Side weirs have many possible applications in the field of hydraulic engineering. They are also considered an important structure in hydro systems. In this study, the support vector machine (SVM) technique was employed to predict the side weir discharge coefficient. The performance of SVM was compared with other types of soft computing techniques such as artificial neural networks (ANN) and adaptive neuro fuzzy inference systems (ANFIS). While ANN and ANFIS models provided a good prediction performance, the SVM model with a radial basis function kernel function outperforms them. The best SVM model was developed with a gamma coefficient and epsilon of 15 and 0.3, respectively. The SVM yielded a coefficient of determination (R2) equal to 0.96 and 0.93 for the training and testing data. Sensitivity analyses of the ANN, ANFIS and SVM models showed that the Froude number and ratio of weir length to the flow depth upstream of the weir are the most effective parameters for the prediction of the discharge coefficient.

INTRODUCTION

Modeling hydraulic structures has received much attention in recent years due to their effects on hydro-system performances. Weirs are the most common structures, which are used in most water engineering projects including hydro power systems, irrigation and drainage networks and sewage networks. A side weir is a hydraulic structure placed on the side of the channel, and sometimes it has been used as a water surface control structure in dams and irrigation projects, whereas the main task of this structure is to remove the excess flow from hydro systems (Haddadi & Rahimpour 2012; Bagheri et al. 2014). Side weirs have many possible uses in the field of hydraulic engineering, and have been investigated as an important structure in water resource projects as well. A study on the hydraulic properties of side weirs has been conducted using physical and numerical modeling (Namaee & Shadpoorian 2015). In the field of physical modeling, researchers have attempted to improve the performance of side weirs by proposing various forms for the weir crest. All different forms have been compared with the standard side weir, which has a rectangular shape. Based on physical modeling research, by improving the shape of the side weir crest, the side weir discharge coefficient can be increased about three times (Singh et al. 1994; Coşar & Agaccioglu 2004; Borghei & Parvaneh 2011; Emiroglu & Kaya 2011; Kabiri-Samani et al. 2011; Kaya et al. 2011; Rahimpour et al. 2011). Due to the high cost of physical modeling and laboratory equipment, investigators have been encouraged to use numerical approaches to determine the properties of this type of weir (Parsaie et al. 2015a, 2015b). Numerical studies of side weirs include two main parts, the first is to solve the governing equation, which is usually a differential equation, by discrete methods such as finite difference, finite volume and finite element. This field of numerical studies is called computational fluid dynamic (CFD) modeling. The other part is related to implementing soft computing approaches, which have been widely used for modeling and prediction of hydraulic phenomena. Researchers using the CFD techniques have attempted to define the water surface profile, velocity distribution and flow pattern along the length of the weir. Another approach in the field of numerical modeling is using the soft computing techniques. Using soft computing techniques helps to model the hydraulic properties accurately. The side weir discharge coefficient has been predicted by the adaptive neuro fuzzy inference system (ANFIS), the group method of data handling, gene expression programing and the multilayer perceptron (MLP) artificial neural network (ANN) (Emiroglu et al. 2011a, 2011b; Emiroglu & Kisi 2013; Ebtehaj et al. 2015a, 2015b). In addition to accurate modeling, the merit of ANN models is that they can be used to improve the accuracy of numerical solutions (Parsaie & Haghiabi 2014, 2015a, 2015b). Based on this work, ANNs have high ability for modeling systems which are based on the data set and for this reason they have become one of the most popular modeling tools (Abrahart et al. 2004; Govindaraju & Rao 2013). The idea of ANN development came from the neural system of the human brain. Choosing a sufficient number of neurons ensures the suitable performance of the model for modeling the phenomena based on the range of the relative available data set, but this characteristic does not guarantee the suitable performance of the model for predicting the phenomena outside the data range. Based on the reports, increasing the complexity of a model does not ensure a better predictive power whereas sometimes adding more complexity (increasing the number of neurons) may reduce the performance of the model (Cartwright 2015). This can be assessed from two aspects, on the other hand, any decrease in the performance of the model on a validation data set can be considered as indicative that the ANN has been over-fitted. The ANN training should not be performed too far with regard to the specific number of ANN parameters. One of the stopping criteria may be adjusted using the cross validation procedure. Alternatively, the model can be considered as over-parameterized: i.e. the available data set is insufficient for training the parameters of the designed model. The complexity of the designed model (number of parameters may have to be adjusted) with regard to the model results at the training stage should be balanced against its performance in modeling the unseen available data set. There is no direct way to determine the optimal structure (number of hidden layers and neurons) for the ANN and there is a risk of selecting ‘over-parameterized’ models: i.e. models so complex that they cannot be considered reliable when applied further than the training data set (Gaume & Gosset 2003). In this study, the support vector machine (SVM) technique is used to predict the discharge coefficient of standard rectangular side weirs. The SVM model was chosen because of its high ability in pattern recognition (Azamathulla & Wu 2011; Azamathulla & Zahiri 2012; Najafzadeh & Azamathulla 2013; Tayfur 2014; Zahiri & Azamathulla 2014; Najafzadeh & Sattar 2015). To evaluate the performance of SVM, other types of soft computing techniques (i.e. ANN and ANFIS) are developed.

MATERIAL AND METHODS

The side weir discharge coefficient is a function of hydraulic properties, fluid characteristics and main channel geometries. A schematic of a side weir is given in Figure 1. Equation (1) gives the most important parameters, which affect the side weir discharge coefficient (). 
formula
1
where, is the flow velocity, L is the side weir length, is the diversion angle of the flow, P is the weir height and S0 is the longitudinal slope of the channel.
Figure 1

Sketch of side weir at subcritical flow condition.

Figure 1

Sketch of side weir at subcritical flow condition.

Using the Buckingham theory as representative of dimensional analysis leads to the derivation of a dimensionless parameter, which helps to develop an optimal structure for empirical formulae and soft computing models. Results of the Buckingham theory are given in Equation (2). 
formula
2
Equation (2) has been used for developing the empirical formula in addition to soft computing techniques. The right side of Equation (2) is considered as input parameters for developing empirical formulae and the soft computing techniques, and the is considered as the desired or output parameter. The most well-known empirical formulae which have been proposed by investigators for the prediction of a standard rectangular side weir discharge coefficient are given in Table 1.
Table 1

Well-known empirical formulae for calculation of the side weir discharge coefficient

SVM

SVMs are a set of related supervised learning methods used for classification and regression. In many applications, a non-linear classifier provides better accuracy. In SVM, the input x is first mapped onto an m-dimensional feature space using some fixed (nonlinear) mapping, and then a linear model is constructed in this feature space. The naive way of making a non-linear classifier out of a linear classifier is to map the data from the input space X to a feature space F using a non-linear function . In the space F, the discriminant function is: 
formula
3
Using mathematical notation, the linear model (in the feature space) f(x, w) is given by: 
formula
4
 
formula
5
 
formula
6
In the feature space, F, this expression takes the following form: 
formula
7
 
formula
8
 
formula
9
There are many kernel functions in SVM, so how to select a good kernel function is also a research issue. However, for general purposes, there are some popular kernel functions.
  • I. Linear kernel:

  • II. Polynomial kernel:

  • III. RBF kernel:

  • IV. Sigmoid kernel:

Here C, γ and r and d are kernel parameters. It is well-known that SVM generalization performance (estimation accuracy) depends on a good setting of meta-parameters, parameters C, γ and r and the kernel parameters. The choices of C, γ and r control the prediction (regression) model complexity. The problem of optimal parameter selection is further complicated, because the SVM model complexity (and hence its generalization performance) depends on all three parameters. Kernel functions are used to change the dimensionality of input space to perform the classification (Azamathulla & Wu 2011; Parsaie et al. 2015a, 2015b).

ANN

ANN is a popular soft computing technique which has been used in the broad field of engineering. The idea of ANN development came from the biological neurons of the human brain. A typical neuron in the neural network is shown in Figure 2. Each input into a neuron in a hidden or output layer is multiplied by a corresponding interconnection weight () and summed by a threshold constant value named bias (). Equation (10) shows the mathematical form of the addition and multiplication operation in each neuron. 
formula
10
Figure 2

A typical neuron.

Figure 2

A typical neuron.

Then the result of is passed through the transfer function. Various types of transfer function have been proposed, the most well-known are given below.

  • I. Gaussian:

  • II. Sigmoidal:

  • III. Tansing:

After performing the transfer function on the inputs (i.e. ) the output of each neuron is computed as Equation (11): 
formula
11

This process (preparation of a neuron in the network) is carried out on all the neurons in the first hidden layer. In order to develop the second hidden layer, the output of the first hidden layer is considered for the neurons. This process is continued until the ANN structure based on the desire of the designer has been formed. As noted during the ANN preparation a network could have one or more hidden layers in this type of neural network named MLP. The MLP is a common type of ANN which is used widely for modeling and predicting engineering problems. The values of the weights and biases are defined during the training process which is called the learning stage. Learning means justifying the values of weights and biases so that the output of the network has minimum error in comparison with the observed values. Several algorithms have been proposed for training the ANNs, such as the Levenberg–Marquardt algorithm, but recently modern optimization algorithms such as the genetic algorithm (GA) and the particle swarm optimization (PSO) have been proposed (Parsaie & Haghiabi 2015a, 2015b).

ANFIS

ANFIS is a powerful tool for modeling complex systems based on input and output data. ANFIS is realized by an appropriate combination of neural and fuzzy systems. This combination enables the numeric power of both intelligent systems to be used. In fuzzy systems, different fuzzification and defuzzification strategies with different rules are considered for input parameters. Three stages should be considered for implementing fuzzy logic on the input data. First, selecting the membership function (MF) for each input variable. In this stage maybe a Gaussian function is considered for each of the input variables. Figure 3(a) shows a fuzzy reasoning process. For simplicity of illustration, a fuzzy system with two input variables and one output was considered. Suppose that the rule base contains two fuzzy if-then rules. 
formula
 
formula
Figure 3

The architecture of an ANFIS model.

Figure 3

The architecture of an ANFIS model.

Where, A1; A2 and B1; B2 are the MFs for inputs x and y; respectively; p1; q1; r1 and p2; q2; r2 are the parameters of the output function. The ANFIS architecture is presented in Figure 3(b). In the first layer, all input variables give the grade of membership with membership function, in layer 2, all the membership grades are multiplied together, in layer 3, all the membership grades are normalized, normalized and in layer 4 the contribution of all the rules is computed. In the last layer the output variable is computed as the weighted average of membership grade (Riahi-Madvar et al. 2009).

Model development

Preparation of the SVM, ANN and ANFIS include choosing the type of the kernel function called the transfer function in ANN and the membership function in ANFIS, and then setting the internal parameters of the kernel function. Adjusting the internal parameters of the kernel function and in other words, training these methods, can be considered as an optimization problem. This problem can be solved using conventional methods such as the quadratic approach in SVM, the Levenberg–Marquardt technique in ANN and the hybrid algorithm in ANFIS. Recently, advanced optimization such as GA or PSO methods have been implemented for this purpose. Designing the structure of ANNs is a trial and error process. To avoid the ‘over-parameterization’ of ANN, designing the ANN structure is conducted step by step. This means that at the beginning of the design process a small number of neurons and a specific transfer function are considered. Then the model is trained and the results of the model are assessed. After verifying the proper operation of the transfer function, in the next step with regard to the performance of the model, for stages of preparation (training, validation and testing), one or more neurons may be added or removed. For assessing the model performance the results of the model are compared with observed data. It is notable that, to avoid over-training, in all iterations part of the data set is considered for model validation. Validation is a stage between model training and testing for avoiding model over-training. Developing the SVM model is similar to the ANN and ANFIS and is based on data sets. Therefore, 477 data sets related to the side weir discharge coefficient published in creditable journals were collected, the ranges of which are given in Table 2. Data were derived from a number of resources (Subramanya & Awasthy 1972; Singh et al. 1994; Borghei et al. 1999; Emiroglu et al. 2011a, 2011b; Bagheri et al. 2014). To prepare the SVM, ANN and ANFIS models, the data set is divided into three groups: training, validation and testing. The validation data set is considered for avoiding the over-fitting (over-learning) of the ANN model. A random approach is considered when assigning a data set for each stage of model development.

Table 2

Ranges of collected data related to the side weir discharge coefficient

Data rangeFr1P/h1L/bL/h1Cd
Min 0.09 0.03 0.21 0.19 0.09 
Max 0.84 2.28 3.00 10.71 1.75 
Avg 0.43 0.76 1.13 3.87 0.50 
STDEV 0.18 0.43 0.85 3.06 0.17 
Data rangeFr1P/h1L/bL/h1Cd
Min 0.09 0.03 0.21 0.19 0.09 
Max 0.84 2.28 3.00 10.71 1.75 
Avg 0.43 0.76 1.13 3.87 0.50 
STDEV 0.18 0.43 0.85 3.06 0.17 

RESULTS AND DISCUSSION

Except for the Emiroglu formula, other empirical formulae have very high errors when calculating the side weir discharge coefficient on the current database. Results of each empirical formula were plotted versus the measured data and are shown in Figure 4. Standard error indices including the coefficient of determination (R2) and root mean square error (RMSE) were used to assess the accuracy of empirical formulae (see Table 3). As shown in Table 3 and Figure 4, the Emiroglu formula with R2 = 0.64 and RMSE = 0.03 is the most accurate one among the empirical formulae.
Figure 4

Performance of empirical formulae to calculate .

Figure 4

Performance of empirical formulae to calculate .

Table 3

Performance of empirical formulae

AuthorR2RMSE
Nandesamoorthy & Thomson (1972)  0.01 0.00 
Subramanya & Awasthy (1972)  0.01 0.00 
Yu-Tech (1972)  0.01 0.00 
Ranga Raju et al. (1979)  0.01 0.00 
Hager (1987)  0.01 0.01 
Cheong (1991)  0.01 0.01 
Singh et al. (1994)  0.07 0.01 
Jalili & Borghei (1996)  0.06 0.01 
Borghei et al. (1999)  0.11 0.02 
Emiroglu et al. (2011a, 2011b)  0.64 0.03 
AuthorR2RMSE
Nandesamoorthy & Thomson (1972)  0.01 0.00 
Subramanya & Awasthy (1972)  0.01 0.00 
Yu-Tech (1972)  0.01 0.00 
Ranga Raju et al. (1979)  0.01 0.00 
Hager (1987)  0.01 0.01 
Cheong (1991)  0.01 0.01 
Singh et al. (1994)  0.07 0.01 
Jalili & Borghei (1996)  0.06 0.01 
Borghei et al. (1999)  0.11 0.02 
Emiroglu et al. (2011a, 2011b)  0.64 0.03 

Results of ANN

Developing the ANN model (e.g. number of hidden layer(s), type of transfer function, training algorithm, etc.) is a trial and error process. The ANN model contains two hidden layers. The first and second hidden layers contain ten and five neurons, respectively. To choose the transfer function, different types of transfer functions including log-sigmoid (logsig), tan-sigmoid (tansig), linear (purelin) etc. which have been provided in Matlab were tested. During the ANN model development, it was found that tansig has a better performance compared with other transfer functions; therefore, this function was considered as the transfer function for the neurons of hidden layers. The structure of the ANN model is shown in Figure 5. As seen from Figure 5, it was found that the structure of the ANN is not small. The Levenberg–Marquardt technique was used for training the ANN model. Of the data set, 70% was used for training, 15% for validation and the rest (15%) was considered for testing the model. The performance of the ANN model in each stage of development (training, validation and testing) is shown in Figures 68. To assess the performance of this model, error indices for each stage of preparation were calculated and are presented in the these figures. In addition to calculating standard error indices, error distributions for all data used for training, validation and testing was plotted as well. The error histogram was plotted to evaluate the error density. As seen from the histogram, the distribution of error is normal and is more concentrated around zeros. Coefficients of determination of the ANN model were obtained as 0.94, 0.84 and 0.89 for training, validation and testing, respectively. Overall, assessing Figures 68 shows that the accuracy of the ANN model is suitable for prediction of . It is notable that in these figures the target is related to the observed data and output is related to the results of the ANN.
Figure 5

The structure of the ANN model.

Figure 5

The structure of the ANN model.

Figure 6

Performance of the ANN model during the training stage.

Figure 6

Performance of the ANN model during the training stage.

Figure 7

Performance of the ANN model during the validation stage.

Figure 7

Performance of the ANN model during the validation stage.

Figure 8

Performance of the ANN model during the testing stage.

Figure 8

Performance of the ANN model during the testing stage.

Results of ANFIS

Developing the ANFIS model is similar to developing the ANN model, based on a data set. For this purpose, the collected data set, given in Table 2, was used and divided into two groups of training and testing. Choosing the training and testing data sets was based on the randomization approach. Designing the ANFIS is similar to ANN and includes defining the number of hidden layer(s), neurons, membership functions and learning algorithms. The main advantage of the ANFIS model is the utility in the structure designing stage compared with ANN models such as the MLP ANN model. This utility is related to specifying the number of neurons to input variables based on the impacts on the output parameter. In this study, the number of membership functions was added one by one to each input parameter and then the ANFIS model was trained and tested. Results of the ANFIS model to predict the are shown in Figures 9 and 10. The training data set was about 80% of the total collected data and the rest (20%) was used for testing. The structure of the ANFIS, which has the best performance, is given in Table 4. As shown in Table 4, the Gaussian function (gaussmf) has been considered for the membership function and weight average (wtaver) approach has been considered for the defuzzification method. As shown in Figures 9 and 10, the histogram and distribution of the errors are plotted as well for assessing the performance of the ANFIS model in training and testing stages. As is clear from Table 4, Fr1 and P/h1 have more membership functions compared with other parameters. The coefficient of determination of the ANFIS model was 0.98 and 0.86 for the preparation and testing stages, respectively. Overall, as shown in Figures 9 and 10, the ANFIS model's ability is suitable for predicting values of in the training and testing stages and this model has a suitable performance to predict the maximum values of the as well. Comparing the structure and performance of the ANFIS model with the ANN shows that the size of the ANFIS structure is smaller than the ANN and the accuracy of the ANFIS is better than the ANN as well.
Figure 9

Performance of the ANFIS model during the training stage.

Figure 9

Performance of the ANFIS model during the training stage.

Figure 10

Performance of the ANFIS model during the testing stage.

Figure 10

Performance of the ANFIS model during the testing stage.

Table 4

The structure and summary of the ANFIS structures

ParameterNo.MFAND methodOR methodDefuzz MethodAgg MethodTypeR2RMSE
Fr1 gaussmf prod max wtaver max Sugeno train test train test 
P/h1 gaussmf prod max wtaver max 
L/b gaussmf prod max wtaver max 0.96 0.86 0.060 0.085 
L/h1 gaussmf prod max wtaver max      
ParameterNo.MFAND methodOR methodDefuzz MethodAgg MethodTypeR2RMSE
Fr1 gaussmf prod max wtaver max Sugeno train test train test 
P/h1 gaussmf prod max wtaver max 
L/b gaussmf prod max wtaver max 0.96 0.86 0.060 0.085 
L/h1 gaussmf prod max wtaver max      

Results of SVM

Developing the SVM model is similar to the ANN and ANFIS models. In other words, to prepare the SVM, the kernel function and setting the internal coefficients of the kernel function should be considered. Developing the SVM is also based on the data set. For this purpose, the collected data were divided into two groups; training and testing. About 80% of the data set was considered for SVM training and the rest (20%) was considered for model testing. Choosing the kernel function is a trial and error process, however, recommendations of researchers who have conducted similar studies are useful. In this study, the proposed points by Parsaie et al. (2015a, 2015b) were considered. As mentioned in the SVM model introduction section, the four most well-known kernel functions: Linear, Polynomial, RBF and Sigmoid were tested. During the SVM model development, it was found that the RBF function had the best performance compared to other types of kernel functions. Results of the SVM model are shown in Figures 11 and 12. The standard error indices were calculated to define SVM performances during model development. As stated in the model development section, training the SVM model could be considered as an optimization problem; therefore, the developed SVM model was prepared using the quadratic optimization approach. During the SVM model development, 0.3 and 15 were obtained for the sigma and gamma coefficients, respectively. During the SVM model training 0.96 was obtained for the correlation coefficient and 0.02 for the RMSE, and when testing the model, the correlation coefficient was 0.93 and the RMSE was 0.03. To give more information about the SVM model performance, the error distribution and histogram were plotted during both stages of model development. The histogram of the error shows that error distribution through all the data are close to symmetric. The error histogram also shows that most error ranges are around zero, which means that the model accuracy is suitable especially during the training stage. Comparison of the SVM model with empirical formulae shows that it performs better than the ANN and ANFIS models.
Figure 11

Performance of the SVM model during the training stage.

Figure 11

Performance of the SVM model during the training stage.

Figure 12

Performance of the SVM model during the testing stage.

Figure 12

Performance of the SVM model during the testing stage.

Sensitivity analysis of effective parameters

The most effective parameters for prediction of the side weir discharge coefficient by ANN, ANFIS and SVM were defined by a simple approach. This approach describes the effect of each parameter on the model performance of predicting the output. At first all the parameters with regard to Equation (2) were considered as inputs for the ANN, ANFIS and SVM, and then one of the input parameters was removed from the input parameters and again the model with the same structure was prepared. It is notable that preparation of the models was considered with regarding to the model development. Preparation of the models was undertaken as described in the section on model development. In other words, designing the ANN structure was based on the step by step approach and the data set was divided into three groups for training, validation and testing. After adjusting the model structure, the sensitivity analysis of the models to define the most effective parameters began. The performance of the models in the absence of each input parameter was assessed using the calculation of error indices including the and RMSE. Removing one of the input parameters caused a change in model performance. Depending on the degree of change in performance, the effect of each parameter was assessed. The results of the sensitivity analysis of ANN, ANFIS and SVM are given in the Tables 57. As seen in Tables 57, the absence of the Froude number () and ratio of weir length to the upstream flow depth () caused a dramatic decrease in the accuracy of the models, so it was found that these are the most important parameters for modeling the discharge coefficient of a side weir. Another scenario regarding the prediction of the discharge coefficient using these two important parameters ( and ) was considered. This idea has been put into practice for three models (ANN, ANFIS and SVM) and the results of modeling using these two effective parameters for each model are given in the last row of Tables 57. These results show that by using these two important effective parameters, the discharge coefficient of a side weir could be predicted accurately.

Table 5

Results of sensitivity analysis of ANN

ModelAbsentInputsOutputR2RMSE
ANN –   0.89 0.057 
   0.78 0.075 
   0.64 0.168 
   0.86 0.073 
   0.42 0.23 
   0.85 0.096 
ModelAbsentInputsOutputR2RMSE
ANN –   0.89 0.057 
   0.78 0.075 
   0.64 0.168 
   0.86 0.073 
   0.42 0.23 
   0.85 0.096 
Table 6

Results of sensitivity analysis of ANFIS

ModelAbsentInputsOutputR2RMSE
ANFIS –   0.86 0.06 
   0.81 0.08 
   0.76 0.25 
   0.84 0.11 
   0.54 0.34 
   0.83 0.08 
ModelAbsentInputsOutputR2RMSE
ANFIS –   0.86 0.06 
   0.81 0.08 
   0.76 0.25 
   0.84 0.11 
   0.54 0.34 
   0.83 0.08 
Table 7

Results of sensitivity analysis of SVM

ModelAbsentInputsOutputR2RMSE
SVM –   0.96 0.032 
   0.94 0.046 
   0.77 0.091 
   0.86 0.056 
   0.63 0.12 
   0.91 0.057 
ModelAbsentInputsOutputR2RMSE
SVM –   0.96 0.032 
   0.94 0.046 
   0.77 0.091 
   0.86 0.056 
   0.63 0.12 
   0.91 0.057 

CONCLUSION

Prediction of the discharge coefficient in side weirs is an important element of hydraulic structure studies. Side weirs are the most common structure in hydro systems. Therefore, predicting the discharge coefficient of this structure plays an important role in hydro-system management. Results of this study showed that calculating the side weir discharge coefficient with empirical formulae leads to apparently incredible errors in calculation. Based on the obtained results, the SVM model with an RBF kernel function has a suitable capability to predict the side weir discharge coefficient. The SVM model also provides better prediction performance than the ANN and ANFIS models.

REFERENCES

REFERENCES
Abrahart
R.
Kneale
P. E.
See
L. M.
2004
Neural Networks for Hydrological Modeling
.
Taylor & Francis
,
London
.
Azamathulla
H. M.
Zahiri
A.
2012
Flow discharge prediction in compound channels using linear genetic programming
.
Journal of Hydrology
454–455
(
0
),
203
207
.
Bagheri
S.
Kabiri-Samani
A. R.
Heidarpour
M.
2014
Discharge coefficient of rectangular sharp-crested side weirs, Part I: traditional weir equation
.
Flow Measurement and Instrumentation
35
,
109
115
.
Borghei
S. M.
Parvaneh
A.
2011
Discharge characteristics of a modified oblique side weir in subcritical flow
.
Flow Measurement and Instrumentation
22
(
5
),
370
376
.
Borghei
S.
Jalili
M.
Ghodsian
M.
1999
Discharge coefficient for sharp-crested side weir in subcritical flow
.
Journal of Hydraulic Engineering
125
(
10
),
1051
1056
.
Cartwright
H.
2015
Artificial Neural Networks
.
Springer
,
New York
.
Cheong
H.
1991
Discharge coefficient of lateral diversion from trapezoidal channel
.
Journal of Irrigation and Drainage Engineering
117
(
4
),
461
475
.
Coşar
A.
Agaccioglu
H.
2004
Discharge coefficient of a triangular side-weir located on a curved channel
.
Journal of Irrigation and Drainage Engineering
130
(
5
),
410
423
.
Ebtehaj
I.
Bonakdari
H.
Zaji
A. H.
Azimi
H.
Khoshbin
F.
2015a
GMDH-type neural network approach for modeling the discharge coefficient of rectangular sharp-crested side weirs
.
Engineering Science and Technology
18
(
4
),
746
757
.
Ebtehaj
I.
Bonakdari
H.
Zaji
A. H.
Azimi
H.
Sharifi
A.
2015b
Gene expression programming to predict the discharge coefficient in rectangular side weirs
.
Applied Soft Computing
35
,
618
628
.
Emiroglu
M.
Kaya
N.
2011
Discharge coefficient for trapezoidal labyrinth side weir in subcritical flow
.
Water Resources Management
25
(
3
),
1037
1058
.
Emiroglu
M. E.
Agaccioglu
H.
Kaya
N.
2011a
Discharging capacity of rectangular side weirs in straight open channels
.
Flow Measurement and Instrumentation
22
(
4
),
319
330
.
Govindaraju
R. S.
Rao
A. R.
2013
Artificial Neural Networks in Hydrology
.
Springer
,
The Netherlands
.
Haddadi
H.
Rahimpour
M.
2012
A discharge coefficient for a trapezoidal broad-crested side weir in subcritical flow
.
Flow Measurement and Instrumentation
26
,
63
67
.
Hager
H.
1987
Lateral outflow over side weirs
.
Journal of Hydraulic Engineering
113
(
4
),
491
504
.
Jalili
M.
Borghei
S.
1996
Discussion: Discharge coefficient of rectangular side weirs
.
Journal of Irrigation and Drainage Engineering
122
(
2
),
132
.
Kabiri-Samani
A.
Borghei
S. M.
Esmaili
H.
2011
Hydraulic performance of labyrinth side weirs using vanes or piles
.
Proceedings of the ICE-Water Management
164
(
5
),
229
241
.
Kaya
N.
Emiroglu
M. E.
Agaccioglu
H.
2011
Discharge coefficient of a semi-elliptical side weir in subcritical flow
.
Flow Measurement and Instrumentation
22
(
1
),
25
32
.
Najafzadeh
M.
Azamathulla
H. M.
2013
Neuro-Fuzzy GMDH to predict the scour pile groups due to waves
.
Journal of Computing in Civil Engineering
29
(
5
),
04014068
.
Najafzadeh
M.
Sattar
A. A.
2015
Neuro-Fuzzy GMDH approach to predict longitudinal dispersion in water networks
.
Water Resources Management
29
(
7
),
2205
2219
.
Namaee
M. R.
Shadpoorian
R.
2015
Numerical modeling of flow over two side weirs
.
Arabian Journal for Science and Engineering
1
16
.
doi:10.1007/s13369-015-1961-x
.
Nandesamoorthy
T.
Thomson
A.
1972
Discussion of spatially varied flow over side weir
.
Journal of Hydraulic Engineering
98
(
12
),
2234
2235
.
Parsaie
A.
Haghiabi
A.
2014
Predicting the side weir discharge coefficient using the optimized neural network by genetic algorithm
.
Scientific Journal of Pure and Applied Sciences
3
(
3
),
103
112
.
Parsaie
A.
Haghiabi
A.
2015a
Computational modeling of pollution transmission in rivers
.
Applied Water Science
1
10
doi:10.1007/s13201-015-0319-6
.
Parsaie
A.
Haghiabi
A.
Moradinejad
A.
2015a
CFD modeling of flow pattern in spillway's approach channel
.
Sustainable Water Resources Management
1
(
3
),
245
251
.
Parsaie
A.
Yonesi
H.
Najafian
S.
2015b
Predictive modeling of discharge in compound open channel by support vector machine technique
.
Modeling Earth Systems and Environment
1
(
2
),
1
6
.
Rahimpour
M.
Keshavarz
Z.
Ahmadi
M.-m.
2011
Flow over trapezoidal side weir
.
Flow Measurement and Instrumentation
22
(
6
),
507
510
.
Ranga Raju
K. G.
Prasad
B.
Grupta
S. K.
1979
Side weir in rectangular channels
.
Journal of the Hydraulics Division
105
,
547
554
.
Riahi-Madvar
H.
Ayyoubzadeh
S. A.
Khadangi
E.
Ebadzadeh
M. M.
2009
An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS
.
Expert Systems with Applications
36
(
4
),
8589
8596
.
Singh
R.
Manivannan
D.
Satyanarayana
T.
1994
Discharge coefficient of rectangular side weirs
.
Journal of Irrigation and Drainage Engineering
120
(
4
),
814
819
.
Subramanya
K.
Awasthy
S. C.
1972
Spatially varied flow over side-weirs
.
Journal of the Hydraulics Division
98
(
1
),
1
10
.
Tayfur
G.
2014
Soft Computing in Water Resources Engineering: Artificial Neural Networks, Fuzzy Logic and Genetic Algorithms
.
WIT Press
,
Southampton, UK
.
Yu-Tech
L.
1972
Discussion of spatially varied flow over side weir
.
Journal of Hydraulic Engineering
98
(
11
),
2046
2048
.