
e
n
t
0 10 20 30 40 50 60 70 80 90 100
Percentage of impervious area,
Equation 3.5 (Asquith 2011)
C
r 
for 45 Texas watersheds with watershed imperviousness greater than 10%
C values extracted from Jens (1979) for comparison with  Texas C  values
C
r
IMP
estimated from the equation proposed by Schaake et. al (1967) for 45 Texas watersheds 
with watershed imperviousness greater than 10%
EXPLANATION
r
 
Fig. 3.6 Runoff coefficients versus the percentage of impervious area, IMP. 
 88
1
10
100
M
o
d
e
l
e
d
 
Q
p
 
(
m
3
/
s
)
1 10 100
Observed Q
p 
(m
3
/s)
modeled using C
modeled using C
1:1 line
? half of log cycle
800
800
rate
r
 
 
Fig. 3.7 Modeled peak discharges (Q
p
) from rational equation (3.1) using C
r
 and 
watershed-median C
rate
 for 1,500 rainfall-runoff events in 80 Texas watersheds against 
observed peak discharges. 
 
0.01
0.1
1
R
u
n
o
f
f
 
C
o
e
f
f
i
c
i
e
n
t
1 10 100
Area (km
2
)
C
r
watershed-mean C
C
rate 
(all events)
rate
 
Fig 3.8. Rate-based runoff coefficients C
rate
 for all events, watershed-mean C
rate
, and C
r
 
versus drainage area (km
2
) in 80 Texas watersheds.
 89
Chapter 4. Return Period Adjustments for Runoff Coefficients Based on Analysis in 
Texas Watersheds 
 
 
4.1 Abstract 
The rational method for peak discharge (Q
p
) estimation was introduced in the 
1880s. The runoff coefficient (C) is a key parameter for the rational method, and the C 
has been declared a function of return period by various researchers. Rate-based runoff 
coefficients as function of return period, C(T), were developed for 36 undeveloped 
watersheds in Texas using peak discharge frequency from previously published regional 
regression equations and rainfall intensity frequency for return periods T of 2, 5, 10, 25, 
50, and 100 years. The C(T) values developed in this study are most applicable to 
undeveloped watersheds. The C(T) values of this study increase with T more rapidly than 
the increase suggested in prior literature. When the larger frequency factors are applied, if 
any resulting C(T) is greater than unity, C(T) is suggested to be set to 1. 
 
 
4.2 Introduction 
The rational method introduced by Kuichling (1889) is typically used to compute 
the peak discharge, Q
p
 (in m
3
/s in SI units or ft
3
/s in English units) for designing drainage 
structures: 
,CIAmQ
op
?
 
(4.1) 
where C is the runoff coefficient (dimensionless), I is the rainfall intensity (mm/hr or 
in/hr) over a critical period of storm time (typically taken as the time of concentration, T
c
, 
 90
of the watershed), A is the drainage area (hectares or acres), and m
o
 is the dimensional 
correction factor (1/360 = 0.00278 in SI units, 1.008 in English units). From inspection of 
the equation, it is evident that C is an expression of rate proportionality between I and Q
p
. 
Typical ?whole watershed? C values, that is, C values representing the integrated 
effects of various surfaces in the watershed and other watershed properties, are listed for 
different general land-use conditions in various design manuals and textbooks. Examples 
of textbooks that include tables of C values are Chow et al. (1988) and Viessman and 
Lewis (2003). Published C values, C
lit
, were sourced from American Society of Civil 
Engineers (ASCE) and the Water Pollution Control Federation (WPCF) in 1960 (ASCE 
and WPCF 1960). The C
lit
 values were obtained from a response survey, which received 
?71 returns of an extensive questionnaire submitted to 380 public and private 
organizations throughout the United States.? No justification based on observed rainfall 
and runoff data for the selected C
lit
 values was provided in the ASCE and WPCF (1960) 
manual. 
A substantial criticism of the rational method arises because observed C values 
vary from storm to storm (Schaake et al. 1967; Pilgrim and Cordery 1993). The ASCE 
and WPCF (1960) manual, in describing tabulations of rational method C, state ?The 
coefficients on these two tabulations (of C values) are applicable for storms having 5- to 
10-year return periods (0.2 to 0.1 annual exceedance probabilities). Less frequent, higher 
intensity storms will require the use of higher coefficients because infiltration and other 
losses have a proportionately smaller effect on runoff.? Schaake et al. (1967) found that 
the average percentage increase of the coefficient for the 10-year return period C(10) was 
only 10 percent as compared to the coefficient for the 1-year return period C(1), and 
 91
proposed adoption of a single value of C for design circumstances. The C has been 
considered a function of return period by various researchers (Jens 1979; Pilgrim and 
Cordery 1993; Hotchkiss and Provaznik 1995; Titmarsh et al. 1995; Young et al. 2009). 
Considering C as function of return period T, the rational formula can be expressed as 
(Jens 1979; Pilgrim and Cordery 1993): 
Q(T) = C(T) I(T) A = C
lit
 C
f
(T) I(T) A, (4.2) 
where C(T) is C as function of the return period T, I(T) is rainfall intensity as function of 
T, C
lit
 is literature-based C as previously defined herein based on T values ranged from 2- 
to 10- year recurrence intervals from text books (e.g., Viessman and Lewis 2003) or 
design manuals (e.g., TxDOT 2002), and C
f
(T) is a frequency factor or multiplier (DRCG 
1969; Jens 1979). Equation 4.2 implies a conversion of I(T) to Q
p
(T) where T denotes the 
same return period for both I and Q. Equation (4.2) also is the probabilistic interpretation 
of the rational formula and commonly used in design practices (French et. al 1974; 
Pilgrim and Cordery 1993). 
Relatively, few studies have been performed to determine rational C values using 
frequency-based analysis of data (Young et al. 2009). An influential paper by Schaake et 
al. (1967) examined the rational method using experimental rainfall and runoff data 
collected from 20 small urban watersheds [< 0.6 km
2
 (0.23 mi
2
)] in Baltimore, MD. 
Those authors used watershed lag time to compute average rainfall intensity and used a 
frequency-matching approach to compute rate-based C values. Hotchkiss and Provaznik 
(1995) estimated C
rate
 for 24 rural watersheds in south-central Nebraska using event-
paired and frequency-matched data. Young et al. (2009) estimated C
rate
 for 72 rural 
watersheds in Kansas with drainage areas less than 78 km
2
 (30 mi
2
) for different return 
 92
periods. The peak discharge Q(T) for each T was estimated using annual peak frequency 
analysis of observed streamflow records and rainfall intensity obtained from rainfall 
intensity-duration-frequency tables (Young et al. 2009). 
For this study, return-period based runoff coefficients C(T) were computed using 
Q
p
 and I calculated, not from statistical analysis of observed pairing of rainfall and runoff 
data, but from coupling of regional regression equations (Asquith and Slade 1997) and 
rainfall intensity (TxDOT 2002) for 36 undeveloped Texas watersheds. Subsequently, 
frequency factors C
f
(T) = C(T)/C(10) were computed. Using C for the 10-year return 
period as a base value to compute frequency factors C
f
(T) is consistent with literature 
(French et. al 1974; Pilgrim and Cordery 1993; Young et al. 2009).  Results of C(T) and 
frequency factors C
f
(T) were analyzed and compared with previous studies. 
 
4.3 Study Watersheds 
 
The study watersheds comprise 36 undeveloped watersheds in Texas, which have 
been used previously by the authors and associates (Asquith et al. 2004). The 36 
watersheds consist of 20 rural watersheds and 16 suburban watersheds in near four cities: 
Austin, Dallas, Fort Worth, and San Antonio. Locations and geographic distribution of 
the streamflow-gaging stations associated with these watersheds are shown in Figure 4.1. 
The classification scheme of developed and undeveloped watersheds 
accommodates the characterization of watersheds in more than 220 USGS reports of 
Texas data from which the original data for the rainfall and runoff database were obtained 
(Asquith et al. 2004). Although this binary classification seems arbitrary, it was 
purposeful and reflects the uncertainty in precise watershed development conditions for 
 93
the time period of available data (Asquith and Roussel 2007). This same binary 
classification was successfully used to prepare regression equations to estimate the shape 
parameter and the time to peak for regional gamma unit hydrographs for Texas 
watersheds (Asquith et al. 2006). 
The drainage area of study watersheds range from approximately 2.3?320 km
2
 
(0.9?123.6 mi
2
); the median and mean values are 20.7 km
2
 (8 mi
2
) and 56.7 km
2
 (21.9 
mi
2
), respectively.  The stream slopes of study watersheds range from approximately 
0.0022?0.0196 dimensionless; the median and mean values are both 0.0089, respectively.  
Many practitioners could argue that the application of the rational method is not 
appropriate for the range of watershed areas presented in this study. ASCE and WPCF 
(1960) made the following statements when the rational method was introduced for 
design and construction of sanitary and storm sewers: ?Although the basic principles of 
the rational method are applicable to large drainage areas, reported practice generally 
limits its use to urban areas of less than 5 sq miles.? (ASCE and WPCF 1960, p. 32).  
Pilgrim and Cordery (1993) stated that the rational method is one of the three methods 
widely used to estimate peak flows for small to medium sized basins.  ?It is not possible 
to define precisely what is meant by ?small? and ?medium? sized, but upper limits of 25 
km
2
 (10 mi
2
) and 550 km
2
 (200 mi
2
), respectively, can be considered as general guides? 
(Pilgrim and Cordery 1993, p. 9.14).  Young et al. (2009) stated that the rational method 
may be applied to much larger drainage areas than typically assumed in some design 
manuals provided that the watershed is unregulated.  Thompson (2006) stated that 
watershed drainage area does not appear to be an applicable factor for discriminating 
between appropriate hydrologic technologies (such as rational method, regional 
 94
regression equations, and site-specific flood frequency relations), other methods for 
discrimination between procedures for making design-discharge estimates should be 
investigated. 
A geospatial database of properties for the 36 watersheds was developed by 
Roussel et al. (2005). For this paper, basin-shape factor, main channel length and channel 
slope were used to estimate time of concentration T
c
 by Kirpich (1940) for channel flow 
plus travel time for overland flow using Kerby (1959). This combination of methods to 
compute T
c
 is discussed by Roussel et al. (2005) and Fang et al. (2008).  The Kirpich 
equation (1940) was developed from the Natural Resources Conservation Service 
(NRCS) data for rural watersheds with drainage areas less than about 0.45 km
2
.
 
Fang et 
al. (2007, 2008) demonstrated that, for watersheds with large drainage areas, the Kirpich 
equation provides as reliable an estimate of T
c
 as the other empirical equations developed 
for large watersheds and as the NRCS velocity method (Viessman and Lewis 2003).  The 
T
c
 estimated using the Kirpich equation reasonably approximate the average T
c
 estimated 
from observed rainfall and runoff data (Fang et al. 2007). 
 
4.4 Runoff Coefficients for Different Return Periods 
Rate-based C(T) values for the 36 study watersheds in Texas, and corresponding 
frequency factors were determined for various return periods using equation (4.3) 
(Pilgrim and Cordery 1993): 
,
),(
)(
)(
0
ATTIm
TQ
TC
c
?
 (4.3) 
where Q(T), C(T), and I(T
c
, T) are peak discharge, runoff coefficient, and rainfall 
intensity for the recurrence interval T, respectively. In this study, Q(T) for each of the 36 
 95
undeveloped Texas watersheds was estimated by regional regression equations for Texas 
developed by Asquith and Slade (1997) that use contributing drainage area, a basin-shape 
factor, and main channel slope. The basin-shape factor is defined as the ratio of main 
channel length squared to contributing drainage area (sq. mi./sq. mi. or km
2
/km
2
) (TxDOT 
2002). 
The T
c
 for each watershed in Texas was developed using the Kerby-Kirpich 
equation (Roussel et al. 2005; Fang et al. 2008).  Considering the county in which each 
watershed is located, the rainfall intensity, I(T
c
, T), for each return period was estimated 
using rainfall intensity-duration-frequency (IDF) relations (TxDOT 2002) with duration 
T
c
: 
,
)(
),(
g
c
c
fT
e
TTI
?
?
 (4.4) 
where e, f, and g are coefficients for specific frequencies and Texas counties (TxDOT 
2002). With Q(T) from Asquith and Slade (1997) and I(T
c
, T) from TxDOT design 
manual (TxDOT 2002), equation (4.3) was used to compute C(T) for each watershed and 
for each return period of T = 2, 5, 10, 25, 50 and 100 years. 
The C(T) versus T for three undeveloped Texas watersheds are presented as 
illustrative examples in Figure 4.2. For these three watersheds, C(T) increases with 
increasing T. The value of C(100) is 0.6 for Deep Creek, 1.05 for East Elm Creek, and 
1.3 for Escondido Creek. The occurrence of C(T) > 1 could be related to inherent 
uncertainties of Q(T) and I(T
c
, T).  Several studies (French et. al 1974; Pilgrim and 
Cordery 1993; Young et al. 2009) have shown that values of C(T) greater than 1 are 
possible when rate-based C was determined from observed peak discharge and rainfall 
intensity.  So it is not a stretch to see C(T) > 1 for purely statistically independent studies 
 96
of Q and I such as Asquith and Slade (1997) and TxDOT (2002). Analysis of observed 
rainfall and runoff data in 90 Texas watersheds has shown that only the volumetric runoff 
coefficient, C
v
, as the ratio of total runoff depth to total rainfall depth, is less than 1 for all 
storm events (Dhakal et al. 2012). 
Statistical summaries of C(T) are listed in Table 4.1 and corresponding boxplots 
of the distribution are shown in Figure 4.3. The median C values by T, as well as the 
curves shown in Figure 4.2, show that C(T) increases with the increasing recurrence 
interval for undeveloped watersheds in Texas. Ratios of C(T)/C(10) or frequency factors 
C
f
(T) are derived for the Texas watersheds and statistical summaries of the ratios are 
listed in Table 4.2; the mean and median values are of special importance for 
representation of frequency. 
 
4.5 Discussion 
Comparing C(T) and C
f
(T) for Texas Watersheds with Other Studies 
Young et al. (2009) estimated median C(T) from observed data for 72 rural 
watersheds in Kansas and these values are shown in Figure 4.3 for comparison. The 
results of C(T) for Texas watersheds reported in Table 4.1 and Figure 4.3 are consistent 
with results reported by Young et al. (2009). The mean values of C(T) derived from 
observed data for 24 rural watersheds in south-central Nebraska (Hotchkiss and 
Provaznik 1995) are shown in Figure 4.3.  Literature-based C values for the Nebraska 
watersheds are 0.35 for T < 10 years (Hotchkiss and Provaznik 1995). The mean C(T) 
values reported by Hotchkiss and Provaznik (1995) for the Nebraska watersheds are 
 97
larger than not only median C(T) determined for the Texas and Kansas watersheds but 
also C
lit
. 
French et al. (1974) mapped C(10) values in New South Wales, Australia, for 37 
rural watersheds, with drainage area up to 250 km
2
 (96 mi
2
). The relations between 
frequency factors and return period from French et al. (1974), reported by Young et al. 
(2009), Jens (1979) and Gupta (1989), and the results for the Texas watersheds are shown 
in Figure 4.4. The frequency factors C
f
(T) determined (1) for the Texas watersheds, (2) 
by Young et al. (2009), and (3) by French et al. (1974) exceed the textbook values from 
Gupta (1989) and Viessman and Lewis (2003), and exceed TxDOT (2002) values when T 
> 10 years.  The Texas frequency factors C
f
(T) are similar to those determined for Kansas 
watersheds by Young et al. (2009). Lastly, the Texas frequency factors C
f
(T) exceed 
those from watersheds in New South Wales, Australia (French et al. 1974) by about 15 
percent when T > 10 years, and less when T < 10 years. 
The frequency factors C
f
(T) specified for Denver watersheds (DRCG 1969; Jens 
1979) and later published in other textbooks (e.g., Gupta 1989, Viessman and Lewis 
2003) and design manuals (e.g., TxDOT 2002) are listed in Table 4.3. Typically, a 
frequency factor C
f
(T) of 1.0 is used when T < 10 years (Table 3). The frequency factors 
C
f
(T) extracted from the FHWA curve (Jens 1979) for percent impervious area equal to 0 
percent and 65 percent and from the study by French et al. (1974) and Young et al. 
(2009) are listed in Table 4.3 for comparison. Of special note is the observation that the 
Texas frequency factors, C(2)/C(10) and C(5)/C(10) as well as those from French et al. 
(1974) and Young et al. (2009), are not equal to 1.0 as in ASCE and WPCF (1960) [and 
later by Gupta (1989) and Viessman and Lewis (2003) and in the TxDOT (2002)] but 
 98
rather are ratios less than 1.0 (Table 4.2 and 4.3). This means that the C(T) for T < 10 
year (more frequent storms) of Texas, Kansas and Australia is less than C
lit
 commonly 
recommended in the literature. 
The frequency factors C
f
(T) values were extracted from the FHWA curve (Jens 
1979) for percent impervious areas of 65 percent because they are approximately the 
same as frequency factors C
f
(T) values presented in design manuals and textbooks (e.g., 
TxDOT design manual [2002]; Gupta 1989; Viessman and Lewis 2003). The frequency 
factors C
f
(T) presented in design manuals and textbooks are seemingly more appropriate 
for urban watersheds with relatively large percentage of impervious area. 
The larger frequency factors C
f
(T) determined for Texas watersheds and those 
determined for Kansas watersheds (Young et al. 2009) are for undeveloped watersheds 
with impervious cover less than a few percent. The larger frequency factors C
f
(T) of 
Texas are similar to frequency factors C
f
(T) extracted from the FHWA curve (Jens 1979) 
for 0 percent impervious areas (Table 4.3). These frequency factors C
f
(T) were proposed 
by Bernard (1938). The frequency factors C
f
(T) from the FHWA curve (Jens 1979) for 
100 percent impervious area is approximately 1.1 for T of 25, 50, and 100 years. If it is 
assumed that C is 1 for 100 percent impervious areas, then frequency factors C
f
(T) should 
be 1.0 for 100 percent impervious area for any T. Therefore, variable frequency factors 
C
f
(T) as additional function of percent of impervious area (Jens 1979) is a reasonable 
conjecture and supported by Young et al. (2009) as well as this study for Texas 
watersheds.  When frequency factors C
f
(T) is applied and if resulting C(T) is greater than 
1, Jens (1979), Gupta (1989), and TxDOT (2002) indicate C(T) should be set equal to 1. 
 
 99
C for 100-year Return Period 
In an adaption of the rational method, Bernard (1938) proposed that C varied in a 
functional manner with the T-year return period when related to the maximum or limiting 
C values (called C
max
): 
,)100(
max
x
TCC ?  (4.5) 
where x is the exponent and ranges from 0.15 to 0.23 for undeveloped watersheds 
(Bernard 1938). Bernard (1938) assumed the C
max
 value corresponds to C(100). In 
relation to equation (4.5), Jens (1979) proposed C
max
 = C(100) = 1.0 for watersheds with 
any percentage of impervious area for application of the equation (4.5) for the FHWA 
Manual (Jens 1979).  C(100) for the 36 Texas watersheds range from 0.34 to 1.44 with 
mean and median values of 0.86 and 0.94 (Table 4.1).  C(100) for three Texas watersheds 
also are presented as illustrative examples shown in Figure 4.2.  Stubchaer (1975) applied 
the calibrated Santa Barbara Unit Hydrograph (SBUH) method on a 388-acre urban 
watershed and developed C(T) using the frequency analysis of rainfall and simulated 
runoff from the SBUH. The C(100) value determined for the watershed is 0.65 
(Stubchaer 1975). The C(100) values for watersheds with different percentages of 
impervious cover from the Denver Manual (DRCG 1969) range from 0.20 to 0.96 and 
from Chow et al. (1988) range from 0.36 to 0.97; C(100) are consistently less than 1. 
 
4.6 Summary 
The runoff coefficients C(T) for different return periods (T) were developed for 
the 36 undeveloped Texas watersheds using previously published regional regression 
 100
equations of peak discharge and county-based tabulated empirical coefficients for a 
model of rainfall intensities at different T. C(T) values increase with T and these increases 
are more than previously thought. The frequency factors C
f
(T) = C(T)/C(10) determined 
in this study exceed those values in textbooks such as Gupta (1989) and Viessman and 
Lewis (2003) and those from TxDOT (2002) when T > 10 years. The frequency factors 
C(2)/C(10) and C(5)/C(10) for the Texas watersheds (Table 4.2) as well as from French 
et al. (1974) and Young et al. (2009) are not equal to 1 as assumed in ASCE and WPCF 
(1960) and published by Gupta (1989) and Viessman and Lewis (2003) and the design 
manual (TxDOT 2002) (Table 4.3 and Figure 4.4) but less than 1. 
 The frequency factors determined for the 36 Texas watersheds and the 72 Kansas 
watersheds (Young et al. 2009), larger than those mostly found in literature, are for 
undeveloped watersheds with relatively small percent impervious areas. The frequency 
factors mostly found in the literature, smaller than those determined for the 36 Texas 
watersheds, are appropriate for urban watersheds with relatively large percentages of 
impervious area, as supported and presented in literature (e.g., DRCG 1969; Stubchaer 
1975; Jens 1979; Gupta 1989; Viessman and Lewis 2003; TxDOT 2002). Such frequency 
factors are consistent with those proposed by Jens (1979). When the frequency factor is 
applied, if resulted C(T) is greater than unity, Jens (1979), Gupta (1989) and TxDOT 
(2002) suggested setting C(T) equal to 1. 
 
4.7 Acknowledgments 
 
 101
The authors thank TxDOT project director Mr. Chuck Stead, P.E., and project 
monitoring advisor members for their guidance and assistance.  This study was partially 
supported by TxDOT Research Projects 0?6070, 0?4696, 0?4193, and 0?4194. 
 
4.8 Notation 
The following symbols are used in this paper: 
A = watershed area in hectares or acres; 
C
f
 (T) = C(T)/C(10), frequency factor or frequency multiplier; 
C
max
 = maximum runoff coefficient for the return period 100 years; 
C(T) = rate-based runoff coefficient for return period T; 
C
v
 = volumetric runoff coefficient as the ratio of total runoff depth and total rainfall 
depth; 
I = average rainfall intensity (mm/hr or in. /hr) with the duration equal to time of 
concentration; 
m
o
 = the dimensional correction factor (1.008 in English units, 1/360 = 0.00278 in SI 
units); 
Q
p
 = peak runoff rate in m
3
/s or ft
3
/s; 
Q(T) = peak discharge for return period T; 
Q
T
 = regional regression equation for natural basins developed for TxDOT; 
T = recurrence interval or return period in years; 
T
c
 = time of concentration; 
 
 
 102
4.9 References: 
ASCE, and WPCF. (1960). Design and construction of sanitary and storm sewers, 
American Society of Civil Engineers (ASCE) and Water Pollution Control Federation 
(WPCF). 
Asquith, W. H., and Slade, R. M. (1997). "Regional equations for estimation of peak 
streamflow frequency for natural basins in Texas." Water Resources Investigations 
Report 96-4307, U.S. Geological Survey, Austin, Texas, 68 p. 
Asquith, W. H., Thompson, D. B., Cleveland, T. G., and Fang, X. (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, Austin, 
Texas. 
Asquith, W. H., Thompson, D. B., Cleveland, T. G., and Fang, X. (2006). ?Unit 
hydrograph estimation for applicable Texas watersheds.? Texas Department of 
Transportation Research Report 0-4193-4, Austin, Texas. 
Asquith, W. H., and Roussel, M. C. (2007). ?An initial-abstraction, constant-loss model 
for unit hydrograph modeling for applicable watersheds in Texas.? U.S. Geological 
Survey Scientific Investigations Report 2007?5243, Austin, Texas, 82 p. 
Bernard, M. (1938). ?Modified rational method of estimating flood flows.? Natural 
Resources Commission, Washington D.C. 
Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). Applied hydrology, McGraw 
Hill New York. 
Dhakal, N., Fang, X., Cleveland, T.G., Thompson, D.B., Asquith, W.H., and Marzen, L.J. 
(2012). ?Estimation of volumetric runoff coefficients for Texas watersheds using 
land-use and rainfall-runoff data.? Journal of Irrigation and Drainage Engineering, 
138(1):43-54. 
DRCG. (1969). ?Urban Storm Drainage Criteria Manual, Vols. 1 and 2.? Prepared by 
Wright-McLaughlin Engineers for Denver Regional Council of Governments 
(DRCG), Denver, CO. 
Fang, X., Thompson, D. B., Cleveland, T. G., and Pradhan, P. (2007). ?Variations of time 
of concentration estimates using NRCS velocity method.? Journal of Irrigation and 
Drainage Engineering, 133(4), pp. 314?322. 
Fang, X., Thompson, D. B., Cleveland, T. G., Pradhan, P., and Malla, R. (2008). ?Time 
of concentration estimated using watershed parameters determined by automated and 
manual methods.? Journal of Irrigation and Drainage Engineering, 134(2), pp. 202?
211. 
 103
French, R., Pilgrim, D. H., and Laurenson, B. E. (1974). ?Experimental examination of 
the rational method for small rural catchments.? Civil Engineering Transactions, The 
Institution Of Engineers, Australia CE16(2), pp. 95?102. 
Gupta, R. S. (1989). Hydrology and hydraulic systems, Prentice Hall, Englewood Cliffs, 
New Jersey. 
Hotchkiss, R. H., and Provaznik, M. K. (1995). ?Observations on the rational method C 
value.? Watershed management: planning for the 21st century; proceedings of the 
symposium sponsored by the Watershed  Management Committee of the Water 
Resources, New York. 
Jens, S. W. (1979). Design of Urban Highway Drainage, Federal Highway 
Administration (FHWA), Washington DC. 
Kerby, W. S. (1959). ?Time of concentration for overland flow.? Civil Engineering, 
29(3), 174 p. 
Kirpich, Z. P. (1940). "Time of concentration of small agricultural watersheds." Civil 
Engineering, 10(6), 362. 
Kuichling, E. (1889). ?The relation between the rainfall and the discharge of sewers in 
populous areas.? Transactions, American Society of Civil Engineers 20, pp. 1?56. 
Pilgrim, D. H., and Cordery, I. (1993). ?Flood runoff.? In Handbook of Hydrology, D. R. 
Maidment, ed., McGraw-Hill, New York, 9.1?9.42. 
Roussel, M. C., Thompson, D. B., Fang, D. X., Cleveland, T. G., and Garcia, A. C. 
(2005). ?Timing parameter estimation for applicable Texas watersheds.? 0?4696?2, 
Texas Department of Transportation, Austin, Texas. 
Schaake, J. C., Geyer, J. C., and Knapp, J. W. (1967). ?Experimental examination of the 
rational method.? Journal of the Hydraulics Division, ASCE, 93(6), pp. 353?370. 
Stubchaer, J. M. ?The Santa Barbara urban hydrograph method.? National Symposium of 
Urban Hydrology and Sediment Control, University of Kentucky, Lexington, KY. 
Thompson, D. B. (2006). "The rational method, regional regression equations, and site-
specific flood frequency relations." FHWA/TX-06/0-4405-1, Center for 
Multidisciplinary Research in Transportation, Texas Tech University, Lubbock, 
Texas. 
Titmarsh, G. W., Cordery, I., and Pilgrim, D. H. (1995). ?Calibration Procedures for 
Rational and USSCS Design Flood Methods.? Journal of Hydraulic Engineering, 
121(1), pp. 61?70. 
 104
TxDOT. (2002). Hydraulic design manual, The bridge division of the Texas Department 
of Transportation (TxDOT), 125 East 11th street, Austin, Texas 78701. 
Viessman, W., and Lewis, G. L. (2003). Introduction to hydrology, 5th Ed., Pearson 
Education, Upper Saddle, N.J., 612 p. 
Young, C. B., McEnroe, B. M., and Rome, A. C. (2009). ?Empirical determination of 
rational method runoff coefficients.? Journal of Hydrologic Engineering, 14, 1283 p. 
 105
Table 4.1 Statistical Summary of C(T) for Select Return Periods (T) for Texas 
Watersheds. 
 2 years 5 years 10 years 25 years 50 years 100 years 
Minimum 0.08 0.14 0.18 0.24 0.30 0.34 
Maximum 0.39 0.64 0.77 0.97 1.14 1.44 
25
th
 percentile 0.12 0.29 0.39 0.48 0.50 0.55 
Median 0.15 0.32 0.43 0.62 0.77 0.94 
75
th
 percentile 0.22 0.37 0.50 0.70 0.91 1.12 
Average  0.18 0.33 0.44 0.59 0.73 0.86 
Standard deviation 0.075 0.091 0.110 0.163 0.238 0.319 
 
 
 
Table 4.2 Statistical Summary of the Frequency Factors C(T)/C(10) for Texas 
Watersheds. 
 C(2)/C(10) C(5)/C(10) C(25)/C(10) C(50)/C(10) C(100)/C(10) 
Minimum 0.25 0.63 1.10
 
1.14 
 
1.16 
Maximum 0.69 0.89 1.55 
 
2.45 
 
3.14 
25
th
 percentile 0.30 0.68 1.23 
 
1.34 
 
1.47 
Median 0.35 0.72 1.42
 
1.66 
 
1.93 
75
th
 percentile 0.51 0.85 1.49 
 
2.06 
 
2.54 
Average  0.41 0.75 1.36 
 
1.68 
 
1.98 
Standard deviation 0.14 0.09 0.14 
 
0.39 
 
0.578 
 
 
 
Table 4.3 Frequency Factor or Multiplier for Literature-based Rational Runoff 
Coefficient C from Different Sources. 
Return period, 
T, years 
Frequency factor, C
f
(T), C(T)/C(10) 
Gupta (1989)
1 
0 % IMP
2
65 % IMP
2
Young et al. (2009) Texas watersheds 
2 1.0 0.48 0.69 0.45 0.41 
5 1.0 0.77 0.87 0.77 0.75
10 1 1 1 1 1 
25 1.1 1.22 1.15 1.30 1.36 
50 1.2 1.40 1.22 1.54 1.68
100 1.25 1.60 1.30 1.77 1.98 
1 
C
f
(T)  from the Denver material (DRCG 1969; Jens 1979) and later published in other 
textbooks (e.g., Gupta 1989, Viessman and Lewis 2003) and design manuals (e.g., 
TxDOT 2002)  
2
 From Jens (1979) 
 106
 
Fig. 4.1 Map showing the locations of U.S. Geological Survey streamflow-gaging 
stations in Texas associated with the 36 undeveloped watersheds considered for this study 
(two stations are very close and overlapped each other). 
 107
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
R
u
n
o
f
f
 
C
o
e
f
f
i
c
i
e
n
t
,
 
C
(
T
)
0 10 20 30 40 50 60 70 80 90 100
Return Period, T(years)
08139000 Deep Creek Sub. 3 near Placid, Texas (Area = 8.86 km   )
08178465 East Elm Creek at San Antonio, Texas  (Area = 6.03 km   )
08187000 Escondido Creek Sub. 1 near Kenedy, Texas (Area = 8.52 km   )
U.S. Geological Survey streamflow-gaging stations
2
2
2
 
Fig. 4.2 Estimation of C(T) versus T for three undeveloped Texas watersheds. 
 108
 
Fig. 4.3 Box plot for the distribution of runoff coefficients for different return periods 
from Texas watersheds, from Hotchkiss and Provaznik (1995) and from Young et al. 
(2009). 
 109
 
Fig. 4.4 Frequency factors for different return periods from Texas watersheds, Young et 
al. (2009), French et al. (1974) and FHWA (Jens 1979) and Gupta (1989). 
 110
Chapter 5. Modified Rational Unit Hydrograph Method and Applications in Texas 
Watersheds 
 
 
5.1 Abstract 
The modified rational method (MRM) is an extension of the rational method to 
develop simple runoff hydrographs. The hydrographs developed using the MRM can be 
considered application of a special unit hydrograph (UH) that is termed the modified 
rational unit hydrograph (MRUH) in this paper. Being a UH, the MRUH can be applied 
to nonuniform rainfall distributions and for watersheds with drainage areas greater than 
typically used for the rational method (a few hundred acres). The MRUH was applied to 
90 watersheds in Texas using 1,600 rainfall-runoff events. Application of the MRUH 
involved three steps: (1) determination of rainfall excess using the runoff coefficient, (2) 
determination of the MRUH using drainage area and time of concentration, and (3) 
applying the unit hydrograph convolution. Times of concentration for the study 
watersheds were estimated using four empirical equations. Runoff coefficients used were 
estimated using two methods (literature-based from land-use information and back-
computed from observed rainfall and runoff data). The Gamma UH, Clark-HEC-1 UH, 
and NRCS dimensionless UH were also used to predict peak discharges of all events in 
the database.  The MRUH performed about as well as these UH methods when the same 
rainfall loss model was used. 
 
5.2 Introduction 
 111
The rational method was originally developed for estimating peak discharge, Q
p
, 
for sizing drainage structures, such as storm drains and culverts. The peak discharge, Q
p
, 
(in m
3
/s in SI units or ft
3
/s in English units) is computed using: 
,CIAmQ
op
?
 
(5.1) 
where C is the runoff coefficient (dimensionless), I is the rainfall intensity (mm/hr or 
in./hr) over a critical period of storm time (the time of concentration, T
c
), A is the 
drainage area (hectares or acres), and m
o
 is the dimensional correction factor (1/360 = 
0.00278 in SI units, 1.008 in English units). Kuichling (1889) and Llyod?Davies (1906) 
are credited with independent development of the rational method (Singh and Cruise 
1992). Texas Department of Transportation (TxDOT) guidelines for drainage design 
recommend use of the rational method for watersheds with drainage areas less than 0.8 
km
2
 or 200 acres (TxDOT 2002). 
Incorporation of detention basins to mitigate effects of urbanization on peak flows 
requires design methods to include the volume of runoff as well as the peak discharge 
(Rossmiller 1980). To use the rational method for hydraulic structures involving storage, 
the modified rational method (MRM) was developed (Poertner 1974). The term 
?modified rational method analysis" refers to ?a procedure for manipulating the basic 
rational method techniques to reflect the fact that storms with durations greater than the 
normal time of concentration for a basin will result in a larger volume of runoff even 
though the peak discharge is reduced" [p. 54 Poertner (1974)]. Emil Kuichling (1889) 
stated: ?in drainage areas of moderate size, the heaviest discharge always occurs when 
the rain lasts long enough at its maximum intensity to enable all portions of the area to 
contribute to the flow.? The MRM is based on the same assumptions as the conventional 
 112
rational method and is a conceptual extension of the rational method (Viessman and 
Lewis 2003). 
The MRM was revisited and reevaluated in this study. The hydrographs 
developed using the MRM represent application of a special unit hydrograph (UH) that 
will be termed the modified rational unit hydrograph (MRUH) in this study. The MRUH 
and unit hydrograph convolution were used to compute the direct runoff hydrographs for 
1,600 rainfall-runoff events for 90 Texas watersheds. The objectives of the MRUH 
application were (1) to evaluate the applicability of the method if blindly applied to 
watersheds of size greater than typically used with either the rational method or the 
modified rational method (that is, a few hundred acres), and (2) to study the effects of the 
runoff coefficient and the time of concentration on prediction of runoff hydrographs 
using MRUH. In addition, three other unit hydrograph models?Clark unit hydrograph 
developed for HEC?1?s generalized basin (Clark 1945; USACE 1981), Gamma unit 
hydrograph for Texas watersheds (Pradhan 2007) and Natural Resources Conservation 
Service Dimensionless (NRCS) unit hydrograph (NRCS 1972)?were used to compute 
the direct runoff hydrograph for each of the 1,600 rainfall-runoff events in 90 Texas 
watersheds for comparison of the results derived from application of the MRUH with 
these unit hydrograph methods. 
 
Revisit of MRM 
For the MRM, an urban stormwater runoff hydrograph resulting from the design 
storm is approximated as being either triangular or trapezoidal in shape (Smith and Lee 
1984; Walesh 1989; Viessman and Lewis 2003), depending on the relation between the 
 113
storm duration (D) and the time of concentration (T
c
). The rising and the falling limbs are 
linear because the time-area relation for a watershed is assumed to be linear. If the storm 
duration (D) is equal to time of concentration (T
c
) of a watershed, the resulting 
hydrograph is triangular with a peak discharge of CIAQ
p
?  at time t = T
c
; that is Case 
(A) in Fig. 5.1. If D is greater than T
c
, the resulting hydrograph is trapezoidal with 
uniform maximum discharge, CIAQ
p
?  from time D to T
c
; that is Case (B) in Fig. 5.1. 
The linear rising and falling limbs each has duration of T
c
, as shown in Fig. 5.1 (e.g., 
from Walesh 1989; Viessman and Lewis 2003).  If the storm duration D is less than T
c
, 
then the resulting hydrograph is trapezoidal with a maximum uniform discharge of Q
p
? 
(Eq. 5.2) from the end of the storm (D) to the time of concentration T
c
. The linear rising 
and falling portions of the hydrograph each has duration of D < T
c
 as shown in Case (C) 
in Fig. 5.1. Smith and Lee (1984) and Walesh (1989) reported the modified rational 
hydrograph for the case when D is less than T
c
 and stated that Q
p
? can be calculated by: 
?
?
?
?
?
?
?
c
p
T
D
CIAQ
'
 (5.2) 
Chien and Saigal (1974) used a linearized subhydrograph approach to derive three 
runoff hydrographs depending on rainfall duration and time of concentration, although 
there was an error for the case of D < T
c
 as reported by Walesh (1975). Wanielista (1990) 
discussed the rational hydrograph in the context of the contributing area and assumed that 
the contributing area varies linearly with time. He derived a triangular hydrograph for D 
= T
c
 and a trapezoidal hydrograph for D > T
c
 from the rational method, similar to results 
by Chien and Saigal (1974) and Walesh (1975). 
 114
Smith and Lee (1984) examined the rational method as a unit hydrograph. They 
noted that if the rate of change of the contributing area is constant so that the accumulated 
tributary area increases and decreases linearly and symmetrically with the time, then the 
instantaneous unit hydrograph (IUH) response function, u(t), is of rectangular shape 
given by: 
c
T
A
dt
dA
tu ??)(  (5.3) 
Using the rectangular response function (Eq. 5.3) in conjunction with a uniform rainfall 
intensity, Smith and Lee (1984) derived the resulting direct runoff hydrographs, q(t) (in 
watershed depth per time), by convolution as: 
?
?? dtuitq
t
e
)()()(
0
??
?
 (5.4) 
where ? is the time with respect to which the integration is carried out and Cii
e
?)(?  is 
the effective rainfall intensity. Two types of outflow hydrographs, triangular and 
trapezoidal shape (Fig. 5.1), were obtained from Eq. (5.4), depending on the duration of 
rainfall. Similar to Smith and Lee (1984), Singh and Cruise (1992) assumed the 
watershed is represented as a linear, time-invariant system whose instantaneous unit 
hydrograph (IUH) is a uniform rectangular distribution of base time equal to the time of 
concentration of the watershed. They used convolution to derive the S-hydrograph and D-
hour unit hydrograph from application of the rational method. The unit hydrograph 
developed thereby is called the modified rational unit hydrograph (MRUH) in this study.  
The MRUH is trapezoidal in shape and three examples of the MRUH used in this study 
are shown in Fig. 5.2.  Cases (A), (B), and (C) of the MRM in Fig. 5.1 are runoff 
 115
hydrographs and none is a unit hydrograph, although Cases (B) and (C) have the same 
shape as MRUH in Fig. 5.2. 
Guo (2000, 2001) developed a rational hydrograph method (RHM) for 
continuous, nonuniform rainfall events. The RHM was used to extract the runoff 
coefficient and the time of concentration from observed rainfall and runoff data through 
optimization. The RHM developed by Guo (2000, 2001) is not unit hydrograph method, 
but is a practical procedure to compute the moving average rainfall intensity for 
application of the MRM. Guo (2000, 2001) used a linear approximation from the 
discharge Q(T
d
) at the end of the rainfall event to zero at the time T
d
 + T
c
 for RHM. We 
checked that, for nonuniform rainfall events, this approximation is incorrect because it 
violates the conservation of mass between the rainfall excess and the runoff hydrograph. 
Treating the MRM as a unit hydrograph method, such as the MRUH, will always 
conserve mass. 
Bennis and Crobeddu (2007) developed an improved rational hydrograph (IRH) 
method for small urban catchments using a rectangular impulse response function (Singh 
and Cruise 1992; Smith and Lee 1984). They considered impervious and pervious areas 
separately.  The IRH method was calibrated and validated using ten rainfall events from 
two urban catchments.  However, the Bennis and Crobeddu (2007) IRH is not a unit 
hydrograph method. 
The unit hydrograph for a watershed can be used to predict the direct runoff 
hydrograph for any given rainfall excess hyetograph (uniform or non-uniform 
distribution) using unit hydrograph convolution (Chow et al. 1988; Viessman and Lewis 
2003). If the MRM is application of a unit hydrograph method, then the approach 
 116
establishes a continuity of hydrograph-development methods from very small watersheds 
to relatively large watersheds. For the MRUH, the assumption and restriction of the 
MRM to uniform rainfall distributions, as stated by Rossmiller (1980) and others, is not 
necessary. The MRUH can be applied to nonuniform rainfall hyetographs to obtain direct 
runoff hydrographs using convolution similar to application of other unit hydrograph 
methods. 
The D-hr MRUH results from a uniform excess rainfall intensity of 1/D in./hr 
over D hrs and has a peak discharge of A/T
c
 in ft
3
/s when drainage area A is in acres and 
T
c
 is in hours (taking into account that one-acre inch per hour is nearly equal to one cubic 
foot per second). If SI units are used (drainage area A in hectare and rainfall intensity in 
mm/hr), the peak discharge from the MRUH should be equal to A/(360T
c
) in m
3
/s.  The 
MRUH has only one control parameter?time of concentration of the watershed. The 
runoff coefficient, C, for the rational method is not a control parameter of the MRUH. 
This is because the MRUH results from one unit of rainfall excess depth and the runoff 
coefficient is actually used to determine rainfall excess, not for transformation of 
effective rainfall to direct runoff hydrograph (DRH) through application of the MRUH. 
Application of the MRUH is straightforward and similar to application of other 
unit hydrograph methods. Convolution of the unit hydrographs with the rainfall excess is 
applied to obtain the direct runoff hydrograph for each storm event.  The excess rainfall 
or the net rainfall is obtained from the product of the incremental rainfall and C, similar 
to Smith and Lee (1984). The MRUH was first tested using data obtained for concrete 
surfaces from Yu and McNown (1964). The first dataset was based on a test bed with an 
area of 152.4 m by 0.3 m (500 ft by 1 ft), surface slope of 0.02, and a uniform rainfall 
 117
intensity of 189 mm/hr. The second dataset was based on a test bed with an area of 76.8 
m by 0.3 m (250 ft by 1 ft), surface slope of 0.005, and a variable rainfall intensity with 
an initial rate of 43.2 mm/hr, increasing to 96 mm/hr at t = 6 minutes, decreasing to 45 
mm/hr at t = 18 minutes, and ending at t = 32 minutes. The T
c
 of about 5 minutes was 
computed using the Kirpich method (Kirpich 1940) for both experiments. A trapezoidal 
1-minute MRUH was developed for each experiment (Fig. 5.2A). The runoff coefficient 
was taken to be unity. The time interval used for unit hydrograph convolution was 1 
minute. Predicted and observed hydrograph ordinates used the same time interval. For 
both cases, the modeled results match the observed results well (Fig. 5.3). 
The Nash-Sutcliffe efficiency, EF, is a parameter to measure goodness-of-fit 
between modeled and observed data (Legates and McCabe 1999) and is defined by Eq. 
(A.3) in Appendix A. For hydrograph simulation, a good agreement between the 
simulated and the measured data is reached when EF is higher than 0.7 (Bennis and 
Crobeddu 2007). For the experiment using the uniform rainfall intensity (Fig. 5.3A), EF 
was 0.93 and for the experiment using the nonuniform rainfall intensity (Fig. 5.3B), EF 
was 0.80, which are greater than 0.7 for both cases indicating a good fit. 
 
5.3 Applications of MRUH in Texas Watersheds 
 
Watersheds Studied and Rainfall-Runoff Database 
 
Watershed data taken from a larger dataset (Asquith et al. 2004) accumulated by 
researchers from the U.S. Geological Survey (USGS) Texas Water Science Center, Texas 
Tech University, University of Houston, and Lamar University were used for this study. 
The dataset comprises 90 USGS streamflow-gaging stations in Texas, each representing a 
 118
different watershed (Fang et al. 2007, 2008). Location and geographic distribution of the 
stations are shown in Fig. 5.4. There are 29, 21, 7, 13 watersheds in Austin, Dallas, Fort 
Worth, and San Antonio areas, respectively. The remaining 20 watersheds are small 
watersheds located in rural areas of Texas. The drainage areas of study watersheds ranged 
from approximately 0.8 to 440.3 km
2
 (0.3 to 170 mi
2
), with median and mean values of 
17.0 km
2
 (6.6 mi
2
) and 41.1 km
2
 (15.9 mi
2
), respectively.  There are 33, 57, and 80 study 
watersheds with drainage areas less than 13 km
2
 (5 mi
2
), 26 km
2
 (10 mi
2
), and 65 km
2
 (25 
mi
2
), respectively.  The stream slope of study watersheds ranged from 0.0022 to 0.0196, 
with median and mean values of 0.0075 and 0.081, respectively.  The percentage of 
impervious area (IMP) of study watersheds ranged from approximately 0.0 to 74.0, with 
median and mean values of 18.0 and 28.4, respectively. 
 The rainfall-runoff dataset comprised 1,600 rainfall-runoff events recorded during 
1959?1986. The number of events available for each watershed varied?for some 
watersheds less than 4 events were available, whereas for others as many as 50 events 
were available (Cleveland et al. 2006). Rainfall depths ranged from 3.56 mm (0.14 in.) to 
489.20 mm (19.26 in.), with median and mean values of 57.66 mm (2.27 in.) and 66.8 
mm (2.63 in.), respectively. Maximum rainfall intensities calculated using the time of 
concentration ranged from 0.01 mm/min (0.03 in./hr) to 2.54 mm/min (6.01 in./hr), with 
median and mean values of 0.25 mm/min (0.58 in./hr) and 0.30 mm/min (0.72 in./hr), 
respectively. 
 Roussel et al. (2005) developed a geospatial database containing watershed 
drainage area, longitude and latitude of the USGS streamflow-gaging station, (which was 
treated as the outlet of the watershed), and 42 watershed characteristics of each individual 
 119
watershed. These watershed parameters were used to estimate time of concentration for 
study watersheds using four empirical methods (Roussel et al. 2005; Fang et al. 2008). 
 
Time of Concentration and Runoff Coefficients 
Time of concentration, T
c
, and the runoff coefficient, C, are the required 
parameters for application of the MRUH. The T
c
 values for the study watersheds were 
estimated by Fang et al. (2008) using four empirical equations: (1) Williams equation 
(1922) developed from data for watersheds with drainage areas less than 129.5 km
2
 (50 
mi
2
), (2) Kirpich equation (1940) developed from NRCS data for rural watersheds with 
drainage areas less than 0.45 km
2
, (3) Johnstone?Cross equation (1949) developed from 
data for watersheds with drainage areas between 65 and 4206 km
2
 (25?1620 mi
2
), and (4) 
Haktanir?Sezen equation (1990) developed from data for watersheds with drainage areas 
from 11 to 9867 km
2
 (4 to 3811 mi
2
). For large watersheds, Fang et al. (2007, 2008) 
demonstrated that the Kirpich equation provides as reliable an estimate of T
c
 as other 
empirical equations and the NRCS velocity method (Viessman and Lewis 2003).  T
c
 
estimated using the Kirpich equation reasonably approximates the average T
c
 estimated 
from observed rainfall and runoff data (Fang et al. 2007). Application of the four 
empirical equations requires watershed parameters watershed drainage area, channel 
length, channel slope, and watershed shape (Fang et al. 2008). For the present study, T
c
 
was estimated using watershed parameters developed by USGS researchers through 
automated watershed delineation using digital elevation models and geographic 
information system (GIS) software (Fang et al. 2008). The range of T
c
 values for the 
 120
study watersheds, along with median and mean values, estimated using the four empirical 
equations are presented in Table 5.1. 
 Rainfall excess was computed using the volumetric interpretation of the rational 
runoff coefficient. Wanielista et al. (1997) showed that rainfall loss for a uniform rainfall 
input (intensity of i) is equal to (1
 
-
 
C)
 
iDA and rainfall excess is equal to CiDA.  The 
rational rainfall loss method is the constant fractional loss model?in which it is assumed 
that the watershed immediately converts a constant fraction (proportion) of each rainfall 
input into an excess rainfall fraction (McCuen 1998). 
 Two estimates of the runoff coefficient were examined for the application of 
MRUH. The first is a watershed composite, literature-based coefficient (C
lit
) derived 
from land-use information for the watershed and published C
lit
 values for appropriate 
land-uses (Dhakal et al. 2011). The composite C assigned to a watershed is the area-
weighted mean C derived from the land-use classes in the watershed.  Values of C
lit
 for 
the study watersheds ranged from 0.29 to 0.63, with median and mean values of 0.50 and 
0.47, respectively. 
 The second runoff coefficient is a back-computed, volumetric runoff coefficient, 
C
vbc
, determined by preserving the runoff volume using observed rainfall and runoff data.  
C
vbc
 was estimated by the ratio of total runoff depth to total rainfall depth for individual 
observed storm event. Computed C
vbc
 ranged from 0.001 to 0.99, with median and mean 
values of 0.29 and 0.33, respectively, for 1,600 rainfall events in the study watersheds. 
The determination and comparison of C
lit
 and C
vbc
 for the study watersheds was 
documented by Dhakal et al. (2011). 
 
 121
Estimated Runoff Hydrographs Using the MRUH 
For the 90 Texas watersheds, observed rainfall hyetograph and runoff hydrograph 
data were tabulated using a time interval of five minutes. Therefore, the five-minute 
MRUH was developed for each of the 90 study watersheds. The five-minute MRUH 
duration is less than the time of concentration for all study watersheds. The basic time 
interval used for unit hydrograph convolution and hydrograph ordinates was five minutes. 
Comparison between observed and simulated peak discharges and time to peak are 
presented in Figs. 5.6 and 5.7. 
The results for the event on 07/08/1973 at the USGS streamflow-gaging station 
08157000 Waller Creek, Austin, Texas are presented in Fig. 5.5 as an illustrative 
example. The watershed drainage area is 5.72 km
2
 (2.21 square miles). The back-
computed volumetric runoff coefficient, C
vbc
, is 0.29. The T
c
 values estimated using 
Kirpich, Haktanir-Sezen, Johnstone-Cross, and Williams equations are 1.7, 2.2, 1.4, and 
3.4 hours, respectively. Peak discharge of the 5-minute trapezoidal unit hydrograph (Fig. 
5.2B) is 24 m
3
/s (cms). Duration of the rainfall event was 19 hours. Three distinct rainfall 
episodes resulted in three distinct discharge peaks. These were reasonably represented by 
results from the MRUH using T
c
 estimated by Kirpich, Haktanir-Sezen, and Johnson-
Cross equations. Results developed from the Williams equation appear to over-estimate 
time of concentration for the watershed and peak discharges were then underestimated 
(Fig. 5.5). The Nash-Sutcliffe efficiencies from MRUH model results using T
c
 values 
estimated from Kirpich, Haktanir-Sezen, Johnstone-Cross, and Williams equations are 
0.83, 0.86, 0.70, and 0.63, respectively. Simulated time to peak agrees reasonably well 
with observed values (Fig. 5.5) when using T
c
 estimated by Kirpich, Haktanir-Sezen, and 
 122
Johnson-Cross equations. However, using T
c
 estimated by Williams equation resulted in 
the computed time to peak exceeding the observed time to peak. Although the drainage 
area of Waller Creek watershed exceeds that usually accepted for MRM application, 
results from application of the MRUH reasonably approximate watershed behavior. 
 Different combinations of T
c 
and C were used for applications of MRUH to predict 
the direct runoff hydrographs for 1,600 rainfall-runoff events in 90 Texas watersheds to 
determine the sensitivity of the peak discharges and time to peak to different T
c
 and C 
values. Five combinations of T
c
 and C were used: 
(A) T
c
 estimated using Haktanir-Sezen equation and C
vbc
, 
(B) T
c
 estimated using Johnstone-Cross equation and C
vbc
, 
(C) T
c
 estimated using Williams equation and C
vbc
, 
(D) T
c
 estimated using Kirpich equation and C
vbc
, and  
(E) T
c
 estimated using Kirpich equation and C
lit
. 
 Figure 5.6 is a plot of the observed and computed peak discharges using C
vbc
 and T
c
 
values calculated using the four different empirical equations (Fang et al. 2008). In 
comparison to observed peak discharges, modeled peak discharges using T
c
 estimated 
from the Haktanir-Sezen, Johnstone-Cross and Kirpich equations not only graphically 
look alike (Fig. 5.6) but also are similar with respect to four statistical parameters (Table 
1): relative root mean square error, RRMSE (Eq. A.1 in Appendix A); coefficient of 
determination, R
2
 (Eq.. A.2); Nash-Sutcliffe efficiency, EF (Eq. A.3); and peak relative 
error, QB (Eq. A.4). The results for EF and RRMSE using the Williams equation seem to 
be inferior to others. The fraction of modeled peak discharges that are within 1/3 of a log-
 123
cycle from the 1:1 line are summarized in Table 1 and ranged from 67.7% (Williams 
equation) to 89.1% (Kirpich equation). Fraction of storms with peak relative error (QB) 
less than ?25% and ?50% is listed in Table 5.1 for applications of MRUH with different 
combinations of T
c
 and C. Applications of MRUH with T
c
 estimated from Kirpich 
equation and back-computed C
vbc
 resulted in 73% of storms with QB less than ?50%.  
Use of C
vbc
 results in preservation of event runoff volume. Ideally, computed and 
observed peaks should plot precisely along the equal value line (black line in Fig. 5.6). 
However, the unit hydrograph is a mathematical model that is an incomplete description 
of the complexity of the combination of the rainfall-runoff process and runoff dynamics. 
Therefore, the relatively simple approach cannot fully capture the nuances of watershed 
dynamics and deviations from this ideal (the equal-value line) are expected. For example, 
Asquith and Roussel (2009) computed mean residual standard error about 1/3 of a log 
cycle for annual peak discharges at 638 streamflow gauging stations in Texas. 
 Figure 5.7 is a plot of the observed time to peak (T
p
) and computed T
p
 values 
predicted using C
vbc
 and T
c
 values calculated using the four different empirical equations 
(Fang et al. 2008). For T
p
, application of MRUH using T
c
 estimated from the Haktanir-
Sezen, Johnstone-Cross and Kirpich equations produces the similar values of the 
quantitative measures: median value of T
p
 relative error (TB) and fraction of storms with 
TB less than ?25% and ? 50% (Table 5.2), and TB equal to ? 50% is graphically 
presented in Fig. 5.7 also. The T
p
 results using the Williams equation seem to be slightly 
inferior to others with respect to TB (Table 5.2). In summary, for predicting peak 
discharge and time to peak, use of T
c
 estimated from Williams equation with the MRUH 
 124
produces less accurate results than those computed using the Kirpich, Haktanir-Sezen and 
Johnstone-Cross equations. 
 Simulated peak discharges derived using MRUH with the forward-computed 
(literature-based) runoff coefficient (C
lit
) are compared against the results obtained from 
the back-computed (C
vbc
) runoff coefficient (Fig. 5.8) when T
c
 values were estimated 
using the Kirpich equation. For the peak discharges predicted using C
lit
, most of the 
values are above the equal value line (1:1 line). Based on visual inspection, about one-
third of the peak discharges from C
lit
 (triangles) are quite far from the peak discharges 
using C
vbc
 (black circles). Peak discharges computed using C
vbc
 are superior to those 
using C
lit
 with respect to all statistical measures used to assess goodness of fit (Table 5.3). 
Therefore, use of literature based values (C
lit
) will tend to generate estimates of peak 
discharge that exceed expected values (observations) when the C
lit
 values are interpreted 
as volumetric coefficients. Furthermore, literature-based estimates (C
lit
) of the runoff 
coefficient yield results that do not preserve runoff volume when applied to measured 
rainfall-runoff events. In contrast, there is no difference in quantitative measures between 
the observed and predicted time to peak values, regardless of which runoff coefficient is 
used (Table 5.3). This is because T
p
 is controlled by time of concentration and rainfall 
hyetograph and the same T
c
 values were used with different runoff coefficients (Fig. 5.8) 
for each of 90 Texas watersheds. 
 Hence, the simulation results of peak discharge are more sensitive to the choice of 
the runoff coefficients (C) or rainfall loss model. Furthermore, the time to peak results are 
not related to C when MRUH was used. 
 
 125
5.4 Estimated Runoff Hydrographs from Different Unit Hydrograph Methods 
 In addition to application of the MRUH for 90 Texas watersheds, three other unit 
hydrograph models?the unit hydrograph developed using the Clark method (Clark 1945) 
with HEC?1?s generalized basin shape (USACE 1981), the NRCS unit hydrograph 
(NRCS 1972), and the Gamma unit hydrograph (GUH) for Texas watersheds (Pradhan 
2007)?were used to develop the direct runoff hydrograph for each rainfall-runoff event 
in the database. The GUH used in this study was that developed by researchers at Lamar 
University for TxDOT project 0?4193 ?Regional Characteristics of Unit Hydrographs.? 
Linear programming was used to develop unit hydrographs from observed rainfall 
hyetographs and runoff hydrograph, and the GUH was fitted to each derived unit 
hydrograph.  Regression equations were developed for five-minute GUH parameters: 
peak discharge Q
p
 (in cfs) and time to peak T
p
 (in hours) (Pradhan 2007), 
,55075.0
06032.042612.026998.0 ?
? SLAT
p
  (5.5) 
,22352.93
5.0326.083576.0
SLAQ
p
?
?
 (5.6)
 
where A is drainage area in square miles, L is main channel length in miles, and S is main 
channel slope (ft/mile, elevation difference in feet divided by main channel length in 
miles). The ordinates of the GUH can be obtained from (Viessman and Lewis 2003): 
,
)](1[
?
?
p
T
t
p
p
e
T
t
QQ
?
?
?
?
?
?
?
?
 
(5.7) 
where Q is the discharge ordinate at time t and ? is the shape parameter of GUH. Two of 
the three GUH parameters (Q
p
, T
p
, and ?) are independent and the shape factor is 
determined from Q
p
 and T
p
 (Aron and White 1982). 
 126
 Clark?s (1945) instantaneous unit hydrograph (IUH) method is based on the time-
area curve method (Bedient and Huber 2002). The Clark IUH method is one of the unit 
hydrographs available in the flood hydrograph package HEC-1 (USACE 1981) and the 
hydrologic modeling system HEC-HMS (USACE 2000).  A synthetic time-area curve 
derived from a generalized basin shape is used to implement Clark?s IUH in HEC-1 and 
HEC-HMS.  The equations for the time-area curve are 
AI = 1.414 TI
1.5
,                      0 ? TI ? 0.5 (5.8) 
1- AI = 1.414 (1-TI)
 1.5
,           0.5 < TI <1 (5.9) 
where AI is the cumulative area as a fraction of watershed area and TI is fraction of time 
of concentration. These equations are applicable to most basins (Bedient and Huber 
2002).  In the Clark method and HEC-1/HEC-HMS programs, the resulting hydrograph is 
routed through a linear reservoir at the outlet of a watershed.  The linear reservoir routing 
was not implemented because this study is to only compare MRUH resulted from an 
equal time-area curve (such as a rectangular watershed) and Clark-HEC-1 UH resulted 
from a generalized watershed shape (ellipse-like shape having a non-uniform time-area 
curve).  Both MRUH and Clark-HEC-1 UH implemented without routing assume that the 
outflow hydrograph results from pure translation of direct runoff to the outlet. 
 The NRCS dimensionless unit hydrograph was developed in the late 1940s (NRCS 
1972). NRCS personnel analyzed a large number of unit hydrographs for watersheds of 
different sizes and in different geographic locations to develop a generalized 
dimensionless unit hydrograph in terms of t/T
p
 and q/Q
p
. The peak flow, Q
p
, for the unit 
hydrograph is computed by approximating the unit hydrograph with a triangular shape 
having base time of 8/3T
p
 and unit area (Viessman and Lewis 2003): 
 127
p
p
T
A
Q
484
? , (5.10) 
where Q
p
 is the peak discharge (cfs) and A is the drainage area (mi
2
). 
 Unit hydrographs developed using all four UH models, including the MRUH, for the 
watershed associated with the USGS streamflow-gaging station 08048520 Sycamore 
Creek in Fort Worth are shown in Fig. 5.9 (A). The purpose of this example is to 
illustrate the differences and similarities of the UH models. The shape of the MRUH is 
trapezoidal. Unit hydrographs from the Clark-HEC-1, the Gamma, and the NRCS 
methods are curvilinear. The unit hydrograph peak discharge from each model is 
different (Fig. 5.9A). However, the area under the UH curves is the same.  This is 
because each UH corresponds to 1 inch of a uniform excess rainfall over 5-minute 
duration (one impulse). 
 Gamma, Clark-HEC-1, and NRCS unit hydrographs developed for each watershed 
were applied to the 1,600 rainfall-runoff events in the database to generate direct runoff 
hydrographs. The constant fraction rainfall loss method (rational method) was used to 
estimate rainfall excess for each rainfall event. The runoff coefficient C
vbc
 determined for 
each event was used. T
c 
determined using the Kirpich method (1940) was used for those 
methods that require T
c
. As an illustrative example, the results for the observed and 
simulated direct runoff hydrographs for the rainfall event on 07/28/1973 at the USGS 
streamflow-gaging station 08048520 (Sycamore Creek in Fort Worth, Texas) by the four 
models (base flow was assumed to be zero) is presented in Fig. 5.9 (B). The watershed 
area is of 45.66 km
2
 (17.63 square miles), the time of concentration is 3.96 hours from 
the Kirpich method, and the back-computed volumetric runoff coefficient, C
vbc
, is 0.20. 
 128
Simulated peak discharges from the four UH methods are different, but comparable. For 
the particular example shown in Fig. 5.9 (B), the MRUH and the Clark-HEC-1 model 
appear to perform better than the other models with regard to prediction of peak 
discharge. For the time to peak, simulated values using the four methods agree 
reasonably well with the observed value (Fig. 5.9). Additionally, the area under the four 
simulated hydrographs matches the observed curve because event C
vbc
 was used. 
Although the drainage area of Sycamore Creek watershed exceeds that usually accepted 
for rational method application, results from the MRUH reasonably approximate 
watershed behavior. 
 The observed and modeled peak discharges from all four UH models developed 
using back-computed runoff coefficient, C
vbc
 and time of concentration from the Kirpich 
method are presented in Fig. 5.10 for 1,600 rainfall and runoff events. Modeled peak 
discharges from all the four UH models are similar (Fig. 5.10). Four statistical parameters 
RRMSE, R
2
, EF, and QB (Appendix A) between the observed and modeled peak 
discharges were computed for evaluation of model performance (Loague and Green 
1991; Cleveland et al. 2006) and are listed in Table 5.4. Based on the statistical measures, 
all the four UH models perform similarly. However, the GUH developed for Texas 
watersheds perform somewhat worse than the other three UH models (Table 5.4). 
Fractions of storms for each model meeting the acceptance tolerance of QB and TB are 
also listed in Table 5.4 (Eqs. A.4 and A.6). Using this acceptance approach, again all the 
models perform similarly. 
 
5.5 Sensitivity of the MRUH to unit hydrograph duration 
 129
 A sensitivity analysis was performed for peak discharges derived from application of 
MRUH using different unit hydrograph durations. The simulated runoff hydrographs 
were obtained for unit hydrograph durations of 5-, 10-, 20-, 30-, 40- and 50-minutes. To 
minimize error in developing discrete MRUH and DRH, a time interval of 10 minutes 
was used for hydrograph convolution when unit hydrograph durations exceeded 10 
minutes. 
 Predicted runoff hydrographs for the rainfall event on 07/28/1973 for the watershed 
associated with the USGS streamflow-gaging station 08178600 Salado Creek San 
Antonio are shown in Fig. 5.11 as an illustrative example. No noticeable differences were 
visible both in terms of the peak and the shape of the hydrographs regardless of the unit 
hydrograph duration. The Nash-Sutcliffe efficiency and relative change in the peak 
discharge for the simulated results are presented in Table 5.5. Based on review of 
statistical measures, results are not sensitive to changes in unit hydrograph duration 
(Table 5.5, EF is derived from runoff hydrograph ordinates). 
 For the 1,600 rainfall events in the database, the median relative change in the peak 
discharge (Q
RE
) for the changes in the UH durations are all 0% (Table 5.6). Fractions of 
storms with Q
RE
 less than ?5% and ?10% are listed in Table 5.6, and almost all of rainfall 
events (95 to 99%) has Q
RE
 less than ?10%. The application of MRUH is not sensitive to 
the selection of the unit hydrograph duration so long as the same time interval is used for 
hydrograph convolution. 
 
5.6 Discussion and Summary 
 
 130
 The modified rational method, MRM, is an extension of the rational method to 
produce simple runoff hydrographs for applications that do not warrant a more complex 
modeling approach. In this study, the MRM was revisited. The hydrographs developed 
using MRM can be considered an application of a special unit hydrograph termed the 
modified rational unit hydrograph, MRUH.  The MRUH method was applied to develop 
unit hydrographs for 90 watersheds in Texas. Unit hydrograph convolution was used to 
determine the direct runoff hydrograph for 1,600 rainfall-runoff events associated with 
the Texas database. The purposes were (1) to evaluate the applicability of the method if 
blindly applied to watersheds of any size, and (2) to study the effects of runoff coefficient 
and the time of concentration on prediction of runoff hydrograph using the MRUH. 
 Runoff coefficients estimated using two approaches by Dhakal et al. (2011) were 
examined for application with the MRUH. The first was a watershed composite 
literature-based coefficient (C
lit
) derived using the land-use information for the watershed 
and published C
lit
 values for various land-uses. The second was a back-computed 
volumetric runoff coefficient (C
vbc
) determined by preserving the runoff volume and 
using observed rainfall and runoff data. Times of concentration for study watersheds 
were estimated by Fang et al. (2008) from four empirical equations, which were based on 
several watershed characteristics. Predicted and observed discharge hydrographs were 
reported and compared. Simulated peak discharges and times to peak from MRUH agree 
reasonably well with observed values. The drainage area of the study watersheds 
(average 440 km
2
 or 15.6 mi
2
) is greater than that usually accepted for rational method 
application (0.8 km
2
 or 0.3 mi
2
), yet results from the MRUH reasonably approximate 
watershed behavior regardless of watershed size. Simulated peak discharges are more 
 131
sensitive to the choice of the runoff coefficient than the time of concentration. Simulated 
times to peak are moderately sensitive to the time of concentration but independent of the 
runoff coefficient. A sensitivity analysis of the MRUH to the unit hydrograph duration 
was performed. The MRUH is not sensitive to the selection of the unit hydrograph 
duration so long as the same time interval is used for hydrograph convolution. 
Three other unit hydrograph models, the Clark (using HEC?1?s generalized basin 
equations), the Gamma, and the NRCS unit hydrographs were also used to compute the 
direct runoff hydrograph for each rainfall-runoff event in the database. Runoff 
hydrographs simulated using all four methods were similar. Simulated peak discharges 
for all events in the database were similar regardless of statistical or quantitative 
measures used for comparison. For time to peak, simulated values using all four models 
agree reasonably well with observed values. The four UH models produce similar values 
of statistical and quantitative measures for both peak discharges and time to peak. 
 Three general conclusions for MRUH are: (1) Being a unit hydrograph, it can be 
applied to nonuniform rainfall distributions and for watersheds with drainage areas 
greater than typically used with either the rational method or the modified rational 
method (that is, a few hundred acres). (2) The MRUH performs about as well as other 
unit hydrograph methods used in this study for predicting the peak discharge and time to 
peak of the direct runoff hydrograph, so long as the same rainfall loss model is used. (3) 
Modeled peak discharges from application of the MRUH are more sensitive to the 
selection of runoff coefficient, less sensitive to T
c
, and not sensitive to the selection of the 
unit hydrograph duration. In predicting peak discharges and runoff hydrographs for 
 132
engineering design, rainfall loss estimation results in greater uncertainty and contributes 
more model errors than variations of UH methods and model parameters for UH. 
 
5.7 Notation 
The following symbols are used in this paper: 
? = shape parameter of gamma unit hydrograph;  
A = drainage area in hectares or acres; 
AI = cumulative area as a fraction of watershed area;  
C = runoff coefficient; 
C
lit
 = composite literature-based runoff coefficient; 
C
vbc 
= back-computed volumetric runoff coefficient; 
D = storm duration; 
EF = Nash-Sutcliffe efficiency; 
I = average rainfall intensity (mm/hr or in. /hr) with the duration equal to time of 
concentration; 
L = main channel length in mile; 
m
o
 = the dimensional correction factor (1.008 in English units, 1/360 = 0.00278 in SI 
units); 
Q
p
 = peak runoff rate in m
3
/s or ft
3
/s; 
Q
p
? = peak runoff rate of the modified rational hydrograph for the case when the storm 
duration is less than the time of concentration of the drainage area; 
QB = peak relative error between the observed and simulated peak discharges; 
q(t) = direct runoff hydrographs (in watershed depth per time) by convolution; 
 133
R
2
 = coefficient of determination; 
RRMSE = relative root mean square error; 
S = main channel slope (ft/mile); 
t = shape parameter of gamma unit hydrograph; 
TB = relative bias of direct runoff hydrograph time to peak; 
TI = fraction of time of concentration;  
T
c 
= time of concentration; 
T
p
 = time to peak; 
u(t) = rectangular instantaneous unit hydrograph response function; 
 
5.8 Appendix A: Statistical Measures to Evaluate Model Performance 
 Four statistical measures were used to analyze model results. They are the relative 
root mean square error (RRMSE), the coefficient of determination (R
2
), the Nash-Sutcliffe 
efficiency (EF), and the peak relative error (QB) between the observed and simulated 
peak discharges and times to peak (Loague and Green 1991; Feyen et al. 2000; Cleveland 
et al. 2006). The equations used to compute these measures are: 
,
)(
5.0
1
2
O
n
OP
RRMSE
n
i
ii
?
?
?
?
?
?
?
?
?
?
 (A.1) 
,
)()(
))((
2
1
22
1
12
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
?
??
?
??
?
n
i
i
n
i
i
n
i
ii
PPOO
PPOO
R
 (A.2)  
 134
and,
)(
)()(
1
2
11
22
?
??
?
??
?
?
?
?
?
?
?
???
?
n
i
N
I
n
i
iii
OOi
OPOO
EF  (A.3) 
,
i
ii
O
OP
QB
?
?  (A.4) 
where, P
i
 are the simulated peak discharge values, O
i
 are the observed peak discharge 
values, n is the number of observations and O  is the mean of the observed peak 
discharge values. RRMSE is a measure of overall spread of the residuals with respect to 
the mean observed value. The target RRMSE value is 0 for acceptance of a model (Feyen 
et al. 2000). The coefficient of determination R
2
 is a measure of the proportion of the total 
variability in observed data that can be explained by the model and ranges from 0 to 1. 
According to Moriasi et al. (2007) values of R
2
 greater than 0.5 are considered acceptable 
for a model. Although R
2
 and EF values are used often for the evaluation of a model, 
Legates and McCabe (1999) suggested that EF is a more appropriate measure for 
goodness-of-fit. For hydrograph simulation, a good agreement between the simulated and 
the measured data is reached when EF is higher than 0.7 (Bennis and Crobeddu 2007). 
Peak relative error, QB, is the difference in magnitude between the modeled and observed 
peak divided by observed peak discharge. Similar to RRMSE, values of QB near to 0 
indicate correspondence between modeled and observed values. Cleveland et al. (2006) 
suggested the following range of the QB for the acceptance of model performance: 
25.025.0 ??? QB  (A.5) 
 For evaluation of the time to peak results, relative bias of direct runoff hydrograph 
time to peak (TB) was estimated (Zhao and Tung 1994) for each storm event using: 
 135
,
po
popm
t
tt
TB
?
?  (A.6) 
where, t
pm
 is the modeled time to peak and t
po
 is the observed time to peak. A positive TB 
indicates that the observed peak occurs sooner (smaller) than the modeled peak (i.e. the 
model predicts a late peak). Similarly, a negative TB indicates that the observed peak 
occurs later (larger) than the modeled peak (i.e. the model predicts an early peak) 
[Cleveland et al. 2006]. 
 
5.9 References: 
Aron, G., and White, E. L. (1982). "Fitting a gamma distribution over a synthetic unit 
hydrograph." Water resources Bulletin, 18(1), 95-98. 
Asquith, W. H., Thompson, D. B., Cleveland, T. G., and Fang, X. (2004). "Synthesis of 
rainfall and runoff data used for Texas department of transportation research 
projects 0-4193 and 0-4194." Austin, Texas 78754-4733. 
Asquith, W. H., and Roussel, M. C. (2009). "Regression equations for estimation of 
annual peak-streamflow frequency for undeveloped watersheds in Texas using an 
L-moment-based, PRESS-Minimized, residual-adjusted approach." U.S. 
Geological Survey Scientific Investigations Report 2009-5087, 48 p. 
Bedient, P. B., and Huber., W. C. (2002). Hydrology and Floodplain Analysis, Prentice 
Hall, Upper Saddle River, NJ 07458. 
Bennis, S., and Crobeddu, E. (2007). "New runoff simulation model for small urban 
catchments." Journal of Hydrologic Engineering, 12(5), 540-544. 
Chien, J.-S., and Saigal, K. K. (1974). "Urban runoff by linearized subhydrograph 
method." ASCE Journal of the Hydraulics Division, 100(HY8), 1141-1157. 
Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). Applied hydrology, McGraw 
Hill, New York. 
Clark, C. O. (1945). "Storage and the unit hydrograph." Transactions of American 
Society of Civil Engineering, 110, 1419-1446. 
 136
Cleveland, T. G., He, X., Asquith, W. H., Fang, X., and Thompson, D. B. (2006). 
"Instantaneous unit hydrograph evaluation for rainfall-runoff modeling of small 
watersheds in north and south central Texas." Journal of Irrigation and Drainage 
Engineering, 132(5), 479-485. 
Dhakal, N., Fang, X., Cleveland, T.G., Thompson, D.B., Asquith, W.H., and Marzen, L.J. 
(2012). ?Estimation of volumetric runoff coefficients for Texas watersheds using 
land-use and rainfall-runoff data.? Journal of Irrigation and Drainage Engineering, 
138(1):43-54. 
Fang, X., Thompson, D. B., Cleveland, T. G., and Pradhan, P. (2007). "Variations of time 
of concentration estimates using NRCS velocity method." Journal of Irrigation 
and Drainage Engineering, 133(4), 314-322. 
Fang, X., Thompson, D. B., Cleveland, T. G., Pradhan, P., and Malla, R. (2008). "Time 
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and manual methods." Journal of Irrigation and Drainage Engineering, 134(2), 
202-211. 
Feyen, L., V?zquez, R., Christiaens, K., Sels, O., and Feyen, J. (2000). "Application of a 
distributed physically-based hydrological model to a medium size catchment." 
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Guo, J. C. Y. (2001). "Rational hydrograph method for small urban watersheds." Journal 
of Hydrologic Engineering, 6(4), 352-357. 
Haktanir, T., and Sezen, N. (1990). "Suitability of two - parameter gamma and three - 
parameter beta distributions as synthetic unit hydrographs in Anatolia." 
Hydrological Sciences, 35(2), 167-184. 
Johnstone, D., and Cross, W. P. (1949). Elements of applied hydrology, Ronald Press, 
New York. 
Kirpich, Z. P. (1940). "Time of concentration of small agricultural watersheds." Civil 
Engineering, 10(6), 362. 
Kuichling, E. (1889). "The relation between the rainfall and the discharge of sewers in 
populous areas." Transactions, American Society of Civil Engineers 20, 1?56. 
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 137
Lloyd-Davies, D. E. (1906). "The elimination of storm water from sewerage systems." 
Minutes of Proceedings, Institution of Civil Engineers, Great Britain, 164, 41. 
Loague, K., and Green, R. E. (1991). "Statistical and graphical methods for evaluating 
solute transport models: overview and application." Journal of Contaminant 
Hydrology, 7(1-2), 51-73. 
McCuen, R. H. (1998). Hydrologic analysis and design, Prentice-Hall, Inc., Upper 
Saddle River, N.J. 
Moriasi, D. N., Arnold, J. G., Van Liew, M. W., Bingner, R. L., Harmel, R. D., and 
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NRCS. (1972). "Hydrology", National Resource Conservation Service (NRCS), U.S. 
Dept. of Agriculture, Washington D.C. 
Poertner, H. G. (1974). "Practices in detention of urban stormwater runoff: An 
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77710. 
Rossmiller, R. L. (1980). "Rational formula revisited." International Symposium on 
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Roussel, M. C., Thompson, D. B., Fang, D. X., Cleveland, T. G., and Garcia, A. C. 
(2005). "Timing parameter estimation for applicable Texas watersheds." 0-4696-
2, Texas Department of Transportation, Austin, Texas. 
Singh, V. P., and Cruise, J. F. (1992). "Analysis of the rational formula using a system 
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Smith, A. A., and Lee, K. (1984). "The rational method revisited." Canadian Journal of 
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TxDOT. (2002). Hydraulic design manual, The bridge division of the Texas Department 
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U.S. Army Corps of Engineers (USACE), Hydrologic Engineering Center (HEC), 
Davis, CA. 
 138
USACE. (2000). ?Hydrologic modeling system HEC-HMS.? U.S. Army  Corps of 
Engineers (USACE), Hydrologic Engineering Center (HEC), Davis, California. 
Viessman, W., and Lewis, G. L. (2003). Introduction to hydrology, Pearson Education, 
Upper Saddle River, N.J. 
Walesh, S. G. (1975). "Discussion of Chien and Saigal (1974)." ASCE Journal of the 
Hydraulics Division, 101(HY11), 1447-1449. 
Walesh, S. G. (1989). Urban Water Management, Wiley, New York. 
Wanielista, M. P. (1990). Hydrology: Water quantity and quality control, John Wiley & 
Sons, Inc. 
Wanielista, M. P., Kersten, R., and Eaglin, R. (1997). Hydrology: Water quantity and 
quality control, John Wiley & Sons, Inc. 
Williams, G. B. (1922). "Flood discharges and the dimensions of spillways in india " 
Engineering (London), 134, 321. 
Yu, Y. S., and McNown, J. S. (1964). "Runoff from impervious surfaces " Journal of 
Hydraulic Research, 2(1), 3-24. 
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programming." Water Resources Management, 8(2), 101-119. 
 139
Table 5.1 Quantitative Measures of the Success of the?Peak Discharges Modeled Using MRUH 
with Back-Computed Volumetric Runoff Coefficient (C
vbc
) and Time of Concentration (T
c
) 
Estimated Using Four Equations. 
 
Statistical Parameters 
Haktanir-
Sezen 
equation 
1
 
Johnstone-
Cross equation 
2
 
Williams 
equation 
3
 
Kirpich 
equation 
4
 
RRMSE (Eq. A.1) 
5
 0.87 0.78 1.03 0.75
R
2
 (Eq. A.2) 0.66 0.72 0.70 0.75 
EF (Eq. A.3) 0.64 0.71 0.49 0.73 
Median value of QB (Eq. A.4) -0.18 0.04 -0.40 -0.09 
Fraction of storms with -0.25 ? QB ? 
0.25 
0.35 0.39 0.25 0.40 
Fraction of storms with -0.5 ? QB ? 0.5 0.68 0.70 0.60 0.73 
% of events within ? 1/3 of a log cycle 81.5 86.4 67.7 89.1 
1
 T
c
 computed using Haktanir-Sezen equation ranged from 0.8 to 17.6 hours in the study 
watersheds, with median and mean values of 2.7 hours and 3.7 hours, respectively. 
2
 T
c
 computed using Johnstone-Cross equation ranged from 0.7 to 7.7 hours in the study 
watersheds, with median and mean values of 1.9 hours and 2.3 hours, respectively 
3
 T
c
 computed using Williams equation ranged from 1.2 to 31.4 hours in the study watersheds, 
with median and mean values of 4.0 hours and 5.9 hours, respectively 
4
 T
c
 computed using Kirpich equation ranged from 0.6 to 16.2 hours in the study watersheds, with 
median and mean values of 2.3 hours and 3.2 hours, respectively 
5
 Statistical parameters are defined in Appendix A. 
 
 
Table 5.2 Quantitative Measures of the Success of the?Time to Peak Modeled Using MRUH with 
Back-Computed Volumetric Runoff Coefficient (C
vbc
) and Time of Concentration Estimated 
Using Four Equations. 
 
Statistical Parameters 
Haktanir-Sezen 
equation 
Johnstone-
Cross 
equation 
Williams 
equation 
Kirpich 
equation 
Median value of TB (Eq. A.6) 0.00 -0.07 0.09 -0.03 
Fraction of storms with -0.25 ? TB ? 
0.25 
0.52 0.52 0.46 0.53 
Fraction of storms with -0.5 ? TB ? 
0.5 
0.70 0.71 0.64 0.71 
 
 140
Table 5.3 Quantitative Measures of the Success of the?Peak Discharges and Time to Peak 
Modeled Using MRUH with Time of Concentration Estimated Using Kirpich Equation and 
Runoff Coefficients Estimated Using Two Different Methods (C
vbc
 and C
lit
). 
 
Statistical Parameters C
vbc
 C
lit
 
RRMSE (Eq. A.1)  0.75 1.07 
R
2
 (Eq. A.2) 0.75 0.48 
EF (Eq. A.3) 0.73 0.45 
Median value of QB (Eq. A.4) -0.09 0.44 
Fraction of storms with -0.25 ? QB ? 0.25 0.40 0.22 
Fraction of storms with -0.5 ? QB ? 0.5 0.73 0.46 
% of events within ? 1/3 of a log cycle 89.1 63.3 
Median value of TB (Eq. A.6) -0.03 -0.03 
Fraction of storms with -0.25 ? TB ? 0.25 0.53 0.53 
Fraction of storms with -0.5 ? TB ? 0.5 0.71 0.71 
 
 
Table 5.4 Quantitative Measures of the Success of the Peak Discharges Modeled Using Four Unit 
Hydrograph Models for 1,600 Rainfall-Runoff Events in 90 Texas Watersheds. 
Statistical Parameters MRUH Gamma UH 
Clark-HEC-1 
UH 
NRCS 
UH 
RRMSE (Eq. A.1) 0.75 0.87 0.74 0.72 
R
2
 (Eq. A.2) 0.75 0.80 0.74 0.78 
EF (Eq. A.3) 0.73 0.65 0.74 0.76 
Median value of QB (Eq. A.4) -0.09 -0.31 0.04 -0.10 
Fraction of storms with -0.25 ? QB ? 
0.25 
0.40 0.32 0.39 0.39 
Fraction of storms with -0.5 ? QB ? 
0.5 
0.73 0.71 0.70 0.76 
% of events within ? 1/3 of a log 
cycle 
89.1 81.2 87.0 90.6 
Median value of TB (Eq. A.6) -0.03 0.03 -0.04 0.00 
Fraction of storms with -0.25 ? TB ? 
0.25 
0.53 0.53 0.53 0.53 
Fraction of storms with -0.5 ? TB ? 
0.5 
0.71 0.72 0.74 0.73 
 
 141
Table 5.5 Sensitivity of Peak Discharges Modeled Using MRUH on Unit Hydrograph Duration 
for the Rainfall Event on 07/28/1973 for the Watershed Associated with the USGS Streamflow-
gaging Station 08178600 Salado Creek, San Antonio, Texas. 
UH Duration EF
1
 
Change in UH duration 
(minutes) Relative (%) change in Q
p
2
 
5-min 0.89 
10-min 0.90 5 to 10 0.00 
20-min 0.91 10 to 20  -0.30 
30-min 0.92 20 to 30  -2.42 
40-min 0.91 30 to 40  1.27 
50-min 0.92 40 to 50  -2.07 
1
 Nash -Sutcliffe efficiency (Eq. A.3) derived from runoff hydrograph ordinates 
2
 Relative change in Q
p
 = Q
RE
 (%) = (Q
p10
 ? Q
p5
)/Q
p5 
?
 
100%, where Q
p10
 and Q
p5
 are peak 
discharges calculated using unit hydrograph durations of 10 and 5 minutes, respectively. Q
RE
 for 
other UH durations is calculated in a similar way. 
 
 
Table 5.6 Sensitivity of Peak Discharges Modeled Using MRUH on Unit Hydrograph Duration 
for 1,600 Rainfall Events in 90 Texas Watersheds. 
Change in UH 
duration 
(minutes) 
Median value of 
Q
RE
 1
 
Fraction of storms with 
-5% ? Q
RE
 ? 5% 
Fraction of storms with 
-10% ? Q
RE
 ? 10% 
5 to 10 0.00 0.93 0.99 
10 to 20 0.00 0.98 0.99 
20 to 30 0.00 0.96 0.99 
30 to 40 0.00 0.90 0.96 
40 to 50 0.00 0.87 0.95 
1
 Q
RE
 is defined in Table 5. 
 
 
 
 
 142
 
Fig 5.1 The modified rational hydrographs for three different cases: (A) Duration of 
rainfall (D) is equal to time of concentration (T
c
), (B) Duration of rainfall is greater than 
T
c
, and (C) Duration of rainfall is less than T
c
. 
 143
0
5
10
15
20
25
30
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0.000
0.001
0.002
0.003
0.004
0.005
0.006
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Ti m e  ( h r )
0 1 2 3 4 5 6
Ti m e  ( m i n u t e s )
for Waller Creek, Austin, Texas
for 152.4 m x 0.3 m (Yu and McNown 1964)
for 76.8 m x 0.3 m (Yu and McNown 1964)
(A)
(B)
 
Fig. 5.2 The modified rational unit hydrographs (MRUH) developed for: (A) two lab 
settings from Yu and McNown (1964) and (B) for the watershed associated with USGS 
streamflow-gaging station 08157000 Waller Creek, Austin, Texas. 
 144
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0.000
0.001
0.002
0.003
0.004
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0.0
2.0
4.0
0.0
4.0
8.0
0 4 8 12 16 20 24 28 32 36 40
Ti m e  ( m i n u t e s )
0 4 8 12 16 20 24 28 32 36 40
Modeled discharge
Observed discharge
Incremental rainfall
I
n
c
r
e
m
e
n
t
a
l
 r
ain
f
all
 (
m
m
)
(A)
(B)
 
Fig. 5.3 Incremental rainfall hyetograph and observed and modeled runoff hydrographs 
using the MRUH for the two lab tests on concrete surfaces: (A) 152.4 m
 
?
 
0.3 m and (B) 
76.8 m
 
?
 
0.3 m reported by Yu and McNown (1964). 
 145
 
 
 
 
Fig. 5.4 Map showing the U.S. Geological Survey streamflow-gaging stations (dots) 
associated with the watershed locations in Texas.
 146
0
1
2
3
4
5
6
7
8
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0
2
4
6
8
5 10 15 20 25 30
Ti m e  ( h r )
Observed discharge
Incremental rainfall
T
c
 (Haktanir-Sezen Equation)
T
c
 (Johnstone-Cross Equation)
T
c
 (Williams Equation)
T
c
 (Kirpich Equation)
I
n
cr
e
m
en
t
a
l
 ra
i
n
f
a
l
l
 (
m
m)
 
 
Fig. 5.5 Sensitivity of Peak Discharges Modeled using MRUH on Unit Hydrograph 
Duration for the Rainfall Event on 07/28/1973 for the Watershed Associated with the 
USGS Streamflow-gaging Station 08178600 Salado Creek, San Antonio, Texas. 
 147
1
10
100
M
o
d
e
l
e
d
 
Q
p
(
c
m
s
)
1
10
100
M
o
d
e
l
e
d
 
Q
p
(
c
m
s
)
1 10 100
Observed Q
p
(cms)
1 10 100
Observed Q
p
(cms)
T
c
 (Williams Equation)
T
c
 (Haktanir-Sezen Equation)
1:1 line
? 1/3 of a log cycle
T
c
 (Kirpich Equat ion)
T
c
 (Johnstone-Cross Equation)
(A)
(C)
(B)
(D)
 
Fig. 5.6 Modeled versus observed peak discharges for 1,600 rainfall-runoff events in 90 
Texas watersheds. Modeled results were developed from MRUH using event C
vbc
 and T
c
 
estimated using four different methods: (A) Haktanir-Sezen equation, (B) Johnstone-
Cross equation, (C) Williams equation and (D) Kirpich equation. 
 148
0.01
0.1
1
10
100
M
o
d
e
l
e
d
 
T
p
(
h
r
)
0.01
0.1
1
10
100
M
o
d
e
l
e
d
 
T
p
(
h
r
)
0.01 0.1 1 10 100
Observed T
p
(hr)
0.01 0.1 1 10 100
Observed T
p
(hr)
T
c
 (Williams Equation)
T
c
 (Haktanir-Sezen Equation)
1:1 line
? 0.5 TB
T
c
 (Kirpich Equat ion)
T
c
 (Johnstone-Cross Equation)
(A)
(C)
(B)
(D)
 
 
Fig. 5.7 Modeled versus observed time to peak for 1,600 rainfall-runoff events in 90 
Texas watersheds. Modeled results were developed from MRUH using event C
vbc
 and T
c
 
estimated using four different methods: (A) Haktanir-Sezen equation, (B) Johnstone-
Cross equation, (C) Williams equation and (D) Kirpich equation. 
 149
1
10
100
M
o
d
e
l
e
d
 
Q
p
(
c
m
s
)
1 10 100
Observed Q
p
(cms)
C
vbc
C
lit
1:1 line ? 1/3 of a log cycle
 
Fig. 5.8 Modeled versus observed peak discharges developed from MRUH using C
vbc
 
(triangles) and C
lit
 (circles) with time of concentration estimated using Kirpich equation 
for 90 Texas watersheds. 
 150
0
10
20
30
40
50
60
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0
6
12
18
24
I
n
c
r
e
m
e
n
t
a
l
 
R
a
i
n
f
a
l
l
 
(
m
m
)
0
20
40
60
80
100
120
140
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
18 20 22 24 26 28 30 32 34 36 38 40 42
Time  ( hr)
0 1 2 3 4 5 6 7 8 9 10 11
Observed
Modeled (MRUH)
Modeled (Clark-HEC-1)
Modeled (Gamma)
Modeled (NRCS)
Incremental Rainfall
MRUH
Clark-HEC-1 UH
Gamma UH
NRCS UH
(A)
(B)
 
Fig. 5.9 (A) Modified rational, Gamma, Clark-HEC-1, and NRCS unit hydrographs 
developed for the watershed associated with USGS streamflow-gaging Station 08048520 
Sycamore, Fort Worth, Texas; and (B) Rainfall hyetograph, observed and modeled runoff 
hydrographs using the four different unit hydrographs for the rainfall event on 
07/28/1973 for the same watershed. 
 151
1
10
100
M
o
d
e
l
e
d
 
Q
p
(
c
m
s
)
1
10
100
M
o
d
e
l
e
d
 
Q
p
(
c
m
s
)
1 10 100
Observed Q
p
(cms)
1 10 100
Observed Q
p
(cms)
Clark-HEC-1 UH
MRUH
1:1 line
? 1/3 of a log cycle
NRCS UH
Ga mma  UH
(A)
(C)
(B)
(D)
 
 
Fig. 5.10 Modeled versus observed peak discharges using: (A) MRUH, (B) Gamma UH, 
(C) Clark-HEC-1 UH, and (4) NRCS UH for 1,600 rainfall-runoff events in 90 Texas 
watersheds. 
 152
0
10
20
30
40
D
i
s
c
h
a
r
g
e
 
(
c
m
s
)
0
4
8
12
0 2 4 6 8 10 12 14 16 18 20
Ti m e  ( h r )
Observed
5-min
10-min
20-min
30-min
40-min
50-min
Incremental rainfall
I
n
c
r
e
m
e
n
t
a
l
 r
a
in
f
a
ll 
(
m
m
)
 
Fig. 5.11 Observed and modeled runoff hydrographs using MRUH with six unit 
hydrograph durations (5, 10, 20, 30, 40 and 50 minutes) for the rainfall event on 
07/28/1973 for the watershed associated with the USGS streamflow-gaging station 
08178600 Salado Creek, San Antonio, Texas. 
 153
Chapter 6 Conclusions and Recommendations 
 
 
6.1 Conclusions 
This research work is a part of TxDOT Project 0-6070 ?Use of the Rational and 
the Modified Rational Methods for TxDOT Hydraulic Design?. The objective of the 
project is to evaluate appropriate conditions for the use of the rational method and 
modified rational methods for designs on small watersheds; evaluate and refine, if 
necessary, current tabulated values of the runoff coefficient and construct guidelines for 
TxDOT analysts for the selection of appropriate parameter values for Texas conditions. 
The objective is achieved through four different phases of work. 
For the first phase of our study, volumetric runoff coefficients were estimated for 
90 Texas watersheds using three different methods?(1) a watershed composite, literature-
based coefficient (C
lit
) was derived from land-use information for the watershed and 
published C
lit
 values for appropriate land-uses, (2) back-computed volumetric runoff 
coefficient (C
vbc
) was estimated by the ratio of total runoff depth to total rainfall depth for 
individual storm events and, (3) rank-ordered volumetric runoff coefficient (C
vr
) was 
determined from the rank-ordered data; similar to the procedure used by Schaake et al. 
(1967). The median value of C
lit
 for developed watersheds exceeds that for undeveloped 
watersheds. Watershed-median values of C
vbc
 for 45 developed watersheds in Texas with 
percent imperviousness greater than 15% are consistent with median values of runoff 
coefficient R
v
 reported for 60 NURP watersheds by the USEPA. 
 154
The key conclusions of this phase of study are: 
? Published runoff coefficients, even though they were not developed from 
observed rainfall-runoff measurements and instead resulted from a survey on 
engineering practices in 1950s, reflect the physical meanings of the original 
runoff coefficients introduced by Kuichling in 1889 ? the runoff coefficient
is related to the percent impervious area within the watershed. Therefore, 
published runoff coefficients remain useful for engineering design of drainage 
systems. 
? The distribution of C
vr
 is different from that of C
lit
 with about 80 percent of C
lit
 
value greater than C
vr
 value. This result might indicate that literature-based runoff 
coefficients overestimate peak discharge for drainage design when used with the 
rational method. 
? Volumetric runoff coefficients are useful in transforming rainfall depth to runoff 
depth such as is done in the curve number method (SCS 1963) and for watershed 
rainfall-runoff modeling, e.g., the fractional loss model (McCuen 1998, p. 493). 
For the second phase of our study, rate-based runoff coefficients, C
rate
, were 
estimated using two different methods. First, C
rate
 was estimated using rational equation; 
the rainfall intensity was computed as the maximum intensity for a moving time window 
of duration T
c
 before and up to the time to peak, T
p
. Second, the frequency-matching 
approach (Schaake et al., 1967) was used to extract a representative runoff coefficient 
(C
r
) for each watershed. The key conclusions of this phase of study are: 
 155
? The rate-based C is dependent on rainfall intensity averaging time t
av
 used for the 
study, because estimates of the runoff coefficient based on observed data cannot 
be decoupled from the selection of the time-response characteristic.  
? The distributions of the watershed-average and watershed-median C
rate
 are 
similar. The C
r
 values for the developed watersheds are consistently greater than 
those for the undeveloped watersheds. 
? The values of C
r
 and C
rate
 were compared with the literature based runoff 
coefficients (C
lit
) developed from land-use data for these study watersheds. About 
75 percent of C
lit
 values are greater than C
r
. 
? For typical applications of the rational method in urban watersheds, watershed C
lit
 
is less than C
r
; using smaller C
lit
 would underestimate Q
p
 for design. 
For the third phase of our study, the runoff coefficients C(T) for different return 
periods (T) were developed for the 36 undeveloped Texas watersheds using previously 
published regional regression equations of peak discharge and county-based tabulated 
empirical coefficients for a model of rainfall intensities at different T.  The frequency 
factors C
f
(T) = C(T)/C(10) determined in this study exceed those values in textbooks such 
as Gupta (1989) and Viessman and Lewis (2003) and those from TxDOT (2002) when T 
> 10 years. The key conclusions of this phase are: 
? C(T) values increase with T and these increases are more than previously thought. 
? The frequency factors determined for the 36 Texas watersheds and the 72 Kansas 
watersheds (Young et al. 2009), larger than those mostly found in literature, are 
for undeveloped watersheds with relatively small percent impervious areas.  
 156
? The frequency factors mostly found in the literature, smaller than those 
determined for the 36 Texas watersheds, are appropriate for urban watersheds 
with relatively large percentages of impervious area, as supported and presented 
in literature (e.g., DRCG 1969; Stubchaer 1975; Jens 1979; Gupta 1989; 
Viessman and Lewis 2003; TxDOT 2002). 
? When the frequency factor is applied, if resulted C(T) is greater than unity, Jens 
(1979), Gupta (1989) and TxDOT (2002) suggested setting C(T) equal to 1. 
 
 The modified rational method, MRM, is an extension of the rational method to 
produce simple runoff hydrographs for applications that do not warrant a more complex 
modeling approach. For the fourth phase of our study, the MRM was revisited. The 
hydrographs developed using MRM can be considered an application of a special unit 
hydrograph termed the modified rational unit hydrograph, MRUH.  The MRUH method 
was applied to develop unit hydrographs for 90 watersheds in Texas. Runoff coefficients 
estimated using two approaches were examined for application with the MRUH. The first 
was a watershed composite literature-based coefficient (C
lit
) derived using the land-use 
information for the watershed and published C
lit
 values for various land-uses. The second 
was a back-computed volumetric runoff coefficient (C
vbc
) determined by preserving the 
runoff volume and using observed rainfall and runoff data. Times of concentration for 
study watersheds were estimated by Fang et al. (2008) from four empirical equations, 
which were based on several watershed characteristics. Simulated peak discharges and 
times to peak from MRUH agree reasonably well with observed values. The drainage 
area of the study watersheds (average 440 km
2
 or 15.6 mi
2
) is greater than that usually 
 157
accepted for rational method application (0.8 km
2
 or 0.3 mi
2
), yet results from the MRUH 
reasonably approximate watershed behavior regardless of watershed size. Simulated peak 
discharges are more sensitive to the choice of the runoff coefficient than the time of 
concentration. Simulated times to peak are moderately sensitive to the time of 
concentration but independent of the runoff coefficient. The MRUH is not sensitive to the 
selection of the unit hydrograph duration so long as the same time interval is used for 
hydrograph convolution. Three other unit hydrograph models, the Clark (using HEC?1?s 
generalized basin equations), the Gamma, and the NRCS unit hydrographs were also used 
to compute the direct runoff hydrograph for each rainfall-runoff event in the database. 
Runoff hydrographs simulated using all four methods were similar. Simulated peak 
discharges for all events in the database were similar regardless of statistical or 
quantitative measures used for comparison. For time to peak, simulated values using all 
four models agree reasonably well with observed values. The four UH models produce 
similar values of statistical and quantitative measures for both peak discharges and time 
to peak. Three key conclusions for MRUH are: 
? Being a unit hydrograph, it can be applied to nonuniform rainfall distributions and 
for watersheds with drainage areas greater than typically used with either the 
rational method or the modified rational method (that is, a few hundred acres).  
? The MRUH performs about as well as other unit hydrograph methods used in this 
study for predicting the peak discharge and time to peak of the direct runoff 
hydrograph, so long as the same rainfall loss model is used. 
? Modeled peak discharges from application of the MRUH are more sensitive to the 
selection of runoff coefficient, less sensitive to T
c
, and not sensitive to the 
 158
selection of the unit hydrograph duration. In predicting peak discharges and 
runoff hydrographs for engineering design, rainfall loss estimation results in 
greater uncertainty and contributes more model errors than variations of UH 
methods and model parameters for UH. 
 
6.2 Recommendations 
Based on the analysis of the volumetric runoff coefficients for 45 developed 
watersheds in Texas and 60 NURP watersheds, a polynomial regression equation was 
recommended which can be used to estimate volumetric runoff coefficients for developed 
urban watersheds that are similar to the 45 developed watersheds in Texas:  
04.030.111.266.1
23
???? IMPIMPIMPC
v
 (6.1) 
 
Above equation is useful mainly for the urban (developed) watersheds. Further study is 
recommended in future to correlate runoff coefficients for undeveloped watersheds to soil 
types and other watersheds characteristics like slope, initial soil moisture condition and 
land-use. 
A single equation was recommended to estimate the rate-based runoff coefficient 
C* (?C-star?) for the unified rational method (URAT) developed for TxDOT: 
15.085.0 ?? IMPC  (6.2) 
The above equation is consistent with the Kuichling?s original idea of the runoff 
coefficient as the amount of imperviousness of the drainage area. Kuichling (1889) 
concluded that the percentage of the rainfall discharged for any given watershed studied 
is nearly equal to the percentage of impervious surface within the watershed. Several 
researchers [Longobardi et al. (2003); Merz and Bl?schl (2009)] have shown that runoff 
 159
coefficients are strongly correlated with the initial soil moisture condition. With the 
increase of the rainfall duration, the degree of land saturated also increases and the runoff 
coefficient increases (or the imperviousness as proposed by Kuichling increases). So for 
future study, it is recommended to study the variation of the runoff coefficient with the 
degree of saturation of the land (temporal variation of C). 
Many would argue that the application of the rational method is not appropriate 
for the range of watershed areas presented in this study. The TxDOT guidelines 
recommend the use of the rational method for watersheds with drainage areas less than 
0.8 km
2
 (200 acres) (TxDOT 2002). However our study showed that there is no 
demonstrable trend in runoff coefficient with drainage area. We applied rate-based C to 
estimate the peak discharge for the study watersheds and found out that the differences 
between the observed and modeled Q
p
 are generally within the expected errors from 
typical hydrologic analysis. We do not advocate any specific limits that should be 
imposed on drainage area for application of the rational method. However further study is 
recommended in other watersheds and with more extensive database, to determine what 
is the reasonable size that can be used with the rational method for the hydrologic design. 
It is observed that application of the MRUH is simple and straightforward. Like 
other UH methods, MRUH can be applied to large watersheds with non-uniform rainfall 
distribution. However, using the runoff coefficient for the rainfall loss estimation doesn?t 
account for the initial moisture condition of the watershed. We concluded that in 
predicting peak discharges and runoff hydrographs for engineering design, rainfall loss 
estimation results in greater uncertainty and contributes more model errors than 
variations of UH methods and model parameters for UH. So for future study it is 
 160
recommended to incorporate the runoff coefficient with another loss parameter which 
accounts for the initial moisture condition of the watershed for rainfall loss estimation in 
application of the MRUH. 
 
 
6.3 References 
 
DRCG. (1969). ?Urban Storm Drainage Criteria Manual, Vols. 1 and 2.? Prepared by 
Wright-McLaughlin Engineers for Denver Regional Council of Governments 
(DRCG), Denver, CO. 
Fang, X., Thompson, D. B., Cleveland, T. G., Pradhan, P., and Malla, R. (2008). ?Time 
of concentration estimated using watershed parameters determined by automated and 
manual methods.? Journal of Irrigation and Drainage Engineering, 134(2), pp. 202?
211. 
Gupta, R. S. (1989). Hydrology and hydraulic systems, Prentice Hall, Englewood Cliffs, 
New Jersey.  
Jens, S. W. (1979). Design of Urban Highway Drainage, Federal Highway 
Administration (FHWA), Washington DC. 
Kuichling, E. (1889). ?The relation between the rainfall and the discharge of sewers in 
populous areas.? Transactions, American Society of Civil Engineers 20, pp. 1?
56McCuen, R. H. (1998). Hydrologic analysis and design, Prentice-Hall, Inc., 
Upper Saddle River, N.J. 
Longobardi, A., Villania, P., Graysonb, R., and Westernb, A. (2003). "On the relationship 
between runoff coefficient and catchment initial conditions." Proceedings of 
MODSIM 2003, 867?872. 
Merz, R., and Bl?schl, G. (2009). "A regional analysis of event runoff coefficients with 
respect to climate and catchment characteristics in Austria." Water Resources 
Research, 45(1), W01405. 
Schaake, J.C., Geyer, J.C., and Knapp, J.W. (1967). ?Experimental examination of the 
rational method.? Journal of the Hydraulics Division, ASCE, 93(6), pp. 353?370. 
Stubchaer, J. M. ?The Santa Barbara urban hydrograph method.? National Symposium of 
Urban Hydrology and Sediment Control, University of Kentucky, Lexington, KY. 
TxDOT. (2002). Hydraulic design manual, The bridge division of the Texas Department 
of Transportation (TxDOT), 125 East 11th street, Austin, Texas 78701. 
 161
Viessman, W., and Lewis, G.L. (2003). Introduction to hydrology, 5th Ed., Pearson 
Education, Upper Saddle, N.J., 612 p.  
Young, C. B., McEnroe, B. M., and Rome, A. C. (2009). ?Empirical determination of 
rational method runoff coefficients.? Journal of Hydrologic Engineering, 14, 1283 p.