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ISH Journal of Hydraulic Engineering

ISSN: 0971-5010 (Print) 2164-3040 (Online) Journal homepage: http://www.tandfonline.com/loi/tish20

Prediction of energy dissipation on the steppedspillway using the multivariate adaptive regressionsplines

Abbas Parsaie, Amir Hamzeh Haghiabi, Mojtaba Saneie & Hasan Torabi

To cite this article: Abbas Parsaie, Amir Hamzeh Haghiabi, Mojtaba Saneie & Hasan Torabi(2016): Prediction of energy dissipation on the stepped spillway using the multivariate adaptiveregression splines, ISH Journal of Hydraulic Engineering, DOI: 10.1080/09715010.2016.1201782

To link to this article: http://dx.doi.org/10.1080/09715010.2016.1201782

Published online: 06 Jul 2016.

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ISH Journal of HydraulIc EngInEErIng, 2016http://dx.doi.org/10.1080/09715010.2016.1201782

Prediction of energy dissipation on the stepped spillway using the multivariate adaptive regression splines

Abbas Parsaiea, Amir Hamzeh Haghiabia, Mojtaba Saneieb and Hasan Torabia

aWater Engineering department, lorestan university, Khorramabad, Iran; bSoil conservation and Watershed Management research Institute, Tehran, Iran

ABSTRACTIn this study, energy dissipation of flow over stepped spillways in skimming flow condition was modelled and predicted using multivariate adaptive regression splines (MARS) techniques. To develop MARS model-related data-set published in the literature was applied. To compare the performance of MARS model with other types of soft computing techniques, multilayer perceptron neural network (ANN) was prepared as a common type of soft computing method. To define the most effective parameters on energy dissipation, the Gamma Test and sensitivity analysis of ANN along with the development of MARS model were carried out. Results showed that the drop number and ratio of critical depth to the height of step are the most important parameters in energy dissipation. Results of modelling and predicting energy dissipation using MARS and ANN showed that both models are accurate, whereas MARS model is a bit more accurate compared to ANN.

© 2016 Indian Society for Hydraulics

KEYWORDSEnergy dissipation; soft computing; drop number; spillways

ARTICLE HISTORYreceived 27 January 2016 accepted 12 June 2016

CONTACT amir Hamzeh Haghiabi haghiabi.a@lu.ac.ir

1. Introduction

Stepped spillway is an effective structure for dissipating energy of flow that conveys water from higher to lower elevations. Stepped spillway is a spillway which includes a number of steps on its face. Steps on the face of spillway play the role of big roughness in energy dissipation mechanism (Chanson 2002; Chen 2015). Dissipating energy on the face of spillway led to significant reduction of the length or eliminated energy dis-sipating structures at downstream of spillway (Babaali et al. 2015; Felder and Chanson 2009; Parsaie 2016a; Parsaie 2016b). Due to high performance of stepped spillways for energy dis-sipation, investigators have studied these hydraulic properties since four last decades. Study of hydraulic properties of stepped spillways has been conducted using physical and numerical modelling. Flow structure on stepped spillways is a complex phenomenon, since flow on steps is almost a two-phase flow (Dehdar-behbahani and Parsaie 2016; Hunt and Kadavy 2011; Pfister and Hager 2011). Using physical modelling of flow char-acteristics, investigators have defined three types of flow as napped, transition and skimming flows on stepped spillways. Napped flow is shown in Figure 1(a), transition in Figure 1(b) and skimming flow is shown in Figure 1(c). Napped regime occurs in small amounts of discharge and skimming regime occurs after a certain amount of discharge (Chanson 1994a; Chanson 1994b; Haghiabi 2012; Heidarpour et al. 2008). In napped flow condition, flow leaves the step up and hits the step down as breakup and falling jet. In napped flow regime, energy is dissipated via mixing jets, partially of fully hydraulic jump. In skimming flow as shown in Figure 1(c), stable recirculation vortices are created and trapped between steps and main flow and caused to make a virtual bed on steps and main flow. In skimming regime, water flows down as a coherent stream (Boes

et al. 2000; Chamani and Rajaratnam 1999). Energy of flow is dissipated in skimming regime by momentum transferring between recirculation vortex trapped among steps and main flow. Transition regime is an unstable condition variety between napped and skimming flows (Chanson 1994a; Christodoulou 1993). There is little information about transient regime and it is also notable that designing stepped spillways is based on probable maximum flood (PMF) and creation of skimming flow on stepped spillways. Chanson (1994b) studied hydrau-lic properties of alternative energy dissipaters proposed for Monksville dam. He assessed performance of stilling basin, flip bucket and stepped spillway to choose the best option between them due to economic and energy dissipation rate. To this pur-pose, three hydraulic models were construed and it is notable that steps’ edge is in contact with the ogee profile. He found that performance of stepped spillways regarding energy dissipation and economic problems is better compared to others. Several studies have been conducted on flow properties on stepped spillways including defining flow regime and energy dissipation mechanisms. Sorensen (1985) studied flow transmission from ogee crest to the steps with physical modelling. He stated that ogee profile before steps causes smooth motion of flow from the crest to the first steps. Pegram et al. (1999) studied the geometry of steps and scale effects on flow regime and energy dissipation. To this purpose, two laboratory models with scales of 1:10 and 1:20 were constructed. They compared performance of stepped spillways with smooth ogee spillway. They stated that energy dissipation on stepped spillways is much more compared to smooth ogee spillway. Rajaratnam (1990) studied the properties of skimming regime and energy dissipation of flow on stepped spillways and have presented an equation for energy dissipation regarding fluid friction coefficient. Felder

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and Chanson (2011) have conducted experimental inves-tigation of stepped spillways. They evaluated five different non-uniform step heights on energy dissipation. They indi-cated that there are minor differences between uniform and non- uniform steps for dissipating energy. Frizell et al. (2013) studied the potential of stepped spillways for creation of cavi-tation. Due to high cost of laboratory experiments, researchers have been encouraged to use numerical methods for modelling and predicting flow properties on stepped spillways (Parsaie and Haghiabi 2015b; Parsaie et al. 2015a) using the numerical modelling including two fields of computational fluid dynamic (CFD) and soft computing techniques (Dehdar-behbahani and Parsaie 2016; Parsaie et al. 2015a). Mohammad Rezapour Tabari and Tavakoli (2016) used Flow-3D software as CFD tool for modelling flow characteristics on stepped spillways. They applied K − ɛ turbulence model during software setting for simulation of flow over stepped spillways. They stated that Flow-3D has a high ability to simulate the flow over stepped spillways and its properties. Recently, by advancements in soft computing techniques in most areas of water resource engi-neering (Habibzadeh et al. 2011; Parsaie and Haghiabi 2015a; Parsaie and Haghiabi 2015c; Parsaie et al. 2015b; Vatankhah 2012; Vatankhah 2013a; Vatankhah 2013b), researchers have attempted to implement these techniques for modelling and predicting hydraulic phenomena, especially for predicting energy dissipation along stepped spillways. Roushangar et al. (2014) conducted an experimental study on energy dissipa-tion on stepped spillways. Using the dimensional analysis, they have attempted to derive dimensionless parameters effective on energy dissipation and then using genetic expression program-ming (GEP), they have attempted to explain the relationship between independent parameters and dependent variables (Najafzadeh et al. 2013; Najafzadeh et al. 2016; Najafzadeh and Sattar 2015; Najafzadeh and Tafarojnoruz 2016; Noori et al. 2009; Noori et al. 2016). In other words, they modelled the relation between independent parameters and dependent variable using GEP technique. Salmasi and Özger (2014) used adaptive neuro fuzzy inference system (ANFIS) for predict-ing energy dissipation of flow over stepped spillways. In this paper, multivariate adaptive regression splines (MARS) tech-nique was used for modelling and predicting energy dissipation over stepped spillways and then to compare the performance of MARS model with other types of soft computing techniques, multilayer perceptron neural network and ANFIS model were developed. MARS technique has been successfully applied for

predicting scour depth at downstream of spillways, river dis-charge forecasting, rainfall-runoff modelling, etc. (Samadi et al. 2015; Sharda et al. 2008; Zhang and Goh 2014).

2. Materials and methods

2.1. Review of effective parameters

To calculate energy dissipation on stepped spillways, the Bernoulli Equation is applied between upstream and down-stream of spillways (Figure 2). As presented in Equation (1), total upstream energy is defined with E0 and as presented in the Equation (2), downstream total energy is defined with E1.

Total energy dissipation or head loss is calculated as Equation (3).

To define the effects of geometrical parameters of stepped spill-ways on energy dissipation, a dimensional analysis is conducted on main parameters involved in energy dissipation. Equation (4) brings together the main parameters on head loss of flow over stepped spillways.

where q is the discharge of flow per cross length of weir, Hw is height of dam, l and h is are the length and height of steps. g is acceleration gravity and N is the number of steps. Using the Buckingham Π theory, the most effective parameters on energy dissipation are derived as Equation (5).

Assuming DN =q2

gH3w

and S =h

l, Equation (5) can be rewritten

as Equation (6).

(1)E0 = Hw + y0 +V 2

0

2g= Hw + y0 +

q2

2g(Hw + y0

)

(2)E0 = y1 +V 2

1

2g= y1 +

q2

2gy21

(3)ΔE

E0

=E0 − E1

E0

= 1 −E1

E0

(4)ΔE

E0

= f(q, l, h,Hw , g ,N

)

(5)ΔE

E0

= f

(q2

gH3w

,h

l,N ,

ych, Fr1

)

Figure 1. Schematic shape of flow on stepped spillways; (a): napped flow, (b): transition flow, (c): skimming flow.

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DN named drop number. To develop soft computing tech-niques (MARS, ANFIS and artificial neural network [ANN]) data-sets published by Salmasi and Özger (2014) were collected and presented in Table 1. To define the most effective parame-ters on energy dissipation, the Gamma test is applied.

2.2. Gamma Test

The Gamma Test is used to examine the relationship between inputs and outputs in numerical data-sets without the need to construct prediction model. The Gamma Test is used for esti-mating variance of the output before modelling, even though the model is unknown. This error variance estimate presents a target mean squared error that any smooth non-linear func-tion should attain on unseen data. Suppose we have a set of observed data represented by:

where, the vector X =(x1,… , xM

) is input, confined to a closed

bounded set C ∊ RM and scalar y is the corresponding output, without loss of generality. The only assumption made is that the relationship of system is in the following form:

where f presents a smooth function and r denotes an inde-terminable part, which may be due to real noise lack of func-tional determination in assumed input/output relationship. The Gamma Test is used to return a data-derived estimate for Var(r) without knowing the underlying function f, just directly from data. Estimate of model’s output variance called Gamma statistic represented by Γ cannot be accounted for by a smooth data model. The Gamma Test is derived from Delta function of input vectors:

(6)ΔE

E0

= f(DN, S,N ,

ych, Fr1

)

(7)((x1,… , xM

), y)=(x, y

)

(8)y = f(x1,… , xM

)+ r

(9)�M(k) =1

M

M∑

i=1

|||xN[i,k]

− xi|||

2

where xN[i,k] denotes index of the kth nearest neighbour to xi, and |.| denotes Euclidean distance. Thus, δM(k) is the mean square distance to the kth nearest neighbour. The correspond-ing Gamma function of the output values is:

The Gamma Test computes mean squared kth nearest neigh-bour distances δ(k),

(1 ≤ k ≤ kmax

) and the corresponding

γ(p)2. In order to compute Γ, the best line is constructed for p points

(�M(k), �M(k)

), and the vertical intercept, Γ is returned

as the gamma value. Regression line slope is also returned to show the complexity of model f. Vratio is the standardized results considering Γ∕Var(y). It returns a scale invariant noise estimate which normally lies between zero and one.

2.3. Review of ANNs

ANN is a popular soft computing technique used in broad fields of engineering. The idea of ANN development was given from biological neurons of human brain. Each input in neuron in hidden or output layer multiplied by a corresponding inter-connection weight (wij) and summed by a threshold constant value named bias (θj). Equation (11) shows mathematical form of addition and multiplication operation in each neuron.

Then, results of Ij are passed through transfer function, vari-ous types of transfer functions have been proposed, the most famous of which are given as follows.

(1) Gaussian: F(x) = a exp(−

(x−b)2

c2

)

(2) Sigmoidal: F(x) = 1

1+exp (−x)

(3) Tansing: F(x) = 2

(1+exp (−2x))− 1

After acting transfer function on inputs (i.e. F(Ij

)), the out-

put of each neuron is computed as Equation (12):

(10)�M(k) =1

2M

M∑

i=1

(yN[i,k]− yi)

2

(11)Ij =

(∑

i

wij + �j

)

(12)Oj = F

(Ij

)

Figure 2. Main involved parameters on energy dissipation in skimming flow over stepped spillways.

Table 1. results of gamma Test analysis in absence of one variable.

Row Absence Inputs output Gammas Gradient Standard error V-ratio1 – DN, S,N, yc∕h, Fr1 ΔE∕E

00.0103 0.1149 0.0052 0.0412

2 dn S,N, yc∕h, Fr1 ΔE∕E0

0.0165 0.2955 0.0084 0.06583 S DN,N, yc∕h, Fr1 ΔE∕E

00.0094 0.1167 0.0035 0.0376

4 N DN, S, yc∕h, Fr1 ΔE∕E0

0.0185 0.1289 0.0068 0.07395 yc∕h dn, S, N, Fr1 ΔE∕E

00.0184 0.1782 0.0038 0.0738

6 Fr1 DN, S,N, yc∕h ΔE∕E0

0.0083 0.2718 0.0051 0.0331

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where Y is the output parameters, (β0) is the constant value, M is the number of functions, hm(x) is the Mth basic function and (βm) is the corresponding coefficient of hm(x). Developing MARS includes two steps. In the first step, all the basic func-tions are prepared. In this step, overfitting may occur. Therefore, in the next step, to prevent overfitting, less important basic functions are pruned with generalized cross-validation (GCV) criteria that is given in Equation (15).

where n denotes the number of observations and C(B) denotes a complexity criteria, which increases by the number of basic functions (Equation (16)).

3. Results and discussion

3.1. Results of Gamma Test

In this study, to define the most important affective param-eters on energy dissipation, the Gamma Test was applied. To perform the GT, the winGamma software was used. Five sce-narios were considered to evaluate the effect of each input on output. In each scenario, the effect of one of input var-iables is evaluated. First, in scenario number one, all input variables were used for the GT and in the next scenarios, one of input variables was removed and the GT was analysed again. Results of scenarios are given in Table 2. The GT parameters such as gamma, gradient, standard error and V-ratio were chosen as criteria to define the most important parameters. Variation of V-ratio is between 0 and 1. This point is notable that whenever this factor is close to zero, it shows that the related scenario could accurately predict output regarding related input parameters.

Reviewing Table 2 shows that scenario number (1) which involves total input variables, has minimum value for the GT parameters. Table 2 shows that removing parameters DN, N and Fr1 causes a significant increase in the Gamma value. Therefore, it was found that these parameters are the most important parameters on energy dissipation. Variation of gamma along standard error values for data-set based on all input variables is given and shown in Figuer 3. Figure 3 shows that standard error and gamma curves are almost flat after point 110. It implies that for modelling discharge coeffi-cient regarding collected data-set, qualification of 110 data-sets (totally 70%) is enough.

(15)GCV =

1

n

n∑

i=1

�yi − f

�xi��2

�1 −

�C(B)

n

��2

(16)C(B) = (B + 1) + dB

This process (preparing a neuron in network) is continued for all neurons in the first hidden layer for developing the second hidden layer in output of the first hidden layer is considered for neurons. This process is continued until ANN structure forms based on the desire of designer. As noted during ANN preparation, a network could have one or more hidden layers and this type of neural network is called multilayer percep-tron (MLP). MLP is a common type of ANN used widely for modelling and predicting engineering problems. Values of weights and biases are defined during training process called learning stage. Learning means justifying the values of weights and biases where the output of network has minimum error compared to observed values. Several algorithms have been proposed for training ANNs, such as Levenberg–Marquardt algorithm (LMA). However, recently using modern optimiza-tion algorithms such as genetic algorithm (GA) and particle swarm optimization (PSO) have been proposed (Azamathulla et al. 2016; Parsaie and Haghiabi 2015c; Parsaie and Haghiabi 2015b).

2.4. Multivariate adaptive regression splines (MARS)

MARS is a novel approach in the field of soft computing which applies a series of simple linear regressions. As mentioned in the introduction section, MARS was introduced by math-ematician Friedman (1991). It is a high precision technique for modelling systems which is based on the data-set. This approach divided the computational space into sub-ranges of input variables (predicting parameters) and defined the rela-tionship between input parameters and output variable. In other words, this technique has a high ability to characterize the relationship between independent variables and dependent variable in each desired phenomena. This process was carried out by fitting a simple regression on each input parameter for predicting output. MARS divided the space of inputs param-eters into various unites and then fitted a spline function on these unites. These elements of regressions are called basic function of MARS methods. One of the main advantages of MARS method is highlighting the input parameters which have more effect on output parameter. This method could be used for small and big data-sets. A basic function gives information about the relationship between inputs and output parameters and is defined as Equation (13).

where h is the basic function, x is the input parameter, C is the threshold value of the independent (input) parameter of x. The general form of MARS is introduced as Function (14).

(13)hm(x) = max(0,C − x) Or hm(x) = max(0, x − C)

(14)Y = f (x) = �0 +

M∑

m=1

�mhm(x)

Table 2. Summary of performance of ann model during development stage.

note: n-H-l: number of hidden layer(s), f-Hl&Tf: first hidden layer and transfer function.*Error indices of MlP during training stage.**Error indices of MlP during testing stage.

Row N-H-l f-Hl&Tf R2* RMSe* R2** RMSe**1 1 5-Purelin 0.63 8.47 0.57 10.852 1 5-radbas 0.84 6.58 0.81 7.363 1 5-logsig 0.93 4.83 0.92 5.254 1 5-tansig 0.987 2.65 0.978 3.985 1 9-tansig 0.99 2.52 0.99 3.68

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performance of model, for stages of preparation (training, validation and testing), one or more number of neurons may be added. To assess model performance, results of model are compared with observed data. It is notable that to avoid over-training, in all iterations, a numbers of the data-sets are always considered for model validation. Validation is a stage between model training and testing to avoid model overtraining. To prepare ANN, the total data-set is divided into three groups as training, validation and testing. The amount of data assigned to training, validation and testing stages was considered regarding results of GT. As shown in Figure 3, about 110 data-sets are enough for training ANN. Results of ANN development are shown in Figures 4–6. To provide more information about the performance of ANN, error distribution and histogram of error are plotted as well. Histogram of error of ANN results shows that concentration of error focuses around zero and is almost symmetrical. As mentioned, designing ANN is a trial and error process; therefore, in the first stage, one hidden layer including five neurons was considered and three transfer functions were tested. Table 3 presents a summary of trial and error process for developing ANN. In the next step, after perfect choosing of transfer function, a number of neurons are added one by one.

3.3. Sensitivity analysis of effective parameters

To define the most effective parameters for predicting energy dissipation using ANN, a sensitivity analysis was conducted. To this purpose, similar to the method applied in the GT, the importance of each input was evaluated. To evaluate the importance of each input parameter, first, regarding developed ANN model structure (Row 4 in Table 3) was considered. In the next step, one of the input parameters is removed from input variables and again ANN model is evaluated. Obviously,

3.2. Results of ANN

Preparation of ANN includes choosing the type of network and transfer function and then setting the internal parameters of transfer function. Multilayer perceptron (MLP) neural network is a common type of ANNs which has been widely used in most areas of ANN modelling. Adjusting internal parameters of transfer function is justified in training stage. Training ANNs can be considered as an optimization problem. This process can be carried out using conventional method such as Levenberg–Marquardt technique. Recently, advanced optimization such as GA or PSO methods, etc. have been implemented to this purpose. Designing the structure of ANNs is a trial and error process. To avoid ‘over-parameterization’ of ANN, designing ANN structure is conducted step by step. This implies that at the beginning of design process, a few numbers of neurons and specific transfer functions are considered. Then, the model is trained and results of model are assessed. After verifying proper operation of transfer function, in the next step regarding the

Figure 3. Variation of gamma Test and standard error with unique data points.

Figure 4. Performance of ann preparation in training stage.

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models in absence of each input parameter is assessed using calculation of error indices including R2 and RMSE.

As seen in Table 4, absence of drop number (DN) and num-ber of steps (N) and ratio of critical depth to the height of

removing each parameter causes a change in performance of ANN. According to severity of model performance, the impor-tance of each input parameters is assessed. Results of ANN sensitivity analysis are presented in Table 4. Performance of

Figure 5. Performance of ann preparation in validation stage.

Figure 6. Performance of ann preparation in testing stage.

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4). Results of developing ANN based on the most effective parameters are shown in Figures 7–9.

3.4. Results of MARS

Preparation of MARS model for predicting energy dissipation of flow over stepped spillways is based on the data-set similar

steps causes a dramatic decrease in the accuracy of models. Therefore, it was found that these parameters are the most effective parameters for modelling energy dissipation of flow over stepped spillways. Another simulation (Scenario 2) based on the most effective parameters was considered. It is notable that the structure of tested model is similar to chosen model developed for modelling the energy dissipation (Row 4 in Table

Table 3. results of ann sensitivity analysis.

Row Absence Inputs output R2 RMSe1 – DN, S,N, yc∕h, Fr1 ΔE∕E

00.978 3.98

2 dn S,N, yc∕h, Fr1 ΔE∕E0

0.768 9.453 S DN,N, yc∕h, Fr1 ΔE∕E

00.952 5.86

4 N DN, S, yc∕h, Fr1 ΔE∕E0

0.846 7.765 yc∕h dn, S, N, Fr1 ΔE∕E

00.813 8.21

6 Fr1 DN, S,N, yc∕h ΔE∕E0

0.948 6.34

Table 4. Basic function and related coefficient of MarS model.

note: x1: Fr1, x2: yc∕h, x3: dn, x4: N, x5: S.

Basic function equation coefficient (βm)h1(x) Bf1 = max(0, x3 − 0.0000056) 15797.92

h2(x) Bf2 = Bf1 × max(0, x1 − 0.05435) 448412.50

h3(x) Bf3 = Bf1 × max(0, 0.05435 − x1) −540389.18

h4(x) Bf4 = max(0, x1 − 0.05435) −27.71

h5(x) Bf5 = max(0, 0.05435 − x1) × max(0, 0.000074 − x3) −109473.14

h6(x) Bf6 = max(0, 0.000074 − x3) 2637.34

h7(x) Bf7 = max(0, 0.05435 − x1) × max(0, x3 − 0.000250) −121424.07

h8(x) Bf8 = max(0, 0.05435 − x1) × max(0, 0.000250 − x3) 137121.87

h9(x) Bf9 = max(0, x3 − 0.00003) −3773.34

h10(x) Bf10 = max(0, x3 − 0.0000015) 44614.44

h11(x) Bf11 = max(0,0.00003 − x3) × max(0, x1 − 0.02768) 713501.03

h12(x) Bf12 = max(0, 0.00003 − x3) × max(0, 0.02768 − x1) −829267.89

h13(x) Bf13 = max(0, x3 − 0.000011) 6210.89

h14(x) Bf14 = max(0, x3 − 0.00049) −52009.18

h15(x) Bf15 = max(0, 0.00049 − x3) 52228.68

h16(x) Bf16 = max(0, x3 − 0.00016) 586.94

h17(x) Bf17 = Bf9 × max(0, x1 − 0.03674) −688291.34

h18(x) Bf18 = Bf9 × max(0, 0.03674 − x1) 673287.51

Figure 7. Performance of ann preparation in training stage (Scenario 2).

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Input parameters were considered regarding Equation (6). This implies that DN, S,N , yc∕hand Fr1were considered as inputs and ΔE∕E0 as output. Results of MARS model during model development (preparation and testing) are shown in Figures 10 and 11. As seen from these figures, MARS model has a high ability for modelling energy dissipation over stepped spillways. Comparing the results of MARS and ANN shows that MARS is a bit more accurate compared to ANN. Reviewing

(17)Q = −25.34 +

18∑

M=1

�mhm(x)

to other types of soft computing techniques. This implies that before using MARS technique, collected data should be prepared. To compare the performance with ANN, training and testing data-sets (validation and testing) applied to ANN development were used. As mentioned, training data-set was about 70% and the rest were considered for testing. During MARS model development, in the first step, 30 basic func-tions were considered and in the second step (pruning step) 12 basic functions were pruned. At the end, optimal MARS model with 18 basic functions was derived. The general form of obtained MARS model is given in Equation (17). Extended form of MARS model is given in Table 5.

Figure 8. Performance of ann preparation in validation stage (Scenario 2).

Figure 9. Performance of ann preparation in testing stage (Scenario 2).

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Table 5 shows that DN is the most important parameter for modelling and predicting energy dissipation of flow over stepped spillways. Results of MARS model uphold the GT results.

Another scenario is developing MARS model based on the most effective parameters derived from GT analysis. Results of MARS development based on the most effective parame-ters are shown in Figures 12 and 13. Equation derived from development of MARS model is given in Equation (6) and its extended form is given in Appendix 1 (Table A1). As presented in Appendix 1 (Table A1), both parameters (yc∕h and DN) have the most repetition in Appendix 1 (Table A1). Therefore, it was found that these parameters are the most effective ones. As shown in Figures 12 and 13, performance of modelling energy dissipation using MARS based on the most effective parameters is suitable and acceptable. Comparison of modelling energy dissipation based on the most effective parameters using ANN and MARS shows that performance of both models is suitable and accuracy of both models is almost equal.

(18)Q = 1.0537 +

9∑

M=1

�mhm(x)

Table 5. Basic function and related coefficients of MarS model.

note: x1: yc∕h, x2: dn, x3: N.

Basic function equation coefficient (βm)h1(x) Bf1 = max(0, x2 − 0.0001) 4279.933

h2(x) Bf2 = max(0, 0.0001 − x2) × max(0, x1 − 0.0121) 269512.1

h3(x) Bf3 = max(0, 0.0001 − x2) × max(0, x1 − 0.00585) −359,070

h4(x) Bf4 = max(0, 0.0001 − x2) × max(0, 0.00585 − x1) 1,622,030

h5(x) Bf5 = max(0, x2 − 0.0000004) −4399.72

h6(x) Bf6 = max(0, x1 − 0.0073) −2.9454

h7(x) Bf7 = max(0, 0.0073 − x1) −116.663

h8(x) Bf8 = max(0, x3 − 0.2071) 0.7689

h9(x) Bf9 = max(0, 0.2071 − x3) −1.2414

Figure 10.  Performance of MarS model for predicting energy dissipation in preparation stage.

Figure 11.  Performance of MarS model for predicting energy dissipation in testing stage.

Figure 12. Performance of MarS preparation in development stage (Scenario 2).

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Chen, S.H. (2015). Hydraulic structures, Springer Berlin Heidelberg, Berlin.Christodoulou, G.C. (1993). “Energy dissipation on stepped spillways.” J.

Hydraul. Eng., 119(5), 644–650.Dehdar-behbahani, S., and Parsaie, A. (2016). “Numerical modeling of

flow pattern in dam spillway’s guide wall. Case study: Balaroud dam, Iran.” Alexandria Eng. J., 55(1), 467–473.

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Parsaie, A. (2016a). “Analyzing the distribution of momentum and energy coefficients in compound open channel.” Model. Earth Syst. Environ., 2(1), 1–5.

Parsaie, A. (2016b). “Predictive modeling the side weir discharge coefficient using neural network.” Model. Earth Syst. Environ., 2(2), 1–11.

Parsaie, A., and Haghiabi, A. (2015a). “The effect of predicting discharge coefficient by neural network on increasing the numerical modeling accuracy of flow over side weir.” Water Resour. Manage., 29(4), 973–985.

Parsaie, A., and Haghiabi, A.H. (2015b). “Computational modeling of pollution transmission in rivers.” Appl. Water Sci., in press. 1–10. doi: http://dx.doi.org/10.1007/s13201-015-0319-6.

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4. Conclusion

Modelling and predicting of phenomena using soft computing techniques as powerful tools leads to accurate presentation of the results of laboratory studies. In this research, energy dissi-pation of flow over stepped spillways was predicted using arti-ficial neural network (ANN) model and MARS. MARS model is a novel approach of soft computing in the field of water engi-neering studies. The most effective parameters on energy dis-sipation were derived using GT and MARS model and senility analysis of ANN. Results showed that the drop number and ratio of critical depth of flow to the height of steps are the most effective parameters for modelling energy dissipation of flow over stepped spillways. Results of MARS model indicated that this model with determination of coefficient (R2 = 0.99) has a suitable performance for predicting energy dissipation of flow over stepped spillways. The advantage of MARS compared to ANN is defining the most effective parameter during model development. Comparing performance of ANN and MARS shows that MARS model is more accurate.

Disclosure statementNo potential conflict of interest was reported by the authors.

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Figure 13. Performance of MarS preparation in testing stage (Scenario 2).

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Appendix 1Table A1. Experimental results of stepped spillways (Salmasi and Özger 2014).

Row ΔE/E0 Fr1 yc/hq2

gH3

w

N S Row ΔE/E0 Fr1 yc/hq2

gH3

w

N S

1 61.52 4.71 0.61 0.0011 5 45 78 21.72 3.719 2.88 0.0501 5 452 60.53 4.88 0.60 0.0010 5 45 79 32.94 3.44 2.582 0.0361 5 453 63.39 4.76 0.57 0.0009 5 45 80 18.07 5.32 1.71 0.0105 5 454 65.93 4.73 0.52 0.0007 5 45 81 74.95 3.549 0.709 0.0007 5 455 70.31 4.66 0.46 0.0005 5 45 82 86.36 4.415 0.281 0.0000 5 456 78.16 3.93 0.40 0.0003 5 45 83 24.64 3.232 6.885 0.0846 10 457 70.20 6.22 0.31 0.0002 5 45 84 16.69 4.086 5.428 0.0414 10 458 84.57 4.53 0.23 0.0001 5 45 85 19.02 4.039 5.3 0.0386 10 459 60.49 4.81 1.23 0.0011 10 45 86 39.91 4.382 3.083 0.0076 10 4510 61.31 4.98 1.15 0.0009 10 45 87 54.5 6.423 1.28 0.0005 10 4511 67.19 5.28 0.88 0.0004 10 45 88 93.84 1.455 0.64 0.0001 10 4512 80.72 3.64 0.78 0.0003 10 45 89 22.92 3.249 10.6 0.0923 15 4513 65.08 7.36 0.60 0.0001 10 45 90 23.78 3.791 8.04 0.0402 15 4514 84.38 3.65 0.62 0.0001 10 45 91 31.23 4.566 5.172 0.0107 15 4515 81.23 5.35 0.47 0.0001 10 45 92 42.2 9.339 1.475 0.0002 15 4516 86.79 4.09 0.45 0.0001 10 45 93 23.39 3.692 6.333 0.0470 15 2517 87.07 4.10 0.44 0.0001 10 45 94 29.8 3.928 5.032 0.0236 15 2518 84.82 5.51 0.36 0.0000 10 45 95 39.45 4.29 3.594 0.0086 15 2519 53.40 5.35 1.96 0.0013 15 45 96 68.12 3.822 1.916 0.0013 15 2520 60.26 4.82 1.88 0.0011 15 45 97 75.38 4.321 1.225 0.0003 15 2521 62.14 4.81 1.78 0.0010 15 45 98 79.24 3.956 1.132 0.0003 15 2522 70.89 4.42 1.48 0.0005 15 45 99 95.73 0.234 0.284 0.0000 15 2523 70.06 5.17 1.24 0.0003 15 45 100 96.44 0.307 0.278 0.0000 15 2524 64.96 6.34 1.12 0.0002 15 45 101 22.52 3.382 2.334 0.0772 5 2525 71.52 5.88 0.99 0.0002 15 45 102 23.89 3.889 1.798 0.0353 5 2526 83.99 4.28 0.80 0.0001 15 45 103 34.14 3.903 1.451 0.0185 5 2527 88.53 3.76 0.65 0.0000 15 45 104 55.41 3.945 0.858 0.0038 5 2528 92.40 3.75 0.43 0.0000 15 45 105 72.67 2.791 0.728 0.0023 5 2529 56.41 5.16 2.55 0.0012 20 45 106 93.74 0.341 0.166 0.0000 5 2530 54.59 5.39 2.51 0.0011 20 45 107 95.73 0.258 0.094 0.0000 5 2531 57.65 5.16 2.47 0.0011 20 45 108 32.76 3.246 4.297 0.0479 10 2532 57.77 5.25 2.40 0.0010 20 45 109 34.48 3.817 3.217 0.0201 10 2533 54.75 5.62 2.37 0.0009 20 45 110 56.07 3.59 2.065 0.0053 10 2534 59.97 5.20 2.29 0.0008 20 45 111 87.76 2.395 0.733 0.0002 10 2535 60.96 5.24 2.20 0.0008 20 45 112 15.14 3.484 7.144 0.1043 15 1536 62.96 5.23 2.08 0.0006 20 45 113 13.15 3.811 6.327 0.0725 15 1537 69.45 4.67 1.94 0.0005 20 45 114 26.78 3.51 5.552 0.0490 15 1538 65.44 5.43 1.83 0.0004 20 45 115 34.04 3.676 4.416 0.0246 15 1539 71.84 5.08 1.59 0.0003 20 45 116 32.18 3.787 4.387 0.0242 15 1540 77.83 4.57 1.40 0.0002 20 45 117 41.18 3.682 3.75 0.0151 15 1541 90.29 2.67 1.06 0.0001 20 45 118 45.07 3.705 3.388 0.0111 15 1542 91.47 3.42 0.71 0.0000 20 45 119 48.37 4.225 2.592 0.0050 15 1543 95.71 3.24 0.37 0.0000 20 45 120 61.49 3.662 2.192 0.0030 15 1544 42.44 6.28 4.48 0.0013 35 45 121 61.85 4.369 1.716 0.0014 15 1545 46.10 6.05 4.38 0.0013 35 45 122 81.98 2.512 1.382 0.0008 15 1546 45.40 6.27 4.22 0.0011 35 45 123 89.19 2.46 0.802 0.0001 15 1547 44.75 6.53 4.04 0.0010 35 45 124 22.38 3.342 13.78 0.0827 30 1548 51.02 6.09 3.89 0.0009 35 45 125 25.74 3.337 12.86 0.0671 30 1549 53.86 6.08 3.64 0.0007 35 45 126 25.17 3.548 11.76 0.0514 30 1550 58.27 5.88 3.40 0.0006 35 45 127 30.23 3.763 9.622 0.0282 30 1551 63.57 5.57 3.14 0.0005 35 45 128 36.84 3.982 7.663 0.0142 30 1552 68.90 5.81 2.50 0.0002 35 45 129 53.03 4.906 3.9 0.0019 30 1553 83.29 4.18 1.94 0.0001 35 45 130 57.67 5.151 3.227 0.0011 30 15

(Continued)

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Table A1. (Continued)

Row ΔE/E0 Fr1 yc/hq2

gH3

w

N S Row ΔE/E0 Fr1 yc/hq2

gH3

w

N S

54 91.52 2.93 1.41 0.0000 35 45 131 95 1.428 1.043 0.0000 30 1555 94.44 2.92 0.91 0.0000 35 45 132 26.39 3.287 2.189 0.0694 5 1556 46.56 6.27 6.04 0.0010 50 45 133 29.01 3.289 2.064 0.0581 5 1557 68.39 5.27 4.25 0.0004 50 45 134 32.89 3.397 1.798 0.0384 5 1558 67.24 5.43 4.25 0.0004 50 45 135 45.87 3.232 1.418 0.0188 5 1559 55.56 7.03 4.17 0.0003 50 45 136 37.73 3.798 1.36 0.0166 5 1560 58.33 6.77 4.10 0.0003 50 45 137 48.17 3.971 0.998 0.0066 5 1561 63.23 6.20 4.03 0.0003 50 45 138 56.69 3.47 0.956 0.0058 5 1562 58.14 6.88 4.03 0.0003 50 45 139 61.17 3.711 0.764 0.0030 5 1563 69.06 5.41 4.00 0.0003 50 45 140 57.29 4.078 0.756 0.0029 5 1564 68.07 6.01 3.60 0.0002 50 45 141 66.84 3.952 0.584 0.0013 5 1565 79.53 4.28 3.44 0.0002 50 45 142 68.88 4.275 0.49 0.0008 5 1566 76.89 4.91 3.30 0.0002 50 45 143 83.1 4.53 0.233 0.0001 5 1567 71.33 6.50 2.88 0.0001 50 45 144 20.52 3.244 5.008 0.1087 10 1568 86.00 3.99 2.49 0.0001 50 45 145 27.13 3.277 4.263 0.0670 10 1569 90.11 3.26 2.16 0.0000 50 45 146 32.46 3.349 3.655 0.0423 10 1570 95.10 1.67 1.80 0.0000 50 45 147 34.18 3.74 2.964 0.0225 10 1571 92.40 3.08 1.74 0.0000 50 45 148 44.37 3.742 2.349 0.0112 10 1572 94.63 2.72 1.38 0.0000 50 45 149 44.11 4.156 2.03 0.0072 10 1573 96.58 2.17 1.05 0.0000 50 45 150 51.19 4.42 1.568 0.0033 10 1574 20.74 3.63 1.18 0.0602 3 45 151 56.82 4.907 1.172 0.0014 10 1575 28.30 3.90 0.90 0.0268 3 45 152 80.93 3.253 0.776 0.0004 10 1576 47.73 3.83 0.60 0.0079 3 45 153 95.48 0.598 0.298 0.0000 10 1577 62.85 4.36 0.33 0.0013 3 45 154 96.22 0.686 0.259 0.0000 10 15

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