ARTICLE IN PRESS

Engineering Applications of Artificial Intelligence 22 (2009) 216–223

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence

0952-19

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/engappai

Application of support vector machines in scour prediction ongrade-control structures

Arun Goel �, Mahesh Pal

Department of Civil Engineering, National Institute of Technology, Kurukshetra, 136119 Haryana, India

a r t i c l e i n f o

Article history:

Received 14 September 2007

Received in revised form

2 May 2008

Accepted 30 May 2008Available online 6 December 2008

Keywords:

Grade-control structures

Scour

Modeling

Support vector machines

Back propagation neural network

76/$ - see front matter & 2008 Elsevier Ltd. A

016/j.engappai.2008.05.008

esponding author.

ail address: drarun_goel@yahoo.co.in (A. Goel

a b s t r a c t

Research into the problem of predicting the maximum depth of scour on grade-control structures like

sluice gates, weirs and check dams, etc., has been mainly of an experimental nature and several

investigators have proposed a number of empirical relations for a particular situation. These traditional

scour prediction equations, although offer some guidance on the likely magnitude of maximum scour

depth, yet applicable to a limited range of the situations. It appears from the literature review that a

regression mathematical model for predicting maximum depth of scour under all circumstances is not

currently available. This paper explores the potential of support vector machines in modeling the scour

from the available laboratory and field data obtained form the earlier published studies. To compare the

results, a recently proposed empirical relation and a feed forward back propagation neural network

model are also used in the present study. The outcome from the support vector machines-based

modeling approach suggests a better performance in comparison to both the empirical relation and

back propagation neural network approach with the laboratory data. The results also suggest an

encouraging performance by the support vector machines learning technique in comparison to both

empirical relation as well as neural network approach in scaling up the results from laboratory to field

conditions for the purpose of scour prediction.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Scour is the removal of sediments around or near structureslocated in flowing water. It means lowering of bed level by watererosion. The grade-control structures like sluice gate, weirs, dams,etc., help in preventing excessive bed degradation in the alluvialchannels. The safety of downstream apron is jeopardized due tothe downstream scour caused by erosive action of fluid jets whichmay lead to failure of the hydraulic structures, hence adequateprotective measures can only be designed with full understandingand estimation of the magnitude and location or extent of thedownstream scour. The scouring phenomenon downstream of anapron rigid or flexible is quite complex in nature due to the abruptchange in the flow characteristics on the sediment bed with time.The maximum depth of scour is a function of many parameterslike jet angle, flow rate, drop height, tail water, type of material,run time, width of channel, diffusion length and many more. Thescour initiate at the downstream edge of the apron when the bedshear stress exerted by the submerged jet exceeds the critical bedshear stress for the bed sediments. The evolution of the vertical

ll rights reserved.

).

dimensions of scour hole is much faster than longitudinal one. Inthe initial stage, the suspension of sediment is the only means ofsediment transport. But with the development of the verticaldimensions of the scour hole, the mode of sediment transportchanges to a combination of suspended and as well as bed loads.The equilibrium scour conditions are reached asymptotically asrate of increase of scour approaches zero. The prediction of scouris difficult because flow in open channel is turbulent, geometry isirregular and vary with time and sediment transport is of complexin nature (Dey and Sarkar, 2006).

Prediction of maximum depth of scour with a reasonableaccuracy is of immense importance for proper planning, designand management of hydraulic structures. Scouring is a complexand dynamic phenomenon affected by many parameters thoseare often interrelated and difficult to understand. However,deterministic models of varying degree of complexity have beenemployed in the past for modeling the scouring process, withvarying degree of accuracy. A number of investigators (Bormanand Julien, 1991; Chee and Padiyar, 1969; D’Agostino and Ferro,2004; Mason and Arumugam, 1985; Mossa, 1998) have studiedthe scour downstream of a grade-control structure and proposedempirical equations of scour prediction on the basis of laboratory/field experiments under different conditions of flow, time,material, tail water, type of structure, etc. These researchers have

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Nomenclature

A50 dimensionless group associated to grain size d50 anda1, a2, a3, a4, a5, a6, a7 numerical constants

B channel widthb weir widthC regularization parameterd biasd* degree of polynomiald50 standard sieve diameter for which 50% of particles are

finerdm mean particle sized90 bed grain size for which 90% of sampled particles are

finerg acceleration due to gravity

H difference in height in water level upstream weir tothe tail water level

h tail water depth or water depth above the unerodedbed level

h0 total head above the weir crestQ water dischargeq ¼ Q/B water discharge per unit widths maximum scour depthz fall heightl downstream face angle of the grade-control structureg Kernel widthx, x0 slack variableski, l

0

i;Zi;Z0i Lagrange multiplierse error probability

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223 217

mainly relied on the conventional experimental approach to studythe scouring by way of physical modeling.

Recently, D’Agostino and Ferro (2004) have suggested anempirical relationship based on the incomplete self-similaritytheory using laboratory as well as field data and compared itsperformance with another empirical relation as proposed byMason and Arumugam (1985). However, a study by Liriano andDay (2001) reported the improved performance by a neuralnetwork approach in comparison to the exiting empirical relationsfor the scour prediction near a culvert. Three-dimensionalcomputational fluid dynamics model (FLUENT) has also beenused for local scour estimates around a bridge pier as well as tocompare its performance with laboratory data. The study hasshown good results obtained by computational fluid dynamicsmodeling technique in simulating complex flow field andinvestigating the scale effects (Yang, 2005).

The artificial neural networks (ANN) adapt itself to reproducethe desired output when presented along with input data sets.Most of the studies employing neural networks for water resourceproblems have used back propagation and radial basis functiontypes of neural networks. A neural network-based modelingapproach requires setting up several user-defined parameterslike learning rate, momentum, optimal number of nodes in thehidden layer and the number of hidden layers, so as to have a lesscomplex network with a relatively better generalization capabil-ity. Further, training a neural network requires a number ofiterations and a large number of training iterations may force ANNto over train, which may affect the predicting capabilities of themodel. Keeping in view these limitations of ANN, this studyexplores the potential of another machine learning approachcalled support vector machines (SVMs) in predicting the max-imum depth of scour using laboratory as well as field data.

The modeling techniques like SVMs have the capability toreproduce the unknown relationship present between a setof input variables and the output of the system. SVMs perfor-mance was found to be better due to its use of the structuralrisk minimization principle in formulating cost functions andof quadratic programming during model optimization. Theseadvantages lead to a unique optimal and global solution ascompared to conventional neural network models.

So far, several studies have reported the use of SVMs in waterresource management but no study has reported its use for thescour prediction. Pal and Goel (2006, 2007) have successfullyapplied SVMs in predicting end-depth ratio and discharge forcircular, semi-circular and trapezoidal channels and found itperforming well in comparison to the empirical relations.Pao-shan et al. (2006) have successfully explored the usefulness

of SVMs-based modeling technique for predicting of real-timeflood stage forecasting on Lan-Yang River in Taiwan 1–6 h ahead.Khan and Coulibaly (2006) has used SVMs in the Lake Erie for thepurpose of water level prediction and compared the results withback propagation neural network model called as multilayerperceptron (MLP). The SVMs showed good performance and isproved to be competitive with MLP. Recently, study by Singh et al.(2008) also suggests a better performance by SVMs in comparisonto the back propagation neural network in estimation of removalefficiency for settling basins.

Keeping in view all these findings, the present study aims toexplore the utility of the SVMs for the scour modeling onlaboratory and field data set and comparing its performance withthe empirical relation proposed by D’Agostino and Ferro (2004) aswell as with a back propagation neural network. Further, itsusefulness is also tested in predicting the scour from a reducedmodel scale (laboratory data) to the large-scale field data.

2. Support vector machines (SVMs)

SVMs are classification and regression methods, which havebeen derived from statistical learning theory (Vapnik, 1995, 1998).The SVMs classification methods are based on the principle ofoptimal separation of classes. If the classes are separable—thismethod selects, from among the infinite number of linearclassifiers, the one that minimize the generalization error, or atleast an upper bound on this error, derived from structural riskminimization. Thus, the selected hyper plane will be one thatleaves the maximum margin between the two classes, wheremargin is defined as the sum of the distances of the hyper planefrom the closest point of the two classes (Vapnik, 1995).

If the two classes are non-separable, the SVMs tries to find thehyper plane that maximizes the margin and that, at the same time,minimizes a quantity proportional to the number of misclassifica-tion errors. The trade-off between margin and misclassificationerror is controlled by a positive constant that has to be chosenbeforehand. This technique of designing SVMs can be extended toallow for non-linear decision surfaces. This can be achieved byprojecting the original set of variables into a higher-dimensionalfeature space and formulating a linear classification problem in thefeature space (Vapnik, 1998). SVMs can be applied to regressionproblems and can be formulated as given below:

Vapnik (1995) proposed e-support vector regression (SVR) byintroducing an alternative e-insensitive loss function. The purposeof the SVR is to find a function having at most e deviation from theactual target vectors (yi) for all given training data and have to be

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A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223218

as flat as possible (Smola, 1996). This can be put in other words asthe error on any training data has to be less than e. For a giventraining data with k number of samples, represented by (x1, y1),(xk, yk) and a linear function

f ðx; aÞ ¼ hw; xi þ d (1)

where wARN and dAR. /w, xS represents the dot product in spaceRN and N is the dimension of input space. A smaller value of windicates the flatness of Eq. (1), which can be achieved byminimizing the Euclidean norm as defined by wk k2 (Smola, 1996).Thus, an optimization problem for this can be written as

minimize1

2wk k2

subject toyi � hw;xii � dp�

hw;xii þ d� yip�

((2)

The optimization problem in Eq. (2) is based on the assumptionthat there exists a function that provides an error on all trainingpairs which is less than e. In real life problems, there may be asituation like one defined for classification by Cortes and Vapnik(1995), which involves in finding a separating hyperplane thatgeneralize well. Further, the dimensionality of the feature spacewill be very large and not all hyperplanes that separate thetraining data will generalize well. Thus, Cortes and Vapnik (1995)suggested the concept optimal hyperplane that separate the twoclasses by maximum margin.

So, to allow some more error in the design of SVMs-basedregression, slack variables x, x0 can be introduced and theoptimization problem defined in Eq. (2) can be written as

minimize1

2wk k2 þ C

Xk

i¼1

ðxi þ x0iÞ

subject toyj � hw;xji � dp�þ xj

hw;xji þ d� yip�þ x0j(3)

and xi; x0

jX0 for all i ¼ 1,2, yy, k.The parameter C is determined by the user and it determines

the trade-off between the flatness of the function and the amountby which the deviations to the error more than e can be tolerated.The optimization problem in Eq. (3) can be solved by replacing theinequalities with a simpler form determined by transforming theproblem to a dual space representation using Lagrange multipliers(Leunberger, 1984).

The Lagrangian is formed by introducing positive Lagrangemultipliers li, l

0

i;Zi;Z0i; i ¼ 1, y, k and multiplying the constraintequations by these multipliers, and finally subtracting the resultsfrom the objective function (i.e., wk k2). The Lagrangian for Eq. (3)can be written as

L ¼1

2wk k2 þ C

Xk

i¼1

ðxi þ x0Þ �Xk

i¼1

lið�þ xi � yi þ hw;xii þ dÞ

�Xk

i¼1

l0ið�þ x0i þ yi � hw;xii � dÞ �Xk

i¼1

ðZixi þ Z0ix0

iÞ (4)

The solution of this optimization problem can be obtained bylocating the saddle point of the Lagrange function defined inEq. (4). The saddle points of Eq. (4) can be obtained by equatingpartial derivative of L with respect to w, d, xi and x0i to zero andgetting:

w ¼Xk

i¼1

ðl0i � liÞxi

Zi ¼ C � li

Z0i ¼ C � l0i (5)

Substituting Eq. (5) in Eq. (4), results in the optimization problemof maximizing:

�1

2

Xk

i¼1

Xk

j¼1

ðl0i � liÞðl0

j � ljÞðxixjÞ � �Xk

i¼1

ðl0i þ liÞ þXk

i¼1

yiðl0

i � liÞ

subject toXk

i¼1

ðl0i � liÞ ¼ 0 and 0pl0i; lipC. (6)

Eq. (6) is a quadratic programming problem and can be solved toget the values of l0i and li. The prediction problem in Eq. (1) cannow be written as

f x;að Þ ¼Xk

i¼1

ðl0i � liÞhxi;xi þ d (7)

The techniques discussed above can be extended to allow for non-linear SVR by introducing the concept of the kernel function(Vapnik, 1995). This is achieved by mapping the data into ahigher-dimensional feature space, thus performing linear regres-sion in feature space. The regression problem in feature space canbe written by replacing xi �xj in Eq. (6) with U(xi) �U(xj). Thus, theoptimization problem for this can be written as

maximize �1

2

Xk

i¼1

Xk

j¼1

ðl0i � liÞðl0

j � ljÞKðxi � xjÞ

� �Xk

i¼1

ðl0i þ liÞ þXk

i¼1

yiðl0

i � liÞ

subject toXk

i¼1

ðl0i � liÞ ¼ 0 and 0pl0i; lipC.

Where

Kðxi;xjÞ � UðxiÞUðxjÞ

The regression function for this can be written as

f ðx;aÞ ¼Xk

i¼1

ðl0i � liÞhxi;xi þ d (8)

In this optimization problem, the kernel function is computedrather than U(x). This approach consists in moving the trainingdata set into a higher-dimensional feature space where thetraining data may be spread further apart and a larger marginmay be found for the optimal hyperplane. A number of kernelfunctions can be used with SVMs. Vapnik (1995) discusses kernelsand the parameters of the kernel function in detail. For this study,polynomial ((x.y)+1)d and radial basis kernelðe�g x�yj j

2

Þ were usedfor different data combinations. In all figures, smo_poly andsmo_rbf represents polynomial and RBF kernel-based SVMsapproach, respectively. In present study, SVMs and back propaga-tion neural network algorithms as implemented by Witten andFrank (2005) are used.

3. Data set used

To assess the usefulness of SVMs for scour modeling ongrade-control structures, available laboratory and field data sets(Table 1) as reported by Borman and Julien (1991) and D’Agostinoand Ferro (2004) were used in the present study.

Laboratory data set of a total of 114 runs with a channel width(B) of 0.5 m having two different values of b/B 0.3 and 0.6 (b isweir width,) with two weir heights (0.41 and 0.71 m) were takenfrom the work reported by D’Agostino and Ferro (2004). Differentgrain-size distributions were used as a bed material (d50 ¼ 4.1 and11.5 mm, d90 ¼ 7.0 and 17.6 mm) with unit water dischargeranging from 0.0167 to 0.167 m2 s�1. Borman and Julien (1991)

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Table 1Experimental data sets used in the present study

S. no. Name of investigator Range of discharge (m2 s�1) Parameters used to predict

scour

Field/lab or large-scale

data

Laboratory data

1 D’Agostino and Ferro (2004) 0.0167–0.167 B, z, h0, d50, d90, h, Q/B Laboratory

2 Borman and Julian (1991) 0.3–2.5 h0, d50, d90, l, B, z, h, Q/B Large-scale laboratory

data

3 Lenzi et al. (2000) 0.0067–0.0292 h0, d50, z, h, Q/B, B Small scale Laboratory

4 Mossa (1998) 0.0045–0.0148 h0, d50, l, Q/B, z, h, B Laboratory

5 Veronese (1937) 0.001–0.083 h0, dm, z, h, Q/B, B Laboratory

Field data

6 Falciai and Giacomin (1978) 2.2–13.4 h0, d50, z, h, Q/B, B Field data

7 Missiga stream (D’Agostino and Ferro, 2004) 0.375–0.531 h0, d50, z, h, Q/B, B Field data

8 Veronese (1937), Scimemi, (1939), Whittaker

and Schleiss (1984)

4.571–275 h0, d50, z, h, Q/B, B Large-scale field data

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223 219

carried out a large-scale experiment with unit discharge rangingfrom 0.3 to 2.5 m2 s�1. All the experiments (88 numbers) werecarried out in a large outdoor flume with channel width B ¼ 0.91m and b/B ¼ 1. Three downstream angles of the structures(l ¼ 1.57, 0.79, and 0.32 rad) and two different mixtures for thebed material (d50 ¼ 0.3 and 0.45 mm, d90 ¼ 1.58 and 1.71 mm)were used.

Lenzi et al. (2000) used a small-scale laboratory installationwith a channel of bed width equal to 0.6 m. They carried out atotal of 13 experiments with a bed of grain-size distributionof d50 ¼ 0.0085 m while Mossa (1998) used a channel of 0.3 mwidth with b/B ¼ 1 having two different downstream angles(l ¼ 0.79 and 0.32 rad). The discharge was varied from 0.0045 to0.0148 m2 s�1 with bed having a single grain-size distribution(d50 ¼ 2 mm). Veronese (1937) used b/B ¼ 1 and H ¼ 1 (differencein upstream and downstream water level) to carry out a total of 36experiments. The discharge was varied from 0.001 to 0.083 m2 s�1

with four-grain-size distribution (dm ¼ 36.2, 21.0, 14.2, and9.1 mm) as a bed material. However, in present study meandiameter as represented by d50 has been used.

Missiga stream data as reported in D’Agostino and Ferro (2004)was collected by taking a set of 13 field measurements. Thechannel slope in the Missiga stream is about 0.15 and streamdegradation is controlled by using 13 consecutive check damswith different weir width b varies from 7.5 to 10.8 m and fallheights z varies from 0.55 to 1.35 m. All the scour measurementswere taken with a peak discharge of 3.98 m3 s�1. The discharge inthe stream was calculated by using a slope-area method andassuming a critical water depth in the grade-control structure.Bed grain-size distribution was sampled between consecutivecheck dams and the grain size was calculated. Falciai andGiacomin (1978) surveyed 29 check dams located in differentstreams of the Sieve and Casentini valleys in Italy. They observedscour due to the flood events with a return period of 10 years.Rainfall-runoff modeling was used to calculate the flood dischargewhile a weighted sieve analysis was carried out to determine thebed grain-size distribution.

For large-scale scour data as mentioned by D’Agostino andFerro (2004), Veronese (1937) has carried out field observationson the Rocchetta dam (Italy) with a uniform bed grain size(d50 ¼ 100 mm) while Scimemi (1939) used Conowingo dam onthe Susquchanna river to measure the downstream scour.

Whittaker and Schleiss (1984) carried out scour measurementsdownstream of Cabora-Bassa dam in Mozambique. A middle levelsluice gate was used to measure the discharge, while the totalhead above broad-crested weir was calculated using a relation fora broad-crested weir across a rectangular channel.

4. Performance evaluation

The data sets mentioned in Section 3 were used in the presentstudy for model building and validation to assess the potential ofSVMs in predicting the scour on the grade-control structure. Thecorrelation coefficient (CC) and root mean square error (RMSE)values are used for the performance evaluation of models andcomparison of the results for prediction of scour using SVMs. Thehigher value of correlation coefficient and a smaller value RMSEmean a better performance of the model. Further, measured scourvalues were plotted against the computed values obtained withSVMs algorithm. To study the scatter around the line of perfectagreement (i.e. a line at 451), two lines at 725% and 730% errorwere also plotted for laboratory-to-laboratory and laboratory-to-field data, respectively, in each plot. Due to the availability ofsmall data sets, a cross-validation was used to test the perfor-mance of the SVMs. The cross-validation is a method of estimatingthe accuracy of a classification or regression model. The input dataset is divided into several parts (a number defined by the user),with each part in turn used to test a model fitted to the remainingparts. In addition to the empirical relation:

s

z¼ 0:540

b

z

� �0:593 h

H

� ��0:126

ðA50Þ0:544 d90

d50

� ��0:856 b

B

� ��0:751

(9)

as proposed by D’Agostino and Ferro (2004) and a backpropagation neural network was also used to compare theperformance of SVMs.

Data sets as provided by D’Agostino and Ferro (2004) andBorman and Julien (1991) were used to assess the performance ofSVMs under laboratory conditions. While all five laboratory datasets (Table 1, serial numbers 1–5) were used to create the modelsin upscaling results from laboratory to the field conditions.

5. Analysis of data

Much success has already been achieved using neural networkalgorithms in other applications, such as rainfall-runoff modeling,stage-discharge analysis and prediction of dimensions of scourhole downstream of a ski jump bucket (ASCE task committeeon application of ANNs I and II, 2000a, b; Bhattacharya andSolomatine, 2000; Azmathullah et al., 2005). Neural networks arenow being applied to several problems related to the hydraulicsand hydrologic modeling, while the use of SVMs is comparativelynew to the field of water resource engineering (Khan andCoulibaly, 2006; Pal and Goel, 2006, 2007; Pao-shan et al.,2006; Singh et al., 2008). One of the important factors in using

ARTICLE IN PRESS

-25% line

+25% line

0

0.1

0.2

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3Actual scour

Pred

icte

d sc

our

smo_poly

smo_rbf

Equation 9

NN

Fig. 1. Comparison of actual and predicted scour based on the laboratory data by

D’Agostino and Ferro (2004).

+25% line

-25% line

1

1.5

2

2.5

3

Pred

icte

d sc

our

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223220

SVMs for prediction of scour is that it requires setting up of thefew user-defined parameters. The SVMs, in addition to the choiceof kernel require setting up of kernel-specific parameters. Theoptimum values of the regularization parameter C and the size ofthe error-insensitive zone e need to be determined. To select user-defined parameters, i.e. (C, g, and d*) a large number of trials werecarried out by using different combination of these parameters oneach of the data sets. Similarly, a number of trials were alsocarried out to find a suitable value of e (error-insensitive zone)with a fixed value of C and kernel-specific parameters. For thisstudy, a value of 0.0010 was found to be working well for differentdata sets. To reach at a suitable choice of these parameters, thecorrelation coefficients (CC) and RMSE were compared and acombination of parameters providing smallest value of RMSE andthe highest value of correlation coefficient was selected for thefinal results. A number of trials were carried out with differentdata set to select a suitable value of regularization parameter C.Variation in the error-insensitive zone e have no effect onthe predicted scour so a value of 0.0010 was chosen for allexperiments. Similarly a number of trials were carried out to findout the optimal values of user-defined parameters (learning rate,momentum, number of iterations, number of hidden layers, andnodes in each hidden layer) with back propagation neuralnetwork also.

In this study, two different sets of experiments were carriedout. First experiments used the laboratory data for both creatingand testing the models while second set of experiments usedlaboratory data to create the model and the field data sets tovalidate the model. The choice of input parameters used inmodeling the scour may influence the predicting capabilities ofthe SVMs so all the parameters mentioned in Eq. (9) were used inthe present study for comparison of results. These parametersinclude channel width (B), total head above the weir crest (h0),bed grain size (i.e. d50 or dm), tail water depth (h), fall height (z)and discharge per unit width of the channel (q ¼ Q/B).

00 0.5 1 1.5 2 2.5 3

0.5

Actual scour

smo_polysmo_rbfEquation 9NN

Fig. 2. Comparison of actual and predicted scour based on the laboratory data by

Borman and Julien (1991).

5.1. Prediction of scour from laboratory to laboratory data

The first set of analysis was carried out by using SVMs withlaboratory-to-laboratory data for predicting the scour on grade-control structures. The SVMs determine a relationship (i.e. createa model) between the input and the output of the available dataset of any system. These models are then used to predict theoutput from the known input values of the same system, thusrequiring sufficient number of data to create and test the models.Six parameters, i.e. B, z, h0, h, Q/B, d50 from the data sets providedby the studies carried out by D’Agostino and Ferro (2004) andBorman and Julien (1991) were used to predict the scour usinglaboratory data. A 10-fold cross-validation was used to create andtest the models. A number of trials were carried out to reach atthe various user-defined parameters required for the SVMs usingboth polynomial and RBF kernels and neural network-basedalgorithms (i.e. back propagation). Measured versus calculatedvalues of the maximum scour depth are plotted as given in Figs. 1and 2 for both the data sets. Fig. 1 provides results for the data byD’Agostino and Ferro (2004) whereas Fig. 2 provides results forBorman and Julien (1991).

To compare the performance of SVMs algorithms, the resultsobtained by Eq. (9) are also plotted in Figs. 1 and 2. Table 2(a)provides the value of correlation coefficients, RMSE as well asdifferent user-defined parameters are used for D’Agostino andFerro (2004) data set. A correlation coefficient of 0.926 and 0.930(RMSE ¼ 0.0215 and 0.0220) are obtained by using polynomialand RBF kernels, respectively, in comparison to a value of 0.81(RMSE ¼ 0.0355) achieved by using Eq. (9) (Table 2(b)). Further, it

is evident from Fig. 1 that more number of points is lying outsidethe 725% error line when Eq. (9) was used to predict the scour incomparison to the SVMs-based algorithm. Thus, suggesting abetter performance by SVMs modeling technique for scourprediction using laboratory data. In comparison to SVMs, theneural network approach provides a correlation coefficient valueof 0.91 (RMSE ¼ 0.024) with this data set (Table 2(c)), thussuggesting the improved performance by SVMs approach (usingboth polynomial and RBF kernel) with this data set as well.

Another set of analysis involved using laboratory data byBorman and Julien (1991) in creating and testing the modelswith SVMs as well as neural network approach and Eq. (9). Fig. 2provides a plot between actual and predicted scour values byusing SVMs, Eq. (9) and neural network approach. Table 2(a)provides the correlation coefficients and RMSE values obtained byusing SVMs with this data set. A higher value of correlationcoefficient of 0.821 and 0.78 (RMSE ¼ 0.23 and 0.25) are achievedwith polynomial and RBF kernels, respectively. In comparisonto this, Eq. (9) provides a correlation coefficient of 0.63(RMSE ¼ 0.997) Table 2(b)). Table 2(c) provides the value of

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Table 2

(a) Correlation coefficient and RMSE values obtained with SVMs (RBF and polynomial kernels) using laboratory data

Kernel used D’Agostino and Ferro (2004) Borman and Julian (1991)

C g/d* Correlation coefficient RMSE C g/d* Correlation coefficient RMSE

Radial basis kernel 5 1 0.930 0.022 0.1 2 0.78 0.25

Polynomial kernel 5 2 0.926 0.0215 0.1 3 0.821 0.23

(b) Correlation coefficient and RMSE values obtained using Eq. (6) with laboratory data

D’Agostino and Ferro (2004) Borman and Julian (1991)

Correlation coefficient RMSE Correlation coefficient RMSE

0.81 0.0355 0.63 0.997

(c) Correlation coefficient and RMSE values obtained with ANN using laboratory data

Hidden nodes Momentum Learning rate Iterations Correlation coefficient RMSE

D’Agostino and Ferro (2004)

8 0.1 0.2 500 0.91 0.024

Borman and Julian (1991)

8 0.1 0.2 1000 0.78 0.260

-30% line

+30% line

00 1 2 3 4

1

2

3

4

Actual scour

Pred

icte

d sc

our

smo_poly

smo_rbf

Equation 9

NN

Fig. 3. Comparison of actual and predicted scour based on parameters B, z, h0, h, Q/

B and d50 using Falciai and Giacomin (1978) field data.

-30% line

+30% line

00 0.5 1 1.5 2

0.5

1

1.5

2

Actual scour

Pred

icte

d sc

our

smo_polysmo_rbfEquation 9NN

Fig. 4. Comparison of actual and predicted scour based on parameters B, z, h0, h,

Q/B and d50 using Missiga stream field data.

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223 221

correlation coefficient (i.e. 0.78) and RMSE (i.e. 0.26) with thisdata using a back propagation neural network. A perusal of theFig. 2 indicates a better performance by polynomial kernel-basedSVMs for this data set. Results from Figs. 1 and 2 as well asfrom Table 2 indicate a data-dependent nature of SVMs-basedmodeling algorithm.

5.2. Prediction of scour from laboratory to field data

This part of the work involves studying the effectiveness ofSVMs algorithm in scaling up the scour prediction results fromlaboratory to field data. To create the models, five laboratory datasets (Table 1, serial numbers 1–5) as reported in D’Agostino andFerro (2004) and Borman and Julien (1991) were used. Alllaboratory data (i.e. a total of 270 data set) were obtained underdifferent experimental conditions (i.e. different discharge ranges,weir width, bed grain size, tail water depth, fall height, anddifference in height from water level upstream of the weir top thetail water depth). The field data set by Falciai and Giacomin(1978), Missiga stream data as well as large-scale data set

obtained form different field measurements (as provided inD’Agostino and Ferro (2004)) were used to validate these models.Figs. 3–5 present the results with data set by Falciai and Giacomin(1978), Missiga stream and large-scale data, respectively, usingSVMs and neural network-based modeling technique as well as byEq. (9). The correlation coefficients and RMSE values obtained byusing SVMs, feed forward back propagation neural network andEq. (9) were also calculated and furnished in Table 3. With Falciaiand Giacomin (1978) data set, the value of correlation coefficientand RMSE for polynomial kernel-based SVMs were 0.71 and 0.48in comparison to the value of 0 and 1.39 with RBF kernel-basedSVMs (Table 3(a)). While, Eq. (9) (Table 3(b)) and neural network(Table 3(c)) provide a correlation coefficient of 0.66 and 0.34(RMSE ¼ 0.67 and 0.67) with this data set suggesting a betterperformance by polynomial kernel-based SVMs for this data set.Fig. 3 suggests that Eq. (9) over predict the maximum scour depthin comparison to polynomial kernel-based SVMs algorithm asmost of the points lies above the line of perfect agreement.

For Missiga stream data, a correlation coefficient of 0.67 and�0.64 (RMSE ¼ 0.126 and 0.912) (Table 3(a)) were achievedby using RBF and polynomial kernel-based SVMs. In comparison,

ARTICLE IN PRESS

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223222

Eq. (9) (Table 3(b)) and the ANN (Table 3(c)) provide a correlationcoefficient value of –0.48 and 0.66 (RMSE ¼ 0.214 and 0.53),respectively. Results indicate an improved performance isachieved by using RBF kernel-based SVMs with this data set interms of correlation coefficient and RMSE. On the other hand, theplot between actual and predicted scour (Fig. 4) suggests that themost of the predicted values are falling on a straight line, thusindicating a poor performance in upscaling the results for thisdata set by SVMs, Eq. (9), and ANN.

For a large-scale data (Table 1, serial number 8), a correlationcoefficient of 0.0 and 0.99 (RMSE ¼ 16.44 and 1.42) were achievedby using RBF and polynomial kernel-based SVMs, thus suggestingan improved performance by polynomial kernel-based SVMs withthis data set. In comparison, Eq. (9) (Table 3(b)) and the ANN(Table 3(c)) provides a correlation coefficient value of 0.99 and0.67 (RMSE ¼ 2.93and 15.12), respectively. For a large-scale data

-30% line

+30% line

0 5 10 15 20 25 300

10

20

30

Actual scour

smo_poly

smo_rbf

Equation 9

NN

Pred

icte

d sc

our

Fig. 5. Comparison of actual and predicted scour based on parameters B, z, h0, h, Q/

B and d50 using large-scale field data by Veronese (1937), Scimemi (1939), and

Whittaker and Schleiss (1984).

Table 3

(a) Correlation coefficient and RMSE values obtained with SVMs for upscaling the scour fro

Kernel used Falciai and Giacomin (1978) Missiga stream (D’Agostino a

2004)

C g/d* CC RMSE C g/d* CC

Radial basis kernel 0.001 0.7 0 1.39 1 0.01 0.67

Polynomial kernel 0.001 1 0.71 0.48 1 1 �0.64

(b) Correlation coefficient and RMSE values obtained with Eq. (6) for upscaling the scour fr

Using Eq. (9) Falciai and Giacomin (1978) Missiga stream (D’Agostino and Ferro, 2004

Correlation coefficient RMSE Correlation coefficient RMSE

0.66 0.67 �0.48 0.214

(c) Correlation coefficient and RMSE values obtained with ANN for upscaling the scour from

Hidden nodes Momentum Learning rate

Falciai and Giacomin (1978)

10 0.1 0.2

Missiga stream data

8,6 0.1 0.2

Large-scale data

20 0.1 0.2

CCcorrelation coefficient.

set (Fig. 5), Eq. (9) works well and the predicted scouris comparable to that achieved by the SVMs. A comparisonof correlation coefficients values as well as RMSE values (Table 3)as well Figs. 3–5 suggests a better performance by SVMs incomparison to neural network and empirical Eq. (9) in predictingthe scour from laboratory to field data. Further, with SVMs the RBFkernel performs well when applied to Missiga stream datawhereas polynomial kernel works well with Falciai and Giacomin(1978) and large-scale data set.

Further, a comparison of Figs. 3–5 suggests that most of thepoints lie within 730% error lines using SVMs algorithm using allthree field data sets to test the models created by using laboratorydata. Even the performance of SVMs seems to be encouraging inupscaling the scour; its performance is affected by the choice ofkernel. For different data sets either polynomial or RBF kernels areperforming well thus suggesting a careful implementation ofSVMs in upgrading the scour from laboratory to filed conditions.

5.3. Sensitivity analysis on Missiga stream data

It can be observed from aforementioned results that most ofthe predicted values especially for Missiga stream data set arefalling on a straight line showing the failure of SVMs (bothpolynomial and RBF kernel), Eq. (9) and the neural network-basedmodeling approach in upscaling the results from laboratory tofield conditions. As SVMs (polynomial kernel-based) provideencouraging performance in upscaling results for Falciai andGiacomin (1978) and large-scale data set, several trials werecarried out to study the effect of different input parameters inpredicting the scour depth for Missiga stream data. Different inputcombinations (e.g. z, h0, h, Q/B, d50 and B, z, h0, h, Q/B and z, h, Q/B,d50 and h0, h, Q/B, d50) were tried to check if any of the inputparameter combination affects the performance of SVMs inupscaling the results for this data set.

Fig. 6 provides graphs between actual and predicted values usingtwo different input combinations (z, ho, h, Q/B, d50 and z, h, Q/B, d50)for comparison of performance by SVMs (polynomial and RBF

m laboratory to field data

nd Ferro, Large-scale data (Veronese, 1937; Scimemi, 1939; Whittaker and

Schleiss, 1984)

RMSE C g/d* CC RMSE

0.126 0.0038 1 0 16.44

0.912 0.0038 1 0.99 1.42

om laboratory to field data

) Large-scale data (Veronese, 1937; Scimemi, 1939; Whittaker and Schleiss, 1984)

Correlation coefficient RMSE

0.99 2.93

laboratory to field data

Iterations Correlation coefficient RMSE

1000 0.34 0.67

500 0.66 0.53

500 0.67 15.12

ARTICLE IN PRESS

+25% line

-25% line

00 0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

1.2

Actual scour

Pred

icte

d sc

our

smo_poly

smo_rbf

-25% line

+25% line

00 0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

1.2

Actual scour

Pred

icte

d sc

our

smo_polysmo_rbf

Fig. 6. (a) Comparison of actual and predicted scour based on parameters z, h0, h,

Q/B and d50 using Missiga stream field data. (b) Comparison of actual and predicted

scour based on parameters z, h, Q/B and d50 using Missiga stream field data.

A. Goel, M. Pal / Engineering Applications of Artificial Intelligence 22 (2009) 216–223 223

kernels). The perusal of Figs. 6(a) (with data set z, h0, h, Q/B, d50) and6(b) (with data set z, h, Q/B, d50) suggests that most of the predictedvalues obtained by using RBF kernel-based SVMs lies around the lineof perfect agreement as compared to Fig. 4 on similar data set butwith different input parameters (i.e. B, z, h0, h, Q/B, d50). Incomparison, polynomial kernel-based SVMs fails to perform wellfor this data set. A comparison of Figs. 4 and 6 for the Missiga streamsuggests an improved performance by input combination of (z, h,Q/B, d50) in upscaling the scour, thus suggesting that upscaling of theresults from laboratory data to field conditions is greatly influencedby the input parameters combination.

6. Conclusions

This study was carried out to judge the suitability of SVMs inmodeling the scour on grade-control structures using laboratoryand field data sets. A conclusion from this study is that both theRBF and polynomial kernel-based SVMs perform well in compar-ison to back propagation ANN and empirical Eq. (9) withlaboratory data as indicated by the higher values of correlationcoefficients and smaller values of RMSEs for both D’Agostino andFerro (2004) and Borman and Julian (1991) data sets.

In case of upscaling the scour from laboratory to fieldconditions the RBF-based SVMs is found to be working well forMissiga stream data only whereas polynomial kernel-based SVMswork better for large-scale (Table 1, serial number 8) as well as

Falciai and Giacomin (1978) data set in comparison to the backpropagation ANN and empirical Eq. (9). Inspite of an encouragingperformance by SVMs in upscaling the scour prediction, one hasto be careful in using this technique as the choice of kernel and acombination of input parameter are found to affect the upscalingto a large extent. Further, in upscaling, a large variation in theflow, geometry, and material conditions may influence theprediction of scour depth.

However, the findings of this study, encourages the use ofSVMs approach in predicting the scour on grade-control struc-tures in water resources projects.

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