A Rainfall-Runoff Simulation 

Model for Estimation of Flood 

Peaks for Small Drainage Basins 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 506-B 

A Rainfall-Runoff Simulation 

Model for Estimation of Flood 

Peaks for Small Drainage Basins 

By DAVID R. DAWDY, ROBERT W. LICHTY, and JAMES M. BERGMANN 

SYNTHESIS IN HYDROLOGY 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 506-B 

A parametric rainfall-runoff simulation model 

is used with rainfall data and daily potential 

evaporation data to predict flood volume and 

peak rates of runoff for sma!l drainage areas 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1972 
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary 

GEOLOGICAL SURVEY 

V. E. McKelvey, Director 

Library of Congress catalog-r. rd No. 78-185756 

First printing 1972 

Second printing 1972 (with minor revisions') 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

W-ashington, D.C. 20402 - Price 40 cents (paper cover) 

Stock Number 2401-1226 
CONTENTS 

ALbstract -----------------------------------------

Introduction ------------------------------------- Historical development of parametric rainfallrunoff models ------------------------------ Transferability of results of modeling _________ _ 

Aldvantages and disadvantages of parametric 

simulation ---------------------------------

ALcknowledgments ----------------------------

Structure of the model ----------------------------

General structure ----------------------------

Infiltration component ------------------------ Soil-moisture-accounting component ___________ _ 

Surface routing component --------------------

System identification ------------------------------

Response of the model ------------------------

Simulation model studies -----------------------

Santa ALnita basin --------------------------------

General physiography -------------------------

Precipitation ---------------------------------

Streamflow ----------------------------------

Evaporation ---------------------------------

Data screening -------------------------------

Parameter definition --------------------------

Page 

B1 

1 

2 

3 

3 

3 

3 

3 

5 

7 

8 

8 

10 

12 

12 

12 

12 

13 

14 

14 

14 

Santa Anita basin-Continued 

Parameter sensitivity -------------------------

ALnalysis of results ---------------------------

Parameter values ----------------------------

Fitting errors --------------------------------

Effect of screened data ------------------------

Accuracy of simulation for Santa ALnita basin __ 

Beetree basin ------------------------------------

General physiography ------------------------

Precipitation ---------------------------------

Streamflow ----------------------------------

Evaporation ---------------------------------

Parameter definition --------------------------

Results and conclusion ------------------------

Little Beaver basin -------------------------------

General physiography -------------------------

Precipitation ---------------------------------

Streamflow ----------------------------------

Evaporation ---------------------------------

Parameter definition --------------------------

Comparison of derived parameter values ------------

Sources of error and their impact ------------------

Conclusions --------------------------------------

References cited ----------------------------------

ILLUSTRATIONS 

Page 

B14 

15 

15 

16 

18 

18 

18 

18 

19 

19 

19 

19 

21 

24 

24 

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26 

27 

28 

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Page 

FIGURE 1. Schematic outline of the model, showing components, parameters, and variables -------------------- B4 

2. Schematic diagram of the two-layered soil-moisture profile used with the infiltration equation in the 

model ------------------------------------------------------------------------------------- 6 

3. Graph showing the relation which determines the effective value of soil-moisture potential for use in 

the infiltration equation -------------------------------------------------------------------- 7 

4. Graph showing the relation which determines rainfall excess as a function of maximum-infiltration 

capacity and supply rate of rainfall -------------------------------------------------------- 7 

5. Schematic flow chart of the flood-hydrograph simulation program ---------------------------------- 7 

6. Schematic drawing of- the Clark unit hydrograph used in the surface-runoff routing component ------ 9 

7. Map of the Santa Anita Creek basin, above the stream-gaging station near Pasadena, Calif__________ 12 

8. Graph showing the relation of mean annual rainfall and of area in the basin to elevation for the 

Santa Anita Creek basin above the gaging station near Pasadena, Calif________________________ 13 

9. Typical scatter diagrams for simulation results in the Santa ALnita Creek basin ----------------------- 16 

10. Typical response curves showing sensitivity of fitting criterion to percentage changes in parameter 

values ------------------------------------------------------------------------------------- 17 

11. Map of the Beetree Creek basin, above the stream-gaging station near Swannanoa, N.C.____________ 20 

12. Graphs showing response of objective function during optimization with the split sample for Beetree 

Creek basin ------r------------------------------------------------------------------------ 21 

13. Scatter diagrams for simulation results in the Beetree Creek basin --------------------------------- 22 

14. Graph showing response of the objective function to change from the optimum value of parameter RR_ 23 

15. Map of the Little Beaver Creek basin above the stream-gaging station near Rolla, Mo________________ 24 

III 
IV CONTENTS 

TABLES 

Page 

TABLE 1. The eight model parameters and their application in the modeling process ---------------------------- B5 

2. Qualitative comparison of errors involved in hydrologic modeling with analogous errors resulting from 

standard linear statistical analysis ---------------------------------·-------------------------- 10 

3. Results of an empirical study of the response of the model to input and output errors__________________ 11 

4. Mean annual rainfall, Santa Anita Creek basin, 1949-62·---------------------------------------------- 12 

5. Simulated peak discharges, using fitted parameters ------------------------------------------------ 13 

6. Fitted-parameter values -------------------------------------------------------------------------- 15 

7. Storm-period data ------------------------------------------------------------------------------- 19 

8. Results of fitting of model parameters to data and of split-sample testing for Beetree Creek near 

Swannanoa, N.C------------------------------------------------------------------------------ 21 

9. Results of fitting of model parameters to data for Little Beaver Creek near Rolla, Mo., using the Rolla 

3-W rain gage ___________________________ _:_________________________________________________ 25 

10. Results of fitting of model parameters to data for Little Beaver Creek near Rolla, Mo., using the Sch'ool 

of Mines rain gage -------------------------------------------------------------------------- 25 

11. Comparison of estimates for flood volumes and peaks for Little Beaver Creek by the use of the two rain 

gages --------------------------------------------------------------------------------------- 25 

12. Summary of results of optimization for the three study basins --------------------------------------- 26 
SYNTHESIS IN HYDROLOGY 

A RAINFALL-RUNOFF SIMULATION MODEL FOR ESTIMATION OF 

FLOOD PEAKS FOR SMALL DRAINAGE BASINS 

By DAVID R. DAWDY, ROBERT W. LICHTY, and JAMES M. BERGMANN 

ABSTRACT 

A parametric rainfall~runoff simulation model is used with 

dat~ from a point rainfall gage and data on daily potential 

evapotranspiration to predict flood volume and peak rates of 

runoff for small drainage areas. The model is based on bulkparameter approximations to the physical laws governing 

infiltration, soil-moisture accretion and depletion, and surface streamflow. Three case studies are presented in which 

an objective fitting method is used for determining optimal 

best-fit sets of parameter values for the data available for 

use in predicting flood peaks. Errors of prediction result both 

from errors in rainfall input and from lack of model equivalence to the physical prototype. These two sources of error 

seem to be of the same order of magnitude for a model of 

the level of simplicity of that present~d. Major gains in 

accuracy of simulation will require improvements in both 

data and model. The limit of accuracy of prediction of flood 

peaks by simulation with a bulk-parameter model using data 

obtained from a single rain gage seems to be on the order of 

25 percent. 

INTRODUCTION 

The development of the digital computer has 

added a new dimension to hydrology. Solutions to 

problems took hours with pen and pencil but now 

they take seconds with the computer. In addition, 

much more complex methods of analysis are feasible because of the speed of solution by the computer. The impact of the computer has been particularly great in the area of rainfall-runoff modeling. 

Historically, surface-water hydrology has been concerned with modeling, for flood routing and unithydrograph analysis are mathematical modeling. 

Complete rainfall-runoff simulation models date 

back at least to the 1920's. However, the present 

burst of activity in hydrologic simulation is a direct 

result of widespread availability of the computer. 

Computers have made rainfall-runoff simulation 

on a large scale economically feasible. Practicality, 

however, depends upon applicability and accuracy 

of the simulation results. Simulation may be practical if one of the following applications is realized. 

1. A rainfall record can be used to supplement a 

streamflow record having a shorter period of 

record than the rainfall record. 

2. Model parameters for ungaged sites can be estimated on the basis of the parameters derived 

for gaged sites, and information can be gained 

at the ungaged sites through the use of recorded or simulated rainfall data and the use 

of estimated parameters at the ungaged sites. 

3. The effect of man-made changes on a basin can 

be related to changes in model parameters, so 

that measured "before" conditions can be 

compared with simulated "after" conditions 

of sufficient accuracy for planning purposes. 

Predictions obtained from rainfall-runoff simulation models are successfully applied in any of the 

above determinations only where the level of accuracy of the predictions is known. Measures of accuracy must be presented to the user in understandable terms. Accuracy should be measured in terms 

of prediction, rather than in terms of fitting. Accu-

-racy of fitting indicates only how well the model 

can reproduce a set of data from adjusted model 

parameters. Accuracy of prediction indicates how 

well the model can reproduce a set of data that was 

not used to derive the parameter values. Therefore, 

prediction involves an independent test of accuracy 

of the model. 

The U.S. Geological Survey research program is 

developing rainfall-runoff simulation models. Research emphasis has been on the utility of the 

models for practical field application to current 

projects and has centered upon both development 

of models and testing of their accuracy of predicBl 
B2 SYNTHESIS IN HYDROLOGY 

tion. This report is a statement of progress to date 

(1971) on model development, including illustrative 

examples of the results of prediction for several 

basins in various hydrologic settings. 

Hydrologic models have been developed in response to hydrologic needs. The use of computers 

has led to the development of more sophisticated 

hydrologic models. The more sophisticated models 

should be more accurate to justify their existence, 

and their accuracy must be measured in terms of 

their ultimate use. 

The models discussed in this report are parametric models, or models that try to simulate physical conditions by a deterministic mathematical 

description, which includes, as much as possible, 

approximations to the physical laws governing 

surface-water hydrology. Wherever possible, a 

physical interpretation is placed upon the parameters used in the models. A separate field of modeling not included in this study is that of stochastic 

simulation. These models describe the hydrologic 

record in statistical terms and use that statistical 

description to generate synthetic "equally likely" 

records. Each type of model has its advantages and 

disadvantages for application to meet a particular 

·need. 

The derivation of a set of optimum parameters 

representing the hydrology of a basin must be based 

on data. A parametric model requires both streamflow and rainfall data and, perhaps, other hydrologic data. Certain data other than that on streamflow also contain streamflow information, and use 

of this additional information should reduce the 

time required to collect streamflow data necessary 

to achieve a given level of accuracy of prediction. 

Most studies involving rainfall-runoff models include the assumption of a stationary time series, 

at least during a period of calibration. Thus, the 

model parameter values can not change with time. 

Often, an assumption is made that, if parameters 

do change, any such changes can be related to 

physical changes on the drainage basin, particularly to man-made changes. 

HISTORICAL DEVELOPMENT OF PARAMETRIC 

RAINFALL-RUNOFF MODELS 

Parametric hydrology is that field of mathematical hydrology which attempts to synthesize a model 

of the land phase of the hydrologic cycle, by approximating the physical laws governing the various components of the rainfall-runoff system. Infiltration, soil-moisture storage, percolation to ground 

water, evapotranspiration, and surface- and subsurface-flow routing are modeled by sets of equations that, hopefully, give a response equivalent to 

the response of the component modeled. The components and all necessary interrelations among 

components are described by means of parameters, 

some of which are empirical, and some of which 

have a physical interpretation. 

One of the earliest overall models of the hydrologic cycle was developed by Folse (1929). Development of that model was begun in 1916 and 

continued through the 1920's. During the 1930's, 

advances were made in the description of all components of the hydrologic cycle. Sherman ( 1932) 

introduced the theory of the unit hydrograph, which 

led to a flurry of developments, culminating with 

Dooge's general linear theory of flood-flow routing 

( 1959). The Horton ( 1939) infiltration equation 

was an empirical attempt to describe unsaturated 

flow. Philip (1954) extended this by deriving an 

approximation based upon the Darcy equation for 

infiltration at a point. Theis (1935) showed the 

analogy of Darcy's equation for flow through saturated porous media to the heat-flow equation. Many 

simplifications for specified boundary conditions 

were subsequently developed and became the basis 

for routing of ground-water discharges, such as the 

equation of Kraijenhoff (1958) for instantaneous 

recharge in two-dimensional flow. 

The digital computer made it possible to combine 

these many approximations into one overall approximation describing the operation of the land phase 

of the hydrologic cycle. Linsley was the first to take 

advantage of this possibility, and his efforts led to 

the development of the Stanford Watershed Model 

(Crawford and Linsley, 1966). Similar models have 

been developed at many universities and in government agencies, both in the United States and abroad. 

The many models currently available or being 

developed must meet certain criteria to be useful 

in practical application. They must: 

1. Require only input data that are generally available. 

2. Be simple enough for the user to operate and to 

understand. 

3. Provide the output desired at an acceptable level 

of accuracy for the application for which it 

is used. 

The U.S. Geological Survey flood-hydrograph 

simulation model follows directly from the historical developments previously described, and is designed to meet the criteria outlined above. 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B3 

TRANSFERABILITY OF RESULTS OF MODELING 

For modeling results to be transferable, the 

parameters derived from simulation studies at 

measured sites must be constant or must possess 

invariant relations with physical variables which 

can be measured in other basins. Time invariance 

is required, or else any changes in time must be 

the result of measureable physical changes within 

the basin. Certain types of information may be 

transferable without the use of simulation. For 

instance, Benson (1962, 1964) showed that rawdata analysis can lead to regionalization of floodfrequency characteristics for a region. Simulation 

might aid such a study by extending the data base 

available for analysis. In addition, simulation is 

necessary if the time sequence of flows, rather than 

just their frequency of occurrence, is needed. 

Parametric simulation is so structured as to include those parameters related to physical measures 

of the basin. Therefore, transferability is implicit 

in parametric simulation. To date (1970), however, 

no studies have presented results leading to regionalization of the parameters of either stochastic or 

parametric simulation, although some thought has 

been given to the problem (Benson and Matalas, 

1967; Matalas and Gilroy, 1968). Feasibility of any 

such regionalization of parameters can be tested by 

comparing the sensitivity of results of simulation 

with the accuracy of the parameter estimates. That 

a statistical or physical parameter can be related to 

some characteristic of a basin is of no value if thestandard error of estimate of the resulting relation 

is such that the simulation would be grossly in 

error. Therefore, .the transferability of parameters 

is limited by the sensitivity of the modeling results 

to the magnitude of the errors in parameter values. 

ADVANTAGES AND DISADVANTAGES OF 

PARAMETRIC SIMULATION 

Rainfall-runoff models, in general, are lumpedparameter models, although often the surfacestreamflow routing is accomplished by the use of a 

finite-difference approximation to the drainage system. A lumped-parameter model attempts to use a 

single parameter value to represent a physical 

measure that also has spatial variability. The 

models are, therefore, at least one step removed 

from simulating the actual flow mechanics at each 

point in the watershed. Derived parameter values 

are, at best, average values for the basin, and are 

an index to, rather than a measure of, the underlying physical system. This approximation introduces a major source of error in a lumped-parameter model and limits the accuracy of prediction 

obtained by the use of the model. 

The parameters in parametric-simulation models 

should require a shorter period of record in order 

to be as well defined as those for either deterministic or stochastic black-box models. This has advantages when data must be collected and the analysis 

postponed until sufficient data are available. Transferability should be easier for parametric models, 

although this is yet to be demonstrated. Parametric 

models require more types of data for each event 

modeled, both for system identification (fitting of 

parameters) and for simulation of synthetic records. 

The emphasis of the models presented in this 

study is on flood-hydrograph simulation for small 

drainage areas. Generally, there is little or no data 

on small drainage areas ; therefore, results must be 

obtained on the basis of short records. In addition, 

only a small percentage of smaller basins can be 

gaged. Consequently, results must have transfera hili ty if the smaller, ungaged basins are to be 

simulated. Concentration on the development of a 

parametric model, thus, seemed to be warranted. 

ACKNOWLEDGMENTS 

The present report represents several years' research by the authors. The project was begun as 

the result of the efforts of R. W. Carter, U.S. Geological Survey, and his helpful encouragement 

throughout was important. Terence O'Donnell, Imperial College, London, joined the Survey's research 

group during his year's sabbatical leave from 

Imperial College, and his continued interest has 

contributed to the development of ideas, particularly on the use of objective fitting methods. Jaime 

Amorocho, University of California at Davis, 

Chester Kisiel, University of Arizona, and Jacob 

Rubin, U.S. Geological Survey, Menlo Park, Calif., 

all helped encourage the authors to "say what they 

mean, and mean what they say." 

STRUCTURE OF THE MODEL 

GENERAL STRUCTURE 

The rainfall-runoff model described in this report 

deals with three components of the hydrologic cycle 

-antecedent moisture, infiltration, and surface runoff. Structure of the model is shown in figure 1. 

Particular effort was made to design a model with 

a degree of equivalence to the physical system. 

Therefore, this model should be very similar in 

structure to any other bulk-parameter model for 

rainfall-runoff simulation. The antecedent-moisture-
B4 SYNTHESIS IN HYDROLOGY 

ANTECEDENT-MOISTURE 

ACCOUNTING COMPONENT 

INFILTRATION 

COMPONENT 

ROUTING 

COMPONENT 

Saturated-unsaturated soil 

moisture regimes 

Philip infiltration 

equation 

Clark instantaneous 

unit hydrograph 

Parameter Variable 

EVC BMS 

RR SMS 

BMSM 

DRN 

Daily rainfall 

Daily pan evaporation 

Initial condition 

Parameter 

SWF 

KSAT 

RGF 

Variable 

BMS 

SMS 

INPUT DATA 

Unit rainfall 

"BMS" 

"SMS" 

OUTPUT DATA 

1---_.:._ ~ D Ll-A 

Parameter Variable 

KSW SW 

Time-area curve 

Rainfall excess 

BMS 

SMS Rainfall excess Discharge 

FIGURE !.-Schematic outline of the model, showing components, parameters, and variables. 

accounting component is a more sophisticated version of the antecedent-precipitation index (API), 

which is designed to determine the initial infiltration rate for a storm. The infiltration component 

uses the Philip equation, which is believed to be a 

somewhat better approximation to the differential 

equation for unsaturated flow than the classical 

Horton exponential-decay-infiltration equation. Surface routing is based on a linear approximation 

developed more than 20 years ago (Clark, 1945). 

The operation of the antecedent moisture accounting component is designed to simulate moisture redistribution in the soil column and evapotranspiration from the soil. It contains four 

parameters: EVC, a pan coefficient converting 

measured pan evaporation to potential evapotranspiration; RR, a coefficient that determines the 

relative amounts of infiltration and surface runoff 

for periods with daily rainfall input; BMSM, a 

maximum effective amount of base moisture storage, 

and DRN, a coefficient controlling the rate of drainage of the infiltrated soil __ :rnoisture. The input to 

this latter component is daily rainfall and daily 

pan evaporation .. The output is the amount of base-, 

moisture storage (BMS) and of infiltrated surfacemoisture storage (SMS). BMS represents a uniform 

antecedent moisture content of the active soil column, and its range of values should simulate the 

moisture range from wilting-point conditions to field 

capacity. SMS represents the moisture content of 

the surface layer that forms during infiltration. 

The infiltration component is based on an approximation to the differential equation for unsaturated flow (Philip, 1954). The equation is based 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B5 

on a two-part accounting of the soil moisture, with 

a wetting layer overlying a layer of uniform moisture content (determined by antecedent events). 

The infiltration component contains three parameters: 

SWF, which represents the combined effects of moisture content and capillary potential at the wetting 

front for field-capacity conditions; RGF, a parameter 

that varies the effective value of SW F as a function of 

BMS; and KSAT, the hydraulic conductivity of the 

transmission zone. Inputs to this component are unit 

rainfall data and the values of BMS and SMS derived 

from previous times. The output is rainfall excessthat is, the remaining rainfall after abstractions by 

infiltration. 

Surface-runoff routing is based on the Clark form 

of the instantaneous unit hydrograph. The single 

parameter is a linear reservoir routing coefficient 

(KSW). In addition, a time-area curve is derived 

which distributes the excess rainfall into a translation hydrograph. The input to this component is 

the rainfall-excess output computed from the infiltration component, and the output is the stormrunoff hydrograph. Table 1 summarizes the eight 

model parameters. 

TABLE 1.-The eight model parameters and their application 

in the modeling process 

Parameter 

identifier 

code1 

SWF ---------

RGF --------

KSAT --------

BMSM-------· 

EVC ________ _ 

DRN ---------

RR-----------

Units 

Inches 

Inches per 

hour. 

Inches--------

Inches per 

hour. 

Application 

Effect of moisture content and capll· 

lary potential at the wetting front 

for field-capacity conditions. 

Varies the effective value of 8WF as 

a function of BM8. 

The minimum (saturatedf value of hydraulic conductivity used to determine infiltration soil rates. 

Soil moisture-storage volume at field 

capacity. 

Coefficient to convert pan evaporation 

to potential evapotranspiration values. 

A constant drainage rate for redistribution of soil moisture. 

Proportion of daily rainfall that infiltrates the soil. 

KSW --------- Hours________ Time characteristic for linear reservoir 

routing. 

1 For explanation of the parameter identifier codes, see preceding text. 

The output from one component is the input to 

the next. Even a model as simple as this one has 

many interactions among the pan.meters. This is 

particularly true of the antecedent-moistureaccounting and infiltration components. Often, adjustments of a parameter in one component can be 

compensated for by an adjustment in a different 

component in another parameter. Over some error 

range, many sets of parameter values may fit a 

given set of data equally well. Even though the 

parameters of the model are chosen so as to be 

analogous to physical parameters in a basin, the 

degree of similarity in the optimum set of derived 

parameter values may mask the relation of the 

values to their supposed physical prototype. Thus, 

the conceptual physical equivalence of the model 

may be lost in the fitting process. (This point is 

discussed more fully in the section entitled "Fitting 

Errors.") 

INFILTRATION COMPONENT 

"Infiltration" is the term used to describe the 

entry into the soil of water available at the soil 

surface. When rain falls on a soil it either infiltrates, goes into detention storage, or becomes 

surface runoff. The rate of infiltration into the soil 

is, of course, limited by the supply rate of rainfall. 

Darcy's law describes flow of a liquid in a homogeneous porous medium and is the basic mathematical 

description of the infiltration process. 

Many empirical equations have been used to approximate the infiltration process. One of the more 

physically meaningful equations is that of Philip 

(1954; Green and Ampt, 1911), which has been 

used as the basis for the infiltration component in 

the flood hydrograph synthesis program. The Philip 

equation assumes a two-part soil moisture distribution, as shown in figure 2. 

A soil column of initial moisture content, mo, is 

infiltrated by water which wets a thickness of soil, 

x, to a uniform liquid content, m. Both m and mo 

are relative moisture contents of their respective 

soil columns, with m representing moisture content 

near saturation. 

The wetting front is at the depth, x, below the 

soil surface. The equation assumes that both the 

velocity of flow throughout the wetted column and 

the soil suction at the wetting front are constant. 

The capillary potential of an unsaturated soil acts 

to move moisture from wetter to drier portions of 

a soil column. 

With these assumptions, Darcy's law reduces to 

Vx k; = P+x+H, 

or 

(1) 

where V is the downward velocity of flow in the 

infiltrating column (units of L/T) ; kh is the capillary conductivity (units of LIT) at soil moisture 
B6 SYNTHESIS IN HYDROLOGY 

LIJ 

() 

c( 

SOIL-MOISTURE CONTENT 

:l: X - - - - --+L-L-L-L.....L...-.I.:.....L....<~L-L--L-L--L-.L.....L...-.~.~'----.L-..1 

::::> 

(/) 

~ 

0 

...J 

LIJ 

m 

J: 

li: 

LIJ 

0 

0 m 

VOLUMETRIC SOIL-MOISTURE CONTENT 

FIGURE 2.-Schematic diagram of the two-layered soilmoisture profile used with the infiltration equation 

in the model. The initial uniform soil-moisture content, stated as a proportion of total volume, is fn<J, 

and the depth of the wetted layer is (1}. The amount 

of infiltrated moisture (SJJS) is (m-fn<J) • (1}, and 

is shown as the hachured area. The amount of 

antecedent moisture (BMS) is shown as the unhachured area. That portion of antecedent moisture 

contained in the wetted layer is fn<J • (1}, and is 

shown as the unhachured area above the· depth (I} in 

the profile. 

m; P is capillary potential at the wetting front 

(units of L) ; and H is the depth of ponded water 

at the surface (units of L). The capillary potential 

generally is several orders of magnitude larger than 

the depth of ponded water, so that the H term may 

be ignored. Because 

v = di/dt, 

and 

x = i/(m -mo), 

equation 1 becomes 

(2) 

(3) 

~; = k• ~ + P(m i---mo~ ' (4) 

where i is the accumulated infiltration in the wetting column (denoted by patterned area in fig. 2). 

The mnemonic identifiers used to designate equation 4 in the computer program and in this paper 

are 

where 

FR = KSAT ~+ ::~ , (5) 

FR = di/dt (units of L/T), 

KSAT = kh (units of L/T), 

PS = P(m -mo) I effective (units of L), 

and 

SMS = i (units of L). 

The capillary potential at the wetting front is not 

a constant, but varies according to the initial soilmoisture condition. Colman and Bodman (1944) 

stated (in a paper used by Philip for some of the 

justification for his equation) that "of the changed 

conditions brought about by using moist rather 

than air-dry soils, the observed results indicate the 

particular importance of the lowered potential 

gradient at the wet front." However, neither Philip 

nor Colman and Bodman gave a method for determining the variation of the potential. The floodhydrograph simulation program determines the 

effective value of PS as varying linearly between 

a value at plant wilting point and a value at field 

capacity. This requires two parameters. The first 

is the effective value of the product P ( m -mo) at 

field capacity (SWF). The other· is the ratio 

(RGF) of the product at wilting point to that at 

field capacity. The effective value of the product of 

capillary potential and soil-moisture deficit is described by a linear relation to soil moisture deficit 

and is computed as 

PS = SWF rGF- (RGF -1) ::s~ , (6) 

where BMS is the beginning soil-moisture storage 

in the soil column, and BMSJU is the maximum 

moisture storage in the soil column at field capacity. This. relationship i& shown in figure 3. 

Equations 5 and 6 represent the approximation 

used for infiltration at a point. Equation 5 is a differential equation with a variable coefficient because 

the soil-suction coefficient, PS, is a function of soil 

moisture, as shown in equation 6. Infiltration occurs 

throughout a basin at varying rates; however, the 

flood-hydrograph synthesis program uses a scheme 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B7 

0 

~ 

I 

~ 

~ RGF X SWF PS= SWF [RGF- (RGF-1) BMS ] BMSM 

~ 1.1.. 

0 

UJ 

:::;) 

...J 

~ 

UJ 

> 

i= 

u 

UJ 

1.1.. 

1.1.. 

UJ 

SWFL---------------------~~ 

Wilting 

point 

(BMS=O) 

Field 

capacity 

(BMS=BMSM) 

SOIL-MOISTURE CONTENT 

FIGURE 3.-The relation which determines the effective 

value of moisture content and capillary potential (PS) 

for use in the infiltration equation. 

first presented by Crawford and Linsley ( 1966, 

p. 210) to convert point potential infiltration to net 

infiltration over a basin. Letting SR represent the 

supply rate of rainfall for infiltration, and QR represent the rate of generation of excess precipitation 

that does not infiltrate, the equations are 

QR = SJl2!2FR 

QR = SR-(FR/2) 

SR<FR 

SR>FR 

(7a) 

(7b) 

The schematic representation of the relations is 

shown in figure 4. The relation may be interpreted 

as describing the probability distribution of potential infiltration throughout the basin by a straight 

line, with net infiltration being the average throughUJ 

~ 0:: 

> 

r1.sR 

a.. 

:::;) 

en 

...J 

...J 

~ z 

< 0:: 

Rainfall excess 

(QR) Infiltration 

100 

PERCENTAGE OF AREA WITH INFILTRATION CAPACITY 

EQUAL TO OR LESS THAN INDICATED VALUE 

FIGURE 4.-The relation which determines rainfall excess 

( QR) as a function of maximum-infiltration capacity 

(FR) and supply rate of rainfall (SR). 

out the basin. However, no claim is made that equation 7 actually is a representation of the probability 

distribution of potential infiltration. Certainly, such 

a distribution would not be uniform, as implied by the 

equation, nor would its shape be similar in time. 

Rather, equation 7 is an empirical tool which eliminates the absolute threshold value for infiltration. 

Thus, there is some runoff from any volume of rainfall, although for low-intensity rains where soil 

conditions are dry, the runoff is very small. The 

major justification for equation 7 is that it aids the 

modeling of the runoff volumes for the smaller, lowintensity storms. 

Equations 5, 6, and 7 together describe the infiltration component. The flow chart for the infiltration component is shown in figure 5. 

RAINFALL 

INPUT 

1 

EVAPO- COMPUTE: 

TRANSPIRATION INFILTRATION 

1 

ACCRETION TO: 

SMS 

l 

USE BMS 

DRAINAGE TO: f.--- WITH RGF 

BMS TO COMPUTE PS 

1 

SPILL TO: 

DEEPER 

STORAGE 

FIGURE 5.-Schematic :flow chart of the flood-hydrograph 

simulation program. 

SOIL-MOISTURE-ACCOUNTING COMPONENT 

The soil-moisture-accounting component in a rainfall-runoff simulation model determines the effect 

of antecedent conditions on the infiltration component. Although the moisture accounting system in 
B8 SYNTHESIS IN HYDROLOGY 

this model was designed to represent the physical 

process to a large degree, the lack of full physical 

equivalence, in application, may result in a curvefitting process, so that the fitted parameter values 

have more apparent than real physical meaning. In 

addition, there is a necessary constraint that the 

soil-moisture accounting component must be compatible with the infiltration component, if a water 

budget is to be maintained throughout the system. 

These two facts should be kept in mind throughout 

the description of the soil-moisture accounting 

component. 

The soil moisture component in the flood hydrograph simulation program is based upon the Philip 

scheme described for the infiltration component. 

The total moisture in storage in the soil column is 

divided into two parts. The first is contained in a 

base moisture storage (BMS) at a soil moisture 

which can vary from field capacity to wilting-point 

conditions. The second is a surface moisture storage 

(SMS) near saturation. Thus, the total infiltrated 

column is assumed to be near saturation capacity. 

This assumption is based upon the results shown by 

Colman and Bodman (1944). A schematic diagram 

of the soil moisture accounting is shown in figure 2. 

SMS depicts accumulated infiltration, and all infiltration during storm periods is added to SMS. 

BMS, on the other hand, is used to compute the 

relative soil moisture deficit. The unpatterned area 

in figure 2 represents BMS. BMS and the ratio 

RGF are together used to compute PS, the effective 

value of the product of the capillary potential and 

the soil moisture deficit, also a part of the infiltration equation. 

Evapotranspiration losses are assumed to occur 

at the potential rate. All evapotranspiration demand is met from SMS, if possible. When storage 

in SMS is zero, the evapotranspiration demand i~ 

met from BMS. 

Drainage occurs from SMS to BMS at a constant 

rate as long as storage exists in SMS. Storage in 

BMS has a maximum value (BMSM) equivalent to 

the field-capacity moisture storage of an active soil 

zone. Zero storage in BMS is assumed to correspond 

to wilting-point conditions in the active soil zone. 

When storage in BMS exceeds BMSM, the excess is 

spilled to deeper storage. The spills could be the 

basis for routing interflow and base-flow components, if desired. However, these components of 

streamflow are not modeled in the flood-hydrograph 

simulation program. If other components of flow 

make up a significant part of the flood peak, a routing of these spills would be necessary. 

SURFACE ROUTING COMPONENT 

The excess precipitation generated in the floodhydrograph simulation program must be converted 

into a flood hydrograph by a routing method. The 

Clark flood-routing method (1945) is used to develop the unit hydrograph for the basin. The Clark 

method has two parts. First, the excess precipitation is converted into a translation hydrograph 

representing the effect of varying traveltimes in 

the basin. The translation hydrograph for the basin 

is represented by a time-discharge histogram. The 

time-discharge histogram is developed from the 

distance-area histogram for the basin. In essence, 

the derivation assumes that distance and traveltime 

are directly proportional. Because of variation both 

in resistance to flow and in channel slope throughout the basin, the assumption of proportionality of 

distance and traveltime does not necessarily hold. 

Therefore, a comparison of the shapes of simulated 

and observed hydrographs for several flood events 

can be used to revise the time-area histogram to a 

more appropriate shape for a study basin. 

The translation hydrograph must be routed 

through some element representing storage in the 

basin. For an instantaneously developed excess precipitation of 1 inch, this results in the instantaneous 

unit hydrograph. The Clark method assumes a linear 

time-invariant storage. Dooge ( 1959) presented an 

excellent discussion on unit hydrograph methods 

and the place of the Clark method in the general 

theory. Figure 6 illustrates the operation of the 

Clark method. 

SYSTEM IDENTIFICATION 

The method of determining optimum parameter 

values is based on an optimization technique devised 

by Rosenbrock (1960). Wilde (1964) referred to 

the method as "the method of rotating coordinates." 

It is a hill-climbing procedure that does not require 

evaluation of partial derivatives of the objective 

function with respect to the parameters. All parameters must be bounded for the method to be used. 

Thus, parameter values may be constrained to a 

range of "reasonable" values if desired. The utility 

of· the procedure, as related to system identification 

in hydrologic modeling, was discussed by Dawdy 

and O'Donnell (1965). 

The method revises the parameter values and 

recomputes the objective function, using the revised 

set of values. If the result is an improvement, the 

revised set is accepted ; if not, the method returns 

to the previous best set of parameters. The objec-
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B9 

Input of 

excess 

precipitation 

Translation into 

time-discharge 

histogram 

Attenuation 

by linear 

storage 

Output flood 

hydrograph 

FIGURE 6.-Schematic drawing of the Clark unit hydrography used in the surface-runoff routing component. 

tive function, or U-function, throughout this study 

is based upon the sum of the squared deviations of 

the logarithms of peak flows, storm volumes, or 

some combination of both. Thus, the fitting procedure develops a nonlinear least-squares solution. 

The logarithms of flows are used because streamflow errors are generally more nearly equal in percentage than they are in absolute terms. Thus, if a 

peak of 1,000 cfs (cubic feet per second) is estimated in error by an average of 100 cfs (10 percent), a peak of 5,000 cfs will have a greater 

probability of an average error of 500 cfs (10 percent) than of 100 cfs (2 percent). The logarithmic 

transformation is meant to make the error of estimation more commensurable for the large and the 

small peaks. The .sum of the squared errors is used 

as an objective function because of the mathematical property that it is a convex function, and because of its direct analogy to least-squares fitting 

in standard linear statistical theory. More concerning this point will be discussed in the .section entitled "Response of the Model." 

Rosenbrock's method of optimization proceeds by 

stages. During the first stage, each parameter represents one axis until arbitrary end-of-stage criteria 

are satisfied. At the end of each stage, a new set of 

orthogonal directions is computed, based on the 

experience of parameter movement during the preceding stage. The major feature of this procedure 

is that, after the first stage, one axis is alined in a 

direction reflecting the net parameter movement 

experienced during the previous stage. 

To .start the fitting process, the hydrologic model 

is assigned an initial set of parameter values, and 

the resulting simulated flood-hydrograph response 

is computed. The objective function is calculated 

and then stored in the computer memory bank as 

a reference value; later, this reference value is used 

to evaluate the results of subsequent trials. A step 

of arbitrary length is attempted in the first-search 

direction. If the resulting value of the objective 

function is less than or equal to the reference value, 

the trial is registered as a success, and the appropriate step-size, e, is multiplited by o> 1. If a failure 

results, the step is not allowed and e is multiplied 

by -{3, where 0<{3<1. An attempt is made in the 

next search direction, and the process continues until the end-of-stage criteria are met. At this point, a 

new search pattern is determined, and another stage 

of optimization undertaken. Only a limited amount 

of information is output during optimization. The 

U-function value and associated parameter values 

are printed for each successful trial. Also, a listing 

by flood event of the simulated hydrologic response 

and of observed data are output at the start of 

each stage. 

Note particularly that the concept of automatically determining optimum model parameters requires the objective function to be compatible with 

the intended use for which the fitting is undertaken. 

In order to give weight to both the volume and the 

shape characteristics of the flood hydrograph, a 

weighted objective function (U3)-including both 

peak and volume error-was used. One component 

of the objective function used in optimization is the 

sum of squared log deviations between recor~ed and 

simulated flood peaks ( U1). Another component, 
BlO SYNTHESIS IN HYDROLOGY 

( U2), is the sum of squared log deviations between 

estimated and simulated surface runoff for each 

storm period. 

Estimated surface runoff is calculated by a crude 

hydrograph-separation technique that integrates the 

volume of runoff under the flood hydrograph, from 

the start of the storm period through the period of 

rise, and for a duration of recession after the peak. 

The contribution from base flow is deducted and is 

assumed to be equal to the volume derived by projecting the level of discharge from the start of the 

rise through the period of integration. Recorded 

flood peaks are similarly reduced by the antecedent 

discharge level to account for the contribution from 

base flow. 

RESPONSE OF THE MODEL 

The "game" of hydrologic simulation is based 

upon engineering approximations. Approximations 

introduce errors into simulation results. To properly 

utilize a model, therefore, there must be some understanding of the magnitude of errors produced by 

use of the model. 

Errors in data are reflected in errors in the fitted 

parameters in a simulation model. If perfect input 

data are routed through a perfect model, the output 

produced would agree perfectly with an error-free 

output record. If errors are introduced into the input or output record, or both, the output results 

will not be exactly reproduced even from a perfect 

model. If a fitting process is used, the parameters 

will deviate from their true values in order to minimize the deviations between the simulated and recorded traces, as specified in the objective function. 

The "optimal" set of parameters will now be in 

error, and the value of the objective function after 

fitting will be less than its "true" value. This is so 

because the value has been derived by a method 

used to find the minimum value for the objective 

function. 

The fitting process is analogous to a statistical 

least-squares analysis. The fitted parameters deviate from their population values because of random 

errors in the data. The standard error of estimate 

is a measure of error in the data. The standard 

error of prediction, however, is somewhat greater 

than the standard error of estimate, for it includes 

both the measure of lack of fit of the data used to 

calibrate the model and the measure of error in the 

fitted parameters. These relationships are given in 

table 2. 

TABLE 2.-Errors involved in hydrologic modeling qualitatively compared with analogous errors resulting from 

standard linear statistical analysis 

Source of error 

Qualitative 

size of error 

variance 

Statistical analog 

Measured data ------------------- a Measurement and sampling error variance. 

Differences between measured and a - b Square of standard ersimulated flows during the cali- ror of estimate. 

bration period. 

Differences between measured and a + c Square of standard ersimulated flows uutside the cali- ror of prediction. 

bration period. 

If the assumptions of regression theory were 

valid, for a linear model with normally distributed 

and homoscedastic errors of the dependent variable 

(that is, the variance about the regression is independent of the independent variables), the standard 

error of prediction could be computed from the 

standard error of estimate, the deviations of the 

independent variables from their mean, and the 

error in the coefficients for the independent variables. These assumptions seldom hold, however, so 

that competent statisticians often resort to splitsample testing. The assumptions also fail for hydrologic simulation, and the models are nonlinear, as 

well; hence, there is no theory by which to compute 

the error of prediction. Therefore, split-sample testing must be used in hydrologic simulation modeling 

whenever possible. 

At present, nonlinearity of the hydrologic process 

precludes any theoretical description of the mechanism by which errors in data are transferred to 

model parameters and, in turn, are combined with 

input errors in the test period to produce errors in 

the simulated streamflow. An empirical study for 

the response of the model to input and output errors 

is shown in table 3. 

A recorded rainfall trace was assumed to be error 

free and was routed through an optimized set of 

parameters for the Little Beaver Creek basin near 

Rolla, Mo. · The optimized parameter values were 

assumed to be the correct values to obtain a "true" 

streamflow trace. Then, a random error with mean 

zero and standard deviation of 10 percent was applied to all rainfall values. These "erroneous" rainfall values were then routed through the model with 

the "true" parameter values, and the resulting value 

of the objective function for the simulated streamflow trace computed. Next, an optimization run was 

made which adjusted the parameters to minimize 

that value. The "optimized" set of parameters is 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B11 

TABLE 3.-Results of an empirical study of the response of the model 

to input and output errors 

Values optimized to Values optimized to 

Parameter 

identifier 1 

Assumed true 

values 

rainfall errors of: streamflow errors of: 

SWF -------------------(in.)--

KSAT ------------(in. per hr) __ 

KS W --------------------( hr) -- EVC -------------------------- BMSM ------------------ (in.) __ 

RGF -------------------------- RR ---------------------------- DRN -------------(in. per hr) -- U: 

3.6 

.063 

1.0 

.56 

4.0 

12.0 

.8 

.020 

10 

percent 

3.6 

.063 

1.04 

.57 

4.02 

11.9 

.796 

.018 

20 

percent 

3.8 

.06 

1.06 

.58 

3.98 

11.94 

.8 

.017 

5 10 

percent percent 

3.7 3.7 

.063 .061 

.98 .98 

.559 .56 

4.04 4.04 

12.12 12.21 

.796 .796 

.020 .019 

Pd 2 ------------------------

pd ------------------------- pD ------------------------- Test 4 ----------------------

3 .0150 (12) .0538(23) .00233 (4.8) .00915 (9.6) 

.0097 (9.9) .0493(22) .00170 (4.1) .00708 (8.3) 

.0039 (6.3) .0152 (12) ---------- ---------- .0196 (14) .0890(30) ---------- ---------- 1 For explanation of parameter identifier codes, seep. B3-B5. 2 P, true parameters; p, optimized parameters; D, correct data; d, erroneous data. 3 First value, shown without parentheses is the average of two-thirds of the squares of natural 

logarithms of the sample peaks plus one-third of the squares for the sample storm volumes. The second 

value, shown in parentheses, converts the first value to an equivalent "percent standard error" by 

SE=antilog U, and averaging plus and minus percentages. 4 Average of nine separate test runs. 

shown, along with the resulting ·value. The "true" 

rainfall trace was then routed through the new 

optimized parameters and the objective function 

evaluated. Assuming independence of the two 

sources of error--one in the input data, and the 

other in the model parameters--the error of prediction should be approximately equal to the square 

root of the sum of the squares of the two separate 

estimates of error. To test this relation, nine independent sets of random errors were applied to the 

rainfall values and routed through the model using 

the optimized parameter values. The average U 

value for the nine test runs is also given in table 3. 

For the case of 10-percent rainfall errors, the 

error introduced by data errors (Pd) is 0.0150, 

while that for parameter errors (pD) is 0.0039. 

The sum of these is 0.0189 which is to be comp~red 

with 0.0196 (U test). The comparison of the error 

of prediction, based upon the variance given above, 

is 13.8 percent (Pd + pD), as compared with 14.3 

(U test). 

Similar results are shown for input rainfall errors with a 20 percent standard error. As also true 

of the 10-percent errors, the error in simulated output was magnified so that it is about 20 percent 

greater than the rainfall error (Pd is 23 percent as 

compared with the previous value of 12 percent). 

Once again Pd + pD should combine to produce a 

value comparable to that for the test results, and 

0.0538 + 0.0152 = 0.0690 is to be compared with 

0.0890. The respective percentages are 26 and 30 

for estimates of the error of prediction. 

Errors in streamflow measurement are transferred to model parameters in the fitting process. 

An example of this is shown in table 3. Errors of 5 

and 10 percent were introduced into runoff estimates, and a set of best-fit parameters derived. The 

rainfall and runoff errors are independent in this 

study, so that the square of the error of prediction 

for 10-percent runoff errors and 20-percent rainfall 

erro-rs would be on the order of the sum of the two 

variance terms, 0.0890 for rainfall errors and 

0.00915 for runoff errors, or a 32-percent error for 

the two combined, as compared with 30 percent for 

rainfall alone. 

Two points are particularly noteworthy in the 

above results. First, rainfall errors have a magnified effect on the simulated streamflow for basins 

similar to the one chosen for the study. This probably is true· for most basins with drainage area less 

than 10 square miles. Therefore, rainfall errors 

probably are the controlling factor determining accuracy of streamflow simulation. Second, the response of this wholly nonlinear hydrologic model 

is approximately linear for errors in rainfall on the 

order of magnitude investigated, which probably 

are on the order of magnitude generally found in a 

field study. One would expect that errors in rainfall 

input would result in greater proportional errors in 

predicted streamflow, because streamflow is a residual after abstractions. However, amounts of excess 

rainfall for different time periods from different 

parts of the basin are combined in the translation 

routing, and the storage routing attenuates errors 

by averaging by means of the storage process. 

Therefore, the relative size of rainfall errors and 

of errors of estimated streamflow depends upon the 

extent to which the model of the routing process 

attenuates the magnification of errors produced in 

model estimates of excess rainfall. 

That the errors of streamflow estimates are approximately linearly related to the errors of rain-
B12 SYNTHESIS IN HYDROLOGY 

fall input data is particularly important. The linearity of errors indicates that there may be some 

hope for the derivation of a theory of errors for 

streamflow simulation. In addition, the linearity 

gives some post hoc justification for the nonlinear 

least square fitting technique used in the fitting 

process. 

SIMULATION MODEL STUDIES 

The proof of the pudding is in the eating thereof. 

The empirical study described in the preceding 

section does give insight into the modeling process 

and, in particular, into the operation of the model. 

To further illustrate the utility of the. model in 

field application, three basin studies are presented. 

They represent a range in location and hydrology. 

The basins are Santa Anita Creek near Pasadena, 

Calif., a semiarid basin; Beetree Creek near Swannanoa, N.C., a humid basin; and Little Beaver Creek 

near Rolla, Mo., a basin in which hydrology is typical of the interior United States. All three basins 

have pronounced relief. 

The data available varied from basin to basin. 

Also, the relative stage of development of the model 

led to the emphasis of different research goals during the analysis of the different areas. Sufficient 

rainfall data were available for the Santa Anita 

Creek basin for a study to be made of the effect of 

bias of rainfall measurements and of the effect of 

time and space variability of rainfall on modeling 

results. Beetree Creek basin was used to study the 

effects of split-sample testing to study the methodology of the use and limitations of the objective 

curve-fitting method. Each basin will be discussed 

separately, and a discussion of the overall results 

and of the problems encountered will be presented. 

SANTA ANITA BASIN 

GENERAL PHYSIOGRAPHY 

The Santa Anita Creek drainage basin is a 9. 7 

square mile (25 sq km) area of the San Gabriel 

Mountains in southern California. The rugged topography ranges in elevation from 1,500 to 5, 700 

feet ( 460 to 1, 700 m above sea level with the mean 

elevation about 3,600 feet (1,100 m). Thin porous 

soils covering a highly fractured bedrock combine 

to give the basin high moisture-retention and absorption properties. The southerly facing basin receives about three-fourths of its rainfall during the 

cool winters. The climate and soils support a thin 

to dense growth of chaparral native to the area. 

PRECIPITATION 

The precipitation-measuring network on the 

Santa Anita Creek basin consists of 6 stations for a 

14-year period ending with the 1962 water year. 

The six sites provide good areal (fig. 7) and elevation (table 4) coverage of the annual precipitation 

338 ../"·. 

Mount Wilson/··-··-· . ._/·· \ 

Elev 5710 o/ 58 ( 

: Sturdavent ·.'-

tr Elev 3225 ft · '""'-..._ 

·. 0 ~ ·. 

( ~ --... 60 ll \ 477 "--··-..... 

oegees Camp Ivy Spring camp• '\ 

Elev 2500 ft Elev 4670 ft ,,/· 

~.,__./· 

/.. N 

•'""'--"'. 

63 

o Big Santa Anita dam 

Rainfall, nonrecording station Elev 1400 ft 

• 

Rainfall, recording station 

0 2 MILES 

Streamflow, recording station 

FIGURE 7 .-Santa Anita Creek basin, above the streamgaging station near Pasadena, Calif. 

TABLE 4.-Mean annual rainfall, Santa Anita Creek basin, 

California, 1949-62' 

Station 

No. 

Station 

name Elevation 

(ft) 

58_________ Sturdavent ----------- 3,225 

60_________ Hoegees Camp Ivy ---- 2,500 

63_________ Big Santa Anita dam__ 1,400 

338________ Mount Wilson --------- 5, 710 

432________ Fern Lodge ----------- 2,035 

477-------- Spring Camp --------- 4,670 

Basin mean 

Thiessen method: 

All stations ------- Thl'ee stations a ___ _ 

Elevation-area method __ 

1 Basin mean used was 29.5 inches. 2 Adjusted on the basis of double-mass analysis. a Stations 60, 338, and 477. 

Mean 

annual 

rainfall 

(in.) 

30.57 

30.44 

19.47 

25.21 

26.30 2 28.89 

28.3 

29.5 

29.5 

Deviation 

from 

basin 

mean 1 

(percent) 

-+-3~2 

-13.5 

-+o~s 

in the basin. A double-mass analysis of the 14-year 

annual precipitation values show fair measurement 

consistency among the 6 stations. Three sites with 

continuous recorders (stas. 477, 60, 338) provided 

the rainfall data required by the simulation mode~. 

Two methods were used to determine the basin 

mean annual rainfall during the 14-year period 

(table 4). First, the standard Thiessen method gave 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B13 

PERCENTAGE OF AREA WITH ELEVATION LE:.SS THAN INDICATED VALUE 

0 10 20 30 40 50 60 70 80 90 100 sooo~------~------~--------~-------.--------.--------,--------.-------.--------,------~ 

...J 

LLI 

> 

~5000~--------------~-----------------+----------------~~--------------~----------~r-~ 

<( 

LLI 

en 

z 

<( 

LLI 

~4000~---------------4-----------------+-----------------t~~~~T-------~--------------~ 

LLI 

> 

0 

til 

<( 

1-

LL.I 

~ 3000~--------------~~~-------------+-----------------r----------~----~--------------~ 

~ 

z 

0 

f= 

~ 2000~~~-----------4-----------------+------~~~~---r----------------~----------------1 LLI 

...J 

LLI 

100~L6------~18--------~20--------~22--------2~4--------~26 ________ 2L8------~3~0------~32~----~3~4------~36. 

MEAN ANNUAL (1949-62) RAINFALL, IN INCHES 

FIGURE 8.-The relation of mean annual rainfall and of area in the basin to elevation for the Santa Anita Creek 

basin above the stream-gaging station near Pasadena, Calif. 

a mean of 28.3 inches (71.9 em) when all six stations were considered and a mean of 29.5 inches 

when only the three recording sites were used. The 

second method, a numerical integration of the relationships for elevation-percent area obtained from 

topographic maps and elevation-annual rainfall defined by the six gage records (fig. 8), gave an annual mean of 29.5 inches. A value of 29.5 inches was 

chosen as the estimate of the mean annual rainfall 

and was used to evaluate the relation of individualstation rainfall to basin-mean rainfall. 

The 24 storm periods selected for simulation had 

both complete records of the rainfall occurring at 

the three recording stations and a significant rise in 

stream discharge. The records for the storm periods were converted to discharge volumes for 15-

minute time intervals. Daily rainfall records were 

used between storms. The storm data were compiled from gage charts provided by the Los Angeles 

County Flood Control District. 

STREAMFLOW 

The streamflow data used for fitting the model to 

the Santa Anita Creek basin were those for the 

U.S. Geological Survey gaging station near Pasadena, Calif. The site has been gaged since 1916. 

The mean flow has been 5.5 cfs (0.16 m3/sec) or 7.7 

inches (0.2 m) (meters) per year. The maximum 

flow of about 5,200 cfs (147 m3jsec) occurred in 

March 1938. The peak discharges during the storm 

periods selected for study ranged from 17 to 2,530 

cfs (0.5 to 71.6 m3jsec). (See table 5.) 

TABLE 5.-Simulated peak discharges, using fitted parameters 

[Observed and simulated discharges, reported in cubic feet per second (cfs), 

do not include base flow] 

Simulated discharge (cfs) with adjusted data (C)l 

Observed 

Storm discharge Station Mean No. (cfs) discharge 2 

477 60 338 (cfs) 

1 ------ 2,529 2,506 2,318 2,722 2,515 2 3 ----- 1,472 2,846 1,667 1,198 1,904 3 3 ----- 338 917 296 346 520 

4 194 224 221 225 223 

5 342 532 405 454 464 

6 ------ 45.8 44.4 53.4 63.8 53.9 

7 ------ 30.6 37.2 38.5 35.2 37.0 

8 ------ 108 184 146 148 159 9 3 --·-- 34.1 35.9 21.5 30.3 29.2 

10 3 ----- 50.1 51.9 14.5 27.3 31.2 

11 660 601 618 472 547 

12 150 121 149 130 133 

13 156 166 190 129 162 

14 111 113 99.6 96.6 103 

15 361 283 417 318 339 

16 332 238 228 253 240 

17 ------ 837 779 696 512 662 

18 3 ----- 709 372 97.9 42.3 171 

19 ------ 243 190 196 161 182 

20 ------ 15.4 31.6 28.9 35.4 32.0 

213 ----- 58.4 18.3 36.2 14.3 22.9 

22 55.0 28.1 29.7 21.9 26.6 

23 91.2 54.] 45.9 69.2 56.4 

24 ------ 1,235 1,238 1,303 1,362 1,301 

U1 4 ----------- ~7 ~6 ~3 ~ 

1 Set C is data for each storm event adjusted to average basin volume. 2 Average of the simulated peaks for the three stations. 3 Not a component of the objective function used in optimization. 4 U1 is the sum of squared log deviations between recorded and simulated 

flood peaks. 
B14 SYNTHESIS IN HYDROLOGY 

EVAPORATION 

Daily values of pan evaporation at Tanbark Flat 

were obtained from the Pacific Southwest Forest 

and Range Experiment Station, U.S. Department 

of Agriculture. The Tanbark Flat climatic station 

is located in the San Dimas Experimental Forest 

about 10 miles (16 km) east of the Santa Anita 

Creek basin and is at an elevation of 2,800 feet 

(850 m). The mean annual evaporation from a 

standard [U.S.] Weather Bureau pan is in excess 

of 60 inches ( 1.5 m) . 

DATA SCREENING 

The amount of rainfall data available for the 

Santa Anita Creek basin was sufficient to investigate the effects of variability of measured rainfall 

upon simulation results. The record of several storm 

events indicated a large spatial variation in total 

storm rainfall over the basin, as indicated by the 

deviation in percent of measured storm volume at 

each site from the weighted mean for each storm. 

Several storms also appeared to have a large spatial variation in total storm rainfall over the basin, 

as indicated by the deviation in percent of measured 

storm volume at each site from the weighted mean 

for each storm. Several storms also appeared to 

have a large spatial variation in rainfall intensities 

over the basin. On the basis of preliminary screening, six storms of the 24 available for analysis were 

not used in fitting the parameters. However, these 

peaks were simulated, and results are shown in the 

scatter diagrams. Only one of the excluded storms 

might have significantly changed the results. The 

records for that storm show extremely high intensities for very short periods of time; the 15-minute 

time interval used to define the rainfall records 

appears to be inadequate for an accurate simulation for that storm. The purpose of screening is to 

eliminate storms with extreme errors in data input, 

so as to minimize the effect of data errors on the 

fitting process. 

PARAMETER DEFINITION 

Nine model parameter determ.inations were made 

as a series of three fittings for each of the three 

rainfall stations (stations 60, 338, 477). The first 

in the series of three fittings was made by using the 

data as recorded at the stations (set A). These 

results are analogous to those for simulation studies for which a single recording rain gage is available in a basin, and for which there is no basis for 

adjusting the record to obtain a better estimate of 

the mean basin rainfall. 

The second series of parameter determinations 

was made for each station by adjusting the recorded 

storm volumes by a constant station factor (set B). 

These factors were computed in order to adjust the 

mean annual depth at the station site to 29.5 inches 

(computed for basin mean annual rainfall, as explained earlier) . These results are analogous to 

those for simulation studies for which a recording 

gage is available in a basin, and supplementary data 

are available to determine an average annual rainfall on the basin and at the gaged site. 

The third fit was made to the data with the storm 

volumes adjusted to a three-station Thiessen weighted 

mean for each event (set C)-that is, the mean 

basin volume was distributed in time in accordance 

with the rainfall-intensity pattern for each individual station. These results are analogous to those for 

simulation studies for which a recording gage plus 

several nonrecording rain gages are available in a 

basin. Thus, a weighted mean basin rainfall for 

each storm can be derived. 

To summarize, the various rainfall intensities are 

adjusted as follows: 

Set /'.. Adjustment 

A -------------- Ri1 = R,1 ~ 

B -------------- Rij = a. i • RH ~ 

c -------------- Rij = Uij • RiJ 

where Rii is the measured intensity for period i at 

station j; 'f!f:i is the adjusted intensity used in the 

given simulation set; a. i is an average adjustment, 

which is the ratio of mean annual rainfall over the 

basin to the mean annual rainfall measured at 

station j; and aii is the ratio of average rainfall 

over the basin for storm i to that volume measured 

for storm i at station j. 

The results of the nine fittings are given in 

table 6. In addition to parameter values the average 

squares of deviations between logarithms of simulated and observed peaks, U1, is given for each set 

of parameters. Table 5 shows the values of the 

simulated peaks for data-set C. Figure 9 shows 

typical scatter diagrams for data-set A. 

PARAMETER SENSITIVITY 

The sensitivity of the goodness of fit criterion to 

changes in parameter values is helpful in discussing 

parameter importance and simulation results. An 

expression of sensitivity of the error criterion to 

given parameters can be obtained by performing 

repeated simulations while incrementing the parameters, holding all other parameters to their fitted 

value, and observing the change in value of the fit-
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B15 

TABLE 6.-Fitted parameter values 

[Input series: A, rainfall data as. recorded; B, rainfall volumes adjusted by mean !'Lnnual factors; C, ratnfall volumes adjusted by mean storm factors. F1gures shown m parentheses are average errors, in percent] 

Parameter Station 

identifier 1 Input 

series 477 60 338 

SWF ------------------- A 21 22 11 B 20 20 16 c 20 18 17 

RGF -------------------- A 7.5 6.1 4.4 B 6.7 5.9 5.5 c 5.6 6.5 6.1 

KSAT ------------------ A .32 .32 .31 B .31 .32 .25 c .32 .32 .31 

BMSM ------------------ A 4.1 3.4 2.1 B 4.0 3.4 3.6 c 3.5 3.5 3.5 

EVC -------------------- A .52 .73 .52 B .59 .71 .80 c .74 .72 .74 

DRN -------------------- A .049 .058 .030 B .045 .057 .043 c .056 .059 .057 

RR --------------------- A 1.14 .98 1.03 B 1.08 1.00 .90 

c .96 1.01 .95 

KSW -------------------- A 2.4 2.8 2.2 B 2.5 2.7 2.3 c 2.6 2.8 2.3 

U1, fit criteria 2 --------- A .097 (32) .123(35) .440( __ ) 

B .100 (32) .122 (35) .438( __ ) 

c .115(35) .098(32) .153(40) 

1 For explanation of parameter identifier codes, see p. B3-B5. 2 A component of the objective function used in optimization. U1 is 

the average of squared log deviations between recorded and simulated flood 

peaks. 

ting criterion. This gives no measure of interaction 

of the parameters but is a simple measure of how 

critically the simulation results are dependent upon 

the individual parameters. The results of this procedure as applied to data set C for station 60 

(table 6) are shown graphically in figure 10. The 

figure is a plot of criterion value versus the percent change in parameter values. Applying this 

procedure to the other data sets produced similar 

relationships. 

ANALYSIS OF RESULTS 

The results of the nine separate optimization runs 

-three for each of the three stations-are summarized in table 6. Shown are fitted parameter 

values and the resulting goodness of fit, U1, which 

is the peak-simulation part of the total fitting criterion used. The representativeness of the rainfall 

data is least for inputs A, and most for inputs C. 

As the data for the three records became more 

nearly similar to each other, the fitted parameter 

values would be expected to converge to common 

values for the three stations. As the data become 

~ore representative, the accuracy of fitting shoqld 

Increase, and the U1 values should decrease. The 

effects of the various· components of error can be 

seen by comparing the variability of parameter 

values and the goodness of fit between stations for 

a given input set. 

PARAMETER VALUES 

Prediction depends upon the fitted parameter 

values for the model, as well as upon the data used 

for the prediction period. The more stable the estimated parameter values, the better the possibility 

of relating the fitted values to measures of the 

basin. Thus, variability of fitted parameters for the 

nine optimization runs may give insight into the 

degree to which model parameters are influenced by 

data errors. 

A wide range in fitted parameter values resulted 

when the data were used as recorded (input series 

A) at the three stations. None of the three sets of 

parameters can· be considered unlikely when viewed 

individually; however, together the sets of values 

illustrate a possible range, depending on the data 

representativeness. In practical application, the 

available data may consist of only one record, which 

must be used without knowledge of its degree of 

representativeness. The variability of fitted parameter values, such as those for series A, will affect the 

feasibility and accuracy of any regionalization of 

parameter values. 

Input series B contains both time-distribution and 

total-volume errors but has been adjusted to reduce 

the gaging bias resulting from errors in the estimated mean annual precipitation over the basin. 

The reduced range in parameter values, except for 

KSAT and RR, indicates a better estimation than 

was obtained in series A. The relative insensitivity 

shown for KSAT in figure 10 is for the independent effect of KSAT in the first term of the Philip 

infiltration equation. Accurate determination is not 

possible and may not be important. The range of 

RR values between series A and B are about equal. 

Values· of RR greater than 1.0 reflect curve fitting 

in the model, and result, in part, from the differences between rainfall measured at a given point 

and average rainfall over the basin. No constraint 

was placed on the value of RR for these optimization runs. 

Input series C has the same estimate for storm 

volume at all stations for each storm. The only 

variability is that introduced by the different time 

distribution during a given storm, as recorded at 

the three stations. All parameters have relatively 

stable fitted values. The variation has been reduced 

to within +-10 percent for all parameters, with only 

the infiltration parameters (SWF and RGF) and 

the routing parameter (KSW) varying by more 

than 5 percent. The overall correspondence of station 477 to station 60 in series Cis very close, espe-
B16 SYNTHESIS IN HYDROLOGY 

0 

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500 1000 5000 

OBSERVED PEAK DISCHARGE, IN CUBIC FEET PER SECOND 

FIGURE 9.-Typical scatter diagrams for simulation results in the Santa Anita Creek basin. Results shown are for 

optimum fits, using rainfall data without adjustment. 

cially when the direct interaction of SWF and RGF 

is considered. 

FITTING ERRORS 

The measure of goodness of fit, 'Ul (table 6) , is 

the average of the squared deviations between logarithms of computed and simulated peaks and is 

analogous to a variance or the square of a standard 

error. 

Several components of error occur in the simulation results. On the assumption that the errors are 

independent, the results of a simulation run, stated 

in terms of components of variance, can be represented as: 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS Bl7 

Q + M + R + V + T- C = U, (8) 

where Q is the variance of error in the computation 

of discharge that results from measurement error, 

from error in rating analysis, and from undefined 

rating changes. M is the variance resulting from 

the approximations used in the model that results 

from the fact the physical laws are not exactly 

known; where known, these laws may be approximated for convenience or speed in computation. 

Both Q and M remain the same for all three sets 

of data. R is the bias error resulting from the use 

of incorrect mean annual rainfall values for the 

basin. The purpose of the adjustments for data set 

B was to minimize this bias as- much as possible for 

the given amount of data. This was accomplished, 

as stated, by using all the data to estimate mean 

basin rainfall and then adjusting each measured 

station mean to the estimate of the basin mean. 

Vis error introduced owing to the fact that a point 

measurement of volume for a given storm differs 

from the mean basin volume for that storm. Adjustments made to obtain input set C were intended to 

minimize this error component. This was accomplished by using all data available to estimate mean 

storm volume for each storm. T is error introduced 

by the fact that point measurements of time variability of intensity during a storm differ, and any 

point measurement differs from an "effective time 

distribution" which best represents average conditions over the basin for simulation purposes. Probably, the only way to minimize the component V 

would be to use an input that varies over the basin. 

C is the curve-fitting error introduced into the 

model parameters by a fitting process. The parameter values are perturbed from a global "best" set 

of values in order to minimize the fitting criterion, 

U, so that C is negative in sign. For use of the 

model in prediction, the curve fitting adds to the 

error. (See table 2.) 

The fitted-error criteria of set A for all three 

stations are closely similar to those for set B, although rainfall values for set A are not adjusted to 

mean basin conditions. The bias in the recorded 

rainfall at each station was compensated for by the 

curve-fitting ability of the model to adjust parameter values. On the basis of these data, bias in 

amount of recorded rainfall affects the resulting 

fitted-parameter values, rather than the accuracy of 

fit. Although the change in value of the fit criterion 

was less than 1 percent, the parameter values for 

station 338 changed so much that the parameter I 

values for set B have a maximum of 1.36 for the 

-40 -20 0 20 40 ~------~------~------~------~ 

A 

1.5 L__--------'------------'---------'----------' 

;:::::;- 2.0 1---------+-~-----t----------,~--7""''--------1 

~ 

z 

Q 

et: 

~ B 5 1.5 L---------l....---------'----------'------------' 

(!) 

z 

~ Li: 

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IJ.J 

:::::> 

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c 1.5 L__ ______ __,__ ______ --'-----------'-----------' 

D 1.5 L----------l....-----------'------------'-----------' 

-~ -~ 0 ~ ~ 

CHANGE IN FITTED PARAMETERS, IN PERCENT 

FIGURE 10.-Typical response curves, showing sensitivity of fitting criterion to percentage changes in 

parameter values. Results are for station 60 time 

distribution applied to Thiessen weighted stormrainfall volumes. 
B18 SYNTHESIS IN HYDROLOGY 

ratio of highest to lowest value, the ratio for parameter EVC. For set A, five parameters (SWF, 

RGF, BMSM, EVC, and DRN) had ratios greater 

than 1.36. The fitted-parameter value for station 

338 is one of the extreme values for each of those 

five parameters in both sets A and B. Thus, the 

errors seem to be transferred from the data to 

the parameters, as is particularly evident for station 338. 

Input set C contains variability among the three 

inputs only in the time distribution of rainfall. The 

goodness of fit for this set ranged from 0.098 for 

station 60 to 0.153 for station 338. Converting the 

range value of 0.055 to an average percentage error 

for the peak discharges yields an estimated 23-

percent error in peak-discharge reproduction, introduced by time variability alone. The fitted parameter for a basin having this degree of variation 

in rainfall patterns also reflects the relative smoothing action introduced by the model and, hopefully, 

by the hydrology ; nonetheless, an average error 

of as much as 20 percent for simulated flood peaks 

can be introduced by the time-distribution error 

alone. Considering only the two better, or seemingly more representative gages, the difference in 

fitted U1 values is 0.017, which gives an average 

percentage error of 13 percent, introduced by timedistribution error in a good record. 

In set C, the most representative gage, in terms 

of goodness of fit, was that closest to the center of 

the basins; the least representative was that on the 

perimeter and at the highest elevation of the basin. 

Therefore, relative representativeness was found to 

be about as expected. 

Input set B contains both time-distribution errors within a given storm and storm-volume errors. 

The records were adjusted to minimize only the 

station bias in relation to basin mean annual rainfall. The results of input set B runs indicate that 

station 477 probably is the most representative 

station for predicting storm volumes, just as results 

of input set C runs indicate that station 60 probably is the most representative for time distribution of rainfall during a storm. 

An estimate of the volume-error component for 

station 60 should be, approximately, the sum of the 

differences between the values of the objective functions for the runs of input sets B and C for the two 

stations. This follows from the fact that the B runs 

contain both volume and time errors. Therefore, 

other errors being constant, 

V6o- V477 = U6o- U477 + T477- Tao= 0.022 

+ 0.017 = 0.039 

yields an estimate for the volume-error component. 

Thus, volume errors can introduce as much as 0.04 

to U1, which is on the order of 20 percent errors. 

The compounding of the time-distribution errors of 

station 477 and the storm-volume errors of station 

60 would give a U1 value of 0.057, which leads to a 

possible combined rainfall-data-error component on 

the order of a 24-percent standard error. 

EFFECT OF SCREENED DATA 

All data used in fitting was screened for gross 

flyers, or outliers. The fitted parameters will predict 

within the indicated range of accuracy for other 

data containing the same range of errors as in the 

screened data. The screened data used for fitting 

contain the usual range of errors normally introduced. However, grossly inadequate or unrepresentative data will produce outliers well beyond the 

errors of the indicated prediction. If data are grossly 

in error, modeling results using such data should 

also be expected to be in error. 

ACCURACY OF SIMULATION FOR SANTA ANITA BASIN 

In general, accuracy of simulation of flood peaks 

for the 18 peaks used in the analysis was on the 

order of a standard error of 32 to 35 percent. Errors 

introduced by rainfall variability over the basin 

were on the order of 24 percent. Assuming that data 

errors and model errors were independent, other 

sources of error are believed to have contributed 

about the same amount to the total error. This follows from the fact that, with independence of 

errors, variances should be additive. Therefore, the 

variance contributed by data error (242 = 576) plus 

that contributed by model approximations (M) is 

equal to the total variance (342 = 1,156). To reduce errors of simulation on this basin, the rainfall 

input must be refined by the use of information 

from more than one gage or by some means of using 

estimates of areal variability in the model other than 

by the assumption of uniform-rainfall distribution, 

as is assumed in the model. 

BEETREE BASIN 

GENERAL PHYSIOGRAPHY 

Beetree Creek drains an area of 5.41 square miles 

( 14 sq km) of rough terrain near Swannanoa, N.C., 

on the western slope of the Great Craggy Mountains in the Blue Ridge province of the Appalachian 

Highlands (Fenneman, 1938). Land and channel 

slopes are steep, with elevations ranging from 2, 700 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B19 

feet (820 m) at the stream-gaging station to 5,600 sec) occurred August 13, 1940. The peak discharges 

feet (1,700 m) at the headwater-drainage divide. during the periods selected for study ranged from 

Th~ basin is approximately rectangular, having a 82 to 1,370 cfs (2.3 to 39m3 per sec), as shown .in 

ma1n channel length of about 3.2 miles ( 5.1 km) table 7. 

TABLE 7 .-Storm-period data 

and an average width of about 1.5 miles (2.4 km). 

The index of channel slope, given by the ratio of fall 

over the reach of channel from 0.1 to 0.85 of main s~!~ 

channel length, is 490 feet per mile (0.00928 ft per 

Date 

Storm 

rainfall 

(in.) 

Sample A 

Peak 

discharge (cfs) 

Surface 

runoff 

(in.) 

ft) . The predominant soil is ma:pped as "stony 

rough land of Porters soil material" and described 

as a gray-brown podzolic type derived from granite, 

gneiss, and schist (Goldstone and others, 1954). 

L----- Apr. 4, 5, 1936 ___ _ 3_...:.____ Nov. 14, 15, 1938 __ 5______ Aug. 17, 18, 1939_ 

7------ Aug. 29, 30, 1940_ 

2.08 

2.29 

2.49 

7.36 

220 

82 

236 

1,180 

0.66 

12 

.50 

4.28 

Practically all the land supports native forest, with 9______ Aug. 24, 25, 1941_ 1.22 

small areas of pasture at lower elevations. 11------ Mar. 8' 9' 1

9

42--- 1.27 

94 

151 

115 

117 

.15 

.43 

.09 

.39 

PRECIPITATION 

The Tennessee Valley Authority has operated a 

recording rain gage since 1935 at the Beetree Dam, 

4,000 feet (1,200 m) downstream from the streamgaging station. For the period 1935-59 the mean 

annual precipitation was measured as 46.4 inches 

(1.18 m) (Tennessee Valley Authority, 1961). In 

1948 new equipment was installed for the recording 

gage, and problems of calibration caused the installation of a nonrecording ·rain gage beside the recording gage. In addition, a recording gage has been 

maintained at various points in the upper area of 

the basin, as indicated in figure 11. 

Data for 40 floo<;l events that occurred during the 

period from April1936, through October 1964 were 

assembled by personnel of the U.S. Geological Survey from published records and copies of original 

recording charts. Storm-period rainfall data were 

compiled on the basis of 15-minute time intervals. 

An analysis of annual rainfall data indicated that 

an inconsistency occurred in the Beetree Creek dam 

record in 1949. A review of the history of the rain 

gage showed that a change in instrumentation was 

made in July 1948, when the originally installed 

Ferguson recording gage was replaced by a Universal recording gage. On the basis of this information, 16 flood events prior to July 1948 were selected for detailed study. 

STREAMFLOW 

The streamflow data used for fitting the model to 

the Beetree Creek basin were those for the U.S. 

Geological Survey gaging station near Swannanoa, 

N.C. The site has been gaged since 1926. The mean 

discharge during the period 1926-60 was 10.4 cfs 

( 0.29 m3 per sec) or 25 inches over the basin ( 0.64 

m). The maximum flow of 1,370 cfs (39 m3 per 

13______ Sept. 20, 21, 1944_ 1.42 

15______ Oct. 5, 6, 1945____ 2.22 

Sample B 

2______ Oct. 15, 16, 1936__ 3.08 

4______ Jan. 29, 30, 1938__ 1. 7 4 

6______ Aug. 11, 13, 1940_ 10.33 

8______ De~ 27, 28, 1940__ ~59 

10______ Feb. 16, 17, 1942__ 1.72 

12______ Dec. 29, 30, 1942__ 2.06 

14______ Mar. 26, 27, 1945_ 1.88 

16______ Feb. 10., 11, 1946__ 1.82 

EVAPORATION 

218 

167 

1,370 

263 

107 

208 

100 

141 

0.62 

.43 

4.42 

.59 

.26 

.74 

.28 

.41 

Daily values of pan evaporation were obtained 

from the Tennessee Valley Authority, which maintains a climatic station 4,000 feet (1,200 m) downstream from the gaging station, at an elevation of 

2,540 feet ( 770 m). The evaporation record has 

- been collected since 1935, and during the period 

1935-59, the average annual pan evaporation was 

39.9 inches (1.01 m). 

PARAMETER DEFINITION 

To facilitate a split-sample comparison of the results of simulation, the screened test sample of 16 

storms was divided into two sets of eight storm 

events each. To achieve an approximate balance ip 

the range in magnitude of peak-discharge rates represented in each sample, the odd-numbered events 

were selected to make up sample A, and the evennumbered events were assigned to sample B. A 

summary of the storm-period data appears in 

table 7. 

Three types of optimization were performed on 

the pre-1948 flood events. First, sample A was used 

for fitting, and optimum model parameters were 

derived to predict the events of sample B. In the 

second, sample B was used for fitting to produce a 

set of optimum parameters used to predict sample A. In the third, all 16 events were used to de-
B20 SYNTHESIS IN HYDROLOGY 

/'··--··-··-··-··-··~ Beetree Gap ......_ /

. 1934-48. .1948-50 • '. 

;· ~ 

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( • C'a~;~nob \. 

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EXPLANATION 

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Streamflow, recording station 

• 

Rainfall, recording station 

0 

Evaporation gage 

I 

I 

.) 

t 

) 

/ 

.. -··-··--............_ \ .. ........__ / ··--·· 

0 1 MILE 

0 KILOMETER 

.268 

Beetree Dam 

0 (Established 1927) 

FIGURE 11.-Beetree Creek basin, above the stream-gaging station 

near Swannanoa, N.C. 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B21 

termine the best-fit parameters for the pre-1948 

record. 

In each optimization run, a 5-week period of daily 

rainfall and pan evaporation was monitored, prior 

to the first storm event, to reduce the effect of arbitrarily initializing storage values for SMS and BMS 

(0 and BMSM, respectively). A similar lead-in period was used for all basins and for all results 

shown in this paper. In addition, initial optimization runs for all three types were started with the 

same set of initial-parameter values. These were 

assigned on the basis of ( 1) assumptions about 

average soil characteristics, (2) an estimate of the 

ratio of potential evapotranspiration to pan evaporation, and (3) the recession and timing characteristics of observed flood hydrogra phs. 

Results for the three optimization runs are given 

in table 8. Both the optimum fitted-parameter values 

TABLE B.-Results of fitting of model parameters to data 

and of split-sample testing for Beetree Creek near Swannanoa, N.C. 

[Figures in parentheses are root mean square error presented as average percentage] 

Parameter Optimum fitted-parameter value 

identifier t Sample A Sample B All storms 

SWF --·------(in.)_ 3.36 4.26 3.62 

KSAT _(in. per hr) _ 0.101 0.097 0.095 KSW ________ (hr)_ 4.97 6.24 5.67 

~l'C -------------- 0.597 0.541 0.58 

BMSM ______ (in.) _ - 1.60 1.67 1.87 

RGF -------------- 14.0 8.15 14.0 

RR --------------- 0.78 0.81 0.75 

DRN __ (in. per hr)_ 0.0050 0.0051 0.0048 

U3: 

A ------------ 0.069(27) 2 0.191 0.074(27) 

B ------------- 2 0.132 0.099(32) 0.107(33) 

All ----------- 0.101(32) 2 0.145 0.090(30) 

Test adjusted 3 _ 0.079(28) 0.098(32) --------- 1 For explanation of parameter identifier codes, see p. B5; for explanation 

of U3, see p. B9. 2 Average error not computed. 8 Peak most in error is removed from the predicted set. 

and the fitted-objective-function values are shown. 

In addition, for each set of eight peaks used for 

fitting, the remaining set of eight peaks is used as a 

test sample, and the accuracy of prediction is shown. 

An adjusted accuracy of prediction is also given, in 

which the peak value most in error is removed from 

the predicted set, to give some indication of the 

effect of extreme errors on the fitting criterion. 

RESULTS AND CONCLUSION 

The response of the objective function during two 

optimization runs is shown in figure 12. Figure 12A 

shows the response with sample A as the control 

used for estimation of parameters, and the corresponding response for the test-sample B used for 

z 

0 

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2.5 r-----,1-----,--~-----.-,----....-,----, 

EXPLANATION' 

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Control-sample A 

Test-sample B -

0 0 

o <>oo 

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NUMBER OF TRIALS 

RESPONSE OF OBJECTIVE FUNCTION DURING 

OPTIMIZATION TO CONTROL-SAMPLE A 

A 

-

80 

co 2.5 r---------..l-----,,.-------.---,------.-,-----, 

0 EXPLANATION 

Control-sample B 

2.0 '- Test-sample A 

.. . . 0 . oo •• 

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40 60 

NUMBER OF TRIALS 

RESPONSE OF OBJECTIVE FUNCTION DURING 

OPTIMIZATION TO CONTROL-SAMPLE B 

B 

-

-

-

I 

80 

FIGURE 12.-Response of objective function during optimization with the split sample for Beetree Creek basin. 

The control sample in each response is included in the 

optimization procedure, and the concurrent value of the 

objective function for the test sample is shown for 

comparison. 

independent prediction of flood peaks. Similarly, 

figure 12B illustrates the results of optimization 

with sample B used as the control. In both responses, the rate of improvement of the objective 

function for the control samples decreased markedly, 
B22 SYNTHESIS IN HYDROLOGY 

with little progress achieved after about 30 trials, 

when a plateau of best fit was noted. Rapid improvement of the objective function during the 

early stage of fitting, followed by an extended period 

of decreasing improvement, is a characteristic of 

the optimization procedure. Figure 12A shows that 

test-sample B is virtually unaffected by, and independent of, parameter adjustments made to improve 

the goodness-of-fit measured over control-sample A. 

However, figure 12B shows that the response of the 

error criterion for test-sample A is strongly related 

to that of control-sample B during the early stage 

of optimization. Eventually, the response diverges, 

becoming progressively worse after a near-optimum 

solution has been achieved for the control sample. 

The degradation of the error criterion measured 

over test-sample A (fig. 12B) can be attributed to 

the influence of episodes of low magnitude that 

produced highly variable simulated-flood runoff in 

response to small changes in the parameters associated with antecedent-moisture accounting. However, 

the variable response of these events does not appear 

to bias the parameters generated from a control 

sample in which they are included. For example, 

the results of simulation for test-sample B, using 

parameter values derived for control-sample A, 

compare favorably with the results based on optimization. Furthermore, the results of simulation for 

test-sample A are similar to those based on optimization, when the influence of those events is discounted. With the exclusion of event 9, for instance, 

the objective function for test-sample A would be 

reduced by about 50 percent and would compare 

favorably with a best-fit results of 0.069, illustrating the fact that an understanding of the distribution of error is important in evaluating the res~lts 

of optimization. 

The simulated response from the split-sample fitting and testing procedure is shown in figure 13. 

Figure 13A is a scatter diagram of observed versus 

simulated flood peaks based on optimization to sample A. Similarly, figure 13B shows the observed 

versus simulated peaks based on optimization to 

sample B. Figure 13C shows the scatter of fit, using 

all 16 events in the optimization. The distribution 

of errors is related both to the approximations and 

simplifications inherent in the hydrologic model and 

to the errors in storm rainfall, known to vary considerably throughout the area. 

The analysis of objective-function response to 

change in optimum-parameter values offers a means 

of evaluating the significance of the optimum solution and illustrates interaction between individual 

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EXPLANATION 

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EXPLANATION 

1000 1--

1-- Control-;ample B 1-- 0 

1- Test-sample A I/ 

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100 B 

I I I I II I I l l l 

EXPLANATION - . -Control-sample, 16 pre-1948 events 

17 

7 . v 

/ 

·/ 

:V . 

v .. v 

v 

v 

100 

I/, 

!/ 

/ 

1000 

1/- / 

/ 

7 

1000 

/:/ ~ 

/ 

1000 

OBSERVED PEAK DISCHARGE ,IN CUBIC FEET PER SECOND 

c 

FIGURE 18.-Scatter diagrams for simulation results in the Beetree Creek basin. In A and B, 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B23 

the results are shown for the split-sample optimization, in which the control sample used for 

the optimization and the test sample are plotted 

together for comparison. In C the results of optimization, using all 16 events with no split-sample 

testing, are shown. 

parameters and groups of parameters. However, 

the objective function will be importantly influenced 

by the nature of the events over which it is computed and may not reflect the overall significance 

of model parameters. For example, figure 14 shows 

the response of the objective function, at 5-percent 

increments from the optimum value of the parameter RR, for both control-samples A and B (RR = 

0.78 and 0.81, respectively). The plots indicate that 

optimization provided best-fit solutions for both 

samples, in that the objective function would be 

degraded by either positive or negative incrementations. However, the objective function computed 

for sample B is much less sensitive to the parameter 

RR than is that for sample A. The sensitivity of 

z 

0 

i= 

() 

z 

2 3.0 

LLI 

> 

~ LLI 

..., 

IXl 

0 2.0 

1.0 

EXPLANATION 

Control-sample A 

Control-sample 8 

FIGURE 14.-Response of the objective function to changes 

~rom the optimum value of parameter RR. Sample A 

Is much more sensitive to changes in RR, indicating 

that antecedent-soil-moisture conditions are more important in the determination of flood volumes and 

peaks for that sample. 

RR for control-sample A results from the critical 

nature of antecedent-soil-moisture conditions in determining the peak of several of the smaller storms. 

The sensitivity for sample A is highly related to 

one event. Deletion of event No. 9 has little effect 

for drier conditions (RR small), but brings controlsamples A and B into relative agreement for wetter 

conditions (RR large). Apparently too high a value 

of RR causes event 9 to be overestimated, and the 

optimum value (0.780) is a result of reducing this 

value sufficiently to estimate event 9 without reducing the accuracy of estimation of other events. 

Note that without event 9, a value of RR of 0.819 

yields a lower error for the remaining eight events 

than does the overall optimum value of 0. 78. 

The final optimization to determine best-fit parameters for the pre-1948 flood events produced an 

objective function of 0.090. Results of the optimization procedure are given in table 8 for several different test runs. With the sample of 16 events, the 

model produces a fit very similar to that achieved 

for the .smaller control samples. For example, the 

magnitude of errors in the optimum solution for all 

storm events was only 8 percent greater than the 

average of the objective functions for the controlsamples A and B. 

Inspection of objective-function sensitivity for 

each of the three control samples indicated a consistent hierarchy of parameter influence. The parameters associated with the method of antecedentmoisture accounting (RR, EVC, DRN) grossly controlled the objective function. The Philip infiltration 

parameters (SWF, KSAT) and the routing coefficient (KSW) were intermediate in importance. The 

range factor (RGF) and field-capacity-moisture 

storage (BMSM) had little influence on the objective function for the various control samples and 

may be poorly identified . 

A sufficient number of events is not the only requirement to obtain a meaningful identification of 

model parameters. Equally important is the need 

for a wide range in both antecedent and stormperiod conditions. For example, if all the flood 

events included in a control sample were associated 

with similar antecedent conditions, then one or 

more of the parameters may exert little influence 

on the results of simulation and be poorly identified, and others may be "overdetermined." In addition, an interpretation of the hierarchy of parameter sensitivity must be tempered by not only an 

understanding of the limitations of the model and 

its lack of equivalence to the physical system, but 

also by careful evaluation of the characteristics of 
B24 SYNTHESIS IN HYDROLOGY 

the criterion used to express the sensitivity. The 

response of low-magnitude events to small changes 

in some parameter values prevents a straightforward assessment of model sensitivity and demonstrates the need for development of alternative 

measures of sensitivity. 

The authors cannot overstress that in the splitsample testing for this station, eight events were 

used to determine eight model parameters. This 

clearly places this study in the area of smallsample theory. The relative consistency of results, 

both in accuracy and in derived parameter values, 

is therefore very encouraging. The various results 

of split-sample testing indicate that the root-meansquare error of prediction is about 30 percent for 

these data, with, apparently, about one small storm 

being grossly in error for each test. 

LITTLE BEAVER BASIN 

GENERAL PHYSIOGRAPHY 

The Little Beaver Creek drainage basin is a 6.41-

square-mile (16.6 sq km) area of the Gasconade 

Hills in the Ozark Mountains, just west of Rolla, Mo. 

The range in elevation is from 790 feet (240m) at 

the U.S. Geological Survey gaging station to 1,180 

feet (360 m). The gently rolling hills are covered 

with a stony porous soil. Rainfall in the southerly 

facing basin is fairly evenly distributed throughout 

the year, although the amounts are somewhat 

greater in the summer than in the winter. 

PRECIPITATION 

The U.S. Geological Survey maintains a recording 

rain gage-the Rolla 3-W gage-near the center 

of the basin. (See fig. 15.) The record obtained 

from that gage was used for simulation of rainfall 

for the entire 1948-64 period of record. In addition, 

a rain gage is maintained at the Missouri School of 

Mines and Metallurgy, about 1 mile east of the east 

boundary of the basin. The average annual rainfall 

during the period 1948-64 was 36.7 inches (0.93 m). 

Data for 29 flood events during the period 1948-

64 were reduced to rainfall intensities for 15-minute 

intervals. These storms were split into a control 

sample of 14 events during the period 1948-53 and 

a test period of 15 events during the period 1954-64. 

STREAMFLOW 

The streamflow data used for fitting the model to 

the Little Beaver Creek basin were those recorded 

at the U.S. Geological Survey stream-gaging staEXPLANATION 

0 1 MILE 

FIGURE 15.-Little Beaver Creek basin, above the streamgaging station near Rolla, Mo. 

tion near Rolla, Mo. The site has been gaged since 

1948. The mean discharge for the period of record 

1948-64 was 3.77 cfs (0.11 m3 per sec) or 11.1 

inches ( 0.28 m) throughout the basin. The maximum flow of 7,420 cfs (210 m3 per sec) occurred 

July 17, 1958. The annual peak discharges during 

the period of study varied from 524 cfs ( 15 m3 per 

sec) to 7,420 cfs (210 m3 per sec). However, individual peaks selected for the present study were as 

low as 200 cfs ( 5.8 m3 per sec) . 

EVAPORATION 

Daily values of pan evaporation, obtained from 

the U.S. Weather Bureau, were recorded at the pan 

evaporation station at Lakeside, Mo., located about 

45 miles west of the Little Beaver Creek basin and 

at an elevation of 595 feet (181m). The average pan 

evaporation during the period 1948-64 was 53 

inches (1.35 m). 

PARAMETER DEFINITION 

Three sets of model-parameter determinations 

were made, using the control period 1948-53. The 

results of these fittings plus two sets of starting 

parameters are given in table 9. The first derivation 

was of set 2 from the starting set 1. The accuracy 

of fit of 0.065 gives a standard error of fit of about 

25 percent. The value of RR of 0.98 seemed to be 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B25 

TABLE 9.-Results of fitting of model parameters to data for Little Beaver Creek near Rolla, Mo., using the Rolla 3-W rain gage 

Start Parameter Optimum Start 

indicator 1 Set 1 Set2 Set3 Set4 

SWF -------------(in.) __ 2.0 2.5 10.1 4.0 

KSAT ------(in. per hr) __ 0.1 0.08 0.07 0.05 KSW _____________ (hr) __ 1.0 21.0 21.0 0.85 

EVC ------------------- 0.7 0.56 2 0.56 0.55 

BMSM -----------(in.) __ 2.0 2.8 2.3 3.0 

RGF ------------------- 10.0 9.4 9.3 10.0 

RR --------------------- 0.8 0.98 2 0.8 0.85 

DRN _______ (in. per hr) __ 0.1 0.02 0.02 0.025 

U1 --------------- 0.065 0.075 0.061 

Standard error (percent) ------- 25 27 25 

1 For explanation of parameter identifier codes, seep. B3-B5; for U1, seep. B9. 2 Parameter values held constant for the run indicated. 

Optimum Test 

(Set 5 

Set5 parameters) 

4.1 4.1 

0.047 0.047 

0.84 0.84 

0.52 0.52 

2.4 2.4 

11.7 11.7 20.85 0.85 

0.022 0.022 

0.055 0.073 

23 27 

high and was believed to be too much of a curvefitting parameter. Therefore, set 3 was derived by 

fixing the evaporation pan coefficient (EVC) at its 

optimum value and the daily-rainfall-infiltration coefficient (RR) at 0.8. A lower limit for RR should 

be 0.7, because the mean annual flow is about 30 

percent of the mean annual rainfall. Therefore, 0.8 

to 0.85 is a reasonable value. The accuracy of fit 

for the parameters for set 3 is 0.075, or about 27 

percent. 

A separate fitting for the Little Beaver Creek 

basin was made to the Missouri School of Mines and 

Metallurgy rain gage, which lies outside the basin. 

The results of the fitting are given in table 10. A 

comparison of rainfall volumes for the two gages 

and of the simulated volumes and peaks is shown in 

table 11. 

On the basis of hydrograph plots for the results 

of set 3, the routing component was recomputed. 

Both the time-area histrogram and the surface routing coefficient (KSW) were revised, and KSW was 

included in the next optimization run. RR was held 

:fixed at 0.85. The fit of set 5 is 0.055, which yields 

about a 23-percent accuracy. The test group of 15 

floods during the period 1954-64 were then simulated with set 5 parameter values. The accuracy. of 

fit for the test-set was 0.073, which yields an estimate of 27 percent for a standard error of prediction. 

TABLE 10.-Results of fitting of model parameters to data for 

Little Beaver Creek near Rolla, Mo., using the Missouri 

School of Mines and Metallurgy.rain gage 

Parameter 

indicator 

SWF -----------------------(in.) __ KSA T ----------------(in. per hr) __ KSW ________________________ (hr) __ 

EVC ------------------------------

BAISM ----------------------(in.) __ 

RGF ------------------------------

RR -------------------------------- DRN ------------------(in. per hr) __ 

U1 (13 events)---------------- Standard error _____ (percent) __ 

U1 (9 events) ________________ _ 

Standard error _____ (percent) __ 

1 Parameter value held fixed for the run. 

Start 1 

4.0 

.05 

.85 

.55 

3.0 

10.0 

.85 

.025 

0.21 

46 

0.121 

35 

TABLE 11.-Comparison of estimates for flood volumes and peaks for Little Beaver Creek 

by the use of the two rain gages 

Rolla3-W Missouri School of Mines and 

Measured rain gage Metallurgy rain gage 

Simulated Measured Simulated 

Measured 

Runoff Peak 1 RF Runoff Peak 1 rainfall Runoff Peak 1 

Date (in.) (cfs) (in.) (in.) (cfs) (in.) (in.) (cfs) 

6-17-48 ------------------- 0.12 376 1.17 0.13 351 0.76 0.21 545 

6-2-49 2 1.05 1,228 2.59 .82 1,328 .89 7-22-49 _:::::::::::::::::: .33 1,199 1.21 .47 1,253 .85 .47 1,247 

10-11-49 ------------------ 2.76 3,121 4.26 2.90 2,848 8 6.05 4.64 3,589 

10-20-49 ------------------ .55 1,142 1.33 .79 1,639 .95 .54 1,321 

1-13-50 ------------------- .64 1,348 1.03 .38 990 1.10 .49 1,124 

4-10-50 ------------------- .24 811 .85 .37 1,053 .88 .52 1,392 

5-19-50 ------------------- 1.06 1,575 1.85 .94 1,446 1.77 .92 1,145 

5-26--50 ------------------- .25 742 1.34 .43 1,167 8 .48 .15 406 

6-9-50 -------------------- 1.78 4,177 3.36 1.73 3,683 8 2.01 1.33 2,461 

6-22-51 ------------------- .31 848 1.16 .38 979 1.08 .58 1,338 6--30-51 1.64 2,079 2.40 1.35 1,514 2.67 1.69 2,218 4-23-53 ::::::::::::::::::: .73 2,054 1.56 .74 1,829 3 2.66 2.19 5,380 

5-17-53 ------------------- .15 416 .56 .10 301 .38 .11 308 

1 Peak rates are surface-runoff rates only; base flow has been subtracted from the measured rate. 2 Not included in the Missouri School of Mines and Metallurgy optimization because the measured storm runoff exceeded the measured rainfall. 8 Missouri School of Mines and Metallurgy gage storm rainfall apparently grossly in error. 

Optimum 2 

1.75 

.063 

.97 

.39 

2.2 

8.0 

1 .85 

.038 

0.19 

44 

0.099 

31 
B26 SYNTHESIS IN HYDROLOGY 

Two conclusions can be drawn from this second 

fitting. First, the School of Mines gage is not an 

adequately representative measure of rainfall on 

the basin, even though it is. just outside the basin. 

The growth of trees in the vicinity of the gage 

created an increasing amount of interference effects. 

In 1956 the gage was moved 100 feet to the south 

to correct the problems of measurement caused by 

the trees. The accuracy of fit is 44 percent. Five 

measured storm volumes are grossly different from 

those measured for Rolla 3-W. One of the five was 

excluded from the fitting, but the other four influenced the fitting and probably caused the higher 

value of KSW and reduced volu1nes of infiltration. 

However, the School of Mines gage does given some 

indication of the effect of variability of storm volume over the basin. For nine of the 14 storms, the 

Rolla 3-W gage· simulation overestimated peaks 

when its measured storm volume exceeded that at 

the School of Mines. This held true for seven of the 

nine peaks above 1,000 cfs, and for nine of the 12 

above 500 cfs. Therefore, although the School of 

Mines gage alone gives much less accurate results 

than those for Rolla 3-W, the two sets of results 

used together could give a better estimate for floodpeak simulation. 

COMPARISON OF DERIVED PARAMETER 

VALUES 

The model is based, at least in part, upon a simulation of the physical processes operating upon the 

basin modeled. The parameter values derived should 

therefore be related to the physical parameters involved. However, the model is a bulk-parameter 

model-that is, it models all the infiltration in the 

basin as if it were uniform over the basin. The 

parameter values derived are in some way optimal 

average values and can be, at best, indices to the 

"true" parameters or to their distribution over the 

basin. 

If the model is to be used in regional studies, it 

can serve either of two purposes : First, it can be 

used to extend a record in time. For that use, the 

most important consideration is the error of prediction. For the three basins for which results are 

presented in this study, a standard error of prediction of about 30 to 35 percent was achieved. This 

was found to be largely dependent upon the accuracy of rainfall measurement. In particular, the use 

of a single rain gage to estimate rainfall variability 

over the basin seems to introduce an error of about 

20 to 25 percent into the simulation. A decision must 

be made as to whether the point rainfall data that 

produce errors of this magnitude add information 

to the record. Second, the model can be used in 

regional studies by relating the derived parameter 

values to physical characteristics measurable in the 

basins which are simulated. The deriv~d relations 

could then be used to estimate parameter values for 

ungaged sites. The accuracy of prediction in this 

use would be a function both of the errors in rainfall input and of the errors in predicted values for 

the model parameters. This accuracy of prediction 

would be compared to the accuracy of floodfrequency methods presently in use. 

The derived parameter values for the three basins 

used in 'this developmental study are shown in 

table 12. All are reasonable values. However, there 

TABLE 12.-Summary of results of optimization for the three study basins 

Basin 

Parameter 

indicator Santa Anita Beetree Little Beaver 

Creek Creek Creek 

SWF --------(in.) __ 20 3.6 4.1 

KSAT __ (in. perhr) __ .32 .1 .05 KSW ________ (hr) __ 2.7 5.7 .84 

EVC --------------- .73 .58 .52 

BMSM _______ (in.) __ 3.5 1.9 2.4 

RGF --------------- 6 14 12 

RR ----------------- 1.0 .75 .85 

DRN ___ (in. per hr) __ .058 .005 .022 L ____________ (rni) __ 4.7 3.2 3.25 

s _______ ( ft per ft) -- .12 .00928 .0124 

L yS --------------- 13.5 33.5 29.3 

1- (RO/RF) ------- .74 .46 .7 

are too few results to draw any general conclusions 

at this time. Each parameter will be discussed as 

to its relation among stations and the reasons for 

variability. RR is a measure of percentage of infiltration for daily rainfall amounts for periods not 

simulated in detail, either because rainfall amounts 

are too small or because records are not accurate 

enough to use for detailed simulation. Also shown 

in table 12 are values of 1 minus the ratio of measured runoff to measured rainfall for each basin 

during the study period. This sets a lower limit on 

RR, and for each basin the fitted value exceeds this 

lower limit. Actually, the lower limit should be 

somewhat higher, because all base flow should be 

subtracted from the runoff to derive the limiting 

value. Beetree Creek basin has the highest base 

flow; thus, the fitted value exceeding the limiting 

value by a relatively large amount is consistent. 

KSAT, SWF, and RGF determine the infiltration 

equation during detailed storm simulation and, 

therefore, should be discussed together. SWF determines the soil-suction characteristics for wet conditions, SWF multiplied by RGF determines them 
SIMULATION MODEL FOR ESTIMATION OF FLOOD PEAKS FOR SMALL DRAINAGE BASINS B27 

for dry conditions, and KSAT represents the soil's 

saturated permeability, or minimum infiltration 

rate. The range of soil suction is from 4 to 50 inches 

(10 to 125 em) for both Beetree Creek and Little 

Beaver Creek basins, and is from 20 to 120 inches 

(50 to 300 em) for Santa Anita Creek basin. Comparable experimental ranges for a sandy loam are 

about 30 to 130 centimeters for Yolo sandy loam, 

and 30 to 200 centimeters for Yolo silt loam (Colman and Bodman, 1944). Seemingly, the minimum 

infiltration rates are anomalous for the measured 

basins, in that 0.3 inches per hour seems to correspond to a sandy loam rate, whereas 0.05 to 0.10 

inches per hour seems to correspond to a rate for a 

silt loam (Musgrave, 1055). Some attempts should 

be made to relate the fitted values to ring infiltrometer or other data collected for study basins. 

BMSM represents an effective maximum soilmoisture retention, and the low values indicate shallow soils. Of the three study basins, Beetree Creek 

basin appears to have the thinnest effective soil 

mantle, and Santa Anita Creek basin, the least shallow. This agrees qualitatively with descriptions of 

the geology and soils. DRN represents the drainage 

rate from the saturated layer to the unsaturated 

layer. This parameter is critical for determining the 

antecedent conditions for some storms, but has no 

effect on most storms. Therefore, it is probably 

poorly defined for all basins. The derived values are 

considerably less than KSAT in each basin (which 

is as expected), but nothing can be said as to the 

reasonableness of the values otherwise. 

EVC should represent an effective average pan 

coefficient for the basin. However, this meaning is 

compounded by the fact that for each basin a correction also must be made to adjust the pan evaporation to average basin conditions. For Little Beaver 

Creek basin, the nearest pan evaporation record was 

45 miles away; for Santa Anita Creek basin it was 

10 miles away; for Beetree Creek basin the evaporation record was nearby, but at a lower elevation. 

All r~cords are for U.S. Weather Bureau Class A 

pans, for which the pan coefficient should range 

from 0.6 to 0.8. EVC should be somewhat lower than 

these values, if an altitude correction is involved. 

Little or no altitude correction should be necessary 

for Santa Anita Creek basin, ·as the pan is at an 

elevation well above the lowest point in the ba~in. 

Both of the other records are for sites at elevations 

below the lowest point in the basin, and for Little 

Beaver Creek basin, considerably lower. Therefore, 

the derived values seem to be of the right order of 

magnitude. 

Neither the hydrograph recession rate (KSW) 

nor the translation hydrograph ordinates enter directly into the fitting process, as both are derived 

from the measured hydrograph shapes. The Little 

Beaver basin has an unusually rapid recession. 

Values of L!VS are shown, where L is the length 

of the main channel, in miles, and S is the slope 

of the basin, in feet per foot, for the reach from 

10 percent to 85 percent of the distance from the 

discharge gaging station to the point on the ridge 

that represents the extension of the main channel 

(Benson, 1962). Although L!VS values for Beetree 

Creek and Little Beaver Creek basins are very 

.similar, the values of KSW differ by a ratio of 7. 

Santa Anita Creek basin is consistent with Beetree 

Creek basin in this regard, in that both L!VS and 

KSW are about half the values for Beetree. The 

reason for the anomalous value for Little Beaver 

Creek basin is unknown, but it may be related to 

the drainage pattern. Both Santa Anita Creek and 

Beetree Creek basins are dendritic, whereas Little 

Beaver Creek basin seems to be more palmate. 

SOURCES OF ERROR AND THEIR IMPACT 

The accuracy of fit for the three basins studied 

was similar. An accuracy of about a 30-percent 

standard error is obtainable. The detailed study for 

Santa Anita Creek basin indicated that about a 

20-percent standard error was attributable to rainfall sampling alone. If the rainfall errors are independent of other modeling errors, then 

RFJ2 + ME2 = TFJ2, 

where RE is the modeling error resulting from 

rainfall-input error, ME is other modeling error, 

and TE is the total error of simulation. For the 

Santa Anita Creek basin, 

202 + MFJ2 = 302 ; ME2 = 500. 

According to Eagleson (1967), if one rain gage 

gives an error of 20 percent, then two properly 

placed rain gages would give an error of about 15 

percent. The use of the information from two gages 

with the present model structure should thus result 

in an error of 

TE2 = 152 + 500 = 725, 

or a standard error of 27 percent, rather than 30 

percent. 

The improvement of the structure of the model 

can also lead to more accurate prediction. If the 
B28 SYNTHESIS IN HYDROLOGY 

model error were cut in half, the resulting standard 

error would be 

TE2 = 2()2 + 250 = 650, 

or a standard error of 25 percent. Thus, to achieve 

any major improvement in the accuracy of simulation, the improvement in both the model and the 

accuracy of rainfall input must be simultaneous. 

Model improvements alone will increase the accuracy of prediction, but there will be a limiting 

accuracy which must be accepted if the constraint 

of a single rain gage is to be maintained. 

The marginal gains in accuracy which should be 

expected from model improvement influence the 

strategy for judging model improvements. Changes 

should be accepted as improvements if they (1) 

add to the simplicity of the model, (2) aid in the 

regionalization of the parameter values, or (3) 

gain accuracy. The search will continue for a better 

model, but, to date, an imperfect model must be 

accepted. 

CONCLUSIONS 

The development of the model demonstrates the 

feasibility of rainfall-runoff simulation. Such simulation is not new, so that such a demonstration of 

feasibility is not unexpected. However, the constraints placed upon the model developed were that 

a single rain gage be used for simulation on a basin. 

This led to the development of a bulk-parameter 

model. Thus, model parameter values are indices of 

average conditions on the basin that only approximate real parameter values. Both the errors of rainfall input and the lack of model equivalence to the 

physical prototype limit the prediction ability of 

simulation. These two sources of error are of similar 

order of magnitude for the basins studied ; hence, 

major gains in accuracy will depend upon simultaneous improvement in both. The limit of accuracy 

of prediction of flood peaks by simulation with a 

single rain gage seems to be on the order of about 

25 percent, and this level of accuracy should be 

understood to have resulted from the imposed constraint. 

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Survey Water-Supply Paper 1580-B, 64 p. 

---1964, Factors affecting the occurrence of floods in 

the Southwest: U.S. Geol. Survey Water-Supply Paper 

1580-D, 72 p. 

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Benson, M. A., and Matalas, N. C., 1967, Synthetic hydrology based on regional statistical parameters: Water 

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Clark, C. 0., 1945, Storage and the unit hydrograph: Am. 

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Philip, J. R., 1954, An infiltration equation with physical 

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505. 

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