Wildcat5 for Windows, 

A Rainfall-Runoff Hydrograph Model: 

User Manual and Documentation

Richard H. Hawkins and Armando Barreto-Munoz

United States Department of Agriculture

Forest Rocky Mountain General Technical

Service Research Station Report RMRS-GTR-334 April 2016
Hawkins, R.H.; Barreto-Munoz, A. 2016. Wildcat5 for Windows, a rainfall-runoff 

hydrograph model: user manual and documentation. Gen. Tech. Rep. RMRS-334. 

Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain 

Research Station. 68 p.

Abstract 

Wildcat5 for Windows (Wildcat5) is an interactive Windows Excel®-based software 

package designed to assist watershed specialists in analyzing rainfall runoff events to 

predict peak flow and runoff volumes generated by single-event rainstorms for a variety of 

watershed soil and vegetation conditions. Model inputs are: (1) rainstorm characteristics, 

(2) parameters related to watershed soil and cover, (3) runoff timing parameters, and 

(4) unit hydrograph shape and scale selections. Many choices are available for each of 

the input categories and guidance is provided for their appropriate selection. The model is 

intended for small catchments responsive to conditions of upland soils and cover. Its peak 

flow estimation techniques are appropriate for projects such as gully control, culvert sizing 

and forest roads, environmental impact analyses, and post-wildfire hydrologic response.

Keywords: hydrology, model, Curve Number, hydrograph, fire, grazing

Authors

Richard H. Hawkins is a professor emeritus, Watershed Resources and Ecohydrology, 

School of Natural Resources, Department of Agricultural and Biosystems Engineering, and 

Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721 

(Emeritus, October 1, 2011).

Armando Barreto-Munoz is a research assistant, Department of Agricultural and 

Biosystems Engineering, University of Arizona, Tucson, AZ 85721.
Contents

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Download Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Purpose of Wildcat5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Applications of Wildcat5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Overview of User Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Features of Wildcat5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Limitations and Omissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Computer Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2: Quick Start Guide and Example . . . . . . . . . . . . . . . . . . . . . 8

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Program Installation and Execution . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Example..............................................12

Chapter 3: Storm Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Distributions in Wildcat5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.21 User Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.22 Standard Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.221 Farmer–Fletcher (Great Basin, UT) . . . . . . . . . 18

3.222 NEH4B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.223 Uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.23 Custom Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Generic Design Rainfall Distribution . . . . . . . . . . . . . . . . . . . . . . 19

3.31 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.32 Application in Wildcat5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Effects of Distribution Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 4: Rainfall Excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Runoff Curve Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.21 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.22 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.23 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.24 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.241 Hydrologic Soil Groups . . . . . . . . . . . . . . . . . . . 26

i
ii

4.25 Effects of Fire on Curve Numbers . . . . . . . . . . . . . . . . . . . 27

4.251 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.252 U.S. Forest Service Tables . . . . . . . . . . . . . . . . 27

4.253 Santa Barbara, CA, Tables . . . . . . . . . . . . . . . . 28

4.254 Easterbrook Estimates . . . . . . . . . . . . . . . . . . . . 29

4.255 Goodrich–Automated Geospatial Watershed 

 Assessment (AGWA) Simulations . . . . . . . . . . . . . 31

4.26 Effects of Grazing on Curve Numbers . . . . . . . . . . . . . . . 31

4.261 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.262 Jornada Experimental Range, NM, 

 Cover Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.263 Badger Wash, CO, Paired Watershed Studies . . 32

4.264 Effects of Vegetation Conversion . . . . . . . . . . . 32

4.265 Pasture–Meadows Studies . . . . . . . . . . . . . . . . . 32

4.266 Australian CNs by Grazing Intensity . . . . . . . . 33

4.27 Curve Number with Ia/S = 0.05 . . . . . . . . . . . . . . . . . . . . 33

4.271 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.272 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Constant Infiltration Capacity: Φ-Index . . . . . . . . . . . . . . . . . . . . 34

4.31 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.32 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.33 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Distributed Infiltration Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.41 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.42 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.43 Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.431 Soil Hydraulic Conductivity . . . . . . . . . . . . . . . 36

4.432 Cover Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.44 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.45 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.46 Other Influences on Loss Rates . . . . . . . . . . . . . . . . . . . . . 39

4.5 Runoff Fraction (Runoff Ratio) . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.51 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.52 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.53 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.6 Distributed Loss Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.61 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.62 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.63 Distributed Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.64 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.65 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.66 Simulating Complacent and Violent Responses . . . . . . . . 42
iii

4.7 Complacent–Violent Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.71 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.72 Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.73 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Curve Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Curve Number with Ia/S = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . 47

φ-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Distributed Infiltration Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 48

Runoff Fraction (Runoff Ratio) . . . . . . . . . . . . . . . . . . . . . . . . . 49

Complacent–Violent Response . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 5: Timing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Choices and Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 6: Unit Hydrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 Triangular Unit Hydrographs with General Geometry . . . . . . . . . 54

6.21 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.22 Background and Description . . . . . . . . . . . . . . . . . . . . . . . 54

6.23 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.24 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.241 Equations for Rising and Falling Limbs . . . . . . 56

6.242 Direct Solutions for HF and “b” . . . . . . . . . . . . 56

6.25 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Broken Triangular Unit Hydrograph . . . . . . . . . . . . . . . . . . . . . . . 56

6.31 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.32 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.4 Curvilinear (SCS–NRCS) Unit Hydrograph . . . . . . . . . . . . . . . . . 57

6.41 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.42 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.43 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Triangular Unit Hydrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Curvilinear Unit Hydrographs . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 7: Output Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.1 Effective Loss Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2 Effective Curve Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.21 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.22 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3 Initial Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4 Post-event Curve Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.41 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.42 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.43 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.44 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.45 Graphical Representations . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.5 Maximum Contributing Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.51 Concepts and Application . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.52 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.6 Transient Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.61 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.62 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.7 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 8: Reservoir Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2 Concepts and Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.3 Application in Wildcat5..................................66

8.4 Technical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.5 Chapter References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

iv
USDA Forest Service RMRS-GTR-334. 2016. 1

Background

The original version of this software was written in 1974 for class use at Utah 

State University on an early Wang desktop computer (Wang Laboratories, Inc., 

Lowell, MA), and programmed in Wang BASIC. Patterned directly after examples in 

the in-service hydrology guide of the U.S. Department of Agriculture (USDA) Soil 

Conservation Service (now the Natural Resources Conservation Service), it used 

runoff Curve Numbers as the rainfall excess mechanism and fixed triangular unit 

hydrographs. It had limited capabilities. It was later rewritten, successively improved, 

and circulated in GW-BASIC® and QuickBASIC® (Microsoft Corp., Redmond, 

WA)1. The source code for these early programs could be contained on two singlespaced pages and was nameless.

In 1978, the Utah Division of Oil, Gas, and Mines contracted with Utah State 

University to reprogram the model in Fortran. It was also made available to the U.S.

Department of the Interior, Bureau of Land Management (BLM) and the USDA Forest 

Service, for use on mainframes. Co-existing with the desktop versions, it was widely 

applied, and incrementally improved. An enhanced version, including graphical outputs, was developed about 1985 by Richard S. Moore, under a contract with the BLM 

Denver Federal Service Center.

In 1989 and 1990, a much-enhanced Microsoft Disk Operating System (DOS) 

desktop version with additional options was constructed at the University of Arizona 

by Richard H. Hawkins and R.J. Greenberg under a contract with the BLM Denver 

Federal Service Center. This version was called Wildcat4 and used the QuickBASIC 

source code. It is still used in compiled form in DOS environments. Its performance 

checks well against the current model.

However, advances in computer technology gradually left DOS software 

stranded, and Wildcat4 is increasingly awkward to use in Microsoft Windows®-based 

systems. In 2005, as a student exercise at the University of Arizona, a version of 

Wildcat4 in Visual Basic® for Windows was contributed by Armando Barreto-Munoz. 

Called Wildcat4W, it is the point of departure for Wildcat5, the current offering.

Disclaimer

Wildcat5 is software in the public domain, and the recipient may not assert any 

proprietary rights thereto nor represent it to anyone as other than a government-produced program. Wildcat5 is provided “as-is” without warranty of any kind, including, 

but not limited to, the implied warranties of merchantability and fitness for a particular 

purpose. The user assumes all responsibility for the accuracy and suitability of this 

program for a specific application. In no event will the U.S. Forest Service or the 

University of Arizona or any of the program and manual authors be liable for any 

damages, including lost profits, lost savings, or other incidental or consequential damages arising from the use of or the inability to use this program.

1 The use of trade or firm names in this publication is for reader information and does not 

imply endorsement by the U.S. Department of Agriculture of any product or service.
2 USDA Forest Service RMRS-GTR-334. 2016.

Download Information

The Wildcat5 program and manual can be downloaded from http://www.stream.

fs.fed.us/publications/software.html.

This software and publication may be updated as features and modeling capabilities are added to the program. Users may wish to periodically check the download 

site for the latest updates. Errors of omission, logic, or miscalculation should be 

brought to the attention of the authors or the National Stream and Aquatic Ecology 

Center.

Wildcat5 is supported by, and limited technical support is available from, the 

U.S. Forest Service, National Stream and Aquatic Ecology Center, Watershed, Fish, 

Wildlife, Air, and Rare Plants Staff, Fort Collins, CO. The preferred method of contact for obtaining support is to send an email to rmrs_stream@fs.fed.us requesting 

“Wildcat5 Support” in the subject line. You may also write to the U.S. Forest Service, 

Rocky Mountain Research Station, National Stream and Aquatic Ecology Center, 

2150A Centre Avenue, Suite 368, Fort Collins, CO 80526-1891, or call 970-295-5986.
USDA Forest Service RMRS-GTR-334. 2016. 3

Chapter 1: Introduction

1.1 Purpose of Wildcat5

Wildcat5 for Windows is a rainstorm-runoff hydrograph model designed to run 

interactively within Microsoft Excel. The user-friendly software program is designed 

to assist watershed specialists in analyzing rainfall-runoff events to predict peak flow 

and runoff volumes generated by single-event rainstorms for a variety of watershed 

soil, vegetation, and land-use conditions, including post-wildfire conditions. 

The general model strategy of Wildcat5 is that of a traditional rainfall-runoff 

model. Necessary model inputs are: (1) rainstorm characteristics of depth, duration, 

and distribution; (2) parameters related to watershed soil and cover to calculate runoff 

depths; (3) runoff timing parameters to define the travel times to the watershed outlet; 

and (4) unit hydrograph shape and scale. Multiple choices are available for each of the 

input categories and guidance is provided for their appropriate selection. The model 

is intended for small catchments responsive to upland soil and vegetation conditions.

Regardless of the application, considerable user judgment or experience is required to 

select appropriate input parameters and obtain reasonable results.

The model is based largely on the U.S. Department of Agriculture (USDA) 

Curve Number method for generating rainfall-runoff, with several other options. It 

also follows USDA’s use of unit hydrographs. Primary technical sources for these 

approaches are two National Engineering Handbooks by the USDA Natural Resources 

Conservation Service (NRCS; formerly the Soil Conservation Service, or SCS) and its 

widely distributed Technical Release 55, hereafter abbreviated as NEH4, NEH630, and 

TR55, respectively. Full citations are given in the Chapter References.

1.2 Applications of Wildcat5

A common problem in applied hydrology is that of estimating rates of runoff 

volume and peak flows of various return periods from ungauged wildland watersheds.

The peak flow estimation techniques in Wildcat5 are applicable to the many kinds and 

complexity of projects on which U.S. Forest Service hydrologists and others typically 

work. Examples of projects requiring peak flows are the design of gully stabilization 

structures, culvert and bridge sizing for low-volume forest roads, flood plain mapping 

in rural areas, environmental impact analysis, and the estimation of peak flows after 

wildfires. In cases involving water storage, such as stock ponds and small reservoirs, 

runoff volume is also required and the entire hydrograph must be developed. More sophisticated methods including unit hydrograph, flood routing, and stochastic frequency 

analysis are available and may be appropriate for projects where failure would cause 

catastrophic property damage or loss of life.

Because Wildcat5 is based on general rainfall-runoff hydrology, it can be applied 

to almost any kind of land use and watershed where model inputs are available and 

where peak flows are due to large rainfall events. Most rainfall-runoff models like 

Wildcat5 have conceptual origins on rain-fed agricultural watersheds, urban areas, 

and rangelands. Thus, most general models, including Wildcat5, do not work as well 
4 USDA Forest Service RMRS-GTR-334. 2016.

in forested watersheds with deep soils and heavy cover. An attempt to bridge this 

gap—with some supporting data—is offered here as the Complacent–Violent option 

for rainfall excess in chapter 4. Transfer of this tool to western wild lands was made 

more in response to a need for a calculation method (despite some loss of validity and 

usefulness) rather than because the methods fit well with western wildland conditions.

Wildcat5 and similar rainfall-runoff models were intended for watersheds where 

flow originates as direct runoff from rainfall. This condition is sometimes satisfied 

after severe wildfires that create extensive hydrophobic conditions. Rainfall-runoff 

models are not well suited to handle situations where maximum runoff includes 

snowmelt or watersheds where runoff may be delayed by heavy forest litter, porous 

topsoil, or lakes and wetlands. Some of these limitations can be overcome by carefully 

adjusting input parameters. In all instances, however, sound judgment is required and 

the user should be aware of the uncertainty associated with model outputs and inputs.

1.3 Overview of User Manual

This manual provides a Quick Start Guide for using the software, including an 

example for ready use of the program. It also describes the fundamental concepts, 

capabilities, limitations, features, input requirements, and output of Wildcat5. The 

manual is organized in the same logical fashion in which the data are entered when 

using the program, as follows: 

Chapter 1: Introduction—this chapter.

Chapter 2: Quick Start Guide and Example—provides a short explanation 

of how to use the program, along with an example for those with experience using 

rainfall-runoff models.

Chapter 3: Storm Rainfall—provides guidance on selecting storm distributions.

Available options are the (1) SCS Type B (the most widely used), (2) Farmer–Fletcher, 

(3) uniform, (4) custom, and (5) generic design storm distribution.

Chapter 4: Rainfall Excess—provides guidance on selecting a conceptual 

model for determining direct runoff (in other words, rainfall excess) from rainstorms. Available options are (1) distributed Curve Number (the default with initial 

abstraction of 0.2), (2) distributed Curve Number with initial abstraction set at 

0.05, (3) exponentially distributed infiltration capacities, (4) distributed loss depth 

(F), (5) lumped constant loss rate (φ- index), (6) lumped constant loss fraction, and 

(7) Complacent–Violent.

Chapter 5: Timing Parameters—provides guidance on timing parameters for 

how quickly rainfall excess becomes runoff in terms of time of concentration or lag.

Available options are (1) user choice override, (2) Kirpich’s equation, (3) Kent’s equation, and (4) Simas’ equation.

Chapter 6: Unit Hydrographs—provides guidance on selecting the form of 

the unit hydrograph for runoff. Available options are (1) the simple triangular unit 

hydrograph (most used), (2) the variable triangular unit hydrograph, (3) the broken triangular unit hydrograph, and (4) the SCS dimensionless curvilinear unit hydrograph.
USDA Forest Service RMRS-GTR-334. 2016. 5

Chapter 7: Output Information—explains the graphical and tabular outputs 

generated by Wildcat5. Output displays are (1) Summary Output Table; (2) Runoff 

hydrograph Table; (3) Outflow Graphs; (4) Cum. Rainfall(P) and Runoff(Q) with Time

and rainfall excess, or Rainfall(P) - Runoff(Q); and (5) Comparative Rainfall(P) - 

Runoff(Q) graph.

Chapter 8: Reservoir Routing—provides guidance for estimating inflow and 

outflow hydrographs due to routing runoff through a storage reservoir.

1.4 Features of Wildcat5

Wildcat5 is a user-friendly, touch-and-feel, follow-your-nose program usable 

by anyone who has experience in Excel and some background in the fundamentals 

of rainfall-runoff models. To these users, most of it should be self-explanatory and 

intuitively obvious. The program offers extensive help options that provide guidance 

for the large number of input options. The most commonly used options are generally 

highlighted as defaults. 

Wildcat5 and this user manual are organized in the same sequence in which 

you would input data into a traditional rainfall-runoff model. The sequence of natural 

processes represented in rainfall-runoff models, the computational steps, and user 

options are shown in figure 1-01. A simple reservoir (pond) routing model based on 

the calculated hydrograph is also included. This manual follows the same sequence.

Internally, Wildcat5 calculations are in English units. If you work with metric units, 

Wildcat5 converts all input and output values internally.

Necessary inputs to the model are:

1. Rainfall characteristics of depth, duration, and distribution. Almost any storm 

distribution can be entered.

2. Parameters related to watershed soil and cover to calculate rainfall excess (runoff 

depths). Usually Curve Numbers are used for this calculation, but other options 

are available.

3. Timing parameters to define the travel times to the watershed outlet. Several 

ways to compute time of concentration are provided.

4. Unit hydrograph shape and scale selections to produce the runoff hydrograph.

Four commonly used choices are included.

Outputs are the calculated hydrograph and a detailed report on all the relevant 

information derived and produced. Similar to all Windows applications, charts and 

tables can be copied and applied to reports and other external files.

1.5 Limitations and Omissions

Although Wildcat5 has many options, it omits several items found in some 

similar models. Some of these options may be available in subsequent versions of 

Wildcat5.

1. It does not contain the Green-Ampt infiltration loss function (either lumped or 

distributed), a popular choice in some models. 
6 USDA Forest Service RMRS-GTR-334. 2016.

Figure 1-01—The sequence of natural processes represented in rainfall-runoff models, computational steps, and options available to 

users of the Wildcat5 model.
USDA Forest Service RMRS-GTR-334. 2016. 7

2. It uses unit hydrographs as the watershed routing devices—a choice appropriate 

to the small watersheds targeted. Thus it contains neither overland flow routing 

(for example, kinematic wave) nor channel routing, for which there are many 

options. It does not contain software to alter the shape and peak flow factor of 

the curvilinear unit hydrograph. However, it does contain reservoir routing for 

the outflow hydrographs; with this feature, advanced users can represent channel 

routing or additional watershed routing, or a combination thereof.

3. Other than the single reservoir case described, it does not account for the 

influence of any additional structures in the watershed.

4. It does not distribute rainfall in space. All rainfall is assumed uniform across the 

watershed.

5. With Curve Number modeling, it does not consider any values of initial 

abstraction (Ia/S = λ) other than 0.20 and 0.05. 

6. Only a single process-group can be represented. For example, the rainfall excess 

cannot be modeled by watershed fractions of Curve Numbers and linear runoff 

ratios at the same time.

7. There is no designated accounting for transmission losses.

8. The time of concentration must be greater than 1/360 of the storm duration. This 

is 4 min in a 24-hr storm.

1.6 Computer Requirements

Wildcat5 is a Windows-based program and requires Microsoft Office Excel 2003 

or later. The program is written within Excel in Visual Basic for Applications. Macros 

must be enabled for the program to work properly. Procedures for enabling macros are 

different for every version of Excel. This manual does not provide a listing of how to 

enable macros for each Excel version. Search “How to enable macros for Excel” for 

your installed version of Excel by using any of the common search engines.

1.7 Chapter References

U.S. Natural Resources Conservation Service [NRCS]. 2003. Updated 2012.

National engineering handbook. Part 630, Hydrology. Washington, DC: U.S. 

Department of Agriculture. directives.sc.egov.usda.gov/viewerFS.aspx?hid=21422.

(March 17, 2015).

U.S. Soil Conservation Service. 1954 [and following]. National engineering handbook. Section 4, Hydrology. Washington, DC: U.S. Department of 

Agriculture.115 p. http://directives.sc.egov.usda.gov/OpenNonWebContent.

aspx?content=18393.wba. (March 19, 2015).

U.S. Soil Conservation Service. 1986. Urban hydrology for small watersheds.

Technical Release 55. Washington, DC: U.S. Department of Agriculture. 164 p. www.

nrcs.usda.gov/Internet/FSE_DOCUMENTS/stelprdb1044171.pdf. (March 18, 2015).
8 USDA Forest Service RMRS-GTR-334. 2016.

Chapter 2: Quick Start Guide and 

Example

2.1 Overview

Wildcat5 for Windows was written for use under the Windows operating system 

using Microsoft Excel spreadsheets and Visual Basic code to carry out the details.

Wildcat5 will operate in Excel 2003 or later. You as the user are assumed to have a basic knowledge of the Windows operating system and to be familiar with the concepts 

of pull-down menus, buttons, scroll bars, opening/closing/moving/resizing windows, 

and so forth. You are also assumed to be acquainted with Excel spreadsheets and their 

use. This application is programmed in Visual Basic but inherits all the characteristics 

and limitations of Excel. Macros must be enabled for all versions of Excel. Another 

reminder is to use the “Enter” key every time you put data into a cell (this step is a 

requirement of Excel). Finally, this program can be used only with a mouse or similar 

pointing device.

The quick start guide in this chapter will allow you to begin to use the program 

within a matter of minutes. The program is intended to be a user-friendly, touch-andfeel, follow-your-nose operation. Users who are familiar with Excel and rainfall-runoff 

models can expect to find most of it self-explanatory and intuitively obvious. It is 

possible to work through the model without reading the instructions, but be alert to the 

cell-cursor phenomenon, and observe the repeated warning about enabling macros.

Numerous information (help) buttons are provided to give background, clarification, and suggested parameter values. Ultimate choices and responsibility for those 

choices are left to the user. In addition, generous navigation buttons are included to get 

you from screen to screen. 

Additional details about specific computation features of Wildcat5 are provided 

in the rest of this manual.

2.2 Program Installation and Execution

Place all of the Wildcat5 files into a single folder that you have created.

Alternatively, download the program and its associated files from the Internet and save 

them in this folder.

The current (April 2015) Wildcat5 program is a file called Wildcat5_

Dec07_2015_64bits.xlsm. Accessory files include storm files *.STM (for the included 

drop-down menu STORM AND STORM DISTRIBUTIONS), *.CST (custom 

storms), and *.GST (generic storms). Default depth and duration information is included in the storm files, but you can alter this information. There are also *.PDF files 

containing the information for the help screens that are found under the “?” buttons.

All of these files are intended for use by the program. 

To run Wildcat5, double click on or load the current Wildcat5_Dec07_2015_64bits.

xlsm file. A security warning at the top of the screen will require you to enable macros.

You must do this every time. Procedures for enabling macros are different for every 
USDA Forest Service RMRS-GTR-334. 2016. 9

version of Excel, so if difficulties arise at this step, we suggest applying any of the 

common search engines for “How to enable macros for Excel xxxx” for your installed 

version of Excel. After macros are enabled, Wildcat5 should run properly.

After you double click the file Wildcat5_Dec07_2015_64bits.xlsm, the main 

screen should appear (fig. 2-01). You can select English or metric units for input and 

output, but this example will be all in English. Note that the current version date (the 

Build) is shown in the lower left-hand corner of the main screen.

Figure 2-01—Screen capture of Wildcat5 main screen.

The main menu offers two major input groups, STORM AND STORM 

DISTRIBUTION and WATERSHED INFORMATION, with subgroups within 

the watershed category: (1) rainfall excess, in other words, how we determine runoff 

from rainfall, such as with Curve Numbers (CNs); (2) watershed timing, and (3) unit 

hydrograph choices. These are roughly in the order that they happen on the watershed, 

and as shown on the process chart (fig. 1-01). 

The model operates by having you select inputs from each group. Click on each 

one, fill out the choices and information, hit the Accept & Continue button, and go on 

to the next input button. 

The STORM AND STORM DISTRIBUTION screen lets you specify the 

duration, storm depth, and distribution. If the distribution is not listed there, then the 

CUSTOM and GENERIC options allow building it and saving it for later use.
10 USDA Forest Service RMRS-GTR-334. 2016.

Under WATERSHED INFORMATION, the Rainfall Excess Method screen 

gives options for both DISTRIBUTED and LUMPED systems. Again, note the 

information buttons on the right of each option. These buttons provide details and 

assumptions, and suggested typical values. If you want runoff based on CNs with 

initial abstraction (Ia) = 0.05S, where S = transient storage, then enter the traditional 

0.20S-based CN values. The equivalent 0.05S CNs and S values are computed internally and then displayed.

The Time of Concentration screen collects specifications on timing for the unit 

hydrograph, and thus the model time step.

The Unit Hydrograph screen has four options, including the common SCS 

triangle and the curvilinear hydrographs from which it was derived. Two other options 

are also available. There are no do-it-yourself options for building custom unit hydrographs beyond altering the shape variable of the triangular hydrograph option.

Each of the four input screens has an Accept & Continue option. The Storm 

Data screen also has Load File and Save File options. From each of the four input 

screens there is an option to return to the main screen.

•  Load File allows you to select a previous input dataset, such as a previously used 

rainstorm. These files are stored with distinctive extensions (*.stm).

•  Save File saves the specified storm on the current screen. You can then load it (see 

above) later if needed. This option saves the contents of the current storm.

•  Accept & Continue does just that. The interface keeps the storm values and characteristics for the hydrographs it will create.

When all four selection groups have been completed, return to the main screen.

Click on the Generate Composite Hydrograph button.

Wildcat5 will then give you an interim panel of Summary Input Data and a last 

check to confirm your inputs. Note the option to cancel and return to the main screen. 

If these values are acceptable, then click on the Calculate Hydrograph button.

Things will happen: The input data will be used to generate a composite hydrograph along with summary tables of input and output details. This step may take a few 

seconds. Be patient, and do not hit the keys during the computations. There may be 

several screens that flash by, and the output screen (fig. 2-02) will be displayed. This 

Summary Preview and Hydrograph screen may provide all the information that you 

require. From this screen additional details of the runoff can be selected with the buttons on the left side.
USDA Forest Service RMRS-GTR-334. 2016. 11

For more details, click on the Summary Output Table button on the top left.

This summary screen gives most of the inputs as well. This is all of the output that 

most users require.

The Hydrograph Table shows the runoff values for each time step. 

The Reservoir Routing is designed to route the hydrograph that was just generated through a reservoir of given surface area and spillway length, with a specified 

broad-crested weir coefficient.

There are more output features, but this should be enough to get started. You are 

encouraged to explore and discover on your own. For example, there are other graphics screens that can also be captured and used outside of the program for presentations 

and reports.

Here is a summary of the entire process:

• First enable the macros. Then go to the main screen.

•  Units Systems gives you options for metric and English units with an information 

button on the main screen. Input can be in either metric or English units, with 

the same choices for outputs, including mixed, such as metric in, English out.

However, the internal program calculations are carried out in English units.

• From the main screen click on the buttons and fill in the choices for the Storm, 

Rainfall Excess, Watershed Information, Time of Concentration, and Unit 

Hydrograph. Input follows the order of the flow chart in figure 1-01. You can 

navigate back and forth by the buttons offered, and easily return to the main screen.

Help buttons containing advice, background, and suggestions are given at many 

locations and in each window. On every screen there is an Accept & Continue

button.

Figure 2-02—Screen capture of Wildcat5 output screen, showing summary results. 
12 USDA Forest Service RMRS-GTR-334. 2016.

• When you have made your selections for these inputs, hit the Generate Composite 

Hydrograph button, which leads to an intermediate screen with a summary check 

of the inputs.

• If OK, then hit the Calculate Hydrograph button, and the calculations begin. The 

ensuing calculations may take several seconds.

• The output screen that first appears gives the Summary Preview and Hydrograph.

Often these results are sufficient for the project.

• For additional outputs, there are buttons on the left side that return to the main 

screen, to the reservoir routing procedure, or to six other output screens. You may 

also return to the main screen and begin anew. The same input values are still there.

• The six other output screens are self-explanatory, and are detailed under the buttons. 

Briefly, they show alternative views of both the inputs and outputs. 

ο The Summary Output Table gives technical details on the inputs, the calculations, and some nontraditional interpretations of the outputs.

ο Four different plots give alternative presentations of the rainfall-runoff event. 

ο The Runoff Hydrograph Table gives calculated values line-by-line, including 

TRANSIENT STORAGE. 

From any of these output screens you may also return to the main screen and 

begin again.

• The Reservoir Routing button (in orange) leads to the reservoir routing option.

This option pertains to the hydrograph just computed, and will require the following information: reservoir surface area, spillway length, and weir coefficient. An 

information button elaborates on the routing process.

• The tables and figures produced can be copied directly for use in other publications 

and reports.

2.3 Example

This is a simple example to get started.

Storm: NEH4 Type B storm of 4 inches in 3 hr

Rainfall excess: 20 ac CN = 90; 200 ac CN = 80; 200 ac CN = 70; 200 ac CN = 60 

Timing: tc = 0.5 hr specified

Unit hydrograph: simple triangular unit hydrograph (standard SCS triangle)

• Go to Storm Data and input Storm Duration = 3 hr, Storm Rainfall = 4 in, 

Storm Distribution = NEH4B. Be sure to use the Enter key. Clicking on Accept 

& Continue will get you back to the main screen, or you may want to hit the Save 

File tab, and save the selection for later use.

• Go to the Rainfall Excess Method screen, and click on Curve Number (default) 

λ = 0.2. Click on the CN Values tab to bring up the Hydrologic Response Units

screen. In the table enter:

20 acres grassland CN = 90

200 acres brush/open CN = 80

200 acres forest CN = 70

200 acres deep forest CN = 60 
USDA Forest Service RMRS-GTR-334. 2016. 13

Note that it calculates the CN based on λ = 0.05 simultaneously. Clicking on 

Accept & Continue gets you back to the main screen. 

• Go to the Time of Concentration screen. Enter Given value TC = 0.5 hours. Click 

on Accept & Continue to return to the main screen.

• Go to the Unit Hydrograph screen. Click on the Simple Triangular Unit 

Hydrograph button, HF=484. Click on Accept & Continue to return to the main 

screen.

• Click on Generate Composite Hydrograph. A summary input screen will come 

up, and if everything is OK, then hit the Calculate Hydrograph button. Screens 

will flash by. Hands off now: wait until you see the output results. It is the same 

Summary Preview and Hydrograph screen (fig. 2-02) as shown previously and 

inserted here (fig. 2-03).

Figure 2-03—Screen capture of Wildcat5 output screen, showing summary results for the step-by-step example. 

• Click on the Summary Output Table tab near the top of the Output Options

screen on the left. It will give you the table in figure 2-04.

• For a line-by-line output, click on Hydrograph Table. It will give you the table in 

figure 2-05. Clicking on the Save to File button will export the page to a TXT file.

• After the Summary Output Table and the main table output are generated, you are 

on your own to explore the other output options. All screens have a button to return 

to the main screen to start a new analysis.
14 USDA Forest Service RMRS-GTR-334. 2016.

Figure 2-04—Screen capture of Wildcat5 summary output table, which also shows input data.
USDA Forest Service RMRS-GTR-334. 2016. 15

Figure 2-05—Screen capture of Wildcat5 table of output data from step-by-step example.
16 USDA Forest Service RMRS-GTR-334. 2016.

• If there is a reservoir at the watershed outlet, Wildcat5 can route a hydrograph 

through it. Click on the Reservoir Routing button on the Output options window, 

and arrive at a new screen. It will ask for the full reservoir surface area (Reservoir 

area, ac or ha) and the Spillway Length (ft or m). A broad-crested weir coefficient 

(Spillway weir coeff) is also required. A typical value in English units for the 

coefficient is 3.0 to 3.1. If you use metric units, Wildcat5 will make conversions 

internally. Click on Execute Routing. For the example here, the assumed surface 

area is 3 ac and the spillway width is 30 ft (fig. 2-06). 

The values for each time step are given in the Calculations Table. A button for 

exporting the tabular results to a TXT file is included on that screen.

Figure 2-06—Screen capture of Wildcat5 input screen for routing a hydrograph through a reservoir.
USDA Forest Service RMRS-GTR-334. 2016. 17

Chapter 3: Storm Rainfall

3.1 Concepts

Rainstorms come in a variety of depths, durations, and distributions. Although 

Wildcat5 allows specifying all three of these variables, only the distribution itself is 

discussed here. The depth (P) and duration (T) together define the frequency, or return 

period of the storm. Rainfall intensities within a storm tend to vary with time. The 

sequence and magnitude of interval intensities within a storm is called its distribution.

While the notion refers to the spread of intensities in a rainstorm, it is common 

to describe the time progress of a storm as a series of break points of cumulate rainfall 

depth P(t) with cumulative time t. The internal interval slope ∆P/∆t is the interval 

intensity.

The storm distribution inputs are standardized to a basis of 0 to 100 percent, in 

both the storm time and cumulative storm depth. Wildcat5 then uses the user-specified 

storm depth (in or mm) and duration (hr) to create the dimensioned storm times and 

depths used in the model simulations. This is done internally.

Dimensionless rainfall distributions have much in common with probability 

distributions or histograms used in statistics. Though not shown here, they can be 

described in terms such as means, medians, modes, and variances, when the interval 

intensities play the role of the histogram columns. The area under the dimensionless 

intensity curve is unity, as is cumulative total.

Graphs of cumulative rainfall depth and storm duration have characteristic 

shapes, and two important attributes stand out: (1) the maximum intensity in terms of 

the average intensity, and (2) the timing of the peak intensity. These characteristics 

are summarized for some of the distributions in Wildcat5 in table 3-01. Sometimes 

these are described by the time-quarter of the storm in which the maximum intensity 

happens, for example, first-quarter storms or third-quarter storms. Design storms are 

usually unimodal: they have only a single peak intensity. 

3.2. Distributions in Wildcat5

3.21 User Choices

In practice, design hydrology applies specific distributions keyed to the local climate and general storm characteristics. These may or may not be events that actually 

occur and cause floods. However, when distributions are used with specific models, 

it is assumed that they will produce return period flood peaks that are consistent with 

regional observations. Often, the distribution is specified by an approving jurisdiction, 

but it may also be chosen by the analyst based on sound judgment, common practice, 

or experience.
18 USDA Forest Service RMRS-GTR-334. 2016.

Table 3-01—Some general characteristics of selected design storms.

Peak intensity Timing of peak within storm

Distribution (% of avg. intensity) % of duration Comments 

Farmer–Fletchera

Great Basin, UT 365 0–10 1st 10 percent of the storm 

Wasatch Front, UT 270 20–30 3rd 10 percent of the storm

NEH4Bb 444 33–41.6 5th 0.5 hr in a 6-hr storm

Uniform 100 No peak intensity Default for interval bursts

Iowa 3-hrc 526 40–53.3 hr 1.2 to 1.6 in 3-hr storm 

Type I (SCS)d 626 42 hr 10 to 11 in a 24-hr storm

Type II (SCS)d 700 44–47 17th 5-min interval in a 3-hr storm

TSMSe 750 2.8–5.6 2nd 5-min interval in a 3-hr storm 

CNphi00 454.7 0–5 1st 9 min of a 3-hr storm

CNphi25 421.1 25–30 45 to 54 min in a 3-hr storm

CNphi50 378.8 45–55 81 to 99 min in a 3-hr storm

CNphi75 424.1 70–75 119 to 135 min in a 3-hr storm

CNphi100 454.7 95–100 last 9 min of a 3-hr storm

a Source: Farmer and Fletcher (1972).

b Source: U.S. Soil Conservation Service (1954).

c Source: Elhakeem and Papanicolaou (2009). 

d Source: U.S. Natural Resources Conservation Service (2003).

e Tucson Stormwater Management System. Source: Simons, Li Associates (1995).

3.22 Standard Distributions

Three of the storm options in table 3-01 are offered in the drop-down menu: (1) 

the Farmer–Fletcher (a first-quarter storm), (2) the NEH4B (a second-quarter storm, 

and also called the SCS Type B or simply the Type B), and (3) the uniform storm.

Simply click on the choice, and the time and intensity calculations are performed 

internally.

3.221 Farmer–Fletcher (Great Basin, UT)

This distribution is claimed to be characteristic of first-quadrant storms in the 

Great Basin area of Utah, and is notable for having the major intensities at the very 

start of the storm. In models, it tends to produce lower flood peaks than storms with 

heavy bursts at the end of the storm. See Farmer and Fletcher (1972). Note that there 

are two separate distributions with the Farmer–Fletcher designation.

3.222 NEH4B

This distribution can be traced to the early version of the NEH4 (U.S. Soil 

Conservation Service 1954) and has been widely used. It was originally specified for 

a storm lasting 6 hr. It has the maximum intensity burst (37 percent of the total storm 

rainfall) in the 5th twelfth of the storm duration (fifth half-hour of a 6-hr storm). It 

can be found in TR-60 (U.S. Natural Resources Conservation Service 1990). It is also 

called the NEH4 Type B, or simply the Type B.
USDA Forest Service RMRS-GTR-334. 2016. 19

3.223 Uniform

This is a constant steady rainfall, the simplest and reference distribution, but it 

is uncommon in recorded flood rainfall. It is also the assumed short-term distribution 

of discrete bursts within a complete storm. From a hydrograph standpoint, it leads to 

minimal flood peaks. There is no change of intensity as the storm proceeds. 

3.23 Custom Distributions

The Custom option allows you to specify the breakpoint coordinates for any 

feasible rainfall distribution. The points must begin at (0, 0), and end at (100,100), 

with all interval point sequences non-diminishing. That is, the distribution cannot have 

any intervals of negative slopes. Thus, any distribution desired or required by local 

practice can be used if the dimensionless coordinates are known. The program can 

accept up to 50 breakpoints. These are saved as *.CST files and can be selected again 

for later use. Some sample-example CST-formatted storms are supplied as files with 

Wildcat5. These are: 

•  SCS Type I and II. These distributions have a large following in the urban hydrology and flood control design community. Coordinates are drawn from http://

hydrocad.net/rainfall/tables.

•  Farmer–Fletcher (Wasatch Front). This is appropriate for the Wasatch Front area 

of Utah, and was issued jointly with the Great Basin distribution. See Farmer and 

Fletcher (1972).

•  Iowa 3-hour. This was used in simulator studies by Elhakeem and Papanicolaou 

(2009), and was extracted as the major rainfall burst from a 24-hr “Type II” storm. 

It is notable that plot-simulator rainfall-runoff data generated with this distribution 

are consistent with the runoff values created following Curve Number [CN] methods in Natural Resources Conservation Service handbooks. 

• TSMS. This distribution was constructed for application to the Tucson [AZ] 

Stormwater Management System (TSMS) hydrology (Simons, Li Associates 1995).

It is very similar to distributions developed from and applied to events at Walnut 

Gulch, AZ.

•  CN–φ distributions. These have the unique property of generating consistent 

relationships between the CN and the time-constant loss rate (φ) for a given storm 

duration (T). Five time-of-peak options are included. These distributions assume 

the timing of the peak intensity within the storm does not destroy the CN-φ relationship: CN = 1200 / (12 + φT), where T is the storm duration in hr, φ is in in/hr, 

and only a single lumped CN is used.

3.3 Generic Design Rainstorm Distribution

3.31 General

Generic rainstorms represent event rainfall distributions (that is, intensity distribution and sequence in time) in functional (algebraic) form. The major descriptors are the 

event depth (P), the event duration (T), the maximum intensity (ix), the minimum intensity during the storm (io)

, and the location of the peak intensity within the storm (tp). Note 
20 USDA Forest Service RMRS-GTR-334. 2016.

that this tp is not the same as the tp used in hydrograph descriptors. Only unimodal storms 

are covered with this option. It was used in several earlier versions of Wildcat.

3.32 Application in Wildcat5

The General choice is offered in the STORM AND STORM DISTRIBUTION

selection as the Generic option. The input screen asks for the minimum and maximum 

rainfall intensities, as a percentage of the average intensity. The average intensity is 

defined as the total storm rainfall depth divided by the storm time, or P/T. It also asks for 

the placement within the storm duration of the maximum intensity as a percentage of the 

duration. For example, if the maximum intensities are to be in the latter part of the storm, 

you may input 80, for 80 percent. If the storm specified was 6 hr long, then the maximum 

intensity would occur at hour 4.8.

 For computational reasons, the minimum specified intensity cannot be zero. But it 

can be approached with a very small number, such as 0.001 percent. A true 0 will cause 

an error message. The exponent “n” is defined by the storm specifications (io and ix) and 

the calculation made internally. The basic algebraic form used is:

i(t) = io + (ip – io)(t/tp)

n for 0 < t < tp

i(t) = io + (ip – io)[(T – t) / (T – tp)]n for tp < t < T

where

i = intensity (length/time)

i

o = minimum intensity at time = 0 (length/time)

i

p = peak intensity at time = tp (length/time)

t = time from beginning of storm (time)

t

p = time of peak intensity during the storm (time)

T = total storm duration (time)

n = a dimensionless exponent.

The exponent “n” is fixed (back-defined) by the other storm specifications, 

and calculated internally as n = (ip – P/T) / (P/T – io). P/T is the mean storm intensity 

(length/time).

The cumulative depths at time t can be determined by integration, or by knowledge of geometry directly. The equations are:

P(t) = t{io + [(ip – io) / (n + 1)](t/tp)

n} 0 < t < tp

P(t) = P – (T – t){[io + [(ip– – io)/(n + 1)][(T – t)/(T – tp)]n} t

p < t < T

P is the total storm depth, and P(t) is the depth at time = t. An illustration is given in 

figure 3-01.

3.4 Effects of Distribution Selection

The choice of a distribution can influence the hydrograph generated. The maximum intensity described by the distribution affects the flood peak, as will the timing 

of the most intense rainfall burst in some cases. This is especially true when using the 

CN method to generate interval rainfall excess. Peak-intensity rainfall bursts early in a 

storm will usually lead to smaller peak flows than will late-storm peak intensities.
USDA Forest Service RMRS-GTR-334. 2016. 21

Figure 3-01—Definition figures, for the case of P = 1 in, T = 1 hr, tp = 0.375 hr, io = 0.20 in/hr, and 

i

p = 4.5 in/hr. For these conditions, n = 4.3. Note that the intensity (A) shows on the cumulative 

rainfall (B) as the slope of the curve. The maximum slope occurs under the peak at 0.375 hr. This 

rainfall distribution is similar to the NEH4B distribution.

3.5 Chapter References

There is a rich literature on storm distributions. The following list is a small 

sample. A useful Web site is http://hydrocad.net/rainfall/tables.

Elhakeem, M.; Papanicalaou, A.N. 2008. Estimation of runoff curve number via 

direct rainfall simulator measurement in the State of Iowa, USA. Water Resources 

Management. DOI: 10.1007/s11269-008-9390-1.

Farmer, E.E.; Fletcher, J.E. 1972. Some intra-storm characteristics of high-intensity 

rainfall bursts. In: Davies, D.A., ed. Geilo Symposium, Distribution of precipitation 

in mountainous areas. Proceedings and key-papers presented during the session. 

Publ. 326. Geneva, Switzerland: World Meteorological Organization: 525-531. 

A

B
22 USDA Forest Service RMRS-GTR-334. 2016.

Simons, Li Associates. 1995. Rev. Existing-conditions hydrologic modeling for 

Tucson Stormwater Management Study, Phase ii, Stormwater Master Plan (Task 

7, Subtask 7A-3). Prepared in association with Camp Dresser & McKee, Lewis & 

Roca, Rillito Consulting Group, SWCA, Inc. 47 p.

U.S. Natural Resources Conservation Service. 1990. Earth dams and reservoirs.

Technical Release 60. U.S. Department of Agriculture. 66 p.

U.S. Natural Resources Conservation Service. 2003. Updated 2012. National 

engineering handbook. Part 630, Hydrology. Washington, DC: Department of 

Agriculture. directives.sc.egov.usda.gov/viewerFS.aspx?hid=21422. (March 17, 

2015).

U.S. Soil Conservation Service. 1954 [and following]. National engineering handbook.

Section 4, Hydrology. Washington, DC: U.S. Department of Agriculture. 115 p. 

http://directives.sc.egov.usda.gov/OpenNonWebContent.aspx?content=18393.wba.

(March 19, 2015).

U.S. Soil Conservation Service. 1986. Urban hydrology for small watersheds.

Technical Release 55. Washington, DC: U.S. Department of Agriculture. 164 p. 

www.nrcs.usda.gov/Internet/FSE_DOCUMENTS/stelprdb1044171.pdf. (March 18, 

2015).
USDA Forest Service RMRS-GTR-334. 2016. 23

Chapter 4: Rainfall Excess

4.1 Concepts

Quantifying rainfall excess is a key step in the modeling process. During a 

rainstorm, rain reacts with the watershed, so it is divided into “losses” that remain on 

the land, and rainfall excess, which becomes runoff. The response of rainfall excess 

is a measure of the hydrologic properties of the uplands, which in turn reflect the land 

use and condition of these lands. Estimating rainfall excess is often the most important 

step in modeling runoff volume or peak flow rates. However, several different mechanisms for generating rainfall excess may be found on a single watershed. The spatial 

and temporal variations of processes—and of the rainfall—are masked by the lumping, 

or assumed uniformity, necessary to apply Wildcat5. 

Professional consensus has not identified a single best technique for estimating 

rainfall excess. One widely applied technique is the Curve Number (CN) method.

Because of its simplicity, popularity, and wide use, it has been highly scrutinized and 

often criticized.

Many factors affect rainfall excess. Several options defining these factors are 

offered in Wildcat5. These options are soil and vegetation properties that either are intrinsically based on rate (driven by infiltration) or on depth (driven by rainfall depth), 

or are spatially lumped or distributed.

4.2 Runoff Curve Numbers 

4.21 General

The CN method is widely used to determine direct runoff (rainfall excess) 

from rainstorms, and is applied throughout the world. Pioneered by the U.S. Soil 

Conservation Service (now the U.S. Natural Resources Conservation Service, or 

NRCS), the technique has been widely used since the late 1950s.

The current reference handbook is NEH630 (U.S. NRCS 2003). Further development and discussion are presented in several sources, such as Hawkins and others 

(2009). Some guidance is given here for wild lands affected by fire and grazing.

This section addresses runoff generation only by the CN method. Several other 

options offered in Wildcat5 and covered in this manual have been long associated with 

the CN method, but are more generally simply “NRCS methods.”

4.22 Concepts

Direct rainfall-runoff is modeled in a lumped form as: 

Q = (P – 0.2S)2 / (P + 0.8S) for P ≥ 0.2S, Q = 0 otherwise (4-01)

where S is a measure of maximum possible difference between P (the rainfall) and 

Q (the runoff, or more appropriately, the rainfall excess). In practice, S is 5/6 of that 

maximum possible difference, between P and Q when the initial abstraction (Ia) of 

0.2S is included. The initial abstraction is the rainfall depth at the onset of the event 

required for runoff to be initiated. For convenience and ease in understanding, S is 

transformed to the coefficient CN by 

CN = 1000 / (10 + S) (4-02)
24 USDA Forest Service RMRS-GTR-334. 2016.

when S, P, and Q are in inches. For most applications, values of CN are found in 

handbook tables (see below) and other agency sources, and vary from 0 (no runoff 

for any storm) to 100 (all rainfall becomes runoff). In Wildcat5 this technique models 

rainfall excess depths Q from rainfall P for a series of time steps within a storm. The 

incremental runoff pulses from each time step are transformed to distributed rates via 

unit hydrographs.

4.23 Use

The original CN technique targeted rain-fed agricultural lands and was based 

on studies on small watersheds throughout the United States. The CN technology was 

subsequently extended to application on urban land, wild land, and disturbed lands.

Success on humid traditional forested watersheds has been limited. Note that in the 

NRCS table (table 4-01) the only forested land use entry is simply “Woods,” a rather 

limited choice given the wide variety of forest types and uses. There are no table entries for “forests” directly; and no adjustments for silvicultural treatments, land use, or 

fire condition are offered. 

4.24 Parameters

In this technique, the most important parameter of interest is the CN, which may 

vary from 0 to 100, though most are in the range of 55 to 95. Several studies have 

shown that the choice of CN is critical. Runoff peaks and volumes are usually more 

sensitive to CN than to rainfall depths or duration.

Handbook tables of CNs for a variety of conditions are given in tables 4-01 

through 4-03. Note that they are defined on the basis of Hydrologic Soil Groups 

(HSGs), cover, land use, and, in some cases, hydrologic condition. The hydrologic 

condition is a description of the surface condition, for example, compacted (poor) or 

well-vegetated (good). Exercise sound judgment when determining the condition; alternatively you may run Wildcat5 for both conditions and report the range of potential 

outcomes. Once you select the CN, Wildcat5 calculates S from equation (4-02), and 

runoff depth from equation (4-01).

An additional approach to CNs for selected wildland settings is given in chart 

form in NEH630 (U.S. NRCS 2003: figs. 9.1 and 9.2). As shown in table 4-01, however, CNs can be represented by functions based on soil, cover density, and vegetation 

type. The general equation is CN = a – (b × percent cover).

Table 4-01—Coefficients for Runoff Curve Numbers 

(Antecedent Runoff Conditions-II) for selected 

western forest-range complexes. Application is 

CN = a – ( b × percent cover).

Type Hydrologic Soil Group a b

Sage-grass B 74 0.46

C 87 0.47

Juniper-grass B 82 0.49

C 90 0.32

Oak-aspen B 73 0.51

C 83 0.48

Herbaceous B 83 0.25

C 90 0.18

 D 95 0.08
USDA Forest Service RMRS-GTR-334. 2016. 25

Table 4-02—Curve Numbers for wildland management conditions for 

Hydrologic Soil-Cover Complexes, Antecedent Runoff Conditions-II, and 

Ia/S = 0.20.

 Treatment Hydrologic Hydrologic Soil Group

Land use or practice conditiona A B C D

Pasture or range Poor 68 79 86 89

Fair 49 69 79 84

Good 39 61 74 80

Contoured Poor 47 67 81 88

Contoured Fair 25 59 75 83

Contoured Good 6 35 70 79

Meadow Good 30 58 71 78

Woods Poor 45 66 77 83

Fair 36 60 73 79

Good 25 55 70 77

Farmsteads 59 74 82 86

Roads (dirt) 72 82 87 89

Roads (hard surface) 74 84 90 92

Herbaceous: mixture of grass, weed, and low-growing brush, with brush the minor 

element

Poor 80 87 93

` Fair 71 81 89

Good 62 74 85

Oak-aspen: mountain brush mixture of oak brush, aspen, mountain mahogany, butter 

brush, maple, and other brush

Poor 66 74 79

Fair 48 57 63

Good 30 41 48

Pinyon-juniper: pinyon, juniper, or both; grass understory

Poor 75 85 89

Fair 58 73 80

Good 41 61 71

Sage-grass: sage with an understory of grass

Poor 67 80 85

Fair 51 63 70

Good 35 47 55

Desert shrub: major plants include saltbrush, greasewood, creosotebush, blackbrush, 

bursage, paloverde, mesquite, and cactus

Poor 63 77 85 86

Fair 55 72 81 86

Good 49 68 79 84

a Poor is <30 percent ground cover (litter, grass, and brush overstory), Fair is 30 to 70 percent 

ground cover, and Good is >70 percent ground cover. Source: excerpted from U.S. NRCS 

(2003: tables 9.1 and 9.2).

The Antecedent Runoff Condition (ARC; formerly Antecedent Moisture 

Condition, or AMC) used in tables 4-01 and 4-02 adjusts CN—and calculated runoff—based on lower (ARC-I), median (ARC-II), and upper (ARC-III) bounds. These 

conditions were originally attributed solely to the site’s soil moisture content at the 

onset of the storm. Condition II is the reference-status CN, which is usually assumed 

for design runoff calculations. Although adjusting for ARC is not recommended here 

or in general practice, you can see how the reference-status CN compares to the CN at 

different ARCs in table 4-03.
26 USDA Forest Service RMRS-GTR-334. 2016.

If the ARC is not specified, it is assumed to be ARC-II. As an alternative to soil 

moisture effects, the variety of CNs—and runoff—has also been described simply 

as “error bands,” and cumulative conditional probabilities of 10, 50, and 90 percent 

estimated for conditions I, II, and III, respectively, for runoff for a given P (Hjelmfelt 

and others 1982).

4.241 Hydrologic Soil Groups

As implied in the above, selection of CN hangs heavily on the HSG. These identities are assigned to soil series in the United States by the NRCS based on soil survey 

criteria and are sometimes adjusted locally by state NRCS offices. Up-to-date HSG 

assignments are available from the NRCS Web Soil Survey at http://websoilsurvey.

nrcs.usad.gov/app/WebSoilSurvey.aspx.

Simpler criteria for assigning HSGs based solely on soil texture are offered 

in the U.S. Soil Conservation Service’s Technical Release 55 (TR55; 1986). These 

categories are taken from an earlier paper by Brakensiek and Rawls (1983). However, 

assignments have been found to be inconsistent when considered internally against 

soil physical properties (Nielsen and Hjelmfelt 1997), and often in error by as much as 

±1 HSG when checked against field data in hydrologic modeling (Sartori and others 

2011; Stewart and others 2010, 2012). 

Table 4-03—Runoff Curve Number (CN) for 

each Antecedent Runoff Condition (ARC).

CN (ARC-II) CN (ARC-I) CN (ARC-III)

100 100 100

95 87 98

90 78 96

85 70 94

80 63 91

75 57 88

70 51 85

65 45 82

60 40 78

55 35 74

50 31 70

45 26 55

0 0 0

Source: condensed from U.S. NRCS (2003: table 10.1). 

Table 4-04—Hydrologic Soil Group (HSG) based on texture.

Texture HSG

Sand, loamy sand, sandy loam A

Silt loam or loama B

Sandy clay loam C

Clay loam, silty clay loam, sandy clay, silty clay, or clay D

a The silt textural classification is missing, but when the above information 

is plotted on a textural triangle, silt is an extension of the B category. 

Source: U.S. Soil Conservation Service (1986).
USDA Forest Service RMRS-GTR-334. 2016. 27

4.25 Effects of Fire on Curve Numbers

4.251 General

Loss of vegetation to wildland fire can dramatically change the hydrologic regime and hence the runoff CN. But unlike research on the effects of cropping patterns, 

urbanization, or grazing, there are no comprehensive studies of the effects of wildland 

fire on CNs. Severe wildfires are unplanned events, and hydrologic instrumentation is 

seldom installed onsite. Furthermore, applying a “hot fire” treatment on a research watershed is difficult for administrative and practical reasons. In addition, recovery times 

are surprisingly short, in the range of 3 to 10 yr, and less than the length of record 

required for hydrologic definition of CNs. Therefore the CNs themselves may change 

quickly. Nonetheless, professional needs have led to pragmatic local practices.

Adjustments to CNs to reflect fire response have been compiled from several 

sources. Tables 4-05 through 4-14 represent values in current practice for a variety 

of conditions. Consider these CNs as suggestions, and draw upon judgment and local 

expertise about local practices and conditions.

4.252 U.S. Forest Service Tables

Table 4-05—Post-fire Curve Numbers 

(CNs) based on fire severity, derived 

from research at Salt Creek Burned 

Area Emergency Response, Uinta 

National Forest (now Uinta-WasatchCache National Forest), UT.

Fire severity Post-fire CNa

High Pre-fire + 15

Moderate Pre-fire + 10

Low Pre-fire + 5

None Pre-fire

a Maximum CN = 100. Sources: Foltz and others 

(2009: 57), Higginson and Jarnecke (2007).

Table 4-06—Post-fire Curve Numbers (CNs) based on fire 

severity or conditions during fire on Santa Fe National Forest, 

NM. 

Fire/condition Post-fire CN

High burn severity with water repellency 95

High burn severity without water repellency 90–91

Moderate burn severity with water repellency 90

Moderate burn severity without water repellency 85

Low burn severity Pre-fire + 5

Straw mulch with good cover 60

Seeding with LEBsa – 1 yr after fire 75

LEBsa without water repellency 85

a Log erosion barriers installed on the contour at the recommended spacing. 

Sources: Foltz and others (2007: 57); Greg Kuyumjian, U.S. Forest Service, 

Okanogan-Wenatchee National Forest, Wenatchee, WA, pers. comm.
28 USDA Forest Service RMRS-GTR-334. 2016.

Table 4-07—Post-fire Curve Numbers (CNs) by 

fire severity or conditions based on research 

on Fishlake National Forest, UT (Foltz and 

others 2009: 58; Solt and Muir 2006).

Fire/condition Post-fire CN

High burn severity 90

Moderate burn severity 85

Low burn severity 80

Unburned and pre-fire 80

Table 4-08—Post-fire Curve Numbers (CNs) by soil group and fire 

severity based on research on the Coronado National Forest, AZ 

and NM (Foltz and others 2009: 58).

Post-fire CN

Hydrologic Soil Pre-fire Low burn Moderate High burn

Group CN severity severity severity

B 56 65 — —

C 67 70 to 75 80 90

D 77 80 to 85 90 95

4.253 Santa Barbara, CA, Tables

Table 4-09—Pre-fire and post-fire Curve Numbers 

(CNs) by pre-fire conditionsa and Hydrologic Soil 

Group in the Santa Barbara Flood Control District, 

CA (Constantine and others 2010; C.R. Constantine, 

Atkins Global Inc., California, pers. comm.)b.

 Land cover type Hydrologic Soil Group

and burn severity A B C D

Forested pre-burn 25 55 70 77

Low 45 66 77 83

Moderate 70 80 88 92

High 70 80 88 92

Scrub/chaparral pre-burn 55 65 77 83

Low 70 77 83 87

Medium 70 80 88 92

High 70 80 88 92

Range/agriculture pre-burn 39 61 74 80

Low 68 79 86 89

Medium 70 80 88 92

High 70 80 88 92

Water–rock pre-burn 100 100 100 100

Low 100 100 100 100

Moderate 100 100 100 100

High 100 100 100 100

Developed pre-burn 72 82 87 89

Low 72 82 87 89

Moderate 72 82 87 89

High 72 82 87 89

a Average antecedent conditions assumed to be ARC-II.

b Fire effects and HSGs are not shown for developed areas or for 

water-rock conditions.
USDA Forest Service RMRS-GTR-334. 2016. 29

4.254 Easterbrook Estimates

The following estimates for CN by cover, fire conditions, and Hydrologic Soil 

Group have been provided for use in geographic information systems (GIS)-based 

models; see Easterbrook (2006). They are presented in tables 4-10 through 4-13 with 

only minor editing.

Table 4-10—Curve Numbers by vegetation type and conditions or fire severity for Hydrologic Soil Group 

A (Easterbrook 2006).

Conditions or fire severity

Prescribed Mod High With hydrophobicity

Vegetation type Good fire Fair Poor burn burn Mod burn High burn

Oak-aspen-mountain brush 20 33 77 77 82 98

Herbaceous-grass-brush 51 65 77 77 82 82

Conifer 27 38 36 45 77 77 82 98

Sagebrush-grass 30 55 77 77 82 82

Oak-woodland 32 47 44 55 77 77 82 98

Pinyon-juniper 30 59 77 77 82 98

Broadleaf chaparral 31 41 40 53 77 77 82 98

Narrowleaf chaparral 55 67 55 70 77 77 82 98

Barren 77 77 77 77 77 82 82

Annual grass 38 51 49 65 77 77 82 82

Table 4-11—Curve Numbers by vegetation type and conditions or fire severity for Hydrologic Soil Group 

B (Easterbrook 2006).

Conditions or fire severity

Prescribed Mod High With hydrophobicity

Vegetation type Good fire Fair Poor burn burn Mod burn High burn

Oak-aspen-mountain brush 30 53 48 66 63 86 72 98

Herbaceous-grass-brush 62 76 74 85 81 86 85 85

Conifer 55 63 60 66 73 86 79 98

Sagebrush-grass 35 60 51 67 65 86 73 73

Oak-woodland 58 69 65 73 80 86 85 98

Pinyon-juniper 41 68 58 75 77 86 82 98

Broadleaf chaparral 57 67 63 70 75 86 81 98

Narrowleaf chaparral 65 77 72 82 77 86 89 98

Barren 86 86 86 86 86 86 89 98

Annual grass 61 74 69 78 83 86 87 87
30 USDA Forest Service RMRS-GTR-334. 2016.

Table 4-12—Curve Numbers by vegetation type and conditions or fire severity for Hydrologic Soil Group 

C (Easterbrook 2006).

Conditions or fire severity

Prescribed Mod High With hydrophobicity

Vegetation type Good fire Fair Poor burn burn Mod burn High burn

Oak-aspen-mountain brush 41 63 57 74 72 91 79 98

Herbaceous-grass-brush 74 86 81 87 89 91 91 91

Conifer 70 75 73 77 84 91 88 98

Sagebrush-grass 47 73 63 80 78 91 83 83

Oak-woodland 72 79 76 82 89 91 91 98

Pinyon-juniper 61 83 73 85 87 91 91 98

Broadleaf chaparral 57 67 63 70 75 86 81 98

Narrowleaf chaparral 71 77 75 80 81 91 82 98

Barren 91 91 91 91 91 91 93 93

Annual grass 75 83 79 86 90 91 92 92

Table 4-13—Curve Numbers by vegetation type and conditions or fire severity for Hydrologic Soil Group 

D (Easterbrook 2006). 

Conditions or fire severity

Prescribed Mod High With hydrophobicity

Vegetation type Good fire Fair Poor burn burn Mod burn High burn

Oak-aspen-mountain brush 48 69 63 79 93 93 94 98

Herbaceous-grass-brush 85 91 89 93 93 93 94 94

Conifer 77 81 79 83 93 93 94 98

Sagebrush-grass 55 78 70 85 93 93 94 94

Oak-woodland 79 84 82 86 93 93 94 98

Pinyon-juniper 71 85 80 90 93 93 94 98

Broadleaf chaparral 78 82 81 85 93 93 94 98

Narrowleaf chaparral 83 87 86 90 93 93 94 98

Barren 93 93 93 93 93 93 94 94

Annual grass 81 87 84 89 93 93 94 94
USDA Forest Service RMRS-GTR-334. 2016. 31

4.255 Goodrich–Automated Geospatial Watershed Assessment 

(AGWA) Simulations

The CNs in table 4-14 were developed from existing CN cover tables and GISbased CN determinations (Goodrich and others 2005). Fire impacts were represented 

by reductions in cover. You might estimate CNs similarly by using table 4-02.

4.26 Effects of Grazing on Curve Numbers

4.261 General

Grazing activities reduce land cover and cause soil compaction, thereby affecting runoff and the CNs that describe it, and are of interest to wildland hydrologists.

Values of CN for pasture, range, and meadow conditions from the NRCS (2012) 

handbook are given in table 4-02. Values of CN as a function of vegetative type and 

ground cover, as recommended in NRCS (2012), are given in table 4-15. Results of 

several studies on grazing and grazing-related impacts are also given as guides (tables 

4-15 through 4-17).

4.262 Jornada Experimental Range, NM, Cover Studies

Rainfall and runoff data were collected on plots at the Jornada Range (NM) 

Long Term Ecological Research Site in a joint study by the National Science 

Foundation and New Mexico State University. The effects of ground cover on CN 

were found to converge to CN ~90 at no cover, and the largest variations with cover 

were found on the sites with the highest percentage of cover. Table 4-15 gives coefficients that approximate the results for the five plot groups using the same equation as 

in table 4-01; see Hawkins and Ward (1998).

Table 4-14—Curve Numbers for Hydrologic Soil Groups, by 

land cover and burn severitya. 

Hydrologic Soil Group

Cover Burn severity A B C D

Shrubland Pre-burn 63 77 85 88

Low 65 79 86 89

Medium 68 82 88 90

High 73 88 91 91

Deciduous forest Pre-burn 55 55 75 80

Low 59 60 78 82

Medium 65 65 80 85

High 70 71 83 87

Coniferous forest Pre-burn 45 66 77 83

Low 49 71 80 85

Medium 55 76 82 88

High 60 82 85 90

Mixed forest Pre-burn 55 55 75 80

Low 59 60 78 82

Medium 65 65 80 85

High 70 71 83 87

a Recommended for Automated Geospatial Watershed Assessment 

simulations.
32 USDA Forest Service RMRS-GTR-334. 2016.

4.263 Badger Wash, CO, Paired Watershed Studies

Data were collected in a U.S. Geological Survey (USGS) study of runoff from 

paired watersheds on shale-derived soils (HSG D) in Badger Wash, CO, during 449 

storms. Curve Numbers were found to be lower on all four ungrazed watersheds than 

on their grazed counterparts (Lusby 1976, Lusby and others 1971): 

• Grazed 92–94 Average CN = 93

• Ungrazed 91–93 Average CN = 92 

4.264 Effects of Vegetation Conversion

University of Arizona studies (Rietz 1999, Rietz and Hawkins 2000) analyzing 

data from the U.S. Department of Agriculture (USDA) and USGS found the following 

effects of land cover change on CN:

• Brush to grass conversion at Boco Mountain, CO Decrease in CN ~18 units

• Mesquite removal at Riesel, TX Increase in CN ~13 units

4.265 Pasture–Meadows Studies

The effects of cover and land use on CNs were examined by comparing USDA 

data from pasture land and meadows in the same watershed (Ohio and Nebraska) 

and in separate watersheds in Texas (Rietz 1999, Rietz and Hawkins 2000). Curve 

Numbers for ungrazed (meadow) and grazed (pasture) lands are shown in table 4-16. 

Table 4-15—Coefficients for estimating Curve Numbers (CNs), where 

CN = a – (b × percent cover), for sites on Jornada Experimental 

Range, NM (Hawkins and Ward 1998). 

Hydrologic 

Site Cover Soil Group a b

Creosote Control Brush B 88.7 0.0790

Creosote Termite Brush B 88.4 0.1367

Creosote Caliche Brush B 92.7 0.0636

Grass Summerford Grass B 83.8 0.3159 

Grass IBP Grass A 87.2 0.4815

Table 4-16—Curve Numbers derived from pasture–meadows comparisons.

Location Meadows (ungrazed) Pasture (grazed) Number of watersheds

Coshocton, OH 70.2–82.6 77.8–88.4 6

Hastings, NE 71 86 1

Riesel, TX 88.3 73.8–96.0 1 meadow, 11 pasture
USDA Forest Service RMRS-GTR-334. 2016. 33

4.266 Australian CNs by Grazing Intensity

The effects of grazing intensity were studied in New South Wales, Australia.

Data were collected from plots of about 1/40 ac after rainstorm events over sample periods varying from 6 to 33 yr and CNs were determined (table 4-17). Average annual 

rainfall was about 23 to 31 in/yr (Cao and others 2011).

Table 4-17—Curve Numbers for two Hydrologic 

Soil Groups, by grazing intensity, New South 

Wales, Australia (Cao and others 2011: fig. 7). 

Grazing intensity

Soils and location Light Medium Heavy

C soils

Cowra 70.0 77.8

Inverell 75.7 72.4

Wagga-Wagga 83.4 87.0

D soils

Gunnedah 72.6 84.5

Scone 76.8 79.4

Burned 80.8 

Wellington 72.6 72.6

4.27 Curve Number with Ia/S = 0.05

4.271 Concepts

This is the Curve Number method, but here Ia/S is set at 0.05 instead of 0.20.

Historically the common practice was to set Ia/S, or λ (lambda) at 0.20, but in some 

cases λ has been found to have other values. Several recent studies (see Chapter 

References, Curve Number with Ia/S = 0.05) have found much smaller values for 

some conditions. Wildcat5 offers the alternative λ = 0.05, which is the consensus value 

from these studies.

Thus, instead of the runoff equation of 

Q = (P – 0.2S)2 / (P + 0.8S) for P > 0.2S (4-03)

the use of Ia/S = 0.05 gives

Q = (P – 0.05S)2 / (P + 0.95S) for P > 0.05S (4-04)

 Because the traditional land use and soils tables of CNs are based on Ia/S = 

0.20, equation (4-03) should more properly have the subscript S0.20, and equation 

(4-04), S0.05. When applying Ia/S = 0.05, you need a different CN. Accordingly, the 

following conversion equation is based on analysis of data from 307 watersheds by 

Jiang (2001):

S0.05 = 1.33S0.20

1.15 (4-05)

This fitting had an r2 of 0.993, and a standard error of 0.36 in for ordered data.

When the original rainfall-runoff data (P >1 in) were backfitted, a higher r2 was 

achieved by using 0.05 in 252 of the 307 cases. With substitution and simplification 

the transfer function becomes:

CN0.05 = 100 /{1.879[(100/CN0.20) –1]1.15 + 1} (4-06)
34 USDA Forest Service RMRS-GTR-334. 2016.

4.272 Application

In Wildcat5, conversions using equations (4-05) and (4-06) are made internally. 

You do not need to input CN0.05 (nor is it possible). Equations (4-05) and (4-06) are 

valid only up to CN0.20 = 98.5. Above that value, CN0.05 = CN0.20.

4.3 Constant Infiltration Capacity: φ-Index

4.31 Concepts

The φ-index (phi-index) method assumes a constant infiltration capacity or 

loss rate (φ) in both time and spatial distribution across the watershed. Assuming a 

watershed time-constant loss rate φ, and a storm described with intensity i, then the 

momentary rainfall excess rate q is:

 q = i – φ for i > φ q = 0 otherwise

and for each interval of time (∆t) within a given storm:

∆Q = q∆t = (i – φ)∆t for i > φ ∆Q = 0 otherwise

For the entire storm: 

Q = Σq∆t = Σ(i – φ)∆t for all i > φ

P = Σi∆t 

The value of φ has also been treated as the overall loss rate in a storm, or 

(P – Q) / duration (Linsley and others 1982).

4.32 Parameter Values

Tables of φ values based on land characteristics have been widely described and 

suggested for hydrologic analysis in design projects, but no authoritative tables of 

φ values are available. For guidance, values are offered in table 4-18. 

Table 4-18—Suggested values for loss rate (φ) on soil with 

and without vegetation, by soil texture (Lee 1980).

Bare soil Vegetated

Texture mm/hr in/hr mm/hr in/hr

Clay 0–5 0–0.20 5–10 0–0.40

Clay loam 5–10 0.20–0.40 10–20 0.40–0.80

Loam 10–15 0.40–0.60 20–30 0.80–1.20

Sandy loam 15–20 0.60–0.80 30–40 1.20–1.60

Sand 20–25 0.80–1.00 40–50 1.60–2.00

Some references, such as Linsley and others (1982), suggest estimating φ from 

observed local events, and extending the values to design situations.

Some land use (grazing) adjustments for observed steady-state infiltration 

rates (approximating φ), are given by Gifford and Hawkins (1978). A summary of 25 

plot studies on several soil types on infiltration for various grazing intensities can be 

grouped into three statistically significant clusters (table 4-19).
USDA Forest Service RMRS-GTR-334. 2016. 35

4.33 Discussion

This is an infiltration-based technique that ignores the observed variation of 

infiltration capacity with time or its surrogate, site wetness. However, the variable 

capacity found in many infiltration equations, such as Horton’s, falls to near-stable 

values in a relatively short time (Overton and Meadows 1976: 33). 

Some references suggest determining the loss rate for current storms for a onetime application. Alternatively, an areally distributed φ-index option is also available 

for an analysis with “Distributed infiltration capacities.”

4.4 Distributed Infiltration Capacity

4.41 General

This method of rainfall excess generates overland flow, assuming that point 

infiltration capacities (f) of a watershed are constant with time but vary by location 

within a watershed. At any point in the watershed the runoff rate q (for example, in/hr) 

is: 

q = (i – f) for i > f, q = 0 otherwise

where both i, the intensity, and f are in units of rate, such as in/hr.

The spatial variation of f is assumed to follow the exponential distribution, 

with its descriptive parameter being the mean point fixed-rate infiltration capacity, μf.

Within Wildcat5, the interval intensities and loss rates are calculated, and applied for 

discrete rainfall intervals.

4.42 Concepts

It is assumed that these time-constant point loss rates f are exponentially distributed with a mean of μf

. The exponential density distribution g(f) and cumulative G(f) 

are given by: 

g(f) = (1/μf)exp(-f/μf) (4-07)

 G(f) = 1 – exp(-f/μf)

With these equations, the areal weighted average rainfall excess rate q for an intensity 

i for a plot or watershed is: 

q = i – μf + μf exp(-i/μf) or q = i – μf [1 – exp(-i/μf)] (4-08)

The characterizing watershed variable μf

 is the spatial mean f, or expected value 

of the time-constant infiltration capacities over the watershed area under rainfall conditions. The variables f, q, and i are in units of rate.

Table 4-19—Observed effects of grazing intensity on 

loss rate.

Grazing intensity Infiltration capacity (average)

Ungrazed 1.60 in/hr (41 mm/hr)

Light to moderate ~1.25 in/hr (32 mm/hr)

Heavy 0.80 in/hr (20 mm/hr)
36 USDA Forest Service RMRS-GTR-334. 2016.

4.43 Parameter Selection

The only parameter needed is μf, the mean point infiltration capacity. Its 

selection for model use is ultimately left to your judgment and experience based on 

knowledge of local conditions. However, selection of a value is approached by the use 

of the “effective” hydraulic conductivity Ke, which depends on soil texture and cover.

Some background follows.

4.431 Soil Hydraulic Conductivity

Water infiltration rate into the soil depends largely on the soil texture and cover.

The literature offers some insights into soil intake rates with soil texture, mainly 

through interpretation and manipulation of the soil hydraulic conductivity, Ks. Refer to 

Clapp and Hornberger (1978) or Rawls and others (1983). Their work is based on ideal 

conditions, which are not usually seen in field situations. Accordingly, an adjusted or 

“effective” value (Ke) is used to account for such real-world conditions as air trapping 

and variable rainfall intensity; sometimes it has been approximated by Ks/2 (Bouwer 

1966). Suggested values for Ks

 and Ke (not μf ) for different textures of bare soils are 

given in table 4-20. 

Table 4-20—Rangeland soil (Ks) and effective (Ke) 

hydraulic conductivity values by soil texture.

Soil texture Ks(mm/hr)a Ks(in/hr)a Ke (in/hr)b

Sand 90.0 3.543 1.772

Loamy sand 30.0 1.181 0.591

Sandy loam 11.0 0.433 0.217

Loam 6.5 0.256 0.128

Silt loam 3.4 0.134 0.067

Silt 2.5 0.098 0.049

Sandy clay loam 1.5 0.059 0.030

Clay loam 1.0 0.039 0.020

Silty clay loam 0.9 0.035 0.018

Sandy clay 0.6 0.024 0.012

Silty clay 0.5 0.020 0.010

Clay 0.4 0.016 0.008

a Sources: Stone and others (1992) and Rawls and others (1983). 

Other values from literature or agency sources might be used as 

alternatives, for example Clapp and Hornberger (1978). 

b The Ke shown is Ks/2.

Furthermore, the state-of-the-technology does not yet accommodate the transfer 

from (1) a soil physics-based Ks to (2) an effective loss rate Ke based on results from 

a controlled rainfall simulator, to (3) Ke determined from natural rainfall events, 

to (4) a spatially varied loss rate for natural rainfall events described by μf. Recent 

research, such as Nearing and others (2011), suggests that the transition from (2) to 

(3) above is a factor between 1/3 and 1⁄2. In the following discussion of parameters, the 

factor is assumed to be 1/3. 
USDA Forest Service RMRS-GTR-334. 2016. 37

4.432 Cover Effects

The above estimation of parameters assumes a bare soil, with no accounting 

for the effects of land use, condition, or cover. The following approach, which considers soil and land cover (Nearing and others 2011), has been developed from the 

Rangeland Hydrology and Erosion Model (RHEM; U.S. Agricultural Research Service 

2013), using rainfall simulator data from 49 rangeland locations in the western United 

States. See Wei and others (2009) for the list of locations. 

First, a baseline loss rate, Keb, is calculated for primary soil and cover effects: 

Keb = (1/3)e(0 .174 – 1.45clay +2.975GC + 0.923CC) (4-09)

where

Keb is the effective baseline conductivity in mm/hr 

clay is the fraction in clay in the top 4 cm of the soil profile, ranging from 0 to 1

GC is the fraction in ground cover (0–1). Ground cover is defined as the sum of 

litter on the ground surface, gravel and rock >5 mm in size, vegetation in 

contact with the ground, and cryptogamic crusts both inside and outside the 

vegetative canopy.

CC is the fraction in canopy cover (0–1). Canopy cover is defined as any standing live or dead vegetative matter not in contact with the ground surface.

In equation (4-09) GC should equal 1 minus the fraction of bare soil. Note that 

it is possible to have both CC = 1.0 and GC = 1.0. As described in Nearing and others

(2011), the factor of 1/3 adjusts Keb computed from rainfall simulator data to Keb

computed for natural rainfall-runoff events. 

Second, the above is adjusted to the effective conductivity in mm/hr, Ke, for 

vegetative types with

 Ke = Keb × vegetative type factor (4-10)

Vegetative type factors are given in table 4-21, and are the only vegetation types 

for which this information is currently available. 

Table 4-21—Vegetative type 

factors for calculating effective 

conductivity in mm/hr (Nearing 

and others 2011). 

Vegetative type Factor

Sod grass 0.80

Bunch grass 1.00

Shrubs 1.20

Ke is the practical loss rate estimated from less-than-extreme natural events, or 

with rainfall simulation. Ke values have also been developed for forest conditions including undisturbed forests and low- and high-severity fires on the U.S. Forest Service 

Water Erosion Prediction Project (WEPP) interface (http://forest.moscow.fsl.wsu.edu).

If you select the Disturbed WEPP interface from this site, specify the soil texture and 

the cover, and then click on the soil texture button above the soil selection box, the 

soil properties for that texture are presented (Elliot 2004). 
38 USDA Forest Service RMRS-GTR-334. 2016.

Third, an estimate of μf is taken from approximate relationships based on rainfall 

simulator data found by Stone and others (2008; J.J. Stone, U.S. Agricultural Research 

Service, Tucson, AZ, 2012, pers. comm.): 

μf = 8e0.0912Ke (4-11)

with μf and Ke in mm/hr. Note that μf is always equal to or greater than Ke.

4.44 Example

Given a site with 20 percent clay in the upper 4 cm, 50 percent ground cover, 

and 20 percent canopy cover, then equation (4-09) becomes: 

Keb = (1/3)e0 .174 – 1.45(0.20)+2.975(0.50) + 0.923(0.20) = 1.596 mm/hr = 0.065 in/hr

Calculating Keb for different vegetative types produces Ke via equation (4-10) 

and the values in table 4-22.

Table 4-22—Adjustment factors and effective 

conductivity (Ke ) by vegetative cover. 

Type Factor Ke(mm/hr) Ke(in/hr)

Sod grass 0.80 1.28 0.050

Bunch grass 1.00 1.60 0.065

Shrubs 1.20 1.92 0.078

If we use shrubs as an example, then calculating μf, from equation (4-11) yields: 

μf = 8e0.0912K

e = 8e0.0912(1.92) = 9.53 mm/hr = 0.375 in/hr

This μf value is suitable for Wildcat5 input. You may also use other values of 

Ke, such as those found in the WEPP interface.

The outcome from the above example is shown in table 4-23 below. It uses 

equation (4-08) directly with μf = 0.375 in/hr, for a 100-ac watershed. 

Table 4-23—Example of distributed lossrate calculation where μf = 0.375 in/hr.

Intensity Runoff rate Acres with

	i (in/hr)	 q (in/hr)	 i ≥ f, q ≥ 0

0 0.000 0.0

0.1 0.012 23.4

0.2 0.045 41.3

0.3 0.093 55.1

0.4 0.154 65.6

0.5 0.224 73.6

0.6 0.301 79.8

0.7 0.383 84.5

0.8 0.469 88.2

0.9 0.559 90.9

1.0 0.651 93.1

1.2 0.840 95.9

1.4 1.034 97.6

1.6 1.230 98.6

1.8 1.428 99.2

2.0 1.627 99.5

2.5 2.125 99.9

3.0 2.625 99.97
USDA Forest Service RMRS-GTR-334. 2016. 39

4.45 Discussion

Use of this method—and μf as its defining parameter—acknowledges that not 

all points in the watershed will be contributing to rainfall excess. That is, μf is the 

average potential infiltration rate for all the points. For a rainfall intensity i = μf , only 

63.2 percent of the watershed will be contributing. In other words, 36.8 percent of 

the points will have an infiltration capacity greater than the intensity i. This general 

distributed loss rate approach to runoff has been reported in several studies (Hawkins 

1981, Hawkins and Cundy 1989, Stone and others 2008).

It bears repeating that with this approach, the point infiltration capacity does not 

vary with time. It is best to think of μf as the mean point infiltration rate after the initial wetting. In most field situations, a nearly time-constant capacity is achieved after 

reasonable durations (0.2 to 0.8 hr). This method does assume that the time-constant 

capacity varies over space. It is not the same at all points in the watershed. Variation is 

assumed to be described by the exponential distribution as stated above. 

4.46 Other Influences on Loss Rates

Some land use (grazing) adjustments for observed steady-state infiltration rates 

are given by Gifford and Hawkins (1978). A summary of about 25 plot studies on infiltration for various grazing intensities aligns into three statistically significant clusters 

(table 4-24).

Table 4-24—Average infiltration capacity associated 

with different grazing intensities (Gifford and 

Hawkins 1978).

Grazing intensity Infiltration capacity (average)

Ungrazed ~1.60 in/hr (41 mm/hr)

Light/moderate ~1.25 in/hr (32 mm/hr)

Heavy ~0.80 in/hr (20 mm/hr)

4.5 Runoff Fraction (Runoff Ratio) 

4.51 Concepts

The runoff fraction is the most simplistic expression of rainfall-runoff. The rainfall excess is a simple linear fraction of the rainfall: 

Q = CP

where C is the “runoff ratio,” 0 ≤ C ≤ 1, and Q and P have units of depth. Despite its 

simplicity, it may be the most appropriate model for some situations. The parameter C 

does have a specialized physical interpretation as the fraction of the watershed impervious area (including water surface).

4.52 Parameters

The parameter C must be between 0 and 1. There are very few studies using 

field data to quantify the runoff ratios or the rational coefficients. Therefore, rely on 

your judgment and experience, and institutional acceptance, when selecting runoff 
40 USDA Forest Service RMRS-GTR-334. 2016.

Table 4-25—Coefficients for runoff fraction and rational equation (Chow 1964: chapters 14 and 21).

Type of drainage area Coefficient “C” Type of drainage area Coefficient “C”

Lawns: sandy soil, flat (2% grade) 0.05–0.10 Apartment dwelling areas 0.50–0.70

Lawns: sandy soil, flat (2–7% grade) 0.10–0.15 Industrial: light industry 0.50–0.60

Lawns: sandy soil, steep (7% grade) 0.15–0.20 Heavy industry areas 0.60–0.90

Lawns: heavy soil, flat (2% grade) 0.13–0.17 Parks, cemeteries 0.10–0.25

Lawns: heavy soil, moderate (2–7%) 0.18–0.22 Playgrounds 0.20–0.35

Lawns: heavy soil, steep (7%) 0.25–0.35 Railroad yard areas 0.20–0.35

Business: downtown areas 0.70–0.95 Unimproved area 0.10–0.30

Neighborhood areas 0.50–0.70 Streets: asphaltic 0.70–0.95

Residential: single family 0.30–0.50 Streets: concrete 0.80–0.95

Multifamily units, detached 0.40–0.60 Streets: brick 0.70–0.85

Multifamily units, attached 0.60–0.75 Drives and walks 0.75–0.85

Suburban 0.25–0.40 Roofs 0.75–0.95

Soil type Cultivated Pasture Woodlands

Above-average infiltration: sandy soil or gravel 0.20 0.15 0.10

Average infiltration: no claypans; loams/similar soils 0.40 0.35 0.30

Below-average infiltration: heavy clay soils, soils 

 with a claypan near the surface, shallow soils above 

 impervious rocks 0.50 0.45 0.40

Table 4-26—Additional rational coefficients by 

vegetation type.

Vegetation type Runoff coefficient

Riparian 0.02

Sagebrush/other shrubs 0.18

Rocks–soils 0.50

Grass–forb 0.20

Conifer 0.10

Aspen 0.08

Pinyon–juniper 0.20

Gambel oak 0.12

ratios. Tables 4-25 and 4-26 present rational coefficients, which are often assumed to 

be equivalent.

Other rational coefficients (table 4-26) are offered for application to wild lands 

as pre-fire runoff coefficients (Easterbrook 2006).

4.53 Discussion

As the most basic “model” for rainfall-runoff response, the runoff fraction, C, is 

intuitively obvious to the point of being seldom mentioned by name in hydrology literature. However, it has a specific interpretation as a direct source area for watersheds 

with conspicuous impervious areas. For small values of C, from about 0.002 to 0.05, 

it represents the Complacent response, which is later offered in Wildcat5 as an option 

under Rainfall Excess Options.
USDA Forest Service RMRS-GTR-334. 2016. 41

The Wildcat5 Summary Output Table and Hydrograph gives the de facto 

values of the found C (= Q/P) for the modeled storm from the rainfall excess used. 

The similarity is apparent between this form and the widely used rational 

equation: 

 qp = Cimax (4-12)

with qp as the peak flow and imax as the maximum storm intensity for a duration equal 

to the time of concentration, both in units of intensity (such as in/hr, or mm/hr). When 

the input is in in/hr and the output is calculated in ft3/sec for a drainage area of A ac, it 

takes the familiar form of 

 qp = CimaxA (4-13)

The units conversion between equations (4-12) and (4-13) for the English system is 121/120, and is usually ignored and treated as 1.00.

4.6 Distributed Loss Depth

4.61 General

This method assumes that losses are not limited by rate, such as with infiltration 

capacities (for example, in/hr), but instead are limited by depth (for example, inches).

That is, after that depth has been filled by rain, all additional rainfall becomes runoff 

(rainfall excess). However, Wildcat5 allows this process to be distributed in space on 

the watershed.

4.62 Concepts

This is not an infiltration rate method, but a loss depth method. It imagines the 

watershed to be composed of distributed points of F, where F is the ultimate retention 

storage at a point potentially satisfied in the rainstorm. It might be seen as a collection 

of open tin cans of variable depth F with each can (point) performing as: 

Q = 0 P < F

Q = P – F P ≥ F

After a “can” has been filled by rain, all additional rain spills and becomes 

rainfall excess draining directly to the outlet. For example, if a point F is 1 inch, then 

no runoff occurs from that point as long as P ≤ 1 in. Above that, all additional rain becomes rainfall excess. This is equivalent to saying Q = 0 for P < 1 inch and Q = P – 1 

for P > 1 in. This is a straight line with an intercept of -1 and a slope of 1.00.

4.63 Distributed Performance

More realistic representations of observed rainfall and runoff behavior can be 

estimated by weighting values of F to different areas (acres) of the watershed. Fi is the 

loss depth F for an individual fractional part in the watershed. The fraction is αi, where 

all the fractions add up to 1.00. The representation of this is: 

Q = ΣαiQi = Σαi(P – Fi) for all i, and for all P > Fi

The robustness and practical possibility of this approach rest in the distributed 

form. The single-point linear all-or-nothing “tin can” runoff process is a greatly 

simplified representation. But by amassing a number of fractional areas of varying 
42 USDA Forest Service RMRS-GTR-334. 2016.

properties, a more realistic overall performance can be simulated. For example, there

is a distribution of F that results in the CN equation of Q with P. Note that no specific 

statistical distribution is suggested here.

4.64 Parameter Values

No authoritative or handbook values of F are available, and parameter choices 

are ultimately the responsibility of the user. However, insofar as F is the maximum 

possible retention of rainfall, it is the approximate cross product of the soil depth and 

the effective porosity (field capacity – ambient soil moisture). Thus, it might be imagined as a simplified representation of Ia + S (= 1.2S) from the CN method. With this in 

mind the values in table 4-27 are given as a guide for those experienced in CN usage.

Keep in mind that 1.2S is the limit, attained as P→∞, and this distributed loss 

depth tactic is not the CN method. Thus lower values—at perhaps 20 to 60 percent of 

the table entries—should be drawn from the above for realistic use as “F” values. The 

relevance of the range of suggested F values (table 4-27) has not been verified in the 

field.

Table 4-27—Suggested potential losses for 

distributed loss depth model.

CN 1.2S (in) Range of suggested F (in)

100 0 0

95 0.632 0.12–0.36

90 1.333 0.27–0.80

85 2.118 0.42–1.28

80 3.000 0.60–1.80

75 4.000 0.80–2.40

70 5.143 1.03–3.09

65 6.462 1.29–3.88

60 8.000 1.60–4.80

4.65 Example

A 500-ac watershed with 100 ac each of F at 0.5, 1.0, 1.5, 2.0, and 2.5 in would 

have no runoff until P = 0.5 in, and the following array of rainfall and runoff depths as 

successive F elements became active by exceeding successive thresholds (table 4-28).

In the above simplified example, the areal fraction α is 0.20 for all cases (not 

to be confused with the 0.2 initial abstraction coefficient used in the CN method). A 

cumulative total can be kept for as many stated points (or continuous distribution) as 

needed. In this example the descriptor F is specified for a user-chosen array of different areas.

4.66 Simulating Complacent and Violent Responses

Complacent and Violent response options are described in the next section, but 

can be modeled by using the distributed loss method. 

You can represent the Complacent response by setting F = 0 for the appropriate 

small fraction, and an absurdly high F value, like 10 in for the remainder of the areas.
USDA Forest Service RMRS-GTR-334. 2016. 43

Thus the following array would simulate the Complacent response Q = 0.02P for 

rainfalls up to 10 in:

α1 = 0.02 F1 = 0 in

α2 = 0.98 F2= 10 in

Violent response (see next section) can be represented similarly, by selecting F2

at the threshold P and an appropriate value of a. For example, assuming

α1 = 0.02 F1 = 0 in

α2 = 0.98 F2 = 2 in 

leads to

Q = 0.02P for P ≤ 2 in 

Q = 0.02P + 0.98(P – 2) for P ≥ 2 in

= P – 1.96

4.7 Complacent–Violent Response

4.71 Concepts

This option is a response pattern found on many forested watersheds. Though 

not unknown in nature, it is generally unappreciated and does not have a long history 

of application or authoritative coefficients. It differs significantly in form and concept 

from the CN approach.

Here the rainfall-runoff process hangs on three identifying elements: (1) a 

low linear response early in the storm (the Complacent phase), (2) a characteristic 

threshold rainfall, and (3) an abrupt change to a high incremental response (Violent 

phase) above the threshold rainfall depth. The two phases may be more than an 

order of magnitude different in converting rainfall to runoff. Wildcat5 represents the 

Complacent–Violent option by the following: 

Complacent behavior Q = CP for P ≤ Pt (4-14)

Violent behavior Q = CP + (b2 – C)(P – Pt) for P ≥ Pt (4-15)

= CPt + b2(P – Pt) for P ≥ Pt

Table 4-28—Example of distributed loss calculation. 

P (in) Q (in) Calculation Percent contributing

0 0 none 0

0.25 0 none 0

0.50 0 none 0

0.75 0.050 0.2(0.75 – 0.50) 20

1.00 0.100 0.2(1.00 – 0.50) 20

1.25 0.200 0.2(1.25 – 0.50) + 0.2(1.25 – 1.00) 40

1.50 0.300 0.2(1.50 – 0.50) + 0.2(1.50 – 1.00) 40

1.75 0.450 0.2(1.75 – 0.50) + 0.2(1.75 – 1.00) + 0.2(1.75 – 1.50) 60

2.00 0.600 0.2(2.00 – 0.50) + 0.2(2.00 – 1.00) + 0.2(2.00 – 1.50) 60

2.25 0.800 0.2(2.25 – 0.50) + 0.2(2.25 – 1.00) + 0.2(2.25 – 1.50) + 0.2(2.25 – 2.00) 80

2.50 1.000 0.2(2.50 – 0.50) + 0.2(2.50 – 1.00) + 0.2(2.50 – 1.50) + 0.2(2.50 – 2.00) 80

2.75 1.250 0.2(2.75 – 0.50) + 0.2(2.75 – 1.00) + 0.2(2.75 – 1.50) + 0.2(2.75 – 2.00) + 0.2(2.75 – 2.50) 100

3.00 1.500 0.2(3.00 – 0.50) + 0.2(3.00 – 1.00) + 0.2(3.00 – 1.50) + 0.2(3.00 – 2.00) + 0.2(3.00 – 2.50) 100
44 USDA Forest Service RMRS-GTR-334. 2016.

where Pt is the threshold rainfall above which the violent condition applies and b2 is 

the runoff rate fraction at rainfalls greater than Pt.

4.72 Parameter Selection

The three parameters in the Complacent–Violent model have some physical 

interpretation, at least in the simplest cases. C is the Complacent coefficient, or Q/P, 

and may be taken as the fraction of the watershed with direct impervious runoff. This 

fraction has been linked to the channel source area (Panky and Hawkins 1983). C is 

also the Constant fraction option offered elsewhere in Wildcat5.

If P is less than Pt, the entire storm runoff will be in the Complacent mode.

Values for C found in data analysis vary from 0 to about 0.06. Values taken from field 

data in the range of 0.005 to 0.02 are common (see table 4-29). 

Pt is the threshold or “tip-over” rainfall. In settings with soils with limited storage and high entry rates, this is the depth of rain needed to fill the canopy and litter, 

and the soil column to the point of soil water movement by gravity. Threshold rainfall 

begins at field capacity (FC), and reaches a maximum condition at saturation, at which 

point surface (overland) flow may occur. Values for Pt

 found by data analysis are typically from 1.5 to 3.0 in, with a cluster near 2 in. It can be less if the soil has a higher 

water content at the start of the storm for pre-wetted conditions. 

With the model described by equation (4-13), most runoff does not occur until 

well into the storm, when Pt has been exceeded. Values for the runoff rate fraction 

found by data analysis are in the range of 0.60 to 1.00. Examples with b2 = 1.00 have 

been calculated in such datasets. By the above reasoning, 0 ≤ C ≤ (C + b2) ≤ 1. 

Insofar as the Violent phase operates at the rarer, larger storms and creates 

unexpected out-of-channel peak flow rates, clean datasets displaying it are not as common as for pure Complacent response. Some suggested typical coefficient values are 

presented in table 4-29.

Table 4-29—Typical values for coefficients used in equations (4-14) and (4-15) for 

predicting Complacent–Violent runoff. 

C 0.001–0.07 Complacent, live channel, some forested watersheds.

0.07–0.30 High linear, for varied sources. “Dry” complacent. 

Pt 1.5–3.0 in In stable watersheds. Much smaller if freshly burned or wetted.

 Values are suggested by the maximum rainfall values in table 4-30.

 Pt

 for these cases should equal or exceed the values shown. 

b2 0.70–1.00 Violent limb, (C + b2 ) ≤ 1.00 in extreme rainstorms or shallow soils, or both.

As with all such efforts, including the CN method, coefficient selection is 

subject to your judgment. Note that either Complacent or Violent behavior can be 

modeled depending on the coefficient values selected. The difference between b2 and 

C is a measure of the behavior change at threshold rainfall Pt. You can represent the 

entire process by manipulating the Distributed F option also offered in Wildcat5.

More recent documentation on the Complacent‒Violent response, with found values of 

C, Pt

, and b2, is given in Hawkins and others (2015).
USDA Forest Service RMRS-GTR-334. 2016. 45

4.73 Discussion

The Complacent–Violent option may be more useful in explaining otherwise 

puzzling on-the-ground or in-channel observations than in direct design or environmental appraisals. A difficulty in using this option is identifying the nature of 

the runoff and the coefficients from field information. Curve Number tables and 

Hydrologic Soil Groups cannot explain Complacent–Violent behavior when it occurs.

An infrequently applied example of a Complacent response is given in Dunne and 

Leopold (1987: 289), a commonly used text that shows rainfall-runoff response from a 

147-ac U.S. Agricultural Research Service watershed at Danville, VT. For 10 events up 

to P = 3 in, the observed runoff conforms closely to Q = 0.06P. Here it can be inferred 

that C = 0.06, and Pt > 3 in. For perspective, a rainstorm of 3 in is at about a 100-yr 

return period for a 3-hr duration for this part of Vermont. 

Though suggested here as a channel interception process, the general low-linear 

rainfall-runoff (Complacent) response is also found in some “dry” conditions in 

wildland and other settings. For example, forest land at Beaver Creek, AZ, which has 

ponderosa pine cover but little or no baseflow, performs as Complacent–Violent with 

the following approximate values: C = 0.07, Pt = 1.80 in, and b2 = 0.94 (Hawkins 

1989). Fully covered sugar cane fields on lateritic (clay) soils in Brazil show a C of 

about 0.008 for storms up to about 3 in (Sartori and others 2011).

Data-based examples of the Violent phase are rarer than for the Complacent 

phase because they occur at higher (rarer) rainfalls, and the extremes sometimes 

exceed flow measurement capabilities. The split Complacent–Violent phenomenon is 

more apparent when data are treated as rank ordered (frequency matched).

Table 4.30 shows Complacent response for many western U.S. wildland watersheds (Springer and others 2005). The Pmax return periods for selected entries in the 

tables are as follows: Arizona: 1-hr duration, 120-yr return interval; 3-hr duration, 

30-yr return interval; Colorado: 1-hr duration, ~20-yr return interval; and Utah: 1-hr 

duration, 30-yr return interval; 3-hr duration, 10-yr return interval. 

Table 4-30—Selected wildland watershed Complacent rainfall-runoff characteristicsa.

Area Pmax r2 Standard

Name State (ac) From To (in) N C (%) error (in) Reference

North Thomas AZ 467 1965 1970 2.35 9 0.0008 79 0.0006 Anderson (1975)

South Thomas AZ 562 1963 1970 2.19 12 0.0010 48 0.0011 Anderson (1975)

Missouri Gulch CO 4,600 1940 1959 1.58 14 0.0030 79 0.0011 Hawkins (1961)

Eggers ID 318 1969 1978 2.31 38 0.0048 87 0.0015 McGurk (1982)

Control ID 401 1969 1976 2.58 32 0.0054 53 0.0030 McGurk (1982)

Cabin ID 271 1970 1978 2.62 43 0.0046 43 0.0032 McGurk (1982)

Ditch ID 252 1969 1975 2.43 27 0.0043 76 0.0024 McGurk (1982)

C Creek ID 460 1970 1978 2.31 29 0.0206 64 0.0126 McGurk (1982)

D Creek ID 292 1969 1978 2.31 30 0.0159 64 0.0105 McGurk (1982)

Murphy ID 306 1967 1977 1.66 27 0.0074 24 0.0105 McGurk (1982)

West Chicken UT 217 1962 1971 1.96 16 0.0096 67 0.0070 Johnson and 

 Doty (1972) 

East Chicken UT 137 1962 1971 1.31 12 0.0048 91 0.0013 Johnson and

 Doty (1972)

Halfway UT 484 1940 1966 1.50 14 0.0113 90 0.0038 Walker (1970)

a C is the least squares fit to Q = CP with natural data; r2 is the variance reduction (percent) achieved by the fitting. Source: Springer 

and Hawkins (2005); some entries have been corrected from the original publication with later data. 
46 USDA Forest Service RMRS-GTR-334. 2016.

4.8 Chapter References

Curve Numbers

Brakensiek, D.L.; Rawls, W.J. 1983. Green-Ampt infiltration model parameters for 

hydrologic classification of soils. In: Borelli, J.; Hasfurther, V.; Burman, D., eds.

Advances in irrigation and drainage, surviving external pressures. Proceedings 

of the American Society of Civil Engineers Specialty Conference. New York: 

American Society of Civil Engineers: 226–233.

Cao, H.; Vervoort, W.; Dabney, S. 2011. Variation in Curve Numbers derived from 

plot runoff data for New South Wales (Australia). Hydrological Processes. 25: 

3774–3789.

Constantine, C.R; Naftaly, N.E.; Frye, J.S. 2010. Fire effects on rainfall-runoff in the 

Santa Barbara area. Southwest Hydrology. 9(1): 10–11.

Easterbrook, R. 2006. Predicting post-fire wildfire watershed runoff using 

ArcGISModelbuilder. Given at ESRI Federal User Conferences, 2006. 35 p. http://

gis.esri.com/library/userconf/feduc06/docs/predicting_wildfire_runoff.pdf. (March 

23, 2015).

Foltz, R.B.; Robichaud, P.R.; Rhee, H. 2009. A synthesis of post-fire road 

treatments for BAER teams, treatment, effectiveness, and decision making tools 

for rehabilitation. Gen. Tech. Rep. RMRS-GTR-228. Fort Collins CO: U.S. 

Department of Agriculture, Forest Service, Rocky Mountain Research Station.

152 p.

Goodrich, D.C.; Canfield, H.E.; Burns, I.S.; Semmens, D.J.; Miller, S.N.; Hernandez, 

M.; Levick, L.R.; Guertin, D.P; Kepner, W.G. 2005. Rapid post-fire hydrologic 

watershed assessment using the AGWA GIS-based hydrologic modeling tool. 

Proceedings of the Watershed Management Symposium, Managing Watersheds for 

Human and Natural Impacts. [CD-ROM]. Reston, VA: American Society of Civil 

Engineers. 13 p. http://www.tucson.ars.ag.gov/agwa/docs/pubs/Goodrich-ASCE05-AGWA.pdf. (March 23, 2015).

Hawkins, R.H.; Ward, T.J. 1998. Site and cover effects on event runoff, Jornada 

Experimental Range, New Mexico. In: Potts, D.A., ed. Proceedings of Specialty 

Conference on Rangeland Management and Water Resources. Herndon, VA: 

American Water Resources Association: 361–370.

Hawkins, R.H.; Ward, T.J.; Woodward, D.E.; Van Mullem, J.A. 2009. Curve Number 

hydrology: state of the practice. Reston, VA: American Society of Civil Engineers. 

106 p.

Higginson, B.; Jarnecke, J. 2007. Salt Creek BAER—2007 Burned Area Emergency 

Response. Specialist Report. Provo, UT: Uinta National Forest. 11 p.

Hjelmfelt, A.T., Jr.; Kramer, L.A.; Burwell, R.E. 1982. Curve Numbers as random 

variables. In: Rainfall-runoff relationship. Littleton, CO: Water Resources 

Publications: 365–370.

Lusby, G.C. 1979. Effects of grazing on runoff and sediment yield from desert 

rangeland at Badger Wash in Western Colorado, 1953–73. USGS Water Supply 

Paper 1532-I. Washington, DC: U.S. Department of the Interior, Geological Survey. 

34 p.
USDA Forest Service RMRS-GTR-334. 2016. 47

Lusby, G.C.; Reid, V.H.; Knipe, O.D. 1971. Effects of grazing on the hydrology and 

biology of the Badger Wash Basin in Western Colorado, 1953–56. USGS Water 

Supply Paper 1532-D. Washington, DC: U.S. Geological Survey. 90 p. 

Nielsen, R.E.; Hjelmfelt, A.T., Jr. 1998. Hydrologic Soil Group assignment. In: Water 

Resources Engineering ’98: Proceedings of the International Water Resources 

Engineering Conference. Reston, VA: American Society of Civil Engineers: 

1297–1302.

Rietz, P.D. 1999. Effects of land use on Runoff Curve Numbers. Tucson, AZ: 

University of Arizona. 115 p. M.S. thesis.

Rietz, P.D.; Hawkins, R.H. 2000. Effects of land use on Runoff Curve Number.

Proceedings, Symposium on Watershed Management. New York: American Society 

Civil Engineers. 11 p.

Solt, A.; Muir, M. 2006. Warm fire—hydrology and watershed report. Richfield, UT: 

U.S. Department of Agriculture, Forest Service, Intermountain Region, Fishlake 

National Forest. 9 p.

U.S. Natural Resources Conservation Service [NRCS]. 2003. Updated 2012. National 

engineering handbook. Part 630, Hydrology. Washington, DC: Department of 

Agriculture. directives.sc.egov.usda.gov/viewerFS.aspx?hid=21422. (March 17, 

2015).

U.S. Soil Conservation Service [SCS]. 1954 [and following]. National engineering 

handbook. Section 4, Hydrology. Washington, DC: U.S. Department of 

Agriculture. 115 p. http://directives.sc.egov.usda.gov/OpenNonWebContent.

aspx?content=18393.wba. (March 19, 2015).

U.S. Soil Conservation Service [SCS]. 1986. Urban hydrology for small watersheds.

Technical Release 55. Washington, DC: U.S. Department of Agriculture. 164 p. 

www.nrcs.usda.gov/Internet/FSE_DOCUMENTS/stelprdb1044171.pdf. (March 18, 

2015).

Curve Number with Ia/S = 0.05

Baltas, E.A.; Dervos, N.A.; Mimikou, M.A. 2007. Technical Note: determination of 

SCS initial abstraction ratio in an experimental watershed in Greece. Hydrology 

and Earth System Science. 11(6): 1825–1829.

D’Asaro, F.; Grillone, G. 2009. Analisi preliminare per la caratterizzazione hidrologica 

dell’Alcantara secondo il modello CN-NRCS [Preliminary analysis for Alcantara 

basin hydrologic characterization, using CN-NRCS method]. Quaderni di 

Idronomia Montana. 28 (2): 491–506.

Elhakeem, M.; Papanicolaou, A.N. 2009. Estimation of the Runoff Curve Number via 

direct rainfall simulator measurements in the State of Iowa, USA. Water Resources 

Management. 23 (12): 2455–2473.

Hawkins, R.H.; Jiang, R.; Woodward, D.E. 2001. Investigation of the Curve Number 

abstraction ratio. Presentation to Annual Meeting of American Water Resource 

Association, Albuquerque, NM, November.
48 USDA Forest Service RMRS-GTR-334. 2016.

Hawkins, R.H.; Van Mullem, J.A.; Woodward, D.E.; Hjelmfelt, A.T., Jr.; Jiang, 

R. 2002. Curve Number method: examination of the initial abstraction ratio. 

Proceedings 2nd Federal Interagency Hydrologic Modeling Conference [CD-ROM]. 

10 p.

Hawkins, R.H.; Ward, T.J.; Woodward, D.E.; Van Mullem, J.A. 2009. State of the 

practice in Curve Number hydrology. Task committee report of the Environmental 

and Water Resources Institute of the American Society of Civil Engineers. Reston, 

VA: American Society of Civil Engineers. 106 p.

Hawkins, R.H.; Woodward, D.E.; Jiang, R. 2001. Investigation of the runoff curve 

number abstraction ratio. Proceedings of the USDA-NRCS Hydraulic Engineering 

Workshop. Washington, DC: U.S. Department of Agriculture, Natural Resources 

Conservation Service. 10 p.

Jiang, R. 2001. Investigation of runoff curve number initial abstraction ratio. Tucson, 

AZ: University of Arizona. 120 p. M.S. thesis.

Shi, Z.H.; Chen, L.D.; Fang, N.F.; Qin, D.F.; Cai, D.F. 2009. Research on SCS–CN 

initial abstraction using rainfall-runoff event analysis in the Three Gorges area, 

China. Catena. 77(1): 1–7. 

Woodward, D.E.; Hawkins, R.H.; Jiang, R.; Hjelmfelt, A.T., Jr. 2003. Runoff curve 

number method: examination of the initial abstraction ratio. In: Bizier, P.; DeBerry, 

P., eds. Proceedings of the World Water and Environmental Resources Conference.

[CD-ROM]. Reston, VA: American Society of Civil Engineers. 

φ-Index 

Gifford, G.F.; Hawkins, R.H. 1978. Hydrologic impact of grazing on infiltration—a 

critical review. Water Resources Research. 14(2): 305–313.

Lee, R. 1980. Forest hydrology. New York: Columbia University Press. 349 p.

Linsley, R.K.; Kohler, M.A.; Paulus, J.L.H. 1982. Hydrology for engineers. 3rd ed.

New York: McGraw-Hill. 508 p.

Overton, D.E.; Meadows, M.E. 1976. Stormwater modeling. New York: Academic 

Press. 358 p.

Distributed Infiltration Capacity

Bouwer, H. 1966. Rapid field measurement of Ari entry value and hydraulic 

conductivity of soil as significant parameters in flow systems analysis. Water 

Resources Research. 2(4): 729–738. 

Clapp, R.B.; Hornberger, G.M. 1978. Empirical equations for some soil hydraulic 

properties. Water Resources Research. 14: 601–604.

Elliot, W.J. 2004. WEPP Internet interfaces for forest erosion prediction. Journal of the 

American Water Resources Association (JAWRA). 40(2): 299–309. 

Gifford, G.F.; Hawkins, R.H. 1978. Hydrologic impact of grazing on infiltration—a 

critical review. Water Resources Research. 14 (2): 305–313.

Hawkins, R.H. 1981. Interpretations of source area variability. In: Rainfall–runoff 

relationships. International Symposium on Rainfall–Runoff Modeling. Chelsea, MI: 

Water Resources Publications: 47–52. 
USDA Forest Service RMRS-GTR-334. 2016. 49

Hawkins, R.H.; Cundy, T.W. 1989. A continuous distributed model of storage 

dominated watershed runoff. Journal of Irrigation and Drainage Engineering 

(American Society of Civil Engineers). 115(2): 305–311.

Nearing, M.A; Wei, H.; Stone, J.J.; Pierson, F.B.; Spaeth, K.E.; Weltz, M.A.; 

Flanagan, D.C.; Hernandez, M. 2011. A rangeland hydrology and erosion model.

Transactions of the ASABE. 54(3): 1–8. 

Rawls, W.J.; Brakensiek, D.L.; Miller, N. 1983. Green-Ampt infiltration parameters 

from soils data. Journal of Hydraulic Engineering (American Society of Civil 

Engineers). 109(1): 62–70. 

Stone, J.J.; Lane, L.J.; Shirley, E.D. 1992. Infiltration and runoff simulation on a plane. 

Transactions of the ASABE. 35(1): 161–170 [Table 1].

Stone, J.J.; Paige, G.B.; Hawkins, R.H. 2008. Rainfall intensity-dependent infiltration 

rate on rangeland rainfall simulation plots. Transactions of the ASABE. 15(1): 

45–53.

U.S. Agricultural Research Service. 2013. Rangeland Hydrology and Erosion Model 

Web tool. http://dss.tucson.ars.ag.gov/rhem/. (March 23, 2015).

Wei, H.; Nearing, M.A.; Stone, J.J.; Guertin, D.P.; Spaeth, K.E.; Pierson, F.B.; 

Nichols, M.H.; Moffet, C.A. 2009. A new splash and sheet erosion equation for 

rangelands. Soil Science Society of America Journal. 73(4): 1386–1392.

Runoff Fraction (Runoff Ratio)

Chow, V.T. 1964. Handbook of applied hydrology: a compendium of water-resources 

technology. New York: McGraw-Hill. 1495 p. 

Chapter 14 is by V.T. Chow, and the first table here is attributed to an earlier report 

by the American Society of Agricultural Engineers and the Water Pollution Control 

Federation. 

Chapter 21 is by H.O. Ogrosky and V. Mockus, and the second table above was 

attributed to an earlier work by M. Bernard (1932), Discussion of runoff: rational 

runoff formulas by R.L. Gregory and C.E. Arnold. Transactions of the American 

Society of Civil Engineers. 96: 161. 

Easterbrook, R. 2006. Predicting post-fire wildfire watershed runoff using ArcGIS 

Modelbuilder. Given at ESRI Federal User Conferences, 2006. 35 p. http://gis.esri.

com/library/userconf/feduc06/docs/predicting_wildfire_runoff.pdf. (March 23, 

2015).

Complacent–Violent Response

Anderson, R.J. 1975. Relationship between rainfall and storm runoff for selected 

Arizona watersheds. Logan, UT: Utah State University. 64 p. M.S. thesis.

Dunne, T.; Leopold, L.B. 1978. Water in environmental planning. San Francisco: W.H. 

Freeman and Co. 818 p.

Hawkins, R.H. 1961. A study to predict storm runoff from storm characteristics and 

antecedent basin conditions. Fort Collins, CO: Colorado State University. 76 p. 

M.S. thesis.
50 USDA Forest Service RMRS-GTR-334. 2016.

Hawkins, R.H. 1992. Variety, classification, and association in rainfall-runoff response. 

Report submitted to and on file with U.S. Environmental Protection Agency, 

Environmental Research Laboratory–Corvallis. 40 p.

Hawkins, R.H. 1993. Asymptotic determination of runoff curve numbers. 1992. 

Journal of Irrigation and Drainage Engineering (American Society of Civil 

Engineers). 119(2): 334–345.

Hawkins, R.H.; Pankey, J.M. 1981. Storm flow as a function of watershed impervious 

area. In: Proceedings Arizona Section American Water Resources Association, and 

Hydrology Section ArizonaNevada Academy of Sciences: 35–37. http://arizona.

openrepository.com/arizona/bitstream/10150/301215/1/hwr_11-035-037.pdf.

(March 23, 2015).

Hawkins, R.H.; Ward, T.J.; Woodward, E.E. 2015. The Complacent‒Violent runoff 

behavior: a departure from tradition. In: Moglen, G.E., ed. Watershed management 

2015: power of the watershed. Proceedings of the ASCE Watershed Management 

Conference. [e-book]. Reston, VA: American Society of Civil Engineers. 12 p.

Johnson, R.S.; Doty, R.D. 1972. Description and hydrologic analysis of two small 

watersheds in Utah’s Wasatch Mountains. Res. Pap. INT-127. Ogden, UT: U.S. 

Department of Agriculture, Forest Service, Intermountain Forest and Range 

Experiment Station. 53 p.

McGurk, B.J. 1982. A comparison of four rainfall-runoff methods. Logan, UT: Utah 

State University. 258 p. Ph.D. dissertation.

Sartori, A.; Hawkins, R.H.; Genovez, A.M. 2011. Reference Curve Numbers and 

behavior for highly weathered soils in Brazil. Journal of Irrigation and Drainage 

Engineering (American Society of Civil Engineers). 137(11): 705–711.

Springer, E.P.; Hawkins, R.H. 2005. Curve number and peakflow response following 

Cerro Grande fire on a small watershed. Presentation at Watershed Management 

Conference. [Abstract in Proceedings CD-ROM]. Reston, VA: American Society of 

Civil Engineers.

Stewart, D.; Canfield, E.; Hawkins, R.H. 2010. A comparison of Curve Number 

determination methods and uncertainty in Hydrologic Soil Groups from semi-arid 

watershed data. Journal of Hydrologic Engineering (American Society of Civil 

Engineers). 17(11): 1180–1187. 

Walker, C.H. 1970. Estimating the rainfall-runoff characteristics of selected small 

Utah watersheds. Logan, UT: Utah State University. 120 p. M.S. thesis. http://

digitalcommons.usu.edu/etd/3109. (March 18, 2015).
USDA Forest Service RMRS-GTR-334. 2016. 51

Chapter 5. Timing Parameters

5.1 Concepts

The timing characteristics of a small watershed control how rapidly generated 

rainfall excess runs off, and thus how “flashy” or delayed the resulting hydrograph 

might be. In Wildcat5 the timing descriptors—time of concentration (tc) or lag (tL)—

are used in scaling the unit hydrographs. From them the unit hydrograph time base 

(tb), time to peak (tp), and the time increment (∆t) used to step through the design 

storm are calculated. Here, as in most professional applications of this kind, ∆t is taken 

to be tp/5.

Timing characteristics also give you an intuitive glimpse of actual watershed 

conditions, allowing inference of an average flow velocity as flow distance/tc. Several 

options—all empirical formulas—are available in Wildcat5, as is a user override.

The most popular notion of watershed timing is tc, defined as the time required 

for water to travel the length of the longest runoff path in the watershed to the outlet. It 

arose from applying the rational equation, equation (4-11), and asserts that during that 

time interval all parts of the watershed contribute runoff simultaneously to the outlet, 

the ideal condition for maximum flood peak generation. 

A centroid-peak lag is also used. It is taken to be the time from the centroid of 

the rainfall excess to the flood peak. This is the usual interpretation when the expression “lag time” is used. 

A third alternative, the centroid-centroid lag, is defined as the time from the 

centroid of the rainfall excess to the centroid of the resulting hydrograph. 

Any one of the three measures above may be used to arrive at the unit hydrograph dimensions tb and tp Results will differ among methods and equations. Note 

that shorter times of concentration or lag can be expected to lead to higher peak flows.

5.2 Choices and Parameter Selection

Four alternatives are offered: (1) a user’s choice override, (2) a direct tc equation, (3) a lag time equation, and (4) an option for the centroid-centroid lag. In 

alternatives 2, 3, and 4 you supply the watershed parameters required, and Wildcat5 

calculates the unit hydrograph (tp) and time step (∆t) parameters. 

1. User choice. This option allows you to specify from judgment or experience tc 

in hours. It also allows you to calculate tc separately from equations not offered 

here.

2. Kirpich’s equation for tc. Although several different forms exist, the original 

equation (Kirpich 1940) is:

t

c = (11.9L3 / H)0.385 

where

t

c is in hr 

L is the length of the longest runoff path (mi)

H is the difference in elevation along the above flowpath (ft)
52 USDA Forest Service RMRS-GTR-334. 2016.

Wildcat5 asks for the information in slightly different form:

 Channel slope (percent), and 

 Channel length (ft) 

The changes are made internally for the calculations.

3. Kent’s equation for lag time. This equation was developed from U.S. Natural 

Resources Conservation Service (NRCS) applications and the definition of 

flood-peak lag:

t

L(ctp) = L0.8(S+1)0.7 /(1900Y0.5)

where 

t

L is the lag time (hr)

L is the hydraulic length of the watershed (ft)

S is 1000/CN – 10 (in). Average Curve Number (CN) is taken from previous steps.

Y is the average watershed land slope (percent)

The time of concentration is then determined from the equation tL = 0.6tc.

4. Simas’ equation. This work deals with the centroid-centroid lag time, and arises 

from work done at the University of Arizona on 31,030 events on 168 small 

watersheds (Simas 1996). It uses CN, watershed length (ft), and watershed width 

(ft). The equation is:

t

L(cc) = 0.0051W0.594Slope-0.150 S0.313

where

t

L(cc) is in hr 

W is the width in feet = area (ft2) / length (ft). Length is the longest flowpath

 (ft) to the highest elevation.

Slope is a channel slope (ft/ft) = the elevation difference along the flowpath

 length

S = (1000/CN) – 10 (in).

With this method, ∆t and tp are calculated internally without using tc, but by 

exploiting the unique geometric characteristics of the unit hydrograph, and proceeding 

on the assumption of ∆t = tp/5.

5.3 Discussion

Numerous other tc and tL equations exist and continue to be developed, as reported in the literature. Thus you are given the option of entering your preferred tc on 

the input page.

One common method recommended in the U.S. Soil Conservation Service and 

NRCS publications is to estimate overland and channel velocities of water as it makes 

its way from the top of the catchment to the outlet, and then to sum up the times for 

each segment of the flowpath. The various methods do not necessarily give the same 

results, just as the above three equations will give different estimates. As most methods predicting peak flow are quite sensitive to time of concentration, take care when 

selecting the methods to predict tc and tL.
USDA Forest Service RMRS-GTR-334. 2016. 53

5.4 Chapter References

Kent, K.M. 1968. A method for estimating volume and rates of runoff in small 

watersheds. Revised April 1973. SCS-TP-149. Washington, DC: U.S. Department 

of Agriculture, Soil Conservation Service. 64 p. ftp://ftp.wcc.nrcs.usda.gov/

wntsc/H&H/TRsTPs/TP149.pdf. (March 17, 2015).

Kirpich, P.Z. 1940. Time of concentration of small agricultural watersheds. Civil 

Engineering. 10(6): 362.

Simas, M.J.C. 1996. Lag time characteristics of small watersheds in the United States.

Tucson, AZ: University of Arizona. 170 p. Ph.D. dissertation.
54 USDA Forest Service RMRS-GTR-334. 2016.

Chapter 6: Unit Hydrographs

6.1 Concepts

Unit hydrograph (UH) methods are widely used for routing periodic pulses of 

rainfall excess from the upland source areas to the watershed outlet. The UH is used as 

a template to generate contributing hydrographs starting from different periods within 

a storm. These hydrographs are lagged in time, superimposed, and summed to create a 

composite storm hydrograph at the outlet. The routing is necessary to model the shortduration surface runoff from the upland watershed sources to the outlet. This process 

of lagging and summing is sometimes called a convolution.

The UH is defined as the characteristic hydrograph for a unit of rainfall excess 

for a subject watershed. It accounts for the watershed properties by its time to peak, 

the total duration, and the shape. These characteristics describe the drainage network and slopes, distribution of source areas, and total drainage area. A UH may be 

visualized as the distribution of travel times from the source points to the watershed 

outlet. By definition and custom, the UH has a volume of 1 unit. The units are linear 

(ordinates are proportional to the volume), time consistent (described by fixed time descriptors such as time to peak and duration), and superimposable (they can be lagged 

and summed). A number of unit hydrograph options are offered in Wildcat5.

 More advanced alternative techniques that achieve similar results are overland 

flow routing by kinematic wave technology, or linear reservoirs or channel routing for 

the drainage network. Neither of these alternatives is offered in Wildcat5 because of 

the unit hydrograph’s comparative simplicity and long-time precedent. 

6.2 Triangular Unit Hydrographs with General 

Geometry

6.21 General

Triangular unit hydrographs are a long-used simplification of the curvilinear 

unit hydrograph that accompanied the emergence in the 1950s of the U.S. Soil 

Conservation Service (SCS) methods, which included the Curve Number (CN) 

method. However, they are not really a part of the CN method, but are a separate 

technology.

6.22 Background and Description

The SCS triangular unit hydrograph has a time to peak of tp, and time of recession of 1.67tp, and thus a base of 2.67tp. Its peak flow is qp (1/T). Note that 2.67 is a 

rounded-off value for 8/3, as 1.67 is for 5/3.

The area of the triangular UH is:

2.67tp qp/2 

which contains the ΔQ from an impulse of rainfall excess. Thus 2.67tp qp/2 = ΔQ. 

Solving for qp, we find:

 qp = ΔQ/(1.333tp)
USDA Forest Service RMRS-GTR-334. 2016. 55

where 

qp is in in/hr

ΔQ is in inches.

To apply this to produce qp in ft3/sec from a watershed with a drainage area of 

A (mi2) requires conversion of inches to feet, square miles to square feet, and hours to 

seconds, leading to: 

qp = 484A ΔQ/tp 

where 

A is in mi2

ΔQ is incremental impulse of rainfall excess (in)

t

p is in hr

qp is in ft3/sec.

In most models ΔQ is the runoff amount for time period of Δt, and Δt is often equal to 

t

p/5, as mentioned in section 5.1 and following.

6.23 General Case

The value 484 is often called the hydrograph factor (HF), or the peak flow 

factor. The term “factor” suggests that the value may be changed. However, doing so 

requires preserving the mass (do not change ΔQ), which can happen only if the relationship between tb and tp changes.

For a general case, consider that the recession (falling) limb is not fixed at 

1.67tp, but rather is generalized as btp. Then the time base of the entire hydrograph is 

(1 + b)tp. Following the example above leads to the general expression: 

 qp = 1290.67AΔQ / [(1+b)tp] (6-01)

with b = 1.67.

Accounting for rounding error, equation (6-01) becomes the familiar qp = 484AΔQ/tp.

6.24 Results

The following results (table 6-01) are for the general case of tr = btp. The 

value “#Δt” is the number of calculations needed for the component hydrographs. Section 4 of the National Engineering Handbook (NEH4; U.S. SCS 1954) 

Table 6-01—Characteristics of variable triangular hydrographsa for the general case 

of tr. 

b = tr/tp	 (1+b)	 HF = 1290.67 / (1 + b)	 #Δt = 5(1 + b)

1 2 = 2.00 645.33 10

3/2 5/2 = 2.5 516.27 12.5→13

5/3 = 1.67 8/3 = 2.67 484 13.3→14 = the traditional case

2 3 430.22 15

7/3 = 2.33 10/3 = 3.33 387.20 16.7→17

8/3 = 2.67 11/3 = 3.67 352 18.4→19

3 4 322.67 20

4 5 258.13 25

5 6 215.11 30

a tp is time to peak; HF is hydrograph factor.
56 USDA Forest Service RMRS-GTR-334. 2016.

recommends Δt ≤ tp/5. The value must be a whole number, so decimal answers are 

rounded up to the next integer.

6.241 Equations for Rising and Falling Limbs

For calculating the q at intervals of t (called τ here) for the component 

hydrographs:

1. For the rising “limb” when τ ≤ tp:

q(τ ) = (τ/tp)qp calculated every Δt from 0 to tp (5 of them)

2. For the declining limb, following tp, or tp ≤ τ ≤ [(1+b)tp = tb] works out to:

q(τ) = (qp/b)[1 + b - τ/tp] calculated every Δt from tp to tb (5b of them) 

6.242 Direct Solutions for HF and “b”

Recall from equation (6-01) that qp = 1290.667A ΔQ / [(1 + b)tp], which is:

qp = 484A × ΔQ/tp when b = 1.6667 or

qp = HF × A × ΔQ/tp

where HF is the hydrograph factor, customarily 484. Thus 

HF = 1290.667 / (1+ b)

 b = 1290.667 / HF – 1

6.25 Discussion

A persistent claim in applied hydrology—much-repeated but from an unknown 

source—is that lag = 0.6tc. Note the dimension of 3/3 for tp and 5/3 for the falling limb 

for the traditional triangular hydrograph. If tp is taken to be the lag, and the falling 

limb time is taken to be the time of concentration (the time it takes to drain from the 

farthest point) and for the instantaneous case Δt→0, then the time to peak is the traditional “lag time.” Then lag = tp= 0.6tc.

6.3 Broken Triangular Unit Hydrograph

6.31 Concepts

The broken triangular unit hydrograph is conceptually similar to the simple 

triangle with two exceptions: it has a change of slope in the recession limb, and the 

time dimensions are fixed. It gives a longer hydrograph (tb = 5tp), a break in slope in 

the recession limb, and a more depressed peak. The breakpoints and dimensions of the 

broken triangular unit hydrograph are as follows (table 6-02):

Table 6-02—Properties of the broken triangular unit 

hydrographa.

 Unit hydrograph Cumulative area

t/tp q/qp ordinate fraction

0 0 0

1 1 5/9 = 0.5555 0.2778 = 5/18

2 0.4 2/9 = 0.2222 0.6667 = 2/3

5 0 0 1.0000

a tp is time to peak; qp is peak flow.
USDA Forest Service RMRS-GTR-334. 2016. 57

The area under this broken triangle is 1.8qpt

b, or 0.36tpqp. Thus the UH peak factor is 

1/1.8 = 5/9 = 0.555. The UH equation for this comes from 5tpqp (1.8/5) = ΔQ, or qp = 

0.555ΔQ/tp.

6.32 Technical Details

The general equation is qp = 358.52AΔQ / tp, where qp is in ft3/sec (cfs), A is 

in mi2, ΔQ is in inches, and tp is in hr. Thus the “Hydrograph Factor” for this case is 

fixed at 358.52. The following equations describing the three straight line portions in 

figure 6-01 apply for UH at time τ:

0 ≤ τ ≤ tp q = (τ/tp)qp rising limb

t

p ≤ τ ≤ 2tp q = (1.6 – 0.6τ/tp)qp first falling limb

2tp ≤ τ ≤ 5tp q = (0.6667 – 0.1333τ/tp)qp second falling limb

If Δt ≤ tp/5, then this configuration requires at least 25 (= 5 × 5) Δt units for 

modeling. The centroid (lag time) comes out to be 47/27tp = 1.7407tp. Thus tp = 

27/47 tL = 0.5744tL.

 

Figure 6-01—Plot of the broken triangular unit hydrograph (UH).

t/tp

q/qp

6.4 Curvilinear (SCS–NRCS) Unit Hydrograph 

6.41 General

The curvilinear unit hydrograph is the hydrograph from which the triangular 

approximations are derived. It is represented in Wildcat5 as a table of 33 points, with 

coordinates found in NEH4 (U.S. SCS 1954) and NEH630, chapter 16 (U.S. Natural 

Resources Conservation Service [NCRS] 2003). Important features of the curvilinear 

(SCS-NRCS) unit hydrograph are given in table 6-03.
58 USDA Forest Service RMRS-GTR-334. 2016.

6.42 Use

Use of the curvilinear hydrograph in lieu of the triangular option may give 

higher flood peaks because the component hydrographs have broader peaks than with 

the pointed triangular shape. It is also a more realistic representation of natural hydrographs. It has an HF of 484.

6.43 Discussion

Inspection of table 6-03 shows that most of the flow contributions—about 98 

percent—have occurred by about t/tp = 3. This may have been the rationale for the 

shortcut time base of 8/3 tp in the triangular unit hydrograph option. Alternative UH 

choices (not given here) ending at t/tp = 3 and suitably adjusted for the small changes 

in volume, should give very similar results in hydrograph models.

A close approximation to the above hydrograph can be taken from the function 

q(t) = (t/tp)

mem(1– t/t

p

)

with m = 3.697. This shape is based on the gamma distribution. However, calculation 

must be limited to t/tp = 5. At that point the function should be truncated and forced to 

q(t) = 0, and the area adjusted to a unit value by proportioning the ordinates. It also has 

an HF of about 484.

6.5 Chapter References 

Triangular Unit Hydrographs

U.S. Natural Resources Conservation Service [NRCS]. 2003. National engineering 

handbook. Part 630, Hydrology. Chapter 16, Hydrographs. Chapter 16, 

Hydrographs. www.wcc.nrcs.usda.gov/hydro/hydro-techref-neh-630.html.

(November 2007). Also see http://directives.sc.egov.usda.gov/viewDirective.

aspx?hid=21422. (March 17, 2015).

Table 6-03—Properties of the curvilinear unit 

hydrograph (UH)a.

t/tp q(t)/qp Q(t)/Qtotal Comments

0 0 0 UH begins

~0.5 47.0 6.5 inflection point

1 100.0 37.5 peak (mode)

~1.17 50.0 median

~1.3 86.0 58.9 centroid (mean)

~1.5 68.0 70.5 inflection point

2 28.0 87.1

3 5.5 97.7

4 1.1 99.7

5 0 100.0 UH ends

a tp is the time to peak and q is the runoff rate (in units of 

length/time); qp is the peak flow and Q is the runoff depth, 

both as a percentage of the maximum value.
USDA Forest Service RMRS-GTR-334. 2016. 59

U.S. Soil Conservation Service [SCS]. 1954 [and following]. National engineering 

handbook. Section 4, Hydrology. Chapter 6. Washington, DC: U.S. Department 

of Agriculture. http://directives.sc.egov.usda.gov/OpenNonWebContent.

aspx?content=18393.wba. (March 19, 2015).

U.S. Soil Conservation Service [SCS]. 1986. Urban hydrology for small watersheds.

Technical Release 55. Washington, DC: U.S. Department of Agriculture. 164 p. 

www.nrcs.usda.gov/Internet/FSE_DOCUMENTS/stelprdb1044171.pdf. (March 18, 

2015).

Curvilinear Unit Hydrographs

U.S. Natural Resources Conservation Service [NCRS]. 1997 [and following]. National 

engineering handbook. Part 630, Hydrology. Chapter 16, Hydrographs. www.

wcc.nrcs.usda.gov/hydro/hydro-techref-neh-630.html. (November 2007). Also 

see http://directives.sc.egov.usda.gov/viewDirective.aspx?hid=21422. (March 17, 

2015).
60 USDA Forest Service RMRS-GTR-334. 2016.

Chapter 7: Output Information

7.1 Effective Loss Rate

A rainstorm is characterized by storm duration, storm depth (P), and runoff (Q), 

as well as by “losses,” and thus a time-based loss rate. This rate is calculated simply 

as:

Loss rate = (P – Q) / Storm Duration (7-01)

Some use this equation as an estimate of the φ loss rate. 

An alternative is to substitute the effective loss rate, using the duration of rainfall 

excess, which is given elsewhere on the output page, in place of the total storm duration in equation (7-01). 

7.2 Effective Curve Number

7.21 Concepts

For any rainfall event with any runoff mechanisms, the de facto Curve Number 

(CN) can be calculated. It does not require adherence to the CN method generating the 

runoff. Wildcat5 performs this calculation.

7.22 Method

The calculation is the solution of the runoff equation for S from observed 

P and Q. For the case of Ia/S = 0.20 it is: 

S = 5[P + 2Q – √(4Q2 + 5PQ)] (7-02)

and CN = 1000 / (10+S) (4-02)

where S is storage, a measure of the maximum possible difference between P (the 

rainfall) and Q (the rainfall excess).

Equations (7-02) and (4-02) are in English units for Ia/S = 0.20, where Ia is 

the initial abstraction, or the rainfall depth required before runoff begins. Wildcat5 

converts to metric units and the 0.05 base internally. This value is then used in a subsequent step to calculate the CN after the storm.

7.3 Initial Abstraction

With the CN method, the Ia is usually taken as 0.2S. Wildcat5 also offers the 

alternative of 0.05S. As shown on the output screen (fig. 2-03), it back-calculated the 

Ia from the effective Curve Number. The actual beginning of runoff in the model—defining the modeled Ia —is when the hydrograph begins to rise from contributions from 

the source areas most prone to runoff (having the highest CN).
USDA Forest Service RMRS-GTR-334. 2016. 61

7.4 Post-event Curve Number 

7.41 General

This output gives the calculated CN after the rainfall event, and exploits the notion that the maximum possible storage (S, the difference between P and Q) at the start 

of the storm is Ia + S. It is approached as a limit as P grows larger, and is 1.2S (or V) 

in current practice. The storm “losses”—the calculated P – Q—remain on the site, and 

reduce the original maximum storage potential (1.2S) by that amount. Several versions 

of this approach are found in the literature and other models. See Hawkins (1958) and 

Williams and others (2012).

If you do not designate the CN method in Wildcat5 to generate rainfall excess, 

the calculation is made by back-calculating the effective start-of-storm CN from the 

storm P and the generated Q. Background and technical development for this approach are given in sections 7.43 and 7.44.

7.42 Calculation

The calculations in Wildcat5 are made as follows (table 7-01). 

Table 7-01—Equations for post-event Curve Numbers (CNs)a.

Case CN2← CN1 Equation 

P = 0 CN2 = CN1

0 < P < 0.2S CN2 = 1200CN1 / (1200 – PCN1)

P = 0.2S CN2 = 600CN1 / (CN1 + 500) 

0 < 0.2S < P CN2 = 100(3P + 24J) / (3P + 24J + 25J2) 

 where J = (100/CN1) –1 = S1/10

CN2 = 1000/[(10 + 25S1

2) / (30P + 24S1)]

a The subscripts 1 and 2 refer to “before” and “after” the event, and not to 

classes of Antecedent Moisture Conditions (AMC) or Antecedent Runoff 

Conditions (ARC). S is in inches.

7.43 Background

The CN runoff equation defines S = lim(P – Ia – Q) as P→∞. Thus the CN value 

hangs on the storage capacity (including Ia) of a site, 1.2S, or V. Though S is often 

cast in the role of a fitting parameter, it has also been interpreted as the sum of the 

available soil retention storage, or (FC – WC) plus the available aboveground interception and depression storage, less Ia, where FC is the profile field capacity and WC 

is current water content.

Note that this site capacity approach differs from the notion of rainfall excess 

generation from infiltration rate processes. However, it is widely used in daily time 

step models. With this approach 1.2S becomes a transient variable subject to rainfall 

recharge and site losses (evapotranspiration + drainage). In terms of conservation of 

mass, this means that from time 1 to time 2

1.2S2 = 1.2S1 – Δ(Storage Capacity)

 = 1.2S1 – (P – Q)
62 USDA Forest Service RMRS-GTR-334. 2016.

Reductions (–) to available storage space V can result from rainfall with or without runoff. Increases (+) to storage capacity V are evapotranspiration and drainage, 

and are ignored in this case.

Note again that this is for onsite storage space capacity, or potential losses, not 

water content. Smaller values of S and V mean more soil water. This approach treats 

the watershed/site as a tank, with the unfilled capacity as V. Recognize also that this 

scheme says nothing about the total site storage, or soil depth, but only the unfilled 

capacity status at the time being considered.

7.44 Development

Four different conditions occur. For generality, between-storm losses via evapotranspiration and drainage are called simply ET. For most event applications ET = 0.

1. No rainfall: Site ET occurs, and ΔStorage is positive. 

1.2S2 = 1.2S1 + ET P = 0, 0 < ET (7-03)

2. Rainfall < Ia: Here all rain remains onsite, reducing V and S, but increasing CN.

1.2S2 = 1.2S1 – P 0 < P < 0.2S (7-04)

3. Rainfall = Ia (= 0.2S): Here P = Ia, and achieves the threshold of runoff. 

1.2S2 = 1.2S1 –Ia = 1.2S1 – 0.2S1 = S1 0 < P = 0.2S (7-05)

4. Rainfall > Ia: Here both runoff and soil/site storage reduction occur; ΔStorage is 

the negative (P – Q). Storage diminishes according to equation (7-06): 

1.2S2 = 1.2S1 – (P – Q) 0 < 0.2S < P (7-06)

These four conditions can be expressed via manipulation of the CN rainfallrunoff equation. Given S2, then CN2 can be determined. Representations of equations 

(7-03) through (7-06) are given in figures 7-01 through 7-03. Algebraic transformation 

of equations (7-03) through (7-06) leads to the post-CN equations given in table 7-01.

7.45 Graphical Representations

Between-storm conditions are considered in the general cases depicted in figures 7-01 through 7-03. Site water losses (gains in V) via ET are considered only in 

figure 7-01. 

As an example of how to use figure 7-03, assume CN = 75 and a storm P of 2 

in. Begin at (0, 75) and follow the dashed line for CN = 75 to P = 2 in. Read across 

to the y-axis to find CN = 83 for x = 0 as the end-of-storm CN value. Then begin at 

(0, 83). For the case of ET = 4 in, follow down the line for CN = 83 to ET = -4 in and 

read across to the y-axis at x = 0 to find CN = 64. Follow this example to begin at (0, 

64) with the next rainfall P, and so forth. The scaled values given in this example are 

approximate.
USDA Forest Service RMRS-GTR-334. 2016. 63

Figure 7-03—Continuous Curve Number 

dynamics on the basis of event rainfall (P, 

inches) and between-storm losses (ET). 

The dashed line sloping to the right and 

starting at (0, 100) represents P = Ia = 

0.2S. ET is shown as a negative value.

Figure 7-01—Dimensionless relationships of post-storm storage (S2) 

to rainfall (P) or losses (ET), scaled 

on S1 (pre-storm storage). P* = P/S1, 

ET* = ET/S1. The ratio S2/S1 might 

be alternatively called S2*.

Figure 7-02— Basic relationships between post-storm (CN2) and pre-storm 

(CN1) Curve Numbers for families of 

event rainfall (P, inches) and betweenstorm losses (ET, inches). 
64 USDA Forest Service RMRS-GTR-334. 2016.

7.5 Maximum Contributing Area

7.51 Concepts and Application

It is widely observed that rainstorm runoff is not uniform across the drainage 

area, especially with smaller storms. Wildcat5 models this feature in several ways, 

simulating different fractions of the watershed generating rainfall excess under different levels of rainfall.

With the DISTRIBUTED Curve Number options, Wildcat5 asks for an array 

of sub-areas with different CNs. The model accounts for the areas with P > Ia for both 

the 0.20 and the 0.05 cases. The total of these areas leads to the calculation of the 

fraction. 

With the Constant Loss Rate (f-index) option there is no direct partial area 

runoff interpretation. It is shown as 100 percent as a default value. 

With the Exponentially distributed loss rate option, the area fraction of i > f is 

calculated by the characteristics of the runoff function and the exponential distribution.

The slope of the intensity‒rainfall excess rate function is the contributing area fraction.

With the Constant fraction (Runoff ratio) option the assumed contributing area 

fraction is taken to be the coefficient C.

With the Distributed F option, Wildcat5 asks for an array of sub-areas with 

different F values. The model accounts for the total area with P > F, thus defining the 

contributing fraction.

7.52 Discussion

In the Distributed CN, Distributed F, Constant fraction, and Complacent–

Violent cases the maximum contributing area is active at the end of the storm. With 

the distributed loss rate, the maximum contributions occur when rainfall is at maximum intensity.

It can also be claimed, and in some instances demonstrated, that the slope of the 

rainfall-runoff function at the defined rainfall status can be interpreted as the fractional 

contributing area. The rainfall-runoff slope will always be greater than 0 and less than 

or equal to 1.

7.6 Transient Storage

7.61 Concepts

Transient storage is the difference between the cumulative rainfall excess and 

the cumulative runoff at any time in a storm. It is an expression of, and exploits, the 

distinction between the two. Though cumulative rainfall excess and cumulative runoff 

are equal in volume for the entire storm, they are time-dependent and not necessarily 

equal while the storm is in progress. Rainfall excess will always exceed or equal the 

runoff.

Rainfall excess is best envisioned as the sum of the processes at all points in the 

watershed that apportion rainfall into losses and (eventual) runoff. Rainfall excess has 

not yet been routed to the outlet.

Runoff is what eventually flows out the bottom of the watershed. There is a 

time lag between the two while the excess is flowing over the land—in transient 
USDA Forest Service RMRS-GTR-334. 2016. 65

storage—and down the channels before reaching the outlet. This lag is expressed by 

the shape and dimensions of the unit hydrographs and the timing of the rainfall excess 

pulses. 

In Wildcat5 the transient storage is graphically represented by the vertical difference between the rainfall excess curve and the runoff curves on the cumulative P:Q 

plots with time. On the Runoff Hydrograph Table it is activated by a drop-down 

menu at the top of the screen, and is a simple subtraction of the column of cumulative 

runoff depth from the column of cumulative rainfall excess.

7.62 Discussion

A large transient storage leads to the question of the distribution of this moving 

water on the watershed at any time on its way to the outlet: Is it flowing overland 

or in the channels? Determination is beyond the scope of this model, but seemingly 

outrageous values may suggest real depths of water in routing processes, incorrect 

routing parameters, or damaging events. For example, a transient storage of 1 inch is a 

spatial average for that moment, but it suggests at least a few much deeper extremes at 

selected points in the watershed.

7.7 Chapter References

Hawkins, R.H. 1978. Runoff Curve Number relationships under varying site moisture 

levels. Proceedings of the American Society of Civil Engineers. 104(IR4): 

389–398.

Williams, J.R.; Kannan, N.; Wang, X.; Santhi, C.; Arnold, J.G. 2012. Evolution of 

the SCS runoff curve number method and its application to continuous runoff 

simulation. Journal of Hydrologic Engineering (American Society of Civil 

Engineers). 17(11): 1221–1229.
66 USDA Forest Service RMRS-GTR-334. 2016.

Chapter 8: Reservoir Routing

8.1 General

As a hydrograph progresses downstream, it encounters storage features such 

as lakes and swamps. The inflow spreads out on the surface: it “piles up,” and is 

temporarily stored while water is simultaneously being released at the outlet. The 

latter occurs in accordance with the outlet controls and water depth. Thus, lakes and 

swamps, flood plains, reservoirs, and even channel stretches reduce the inflow peaks 

and prolong the outflow duration. Even a full reservoir reduces flood peaks through 

this surcharge storage. 

8.2 Concepts and Process

The general logic is conservation of mass over time steps of ∆t: 

Inflow volume – outflow volume = ∆Storage

If the inflow hydrograph is known, as for example from running Wildcat5, it is 

expressed as a series of equally time-spaced flows q (L3/T, where L is a unit of length 

and T is time). Outflows from the reservoir—here taken to be a broad-crested spillway—

are in the same units of L3/T. Storage, which varies with time as outflows occur, is 

that on the reservoir surface (L2) above the spillway lip elevation. Storage volume is 

expressed in L3. The reservoir is assumed to be full of water at the spillway crest when 

the inflow flood occurs.

This is a fairly simple idea, but it does require some algebra to explain; the 

storage-indicator method is described here. Though instantaneous in nature, the process is modeled with finite differences. Over intervals of ∆t from time (1) to time (2), 

and using i and o as subscripts to indicate inflow and outflow, the change in storage 

volume is: 

(Average inflow rate × ∆t) – (average outflow rate × ∆t) = change of storage (S)

or 

(8-01)

Solving equation 8-01 for qo(2) is shown in the following paragraphs.

8.3 Application in Wildcat5 

For reservoir routing, Wildcat5 asks for the spillway length (L, ft or m), the 

reservoir surface areas (ac or ha), and the spillway coefficient (C). This assumes that 

the spillway condition is that of a broad-crested weir of the form q = CLh3/2 (where h 

is water depth above the spillway lip) and that the reservoir surface does not change 

appreciably with depth above the spillway elevation. 

The weir coefficient C (ft1⁄2/ sec) is given in many handbook sources, but reference is made here to the U.S. Department of the Interior, Bureau of Reclamation 

(1987). Coefficient values hover around 3.1 in English units, and the true value 

depends on the exact nature of the spillway section. Consistent use of 3.1 will not 
USDA Forest Service RMRS-GTR-334. 2016. 67

appreciably alter the routing outcomes. No metric options are offered, but use of the 

English values with otherwise metric data here is not harmful. Wildcat5 makes the 

calculations in English units and converts the results to metric units. 

Simply fill in the screen/cell with the choices, and press the Execute Routing

button. The routing will be performed on the current hydrograph, and the results will 

appear in graphical form. If you need details, the Calculations Table button will lead 

to the line-by-line results. Maximum spillway outflow and depth are given in the 

Routing Preview. 

8.4 Technical Solution

The routing product is the end-of-interval outflow rate qo(2) at time t. As you 

can see in equation (8-01), outflow rate depends on the known inflow rates, the storage 

volumes at the start and end of the interval, and the interval start outflow rate qo(1).

Outflow rate and storage volume depend on the depth of water (h) above the spillway 

lip. The storage is expressed simply as:

S = hA (8-02)

where A is the reservoir area (L2), and h is water depth above the spillway lip. The 

outflow rate qo is also a function of h, and is usually taken as a broad-crested weir 

such that:

 qo = CLh3/2 (8-03) 

where L is the spillway length, and C is the weir coefficient (~3.1 in the English 

system).

When equations (8-02) and (8-03) are substituted into (8-01), and simplifications 

are made, the following results:

(8-04)

Note that in equation (8-04) everything on the left-hand side is known at the 

start of an interval, and the desired qo(2) is isolated on the right-hand side. There is 

no direct solution of qo(2). However, a solution can be found sequentially by steps of 

∆t, made possible by assuming qi = qo = 0 at t = 0, and S = 0 and h = 0 at t = 0. As 

the time steps are effected, the just-calculated qo(2), h(2), and S(2) become the known 

values for the next step, the starting conditions “(1),” in equation (8-04). 

An array of possible values of the right-hand side as a function of qo(2) is made 

at the outset, and then qo(2) values are simply interpolated based on the calculated values from the known start-of-interval values on the left-hand side. The storage-indicator 

method has a long history of successful use. Note that the peak of the reservoir 

outflow will always fall on the descending limb of the inflow hydrograph. Several 

channel routing procedures share much of the same logic.

An interesting side note is that equation (8-04) is really a truncated cubic equation in the form [qo(2)]1/3. Rather than solving it as a cubic equation, you may find the 

pre-solution array and interpolation to be more practical.
68 USDA Forest Service RMRS-GTR-334. 2016.

8.5 Chapter References

Reservoir routing is a popular and widely known procedure, and is included in many 

standard texts. It has been programmed for hand-held devices. 

Golding, B.L. 1981. Flood routing program. Civil Engineering. (June): 74–75.

Heggen, R.J. 1983. Flood routing on a small computer. Civil Engineering. (March): 

63–65.

U.S. Department of the Interior, Bureau of Reclamation. 1987. Design of small dams 

3rd ed. Washington, DC. 904 p. http://www.usbr.gov/pmts/hydraulics_lab/pubs/

manuals/SmallDams.pdf. (March 18, 2015). 

Acknowledgments

Support for software development of Wildcat5 and preparation of the user manual was provided through a cost-share agreement between the University of Arizona and 

the U.S. Forest Service, Stream Systems Technology Center (now the National Stream 

and Aquatic Ecology Center). The authors are indebted to John Potyondy, Stream 

Systems Technology Center, Fort Collins, CO; Greg Kuyumjian, U.S. Forest Service, 

Okanogan–Wenatchee National Forest, Wenatchee, WA; and Jeffry J. Stone, USDA 

Agricultural Research Service, Southwest Watershed Research Center, Tucson, AZ, 

for contributions on the distributed infiltration option; and Pablo Garcia, University 

of Arizona, Department of Agricultural and Biosystems Engineering, for graphics and 

early reviews of the software and manual.

To learn more about RMRS publications or search our online titles:

www.fs.fed.us/rm/publications

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