Determination of an optimal unit pulse response function

using real-coded genetic algorithm

Ashu Jaina,*, Sanaga Srinivasalub, Rajib Kumar Bhattacharjyaa

aDepartment of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208 016, IndiabCenter for Spatial Information Technology, Institute of Post Graduate Studies and Research, Jawaharlal Nehru Technological University,

Hyderabad 500 028, India

Received 20 November 2002; revised 9 July 2004; accepted 30 July 2004

Abstract

This paper presents the results of employing a real-coded genetic algorithm (GA) to the problem of determining the optimal

unit pulse response function (UPRF) using the historical data from watersheds. The existing linear programming (LP)

formulation has been modified, and a new problem formulation is proposed. The proposed problem formulation consists of

fewer decision variables, only one constraint, and a non-linear objective function. The proposed problem formulation can be

used to determine an optimal UPRF of a watershed from a single storm or a composite UPRF from multiple storms. The

proposed problem formulation coupled with the solution technique of real-coded GA is tested using the effective rainfall and

runoff data derived from two different watersheds and the results are compared with those reported earlier by others using LP

methods. The model performance is evaluated using a wide range of standard statistical measures. The results obtained in this

study indicate that the real-coded GA can be a suitable alternative to the problem of determining an optimal UPRF from a

watershed. The proposed problem formulation when solved using real-coded GA resulted in smoother optimal UPRF without

the need of additional constraints. The proposed problem formulation can be particularly useful in determining the optimal

composite UPRF from multiple storms in large watersheds having large time bases due to its limited number of decision

variables and constraints.

q 2004 Elsevier B.V. All rights reserved.

Keywords: Unit pulse response function; Unit hydrograph; Non-linear optimization; Linear programming; Real-coded genetic algorithm;

Simplex method; Rainfall–runoff modeling

0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2004.07.014

* Corresponding author. Tel.: C91 512 259 7411; fax: C91 512

259 7395.

E-mail address: ashujain@iitk.ac.in (A. Jain).

1. Introduction

One of the key components of a water resources

management or a design activity, such as flood control

and management, or design of any hydraulic structure,

is a mathematical model of the rainfall–runoff process

of a watershed. The primary interest of a hydrologist

Journal of Hydrology 303 (2005) 199–214

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A. Jain et al. / Journal of Hydrology 303 (2005) 199–214200

handling such a problem is to determine the response

from a watershed in the form of a direct runoff

hydrograph (DRH) when it is subjected to a rainfall

event. The transformation of the effective rainfalls into

the DRH can be mathematically modeled using a

‘kernel function’ or a ‘transfer function operator’. The

most commonly used ‘transfer function’ of a watershed

is the unit pulse response function (UPRF) also known

as a unit hydrograph (UH). The theoretical concept of

the UH was first proposed by Sherman (1932). A D-h

UH is defined as the DRH response at the outlet of a

watershed when it is subjected to a unit effective

rainfall (1 in. or 1 cm) occurring in a specified duration

(D h) such that the distribution of the effective rainfall

is uniform over space and time. The UH theory of

Sherman is based on two important assumptions of

linearity and time invariance. Knowing the UPRF of a

watershed for a specified duration, it can be used to

obtain the DRH response at the outlet of a watershed

from a single storm event of the specified effective

duration or multiple storm events in which each

effective rainfall impulse has duration equal to the

specified duration. This can be accomplished using the

discrete form of the convolution equation as follows

(Chow et al., 1988)

Qn ZXn%M

mZ1

PmUnKmC1 (1)

where Qn is the DRH ordinate at a discrete time step n,

Pm is the effective rainfall impulse at a discrete time

step m, and UnKmC1 is the ordinate of the UPRF at any

discrete time step nKmC1. On the other hand, when

effective rainfall impulses (Pms) and DRH ordinates

(Qns) are known, then the above equation can be used

to determine the ordinates of the UPRF through a

reverse process. This reverse process of determining

the ordinates of the UPRF of a watershed is sometimes

referred to as the de-convolution process. If the number

of effective rainfall impulses is M and the number of

DRH ordinates is N, then there will be NKMC1

ordinates in the UPRF of the watershed. It must be

noted that Eq. (1) represents an over-determined

system of simultaneous linear equations i.e. the

number of equations (N) is more than the number of

unknowns (NKMC1) in the de-convolution process.

Many methods have been proposed to solve the

system of simultaneous linear equations represented

by Eq. (1) to determine the UPRF. All of the methods

can be classified into four different groups: (1) Method

of Successive Substitution (Chow et al., 1988) has the

disadvantage of not being able to use all the data that

are available. (2) Method of Unconstrained Optimiz-

ation. This method is capable of providing a

representative UPRF considering all the data that are

available. The method of least squares falls under this

category. Some other examples of this method include

the method of successive approximations (Collins,

1939; Barnes, 1959; Bender and Roberson, 1961;

Bruen and Dooge, 1984). (3) Method of System

Transformation. In this method, a Fourier transform is

applied for the solution of the problem. The Fourier

method proposed by O’Donnel (1960), and Meixner

method proposed by Dooge and Garvey (1978) fall

under this category. The major disadvantage of the

above three methods is that they may produce a

solution containing negative UPRF ordinates, which

is not acceptable in water resources applications. (4)

Constrained Optimization Method. In this method, an

objective function is formulated using a measure of

the errors between observed and computed DRHs and

then the error function is minimized under certain

constraints. The problem of negative UPRF ordinates

is overcome by posing the non-negativity constraints

on the decision variables. The constraints in the

optimization problem formulation for determining an

optimal UPRF normally include the set of discrete

equations represented by convolution Eq. (1),

constraint of the unit effective rainfall, and the non-

negativity constraints. Sometimes, additional con-

straints are imposed on the UPRF ordinates after the

peak in order to ensure a smoother UPRF. Many linear

programming (LP) formulations and their solutions

have been proposed (Deininger, 1969; Singh, 1976;

Mays and Coles, 1980; Morel-Seytoux, 1982; Singh,

1988). Mays and Taur (1982) presented a non-linear

programming approach for unit hydrogen determi-

nation. More recently, Yue and Hashino (2000) have

proposed a new approach for deriving the UPRF for

quick and slow runoff components of streamflow by

simulating a watershed with three tanks in series and

one tank in parallel.

This study focuses on determining an optimal

UPRF of a watershed through the de-convolution

process using constrained optimization method. The

past attempts at determining an optimal UPRF of

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 201

a watershed have focused on classical optimization

techniques, mainly the LP approach, as mentioned

above. It has been reported that the LP methods

involve many unnecessary decision variables and

constraints, the number of storms may need to be

limited in determining optimal composite UPRFs, and

additional constraints may be needed to ensure

smooth UPRFs (Mays and Coles, 1980). Recently,

genetic algorithms (GAs) have been successfully

employed to solve many optimization problems in

hydrology and water resources. The GAs can over-

come some of the problems associated with LP

methods, but the attempts of determining an optimal

UPRF of a watershed using GAs have been limited.

Further, while developing mathematical models of a

physical system, one needs to evaluate the perform-

ance of the models using certain statistical par-

ameters. Normally, a few standard statistical

measures e.g. root mean square error (RMSE) or its

variations are employed. In order to select the best

model among all the models developed, a wide variety

of statistical parameters need to be considered that are

capable of assessing the efficiency of the models in

estimating different characteristics of the physical

system being modeled.

The objectives of the present study are to: (a)

solve the existing LP formulation for determining an

optimal UPRF using real-coded GA, (b) develop an

optimization problem formulation to have a fewer

decision variables and constraints for ease in solution

procedure, (c) solve the proposed optimization

problem formulation using real-coded GA, and (d)

evaluate the performance of all the models using a

wide variety of statistical parameters. The proposed

optimization formulation, which consists of an

objective function that is non-linear in nature, can

be used to determine an optimal UPRF using data

from an individual storm or an optimal composite

UPRF using data from multiple storms. The penalty

function approach was used to enforce constraints

while solving the optimization problem formulations

using real-coded GA. The proposed methodologies

have been tested using the rainfall and runoff data

from two different watersheds. The results obtained

in this study are compared with those reported earlier

by others using LP methods for the same data sets.

The paper begins with a brief description of the real-

coded GA.

2. Real-coded genetic algorithm

The GA is a search technique based on the concept

of natural selection inherent in the natural genetics,

which combines an artificial survival of the fittest with

genetic operators abstracted from nature (Holland,

1975). The major difference between GA and the

classical optimization search techniques is that the

GA works with a population of possible solutions;

whereas, the classical optimization techniques work

with a single solution. An individual solution in a

population of solutions is equivalent to a natural

chromosome. Like a natural chromosome completely

specifies the genetic characteristics of a human being,

an artificial chromosome in GA completely specifies

the values of various decision variables representing a

decision or a solution. For most GAs, the candidate

solutions are represented by chromosomes coded

using either a binary number system or a real decimal

number system. The GA that employs binary strings

as its chromosomes is called the binary-coded GA;

whereas, the GA that employs real valued strings as its

chromosomes is called the real-coded GA. The real-

coded GAs offer certain advantages over the binary-

coded GAs as they overcome some of the limitations

of the binary-coded GAs (Deb and Agarwal, 1995;

Deb, 2000). Regardless of the coding method used,

the GA consists of three basic operations: selection,

crossover or mating, and mutation.

2.1. Selection

The GA starts with randomly generating an initial

population of possible solutions (chromosomes).

These chromosomes are evaluated based on their

performances (fitness values) in terms of certain

objective function. Then the chromosomes compete

for survival in a tournament selection, in which one

parent is selected having the best fitness value among

two or more randomly picked chromosomes. A second

parent is selected by repeating the same process. This

process of the selection of individual chromosomes

based on their relative fitness is called natural

selection. The tournament selection process can be

used to carry out the selection operator (Goldberg and

Deb, 1991). The chromosomes compete for survival in

a tournament selection, where the chromosomes with

optimal fitness values enter the mating population

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214202

and the remaining ones die off. The selected chromo-

somes form what is known as the mating population on

which the crossover operator is applied.

2.2. Crossover

In applying the crossover operator, the genetic

information to the right of the random crossover

location in a chromosome is simply swapped between

the two parents to create children solutions in the

binary-coded GA. However, in real-coded GAs, the

implementation of crossover operator is not that

simple. Deb and Agarwal (1995) developed Simulated

Binary Crossover (SBX) operator, which simulates

the principle of the single point crossover to create

offspring from the mating population of solutions. The

procedure for computing offspring from two parents

using SBX operator is explained here in brief. First, a

random number (say ui) between 0 and 1 is created.

Then, from a specified probability distribution func-

tion, the ordinate f(ui) is found such that the

cumulative probability for ordinate f(ui) is equal to

ui. After finding the ordinate f(ui), the offspring are

calculated using the following equations

C1 Z 0:5½ð1 C f ðuiÞÞP1 C ð1 K f ðuiÞÞP2� (2)

C2 Z 0:5½ð1 K f ðuiÞÞP1 C ð1 C f ðuiÞÞP2� (3)

where C1 and C2 are children solutions after applying

the crossover operator and P1 and P2 are the parents.

Like the single point crossover in binary-coded GA,

the SBX operator is also applied with a probability of

crossover of Pc. The specified probability distribution

function, which is used to determine f(ui), involves a

parameter (hc), which is a non-negative real number.

A larger value of hc helps in creating ‘near parent’

solutions while a smaller value of hc helps in creating

‘distant parent’ solutions.

2.3. Mutation

If only selection and crossover operators are used

in a GA, then it is possible for GA to converge to a

local optimum. This is because the GA is a very

aggressive search technique and rapidly discards

chromosomes with poor fitness values. In order to

maintain diversity in a population from one gener-

ation to the next, a mutation operator is normally

applied. In a binary-coded GA, mutation is achieved

through a local perturbation (i.e. replacing 0 with 1

and vice-versa) in the binary strings, with a prob-

ability of Pm. The procedure of creating a child from a

parent using the parameter based mutation operator

(Deb, 2001) is similar to the SBX operator in

implementation, where the parent is perturbed by a

specified amount. First, a random number (say ri)

between 0 and 1 is created. Then, from a specified

probability distribution function, the ordinate f(ri) is

found such that the cumulative probability for

ordinate f(ri) is equal to ri. After computing f(ri), the

offspring is calculated using the following equation

C Z P C ðPU KPLÞf ðriÞ (4)

where C is a child, P is a parent, PU and PL are the

upper and lower bounds of the parent, respectively.

The probability distribution used to compute the

perturbation, involves a parameter (hm), which

controls the shape of the distribution and determines

the order of the perturbation (1/hm).

This process of selection, crossover, and mutation

is repeated for many generations with the objective of

reaching the global optimal solution after a sufficient

number of generations. The flow chart for the steps

involved in the real-coded GA procedure employed in

the present study is provided in Fig. 1.

GA has been applied to many problems in

hydrology and water resources. Wang (1991) success-

fully used GA technique to calibrate the conceptual

rainfall–runoff models. Liong et al. (1995) used GA

for peak flow forecasting in a watershed in Singapore.

Cieniawski et al. (1995) used GA to solve a multi-

objective groundwater monitoring problem. Aly and

Peralta (1999) used GA for optimal design of an

aquifer cleanup system under uncertainty. Aral et al.

(2001) used progressive GA for the identification of

contaminant source location and recharge history for

groundwater management. Jain and Srinivasulu

(2002) used real-coded GA for estimating parameters

of infiltration equations. Samuel and Jha (2003)

employed GA to determine aquifer parameters from

pumping test data. More recently, Jain et al. (2004)

used GA for optimal design of composite irrigation

channels; Prasad and Park (2004) used multi-objec-

tive GA for the optimal design of water distribution

networks and Jain and Srinivasulu (2004) used GA

Fig. 1. Flow chart of the genetic algorithm implemented.

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 203

and artificial neural networks to develop improved

methodologies for rainfall–runoff modeling. Some

other notable examples of GA applications in

hydrology and water resources include Yapo et al.

(1998), Reed et al. (2000), Smalley et al. (2000), Yoon

and Shoemaker (2001), Dandy and Englehardt (2001)

and Munavalli and Kumar (2003). However,

the efforts in the area of application of GA for

determining the optimal UPRF have been limited.

3. Model development

Three different problem formulations are being

investigated in the present study for the purpose of

determining an optimal UPRF using the historical

rainfall and runoff data from a watershed. The first

formulation is the existing LP formulation solved by

others using LP methods, and is presented in Section

3.1. The first formulation has been modified to

reduce the number of constraints giving rise to the

second formulation, which is solved using real-coded

GA in this study and is presented in Section 3.2. The

first two formulations consist of objective functions

and constraints that are linear in nature. The third

formulation is the proposed formulation consisting of

a non-linear objective function, which is solved using

real-coded GA and is presented in Section 3.3.

3.1. Existing LP formulation

In the past, the problem formulations for determin-

ing the optimal UPRFs have employed the total sum

of absolute deviations (TSAD) between observed and

modeled DRH ordinates, as the objective function.

The existing LP formulation for the determination of

an optimal UPRF can be described as follows (Chow

et al., 1988)

Minimize E :XN

nZ1

ðqn CbnÞ (5a)

subject to

F1 :Xn%M

mZ1

PmUnKmC1 Kqn Cbn Z Qn;

n Z 1; 2;.;N

(5b)

F2 : DtXNKMC1

rZ1

Ur Z 1 (5c)

F3 : Ur R0; r Z 1; 2;.;N KM C1 (5d)

F4 : qnR0; bnR0; n Z 1; 2;.;N (5e)

where E is the objective function to be optimized,

Fi’s are the constraints to be imposed, qn and bn are

the non-negative decision variables representing

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214204

errors due to over-estimation and under-estimation,

respectively; Ur is the ordinate of the UPRF at

discrete time step r; Dt is the time interval at which

UPRF ordinates are to be determined; N is the total

number of DRH ordinates; M is the total number of

effective rainfall impulses; n, m, r, and nKmC1 are

various indices representing the discrete time

domain; and other variables carry the same meaning

as explained earlier.

The number of decision variables and constraints

for the above formulation will be 3NKMC1 and

4NKMC2, respectively. Out of the 4NKMC2

constraints, 3NKMC1 are inequality constraints

and NC1 are equality constraints. This LP formu-

lation has been solved using LP methods by many

researchers in the past. Singh (1976) solved this LP

formulation to determine optimal UPRF for a single

storm case using the data derived from north branch

Potomac River, Cumberland, MD. Mays and Coles

(1980) solved this LP formulation to find the optimal

composite UPRF using the data from the same

watershed by considering the multiple storms

simultaneously.

3.2. Modified formulation

The existing formulation represented by the set of

Eq. (5) has been modified in this study to reduce the

number of constraints and suit solution using GA.

The GA has an inherent limitation of not being able

to handle equality constraints. Therefore, the equality

constraints need to be converted to inequality

constraints. This was accomplished by the introduc-

tion of a dummy variable 3 in the LP formulation.

The other constraints can be handled by providing

lower bounds to the decision variables in GA. The

resulting modified formulation can be described by

the following equations

Minimize E :XN

nZ1

ðqn CbnÞ (6a)

subject to

F1 : 3 KXn%M

mZ1

PmUnKmC1 Kqn Cbn KQn

!R0;

n Z 1; 2;.;N (6b)

F2 : 3 K 1 KDtXNKMC1

rZ1

Ur

!R0 (6c)

where 3 is an additional dummy variable (having a

small constant value equivalent to the acceptable error)

and all other variables carry the same meaning as

explained earlier. In this formulation, the total number

of decision variables and constraints are 3NKMC1

and NC1, respectively. Thus, the number of con-

straints is reduced by 3NKMC1 in this formulation as

compared to the existing LP formulation. The modified

formulation was solved using real-coded GA and is

referred to as GA1 model in this paper.

3.3. The proposed problem formulation

The existing problem formulation represented by

the set of Eq. (5) is motivated by an inherent desire of

having an optimization problem in which all the

decision variables and constraints are linear. This is

true because an LP problem is easier to solve as

compared to a non-linear programming problem.

However, this leads to an optimization problem in

which the number of decision variables and con-

straints are much more than necessary. Since the

number of decision variables and constraints in

the two formulations presented above depend on the

number of DRH ordinates (N), the problem and the

solution procedure will become extremely complex

with increasing values of N. Thus, for large water-

sheds having large time bases or for the multiple

storm cases, it will be desirable to have a problem

formulation in which the complexity of the

problem does not depend upon N considerably. The

problem formulation proposed in this study is

motivated by considerations to overcome these

limitations of the existing formulations. This can be

easily achieved by embedding the constraint(s) within

the objective function. In the proposed optimization

problem formulation, the objective function is rep-

resented by the sum of the squares of the deviations

between observed and modeled DRH ordinates. The

resulting problem formulation can be described by the

following equations

Minimize E :XN

nZ1

Xn%M

mZ1

PmUnKmC1 KQn

!2

(7a)

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 205

subject to

F1 : 3 K 1 KDtXNKMC1

rZ1

Ur

!R0 (7b)

where all the variables carry the same meaning as

described earlier. This problem formulation consists of

NKMC1 decision variables and only one constraint

corresponding to the effective rainfall being unity. This

problem formulation is solved using real-coded GA

and is referred to as GA2 model in this paper.

The proposed problem formulation avoids the use

of intermediate dummy decision variables (qn and bn)

reducing the number of decision variables by 2N. This

is advantageous in solving an optimization problem

using GA, as the size of the initial population is a

function of the number of decision variables. Fewer

decision variables means smaller size of the population

in GA, which is desirable from the point of view of

computational efficiency of the solution procedure. It is

to be noted that the proposed optimization problem

formulation also avoids the use of constraints

described in Eq. 5(b) of the existing LP formulation.

The reduction in total number of constraints in this

formulation is 4NKMC1 and N as compared to the

existing LP and modified LP formulations described

earlier, respectively. Table 1 summarizes the number

of decision variables and constraints for each of the

problem formulations.

3.4. Model applications

The proposed problem formulation (GA2) for

determining the optimal UPRF presented in Section

3.3 was tested using data from two different

watersheds. The first data set is taken from the north

branch Potomac River near Cumberland, MD, USA.

This data set, directly taken from Mays and Coles

(1980), consists of effective rainfall and DRH

ordinates from three storms A, B, and C. The duration

of effective rainfall is 1 h and the DRH ordinates were

available at 1 h interval. Singh (1976) used this data

set to find optimal UPRF using individual storms A

and B. Mays and Coles (1980) have also used this data

set to determine optimal composite UPRF using

multiple storms A and B simultaneously. In this study,

the models GA1 and GA2 were used to determine

the optimal UPRFs for both individual and multiple

storm cases.

The proposed problem formulation was further

tested using a larger data set from Nenagh watershed of

area 295 km2 in Clariana (taken from Bree, 1978). The

data consists of effective rainfall and DRH ordinates

from 22 storms. The duration of the effective rainfall

for this data set is 3 h and the DRH ordinates were

available at 3 h interval. Zhao and Tung (1994)

employed the same data set to test four different LP

formulations for optimal UH determination. The four

LP formulations of Zhao and Tung (1994) consisted of

different objective functions: sum of absolute devi-

ations (MSAD), sum of weighted absolute deviations

(MWSAD), largest absolute deviation (MLAD), and

range of deviations (MRNG). The effective rainfall and

runoff data from the first 20 storms events were

employed to determine 20 different 3-h UHs using the

GA2 model proposed in this study for the case of

individual storm and results are compared with those

reported by Zhao and Tung (1994). Bree (1978) used a

stochastic approach of linear systems to determine

optimal composite UH using the same data set. Bree

(1978)’s formulation consisted of an objective func-

tion of minimizing the sum of the squares of the

deviations between observed and modeled DRH

ordinates. The data from first 21 storms were employed

to determine optimal composite UPRF using the GA2

model. The results corresponding to the composite

UPRF obtained in this study are compared with those

corresponding to Bree (1978)’s composite UH.

3.5. The solution procedure for GA

The existing formulation presented in Section 3.1

was not solved in this study and the results from

already published works (Singh, 1976; Bree, 1978;

Mays and Coles, 1980; Zhao and Tung, 1994) are

being reproduced here. The modified LP formulation

(GA1) and the proposed problem formulation (GA2)

presented in Sections 3.2 and 3.3, respectively, were

solved using the real-coded GA in this study.

The first step in the solution of an optimization

problem using GA is to generate an initial random

population of the possible solutions. The size of

population depends on the number of decision

variables involved in the problem among many

other factors (Reed et al., 2000). A minimum

Table 1

Comparison of the various optimization formulations

S. No. Model No. of

decision

variables

No. of

constraints

Population size

for NZ17;

MZ6

1 LP 3NKMC1 4NKMC2 –

2 GA1 3NKMC1 NC1 460

3 GA2 NKMC1 1 120

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214206

population size of 10 times the number of decision

variables is normally recommended while solving an

optimization problem using real-coded GA. The size

of initial population was thus taken based on the

number of decision variables, which are different in

different formulations and different storm cases.

The selection process was carried out using the

tournament selection method with a tourney size of

two. SBX crossover and parameter based mutation

operators, described earlier were employed in order to

generate children solutions from parent solutions.

Separate polynomial probability distributions were

employed to carry out SBX crossover and parameter

based mutation operators, which involve a parameter

called distribution index (hc for crossover and hm for

mutation). The distribution indices hc and hm were

tested in the ranges of 0.1–5 and 10–500, respectively,

before selecting the final values of 1.0 and 250 to be

employed in the solution procedure. The value of the

probability of crossover (Pc) of 0.90 and that for

mutation (Pm) of 0.01 was employed for all optimiz-

ation problems solved using real-coded GA. The size

of population was taken 10 times the number of

decision variables involved in the formulation, which

varied for different formulation and different storms.

The process of tournament selection, SBX crossover,

and parameter-based mutation was repeated from one

generation to the next until the convergence in terms

of the fitness value for each starting population was

achieved. Although the global optimal solution can

never be guaranteed, the real-coded GA was applied

to 20 different initial random populations in an

attempt to achieve a close approximation to the global

optimal solution. Out of the 20 runs, the best solution

for each model is being reported here.

The constraints were handled using the penalty

function approach while solving the optimization

problems using real-coded GA. The penalty function

approach can be used to convert any constrained

optimization problem into an unconstrained optimiz-

ation problem as follows

Minimize U Z E CXI

iZ1

RhFii2 (8)

where U is the augmented objective function to be

optimized;R penalty function parameter having a high

positive value; h i operator defined as follows:

hFiiZFi; when Fi is negative, and zero otherwise, i

is an index representing constraint, and I is total

number of constraints imposed on a particular problem

formulation.

A comparison of the three models, LP, GA1, and

GA2, is presented in Table 1 in terms of the number of

decision variables, constraints, and the size of

populations involved. It can be noted from Table 1

that the GA2 model involves the least number of

decision variables (NKMC1) and only one con-

straint. This is the major strength of the GA2 model

proposed in this study to determine an optimal UPRF

as it makes the GA2 model suitable for problems with

greater number of DRH ordinates and for multiple

storm case. The performance of each model investi-

gated was measured in terms of various standard

statistical parameters, which are discussed next.

4. Performance statistics

The performance of all optimization models

investigated in the present study was evaluated

using a wide variety of standard statistical parameters.

A total of nine different standard statistical parameters

were employed. A description of the standard

statistical parameters is provided below.

4.1. Total sum of absolute deviations (TSAD)

The TSAD is defined as the sum of absolute

differences between the computed and observed DRH

ordinates. This can be calculated either for calibration

data or validation data using the following equation

TSAD ¼XN

n¼1

jQ̂n KQnj% (9)

where Qn is the observed DRH ordinate, Q̂n represents

the computed DRH ordinate, and N is the total number

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 207

of DRH ordinates predicted. Smaller TSAD statistic

indicates good model performance and vice-versa.

4.2. Root mean square error (RMSE)

The RMSE statistic is a measure of residual

variance, and is indicative of the model’s ability to

predict high flows (Hsu et al., 1995). The RMSE from

a model can be defined as the square root of the

average value of the squares of the differences

between the computed and observed DRH ordinates

from the model. The RMSE can be calculated from

the following equation

RMSE Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

nZ1

ðQ̂n KQnÞ2

N

vuuut(10)

where the various variables carry the same meaning as

explained earlier. The low RMSE implies good model

performance and vice-versa.

4.3. Average absolute relative error (AARE)

The average absolute relative error (AARE) is a

measure of error in estimating the runoff discharge

(Jain et al., 2001; Jain and Ormsbee, 2002; Jain and

Indurthy, 2003). The AARE is defined as the average

of the absolute relative errors in forecasting a certain

number of DRH ordinates. The AARE can be

calculated using the following equation

AARE ZXN

nZ1

1

N

Q̂n KQn

Qn

�������� (11)

where all the variables carry the same meaning as

explained before. It is clear from their definition,

smaller the value of AARE better is the model

performance.

4.4. Absolute relative errors in DRH response

parameters {ARE(X)}

The parameters of a DRH response, such as,

magnitude of the peak discharge, time to peak

discharge, and the volume of the DRH are important

in characterizing the runoff response from a watershed

to a given rainfall event. The absolute relative error

(ARE) in predicting these DRH response parameters

was considered as performance statistics to quantify

the performance of each model in this study. This can

be computed as follows

AREðXÞ ZX̂ KX

X

�������� (12)

where ARE(X) is the ARE in predicting a particular

DRH response parameter, X is the observed DRH

response parameter, and X̂ is the predicted DRH

response parameter. The DRH response parameters

considered in this study are magnitude of the peak

discharge (Qp), time to peak discharge (tp), and the

volume of the DRH response (V). The corresponding

performance statistics, ARE(Qp), ARE(tp), and

ARE(V), from various model formulations are

presented in Section 5. It is clear from its definition

that smaller values of ARE(X) imply a better

performance from a model and vice-versa.

4.5. Relative bias in DRH response parameters

{BIAS(X)}

While AARE and ARE(X) in estimating a DRH

response parameter give a measure of the magnitude

of the error, they do not provide information on the

direction of the error. That is, whether the model is

over-predicting or under-predicting a particular DRH

response parameter is not clear from the AARE and

AREs of various DRH response parameters. The

information on whether a particular model is biased

towards over-prediction or under-prediction is also

important while evaluating the performance from a

model. This kind of information can be obtained by

computing another error statistic called relative bias in

estimating a particular DRH response parameter

{BIAS(X)}. The relative bias in predicting a particular

DRH response parameter can be calculated using the

following equation

BIASðXÞ ¼XN

n¼1

X̂n KXn

Xn

(13)

where all the variables carry the same meaning as

explained before. The bias statistic for DRH ordinates

(Q), time to peak (tp), and DRH volume (V) were

considered in this study. It is clear from the definition

that positive bias values would mean over-prediction

Table 2

Calibration statistics from north branch Potomac River basin-individual storm case

Model TSAD RMSE ARE(Qp) ARE(tp) ARE(V) BIAS(Q) BIAS(tp) BIAS(V) AARE

(a) Storm-A

LP 0.3809 0.0431 2.27!10K04 0.00 1.85!10K02 K1.2515 0.00 K1.85!10K02 14.14

GA1 0.5631 0.0481 0.00 0.00 9.50!10K03 0.5692 0.00 K9.50!10K03 8.67

GA2 0.4037 0.0329 2.27!10K04 0.00 1.04!10K02 0.2609 0.00 K1.04!10K02 8.75

(b) Storm-B

LP 0.0154 0.0017 6.30!10K04 0.00 9.37!10K04 K0.3328 0.00 K9.37!10K04 3.35

GA1 0.0057 0.0010 0.00 0.00 1.54!10K03 K0.1539 0.00 K1.54!10K03 2.65

GA2 0.0071 0.0010 0.00 0.00 1.14!10K03 0.0798 0.00 K1.14!10K03 2.69

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214208

and negative bias values would indicate under-

prediction.

5. Results and discussions

The results in terms of various performance

statistics from the north branch Potomac River

watershed are presented in Tables 2–5. The results

for the Nenagh River data are presented in Tables 6–8.

The discussion of results has been accordingly divided

into two sections.

5.1. Results from north branch Potomac River

watershed

The discussion of results for north branch Potomac

River watershed has been further divided into two

parts: results corresponding to single storm case and

the results corresponding to the multiple storm case.

The results corresponding to the single storm case are

presented in Tables 2 and 3, and those corresponding

to the multiple storm case are presented in Tables 4

and 5.

Table 3

Validation statistics from north branch Potomac River basin-individual st

Model TSAD RMSE ARE(Qp) ARE(tp) A

(a) Storm-A

LP 0.33!10 0.2107 0.1472 0.00 0.

GA1 0.35!10 0.2107 0.1880 0.00 0.

GA2 0.35!10 0.2107 0.1484 0.00 0.

(b) Storm-B

LP 0.50!10 0.2844 0.2179 0.00 0.

GA1 0.50!10 0.2844 0.2139 0.00 0.

GA2 0.50!10 0.2844 0.2140 0.00 0.

5.1.1. Results for single storm

The calibration statistics from north branch

Potomac River watershed for single storm case are

presented in Table 2. It can be noted from Table 2 that

the GA1 model performed the best in terms of

ARE(Qp), ARE(V), BIAS(V), and AARE; GA2 model

performed the best in terms of RMSE and BIAS(Q);

and LP model performed the best in terms of TSAD

for calibrating the optimal UPRF determined using

the single storm A. Similarly, the GA1 model

performed the best in terms of TSAD, RMSE,

ARE(Qp), and AARE; LP model performed the best

in terms of ARE(V) and BIAS(V); and GA2 model

performed the best in terms of RMSE, ARE(Qp), and

BIAS(Q) for calibrating the optimal UPRF deter-

mined using the single storm B. It is interesting to note

that the time to peak was estimated perfectly for both

storms A and B from all the three models during both

calibration and validation. The validation statistics for

this data set for single storm case are presented in

Table 3. It can be noted from Table 3 that the

performance of all the models was comparable during

validation of the optimal UPRFs determined using the

data from both the storms A and B. The performance

orm case

RE(V) BIAS(Q) BIAS(tp) BIAS(V) AARE

0793 K0.1506 0.00 K0.0793 27.90

0783 K0.1139 0.00 K0.0783 30.30

0910 K0.1068 0.00 K0.0783 30.50

0728 0.5183 0.00 K0.0793 53.17

0827 0.5253 0.00 K0.0783 54.15

0818 0.5265 0.00 K0.0910 54.09

Table 5

Validation statistics from north branch Potomac River basin-multiple storm case

Model TSAD RMSE ARE(Qp) ARE(tp) ARE(V) BIAS(Q) BIAS(tp) BIAS(V) AARE

Storm-C

LP 0.35!10 0.2133 0.1566 0.00 0.0880 0.1484 0.00 K0.0880 29.92

GA1 0.35!10 0.2133 0.1482 0.00 0.0891 0.1460 0.00 K0.0891 30.08

GA2 0.35!10 0.2133 0.1494 0.00 0.0897 0.1810 0.00 K0.0897 31.00

Table 4

Calibration statistics from north branch Potomac River basin-multiple storm case

Model TSAD RMSE ARE(Qp) ARE(tp) ARE(V) BIAS(Q) BIAS(tp) BIAS(V) AARE

(a) Storm-A

LP 0.0721 0.01074 4.54!10K05 0.00 4.44!10K03 K0.2413 0.00 K4.44!10K03 1.46

GA1 0.0469 0.00847 5.11!10K04 0.00 2.12!10K03 K0.2923 0.00 K2.12!10K03 1.50

GA2 0.2321 0.01770 1.27!10K02 0.00 4.46!10K04 0.3855 0.00 K4.46!10K04 3.48

(b) Storm-B

LP 0.5615 0.0504 7.66!10K03 0.00 1.07!10K04 K2.3563 0.00 1.07!10K04 30.55

GA1 0.5862 0.0509 6.88!10K03 0.00 2.88!10K03 K0.8127 0.00 2.88!10K03 30.65

GA2 0.5242 0.0443 1.47!10K02 0.00 5.22!10K03 K1.7121 0.00 5.22!10K03 28.70

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 209

of all the models were identical in terms of RMSE,

ARE(tp), BIAS(tp), and TSAD. The performance of

GA models was either better or comparable in terms

of the remaining performance statistics. Further, all

the models were biased towards under prediction in

estimating runoff volume during both calibration and

validation.

5.1.2. Results for multiple storms

The calibration statistics from north branch

Potomac River watershed for the multiple storm

case are presented in Table 4. It can be noted from

Table 4 that the GA1 model performed the best in

terms of TSAD and RMSE; GA2 model performed

the best in terms of ARE(V); and LP model performed

Table 6

Calibration statistics from Nenagh basin-individual storm case

Statistic Zhao and Tung (1994)’s formulations

MSAD MWSAD M

No. of decision

variables

83 83

No. of constraints 30 30

TSAD 5.93380 6.58640

RMSE 0.44787 0.48899

ARE(V) 0.00149 0.00149

BIAS(V) K0.00019 K0.00019 K

R2 – – –

the best in terms of AARE and ARE(Qp) for

calibrating storm A using the optimal composite

UPRF determined using multiple storms. Similarly,

GA2 model performed the best in terms of TSAD,

RMSE, and AARE; GA1 model performed the best in

terms of ARE(Qp) and BIAS(Q); and LP model

performed the best in terms of ARE(V) and BIAS(V)

for calibrating storm B using the optimal composite

UPRF determined from multiple storms. Interestingly,

the time to peak was estimated perfectly by all the

three models during both calibration and validation

for the results corresponding to multiple storms also.

Further, there was either very little bias shown by

various models investigated in this study, or the bias

was uniform in either direction in estimating runoff

Present study GA2

LAD MRNG

83 25 24

30 60 1

17.5810 17.0300 7.25311

0.6367 0.62505 0.38714

0.00149 0.00149 0.00525

0.00019 K0.00019 K0.00130

– 0.99326

Table 7

Validation statistics from Nenagh basin-individual storm case

Statistic Zhao and Tung (1994)’s formulations Present study GA2

MSAD MWSAD MLAD MRNG

TSAD 60.5020 58.4620 61.3610 62.4330 59.12130

RMSE 3.15110 3.0655 3.2258 3.2238 3.07110

ARE(V) 0.00149 0.00149 0.00149 0.00149 0.00610

BIAS(V) K0.00014 K0.00014 K0.00014 K0.00014 K0.00130

R2 – – – – 0.73064

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214210

volume and discharge while determining the UPRF

using multiple storms. The validation statistics from

north branch Potomac River basin for multiple storms

case are presented in Table 5. It can be noted from

Table 5 that the GA1 model performed the best in

terms of ARE(Qp) and BIAS(Q); LP model performed

the best in terms of ARE(V), BIAS(V), and AARE;

and the performance of all the models were identical

in terms of TSAD, RMSE, ARE(tp), and BIAS(tp) for

validating storm C using the optimal UPRF deter-

mined from multiple storms.

The results in graphical form from the north branch

Potomac River watershed are shown in Figs. 2–5.

Figs. 2 and 3 show optimal UPRF obtained from

different models for single storm (using data for

storm B) and multiple storm cases, respectively. It is

clear that the optimal UPRFs from various methods

for single storm case are almost identical while they

differ slightly in the multiple storm case. Fig. 4 shows

the calibration of the optimal composite UPRFs

obtained from various models using data from

storms B. Fig. 5 shows the validation of the optimal

composite UPRFs obtained from various models

using data from storms C. It has been found that the

GA2 model was able to giver smoother UPRF after

the peak without the need of additional constraints.

Overall, based on the results from the north

branch Potomac River watershed obtained in this

Table 8

Validation statistics from Nenagh basin-multiple storm case

Statistic Bree (1978) GA2

TSAD 35.64 34.4832

RMSE 1.9536 1.8484

ARE(V) 0.001475 0.001429

BIAS(V) 0.001475 K0.001429

R2 0.919 0.923

study, it can be said that the performance of all

the three models is comparable. Thus, the results

are able to demonstrate the proposed problem

formulation solved using real-coded GA can be a

suitable alternative to the existing formulations

solved using LP methods both for single storm and

multiple storm cases. However, the major strength

of the proposed formulation can be evident in a

bigger data set, which involves considering data

from many storm events.

5.2. Results from Nenagh River watershed

The rainfall and runoff data from 22 storms from

the Nenagh River watershed were used to test the

performance of various models investigated in this

study. The statistical results for Nenagh River

watershed are presented in Tables 6–8. Tables 6 and

7 present the performance statistics for single storm

case during calibration and validation, respectively.

The results from GA2 model are compared with those

of Zhao and Tung (1994)’s various formulations. It

can be noted from Table 6 that the GA2 model

Fig. 2. Optimum UPRFs from single storm B: north branch Potomac

River.

Fig. 3. Optimum composite UPRFs: north branch Potomac River.

Fig. 5. Validation of composite UPRFs: north branch Potomac

River.

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 211

consists of fewer decision variables and only one

constraint when compared to the existing LP formu-

lations. It can be noted from Table 6 that during

calibration period, the GA2 model obtained a

correlation coefficient value of 0.99326, which is

very good. It performed the best in terms of RMSE

statistic, better than MLAD and MRNG formulations

for TSAD statistic, and comparable to MSAD and

MWSAD formulations in terms of other statistics.

Similarly, during validation period (Table 7), the GA2

model performed better than MLAD, MRNG and

MSAD formulations in terms of TSAD and RMSE

statistic and the MWSAD formulation was the best

choice among all the formulations though marginally.

However, the performance of GA2 model was

comparable to other formulations for single storm

case for this data set.

Table 8 presents the validation statistics from

Nenagh River watershed for multiple storm case.

Fig. 4. Calibration of composite UPRFs: north branch Potomac

River.

The results of GA2 model are compared to those of

Bree (1978) reported earlier for the same data set. It is

obvious from Table 8 that the GA2 model, which

obtained a correlation coefficient value of 0.923 during

validation, consistently outperformed Bree (1978)’s

formulation in terms of all the statistics. It is

particularly interesting because, as the number of

storms increases, the complexity of the problem of

determining optimal composite UPRF also increases.

Thus, the proposed formulation can be particularly

useful in large watersheds having large time bases for

determining composite UPRF using data from multiple

storms. The graphical results from the Nenagh River

watershed are shown in Figs. 6 and 7. Fig. 6 shows the

validation of optimal composite UPRF obtained from

GA2 model while Fig. 7 shows the validation of

optimal composite UPRF obtained from Bree (1978).

It can be noted from Figs. 6 and 7 that the GA2 model

was able to estimate the peak magnitude perfectly

while Bree (1978) method over-predicts the peak

discharge slightly.

Fig. 6. Validation of composite UPRF using GA2 model: Nenagh

River.

Fig. 7. Validation of composite UPRF using Bree (1978): Nenagh

River.

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214212

Another interesting observation made in this study

was that the existing formulation solved using LP

methods obtained better ARE(V) and BIAS(V)

statistics i.e. the LP approach was able to predict the

runoff volumes with a better accuracy than the other

models. Similarly, the GA2 model obtained better

AARE, RMSE, and ARE(X) statistics indicating that

the GA2 model was able to predict the runoff

magnitudes with better accuracy for both single and

multiple storm cases.

6. Summary and conclusions

This paper presents the findings of employing a real-

coded GA to the problem of determination of an

optimal UPRF. The existing LP formulation has been

modified to have fewer constraints. In addition, a new

problem formulation is proposed consisting of fewer

decision variables, only one constraint, and an objec-

tive function that is non-linear in nature. The proposed

problem formulation can be used to determine an

optimal UPRF from a single storm or an optimal

composite UPRF from multiple storms considered

simultaneously. The data in terms of rainfall excess and

DRH ordinates derived from two different watersheds,

namely, north branch Potomac River watershed,

Cumberland, MD and Nenagh River watershed,

Clariana, were employed for the purpose of calibration

and validation of all the models investigated in this

study. The model performance was evaluated using a

wide variety of standard statistical measures.

The existing LP formulation when solved using

real-coded GA, gave similar results when it was

solved using conventional solution procedures (such

as simplex method) as reported earlier by others. The

results obtained in this study demonstrate that the

proposed problem formulation solved using real-

coded GA can be a suitable alternative to the existing

formulations solved using LP methods for single

storm case. However, the proposed problem formu-

lation coupled with real-coded GA can be preferred

over the existing formulations solved using the

classical optimization methods in determining the

optimal composite UPRF using data from multiple

storms considered simultaneously.

Mays and Coles (1980), who also used the north

branch Potomac River data to determine optimal

composite UPRF, reported that: (a) the model and

data errors may not be absorbed by an LP formulation,

(b) DRH response parameters such as peak discharge

and time to peak should be considered in the model

development and evaluation, (c) additional constraints

need to be considered to ensure smooth response on

the falling limb of a UH, and (d) there is a limit to the

number of rainfall events that can be considered

because of the number of decision variables and

constraints. The problem formulation proposed in this

study coupled with the real-coded GA overcomes

some of these limitations/suggestions reported by

Mays and Coles (1980). For example, DRH response

parameters were given due consideration in the model

evaluation, peak magnitude and time to peak were

estimated with very good accuracy in this study; the

proposed problem formulation when solved using

real-coded GA provides smooth UPRF without the

need of additional constraints; and there is no limit on

the number of rainfall events that can be considered

because of the number of decision variables and

constraints for determining composite UPRF, as the

proposed problem formulation provides a robust

mechanism to handle multiple storms without signifi-

cantly increasing the number of decision variables and

constraints. This is the major strength of the proposed

formulation presented in this study in light of the

problems and improvements suggested over the

existing ones. The flexibility of reduced number of

decision variables and constraints offered by the

proposed problem formulation coupled with the real-

coded GA can be particularly useful in determining

the optimal composite UPRF from multiple storms for

watersheds having DRH with large time bases.

A. Jain et al. / Journal of Hydrology 303 (2005) 199–214 213

Further, the statistical results in terms of different

DRH response parameters (e.g. runoff volume,

discharge, etc.) obtained in this study indicate that

(a) existing formulations solved using LP are suitable

for single storm case when the derived UPRFs are

intended for detention design purposes, where runoff

volumes are important, (b) proposed formulation

coupled with real-coded GA can be preferred for

single storm case when the derived UPRFs are

intended for other hydraulic design and water

resources management applications, where runoff

magnitudes are important, and (c) the proposed

formulation coupled with real-coded GA is preferable

in determining the optimal composite UPRF for

multiple storm case for all water resources design,

operation, and management applications.

Finally, the authors wish to mention that the results

presented in this study are preliminary in nature and

need to be validated by others by applying the proposed

methodologies to different watersheds of varying

climatic and hydrologic conditions. It is hoped that

future research efforts will focus in these directions in

order to improve the accuracy of the process of

determination of an optimal UPRF of a watershed

ultimately resulting in the betterment of the water

resources design, operation, and management systems.

Acknowledgements

The authors wish to thank Dr Kalyanmoy Deb,

Professor, Department of Mechanical Engineering,

Indian Institute of Technology Kanpur, for his

valuable suggestions during the progress of this work.

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