Proceedings of the Institution of Civil Engineers
Water Management 165 October 2012 Issue WM9
Pages 481–493 http://dx.doi.org/10.1680/wama.11.00008
Paper 1100008
Received 11/01/2011 Accepted 14/09/2011
Published online 03/07/2012
Keywords:bridges/design methods & aids/mathematical modelling
ICE Publishing:All rights reserved
Water Management
Volume 165 Issue WM9
Bridge pier scour prediction by gene
expression programming
Khan,Azamathulla,Tufail and Ab Ghani
Bridge pier scour prediction by
gene expression programming
j
1
Mujahid Khan
ME
Lecturer and PhD Scholar,Department of Civil Engineering,
University of Engineering & Technology,Peshawar,Pakistan
j
2
Hazi M.Azamathulla
ME,PhD
Senior Lecturer,River Engineering and Urban Drainage Research
Centre (REDAC),Universiti Sains Malaysia,Penanag,Malaysia
j
3
Mohammad Tufail
MSc,PhD
Assistant Professor,Department of Civil Engineering,University of
Engineering & Technology,Peshawar,Pakistan
j
4
Aminuddin Ab Ghani
MSc,PhD
Professor,River Engineering and Urban Drainage Research Centre
(REDAC),Universiti Sains Malaysia,Pulau Pinang,Malaysia
j
1
j
2
j
3
j
4
Extensive research has been carried out to predict bridge pier scour,with laboratory and ﬁeld data,using different
modelling techniques.This study introduces a new soft computing technique called gene expression programming
(GEP) for pier scour depth prediction using ﬁeld data.A functional relationship has been established using GEP and
its performance is compared with other inductive modelling techniques such as artiﬁcial neural networks (ANNs) and
conventional regressionbased techniques.Field data comprising 370 data sets were collected from the published
literature and divided into calibration and validation (testing) data sets.The performance of GEP was found to be
satisfactory and encouraging when compared with regression and ANN models in predicting bridge pier scour depth.
GEP has the unique capability of providing a compact and explicit mathematical expression for computing bridge
scour.This advantage of GEP over ANN is one of the main motivations for this work.The resulting GEP models add
to the existing literature on artiﬁcial intelligence based inductive models that can be used effectively for bridge scour
modelling.
Notation
b pier width
d
50
mean sediment diameter
d
s
bridge pier scour depth
F
r
Froude number
F
rc
critical Froude number (¼ V
c
=( gY)
1=2
)
g gravitational acceleration
K
s
,K
Ł
coefﬁcient for pier shape and pier alignment
respectively
L length of pier normal to the ﬂow
R
2
coefﬁcient of determination
V approach ﬂow velocity
V
c
critical velocity
Y approach ﬂow depth
Ł
s
Shield mobility parameter
sediment gradation coefﬁcient
1.Introduction
Bridge scour is the result of the erosive action of ﬂowing water,
excavating and carrying away material from the bed and banks of
streams and from around the piers and abutments of bridges.
Bridge scour is one of the biggest causes of bridge failure and a
major factor that contributes to the total construction and
maintenance costs of bridges.Underprediction of bridge scour
designs can lead to costly bridge failures associated with possible
loss of human life.Similarly,overprediction can result in wasting
millions of dollars on a single bridge (FDoT,2005).After
examining more than 500 bridge failures that occurred in the
USA between 1989 and 2000,it was found that 53% were due to
ﬂoods and resulting scour (Wardhana and Hadipriono,2003).
Acknowledging the problems posed by bridge failures due to
scourrelated processes,the subject of bridge scour prediction
requires proper attention and there is a need to enhance current
research in this area,including tools and techniques,to accurately
predict bridge scour.Enhancing developments in these areas aims
to lead to safe,economical and technically sound bridge pier
design (Azamathulla et al.,2010).
The mechanism of scour around bridge piers is often complicated
and it is difﬁcult to establish a general empirical relation for its
prediction under different ﬁeld conditions.In order to understand
the mechanism of bridge scour,many investigators have devel
oped pier scour prediction equations based on conventional
regressionbased techniques using laboratory and ﬁeld data
(Breusers et al.,1977;FDoT,2005;Melville and Coleman,2000;
Richardson and Davis,2001;Sheppard et al.,2004).Most of the
481
developed equations yield good results for laboratory data but fail
to establish a good causeandeffect relationship when applied to
ﬁeld data.Various investigators have compared different local
scour depth prediction equations to evaluate their utility for
predicting bridge pier scour (Breusers et al.,1977;Landers and
Mueller,1996).Recent comparisons of inductive (datadriven)
models include that of Mohamed et al.(2005),who reported that
the Laursen and Toch (1956) formula and the Colorado State
University (CSU) formula (Richardson and Davis,2001) gave
reasonable estimates of bridge pier scour,while the Melville and
Sutherland (1988) and Jain and Fischer (1979) formulae over
predict pier scour based on a comparison of bridge pier scour
formulae using available ﬁeld data.
There are thus a number of empirical equations available based
on the principles of conventional inductive modelling techniques
such as regression analysis of available ﬁeld and laboratory data.
The complexity of bridge scour process requires that prediction
models for pier scour must be based on data sets that include all
relevant decision variables that contribute to the process of bridge
scour.Furthermore,there is also the requirement that the model
ling techniques used to derive empirical models for bridge scour
are effective and accurate and can capture the causeandeffect
relationship of the input and output variables involved in the
process.
In summary,the two major factors affecting the advancement of
bridge scour prediction methods are the availability of sufﬁcient
data (ﬁeld or laboratory) covering all relevant parameters and the
availability and application of effective and efﬁcient modelling
tools that can be applied to the available data to generate accurate
bridge scour prediction models for use by designers in bridge
design.Use of unreliable data and/or ineffective and inefﬁcient
modelling techniques can lead to models that may not properly
and accurately predict bridge scour depth.It is also important to
calibrate existing models with reliable data.However,Yanmaz
(2003) concluded that the calibration of scour prediction models
with ﬁeld data is restricted mainly due to the lack of a relevant
amount and precision of available data.
To address the relevant needs of research in the ﬁeld of bridge
scour modelling,recent research initiatives are exploring ways to
enhance data collection efforts by collecting reliable ﬁeld and
laboratory data sets and/or enhance the available modelling tools
used to ﬁt empirical models to available data sets.In particular,
recent research has made good advances in the development of
modern datadriven modelling tools such as those based on
artiﬁcial intelligence (AI) techniques.Such techniques have found
excellent applications in the ﬁeld of hydraulic and water
resources engineering and have provided more effective model
structures when compared with more conventional techniques
such as those based on multiple linear regression (MLR).Recent
literature reveals that AIbased inductive modelling techniques
are increasingly being used to model complex response functions
(such as bridge scour analysis) due to their powerful and non
linear model structures and their increased capability to capture
the causeandeffect relationship of such complex processes.Such
AIbased techniques include artiﬁcial neural networks (ANNs),
adaptive neurofuzzy inference systems (ANFISs),genetic algo
rithms (GAs) and genetic programming (GP) (ASCE,2000a,
2000b;Azamathulla et al.,2005,2010;Bateni et al.,2007;Lee
et al.,2007) and have been found to provide favourable results in
modelling complex response functions including bridge scour
depth based on available data collected either in the ﬁeld or
laboratory.ANNs have been reported to provide reasonably good
solutions for hydraulic engineering problems in cases of highly
nonlinear and complex response functions (Azamathulla et al.,
2005,2009).
Most recently,a new technique called gene expression program
ming (GEP) was developed,which is an extension of GP (Koza,
1992).It is a search technique that evolves computer programs
(mathematical expressions,decision trees and logical expres
sions).The computer programs of GEP are all encoded in linear
chromosomes,which are then expressed or translated into expres
sion trees.Expression trees are sophisticated computer programs
that are usually evolved to solve a particular problem and are
selected according to their ﬁtness at solving that problem.From
these trees,corresponding empirical expressions can be derived.
GEP has been found to give reasonably good predictions for
sediment load (Ab Ghani and Azamathulla,2011;Ab Ghani and
Azamathulla (2011).
The main objective of this work was to further enhance the
available inductive modelling tools for predicting bridge scour by
developing GEPbased models for pier scour prediction utilising
available ﬁeld data and comparing their performance with ANN
and regressionbased models.A further objective was to evaluate
the utility of GEPbased models for bridge scour prediction with
the aim of providing a compact and explicit empirical expression
that can be used to predict bridge scour.While ANNbased
models are powerful in that they provide a good ﬁt to the data
used in model training and validation,such models often do not
result in compact and explicit equations for use by designers.The
resulting ANNbased model structure is often a long expression
consisting of activation functions with variable complexity de
pending on the number of hidden layers used in the model
structure.The data set used in the development of the bridge
scour prediction models in this study was collected in the ﬁeld by
the United States Geological Survey (USGS),has been previously
applied in various studies and covers a broad spectrum and
variation of the related model input decision variables.
2.Local scour problemaround a pier
The equilibrium local scour depth d
s
around a circular pier
(Figure 1) in a steady ﬂow over a bed of uniform and non
cohesive sediment depends on numerous parameters including
ﬂow,ﬂuid,sediment characteristics and pier geometry.Local
scour depth at piers is a function of the following decision
variables
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Bridge pier scour prediction by gene
expression programming
Khan,Azamathulla,Tufail and Ab Ghani
d
s
¼ f (V,Y,g,d
50
,b,L,)
1:
where V is the approach velocity,Y is the approach ﬂow depth,g
is acceleration due to gravity,d
50
is the particle mean diameter,b
is the pier width,L is the length of the pier and is the standard
deviation of grain size distribution.The ﬁeld data used in this
study for the development of inductive (datadriven) models
comprised 370 data sets and was obtained from Landers and
Mueller (1996).Table 1 summarises the ranges of the variables in
the collected ﬁeld data.
Dimensional analysis of the variables in Equation 1 reduces it to
ﬁve nondimensional parameters
d
s
Y
¼ f F
r
,
d
50
Y
,
b
Y
,
L
Y
,
2:
where F
r
is the Froude number.These ﬁve parameters were used
as decision variables in the development of new MLRbased
equation,ANN and GEP models.
3.Development of datadriven models for
bridge pier scour depth using ﬁeld data
The work described in this paper presents the development of
inductive models for bridge scour prediction using a range of
modelling techniques using ﬁeld data collected by the USGS and
previously used by Landers and Mueller (1996).As noted earlier,
the main aim of this paper is to advance datadriven modelling
techniques by investigating more robust and efﬁcient techniques
to provide accurate and effective inductive models for bridge
scour prediction.In this regard,the current work investigates the
utility of two AIbased models (ANNs and GEP) for bridge scour
prediction and compares their performance with traditional
regressionbased models.The models studied thus comprise
(a) previously developed regressionbased empirical formulae for
bridge scour prediction
(b) new MLR models trained and tested on the same available
ﬁeld data that will be used in the ANN and GEP models.
4.Traditional regression models
Bridge pier scour is dependent on a number of factors as
discussed earlier.Most of the pier scour prediction formulae
available in the literature are based on conventional regression
methods and most overpredict pier scour,resulting in uneconomi
cal bridge foundation design.Under current practice for bridge
scour prediction,an engineer may refer to one of the already
available empirical equations for a particular application or,if
sufﬁcient new data (ﬁeld or laboratory) are available for the
particular application,a new empirical model may be developed
by ﬁtting the available data to conventional empirical model
D
Pier
Water surface
U
Y
d
s
Figure 1.Flow and local scour at pier (Bateni et al.,2007)
Variable Minimum Median Maximum
Training Testing Training Testing Training Testing
b:m 0
.
290 0
.
610 1
.
220 0
.
880 4
.
270 1
.
370
L:m 2
.
440 8
.
530 10
.
520 10
.
670 27
.
430 25
.
300
V:m/s 0
.
150 0
.
190 1
.
370 0
.
940 4
.
480 2
.
590
Y:m 0
.
120 0
.
460 0
.
640 4
.
630 12
.
620 9
.
300
1
.
300 1
.
500 2
.
500 2
.
200 12
.
100 4
.
900
d
50
:m 0
.
000 0
.
000 0
.
001 0
.
001 0
.
108 0
.
072
d
s
:m 0
.
000 0
.
060 0
.
640 0
.
490 7
.
650 1
.
550
F
r
0
.
038 0
.
033 0
.
267 0
.
141 0
.
834 0
.
726
d
50
/Y 0
.
000 0
.
000 0
.
001 0
.
000 0
.
094 0
.
157
b/Y 0
.
044 0
.
073 0
.
369 0
.
190 8
.
667 1
.
326
L/Y 0
.
621 1
.
219 3
.
048 2
.
236 91
.
417 27
.
174
1
.
300 1
.
500 2
.
500 2
.
200 12
.
100 4
.
900
d
s
/Y 0
.
000 0
.
028 0
.
224 0
.
141 3
.
467 0
.
830
Table 1.Ranges of ﬁeld data (Landers and Mueller,1996)
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Bridge pier scour prediction by gene
expression programming
Khan,Azamathulla,Tufail and Ab Ghani
structures such as regressionbased techniques.In the current
study,these two approaches are evaluated by
(a) selecting empirical equations already developed by previous
researchers and comparing their performance on the training
and validation (testing) data set to other models or techniques
(ANN and GEP in this study)
(b) developing new MLRbased empirical models by utilising the
training data set used in the study and subsequently testing
them on the validation data set.
In this way,the performance of existing empirical equations can
be compared with that of those speciﬁcally developed for the data
used in this paper (370 ﬁeld data sets).The performance of both
types of regressionbased models will then be compared to that
of the more complex and nonlinear AIbased models including
ANNs and GEPbased models.
4.1 Regression models previously developed by other
researchers
This study will evaluate the performance of the models (equa
tions) proposed by Breusers et al.(1977),Jain and Fischer
(1979),Froehlich (1988) and Richardson and Davis (2001) (the
CSU formula,or HEC18).These four models were chosen
because they are based on conventional regression techniques and
are often used by researchers and practising engineers.
4.1.1 Breusers et al.(1977) formula
Breusers et al.(1977) developed a formula for the prediction of
local pier scour under clear water and live bed scour conditions
(Gaudio et al.,2010)
d
s
b
¼ 2 2
V
V
c
1
tanh
Y
b
K
s
K
Ł
3:
in which d
s
is the equilibrium scour depth,V
c
is the critical
velocity for sediment motion,K
s
is the pier shape factor and K
Ł
the pier alignment factor.Different values of K
s
and K
Ł
can be
found in Simons and Senturk (1992).V
c
was computed by Neill
(1973) for SI units as
V
c
¼ 31
:
08Ł
s
1=2
Y
1=6
d
1=3
50
where Ł
s
is the Shield mobility parameter given by (Muller and
Wagner,2005)
Ł
s
¼ 0
:
0019d
0
:
384
50
for d
50
,0
:
0009 m
Ł
s
¼ 0
:
0942d
0
:
175
50
for 0
:
0009,d
50
,0
:
02 m
Ł
s
¼ 0
:
047 for d
50
.0
:
02 m
The equations were used as one of the regressionbased models
for comparison with the new MLR,ANN and GEPbased models
in the current study.This formula has been found to under
estimate and overestimate values of pier scour depending on the
data used (Gaudio et al.,2010).
4.1.2 Jain and Fischer (1979) equations
Jain and Fischer (1979) developed the following set of equations
for clear water and live bed scour conditions based on laboratory
experiments.
d
s
¼ 1
:
84b
Y
b
0
:
3
F
0
:
25
rc
for F
r
F
rc
,0 in clear water conditions
4a:
d
s
¼ 2
:
0b
Y
b
0
:
5
(F
r
F
rc
)
0
:
25
for F
r
F
rc
¼ 0
:
2 in live bed conditions
4b:
where F
rc
is the critical Froude number [¼ V
c
=( gY)
1=2
].For
0,F
r
F
rc
,0
.
2 the largest value obtained from Equation 4a
or 4b is taken.
These formulae were recently used by Mohamed et al.(2005)
and Gaudio et al.(2010) in a comparison with other regression
based models and techniques and it was found that they over
predict the local scour depth.
4.1.3 Froehlich (1988) formula
Froehlich (1988) developed the following regressionbased for
mulae for local pier scour under live bed conditions.
d
s
¼ 0
:
32bjF
r
0
:
2
b
e
b
0
:
62
Y
b
0
:
46
b
d
50
0
:
08
5:
where b
e
is the width of the pier projected normal to the
approaching ﬂow and j is a dimensionless coefﬁcient based on
pier nose shape:j ¼1
.
3 for squarenosed piers,1
.
0 for round
nosed piers and 0
.
7 for sharpnosed piers.Gaudio et al.(2010)
reported that this formula overpredicts scour depth but gave
reasonable results when compared with other formulae.
4.1.4 Richardson and Davis (2001) formula
Originally this formula was presented in HEC18 (Richardson
and Davis,2001) and was recently used by Mohamed et al.
(2005) for comparison with other regressionbased equations.
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Water Management
Volume 165 Issue WM9
Bridge pier scour prediction by gene
expression programming
Khan,Azamathulla,Tufail and Ab Ghani
d
s
Y
¼ 2
:
0K
s
K
Ł
b
Y
0
:
65
F
0
:
43
r
6:
Different values for K
s
and K
Ł
are reported by Simons and
Senturk (1992).Equation 6 accordingly is used as one of the
regressionbased equations for comparison with the new MLR,
ANN and GEPbased models.
4.1.5 Comparison
Gaudio et al.(2010) compared six different regressionbased
formulae,including the four presented here,and concluded that
the Richardson and Davis (2001) and Froehlich (1988) formulae
gave the best predictions of scour depth under live bed condi
tions.
Note that for the four existing regression equations considered
here,model training was not performed in the current study since
the formulae have been developed and trained previously.How
ever,the equations were directly applied to both the training and
testing data sets used in this study in order to compare their
performance with the new MLR,ANN and GEPbased models.
Since these equations were previously derived using different data
sets,the current work does not explain the range of applicability
of these equations.For an indepth analysis of how these
equations were derived,the reader is referred to the respective
references.
4.2 New MLRbased datadriven model
The ﬁeld data used by Landers and Mueller (1996) were used for
the development of new MLR,ANN and GEP models.Out of the
total 370 data sets,281 (75%) were used for model training
while the remaining 89 (25%) were used for model validation
or testing.There were ﬁve input variables (F
r
,d
50
/Y,b/Y,L/Y and
) and one output variable (d
s
/Y).This data split and input/output
conﬁguration applies to all three types of models developed in
this study.
A new MLR model was developed for predicting pier scour depth
based on the ﬁve model inputs.The general form of the MLR
model is
BS ¼ a
0
þ a
1
F
r
a
2
d
50
Y
þ a
3
b
Y
þ a
4
L
Y
þ a
5
7:
where a
0
,a
1
,a
2
,a
3
,a
4
and a
5
are regression coefﬁcients,b/Y
(relative pier diameter),d
50
/Y (relative sediment size),L/Y (length
over ﬂow depth), (standard deviation of bed material) and F
r
are the independent decision variables in the regression model
and BS (relative bridge scour) is the dependent variable to be
predicted by the model.The results of the MLRbased modelling
exercise resulted in the following optimal equation for the
prediction of bridge pier scour based on model training
d
s
Y
¼ 0
:
0875 þ0
:
7348F
r
3
:
12
d
50
Y
þ0
:
305
b
Y
þ0
:
0113
L
Y
þ0
:
00532
8:
The results of the new MLR model (Equation 8) in model
training and validation are discussed in Section 9.
5.AIbased datadriven models
An inductive or empirical model is based on data and is often
used to predict,not explain,a system.The equations and
calibrations of inductive models rely (more directly) on ﬁeld/
laboratory data or empirical observations (Tufail et al.,2008).
Numerous empirical formulae or inductive models based on
regression analysis have been proposed to estimate scour depth
at bridge piers under different conditions using laboratory data
as well as ﬁeld data.As noted earlier,most of the regression
based models developed and applied for bridge scour prediction,
including the ones described here,tend to overestimate the
scour depth.This is because such conventional models are too
simple (linear) in their structure thereby failing to accurately
predict scour depth due to its complexity.Accordingly,such
models often fail to accurately model the causeandeffect
relationship between inputs and output variables.Recognising
these difﬁculties and the importance of improving prediction
capabilities,this work investigated the utility of two AIbased
inductive models – ANNs and a relatively new GPbased
technique called GEP – for predicting bridge scour depth using
ﬁeld data.
6.ANN model
An ANN is a mathematical model constructed so as to approx
imate the basic functions associated with a biological neuron.In
other words,it is a digital model of the human brain and it
imitates the way a human brain works.It consists of a highly
interconnected network of several simple processing units called
neurons.ANNs are constructed by creating connections between
a set of digital processing elements (the computer equivalent of
neurons) that consist of an input layer of elements or neurons,
hidden layers of neurons and an output layer of neurons.The
organisation and weights of these connecting elements are
adjusted through a process of ‘training’ in order to calibrate the
model.A wealth of information on the architecture of neural
networks,training and testing can be found in the literature (e.g.
Zurada,1992);the concepts involved behind such training
schemes have also been reported (ASCE,2000a,2000b).The
number of hidden layers varies based on the complexity of the
model,and the number of hidden layers and number of neurons
in each hidden layer are often varied to optimise the performance
of the ﬁnal model.
ANNbased models such as the popular multilayer feedforward
networks have frequently been used to approximate the response
of a particular system by training with available data.ANN
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expression programming
Khan,Azamathulla,Tufail and Ab Ghani
models are often referred to as ‘black box models’ as they are
not primarily used to produce an empirical equation to represent
a process,but are rather used to produce outputs according to
inputs received by the model.Such models generally require
considerable data for training and thus may not be favourable for
applications where the objective is to obtain a simple,easy to use
and functionally compact approximation.
ANNs have been successfully used in modelling of water
resources systems in areas such as stream ﬂow forecasting
(Birikundavyi et al.,2002),rainfall–runoff modelling,reservoir
operations,etc.(Babovic and Bojkov,2001;de Vos and Rientjes,
2005).ANNs have also been applied extensively in the ﬁeld of
hydrology for estimation and forecasting of hydrologic variables
(ASCE,2000a,2000b).Jeng et al.(2006) reported that the neural
network approach has been applied to many branches of science,
including aspects of hydraulic and environmental engineering.
Some of the earliest applications of neural network models in
hydrology and water resources engineering were reported by
Daniell (1991) and other applications have been reported by
Karunanithi et al.(1994),Grubert (1995),Minns (1998),Nagy et
al.(2002),Coppola et al.(2003),Jain and Indurthy (2003) and
Sudheer and Jain (2003).More recent applications of ANNs in
the ﬁeld of hydraulic engineering have been studied by Aza
mathulla et al.(2009),Bateni et al.(2007),Lee et al.(2007),
Jeng et al.(2006) and Azamathulla et al.(2005).Kambekar and
Deo (2003) estimated scour depth around a group of piles using
neural network models.Some of the researchers used ﬁeld data
while others used laboratory data in the development of these
ANN models.
Neuro sort software (Lingireddy et al.,2003) was used for the
development of ANN models in this study.A simple feedforward
type network was trained using the backpropagation technique.
The data were normalised before being fed to the software for
subsequent training and validation.Neural network training was
done using a standard error,supervised backpropagation training
algorithm (Haykin,1994;Rumelhart and Mclelland,1986) with a
learning rate of 0
.
1 and momentum factor of 0
.
4.The learning
rate,also known as step size,is a factor that determines the
amount by which the connection weight is changed according to
error gradient information.The momentum parameter governs the
weight change in the current iteration of the algorithm due to a
change in the previous iteration.The values used for the learning
rate and momentum,0
.
1 and 0
.
4 respectively,were obtained by
the trialanderror method (Haykin,1994;Maier and Dandy,
1998).The backpropagation algorithm (Rumelhart and Mclel
land,1986) used in the current study employs a gradient descent
technique to adopt weights in the ANN structure to minimise the
mean squared difference between ANN output and desired
(actual) output.The number of neurons in the hidden layer was
varied between ﬁve (number of model inputs) and a maximum of
ten.The simple architecture of a generalised feedforward back
propagation neural network reported by Bateni et al.(2007) is
shown in Figure 2.
In all cases,optimal results were obtained by using ﬁve neurons
in the hidden layer;the model results did not vary signiﬁcantly
for ANN models with the number of neurons in the hidden layer
ranging from ﬁve to ten.In hidden and output layers,a sigmoidal
activation function (Equation 9) was used for modelling the
transformation of values across the layers
f (x) ¼
1
1 þe
x
9:
The initial weights used in the ANN model were generated
randomly to values close to zero.The maximum number of
epochs (iterations) in model training was set to 20 000 for all
ANN models developed in this study.The training epochs were
decided based on trials by observing ANN training and validation
(testing) error results simultaneously to locate the optimal
termination.Such simultaneous monitoring of training and testing
model errors is beneﬁcial as it avoids overtraining the models.
Overtraining leads to a further decrease in the training error but
the corresponding error for testing data increases,thereby
reducing the prediction capability of the ANN model.The
statistical measures used by Mohamed et al.(2005) and Aza
mathulla et al.(2010) (i.e.coefﬁcient of determination (R
2
),
mean absolute error (MAE) and root mean square error (RMSE))
were evaluated as a measure of performance of the ANN models
R
2
¼
X
d
s,obser
d
s,pred
X
d
2
s,obser
X
d
2
s,pred
1=2
2
6
4
3
7
5
2
10a:
in which
X
1
X
2
y
Output layer
X
n
Input ayerl
Hidden layer
Figure 2.Architecture of artiﬁcial neural network model (Bateni et
al.,2007)
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expression programming
Khan,Azamathulla,Tufail and Ab Ghani
d
s,obser
¼ (d
s,obser
)
i
d
s,obser
d
s,pred
¼ (d
s,pred
)
i
d
s,pred
where d
s,obser
represents observed values,
d
s,obser
is the mean of
d
s,obser
,d
s,pred
is the predicted value and
d
s,pred
is the mean of
d
s,pred
:
MAE ¼
X
n
i¼1
e
i
j j
n
10b:
RMSE ¼
X
n
i
e
2
i
n
!
1=2
10c:
where e
i
is the difference between the observed and predicted
scour and n is the number of data points used.The results of the
ANN model for training and validation data sets are discussed in
Section 9.
7.Overviewof GEP
GEP is a new evolutionary AIbased technique developed by
Candida Ferreira in 1999 as an extension of GP developed by
Koza (1992).The genome is encoded as linear chromosomes of
ﬁxed length (just as in GAs) that are then expressed as phenotype
in the form of expression trees by GEP.GEP combines the
advantages of both its predecessors,GAs and GP,while eliminat
ing some of the limitations of the two.GEP is a fully ﬂedged
genotype/phenotype system in which both are dealt with sepa
rately.
In GEP,as in other evolutionary methods,the process starts from
the random generation of an initial population that consists of
individual chromosomes of ﬁxed length.The chromosomes may
be unigenic or multigenic.Each individual chromosome of the
initial population is then evaluated and their ﬁtness is computed
using a ﬁtness function based on the mean square error (MSE).
These chromosomes are then selected based on the ﬁtness value
using a roulette wheel selection process,with ﬁtter chromosomes
having an increased chance of selection into the next generation.
After selection,they are reproduced with some modiﬁcations
carried out by genetic operators (e.g.mutation,inversion,trans
position and recombination).Mutation is found to be the most
effective genetic operator and in most cases is found to be the
only operator used to modify the chromosomes.The new
individuals are then subjected to the same process of modiﬁcation
and the process continues until the maximum number of genera
tions is reached or the required accuracy is achieved (Ferreira,
2001a,2001b).In GEP,several genetic operators are used for
genetic modiﬁcation of the chromosomes (Ferreira,2006).
(a) Mutation.This is the most important and inﬂuential of all the
operators.In GEP modelling,mutation can take place at any
position in a genome.However,the structural organisation of
the chromosomes must remain the same;that is,in the head
of a gene,a function can be replaced by either another
function or a terminal but,in the tail of a gene,terminals can
only change into other terminals as there is no function in the
tail.In this way all new individuals produced by mutation are
structurally correct programs.
(b) Inversion.In this operator a sequence within the head of a
gene is selected and is inverted.It randomly chooses the
chromosome,the gene to be modiﬁed and the start and
terminal points of the portion of head to be inverted.
(c) Insertion sequence (IS) transposition.IS elements are short
portions of the genome having a function or terminal at the
ﬁrst position.This operator randomly chooses the
chromosome,the gene to be modiﬁed and the start and end of
the IS element and transposes it to the start of the gene just
after the root.
(d) Root insertion sequence (RIS) transposition.This is a short
fragment of the genome like the IS element with the only
difference being that here the starting point is always a
function.RIS randomly selects the chromosome,the gene to
be modiﬁed and the start and end points of the RIS element
and transposes it to the start point of the gene.
(e) Gene transposition.In gene transposition,an entire gene
works as a transposon and transposes itself to the beginning
of the chromosome.In contrast to the other forms of
transposition,in this operation the transposon (the gene) is
deleted at the place of origin.
( f ) Single or double crossover/recombination.In single cross
over,the parent chromosomes are paired and a common point
is selected.The portion of the gene downstream of the cross
over point is then exchanged between the two chromosomes.
In double crossover,two parent chromosomes are paired and
two points are randomly chosen as crossover points.The
material between the crossover points is then exchanged
between the parent chromosomes,forming two new offspring
chromosomes (Guven and Aytek,2010).
(g) Gene crossover.Entire genes are exchanged between two
parent chromosomes,forming two offspring chromosomes
containing genes from both parents.The exchanged genes are
randomly chosen and occupy exactly the same position in the
parent chromosomes.
Since a random numerical constant is a crucial part of any
mathematical model,it must be taken into account in deriving an
empirical expression for the response function being modelled.
GEP has the ability to handle random numerical constants
efﬁciently given a userdeﬁned range of minimum and maximum
values.
Recent applications of GEP in different ﬁelds include sediment
transport in sewer pipes system (Ab Ghani and Azamathulla,
2011),rainfall–runoff model development (Fernando et al.,
487
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Khan,Azamathulla,Tufail and Ab Ghani
2009),constructing sentence ranking functions (Xie et al.,2004),
stage–discharge relationships (Guven and Aytek,2010),hydraulic
data prediction (Eldrandaly and Negm,2008),time series model
ling (Ba
˘
rbulescu and Ba
˘
utu,2009),hypertension therapy (Hirano
et al.,2008),image compression and multiobjective classiﬁca
tion rule mining (Dehuri and Cho,2008).
8.GEP modelling of bridge pier scour depth
using ﬁeld data
The available data sets (total of 370) were divided into training
(281) and testing (89) data sets.After data division,different
parameters for the model were chosen,as demonstrated in the
following sixstep procedure.
(a) Multigenic chromosomes (consisting of three genes) were
used for the initial population of individuals.Any size can be
used for the initial population but a population in the range
30–100 chromosomes has given good results in the past
(Ferreira,2001b).After several trials,a population size of 50
chromosomes was selected as the optimal size and was
subsequently used in all GEPbased models.
(b) After initialising the population,the individuals were
evaluated and their ﬁtness function computed using the MSE
as the ﬁtness function
f
i
¼ 1000
1
1 þ E
i
for E
i
¼ P
ij
O
j
11:
where P
ij
is the value predicted by individual chromosome i
for ﬁtness case j and O
j
is the observed value for ﬁtness case
j.P
ij
¼O
ij
means that E
ij
¼0,representing a perfect solution
with no error.
(c) After selecting the ﬁtness function,the next step is to decide
the set of terminals and set of functions for each gene of the
chromosome.Here,the four basic arithmetic operators were
used as functions (F ¼{+,,3,4}) while the set of
terminals
T ¼ F
r
,
d
50
Y
,
b
Y
,
L
Y
,,?
was used (terminal?represents the random numerical
constants).Note that the set of terminals was ﬁxed to
represent the decision variables as given in Equation 2.
However,GEP allows the use of a more diverse range of
functions or subfunctions.For instance,these decision
variables can be grouped together to form one parameter for
use in the GEP model;other functions (e.g.SQRT,EXP) can
also be used.The goal in this exercise was to use the basic
parameters by setting them to the decision variables given in
Equation 2 and seek a relatively simple and compact
expression for scour depth.Such a control does,however,
limit the capability of the GEP models and future work will
focus on model sensitivity by varying the complexity of these
functional arrangements.
(d) The next step is to decide on the number of genes and the
length of the head and tail for each gene in a chromosome.
According to Ferreira (2001b),increasing the number of
genes from one to three considerably increases the success
rate;therefore,after some trials,three genes per chromosome
were used.Head length was taken equal to ten (h ¼10) and,
since the maximum number of arguments per function is
equal to two (n
max
¼2),the tail length t can be calculated
from t ¼10 3(2 – 1) + 1,giving t ¼11.To account for the
random numerical constants,an additional domain D
c
of
length equal to the tail of gene was introduced.Five ﬂoating
type random numerical constants will be selected in the range
{10,10}.So,the lengths of the gene equal
10 + 11 + 11 ¼32.Since there are three genes per
chromosome,chromosome length is thus equal to 96.
(e) After ﬁnalising the chromosome architecture,genetic
operators and rates were decided.All genetic operators such
as mutation,inversion,transposition (IS,RIS and gene
transposition),recombination or crossover (onepoint,two
point and gene recombination),and D
c
speciﬁc genetic
operators were used.Two onepoint mutations with mutation
rate of 0
.
044 were used.The rates of the remaining genetic
operators are given in Table 2.
Population size 50
Number of generations 342000
Function set +,,3,4
Terminal set F
r
,d
50
/Y,b/Y,L/Y,,?
Random constant array length 05
Random constant type Floating point
Random constant range [10,10]
Head length 10
Gene length 32
Number of genes 03
Chromosome length 96
Linking function +
Mutation rate 0
.
044
Inversion rate 0
.
1
IS transposition rate 0
.
1
RIS transposition rate 0
.
1
Gene transposition rate 0
.
1
Onepoint recombination rate 0
.
1
Twopoint recombination rate 0
.
3
Gene recombination rate 0
.
3
D
c
speciﬁc mutation rate 0
.
044
D
c
speciﬁc inversion rate 0
.
1
D
c
speciﬁc IS transposition rate 0
.
1
Random constant mutation rate 0
.
01
Table 2.Parameters of the optimised GEP model
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Khan,Azamathulla,Tufail and Ab Ghani
( f ) The last step is to select the linking function.We have three
genes,resulting in three different subexpression trees.To get
the ﬁnal solution,these subexpression trees must be linked
through some linking function.In this study,the addition
operator (+) was used as the linking function.
After all the parameters are deﬁned,the model is simulated.
The powerful soft computing software package GeneXproTools
4.0 (Ferreira,2006) was used to develop GEPbased models
for bridge pier scour depth prediction in this work.This
program provides a compact and explicit mathematical expres
sion for the bridge scour model.The terminating criterion was
the maximum ﬁtness function,which in turn is a function of
the MSE.The program was run for a number of generations
and was stopped when there was no improvement in ﬁtness
function value or coefﬁcient of determination.After some trials
it was found that there was no appreciable change after
342 000 generations.A simple setting for the GEP is shown in
Table 2.
The best of generation gave a ﬁtness value of 970
.
3 for d
s
/Y and
an R
2
value of 0
.
76.The explicit equation obtained from the GEP
model for d
s
/Y is given in Equation 12 and the corresponding
expression trees are shown in Figure 3.
Subexpression tree 1
Subexpression tree 2
Subexpression tree 3
d
1
d
3
c
0
d
4
c
0
c
1
c
0
d
2
c
0
d
2
d
2
d
4
c
1
d
2
d
2
c
0
d
2
d
0
d
0
c
1
c
1
d
4
d
1
Figure 3.Expression trees for the GEP formulation
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Bridge pier scour prediction by gene
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Khan,Azamathulla,Tufail and Ab Ghani
d
s
Y
¼
b
Y
0
:
595 F
r
d
50
Y
b
Y
2
"#
þ F
r
F
r
þ0
:
063
d
50
Y
d
50
Y
2
1
ð Þ
þ F
r
b
Y
F
r
3
:
24d
50
=Y
F
r
1
ð Þ
12:
9.Results and discussion of GEP modelling
The performance of GEP model was evaluated by comparing its
performance with that of other models including conventional
regression and ANN models.The regressionbased empirical
equations used for comparison are those derived by Breusers et
al.(1977) (Equation 3),Jain and Fischer (1979) (Equation 4),
Froehlich (1988) (Equation 5) and Richardson and Davis (2001)
(Equation 6) and the newly developed MLRbased equation
(Equation 8).The statistical measures R
2
,RMSE and AAE
(average absolute error) were calculated for all the models (Table
3).Scatter plots for the models are shown in Figures 4–7.
Table 3 shows that the newly developed MLRbased model
performs better than the four other regressionbased models
considered and that the results of the Richardson and Davis
(2001) model are superior to the other three previously derived
empirical formulae.
The relatively inferior performance of the regressionbased
models further strengthens the notion that such models are not
always suitable for effectively predicting bridge pier scour depth.
Comparison of Figures 4,5 and 6 indicates that the new MLR
based equation performs better than the four previously devel
oped regressionbased equations,but Figures 6 and 7 show that it
is inferior to the AIbased models of ANNs and GEP.
Table 3 shows that GEP performs better than the ANN model:it
produced smaller values for RMSE and AAE and a slightly
greater value of R
2
:The training and testing scatter plots of the
Model Training Testing
AAE RMSE R
2
AAE RMSE R
2
Breusers et al.(1977) 1
.
0717 1
.
8834 0
.
23 0
.
4517 0
.
7288 0
.
15
Jain and Fischer (1979) 0
.
3326 0
.
6556 0
.
52 0
.
2619 0
.
4267 0
.
27
Froehlich (1988) 0
.
6892 0
.
8957 0
.
36 0
.
6720 0
.
4165 0
.
32
Richardson and Davis (2001) 0
.
8340 1
.
4600 0
.
65 0
.
6303 0
.
9652 0
.
13
New MLRbased equation 0
.
1617 0
.
2348 0
.
65 0
.
0778 0
.
1069 0
.
56
ANN 0
.
1420 0
.
2017 0
.
74 0
.
0778 0
.
1122 0
.
65
GEP 0
.
1402 0
.
1947 0
.
76 0
.
0574 0
.
0854 0
.
74
Table 3.Summary of model results based on statistical measures
Predicted relative scour depth:/dY
s
Training
1
0
1
2
3
4
5
6
7
8
0
1
2 3
4 5
6
7
8
Observed relative scour depth:d Y
s
/
Breusers. (1977)et al
Jain and Fischer (1979)
Froehlich (1988)
Richardson and Dani (2001)
Figure 4.Comparison of equations of Breusers et al.(1977),Jain
and Fischer (1979),Froehlich (1988) and Richardson and Davis
(2001) for training data
2
1
0
1
2
3
4
5
6
0
1
2 3
4 5
6
Observed relative scour depth:/d Y
s
Predicted relative scour depth:dY
s
/
Breusers. (1977)et al
Jain and Fischer (1979)
Froehlich (1988)
Richardson and Dani (2001)
Testing
Figure 5.Comparison of equations of Breusers et al.(1977),Jain
and Fischer (1979),Froehlich (1988) and Richardson and Davis
(2001) for testing data
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Water Management
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Bridge pier scour prediction by gene
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Khan,Azamathulla,Tufail and Ab Ghani
new MLRbased method,the ANN and GEP (Figures 6 and 7
respectively) show that GEP performs better than the ANN.GEP
has the unique property of providing an easy to use explicit
expression (Equation 12) and is thus far superior to the ANN.
In summary,the regressionbased equations are of low perform
ance and are not suitable for effective design purposes.Although
the equation proposed by Richardson and Davis (2001) and the
newly developed regressionbased equation perform reasonably
well,they cannot compete with AIbased techniques such as
ANN and GEP.GEP performs better than ANNs with respect to
statistical measures and scatter plots.GEP has the ability to
provide an explicit and compact empirical expression that should
be helpful for designers.
10.Conclusions
Bridge pier scour is a complex phenomenon and scour depths
need to be predicted accurately.The use of new AIbased models
for bridge scour modelling adds to the limited applications that
exist in this area.Bridge scour modelling is challenging owing to
the signiﬁcant variability in the various input decision variables.
This is encountered in both datadriven and processbased
(deductive) modelling approaches.While deductive models may
be preferred owing to their ability to better reﬂect the true
dynamics of the process or processes modelled,there are
scenarios where this is not possible (e.g.computational expense
constraints,lack of extensive knowledge of the process being
modelled,budgetary or other nonmonetary constraints).In such
scenarios,datadriven models can be effectively used to model
bridge pier scour based on available ﬁeld or laboratory data.
This paper investigated the use of MLR and AIbased inductive
models for predicting relative bridge pier scour depth utilising
previously collected ﬁeld data reported by Landers and Mueller
(1996).The paper explored the utility of a range of datadriven
modelling techniques,from simple (MLR) to complex (AIbased)
in nature.In particular,a new AIbased soft computing technique,
GEP,was applied for the prediction of bridge pier scour depth
using ﬁeld data and its performance was compared with regres
sionbased and ANN models.The performance of the optimal
empirical model developed using GEP was found to be signiﬁ
cantly better than all regressionbased models (existing equations
as well as the new MLR model developed here) and slightly
better than the ANN model in terms of statistical measures.Table
3 shows that R
2
,AEE and RMSE for GEP are superior to the
regression models and the ANN model.
GEP has the added advantage that it results in an explicit and
compact equation (Equation 12) that can be used by engineers in
bridge design.This capability of GEP makes it unique and more
effective when compared with the other models evaluated in this
paper.While the statistical measure of performance of the
regression models was inferior to the AIbased models,it should
be noted that such traditional regression models continue to be
popular due to their ease of use and simple model structures.
Furthermore,since AIbased models are not readily available to
all engineers,their use in most cases is restricted to academic and
research purposes.It is hoped that this paper highlights the utility
of AIbased models with a view to increase their usage by
engineers and planners working on bridge scour problems.
The study also validates the promise of GEP as an effective
modelling tool for applications in hydraulic modelling.GEP
comes with the added advantage of providing a simple and easy
to use empirical expression for the response function modelled.
Conversely,ANNbased models require considerable data for
training and are not favourable in applications where the
objective is to obtain a simple,easy to use and functionally
compact approximation.As the number of hidden layers and
number of neurons in each hidden layer increase,the functional
form extracted from these socalled black box models can turn
out to be a long expression (a linear and nonlinear combination
of sigmoidal functions) with numerous terms.
Training
0
1
2
3
4
0
1
2 3
4 5
Observed relative scour depth:/d Y
s
Predicted relative scour depth:/dY
s
New MLR
ANN
GEP
Figure 6.Comparison of new MLR,ANN and GEP models
(training data)
Testing
0
0·2
0·4
0·6
0·8
1·0
0 0·2 0·4 0·6 0·8 1·0
Observed relative scour depth:/d Y
s
Predicted relative scour depth:/dY
s
New MLR
ANN
GEP
Figure 7.Comparison of new MLR,ANN and GEP models (testing
data)
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Water Management
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Khan,Azamathulla,Tufail and Ab Ghani
Acknowledgements
This investigation was carried out at the River Engineering and
Urban Drainage Research Centre (REDAC),Universiti Sains
Malaysia,during a study visit by the ﬁrst author.The authors are
grateful to the anonymous reviewers for their valuable comments.
The authors also thank Ravikanth Chittiprolu for his assistance in
preparation of the manuscript.
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Bridge pier scour prediction by gene
expression programming
Khan,Azamathulla,Tufail and Ab Ghani
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