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: [email protected] (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
www.elsevier.com/locate/jhydrol
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.
References
Aly, A.H., Peralta, R.C., 1999. Optimal design of aquifer cleanup
systems under uncertainty using a neural network and genetic
algorithm. Water Resour. Res. 35 (8), 2523–2532.
Aral, M.M., Guan, J., Maslia, M.L., 2001. Identification of
contaminant source location and release history in aquifers.
J. Hydrol. Eng., ASCE 6 (3), 225–234.
Barnes, B.S., 1959. Consistency in unit hydrographs. Proc. ASCE
85 (HY8), 39–63.
Bender, D.L., Roberson, J.A., 1961. The use of a dimensionless unit
hydrograph to derive unit hydrographs for some Pacific
northwest basins. J. Geogr. Res. 66, 521–527.
Bree, T., 1978. The stability of parameter estimation in the general
linear models. J. Hydrol. 37, 47–66.
Bruen, M., Dooge, J.C.I., 1984. An efficient and robust method for
estimating unit hydrograph ordinates. J. Hydrol. 70, 1–24.
Chow, V.T., Maidment, D.R., Mays, L.R., 1988. Applied
Hydrology. McGraw-Hill International Editions, Singapore.
Cieniawski, S.E., Eheart, J.W., Ranjithan, S., 1995. Using genetic
algorithm to solve a multiobjective groundwater monitoring
problem. Water Resour. Res. 31 (2), 399–409.
Collins, W.T., 1939. Runoff distribution graphs from precipitation
occurring in more than one time unit. Civ. Eng. 9 (9), 559–561.
Dandy, G.C., Englehardt, M., 2001. Optimal scheduling of water
pipe replacement using genetic algorithm. J. Water Resour.
Plan. Manag., ASCE 127 (4), 214–223.
Deb, K., 2000. An efficient constraint handling method for genetic
algorithms. Comput. Methods Appl. Mech. Eng. 186, 311–338.
Deb, K., 2001. Multi-Objective Optimization Using Evolutionary
Algorithms. Wiley, New York.
Deb, K., Agarwal, R.B., 1995. Simulated binary crossover for
continuous search space. Complex Syst. 9, 115–148.
Deininger, R.A., 1969. Linear program for hydrologic analysis.
Water Resour. Res. 5 (5), 1105–1109.
Dooge, J.C.I., Garvey, B.J., 1978. The use of Meixner function in
the identification of heavily damped systems. Proc. R. Irish
Acad., Sec. A 78 (18), 157–179.
Goldberg, D.E., Deb, K., 1991. A comparison of selection schemes
used in genetic algorithms. Found. Genet. Algorithm 1, 69–93.
Holland, J.H., 1975. Adaptation in Natural and Artificial Systems.
University of Michigan Press, Ann Arbor, MI pp. 183.
Hsu, K., Gupta, V.H., Sorooshian, S., 1995. Artificial neural
network modeling of the rainfall–runoff process. Water Resour.
Res. 31 (10), 2517–2530.
Jain, A., Indurthy, S.K.V.P., 2003. Comparative analysis of event
based rainfall–runoff modeling techniques—deterministic, stat-
istical, and artificial neural networks. J. Hydrol. Eng., ASCE 8
(2), 1–6.
Jain, A., Ormsbee, L.E., 2002. Evaluation of short-term water
demand forecast modeling techniques: conventional v/s artifi-
cial intelligence. J. Am. Water Works Assoc. 94 (7), 64–72.
Jain, A., Srinivasulu, S., 2002. Calibration of infiltration parameters
using genetic algorithm. Proceedings of HYDRO 2002:
Conference On Hydraulics, Water Resources and Ocean Eng.,
December 16–17, IIT Bombay, Bombay, India.
Jain, A., Srinivasulu, S., 2004. Development of effective and
efficient rainfall–runoff models using integration of determinis-
tic, real-coded genetic algorithms, and artificial neural network
techniques. Water Resour. Res. 40 (4), W04302.
Jain, A., Varshney, A.K., Joshi, U.C., 2001. Short-term water
demand forecast modeling at IIT Kanpur using artificial neural
networks. Water Resour. Manag. 15 (5), 299–321.
Jain, A., Bhattacharjya, R., Srinivasalu, S., 2004. Optimal design of
composite channels using genetic algorithm. J. Irrig. Drain.
Eng., ASCE 130 (4), 286–295.
Liong, S., Chan, W.T., Shreeram, J., 1995. Peak-flow forecasting
with genetic algorithm and SWMM. J. Hydraul. Div., ASCE
121 (8), 613–617.
A. Jain et al. / Journal of Hydrology 303 (2005) 199–214214
Mays, L.W., Coles, L., 1980. Optimization of unit hydrograph
determination. J. Hydraul. Div., ASCE 106 (HY1), 85–97.
Mays, L.W., Taur, C.K., 1982. Unit hydrograph via non-linear
programming. Water Resour. Res. 18 (4), 744–752.
Morel-Seytoux, H.J., 1982. Optimization methods in rainfall–runoff
modeling, in: Singh, V.P. (Ed.), Rainfall–Runoff Relationships.
Water Resour. Pub., Littleton, CO, pp. 487–506.
Munavalli, G.R., Kumar, M.S.M., 2003. Optimal scheduling of
multiple chlorine sources in water distribution systems. J. Water
Resour. Plan. Manag., ASCE 129 (6), 493–504.
O’Donnel, T., 1960. Instantaneous Unit Hydrograph by Harmonic
Analysis, IASH Pub. No. 51 1960 pp. 546–557.
Prasad, T.D., Park, N.J., 2004. Multi-objective genetic algorithm for
design of water distribution networks. J. Water Resour. Plan.
Manag., ASCE 130 (1), 73–82.
Reed, P.M., Minsker, B.S., Valocchi, A.J., 2000. Cost effective
long-term groundwater monitoring design using a genetic
algorithm and global mass interpolation. Water Resour. Res.
36 (12), 3731–3741.
Samuel, M.P., Jha, M.K., 2003. Estimation of aquifer parameters
from pumping test data by genetic algorithm optimization
technique. J. Irrig. Drain. Eng., ASCE 129 (5), 348–359.
Sherman, L.K., 1932. Stream flow by rainfall by the unit graph
method. Eng. News Rec. 108, 501–505.
Singh, K.P., 1976. Unit hydrographs: a comparative study. Water
Resour. Bull. 12 (2), 381–392.
Singh, V.P., 1988. Hydrologic Systems, vol. 1. Prentice Hall,
Englewood Cliffs, NJ.
Smalley, J.B., Minsker, B.S., Goldberg, D.E., 2000. Risk-based in
situ bio-remediation design using a noisy genetic algorithm.
Water Resour. Res. 36 (20), 3043–3052.
Wang, Q.J., 1991. The genetic algorithm and its application to
calibrating conceptual rainfall–runoff models. Water Resour.
Res. 27 (9), 246–271.
Yapo, P.O., Gupta, H.V., Sorooshian, S., 1998. Multi-objective
global optimization for hydrologic models. J. Hydrol. 204,
83–97.
Yoon, J., Shoemaker, C., 2001. An improved real-coded GA
for groundwater bio-remediation. J. Comput. Civ. Eng. 15 (3),
224–231.
Yue, S., Hashino, M., 2000. Unit hydrographs to model quick and
slow runoff components of streamflow. J. Hydrol. 227, 195–206.
Zhao, B., Tung, Y., 1994. Determination of optimal hydrographs by
linear programming. Water Resour. Manag. 8, 101–119.