Evaluation of GA-SVR method for modeling bed load transport in gravel-bed rivers

Accepted Manuscript Evaluation of GA-SVR Method for Modeling Bed Load Transport in Gravel – Bed Rivers Kiyoumars Roushangar, Ali Koosheh PII: DOI: Reference:

S0022-1694(15)00419-9 http://dx.doi.org/10.1016/j.jhydrol.2015.06.006 HYDROL 20504

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

6 November 2014 7 May 2015 3 June 2015

Please cite this article as: Roushangar, K., Koosheh, A., Evaluation of GA-SVR Method for Modeling Bed Load Transport in Gravel – Bed Rivers, Journal of Hydrology (2015), doi: http://dx.doi.org/10.1016/j.jhydrol. 2015.06.006

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Evaluation of GA-SVR Method for Modeling Bed Load Transport in Gravel – Bed Rivers Kiyoumars Roushangar, Ali Koosheh Department of Civil Engineering, University of Tabriz, Tabriz, Iran Corresponding Author: Kiyoumars Roushangar Postal Address: Department of Civil Engineering, University of Tabriz, Tabriz, Iran Tel: +98 4133392416 Email: [email protected]

1

Abstract

2

The aim of the present study is to apply Support Vector Regression (SVR) method to predict bed load

3

transport rates for three gravel-bed rivers. Different combinations of hydraulic parameters are used as inputs

4

for modeling bed load transport using four kernel functions of SVR models. Genetic Algorithm (GA) method

5

is applicably administered to determine optimal SVR parameters. The GA-SVR models are developed and

6

tested using the available data sets, and consecutive predicted results are compared in terms of Efficiency

7

Coefficient and Correlation Coefficient. Obtained results show that the GA-SVR models with Exponential

8

Radial Basis Function (ERBF) kernel present higher accuracy than the other applied GA-SVR models.

9

Furthermore, testing data sets are predicted by Einstein and Meyer-Peter and Müller (MPM) formulas. The

10

GA-SVR models demonstrate a better performance compared to the traditional bed load formulas. Finally,

11

high bed load transport values were eliminated from data sets and the models are re-analyzed. The

12

elimination of high bed load transport rates improves prediction accuracy using GA-SVR method.

13 14

Key words: Bed load transport; Support vector regression; Genetic algorithm

15 16

1. Introduction

17

The prediction of bed load transport rate is one of the important issues of the hydraulic/hydrologic researchers

18

due to the complex nature of bed load transport and the limitation of measured hydraulic data. Furthermore, bed

19

load transport is applied to solve many environmental, hydrological and geological problems. Since this is a

20

complex and nonlinear phenomenon, the obtained results through employing various approaches may differ

1

21

drastically from each other. There are three main reasons for the difficulty to predict bed load transport: (a) the

22

upstream sediment supply changes dynamically (Gao, 2011), (b) riverbed structure and roughness affect the

23

flow condition and bed load transport (Zhang et al., 2010), and (c) the bed material size varies widely from

24

which size to which size (Zhang et al., 2010). Numerous bed load transport rate formulas have been proposed

25

by many hydraulic researchers based on different concepts. One of the most initial empirical formulas which

26

was proposed by Meyer – Peter and Müller (1948) is a simple approach that has been used widely in many

27

studies. Einstein (1950) described bed load transport according to the particle motion concept which might be

28

affected by hydrodynamic forces and particle weight. Bagnold (1966) introduced an energy concept and related

29

sediment transport rate to the work done by the fluid (van Rijn. 1993). Barry et al. (2004) used bed load

30

transport rates measured in 24 gravel-bed rivers, to compare different bed load transport formulas. The results

31

showed substantial differences in performance of these formulas (Yu et al., 2009). Wang et al. (2001) analyzed

32

several bed load transport formulas with field data and found that all the formulas except for Einstein’s

33

formula gave much lower transport rates compared with measured data (Yu et al. , 2009). Martin (2003)

34

applied Einstein and Meyer–Peter and Müller (MPM) formulas and stream power correlation to predict Vedder

35

River bed load transport. Results showed the superiority of stream power correlation in relation to other

36

formulas. Twelve bed load transport formulas with different data sets were evaluated by Gomez and Church

37

(1989), and it was found that no bed load formula can be applied to all data sets. Roushangar et al. (2011)

38

described a kind of mathematical model to solve the 1D unsteady flow over a coarse bed river, and variation of

39

water level and bed level profiles due to hydrographs are assessed. More and more studies have been carried out

40

and the obtained results from different formulas would often differ from each other and also from observed data

41

and no universal bed load equation was established.

42

During recent years, soft computing techniques, such as Artificial Neural Networks (ANNs), Support Vector

43

Machine (SVM), Adaptive Neuro-Fuzzy Inference System (ANFIS) and Genetic Programming (GP) have been

44

applied in water resources studies. Sasal et al. (2009) used feed forward–backpropagated (Levenberg–

45

Marquardt algorithm) Artificial Neural Network (ANN) for bed load transport estimation, and results

46

showed that

47

methods. Ghani et al. (2011) used ANN approach without any restrictions to an extensive database compiled

48

from measurements in Langat, Muda, Kurau Rivers in Malaysia. The ANN method demonstrated a superior

49

performance compared to other traditional methods. Azamathulla et al. (2009) applied ANFIS method to

50

predict measured bed load data, and it was found that the recommended network can more accurately

the ANN method demonstrated an encouraging performance compared to other standard

2

51

predict the measured bed load data when compared with an equation based on a regression

52

Roushangar et al. (2014a) assessed capability of machine learning approaches (Gene Expression Programming

53

(GEP) and ANN) to predict stepped spillway energy dissipation and they found out reliability of depicted

54

performances. Zakaria et al. (2010) compared the performance of Gene Expression Programming with the

55

traditional total bed material load formulas of Yang and Engelund, and for all data sets the GEP model gives

56

either better or comparable results. Roushangar et al. (2014b) aimed at developing GP based on the formulation

57

of Manning roughness coefficient in alluvial channels. The obtained results revealed the GP capability in

58

modeling resistance coefficient of alluvial channels’ bed. Chang et al. (2012) evaluated the performance of

59

three

60

Networks (FFNN) and Adaptive Neuro-Fuzzy Inference System (ANFIS) in prediction of total bed load for

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three Malaysian rivers, and the results were promising. Roushangar et al. (2013) also applied GEP and ANN

62

approaches to the northwest of Iran to predict daily stream flow of Vaniar River and confirmed that GEP

63

performance was better than ANN. Investigating further, Roushangar et al. (2014c) evaluated the performance

64

of ANFIS and GEP to predict total bed material load. These models were compared with some well-known

65

traditional models. Obtained results showed that the GEP and ANFIS performed better than traditional models.

66

Kitsikoudis et al. (2014a) employed three data driven techniques, namely ANN, ANFIS and Symbolic

67

Regression (SR) based on GP, for the prediction of bed load transport rates in gravel – bed steep mountain

68

streams and rivers in Idaho (U.S.A.). The derived models generated results superior to those of some of the

69

widely used bed load formulas. In a further study, Kitsikoudis et al. (2014b) applied ANN, ANFIS, SR and

70

Support Vector Regression (SVR) to predict sediment transport capacity of flowing water in rivers. These

71

models were robust and the results were encouraging. Kakaei Lafdani et al. (2013) investigated the abilities of

72

Support Vector Machine (SVM) and ANN models to predict daily suspended sediment load in Doiraj River,

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located in Iran. The results showed that ANN models and nu-SVR (with different penalty parameter than

74

common SVR) model using Gamma test for input selection had better performance than regression

75

combination. Kisi (2012) compared Least Square Support Vector Machine (LSSVM) models with those of the

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Artificial Neural Networks (ANNs) and Sediment Rating Curve (SRC) in prediction of suspended sediment

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concentration of upstream and downstream stations. LSSVM and ANN models showed better performance than

78

the SRC model for the upstream station. For the downstream station, however, SRC model outperformed the

79

LSSVM and ANN models. Kalteh (2013) investigated the relative accuracy of ANN and Support Vector

80

Regression (SVR)

soft

method.

computing techniques, namely Gene Expression Programming (GEP), Feed Forward Neural

models coupled

with wavelet

transform

3

in

monthly river flow forecasting. The

81

comparison of results revealed that both ANN and SVR models, coupled with wavelet transform, are able to

82

provide more accurate forecasting results than the regular ANN and SVR models. However, it is found that

83

regular SVR models perform better than regular ANN models.

84

As discussed above, SVM and SVR methods are used for different applications. The main aim of this study is to

85

model bed load transport using SVR technique and to compare the obtained results by using different kernel

86

functions. Additionally, achieved results by SVR method are also compared with the results of traditional bed

87

load formulas. Based on the literature review by the authors, this is the first time of GA-SVR application in

88

modeling bed load transport. Coupling GA would allow us to optimize the SVR parameters more thoroughly in

89

relation to trial and error method. Nevertheless, the analysis of various Kernel functions as well as evaluating

90

various input combinations for distinguishing the most influential hydraulic parameters on bed load transport

91

amount have been made for sensitivity analysis.

92

2. Methods

93 94 95

2.1 Support vector machine

96

This method was proposed by Vapnik (1995) and is known as structural risk minimization. The SVM method is

97

based on the concept of optimal hyperplane that separates samples of two classes by considering the widest gap

98

between two classes. SVM solves a quadratic problem in which the objective function is obtained by combination

99

of loss function and regularization term (Basak et al., 2007). Support Vector Regression (SVR) is an extension of

100

SVM regression. The purpose of SVR is to find a kind of function that has at most ε deviation from the actually

101

obtained objectives for all training data  and at the same time it would be as flat as possible. SVR formulation is

102

as follows:

103

f(x)=wφ(x) + b

104

where

105

b is called the bias. These coefficients are estimated by minimizing regularized risk function as expressed below:

106

min R = C  ∑    ( ,  ) +

107

where

108

 ( ,  ) = 

109

The constant C is the cost factor and determines the trade-off between the weight factor and approximation error.

110

ε is the radius of the tube within which the regression function must lie. The  ( ,  ) represents the loss

The Support Vector Machine (SVM) approach is used in information categorization and data set classification.

(1)

φ(x) denotes a nonlinear function in feature of input x, the vector w is known as the weight factor and





‖‖

(2)

| −  | − : | −  | ≥  0: ℎ

(3)

4

111

function in which  is forecasted value and  is desired value in period i. The ‖‖ is the norm of w vector and

112

the term ‖‖ can be written in the form  ! .w in which  ! represents the transpose form of w vector.

113

According to Eq. (3), the loss will be zero if the forecasted value is within the ε–tube. However, if the value is out

114

of ε–tube then the loss is the absolute value, which is the difference between forecasted value and ε. Since some

115

data may not lie inside the ε–tube, the slack variables (ξ, " ∗) must be introduced. These variables represent the

116

distance from actual values to the corresponding boundary values of ε-tube, and Fig. 1 depicts the situation

117

graphically. Therefore, it is possible to transform Eq. (2) into objective function:

118 119 120 121

∗ min R =C ∑$ % (ξ+ξ ) + ‖w‖ 

(4)

subject to:  -  φ(' ) – b ≤ ε + "  φ(' ) + b -  ≤ ε + "∗ " , "∗ ≥ 0

122

By using Lagrangian multipliers and Karush-Kuhn-Tucher condition in Eq. (4), thus yields the dual Lagrangian

123

form:

124

∗  ∗ max L() ,)∗ )=- ε ∑  ( ) + ) )+∑   () − ) )

125

 ∗ ∗ - ∑   ∑ () − ) ) ()+ − )+ ) K (' , '+ ) 

∗ ∑  () − ) ) = 0

126

subject to:

127

0 ≤ ) ≤ ,

128 129 130

(5)

i=1 , 2 ,… , N

0 ≤ )∗ ≤ , i=1 , 2 ,… , N

where ) and )∗ are Lagrange multipliers and L() , )∗ ) represents the Lagrange function. K (' , '+ ) = φ(' ). φ('+ ) is a kernel function to yield the inner products in the feature space φ(' ) and φ('+ ).

131

Insert Fig. 1.

132

Different kernel functions have been used in SVR problems, such as Gaussian kernel, polynomial kernel and

133

linear kernel. Each kernel function has its own variable parameters which considerably affect the flexibility of

134

regression function. Radial Basis Function (RBF) is one of those popular functions that have been used widely in

135

different studies. When there is no prior knowledge about data characteristics, RBF kernel can be recommended.

136

Moreover, Exponential Radial Basis Function (ERBF) and RBF and polynomial kernel function can be used for

137

nonlinear models, however, linear kernel is used for linear models of SVR. After calculating ) and )∗ , an

138

optimal desired weights vector of regression is

139

∗ - ∗ = ∑  () − ) ) K (' , '+ ).

5

140

Thus, the regression function can be given as:

141 142

∗ f (x) = ∑  () − ) ) K (' , '+ ) + b

(6)

143 144

2.2 Genetic algorithm

145

Genetic algorithms belong to the evolutionary algorithms. The basic concept of GAs is designed to simulate

146

processes in natural systems necessary for evolution, and these algorithms are based on the survival principle of

147

the fittest member in population, which tries to retain genetic information from generation to generation (Patil et

148

al., 2012). One of the most important GAs characteristics is their inherent parallelism regarding other algorithms

149

that are serial. Since, if one path turns out to be a dead end, it would be eliminated and other promising paths will

150

be continued. Other remarkable capability of GAs is finding global optimum among many local optima even on

151

the condition of complex problems. The GA procedure has the ability to move away from local optima if better

152

function value can be found.

153

The process of genetic algorithm is described as follows:

154

Step 1. Initially, random population of chromosomes is generated.

155

Step 2. Fitness of each chromosome is evaluated. In the present study, Efficiency Coefficient (EC) is used as the

156

fitness function

157

2 ∑3 /45 (./ 01/ ) EC=1– 3 ∑/45 (./ 0.̅)2

158

(7)

159

where N represents the total number of testing data and  is the predicted value.  is the observed value and ̅

160

is the mean of the observed values.

161

Step 3. New population is generated using following steps.

162 163

a) Selection: A proportion of the existing population is selected to create new generation. Therefore, in this

164

process most fitted members of the population survive, while the least fitted members are eliminated.

165 166

b) Crossover: This operator is inspired by crossover of DNA strands that occur in reproduction of biological

167

organisms and subsequent creation of new generation through the crossover of current population.

168

Step 4. New population is used for further run of algorithm.

6

169

Step 5. If the stopping condition is satisfied, the best solution is returned in current population, otherwise step 2

170

should be performed again.

171

The applied GA method settings in the present study are shown in Table1.

172 173 174

2.3 GA – SVR model

175

Implementation of SVR requires the allocation of the trade - off different parameters such as, constant C, ε and

176

kernel parameters. In this study, four GA-SVR models were developed by using different kernel functions (Table

177

2) in order to evaluate the performance of each kernel during the prediction of bed load transport rate, for

178

selecting the optimal parameters which produce the most accurate prediction. After generating initial population,

179

the system performs a typical SVR process by using the assigned value of parameters, and the performance of

180

each solution is calculated through the fitness function. Optimal parameters would be selected if the fitness

181

function value satisfies stopping conditions, otherwise the next generation of solutions is generated through

182

genetic operators such as mutation, crossover and selection until stopping conditions are accomplished. It is

183

expected that the average fitness of the population will increase each cycle, and by repeating this process, desired

184

results are obtained. In other words, there are separate roles for GA and SVR in the GA-SVR procedure. The

185

SVR is trained by training data set and predicts bed load transport rate, while GA evaluates SVR prediction by

186

calculating fitness function (EC value) and leads SVR to the best prediction by regulating its parameters and

187

assigning the optimal values for them. Finally, the optimal values of SVR parameters and prediction data set

188

(obtained from optimal parameters assigned) are the GA-SVR output. Fig. 2 represents the GA-SVR procedure.

189

Insert Fig. 2.

190 191

3. Study area and data set

192

The present study covers three gravel - bed rivers, namely, the Oak Creek, Nahal Yatir and Diaoga Rivers. The

193

Oak Creek River is located in McDonald state forest near Corvallis, Oregon, US. The field location is shown in

194

Fig. 3 where drainage area is about 6.73 78 . The Oak Creek data set (66 data) has been extracted from

195

Almedeij reference (2002), that is recorded during 1971 winter. The bed material of stream is gravel, the armor

196

layer has uniform distribution of particles and the material lying below the armor layer is small and well graded

197

(Milhous, 1973). The second case is the Nahal Yatir River with gravel–bed material that drains a water

7

198

catchment of 19 km which lies on the southwestern flank of the Herbon Mountains in the northern Negev

199

Desert, 18 km NE of Beer Sheva (Israel). The bed material mean diameter is 15 mm (Laronne et al., 1992) and

200

data set (74 data) was obtained from Reid and Laronne (1995). The location of Nahal Yatir station is shown in

201

Fig. 4. This channel at the bed load monitoring station is straight, with a 3.5 m width. Banks are near–vertical and

202

channel bed is planar (Laronne et al., 1992). This monitoring station was installed in 1990 and was in operation

203

during the 1990-91 flood season. Available data set for Nahal Yatir River is recorded in 1991 (fall and winter).

204

Located in southern China, Diaoga River is the third case of the present study. Diaoga River is a small mountain

205

stream with a main channel length of 12 km and a drainage area of about 54 78 . Diaoga River data set (86 data)

206

obtained from Yu et al. (2009) was measured during 2006 and 2007. River average gradient is about 9.6% and its

207

location is depicted in Fig. 5. The general characteristics of mentioned rivers are given in Table 3. For all data

208

sets, 70% of data were used for training and the remained others were selected for testing. The selection of data

209

for testing and training was carried out completely in randomized way, obviously regarding the fact that both

210

types of selections (testing and training data) should include appropriate range of whole data set.

211

Insert Fig. 3.

212

Insert Fig. 4.

213

Insert Fig. 5.

214

4. Modeling overview

215

4.1 Input variables

216

Selection of appropriate parameters as inputs for the models is a crucial step throughout the modeling process.

217

River parameters considered in this study are: river water discharge (Q), flow depth (y), median grain size (D50)

218

and the shear stress of river bed (τ). The shear stress can be determined as:

219

;=γ<= S

220

(8)

where γ is fluid specific gravity, <= is river hydraulic radius and S is water surface slope in uniform flow.

221

Applied parameters are bed load transport principal parameters which have been used widely in various studies

222

and formulas in recent decades. Various combinations of these parameters were examined and their influence on

223

bed load was investigated. Table 4 represents suggested input combinations.

224

4.2 Bed load formulas

225

A variety of formulas have been developed to predict bed load transport in gravel-bed rivers, ranging from simple

226

regressions to complex multi-parameter formulations (Barry, 2007). There are different concepts and approaches

8

227

that are used in derivation and extraction process of incipient motion criteria and bed load transport functions and

228

formulas. One of the most prominent approaches is shear stress approach. According to the shear stress approach,

229

there will be no bed load transport until bed shear stress exceeds its critical value. Shields (1936) proposed a

230

formula which introduces Shield’s parameter as critical value for particle motion. Subsequently, Meyer–Peter and

231

Müller (1948) formula was proposed as an energy slope approach to predict bed load transport.

232

But for the first time, probabilistic approach was introduced by Einstein (1950). Einstein described bed load

233

transport rate as a result of complex interaction between fluid and bed load material. Regression approach

234

formulas are based on mathematical concepts. Yalin (1963), Engelund and Hansen (1967) and van Rijn (1993)

235

formulas are the most popular formulas of regression approach. In regression approaches, efficiency criteria are

236

defined as mathematical measures of how a model simulation fits the available observations (Talukdar et al.,

237

2012). In the present research, Einstein and Meyer – Peter and Müller (MPM) formulas were used as reliable

238

formulas to calculate bed load transport rate and moreover, to compare with calculated values by GA-SVR

239

models.

240

4.2.1 Einstein formula

241

Einstein (1950) pioneered sediment transport studies from the probabilistic approach. It is assumed that the

242

probability of the particles movement depends on the probability of exceedance of the fluid forces over the

243

resistive forces. This probability might be considered as a function of particle weight over the average lift on

244

the particle. The Einstein equation has the form @A C 5 BC DC .FGH A

245

>? =

246

and

247

Ѱ=

248

The relation between ф and Ѱ can be expressed as:

249

ф=( − 0.188) 2

250

@A 0@ @

N



ф

(9)

J

K.LM

(10)

H

(11)

Ѱ

where >? is the specific bed load transport rate (kg 80  0 ), Q is fluid specific gravity and Q1 is sediment specific

251

gravity. D is RST of bed material, <= and S are section hydraulic radius and water surface slope, respectively.

252

4.2.2 Meyer - Peter and Müller formula

9

253

Meyer - Peter and Müller (1948) formula is one of the most widely used formulas in laboratory, field investigations

254

and numerical simulations of bed load transport. This formula is obtained from experiments carried out in the

255

laboratory of Hydraulic Research and Soil Mechanics of the Swiss Federal Institute of Technology at Zurich,

256

Switzerland. According to the proposed relation, bed load transport is a function of excess bed shear stress caused by

257

the rivers water flows. Mathematically, it reads:

258 259

>? =8[

VA

VA 0V

] . BV . [Y R Z [\′ ] X

H

\ 2

H

− 0.047 (Y1 − Y)R] 2

where D is the RST of the material,

\

\`

(12)

represents the bed form correction and accounts for the presence of the form

260

resistance in the channel, which reduces the shear stress available for transport (Martin, 2003). Y and Y1 are water and

261

sediment densities, respectively.

262 263 264

5. Results and discussion

265

model performance was assessed by evaluating the scatter plots between the observed and predicted results.

266

Correlation Coefficient (<) and Efficiency Coefficient (EC) were used as statistical parameters which expressed as:

All GA-SVR models were analyzed by applying various kernel functions and mentioned input combinations. Each

267 268

Correlation Coefficient (<) =

e∑d

269

c bc  b =1[>a −>a ][>a −>a ]

∑d

=1

2

2

c bc e d b a −>a ] ∑=1[>a −>a ]

[>

k j ∑3 /45ghi/ 0hi/ l j ∑3 mij l /45ghi/ 0h



(13)

2

2

270

Efficiency Coefficient (EC) = 1 -

271

where N represents the number of test data, >?n is the observed bed load rate (kg/(ms)) , >? is the predicted

272

bed load rate (kg/(ms)) , also:

273

n >m?n = ∑   >?

274

and

(14) o





275

o o >m? = ∑   >?

276

Optimal parameters of models obtained from GA-SVR are provided in Table 5. In terms of polynomial kernel, the

277

Oak Creek River has higher d (degree of function) values than Nahal Yatir and Diaoga Rivers. It is clear from Table 5

278

that among the cases, Oak Creek River has the lowest values of ε. Also the Nahal Yatir River has the highest values of

279

loss function (C). Comparison between ε values and C values indicates that the loss function (C) has a wider range than

280

ε for all cases.





10

281

In case of RBF and ERBF kernels, σ denotes the optimal width of kernel function. RBF and ERBF with large σ

282

allow support vector to have a strong impact over a larger area. To evaluate the accuracy and also the capability of the

283

applied models to predict bed load transport rate, there was a comparison between observed and predicted values that

284

is represented in Fig. 6-11. Also the perfect agreement lines are shown in Figs. 6-11 in which observed and predicted

285

values are equal. Fig. 6 shows scattered plots of observed and predicted bed load transport rate by using GA-SVR

286

model through RBF kernel function for Oak Creek River. It is clear from Fig. 6 that GA-SVR performs better in low

287

bed load transport rate than in high rate. Furthermore, Fig. 7 shows scattered plots of observed and predicted bed load

288

transport rate of GA-SVR model with ERBF kernel for Oak Creek River. For both low and high transport rates, GA-

289

SVR performance was improved by using ERBF kernel. The scatter plots of observed and predicted bed load transport

290

rate using polynomial and ERBF kernels for Nahal Yatir River are shown in Fig. 8 and 9, respectively. As it can be

291

seen from Figs. 8 and 9, polynomial kernel seems to be better than the ERBF kernel in prediction of low bed load

292

transport values. Also, the scatter plots of observed and predicted bed load transport rate using RBF and ERBF kernels

293

are shown in Figs. 10-11 for Diaoga River which have almost similar prediction, especially in high rates of bed load

294

transport.

295

Statistical parameters of GA-SVR models established with whole data sets are given in Table 6. Results comparison of

296

the Oak Creek River reveals the superiority of RBF and ERBF kernels over the other kernels. Using RBF and ERBF

297

kernels improves prediction results about 25%. For input combination 1 and input combination 2, linear and

298

polynomial kernel functions have considerably inferior results than RBF and ERBF kernels. Among RBF kernel

299

results, input combination 1 (input variable: Q) has the lowest EC value and input combination 2 (input variables: Q,

300

y/D50) has the highest EC value for the Oak Creek River. In terms of ERBF kernel, input combination 2 and input

301

combination 3 (input variables: Q, y/D50, τ/QD50) have better prediction accuracy than the other input combinations.

302

Polynomial and linear kernel function results show that using input combination 3 and input combination 4 (input

303

variables: y/RST , τ/ QRST ) as GA-SVR inputs improves (5-7%) the prediction accuracy. Superiority of ERBF kernel

304

over the other kernels is clearly seen from Table 8. It seems that input combination 2 is appropriate for Oak Creek

305

River.

306

Insert Fig. 6.

307

Insert Fig. 7.

308

The second section of Table 6 represents the prediction results of Nahal Yatir River. By using input combination 1

309

(EC=0.89 and R=0.91) and input combination 2 (EC=0.85 and R=0.85) as inputs of GA-SVR model, polynomial

310

kernel function shows better performance than other kernel functions. GA-SVR models with polynomial kernel

11

311

functions (input combination 1 and input combination 2) perform better than GA-SVR models with other kernel

312

functions (input combination 1 and input combination 2). In terms of ERBF kernel, input combination 3 and input

313

combination 4 have better predicting accuracy than the other input combinations, and prediction results were

314

improved about 5%. In case of input combination 3, the EC value increases 5% by using ERBF kernel.

315

Here, R values of input combination 1 are higher than other input combinations except for ERBF and RBF kernels.

316

Results of input combination 3 (input variables: Q, y/D50, τ/ QRST ) in which shear stress is added as an input variable,

317

are better than those of input combination 2. This means that prediction results are improved by adding shear stress

318

parameter. Among sixteen GA-SVR models, the model which has the input combination 4 with ERBF kernel shows

319

the best performance in bed load prediction for the Nahal Yatir River.

320

The prediction results of Diaoga River are shown in the last section of Table 8. It can be clearly seen that RBF and

321

ERBF performances are considerably better than other kernel functions (polynomial and linear). So, RBF kernel

322

increases EC values 5-15% and ERBF kernel increases about 10-20%. It is also clear that input combination 3 and

323

input combination 4 show better prediction accuracy than the other input combinations.

324

In this study, for all cases, using GA-SVR method, the ERBF kernel is the most appropriate kernel to predict bed load

325

transport. Although all rivers have a gravel-bed, selection of best input combination seems to be the complex part of

326

this study. For the Oak Creek River, a collection of flow discharge (Q) and y/D50 seems to be the best input in bed

327

load prediction.

328 329 330

Insert Fig. 8. For the Nahal Yatir River, the dominant parameters seem to be shear stress and median diameter of particles. There

331

is a similarity in terms of Diaoga River and Nahal Yatir River results, in which a mixture of shear stress, median

332

diameter of particles and flow depth improves prediction accuracy. In the case of Nahal Yatir and Diaoga Rivers,

333

obtained results revealed that adding shear stress into the models’ inputs can significantly increase the accuracy of

334

models.

335

The input combinations listed in Table 4 were applied as GA-SVR inputs for all three studied rivers. The kind of

336

river (e.g. mountain stream, alluvial stream) and the kind of flow regime (ephemeral or perennial) have the most

337

effective roles in bed load transport mechanism. Therefore, differences between the characteristics of rivers cause

338

different effective parameters in bed load transport at various flow conditions. Since the studied rivers have different

339

field conditions, so different hydraulic parameters could be obtained as the most effective parameters in bed load

340

transport for them. The Oak Creek is alluvial river, while Diaoga River is mountain stream and Nahal Yatir is semi-

341

arid river. It seems water surface slope (S) is the important factor in mountain streams, which affects shear stress

12

342

directly (;=γ<=S). Also in semi-arid channels and ephemeral rivers (where floods may initiate bed load transport by

343

moving bed materials), shear stress is the most important criterion for incipient motion (Shields parameter).

344

Therefore, it seems, because of having different characteristics (alluvial river and permanent stream), that the Oak

345

Creek River has different effective hydraulic parameters.

346 347 348 349 350

Insert Fig. 9. Insert Fig. 10. Insert Fig. 11. Statistical parameters of bed load transport formulas are provided in Table 7. Traditional bed load formulas clearly

351

showed poor performance in bed load transport rate prediction which made the obtained results to be no reliable

352

enough. Also, it can be seen that prediction results of MPM formula are slightly better (about 1-5%) than those of

353

Einstein formula.

354

Due to stochastic nature of bed load movement along river bed, it is difficult to define precisely at what flow

355

condition a sediment particle will begin to move. There are several classical approaches to study bed load transport

356

in which different opinions have been applied to define incipient motion in rivers. MPM indicated that for a bed

357

load transport rate substantially greater than the threshold value for particle incipient motion, part of the shear

358

stress applied by the flowing water over the bed is absorbed by the particles in bed load transport. As a result, the

359

effective force causing the movement of sediment is reduced. Einstein postulated the probability of particle

360

movement as the probability of lift exceeding the submerged weight of the particle, leading to the transport

361

function being related to characteristics of turbulent flow. As there is no specific way to define incipient motion

362

exactly, different incipient motion definitions may lead classical approaches to imprecise prediction of bed load

363

transport rate. The results of present study show that the GA-SVR method even with flow discharge (simply

364

measured data) as input variable performs considerably better than classic formulas.

365

In this part of study, all models were re-analyzed by eliminating high (extreme) bed load transport rates. Obtaining

366

the optimal value of transport rate which separates high and low transport rate is indeed a complicated process. In

367

the present study, the trial and error method was applied to obtain the optimal values of separators in the rivers.

368

Different values were examined as separators of high and low transport rate and improvement percentages of EC

369

values were evaluated by eliminating high transport values, since

370

improvement for EC value. Table 8 shows improvement percentage of the statistical parameters after eliminating

371

the high (extreme) values of bed load transport. As it can be seen from Table 8, prediction results were improved

372

by eliminating high bed load transport rate, and it means that high bed load transport data can lead calculation to

373

inferior results.

13

the optimal separator gives the highest

374

The percentage of improvement differs for various models. Although both < and EC values increase, elimination of

375

high transport rate effects on < values more than on EC values. The most variations can be observed for Diaoga

376

River where the highest improvement was obtained 10.74%. Also the results of Diaoga River show that the

377

variations of linear and polynomial kernels are higher than of other kernel functions. The highest variation of EC is

378

9.85% (linear kernel) for Oak Creek River, while it is 6.33% (RBF kernel) for Nahal Yatir River. The results of

379

Oak Creek River show that the highest R variation was obtained by ERBF kernel (8.05%).

380

Table 9 shows statistical parameters of GA-SVR models after eliminating the high (extreme) values of bed load

381

transport. It is clear from Table 9 that the obtained results are similar to results of whole data set (before

382

eliminating). The ERBF kernel produces the most accurate prediction for all of the rivers. Also, obtained the most

383

effective input combinations are exactly similar to those of Table 6 (before eliminating high values).

384

Obtained results reveal that SVR shows better performance in low bed load transport rate than in high transport

385

rate. It seems bed load transport to be more complicated in high rate than in low transport rate. Therefore, it is

386

recommended that bed load transport rate is divided into different ranges according to appropriate hydraulic

387

criteria, and more effective parameters should be considered in studies to achieve acceptable prediction accuracy.

388

When using data driven approaches (e.g. SVR), the introduced and applied input parameters might be reduced to

389

some most necessary parameters which fundamentally affect the studied phenomenon. Nevertheless, there are

390

various approaches for selecting the input variables of the data driven approaches, among them, the physical

391

similarity is one of the most common methods. So, in the present study, this method was applied for feeding the

392

SVR model. By using this approach, there will be opportunity to use the minimum required parameters of

393

modeling, to construct the necessary input-output mapping using the data set. Consequently, there is no need to

394

take into consideration the dynamically variable sediment values, bed roughness characteristics, and various

395

diameter sizes (which govern the sediment variations) to get an acceptable level of simulation issue for bed load

396

transport. Instead, the easily recordable/computable parameters, e.g. river flow (Q), flow depth (y), median

397

diameter (D50), and shear stress are applied to feed the SVR model for simulating bed load transport.

398

The GA-SVR method is based on black box concept in which the method solves a problem without considering the

399

nature of problem. Contrarily, other classic methods (e.g. empirical and semi-empirical formulas, stream power

400

correlation) are based on bed load motion theory, therefore, mathematical relation should be set between hydraulic

401

parameters and bed load transport rate and related formulas should be obtained. Therefore, the GA-SVR method,

402

without knowledge about bed load transport mechanism and its complicated nature, is able to apply different

403

combinations of hydraulic parameters and predict bed load transport rate better than classic methods and formulas.

14

404

The effect of each parameter or combination of parameters on bed load transport rate could be investigated by GA-

405

SVR method.

406 407

6. Conclusion In this paper, it was attempted to depict the functionality of SVR method in order to predict bed load transport

408

rate. Three data sets of Oak Creek River, Nahal Yatir and Diaoga Rivers were used to achieve the purpose.

409

Furthermore, four combinations of hydraulic parameters were applied as input variables for models. A genetic

410

algorithm was applied to search optimal parameters for SVR models, since SVR method performance is highly

411

influenced by the good setting of parameters. This study demonstrates a successful application of the GA-SVR

412

modeling concept to bed load transport. In the case of Oak Creek and Diaoga Rivers, ERBF and RBF kernels

413

show better performance than other kernel functions. But regarding Nahal Yatir River, ERBF and polynomial

414

kernels were chosen as appropriate kernel functions. According to the results, in terms of Oak Creek, flow

415

discharge and y/DST have dominant role in bed load transport, while for Nahal Yatir and Diaoga Rivers,

416

τ/QRST has the most effective role in bed load transport. Finally, the tested data sets of all cases were predicted by

417

Einstein and MPM formulas. Results comparison showed that GA-SVR has better bed load predicting capability

418

than the other traditional formulas. Therefore, with promising and acceptable results, GA-SVR can be utilized to

419

provide a reliable solution to predict bed load transport rate of rivers. Furthermore, high bed load transport rate

420

values were eliminated and low transport rate data sets were predicted by GA-SVR method. Obtained results

421

showed that GA-SVR technique performs better in low transport rate than in high transport rate, and bed load

422

transport studies should be split into different ranges of transport rate. Further researches may be carried out for

423

predicting total sediment load in gravel-bed rivers using GA-SVR method. As well as, application of GA-SVR

424

for modeling and predicting incipient motion in rivers might be considered in future studies.

425 426

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427

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428

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429 430

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440 441

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perennial counterparts. Water Resources Research, 31(3), 773-781.

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521

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522 523

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524 525

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526

load in mountain streams. International Journal of Sediment Research, 24(3), 260-273.

527 528

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529

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530

Zhang, K., Wang, Z. Y., & Liu, L. (2010) The effect of riverbed structure on bed load transport in mountain streams. Mouth, 6,

531

0-03.

532

18

533 534

Figure Captions:

535

Fig. 1.

Cost function of SVM model

Fig. 2.

GA-SVR flowchart

Fig. 3.

The location of Oak Creek River

Fig. 4.

The location of Nahal Yatir station

Fig. 5.

The location of Diaoga River

545

Fig. 6.

Predicting using GA-SVR (RBF kernel) against observed bed load rate – Oak Creek River

546 547

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

536 537 538 539 540 541 542 543 544

548 549 550

Fig. 7. River

551 552

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

Predicting using GA-SVR (ERBF kernel) against observed bed load rate – Oak Creek

553 554 555

Fig. 8. River

556 557

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

Predicting using GA-SVR (polynomial kernel) against observed bed load rate –Nahal Yatir

558 559 560

Fig. 9. River

561 562

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

Predicting using GA-SVR (ERBF kernel) against observed bed load rate –Nahal Yatir

563

19

564

Fig. 10.

565 566

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

Predicting using GA-SVR (RBF kernel) against observed bed load rate –Diaoga River

567 568

Fig. 11.

569 570

(a) Input Combination 1, (b) Input Combination 2, (c) Input Combination 3 and (d) Input Combination 4

Predicting using GA-SVR (ERBF kernel) against observed bed load rate – Diaoga River

571 572 573 574

20

575

576 577 578 579

Fig. 1.

580

21

581 582 583 584

Initial SVR Structure Creation (randomized values for C, ε and kernel function parameters are assigned)

585 GA Procedure

586 587

Initial Population (Solutions) Generation

588 589 590

Fitness Calculation and Generation Evaluation (the SVR is trained by available training data set and Efficiency Coefficient (EC) is calculated for predicted data set)

591 592 593 594

Termination Criterion Satisfied? (obtaining the highest EC value is the optimization purpose)

Yes

End

Introducing the Optimal Values for C, ε and Kernel Function Parameters

595 No

596 597

Obtaining Best Prediction Data Set

Selection

598 Crossover

599 600

Mutation

601 602 603 604 605

Fig. 2.

606 607 608 22

609 610

611 612 613 614 615

Fig. 3.

616

23

617 618 619 620

621 622 623 624 625

Fig. 4.

626

24

627

628 629 630 631 632

Fig. 5.

633

25

634

x10-3

1.2

a

0.9

x10-3

b

0.9

0.6

0.6

0.3

0.3

0

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

0

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

Predicted qb (kg/(ms))

636 1.2

x10-3

d

0.9 0.6 0.3 0

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

637 638

Fig. 6.

639

26

Predicted qb (kg/(ms))

Predicted qb (kg/(ms))

1.2

Predicted qb (kg/(ms))

635 1.2

x10-3

c

0.9 0.6 0.3 0

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

640 641 642 x10-3

a

0.9 0.6 0.3

-1.11E-15

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

1.2

x10-3

b

0.9 0.6 0.3

-1.11E-15

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

643 Predicted qb (kg/(ms))

x10-3

1.2

d

0.9 0.6 0.3

-1.11E-15

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

644 645 646

Fig. 7.

647

27

Predicted qb (kg/(ms))

1.2

Predicted qb (kg/(ms))

Predicted qb (kg/(ms))

x10-3

1.2

c

0.9 0.6 0.3

-1.11E-15

x10-3

0

0.3 0.6 0.9 1.2

Observed qb (kg/(ms))

648 649

a

0 1 2 3 4 5 6 7 8 Observed qb (kg/(ms)) Predicted qb(kg/(ms))

651 8 7 6 5 4 3 2 1 0

8 7 6 5 4 3 2 1 0

Predicted qb(kg/(ms))

8 7 6 5 4 3 2 1 0

Predicted qb(kg/(ms))

Predicted qb (kg/(ms))

650

b

0 1 2 3 4 5 6 7 8

Observed qb (kg/(ms))

Observed qb (kg/(ms))

d

Observed qb (kg/(ms))

653 654 655

c

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

652

8 7 6 5 4 3 2 1 0

Fig. 8.

656

28

657

Predicted qb (kg/(ms))

659 8 7 6 5 4 3 2 1 0

a

8 7 6 5 4 3 2 1 0

Predicted qb (kg/(ms))

8 7 6 5 4 3 2 1 0

Predicted qb (kg/(ms))

Predicted qb (kg/(ms))

658

b

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Observed qb (kg/(ms))

Observed qb (kg/(ms))

Observed qb (kg/(ms))

d

Observed qb (kg/(ms))

661 662 663

c

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

660

8 7 6 5 4 3 2 1 0

Fig. 9.

664

29

665 666 667 668

a

4 3 2 1 0 0

1

2

3

4

5

b

4 3 2 1 0 0

Observed qb (kg/(ms))

1

2

3

4

5

d

3 2 1 0 0

1

2

3

4

5

Observed qb (kg/(ms))

671 672

673

c

4 3 2 1 0

1

2

3

4

Observed qb (kg/(ms))

5 4

5

0

Observed qb (kg/(ms))

670 Predicted qb (kg/(ms))

5

Predicted qb (kg/(ms))

5

Predicted qb (kg/(ms))

Predicted qb (kg/(ms))

669

Fig. 10.

674

30

5

a

4 3 2 1 0 0

1

2

3

4

5

Observed qb (kg/(ms))

b

4 3 2 1 0 0

1

2

3

4

5

Observed qb (kg/(ms))

676 Predicted qb (kg/(ms))

5

Predicted qb (kg/(ms))

5

Predicted qb (kg/(ms))

Predicted qb (kg/(ms))

675

d

3 2 1 0 0

1

2

3

4

5

Observed qb (kg/(ms))

677 678 679

c

4 3 2 1 0 0

1

2

3

4

Observed qb (kg/(ms))

5 4

5

Fig. 11.

680

31

5

681 682 683 684 685 686 687 688 689 690 691

Table 1. Genetic algorithm settings Population size Population type Selection function Crossover fraction Crossover function Mutation function

20 Double vector Stochastic uniform 0.8 Scatter Gaussian

692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709

32

710 711 712 713 714

Table 2. Kernel functions Kernel RBF ERBF Polynomial Linear

715

Function K (' , '+) = exp(K (' , '+) = exp(-

Kernel parameter ||r/0rs

||2

t 2 ||r/0rs || t 2

)

K (' , '+) = ((' , '+ ) + 1)

K (' , '+) = (' , '+ )

u

) v

u d -

716 717 718 719 720 721 722 723 724 725 726

Table 3. Characteristics of the studied rivers

River Oak Creek Nahal Yatir Diaoga

Features Alluvial stream Semi- arid Mountain stream

Flow depth (m) 0.12 - 0.53 0.11 - 0.59 0.06 - 0.22

727 728 729 730 731 732 733

33

Discharge (m3/s) 0.153 – 3.398 0.3 - 4.9 0.034 - 1.565

Median diameter (mm) 15 54 6

734 735 736 737 738 739 740 741 742 743

744

Table 4. Applied input combinations Model

Parameter(s)

Input combination 1

Q

Input combination 2

Q,

Input combination 3

Q,

Input combination 4

Jxy

w

w

Jxy w

Jxy

,

z

,

z

@.Jxy

@.Jxy

745 746 747 748 749 750 751 752 753 754 755 756

34

757 758 759 760 761 762

Table 5. Optimal parameters of GA-SVR models Kernel

Input combination

ε

C

{

d

1 2 3 4

0.005 0.005 0.003 0.004

0.18 0.27 0.25 0.35

0.14 0.24 0.24 0.28

-

1 2 3 4

0.006 0.004 0.002 0.004

0.78 0.77 0.25 0.28

0.11 0.16 0.75 0.65

-

1 2 3 4

0.005 0.004 0.004 0.002

0.39 0.32 0.54 0.29

-

1.15 1.3 0.95 1.23

1 2 3 4

0.002 0.005 0.006 0.007

0.22 0.42 0.58 0.21

-

-

1 2 3 4

0.55 0.48 0.77 0.78

3.50 1.91 2.79 2.64

0.72 0.69 0.95 0.96

-

1 2 3 4

0.34 0.33 0.44 0.08

1.77 1.81 2.19 1.67

0.95 0.93 0.87 2.78

-

1 2 3 4

0.99 0.58 0.64 0.7

1.73 0.89 0.04 1.15

-

0.61 0.87 0.75 0.54

1 2 3 4

0.53 0.92 0.68 0.87

0.32 1.76 0.06 0.03

-

-

1 2 3 4

0.012 0.016 0.011 0.022

0.85 0.92 0.77 0.53

0.77 0.87 0.70 0.62

-

1 2 3 4

0.012 0.027 0.009 0.020

0.81 0.87 0.65 0.57

0.84 0.69 0.58 0.82

-

1 2 3 4

0.033 0.045 0.033 0.021

0.33 0.51 0.46 0.41

-

0.93 0.88 0.94 1.06

Oak Creek River

RBF

ERBF

Polynomial

Linear

Nahal Yatir River RBF

ERBF

Polynomial

Linear

Diaoga River RBF

ERBF

Polynomial

35

Linear 1 2 3 4

0.055 0.021 0.062 0.041

763 764

36

0.66 0.74 0.61 0.92

-

-

765 766 767

Table 6.

768 Statistical parameters of GA-SVR models established with whole data Input combination Oak Creek River 1 2 3 4

RBF

ERBF

Polynomial

<

EC

<

EC

0.70 0.83 0.79 0.82

0.69 0.83 0.78 0.76

0.77 0.93 0.89 0.84

0.72 0.85 0.84 0.84

< 0.65 0.75 0.71 0.73

Linear EC

<

EC

0.52 0.49 0.56 0.60

0.63 0.69 0.58 0.63

0.50 0.47 0.55 0.54

Nahal Yatir River 1 2 3 4

0.88 0.85 0.91 0.88

0.85 0.84 0.85 0.83

0.92 0.87 0.94 0.93

0.86 0.83 0.90 0.92

0.91 0.85 0.88 0.88

0.89 0.85 0.86 0.88

0.88 0.84 0.88 0.86

0.86 0.84 0.85 0.85

1 2 3 4

0.75 0.81 0.78 0.80

0.66 0.75 0.71 0.72

0.77 0.82 0.87 0.88

0.71 0.75 0.81 0.80

0.71 0.65 0.71 0.72

0.61 0.58 0.66 0.65

0.55 0.64 0.68 0.63

0.52 0.58 0.64 0.61

Diaoga river

769 770 771 772 773 774 775 776 777 778 779 780 781 782 783

37

784 785 786 787 788 789 790

Table 7. Calculated statistical parameters of bed load formulas Einstein MPM <

Oak Creek Nahal Yatir Diaoga

EC

0.12 0.34 0.14

0.04 0.33 0.07

<

EC

0.18 0.40 0.10

0.09 0.35 0.08

791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806Table 8. 807Improvement percentage of the statistical parameters after eliminating the high (extreme) values of bed load 808transport. Input combination Oak Creek 1

RBF ∆<(%) 7.23

ERBF ∆EC(%) 3.13

∆<(%) 8.05

38

Polynomial

Linear

∆EC(%)

∆<(%)

∆EC(%)

3.25

6.62

3.25

∆<(%) ∆EC(%) 4.23

5.47

2 3 4

4.10 5.52 6.32

2.68 2.98 3.98

1.74 3.91 5.51

4.36 6.11 1.27

2.14 6.68 5.93

1.70 6.81 4.32

8.36 5.23 4.55

9.85 6.32 2.98

1 2 3 4

4.09 4.00 2.47 3.63

3.67 1.39 2.30 6.33

2.85 5.89 1.06 1.66

2.34 2.66 1.84 1.36

2.75 2.24 4.13 1.57

1.01 1.47 1.52 0.74

7.85 5.36 3.21 4.19

5.68 5.98 1.47 3.93

1 2 3 4

7.42 5.87 8.35 7.12

4.81 6.25 5.57 6.44

7.32 6.31 6.65 4.41

5.87 5.93 2.77 3.64

7.80 8.74 10.65 7.41

5.55 6.38 4.88 3.32

10.74 9.87 7.78 9.25

6.69 5.54 4.66 6.51

Nahal Yatir

Diaoga

809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824

39

825 826 827

Table 9.

828 Statistical parameters of GA-SVR models after eliminating the high (extreme) values of bed load transport Input combination Oak Creek River 1 2 3 4

RBF

ERBF

Polynomial

<

EC

<

EC

0.77 0.87 0.84 0.88

0.72 0.85 0.81 0.80

0.85 0.94 0.93 0.89

0.75 0.90 0.90 0.85

< 0.71 0.77 0.78 0.79

Linear EC

<

EC

0.55 0.50 0.63 0.64

0.67 0.77 0.63 0.68

0.55 0.57 0.61 0.57

Nahal Yatir River 1 2 3 4

0.92 0.89 0.94 0.92

0.89 0.85 0.88 0.89

0.94 0.93 0.95 0.94

0.88 0.85 0.92 0.93

0.93 0.87 0.92 0.90

0.90 0.86 0.88 0.89

0.95 0.90 0.91 0.90

0.90 0.90 0.86 0.89

1 2 3 4

0.83 0.87 0.86 0.87

0.71 0.81 0.76 0.78

0.84 0.88 0.93 0.92

0.77 0.81 0.84 0.83

0.79 0.74 0.81 0.79

0.66 0.64 0.71 0.69

0.65 0.74 0.76 0.72

0.59 0.64 0.69 0.67

Diaoga river

829 830

40

831

Highlights

832

•

Bed load transport rates of three gravel-bed rivers were predicted using GA-SVR.

833

•

Different combinations of hydraulic parameters were used as GA-SVR inputs.

834

•

ERBF kernel showed better performance than other kernels in bed load prediction.

835

•

The GA-SVR models were superior to traditional bed load formulas.

836

•

The elimination of high bed load transport rates improved prediction accuracy.

41