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 regression-based 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

scour-related 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

regression-based 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 cause-and-effect 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 (data-driven)

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 cause-and-effect

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 data-driven 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 AI-based 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 cause-and-effect relationship of such complex processes.Such

AI-based techniques include artiﬁcial neural networks (ANNs),

adaptive neuro-fuzzy 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

non-linear 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 GEP-based models for pier scour prediction utilising

available ﬁeld data and comparing their performance with ANN

and regression-based models.A further objective was to evaluate

the utility of GEP-based 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 ANN-based

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 ANN-based 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

482

Water Management

Volume 165 Issue WM9

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 (data-driven) 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 non-dimensional 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 MLR-based

equation,ANN and GEP models.

3.Development of data-driven 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 data-driven 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 AI-based models (ANNs and GEP) for bridge scour

prediction and compares their performance with traditional

regression-based models.The models studied thus comprise

(a) previously developed regression-based 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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Water Management

Volume 165 Issue WM9

Bridge pier scour prediction by gene

expression programming

Khan,Azamathulla,Tufail and Ab Ghani

structures such as regression-based 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 MLR-based 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 regression-based models will then be compared to that

of the more complex and non-linear AI-based models including

ANNs and GEP-based 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 HEC-18).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 regression-based models

for comparison with the new MLR,ANN and GEP-based 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 regression-based 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 square-nosed piers,1

.

0 for round-

nosed piers and 0

.

7 for sharp-nosed 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 HEC-18 (Richardson

and Davis,2001) and was recently used by Mohamed et al.

(2005) for comparison with other regression-based equations.

484

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

regression-based equations for comparison with the new MLR,

ANN and GEP-based models.

4.1.5 Comparison

Gaudio et al.(2010) compared six different regression-based

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 GEP-based 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 in-depth analysis of how these

equations were derived,the reader is referred to the respective

references.

4.2 New MLR-based data-driven 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 MLR-based 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.AI-based data-driven 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 cause-and-effect

relationship between inputs and output variables.Recognising

these difﬁculties and the importance of improving prediction

capabilities,this work investigated the utility of two AI-based

inductive models – ANNs and a relatively new GP-based

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.

ANN-based models such as the popular multi-layer feed-forward

networks have frequently been used to approximate the response

of a particular system by training with available data.ANN

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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 feed-forward

type network was trained using the back-propagation 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 back-propagation 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 trial-and-error method (Haykin,1994;Maier and Dandy,

1998).The back-propagation 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 feed-forward 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 over-training the models.

Over-training 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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Bridge pier scour prediction by gene

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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 AI-based 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 uni-genic or multi-genic.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 cross-over/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 cross-over,two parent chromosomes are paired and

two points are randomly chosen as cross-over points.The

material between the cross-over points is then exchanged

between the parent chromosomes,forming two new offspring

chromosomes (Guven and Aytek,2010).

(g) Gene cross-over.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 user-deﬁned 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.,

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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 multi-objective 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 six-step procedure.

(a) Multi-genic 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 GEP-based 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 sub-functions.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 cross-over (one-point,two-

point and gene recombination),and D

c

-speciﬁc genetic

operators were used.Two one-point 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

One-point recombination rate 0

.

1

Two-point 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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( f ) The last step is to select the linking function.We have three

genes,resulting in three different sub-expression trees.To get

the ﬁnal solution,these sub-expression 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 GEP-based 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.

Sub-expression tree 1

Sub-expression tree 2

Sub-expression 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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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 regression-based 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 MLR-based 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 MLR-based model

performs better than the four other regression-based 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 regression-based

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 regression-based equations,but Figures 6 and 7 show that it

is inferior to the AI-based 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 MLR-based 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 MLR-based 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 regression-based 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 regression-based equation perform reasonably

well,they cannot compete with AI-based 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 AI-based 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 data-driven and process-based

(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 non-monetary constraints).In such

scenarios,data-driven 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 AI-based 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 data-driven

modelling techniques,from simple (MLR) to complex (AI-based)

in nature.In particular,a new AI-based soft computing technique,

GEP,was applied for the prediction of bridge pier scour depth

using ﬁeld data and its performance was compared with regres-

sion-based and ANN models.The performance of the optimal

empirical model developed using GEP was found to be signiﬁ-

cantly better than all regression-based 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 AI-based 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 AI-based 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 AI-based 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,ANN-based 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 so-called black box models can turn

out to be a long expression (a linear and non-linear 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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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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Khan,Azamathulla,Tufail and Ab Ghani

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