(in lingua inglese)

A mathematical model between is generated by Gene expression programming (GEP) for the purpose of predicting local scour depth downstream stilling basin through trapezoidal channel. The stilling basin is provided with current deflector placed at different relative positions to control the local scour depth. In addition, an artificial neural network (ANN) and multiple linear regression (MLR) models are implemented to predict the scour depth downstream the hydraulic structures.

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CIVIL ENGINEERING Modeling of local scour depth downstream

hydraulic structures in trapezoidal channel

using GEP and ANNs Yasser Abdallah Mohamed Moussa * Water Engineering Department, Faculty of Engineering, Zagazig University, Egypt Received 19 November 2012; revised 21 February 2013; accepted 16 April 2013

Available online 5 June 2013 KEYWORDS Local scour;

GEP;

ANN;

Current de'ector;

Trapezoidal channel Abstract Local scour downstream stilling basins is so complex that it makes it difﬁcult to establish a general empirical model to provide accurate estimation for scour depth. Lack estimation of local

scour can endanger to stability of hydraulic structure and can cause risk of failure. This paper pre-

sents Gene expression program (GEP) and artiﬁcial neural network (ANNs), to simulate local scour

depth downstream hydraulic structures. The experimental data is collected from the literature for

the scour depth downstream the stilling basin through a trapezoidal channel. Using GEP approach

gives satisfactory results compared with artiﬁcial neural network (ANN) and multiple linear regres-

sion (MLR) modeling in predicting the scour depth downstream of hydraulic structures. 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. 1. Introduction Scour is a natural phenomenon caused due to the erosive

action of 'owing stream on alluvial beds which removes the

sediment around or near structures located in 'owing water.

In addition, when a hydraulic structure such as dam, regulator,

spillway, or bridge, is placed in a hydraulic/marine environ-

ment, the presence of the structure will change the 'ow pattern in its immediate neighborhood, resulting in these changes

usually cause an increase in the local sediment transport capac-

ity and thus lead to scour. Scour can induce failure of hydrau-

lic and marine structures [1]. So, local scour modeling is an

important issue in environmental/water resources engineering

in order to prevent degradation of river bed and safe the stabil-

ity of grade control structures [2,3]. The river bed in the vicin-

ity of a hydraulic structure is generally protected against

current, waves, and eddies [4''6,1,7]. The effect of baf'es on

scour depth over stilling basis was investigated by many

researchers [8''16]. ANN was used to investigate its possibility

as a modeling tool for simulation of tidal 'ow in two-dimen-

sional 'ow ﬁeld [17]. An ANN has been employed to predict

scour at a culvert outlet [18,19]. The scour depth downstream

hydraulic structures were predicted using artiﬁcial neural net-

work [20]. In addition, it was employed to predict 'ow charac-

teristics in irregular open channel [21] and sediment load [22]. * Tel.: +20 01111890996.

E-mail addresses: Yasser_eng1997@yahoo.com, Yasser_eng1997@

zu.edu.eg. Peer review under responsibility of Ain Shams University.**Production and hosting by Elsevier** Ain Shams Engineering Journal (2013) 4, 717''722 Ain Shams University Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com 2090-4479 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asej.2013.04.005 Moreover, the local scour depth downstream spillway and

ski-jump, was predicted [23,24], respectively. Gene expression

programming, GEP, is used by many researchers, to develop

combined run off [25], prediction of surface roughness [26],

investigate the hydraulic jump characteristics [27]. In addition,

this technique is used as a new algorithm for solving problems

[28]. GEP is used to predict the local scour depth for different

types of hydraulic structures [29''33]. This paper presents the

modeling of local scour depth though the trapezoidal channel

using artiﬁcial techniques; ANNs and GEP. The stilling basin

is provided with current de'ector at different positions from

the sluice gate, to control and minimize the local scour depth

downstream hydraulic structure. 2. Overviews of ANNs and GEP 2.1. The artiﬁcial neural networks ANNs as a technique of prediction could be used to predict the

local scour depth by building a multilayer feed forward net-

work. Such type of ANN consists of several layers (Fig. 1);

each has one or more units (neurons). Each unit of the ﬁrst

layer (input layer) receives the input data of an independent

factor (variable), multiplies each input by the connection

weight, and transmits the result to the corresponding unit in

the hidden layer where the activation function is applied.

The results from the hidden layer are transferred to the output

layer by multiplying the output of each neuron in the hidden

layer by the corresponding connection weight between hidden

and output neurons. The output layer produces the network

output for further processing of the data. At this stage, the net-

work output is compared to the desired (target) output to com-

pute the error. If the error is acceptable, then the output is

assumed to be correct otherwise the weights of the connection

are adjusted starting from the output layer and propagating

backward. Once the weights are updates, a new iteration be-

gins and so on until training is completed. The training is

stopped when the error level is reached or when the number

of iterations is ﬁnished. The basics of the ANNs were intro-

duced by many authors, e.g. [34,17,35]. 2.2. Gene expression programming GEP is considered as an extension of Genetic programming

(GP). Gene expression programming is a full-'edged geno-

type/phenotype system, with the genotype totally separated from the phenotype, where in GP, genotype and phenotype

are mixed together in a simple replicator system. GEP com-

puter program is encoded in linear chromosomes composed

of genes structurally organized in a head and a tail. The

chromosomes function as a genome and are subjected to mod-

iﬁcation by means of mutation, transposition, root transposi-

tion, gene transposition, gene recombination, and one- and

two-point recombination. The chromosomes encode expres-

sion trees which are object of selection. The creation of these

separates entities (genome and expression tree) with distinct

functions allows the algorithm to perform with the high

efﬁciency that greatly surpasses existing adaptive techniques

[28]. The interplay of chromosomes and expression trees in

GEP implies an unequivocal translation system for translating

the language of chromosomes into the language of expression

trees (ETs). The genetic code of Gene expression programming

is very simple: a one-to-one relationship between the symbols

of chromosome and the nodes they represent in the trees.

The rules are also very simple. They determine the spatial

organization of nodes in the expression trees and the type of

interaction between sub-ETs. Therefore, there are two

languages in GEP, the language of genes and the language

of expression trees. This unequivocal bilingual notation is

called karva language. Expression trees and Karva Language

are explained in details by [36,28]. The steps for GEP are

shown in Fig. 2. The process begins with random generation

of chromosome of the initial population. Then, the chromo-

somes are expressed, and ﬁtness of each individual is evalu-

ated. The individuals are then selected according to ﬁtness to Figure 1 Structure shape of ANN. Create Chromosome of Initial Population Express Chromosome Execute Each Program Evaluate Fitness Iterate or

Terminate Keep Best Program Select Program Reproduction steps Prepare New Programs of Next Generation End Iterate Figure 2 Flow chart of a gene expression algorithm, Ferreira 2001. 718 Y.A.M. Moussa reproduce with modiﬁcation, leaving progeny with new traits.

The individuals of this new generation are, in their turn, sub-

jected to the same developmental process: expression of the

genomes, confrontation of the selection environment, and

reproduction with modiﬁcation. The process is repeated for a

certain number of generations or the required accuracy is

achieved [28]. In GEP system, the operators used for the genet-

ic modiﬁcation of chromosomes are explained [36]. 3. Theoretical background Fig. 3 shows a deﬁnition sketch of the experimental model [37].

Using the principles of the dimensional analysis, the following

relationship is obtained; Ds=y1 ¼ fðLg=Lb; F1' ð1' In which, Ds is the local scour depth downstream stilling basin,

y1 is the super critical 'ow depth, Lg is the length from gate to

the beginning of Current de'ector, Lb is the basin length, and

F1 is the initial Froude number at the depth of y1. The exper-

imental data are collected from [37]. The ranges of various

parameters included in the present study are summarized in

Table 1. Figure 3 Deﬁnition sketch for the experimental model Shaheen [37]. Table 1 Ranges of data employed to train and test the GEP, ANN, and MLR. Parameter Range Froude number F1 1.45''8.45 Grain size 1.77 mm Channel side slope 1 (Horizontal): 4 (Vertical) Current de'ector

conﬁgurations Lg /Lb = 0.00''0.74, Height = 2.5 cm, With angle = 10.2 Table 2 Parameters of the GEP models. Function set +, , ·, /, S qrt, Exp, Ln, Sin,

Cos, Atan Mutation rate 0.044 Number of gene 23 IS transportation rate 0.1 Head size 7 RIS transportation rate 0.1 Linking function + Gene transportation rate 0.1 Number of

generation 1000 One-point recombination

rate 0.3 Number of

population 100 Two-point recombination

rate 0.3 Number of best

individuals cloning 10 Gene recombination rate 0.1 Figure 4 Experimental data versus the output of GEP model for train data set. Figure 5 Experimental data versus the output of GEP model for test data set. Figure 6 Experimental data versus the output of ANN model for train data set. Modeling of local scour depth downstream hydraulic structures in trapezoidal channel 719 4. Modeling of local scour depth using ANNs, GEP and MLR 4.1. ANNs model The experimental data are divided into 70% of the data for the

training of the network and the remaining 30% of the data for

testing the network prediction. The Neural Connection Soft-

ware 1998 is used to train the network [38]. All the data are

normalized using the zero-mean-unit-standard deviation. Sev-

eral trials are conducted to have the best structure of the arti-

ﬁcial neural network. The best values of the initial weights

(±0.01), the transfer functions (Sigmoid), the number of hid-

den neurons and the best number of iterations are 4, and

2000 respectively, See Fig. 1. 4.2. GEP model Automatic Problem Solver '' APS 3.0 '' (www.gepsoft.com), GEP a powerful soft computing software package, is used in

modeling the local scour depth downstream stilling basin

through a trapezoidal channel. The previous 70% of data set

is used to build GEP model and the rest of observations for

check the test data set. The parameters used in the GEP mod-

els are given in Table 2. Froude number (F1), and relative po-

sition of current de'ector Lg/Lb are assigned to the columns as

independent input variables while relative local scour depth

Ds/y1 is used as dependent output variable. Therefore, a Ds/

y1 model of output variable is developed by using GEP. 4.3. MLR model The same training data sets for building GEP and ANN mod-

els are used also to build the multiple regression model. The

following equation is obtained to correlate the relative scour

depth with the other independent parameters (Froude number

and relative position of current de'ector). D1=y1 ¼ 0:57 þ 1:6eð5'F1 3:2eð3'ðLg=Lb' þ 1:13eð3'ðLg=Lb' 0:2 ð2' 5. Discussion of results Numerical results using GEP, ANNs, and multiple linear

regressions (MLR) techniques are plotted versus the experi-

mental results; Figs. 4 and 5 show the experimental data versus

numerical results using GEP for both train and test data sets,

respectively. The same plots are prepared for ANNs tech-

niques (Figs. 6 and 7) and MLR (Figs. 8 and 9). The statistical Figure 7 Experimental data versus the output of ANN model for test data set. Table 3 Results of GEP, ANN, and MLR Models. Model Data set R 2 Stander error AMRE ¼ ABS DsðMeasured'Dsðmodeloutput' DSðMeasured' (Absolute mean relative error) GEP Train data 0.86 0.11 0.08 Test data 0.96 0.08 0.07 ANN Train data 0.71 0.25 0.12 Test data 0.74 0.20 0.13 MLR (Eq. (2)) Train Data 0.64 0.50 0.14 Test data 0.67 0.25 0.20 Figure 8 Experimental data versus the output of MLR model for train data set. Figure 9 Experimental data versus the output of MLR model for test data set. 720 Y.A.M. Moussa results of model predictions for training and testing sets are

given in Table 3. It is clear that GEP model predicted the scour

depth for both training and testing set with lower error AMRE

(0.08 and 0.07) and higher accuracy R 2 (0.86 and 0.96), respec- tively. Table 3 shows that the outperforming of the GEP model

is considered the best one compared to ANNs and MLR

models. The numerical results for GEP model are presented versus the experimental data for different relative positions of current

de'ector (Lg/Lb = 0:0.76), Fig. 10. This ﬁgure shows that

GEP expresses well the experimental data for different relative

position of current de'ector. The best location of the current

de'ector to have minimum local scour depth equals (Lg/Lb),

0.37, Fig. 11. 6. Conclusions A mathematical model between is generated by Gene expression

programming (GEP) for the purpose of predicting local scour

depth downstream stilling basin through trapezoidal channel. The stilling basin is provided with current de'ector placed at

different relative positions to control the local scour depth. In

addition, an artiﬁcial neural network (ANN) and multiple lin-

ear regression (MLR) models are implemented to predict the

scour depth downstream the hydraulic structures. From this

study, it is clearly found that the Gene expression programming

model simulate the local scour depth downstream stilling basin

effectively compared to the other models (ANN, MLR). GEP

model is showing well the effect of current de'ector on the scour

formed through the trapezoidal channel section. Acknowledgement The Author would like to express his appreciation to Engineer-

ing Marowa Shaheen, Ministry of Water Resources and Irriga-

tion, for providing the experimental data used in this paper. References [1] Azamathulla HM. Neural networks to estimate scour downstream of ski-jump bucket spillway. PhD Thesis, Indian Institute of

Technology, Bomba, India; 2006. [2] Laucelli D, Giustolisi O. Scour depth modelling by a multi- objective evolutionary paradigm. Environ Modell Softw 2011;26(4):498''509. [3] Abdellateef M, Abdelhaﬁz EA, Khalifa A. A study of scour downstream one vent regulator. In: Albertson, Kia, editors.

Design of hydraulic structures. Rotterdam: Balkema; 1989. [4] Mason PJ, Arumugam K. Free jet scours below dams and 'ip buckets. J Hydraul Eng ASCE 1985;111(2):220''35. [5] Shields FD, Knight JR, Cooper SS. Incised stream physical habitat restoration with stone weirs. Regul Rivers: Res Manage

1995;10:181''98. [6] Hoffmans GJ, Verheij HJ. Scour manual. Rotterdam: Balkema; 1997. p. 68''87. Figure 10 Relationship between F1 and Ds/y1 for GEP simulated model and experimental data for different relative current de'ector positions (Lg/Lb = (a) 0.0, (b) 0.01, (C) 0.06, (d) 0.11, (e) 0.36, and (f) 0.76). Figure 11 Relationship between Lg/Lb and Ds/y1 for GEP predicted data and different Froude numbers. Modeling of local scour depth downstream hydraulic structures in trapezoidal channel 721 [7] Guven A, Gunal M. Prediction of scour downstream of grade- control structures using neural networks. J Hydraul Eng

2008;134(11):1656''60. [8] Osman C, Korkt O, Rifat T. A study of scours at the end of stilling basin and use of horizontal beams as energy dissipation.

Int Comm High Barrages Madrid 1973:23''37. [9] Nettleton Peter C, McCorquodal JA. Radial stilling basin with baf'es. Proc Can Soc Civil Eng Montreal Canada 1983;2:561''70. [10] Nashta A. The proper location of 'oor sill with scour reach downstream of heading-up structure. Bull Fac Eng, Assuite Univ

1995;23(2):11''9. [11] Mohamed MS, Negm AM, EL-Saiad AA. Combined effects of baf'e piers and end sill on scour downstream of hydraulic

structure. Civil Eng Res Mag, Fac Eng, Al-Azhar Univ, Egypt

2000;4:1620''38. [12] Negm AM, Mohamed MS, El-saiad AA. Combined effects of baf'e piers and an end sill on scour downstream of hydraulic

structures. Civil Eng Res Mag (CERM), Al-Azhar Univ, Egypt

2000;22(4):1620''38. [13] Negm AM, Abdel-Aal GM, Elﬁky M, Mohamed YA. Optimal position of curved de'ector to minimize scour downstream multi-

vents regulators. In: 12th International water technology confer-

ence, ITWTC12, Alexandria, Egypt; 2008. [14] EL Masry AA, Sarhan TE. Minimization of scour downstream heading up structures using single lines of angle baf'es. Eng Res J,

Helwan Univ, Fac Eng, Mataria, Cairo, Egypt 2000;69:192''207. [15] EL Masry A. Minimization of scour downstream heading up structures using double lines of angle baf'es. In: Sixth interna-

tional water technology conference, IWTC, Alexandria, Egypt;

2001. [16] El-Gamal MM. Effect of using three-lines of angle baf'es on scour downstream heading-up structures. Mansora Eng J, Fac

Eng, Mansora Univ, Egypt 2001;26(2):73''485. [17] Dibike YB, Minns AW, Abbott MB. Applications of artiﬁcial neural networks to the generation of wave equations from

hydraulic data. J Hydraul Res 1999;37(1):81''97. [18] Liriano SL, Day RA. Prediction of scour depth at culvert outlets using neural networks. J Hydro-Informatics 2001;3(4):231''8. [19] Azamathulla HM, Haque AM. Knowledge extraction from trained neural network scour model at culvert outlets. Neural

Comput Appl 2012(September). http://dx.doi.org/10.1007/s00521-

012-1164-2. [20] Mohamed S. Artiﬁcial neural net work prediction of maximum scour hole downstream hydraulic structures. In: 11th Interna-

tional water technology conference IWTCII, Sharm EL Sheikh,

Egypt; 2007. [21] Abdeen M. Neural network model for predicting 'ow character- istics in irregular open channel. Sci J, Fac Eng-Alexandria Univ,

Alexandria 2001;40(4):539''46. [22] Nagy HM, Watanabe K, Hirano M. Prediction of sediment load concentration in rivers using artiﬁcial neural network model. J

Hydraul Eng 2002;128(6):588''95. [23] Azamathulla H, Abghani A, Zakaria NA. An ANFIS based approach for predicting the scour below Flip-Bucket spillway.

Riverside Kuching, Sarawak, 6''8 Malaysia; 2008. [24] Azmathullah HM, Deo MC, Deolalikar PB. Neural networks for estimation of scour downstream of ski-jump bucket. J Hydraul

Eng 2005;131(10):898''908. [25] Fernando DK, Shamseldin AY, Abrahart RJ. Using gene expression programming to develop a combined runoff estimate

model from conventional rainfall-runoff model outputs. In: 18th

world ImaCS/MODSIM congress, Caims, Australia 13''17; 2009. [26] Yang Y, Xinyu Li, Ping Jiang, Liping Zhang. Prediction of surface roughness in end milling with gene expression program-

ming. In: Proceedings of the 41st international conference on

computers & industrial engineering; 2010. [27] Eldrandaly K, Negm AM. Performance evaluation of gene expression programming for hydraulic data mining. Int Arab J

Inform Technol 2008;5(2). [28] Ferreria C. Gene expression programming: a new adaptive algorithm for solving problems. Complex Syst 2001;13(2):87''129. [29] Mujahid Khan H, Azamathulla HM, Tufail. Gene-expression programming to predict pier scour depth using laboratory data. J

Hydroinformatics 2012;14(3). [30] Azamathulla HM. Gene expression programming for prediction of scour depth downstream of sills. J Hydrol 2012:169''72, 460''

461C. [31] Azamathulla HM. Gene-expression programming to predict scour at a bridge abutment. IWA J Hydroinformatics 2012;14(2):324''31. [32] Azamathulla HM, Haque AA. Prediction of scour depth at culvert outlets using gene-expression programming. Int J Innova-

tive Comput Inform Control (IJICIC) 2012;8(7):5045''54. [33] Mohammadpour R, Ghanib A, Azamathullac HM. Estimation of dimension and time variation of local scour at short abutment. Int

J River Basin Manage 2013. [34] Dibike YB, Abbott MB. Application of artiﬁcial neural networks to the simulation of a two dimensional 'ow. J Hydraul Res

1999;37(4):435''46. [35] Negm AM. Prediction of hydraulic design parameters of expand- ing stilling basins using artiﬁcial neural networks. Egypt J Eng Sci

Technol, EJEST 2002;6(1):1''24. [36] Ferreira C. Gene-expression programming: mathematical model- ing by an artiﬁcial intelligence. Berlin, Germany: Springer; 2006. [37] Shaheen M. Scour DS gates in non-rectangular channels. Thesis MsC. Zagazig university, Egypt; 2009. [38] Neural Connection. Software and manuals. SPSS/Recognition Systems Limited; 1998. Mr. Yasser A.M. Moussa received the M.Sc.

and Ph.D degrees in civil engineering depart-

ment from college of engineering, Zagazig

University, Egypt in 2002 and 2005 respec-

tively. During 2009, Dr Yasser Moussa was a

Post Doctorate fellow with Hydroscience and

Engineering, college of Engineering, Univer-

sity of Iowa, USA. Currently, he is an asso-

ciate professor at college of engineering,

Zagazig University, Egypt. His research interests are open channel hydraulics, stilling basins, sediment transport, simulation models, and artiﬁcial intelligence. 722 Y.A.M. Moussa# Document Outline

Modeling of local scour depth downstream hydraulic structures in trapezoidal channel using GEP and ANNs 1 Introduction 2 Overviews of ANNs and GEP 2.1 The artificial neural networks 2.2 Gene expression programming 3 Theoretical background 4 Modeling of local scour depth using ANNs, GEP and MLR 4.1 ANNs model 4.2 GEP model 4.3 MLR model 5 Discussion of results 6 Conclusions Acknowledgement References

hydraulic structures in trapezoidal channel

using GEP and ANNs Yasser Abdallah Mohamed Moussa * Water Engineering Department, Faculty of Engineering, Zagazig University, Egypt Received 19 November 2012; revised 21 February 2013; accepted 16 April 2013

Available online 5 June 2013 KEYWORDS Local scour;

GEP;

ANN;

Current de'ector;

Trapezoidal channel Abstract Local scour downstream stilling basins is so complex that it makes it difﬁcult to establish a general empirical model to provide accurate estimation for scour depth. Lack estimation of local

scour can endanger to stability of hydraulic structure and can cause risk of failure. This paper pre-

sents Gene expression program (GEP) and artiﬁcial neural network (ANNs), to simulate local scour

depth downstream hydraulic structures. The experimental data is collected from the literature for

the scour depth downstream the stilling basin through a trapezoidal channel. Using GEP approach

gives satisfactory results compared with artiﬁcial neural network (ANN) and multiple linear regres-

sion (MLR) modeling in predicting the scour depth downstream of hydraulic structures. 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. 1. Introduction Scour is a natural phenomenon caused due to the erosive

action of 'owing stream on alluvial beds which removes the

sediment around or near structures located in 'owing water.

In addition, when a hydraulic structure such as dam, regulator,

spillway, or bridge, is placed in a hydraulic/marine environ-

ment, the presence of the structure will change the 'ow pattern in its immediate neighborhood, resulting in these changes

usually cause an increase in the local sediment transport capac-

ity and thus lead to scour. Scour can induce failure of hydrau-

lic and marine structures [1]. So, local scour modeling is an

important issue in environmental/water resources engineering

in order to prevent degradation of river bed and safe the stabil-

ity of grade control structures [2,3]. The river bed in the vicin-

ity of a hydraulic structure is generally protected against

current, waves, and eddies [4''6,1,7]. The effect of baf'es on

scour depth over stilling basis was investigated by many

researchers [8''16]. ANN was used to investigate its possibility

as a modeling tool for simulation of tidal 'ow in two-dimen-

sional 'ow ﬁeld [17]. An ANN has been employed to predict

scour at a culvert outlet [18,19]. The scour depth downstream

hydraulic structures were predicted using artiﬁcial neural net-

work [20]. In addition, it was employed to predict 'ow charac-

teristics in irregular open channel [21] and sediment load [22]. * Tel.: +20 01111890996.

E-mail addresses: Yasser_eng1997@yahoo.com, Yasser_eng1997@

zu.edu.eg. Peer review under responsibility of Ain Shams University.

ski-jump, was predicted [23,24], respectively. Gene expression

programming, GEP, is used by many researchers, to develop

combined run off [25], prediction of surface roughness [26],

investigate the hydraulic jump characteristics [27]. In addition,

this technique is used as a new algorithm for solving problems

[28]. GEP is used to predict the local scour depth for different

types of hydraulic structures [29''33]. This paper presents the

modeling of local scour depth though the trapezoidal channel

using artiﬁcial techniques; ANNs and GEP. The stilling basin

is provided with current de'ector at different positions from

the sluice gate, to control and minimize the local scour depth

downstream hydraulic structure. 2. Overviews of ANNs and GEP 2.1. The artiﬁcial neural networks ANNs as a technique of prediction could be used to predict the

local scour depth by building a multilayer feed forward net-

work. Such type of ANN consists of several layers (Fig. 1);

each has one or more units (neurons). Each unit of the ﬁrst

layer (input layer) receives the input data of an independent

factor (variable), multiplies each input by the connection

weight, and transmits the result to the corresponding unit in

the hidden layer where the activation function is applied.

The results from the hidden layer are transferred to the output

layer by multiplying the output of each neuron in the hidden

layer by the corresponding connection weight between hidden

and output neurons. The output layer produces the network

output for further processing of the data. At this stage, the net-

work output is compared to the desired (target) output to com-

pute the error. If the error is acceptable, then the output is

assumed to be correct otherwise the weights of the connection

are adjusted starting from the output layer and propagating

backward. Once the weights are updates, a new iteration be-

gins and so on until training is completed. The training is

stopped when the error level is reached or when the number

of iterations is ﬁnished. The basics of the ANNs were intro-

duced by many authors, e.g. [34,17,35]. 2.2. Gene expression programming GEP is considered as an extension of Genetic programming

(GP). Gene expression programming is a full-'edged geno-

type/phenotype system, with the genotype totally separated from the phenotype, where in GP, genotype and phenotype

are mixed together in a simple replicator system. GEP com-

puter program is encoded in linear chromosomes composed

of genes structurally organized in a head and a tail. The

chromosomes function as a genome and are subjected to mod-

iﬁcation by means of mutation, transposition, root transposi-

tion, gene transposition, gene recombination, and one- and

two-point recombination. The chromosomes encode expres-

sion trees which are object of selection. The creation of these

separates entities (genome and expression tree) with distinct

functions allows the algorithm to perform with the high

efﬁciency that greatly surpasses existing adaptive techniques

[28]. The interplay of chromosomes and expression trees in

GEP implies an unequivocal translation system for translating

the language of chromosomes into the language of expression

trees (ETs). The genetic code of Gene expression programming

is very simple: a one-to-one relationship between the symbols

of chromosome and the nodes they represent in the trees.

The rules are also very simple. They determine the spatial

organization of nodes in the expression trees and the type of

interaction between sub-ETs. Therefore, there are two

languages in GEP, the language of genes and the language

of expression trees. This unequivocal bilingual notation is

called karva language. Expression trees and Karva Language

are explained in details by [36,28]. The steps for GEP are

shown in Fig. 2. The process begins with random generation

of chromosome of the initial population. Then, the chromo-

somes are expressed, and ﬁtness of each individual is evalu-

ated. The individuals are then selected according to ﬁtness to Figure 1 Structure shape of ANN. Create Chromosome of Initial Population Express Chromosome Execute Each Program Evaluate Fitness Iterate or

Terminate Keep Best Program Select Program Reproduction steps Prepare New Programs of Next Generation End Iterate Figure 2 Flow chart of a gene expression algorithm, Ferreira 2001. 718 Y.A.M. Moussa reproduce with modiﬁcation, leaving progeny with new traits.

The individuals of this new generation are, in their turn, sub-

jected to the same developmental process: expression of the

genomes, confrontation of the selection environment, and

reproduction with modiﬁcation. The process is repeated for a

certain number of generations or the required accuracy is

achieved [28]. In GEP system, the operators used for the genet-

ic modiﬁcation of chromosomes are explained [36]. 3. Theoretical background Fig. 3 shows a deﬁnition sketch of the experimental model [37].

Using the principles of the dimensional analysis, the following

relationship is obtained; Ds=y1 ¼ fðLg=Lb; F1' ð1' In which, Ds is the local scour depth downstream stilling basin,

y1 is the super critical 'ow depth, Lg is the length from gate to

the beginning of Current de'ector, Lb is the basin length, and

F1 is the initial Froude number at the depth of y1. The exper-

imental data are collected from [37]. The ranges of various

parameters included in the present study are summarized in

Table 1. Figure 3 Deﬁnition sketch for the experimental model Shaheen [37]. Table 1 Ranges of data employed to train and test the GEP, ANN, and MLR. Parameter Range Froude number F1 1.45''8.45 Grain size 1.77 mm Channel side slope 1 (Horizontal): 4 (Vertical) Current de'ector

conﬁgurations Lg /Lb = 0.00''0.74, Height = 2.5 cm, With angle = 10.2 Table 2 Parameters of the GEP models. Function set +, , ·, /, S qrt, Exp, Ln, Sin,

Cos, Atan Mutation rate 0.044 Number of gene 23 IS transportation rate 0.1 Head size 7 RIS transportation rate 0.1 Linking function + Gene transportation rate 0.1 Number of

generation 1000 One-point recombination

rate 0.3 Number of

population 100 Two-point recombination

rate 0.3 Number of best

individuals cloning 10 Gene recombination rate 0.1 Figure 4 Experimental data versus the output of GEP model for train data set. Figure 5 Experimental data versus the output of GEP model for test data set. Figure 6 Experimental data versus the output of ANN model for train data set. Modeling of local scour depth downstream hydraulic structures in trapezoidal channel 719 4. Modeling of local scour depth using ANNs, GEP and MLR 4.1. ANNs model The experimental data are divided into 70% of the data for the

training of the network and the remaining 30% of the data for

testing the network prediction. The Neural Connection Soft-

ware 1998 is used to train the network [38]. All the data are

normalized using the zero-mean-unit-standard deviation. Sev-

eral trials are conducted to have the best structure of the arti-

ﬁcial neural network. The best values of the initial weights

(±0.01), the transfer functions (Sigmoid), the number of hid-

den neurons and the best number of iterations are 4, and

2000 respectively, See Fig. 1. 4.2. GEP model Automatic Problem Solver '' APS 3.0 '' (www.gepsoft.com), GEP a powerful soft computing software package, is used in

modeling the local scour depth downstream stilling basin

through a trapezoidal channel. The previous 70% of data set

is used to build GEP model and the rest of observations for

check the test data set. The parameters used in the GEP mod-

els are given in Table 2. Froude number (F1), and relative po-

sition of current de'ector Lg/Lb are assigned to the columns as

independent input variables while relative local scour depth

Ds/y1 is used as dependent output variable. Therefore, a Ds/

y1 model of output variable is developed by using GEP. 4.3. MLR model The same training data sets for building GEP and ANN mod-

els are used also to build the multiple regression model. The

following equation is obtained to correlate the relative scour

depth with the other independent parameters (Froude number

and relative position of current de'ector). D1=y1 ¼ 0:57 þ 1:6eð5'F1 3:2eð3'ðLg=Lb' þ 1:13eð3'ðLg=Lb' 0:2 ð2' 5. Discussion of results Numerical results using GEP, ANNs, and multiple linear

regressions (MLR) techniques are plotted versus the experi-

mental results; Figs. 4 and 5 show the experimental data versus

numerical results using GEP for both train and test data sets,

respectively. The same plots are prepared for ANNs tech-

niques (Figs. 6 and 7) and MLR (Figs. 8 and 9). The statistical Figure 7 Experimental data versus the output of ANN model for test data set. Table 3 Results of GEP, ANN, and MLR Models. Model Data set R 2 Stander error AMRE ¼ ABS DsðMeasured'Dsðmodeloutput' DSðMeasured' (Absolute mean relative error) GEP Train data 0.86 0.11 0.08 Test data 0.96 0.08 0.07 ANN Train data 0.71 0.25 0.12 Test data 0.74 0.20 0.13 MLR (Eq. (2)) Train Data 0.64 0.50 0.14 Test data 0.67 0.25 0.20 Figure 8 Experimental data versus the output of MLR model for train data set. Figure 9 Experimental data versus the output of MLR model for test data set. 720 Y.A.M. Moussa results of model predictions for training and testing sets are

given in Table 3. It is clear that GEP model predicted the scour

depth for both training and testing set with lower error AMRE

(0.08 and 0.07) and higher accuracy R 2 (0.86 and 0.96), respec- tively. Table 3 shows that the outperforming of the GEP model

is considered the best one compared to ANNs and MLR

models. The numerical results for GEP model are presented versus the experimental data for different relative positions of current

de'ector (Lg/Lb = 0:0.76), Fig. 10. This ﬁgure shows that

GEP expresses well the experimental data for different relative

position of current de'ector. The best location of the current

de'ector to have minimum local scour depth equals (Lg/Lb),

0.37, Fig. 11. 6. Conclusions A mathematical model between is generated by Gene expression

programming (GEP) for the purpose of predicting local scour

depth downstream stilling basin through trapezoidal channel. The stilling basin is provided with current de'ector placed at

different relative positions to control the local scour depth. In

addition, an artiﬁcial neural network (ANN) and multiple lin-

ear regression (MLR) models are implemented to predict the

scour depth downstream the hydraulic structures. From this

study, it is clearly found that the Gene expression programming

model simulate the local scour depth downstream stilling basin

effectively compared to the other models (ANN, MLR). GEP

model is showing well the effect of current de'ector on the scour

formed through the trapezoidal channel section. Acknowledgement The Author would like to express his appreciation to Engineer-

ing Marowa Shaheen, Ministry of Water Resources and Irriga-

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Technol, EJEST 2002;6(1):1''24. [36] Ferreira C. Gene-expression programming: mathematical model- ing by an artiﬁcial intelligence. Berlin, Germany: Springer; 2006. [37] Shaheen M. Scour DS gates in non-rectangular channels. Thesis MsC. Zagazig university, Egypt; 2009. [38] Neural Connection. Software and manuals. SPSS/Recognition Systems Limited; 1998. Mr. Yasser A.M. Moussa received the M.Sc.

and Ph.D degrees in civil engineering depart-

ment from college of engineering, Zagazig

University, Egypt in 2002 and 2005 respec-

tively. During 2009, Dr Yasser Moussa was a

Post Doctorate fellow with Hydroscience and

Engineering, college of Engineering, Univer-

sity of Iowa, USA. Currently, he is an asso-

ciate professor at college of engineering,

Zagazig University, Egypt. His research interests are open channel hydraulics, stilling basins, sediment transport, simulation models, and artiﬁcial intelligence. 722 Y.A.M. Moussa

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