Optimal Determination of Loss Rate Functions and Unit Hydrographs

WATER RESOURCES RESEARCH, VOL. 20, NO. 2, PAGES 203-214, FEBRUARY

1984

OptimalDeterminationof Loss Rate Functionsand Unit Hydrographs 0LCAY0NVERANDLARRYW. Mays Department of Civil Engineering, University of Texas at Austin An optimization model is presentedthat can be used in the determinationof lossrate functionsand unit hydrographsusingobservedrainfall and runoff data. Compositeunit hydrographsand lossrate parameterscan be determinedby usingseveralmultiperiodstormssimultaneously or usingindividual multiperiodstorms.The model is a nonlinearprogrammingproblemso that a generalizedreduced gradientmethodis usedas the solutiontechnique.Kostiakov's,Philip's, Horton's, and the •bindex methodsare used to illustrate the model for comparativepurposes.An infiltration equation that includesrainfallintensityis alsointroducedandis comparedwith theothersby usingthe optimization model. The model is appliedto stormsfor a hypotheticalexample, ShoalCreek watershedin Austin, Texas, and Wills Creek watershednear Cumberland,Maryland. A comparisonof the new modelwith earlier optimization proceduresis also presented.

iNTRODUCTION

advance by using an assumed loss rate function. Some of

The use of infiltration equations or loss rate functions is somewhathampered in hydrology practice owing to limited means of efficiently determining the values of the parameters for these functions. The hydrologist commonly analyzes actual storm events and the resulting runoff to determine lossesand expressthese lossesin equationform. At best this is a trial and error procedure consideringvarious values of the parameter(s) of the particular loss rate function. The infiltration-loss

rate

functions

that

are considered

could

range from the well-known •b index method or Horton's infiltration capacity equation to the less commonly used methods such as Kostiakov's equation or Philip's equation. Usually very little effort is given to determine the infiltrationloss rate function that is most appropriatefor the application being considered. This paper develops a procedure based upon nonlinear programming (NLP) to determine optimal loss rate functions for watersheds using data from one or more rainfall-runoff

events.

Optimal loss rate functions are determined as well as unit hydrographs,which continue to be one of the most practical tools available to hydrologists. Because of the widespread use of loss rate functions and unit hydrographs,procedures that can provide their optimal determination are desirable. Both the lossrate and unit hydrographare directly related to rainfall-runoff events for a particular watershed. Determining the loss rate function and unit hydrograph by using recorded

data

is made

even

more

difficult

because

the

hydrologist must consider (and usually simplify) complex multipeakedrunoff hydrographsresulting from multiperiod rainfalls. The processis complicatedeven more becausethe loss rate function and parameter values and the unit hydrograph can vary for different storms on the same watershed. Several methods have been reported in the literature for determiningoptimal unit hydrographs,usingtechniquesthat range from successiveapproximationto least squaresand optimization techniques [Bender and Roberson, 1961;Deininger, 1969;Newton and Vinyard, 1967; $ingh, 1976;Eagleson et al., 1966; Diskin and Boneh, 1975; Bree, 1978; Mawdsley and Tagg, 1981; Boorman and Reed, 1981]. All

thesetechniques requirethattherainfallexcess bedefined in

thesemodelscan considerseveralmultiperiodstormssimultaneously.Mays and Coles [1980] presenteda linear programmingmodel that can be used to derive a compositeunit hydrographby consideringseveralmultiperiodstormssimultaneouslyfor specifiedrainfall excess values. Mays and Taur [1982] presented a new model based upon nonlinear programmingthat can be used to derive a compositeunit hydrographand to determine the rainfall lossesfor each time period without an assumed loss rate function. Several multi-

period rainfall events can be consideredsimultaneouslyin this model. The word compositeis used to denote that the unit hydrographis basedupon more than one rainfall-runoff event. This model is based upon determiningthe rainfall lossesandthebestcompositeunit hydrographby minimizing the deviations between observed and derived direct surface

runoff hydrographs. The new model described herein is an extension of the

Mays and Taur model to determine the optimal parameter valuesof lossrate functionsand the optimalcompositeunit hydrograph. This model can be used to determine the optimal parameter values for infiltration models while at the sametime determiningthe optimalunit hydrographby using rainfall-runoff

data. NLP

MODEL

The NLP model developedby Mays and Taur [1982] is stated as follows: i

N,

Min Z0 = •

•

i=1

(Zi,n• + Vi,no)

(1)

n=l

subjectto the following nonlinear and linear constraints

(Rim- Hi,n)Ul + (Ri,n-I - Hi,n-l)e2 + ' ' ' +

(Ri,n-m+l- Hi,n-m+l)Umq- Zi,n - Vi,n= Qi,n i=

Ri,.-

1,''',I

Hi,n • 0

n=

(2)

1,''',Ni

i = 1,' ' ', I

n = 1, ß ß., Ni (3)

Lt

•] (Ri,n- Hi,n)= Di

i = 1,...,

I

(4)

n=l

Copyright1984by the AmericanGeophysicalUnion. M

Paper number 3W 1823. 0043-1397/84/003 W- 1823505.00

(5) m=l

2O3

204

0NVER AND MAYS: OPTIMAL DETERMINATION OF LOSS RATE FUNCTIONS

and the non-negativity constraintsfor the following decision

where K in (7) is the coefficient to convert the direct runoff

variables:

hydrographvolume (ft3/s x time) to a volumein cubic inches.

Hi,n, Zi,n, Vi,n i=

1,''

',I

n-

1,.'

Um •> O ',Ni

rn-

(6)

SOLUTION TECHNIQUE

1,.''.,M

The optimization model, equations(1)-(6), is a nonlinear programming model with both linear and nonlinear constraintsand with either a linear (/• = 1) or nonlinear(/• :/: 1) z0 objective function; Zi,n deviation of the derived direct runoff hydrograph objectivefunction, requiring a nonlinearprogrammingalgoordinateQi,nbelowthe observeddirectrunoffhydro- rithm. The three most promising approaches for solving graph Qi,nfor the nth ordinateof the ith observed nonlinear problems are generalized reduced gradient, successivelinear programming, and successivequadratic hydrograph; deviation of the derived direct runoff hydrograph programmingalgorithms[Lasdon, 1981].Each techniquehas its advantagesand disadvantages,and no one techniquehas ordinate above the observed direct runoff hydroproven to be the best for all kinds of nonlinearproblems. A graph ordinate; detailedexplanationand comparisonof these algorithmscan a constant; be found in the work of Lasdon [1981] and Lasdon [1982]. R total rainfall over a At hour period; They can easily be adapted to large-scale nonlinear probH rainfall lossesover a At hour period; lems, which may be necessarywhen several rainfall-runoff U unit hydrograph ordinate; events are considered simultaneously at reasonably small D volume of direct runoff; time intervals. At time interval between hydrograph ordinates, which The solution algorithm used in this study is a generalized can also be equal to the unit hydrograph duration, reduced gradient technique called GRG2 [Lasdon et al., where i refers to the observed hydrograph; 1978]. Lower and upper bounds on decision variables are I total number of observed hydrographs; quite easy to handle with GRG2. The GRG2 requires a userordinate number of each observed hydrograph; suppliedsubroutineGCOMP for the purposeof computing total number of ordinates of the ith observed hydrothe constraint and objective function for values of the graph; decision variables. GCOMP can also be used to read in initial tn unit hydrograph ordinate; values of any user-required constants.GRG2 is a modular total number of unit hydrograph ordinates. programwritten to provide dynamic memory allocationwith Whenflow ratesare expressed in ft3/s,the constantK is all arrays set up as portions of one large main array so that expressed as redimensioning of arrays is never required. The GRG2 requiresapproximately 120 K bytes of core on an IBM 370/ 12(3600) At

in which

K =

(7)

(5280)2 A

where At is the time interval in hours between hydrograph ordinates (also the unit hydrograph duration) and A is the

drainageareaof the watershedin mi2. Whenflowratesare expressedin in/h, then K = At. The objective function (1) is to minimize the deviations

betweenthe deriveddirectrunoffhydrograph ordinates Q and the observed direct runoff hydrograph ordinates Q. The deviationsare definedsuchthat if Zi,n > 0, then Vi,n= 0, and

if Vi,n > 0, then Zi,n = 0. The exponent/• in the objective function is equal to 1 if a linear objective equation is desired and 2 if a least squarestype optimization is desired. Constraint equation (2) is derived from the relationshipbetween

158 [Lasdon et al., 1980].

The NLP model formulation given herein includes deviationsZ and V to consideronly positive values. It shouldbe pointedout that this is not necessary,becauseGRG2 allows for both positive and negative values of decision variables. However, many of the nonlinear optimization proceduresdo not have this capability, so the authorschoseto develop and utilize the more general formulation given. NLP

MODEL

WITH

INFILTRATION/LOSS

RATE FUNCTIONS

The rainfall loss,Hcnfor the nth period(At - hr) of the ith rainfall event can be expressedas

Hi,n = Fi,n - Fi,n- I =

Q andQ, expressed as

ftn

f (t)dt

(11a)

tn- I

Qi,n = Qi,n + Zi,n-

Vi,n

(8)

with

if Ri,n >

f (t)dt

n-I

Qi,n = (Ri,n - Hi,n)el q- (Ri,n-I - Hi,n-I)e2

+ ' ' ' + (Ri,n-m+I - Hi,n-m+I)Sm

Itt11

(9)

Constraint equation (3) states that the infiltration amount cannot exceed the total rainfall in any time period. Constraint equation (4) equatesthe direct runoff volume Di to the total precipitation excessfor the ith rainfall event. The direct runoff volume for the ith rainfall event is expressedas

Hi,n = Ri,n if Ri,n -