In this study, modeling of discharge was performed in compound open channels with non-prismatic floodplains (CCNPF) using soft computation models including multivariate adaptive regression splines (MARS) and group method of data handling (GMDH), and then their results were compared with the multilayer perceptron neural networks (MLPNN). In addition to the total discharge, the discharge separation between the floodplain and main channel was modeled and predicted. The parameters of relative roughness coefficient, the relative area of flow cross-section, relative hydraulic radius, bed slope, the relative width of water surface, relative depth, convergence or divergence angle, relative longitudinal distance as inputs, and discharge were considered as models output. The results demonstrated that the statistical indices of MARS, GMDH, and MLPNN models in the testing stage are R2 = 0.962(RMSE = 0.003), 0.930(RMSE = 0.004), and 0.933(RMSE = 0.004) respectively. Examination of statistical error indices shows that all the developed models have the appropriate accuracy to estimate the flow discharge in CCNPF. Examination of the structure of developed GMDH and MARS models demonstrated that the relative parameters: roughness, area, hydraulic radius, flow aspect ratio, depth, and angle of convergence or divergence of floodplain have the greatest impact on modeling and estimation of discharge.

  • Estimation of discharge was performed in compound open channels with non-prismatic floodplains using soft computation models.

  • Discharge separation between the floodplain and main channel was predicted.

  • Effective parameters for determining the discharge are introduced.

  • The accuracy of each of the methods is calculated.

  • The performance of the developed models was compared with the ANFIS model.

ANFIS

adaptive neuro-fuzzy inference system

Ar

relative area

CCNPF

compound open channel with non-prismatic floodplains

fr

relative roughness

GMDH

group method of data handling

MARS

multivariate adaptive regression splines

MLPNN

multilayer perceptron neural networks

R2

coefficient of determination

RMSE

root mean square error

Rr

relative hydraulic radius

S0

bed slope

SVM

support vector machine

xr

relative longitudinal distance

θ

convergence or divergence angle

α

relative width

Dr

relative flow depth

δ*

flow aspect ratio

Rivers are one of the most important resources of water for agriculture, industry, and drinking. Easy access and availability have led to the formation of human civilizations in their neighborhood for a long time (Torfi et al. 2021). Also, some industrial projects have been built near rivers to reduce water transfer costs. Floods are one of the most dangerous natural disasters that occur in river sections. These phenomena with different recurrence periods (from low-flow floods to high-flow floods), can cause very serious damage to human communities, agricultural lands, and industrial projects. The study of river hydraulics is the first important step in the study and control of floods (Graf & Altinakar 1998; Dingman 2009). In estimating damages due to floods, estimating the discharge in the floodplains and the main section is very important. By the estimation of flow discharge in floodplains, the depth, flooding zone, and also the time of submergence of the floodplains can be estimated.

Normally, the stream flows in its main channel in the river, but in flood conditions, the stream leaves its bed and enters the surrounding areas (floodplains). In this condition, the cross-section of the stream consists of the main channel and floodplains. As the flow enters the floodplains, its properties differ from the single channel. For example, the velocity in floodplains is lower than the main channel, and their roughness is higher than the main channel (Bousmar et al. 2006). The difference in velocity in these regions leads to shear stresses and vortices at the boundary between the main channel and the floodplains. In the past two decades, good efforts have been made to more accurately simulate hydraulics of flood in compound sections at rivers. In compound open channel hydraulics, the flow characteristics at the main channel and floodplains as well as their interaction with each other are considered. So far, many types of research have been conducted on the hydraulics of flow in compound open channels (Roushangar et al. 2021). Most of them have focused on determining the relation between stage-discharge, velocity distribution (Knight & Demetriou 1983), and shear stress in the main channel and floodplains, secondary flow, and turbulence characteristics. Sellin (1964) showed that the interaction of the flow in the main channel and the floodplains cause vortices in their common boundary, which cause the head loss of flow, and consequently the total flow rate is reduced. Mohanty et al. (2011) investigated the shear stress changes in the compound open channel focusing on the boundary range between the main channel and the floodplain. Their studies demonstrated that the shear layer depends on the geometric and hydraulic conditions of the flow. By increasing the ratio of floodplain width to the width of the main channel, the shear layer expansion range decreases, and with the decrease of the ratio of the flow depth in the main channel to the floodplain, it will increase. Parsaie et al. (2016) investigated the accuracy of empirical models including the unit cross-section method, divided channel method (DCM), and coherence method using laboratory and field data. They stated that the divided channel method works well by considering the virtual vertical line as the dividing line between the sub-sections (main channel and floodplains) and can be used for practical problems. Changes in the geometry of the flow section along the river's path cause the cross-section to be non-prismatic and change the flow from uniform to non-uniform.

In other words, the hydraulic complexity of the compound open channel flow increases when the width of the cross-section changes and the compound channel section changes from prismatic to non-prismatic. Bousmar et al. (2006) investigated the hydraulic flow in a compound channel with convergent floodplains. Their findings demonstrated that at high relative depths, the lateral mass transfer in the final half of the convergent region is higher than the initial half region.

Rezaei & Knight (2009) investigated the accuracy of the Shiono & Knight's model (SKM) (1988, 1991) in compound channels with non-prismatic sections. They stated that the SKM model does not have an appropriate accuracy for hydraulic modeling of flow in such sections. For this purpose, they modified the SKM model and the modified Shiono Knight model (M-SKM) to estimate the flow parameters including average depth velocity, boundary shear stress, and determining the stage-discharge relation. Naik et al. (2014) conducted a laboratory study of hydraulic flow in a compound channel with non-prismatic floodplains. The results of their research demonstrated that the depth average velocity and boundary shear stress increase during the convergence region. Today, flow characteristics in prismatic compound open channels are estimated using soft computing models and mathematical modeling.

For example, discharge in prismatic compound channels using artificial neural network models, support vector machine models, fuzzy adaptive neural network (Najafzadeh & Zahiri 2015; Parsaie et al. 2017), and multivariate adaptive spline regression method (Parsaie & Haghiabi 2017) and gene expression programming (Azamathulla & Zahiri 2012) have been modeled with great accuracy. Also prediction of depth-averaged velocity distribution of compound channels with converging flood plains using Fluent software and artificial neural network has been performed (Naik et al. 2017). In another study, Singh et al.(2019) used gene expression and a back propagation neural network to investigate the apparent shear stress in an asymmetric compound channel. The results of their research showed that the formulas produced by these two models provide very satisfactory results compared to previous models. Singh & Tang (2020), using the neuro-fuzzy inference system, predicted the apparent shear stress values in an asymmetric compound channel with five geometrical and hydraulic parameters with appropriate accuracy. The discharge in meandering compound channels has also been estimated by Mohanta et al. (2020) and Pradhan & Khatua (2019a) using empirical models. Using soft computing models the research to date has tended to focus on estimating discharge in compound channels with prismatic floodplains rather than non-prismatic floodplains. In other words, little research has been done on non-uniform sections (Pradhan & Khatua 2019b; Mohanta & Patra 2021) or compound open channels with non-prismatic floodplains (CCNPF; Das et al. 2020). Hence, due to the importance of discharge estimation especially in flood conditions, in this study, in addition to estimating the total flow rate, discharge prediction is considered in the floodplain and the main channel.

The present study fills a gap in the literature by using soft computation models. These models include the group method of data handling and multivariate adaptive spline regression method, modeling and estimating discharge in compound CCNPF. Also with using these models, the percentage of participation of discharge in the floodplains and the main channel has been estimated.

This section introduces the important parameters involved in predicting the discharge in non-prismatic compound open channels and then defines the statistical properties of the collected data and related sources. Finally, the soft computing models used in this research, which include multivariate adaptive regression splines (MARS), GMDH, and multilayer perceptron neural networks (MLPNN) will be introduced.

A view of non-prismatic compound open channels including convergent or divergent floodplains is shown in Figure 1. By examining the hydraulics of the flow in non-prismatic compound open channels, the researchers found that the flow in the compound channel depends on the coefficient of friction, area ratio, width ratio, ratio of the hydraulic radius, relative longitudinal distance, relative depth, flow aspect ratio, and bed slope. Hence, in this present study, for developing the MLPNN, GMDH, and MARS models, nine dimensionless input parameters presented in Equation (1) were considered.
(1)
where, fr = fmc/ffp= relative roughness with fmc, ffp = the main channel and floodplain Darcy-Wiesbach friction factor,respectively; Ar = Amc/Afp = relative area with Amc, Afp = the main channel and floodplain cross section, respectively; Rr = Rmc/Rfp = relative hydraulic radius with Rmc, Rfp = hydraulic radius of the main channel and floodplain, respectively; δ* = b/H = flow aspect ratio(main channel width to the total flow depth); α = B/b = relative width(width of the floodplain to the width of the main channel); Dr = (Hh)/H = relative flow depth(h is bankfull depth); xr=l/L = relative longitudinal distance with l, L = distance of any point from beginning of the divergent or convergent and total length of divergent or convergent region, respectively; θ = convergence or divergence angle and S0 = bed slope.
Figure 1

View of a non-prismatic compound channel (a) divergent floodplain (b) convergent floodplain.

Figure 1

View of a non-prismatic compound channel (a) divergent floodplain (b) convergent floodplain.

Close modal

To develop the mentioned artificial intelligence models, laboratory data of Othman (2000), Bousmar (2002), Rezaei & Knight (2009), Yonesi et al. (2013), and Naik & Khatua (2016) have been used. Table 1 shows the statistical characteristics of the collected data.

Table 1

Statistical characteristics of the collected data

frArRrDrS0δ*αxrθQ
Minimum 0.31 0.93 1.70 0.11 0.0009 1.41 1.33 0.00 −13.38 0.01 
Maximum 0.84 22.59 35.09 0.54 0.0020 6.54 3.02 1.00 11.31 0.06 
Range 0.53 21.66 33.39 0.43 0.0011 5.13 1.69 1.00 24.69 0.05 
Mean 0.70 4.37 3.51 0.34 0.0012 4.40 2.10 0.42 −2.17 0.02 
Median 0.71 3.01 2.80 0.33 0.0010 4.19 2.00 0.33 −1.91 0.02 
First quartile 0.65 1.86 2.30 0.25 0.0010 3.77 1.67 0.17 −9.00 0.02 
Third quartile 0.76 5.58 3.73 0.44 0.0011 5.38 2.51 0.67 3.81 0.04 
Variance 0.01 13.86 9.89 0.01 0.0000 1.62 0.30 0.10 48.85 0.00 
Average deviation 0.07 2.70 1.43 0.10 0.0003 1.04 0.47 0.28 6.01 0.01 
Standard deviation 0.09 3.72 3.14 0.12 0.0004 1.27 0.55 0.32 6.99 0.01 
Coefficient of variation 0.13 0.85 0.90 0.35 0.3440 0.29 0.26 0.77 −3.22 0.56 
Skew −1.06 2.09 6.57 0.19 1.2530 −0.61 0.32 0.32 −0.06 1.15 
Kurtosis 2.03 5.58 55.13 −0.99 −0.3250 −0.02 −1.11 −1.13 −1.17 0.52 
frArRrDrS0δ*αxrθQ
Minimum 0.31 0.93 1.70 0.11 0.0009 1.41 1.33 0.00 −13.38 0.01 
Maximum 0.84 22.59 35.09 0.54 0.0020 6.54 3.02 1.00 11.31 0.06 
Range 0.53 21.66 33.39 0.43 0.0011 5.13 1.69 1.00 24.69 0.05 
Mean 0.70 4.37 3.51 0.34 0.0012 4.40 2.10 0.42 −2.17 0.02 
Median 0.71 3.01 2.80 0.33 0.0010 4.19 2.00 0.33 −1.91 0.02 
First quartile 0.65 1.86 2.30 0.25 0.0010 3.77 1.67 0.17 −9.00 0.02 
Third quartile 0.76 5.58 3.73 0.44 0.0011 5.38 2.51 0.67 3.81 0.04 
Variance 0.01 13.86 9.89 0.01 0.0000 1.62 0.30 0.10 48.85 0.00 
Average deviation 0.07 2.70 1.43 0.10 0.0003 1.04 0.47 0.28 6.01 0.01 
Standard deviation 0.09 3.72 3.14 0.12 0.0004 1.27 0.55 0.32 6.99 0.01 
Coefficient of variation 0.13 0.85 0.90 0.35 0.3440 0.29 0.26 0.77 −3.22 0.56 
Skew −1.06 2.09 6.57 0.19 1.2530 −0.61 0.32 0.32 −0.06 1.15 
Kurtosis 2.03 5.58 55.13 −0.99 −0.3250 −0.02 −1.11 −1.13 −1.17 0.52 

Review of MLPNN

A neural network is an advanced non-linear mathematical model that can simulate the relationship between the inputs and outputs of a complex non-linear system. In many simulations that lead to the solution of complex equations to find the relation between the factors affecting the system, a neural network can be used as a suitable alternative. The MLPNN is one of the most widely used types of ANN models. Figure 2 shows a view of this model. As shown in this figure, the inputs of the neural network model are multiplied by coefficients (w: weight) and then those are added by a constant value (b: bias). The transfer function is then added to the result. The design of an MLPNN model consists of several steps, which are: defining the number of model layers including the input layer, hidden layer(s), an output layer, number of neurons in each layer, defining the active transfer function, and finally choosing the training method. Neural network training means determining the values of weights and constants that are multiplied and added to each input. Various methods for neural network training have been proposed, such as reduction gradient methods, Levenberg–Marquardt algorithm, or metaheuristic methods such as genetic algorithm and particle swarm optimization. In the metaheuristic methods, neural network training is assumed as an optimization problem (Sihag et al. 2021).

Figure 2

The structure of the MLPNN developed to estimate the discharge in the compound open channel with non-prismatic floodplains (a) 6 inuts (b) 9 inputs.

Figure 2

The structure of the MLPNN developed to estimate the discharge in the compound open channel with non-prismatic floodplains (a) 6 inuts (b) 9 inputs.

Close modal

Review of GMDH

The GMDH model is one of the a posteriori approaches based on perceptron theory. This approach has been developed to identify, model, and predict complex systems (Najafzadeh & Azamathulla 2015). GMDH is a combination of adlines, and modified versions of this method have been used for various modeling applications. This method has higher accuracy compared with the perceptron-type structure because in this model, the division of information into two categories of useful and non-useful is used. GMDH also requires less observational data (Ivakhnenko 1971). Figure 3 shows the structure of the GMDH network developed with a quadratic polynomial as the activation function (Equation (2)) in this study to model and prediction of flow discharge in the non-prismatic compound open channels (Bhoria et al. 2021).

Figure 3

The structure of the developed GMDH model with 6 inputs to estimate the discharge in a non-prismatic compound channel.

Figure 3

The structure of the developed GMDH model with 6 inputs to estimate the discharge in a non-prismatic compound channel.

Close modal

The network structure is determined by the proposed algorithm as follows: selection of input variables to the model, collecting of a set of observational data related to the subject, dividing of the data into two groups: calibration and validation data, building N-adline for all two pairs of input variables, the transfer function for each neuron, estimate the weighting coefficients for all N-adlines by least squares error method (LSE), calculation of filtration criteria, comparison of calculated criteria for all N-adlines with a set threshold value. If the threshold value is greater than the filtration criteria, the M-adline is removed. Then other adlines remain for the next layer. A fixed threshold value can be considered for all layers or a threshold value for each layer.

Each neuron in the GMDH structure performs a non-linear function of the inputs. This non-linear function is as follows:
(2)
where w0 to w5 is the weight of the parameters and x1, x2 are the input variables. Using the LSE method, six coefficients are calculated for each neuron. The steps of which are as follows: a set of 6 coefficients must be found for each neuron so that the mean squared error between the neurons' outputs is yn and the actual value of is minimum.
(3)
That this equation is written as the following general matrix:
(4)
Matrixes , X and W have dimensions N × 1, N × 6, and 6 × 1, respectively. Normal equations are obtained by multiplying both sides of Equation (4) in the matrix X transposition:
(5)

XTX is a 6 × 6 matrix and the coefficients can be obtained using the inverse method. The W matrix consists of a set of 6 coefficients. This matrix can approximate correct outputs with a minimum mean square error. The above steps are repeated for all the first layer neurons as well as for all subsequent layer neurons. The neuron's performance function is determined after obtaining six coefficients. This performance function is controlled by correlating or calculating the mean square error between the actual data outputs. Finally, to continue the solution, neurons with a performance function higher than the threshold value are selected (Najafzadeh et al. 2013; Najafzadeh & Azamathulla 2015; Najafzadeh & Sattar 2015).

Review of MARS method

The MARS method first was proposed by Friedman (1991) to predict continuous numerical outputs. This model is one of the local non-parametric models. The application of the non-parametric term in this model is that the structure of the model is not known before modeling. Also, the MARS model not only does not use all data together but also divides data into subcategories and then performs the modeling operation for each of these subcategories. The mathematical models of such subcategories are local. In the MARS model, it is possible to find a hidden non-linear pattern in a data set including a large number of variables. In this method, it is possible to define the estimation function and there is no need to combine several statistical methods. This model is based on basis functions, which are defined as follows for each variable.
(6)
where t = a node and in practice is an explanatory variable. These functions are called spline functions, which are reflected in the node t-pair. The general form of the MARS model is as follows :
(7)
where Ŷ= estimate the value of the response variable, X = vector of the explanatory variables, Bk = the basis function, and βk = coefficients that are determined by minimizing the sum of the squares of the residuals.

Each basis function may be a spline linear function or the multiplication of two or more of them that express interactions. MARS divides the space of explanatory variables into specific regions with specific nodes. The MARS model fitting is performed in two stages, forward and backward, respectively. In the forward stage, a large number of basis functions with different nodes are continuously added to the model. In the backward stage, the basis functions that are less important and effective in estimating are removed.

Finally, the best model is selected based on a minimum criterion called ‘generalized cross-validation (GCV)’. GCVk is the kth model in deleting phase. This quantity is defined by the following expression:
(8)
where = estimated model in kth step of backward elimination stage, and C(k)=λ. M = the number of sentences in kth model, M = the number of spline function nodes in the model(2 < M < 4), and λ = cost function criterion.

Modeling strategies

As shown in Equation (1), nine parameters can be used as input variables to model and prediction of flow discharge in a compound channel with non-prismatic floodplains using soft computing models (Figure 4). Therefore, in designing the pattern of input variables, a combination of one to nine parameters can be considered. In other words, it is possible to define the input variable by combining k (1<k<9) and n=9. Different approaches can be used in soft computing models to reduce modeling time and identify the best input combination. The first step is to use the variables that have the highest correlation with the output variable (discharge). For this purpose, the correlation of independent variables and discharge are calculated and are presented in Table 2 and Figure 5.

Table 2

Correlation of variables involved in estimating discharge

frArRrDrSoδ*αxrθQ
fr 1.00          
Ar −0.71 1.00         
Rr −0.79 0.57 1.00        
Dr 0.84 −0.58 −0.49 1.00       
So 0.16 −0.21 −0.16 0.03 1.00      
δ0.12 −0.03 −0.30 −0.16 0.35 1.00     
α 0.41 −0.70 −0.26 0.20 0.27 0.05 1.00    
xr 0.15 −0.14 −0.13 0.11 0.10 0.12 0.10 1.00   
θ −0.09 −0.10 0.16 0.06 −0.25 −0.16 0.05 0.36 1.00  
−0.34 0.22 0.31 −0.21 −0.13 −0.81 −0.14 −0.14 0.10 1.00 
frArRrDrSoδ*αxrθQ
fr 1.00          
Ar −0.71 1.00         
Rr −0.79 0.57 1.00        
Dr 0.84 −0.58 −0.49 1.00       
So 0.16 −0.21 −0.16 0.03 1.00      
δ0.12 −0.03 −0.30 −0.16 0.35 1.00     
α 0.41 −0.70 −0.26 0.20 0.27 0.05 1.00    
xr 0.15 −0.14 −0.13 0.11 0.10 0.12 0.10 1.00   
θ −0.09 −0.10 0.16 0.06 −0.25 −0.16 0.05 0.36 1.00  
−0.34 0.22 0.31 −0.21 −0.13 −0.81 −0.14 −0.14 0.10 1.00 
Figure 4

Input variables for discharge estimation in CCNPF.

Figure 4

Input variables for discharge estimation in CCNPF.

Close modal
Figure 5

Involved parameters versus discharge in CCNPF.

Figure 5

Involved parameters versus discharge in CCNPF.

Close modal

As can be seen from this figure and table, in the data range used, there is the highest positive correlation between Q and Ar and Rr. Also, there is a slight positive correlation with the divergence (convergence) angle. As shown in this table, the discharge is inversely related to other parameters such as the fr and β. For finding the most effective parameters, other methods, such as the gamma test can be used (Das et al. 2020). This method also works based on the correlation coefficient. Such methods are used to design the input variables of the soft computing model, such as MLPNN, adaptive neuro-fuzzy inference system (ANFIS), and SVM. However, in the GMDH and MARS models, the most important influential parameters in determining the output parameter (discharge) are automatically identified in the model development process, and in the final model, the highest weight coefficient is assigned to them. In this study, for the development of the mentioned soft computing models including GMDH, MARS, MLPNN models, the collected dataset (196 datasets) was divided into two categories of training and testing.

The histogram of the data used is shown in Figure 6. Eighty percent of the data were randomly selected to model training, and the remaining 20% was devoted to testing of the model.

Figure 6

Histogram of data used to estimate discharge in compound channels with non-prismatic floodplains.

Figure 6

Histogram of data used to estimate discharge in compound channels with non-prismatic floodplains.

Close modal

The statistical characteristics of the data assigned to each of the training and testing categories are presented in Tables 3 and 4, respectively. The most important point in assigning data to each of the training and testing groups is that the range of data assigned to both categories is close to each other.

Table 3

Statistical characteristics of the data assigned to the training

frArRrDrS0δ*αxrθQ
Minimum 0.38 0.93 1.70 0.11 0.00 1.41 1.33 0.00 −13.38 0.01 
Maximum 0.84 22.59 17.69 0.54 0.00 6.54 3.02 1.00 11.31 0.06 
Range 0.45 21.66 15.99 0.43 0.00 5.13 1.69 1.00 24.69 0.05 
Mean 0.70 4.45 3.30 0.34 0.00 4.35 2.08 0.44 −2.32 0.02 
Median 0.71 3.06 2.79 0.33 0.00 4.10 2.00 0.50 −1.91 0.02 
First quartile 0.65 1.86 2.31 0.26 0.00 3.77 1.67 0.17 −9.00 0.02 
Third quartile 0.76 5.79 3.70 0.49 0.00 5.36 2.51 0.67 3.81 0.04 
Variance 0.01 13.94 3.74 0.01 0.00 1.57 0.29 0.10 48.68 0.00 
Average deviation 0.07 2.75 1.15 0.10 0.00 1.01 0.46 0.28 6.07 0.01 
Standard deviation 0.09 3.75 1.94 0.12 0.00 1.26 0.54 0.32 7.00 0.01 
Coefficient of variation 0.01 13.94 3.74 0.01 0.00 1.57 0.29 0.10 48.68 0.00 
Skew −0.73 1.99 4.03 0.19 1.50 −0.61 0.36 0.24 −0.09 1.10 
Kurtosis 0.77 5.19 23.03 −0.97 0.40 0.07 −1.05 −1.17 −1.24 0.41 
frArRrDrS0δ*αxrθQ
Minimum 0.38 0.93 1.70 0.11 0.00 1.41 1.33 0.00 −13.38 0.01 
Maximum 0.84 22.59 17.69 0.54 0.00 6.54 3.02 1.00 11.31 0.06 
Range 0.45 21.66 15.99 0.43 0.00 5.13 1.69 1.00 24.69 0.05 
Mean 0.70 4.45 3.30 0.34 0.00 4.35 2.08 0.44 −2.32 0.02 
Median 0.71 3.06 2.79 0.33 0.00 4.10 2.00 0.50 −1.91 0.02 
First quartile 0.65 1.86 2.31 0.26 0.00 3.77 1.67 0.17 −9.00 0.02 
Third quartile 0.76 5.79 3.70 0.49 0.00 5.36 2.51 0.67 3.81 0.04 
Variance 0.01 13.94 3.74 0.01 0.00 1.57 0.29 0.10 48.68 0.00 
Average deviation 0.07 2.75 1.15 0.10 0.00 1.01 0.46 0.28 6.07 0.01 
Standard deviation 0.09 3.75 1.94 0.12 0.00 1.26 0.54 0.32 7.00 0.01 
Coefficient of variation 0.01 13.94 3.74 0.01 0.00 1.57 0.29 0.10 48.68 0.00 
Skew −0.73 1.99 4.03 0.19 1.50 −0.61 0.36 0.24 −0.09 1.10 
Kurtosis 0.77 5.19 23.03 −0.97 0.40 0.07 −1.05 −1.17 −1.24 0.41 
Table 4

Statistical characteristics of the data assigned to the testing

frArRrDrS0δ*αxrθQ
Minimum 0.31 0.94 1.72 0.15 0.00 1.44 1.33 0.00 −13.38 0.01 
Maximum 0.84 20.60 35.09 0.53 0.00 6.43 3.02 1.00 11.31 0.06 
Range 0.53 19.66 33.37 0.39 0.00 4.99 1.69 1.00 24.69 0.05 
Mean 0.69 4.05 4.36 0.33 0.00 4.62 2.20 0.34 −1.55 0.02 
Median 0.71 2.85 2.85 0.34 0.00 4.85 2.01 0.25 −1.91 0.02 
First quartile 0.64 1.92 2.26 0.23 0.00 3.84 1.72 0.06 −5.00 0.01 
Third quartile 0.76 4.93 3.84 0.40 0.00 5.59 2.71 0.50 3.81 0.02 
Variance 0.01 13.43 33.74 0.01 0.00 1.73 0.31 0.10 49.03 0.00 
Average deviation 0.08 2.50 2.72 0.10 0.00 1.09 0.50 0.27 5.86 0.01 
Standard deviation 0.11 3.71 5.88 0.12 0.00 1.33 0.57 0.32 7.09 0.01 
Coefficient of variation 0.01 13.43 33.74 0.01 0.00 1.73 0.31 0.10 49.03 0.00 
Skew −1.63 2.73 4.43 0.24 0.58 −0.71 0.18 0.71 0.07 1.47 
Kurtosis 3.56 9.78 21.06 −1.04 −1.71 0.01 −1.30 −0.65 −0.94 1.59 
frArRrDrS0δ*αxrθQ
Minimum 0.31 0.94 1.72 0.15 0.00 1.44 1.33 0.00 −13.38 0.01 
Maximum 0.84 20.60 35.09 0.53 0.00 6.43 3.02 1.00 11.31 0.06 
Range 0.53 19.66 33.37 0.39 0.00 4.99 1.69 1.00 24.69 0.05 
Mean 0.69 4.05 4.36 0.33 0.00 4.62 2.20 0.34 −1.55 0.02 
Median 0.71 2.85 2.85 0.34 0.00 4.85 2.01 0.25 −1.91 0.02 
First quartile 0.64 1.92 2.26 0.23 0.00 3.84 1.72 0.06 −5.00 0.01 
Third quartile 0.76 4.93 3.84 0.40 0.00 5.59 2.71 0.50 3.81 0.02 
Variance 0.01 13.43 33.74 0.01 0.00 1.73 0.31 0.10 49.03 0.00 
Average deviation 0.08 2.50 2.72 0.10 0.00 1.09 0.50 0.27 5.86 0.01 
Standard deviation 0.11 3.71 5.88 0.12 0.00 1.33 0.57 0.32 7.09 0.01 
Coefficient of variation 0.01 13.43 33.74 0.01 0.00 1.73 0.31 0.10 49.03 0.00 
Skew −1.63 2.73 4.43 0.24 0.58 −0.71 0.18 0.71 0.07 1.47 
Kurtosis 3.56 9.78 21.06 −1.04 −1.71 0.01 −1.30 −0.65 −0.94 1.59 
The error indices including coefficient of determination (R2) and root mean square error (RMSE) was used to evaluate the developed models. In addition, the developed discrepancy ratio (DDR) is used for evaluating the characteristics of the developed models. Taylor diagram was used to compare the performance of the developed models. The DDR index shows the ratio of the predicted values to the observed values minus one. If the value of this index is positive, it indicates the property of overprediction, and if it is negative, it has the property of underprediction. Based on this index, in addition to its numerical values for training and test data, it is necessary to draw a histogram and its range.
(9)
(10)
(11)

In Equations (9) and (10), O, P = observational and predicted or output values, respectively. , = Average observational and predicted data, respectively, and n= the number of samples.

Results of the MARS model

The preparation and development of the MARS model as stated in the material and methods section consists of two stages, the first stage includes the growth and development of the model and the second stage includes pruning the model. In the first stage, to explore the model space and fit the regression equations, they are divided into several subspaces, and in the second stage, some of them are pruned to prevent overfitting. To develop the MARS model to estimate the discharge in non-prismatic compound channel, in the first stage, about 30 basis functions were prepared, and in the second stage (pruning stage) 10 of them were removed. The mathematical model obtained from the MARS model is presented in Equation (12). The functions and coefficients of this model are presented in Table 5. Evaluation of the MARS structure shows that: δ*, So, β, Ar, and fr had the greatest effects in the final mathematical model to estimate the discharge in CCNPF. The error statistics of the MARS model developed in the training and testing steps are shown in Figure 7. As shown in this figure, the statistical indices of this model in the training step are R2 = 0.986 and RMSE = 0.002, and in the testing step are R2 = 0.962 and RMSE = 0.003. In this model, the DDR index was also calculated and the corresponding diagram was prepared, which is shown in Figures 8 and 9. It can be seen from these figures, the range of DDR index changes in the training and testing step is between [−0.4, 0.3].

Table 5

The basic function of the developed mathematical model by MARS algorithm

BF functionsCoefficients
BF1 = max(0, δ* − 1.87) −0.035 
BF2 = max(0, 1.87 −δ*) 0.046 
BF3 = BF1 * max(0, 0.0011 −So33.067 
BF4 = max(0, So −0.00099) 16.524 
BF5 = max(0, 0.00099 −So−533.352 
BF6 = BF1 * max(0, β − 0.364) −0.062 
BF7 = BF1 * max(0, 0.364 −Dr) 0.034 
BF8 = max(0, θ +1.91) −0.013 
BF9 = max(0, −1.91 −θ0.006 
BF10 = BF1 * max(0, −3.81 −θ−0.003 
BF11 = max(0, θ −5.71) 0.002 
BF12 = BF4 * max(0, 0.708 −fr44.688 
BF13 = max(0, θ +9) * max(0, δ* − 4.17) −0.002 
BF14 = BF9 * max(0, δ* − 3.96) 0.004 
BF15 = BF9 * max(0, 3.96 −δ*) −0.004 
BF16 = max(0, θ +3.81) 0.011 
BF17 = max(0, 5.71 -θ) * max(0, 0.269 −Dr) −0.003 
BF18 = max(0, θ +9) * max(0, δ* − 3.91) 0.003 
BF19 = BF16 * max(0, 1.28 -Ar0.007 
BF20 = BF16 * max(0, fr −0.82) −0.236 
BF functionsCoefficients
BF1 = max(0, δ* − 1.87) −0.035 
BF2 = max(0, 1.87 −δ*) 0.046 
BF3 = BF1 * max(0, 0.0011 −So33.067 
BF4 = max(0, So −0.00099) 16.524 
BF5 = max(0, 0.00099 −So−533.352 
BF6 = BF1 * max(0, β − 0.364) −0.062 
BF7 = BF1 * max(0, 0.364 −Dr) 0.034 
BF8 = max(0, θ +1.91) −0.013 
BF9 = max(0, −1.91 −θ0.006 
BF10 = BF1 * max(0, −3.81 −θ−0.003 
BF11 = max(0, θ −5.71) 0.002 
BF12 = BF4 * max(0, 0.708 −fr44.688 
BF13 = max(0, θ +9) * max(0, δ* − 4.17) −0.002 
BF14 = BF9 * max(0, δ* − 3.96) 0.004 
BF15 = BF9 * max(0, 3.96 −δ*) −0.004 
BF16 = max(0, θ +3.81) 0.011 
BF17 = max(0, 5.71 -θ) * max(0, 0.269 −Dr) −0.003 
BF18 = max(0, θ +9) * max(0, δ* − 3.91) 0.003 
BF19 = BF16 * max(0, 1.28 -Ar0.007 
BF20 = BF16 * max(0, fr −0.82) −0.236 
Figure 7

The results of soft computing models in estimation of discharge in compound channel with non-prismatic flood plains.

Figure 7

The results of soft computing models in estimation of discharge in compound channel with non-prismatic flood plains.

Close modal
Figure 8

DDR index for models in the training step.

Figure 8

DDR index for models in the training step.

Close modal
Figure 9

DDR index of the models in the testing stage.

Figure 9

DDR index of the models in the testing stage.

Close modal

To determine whether this model has over-fitting or under-fitting properties, the DDR index histogram is plotted (Figures 8 and 9).

Assessment of these figures at the training and testing steps show that: the DDR histogram is not particularly skewed, and the MARS model has no significant overestimation or underestimation.

The survey of statistical indicators shows that the MARS model has a good performance for estimating the discharge in CCNPF. The results of the correlational analysis can be seen in Equation (12).
(12)

GMDH model results

To develop the GMDH model for estimating the discharge in CCNPF, the same data used in the training and testing steps of the MARS model were used. In the GMDH model, only two parameters are assigned to each neuron. There are 36 neurons in the first layer of the GMDH model (Figure 3). Only neurons that have higher accuracy in estimating the discharge are selected and used in the construction of the next layer.

The coefficients of the activated function of the developed GMDH model are presented in Table 6. Evaluating the structure of the developed GMDH model shows that similar to the MARS model, this model has identified the parameters δ*, So, Dr, and fr as the most important parameters in estimating the discharge in CCNPF.

Table 6

The developed GMDH network coefficients to estimate discharge in non-prismatic compound channels

b0b1b2b3b4b5b6
layer-1 −0.203 509.015 −0.047 −157,389.086 0.005 −6.383 0.004 
0.002 0.288 0.009 −0.267 −0.001 −0.039 0.005 
−0.027 0.299 −0.002 −0.209 0.000 −0.018 0.007 
0.093 −0.004 −0.022 0.000 0.001 0.002 0.007 
layer-2 0.099 −0.027 −0.001 0.002 0.000 0.000 0.007 
0.004 0.528 0.155 −17.867 −30.246 53.195 0.004 
0.004 0.461 0.233 −15.991 −30.793 51.799 0.004 
−0.001 0.391 0.598 42.442 44.894 −86.729 0.004 
0.004 0.380 0.305 −5.786 −20.122 31.047 0.004 
−0.002 0.973 0.248 −40.516 −43.986 81.660 0.005 
b0b1b2b3b4b5b6
layer-1 −0.203 509.015 −0.047 −157,389.086 0.005 −6.383 0.004 
0.002 0.288 0.009 −0.267 −0.001 −0.039 0.005 
−0.027 0.299 −0.002 −0.209 0.000 −0.018 0.007 
0.093 −0.004 −0.022 0.000 0.001 0.002 0.007 
layer-2 0.099 −0.027 −0.001 0.002 0.000 0.000 0.007 
0.004 0.528 0.155 −17.867 −30.246 53.195 0.004 
0.004 0.461 0.233 −15.991 −30.793 51.799 0.004 
−0.001 0.391 0.598 42.442 44.894 −86.729 0.004 
0.004 0.380 0.305 −5.786 −20.122 31.047 0.004 
−0.002 0.973 0.248 −40.516 −43.986 81.660 0.005 

But instead of the Ar, the model has chosen the Rr, although there is a direct relationship between the Ar and Rr. The performance of the developed GMDH model in the training and testing steps is shown in Figure 7. As shown in this figure, the statistical indicators of GMDH model in the training step are R2 = 0.935 and RMSE = 0.003 and in the testing step are R2 = 0.930 and RMSE = 0.004, respectively. Comparison of the statistical indicators of the GMDH model with the MARS model shows that the GMDH model is partially less accurate. Also, the DDR index was calculated for this model. Its distribution diagram is plotted on the data used as well as its histogram (Figures 8 and 9). The range of DDR index changes for the GMDH model results in both training and testing steps varies between [−0.4, 0.5].

The scatter range of this model is larger than the MARS model. Examination of the DDR histogram of this model in the training step shows that this model has significantly overestimation. But in the testing step, the MARS model reduces this feature by dividing the space between input variables and regression modeling and provides more acceptable results.

Development of the MLPNN model

For the preparation of the MLPNN model, two strategies were used: considering all input parameters as the first model and then considering the most important parameters specified from the previous models as the input of the second model of input variables. The second strategy involves preparing two hidden layers for the network, each with five neurons. This strategy aimed to evaluate and compare the accuracy of the MLPNN model with the GMDH model. To use this model, a script has been written in Matlab software. To select the activation function, the types of functions embedded in this software were examined and the result showed that the sigmoid tangent function has the best performance. The structure of the developed MLPNN model based on both scenarios is shown in Figure 10. Notably, the same data used for developing the MARS and GMDH models were used to develop the MLPNN model.

Figure 10

Taylor diagram for comparison of performance in models used in the training stage.

Figure 10

Taylor diagram for comparison of performance in models used in the training stage.

Close modal

The performance of the MLPNN model developed at different stages of training and testing is shown in Figure 7. Examination of this figure shows that the statistical indices of error of the MLPNN model based on considering all the involved variables (9 variables) as input in the training stage are R2 = 0.969 and RMSE = 0.002, and in the testing stage are R2 = 0.933 and RMSE = 0.004. Also, the performance of the MLPNN model is calculated based on the most important parameters as input (6 variables) and is shown in Figure 7. Examination of this figure shows that the error indices of the MLPNN model based on the most important influential variables in the training and testing step are R2 = 0.983 and RMSE = 0.002 and R2 = 0.969 and RMSE = 0.002. Examining the performance of the MLPNN model based on both scenarios shows that any additional information as model input not only does not increase the accuracy of the neural network model but also slightly decreases its accuracy. The Taylor diagram was used to comprehensively examine the developed models including MARS, GMDH, and MLPNN (based on both scenarios). The Taylor diagram for the training and testing stages is shown in Figures 10 and 11. As it shows, in the training stage, the performance of all models is almost equal to each other, but in the testing stage based on the taylor diagram that used the R2 and RMSE indices, the performance of the GMDH model is significantly weaker.

Figure 11

Taylor diagram for comparison of performance in models used in the testing stage.

Figure 11

Taylor diagram for comparison of performance in models used in the testing stage.

Close modal

Estimating discharge in main channel and floodplains

As mentioned, in flood engineering studies, it is important to estimate the discharge in the main channel and the floodplains. In this section, the contribution of each of the different parts of the compound channel including the main channel and non-prismatic floodplains of the total flow discharge is presented using the mentioned soft calculation models. Relative roughness parameters (fr), convergence or divergence angle, relative depth, and relative distance were used as inputs of the mentioned models. The structure of the MARS model developed to estimate the share of the main channel of the flow discharge (Qm) is presented in Equation (13) and the extended form of the basis functions and their coefficients are presented in Table 7.
(13)
Table 7

Basis functions and model coefficients developed for estimating Qm

Basic functionsCoefficients
BF1 = max(0, xr −0.25) * max(0, Dr −0.35) −356.244 
BF2 = max(0, xr −0.25) * max(0, 0.35 −Dr1,695.451 
BF3 = max(0, θ −3.81) −1.257 
BF4 = max(0, 3.81 −θ−1.803 
BF5 = max(0, 0.35 −Dr36.839 
BF6 = max(0, xr −0.25) * max(0, θ +3.81) −3.968 
BF7 = max(0, xr −0.25) * max(0, −3.81 −θ−0.902 
BF8 = max(0, −3.81 −θ3.520 
BF9 = max(0, θ +3.81) * max(0, 0.5 −xr2.544 
BF10 = max(0, xr −0.25) * max(0, 0.4 −Dr−741.817 
BF11 = max(0, xr −0.25) * max(0, 2 −fr−5.626 
BF12 = BF5 * max(0, 0.5 −xr−67.410 
BF13 = max(0, xr −0.25) * max(0, 0.3 −Dr−861.265 
BF14 = max(0, xr −0.25) * max(0, Dr −0.25) 429.220 
BF15 = BF4 * max(0, Dr −0.3) −9.656 
BF16 = BF4 * max(0, 0.3 −Dr11.211 
BF17 = BF5 * max(0, −3.81 −θ−15.543 
BF18 = BF3 * max(0, xr +0) 4.091 
Basic functionsCoefficients
BF1 = max(0, xr −0.25) * max(0, Dr −0.35) −356.244 
BF2 = max(0, xr −0.25) * max(0, 0.35 −Dr1,695.451 
BF3 = max(0, θ −3.81) −1.257 
BF4 = max(0, 3.81 −θ−1.803 
BF5 = max(0, 0.35 −Dr36.839 
BF6 = max(0, xr −0.25) * max(0, θ +3.81) −3.968 
BF7 = max(0, xr −0.25) * max(0, −3.81 −θ−0.902 
BF8 = max(0, −3.81 −θ3.520 
BF9 = max(0, θ +3.81) * max(0, 0.5 −xr2.544 
BF10 = max(0, xr −0.25) * max(0, 0.4 −Dr−741.817 
BF11 = max(0, xr −0.25) * max(0, 2 −fr−5.626 
BF12 = BF5 * max(0, 0.5 −xr−67.410 
BF13 = max(0, xr −0.25) * max(0, 0.3 −Dr−861.265 
BF14 = max(0, xr −0.25) * max(0, Dr −0.25) 429.220 
BF15 = BF4 * max(0, Dr −0.3) −9.656 
BF16 = BF4 * max(0, 0.3 −Dr11.211 
BF17 = BF5 * max(0, −3.81 −θ−15.543 
BF18 = BF3 * max(0, xr +0) 4.091 

The performance of the MARS model developed at different phases, including training and testing, is shown in Figure 12, respectively. The statistical indices of the developed model in the training stage are R2 = 0.986 and RMSE = 1.320 and in the testing stage are R2 = 0.979 and RMSE = 1.723.

Figure 12

Performance of models developed to estimate discharge in the main channel.

Figure 12

Performance of models developed to estimate discharge in the main channel.

Close modal

In the following, the performance of GMDH and MLPNN models is presented and evaluated to estimate the contribution of the main channel cross-section in the convey of total flood discharge. The structures of the developed GMDH and MLPNN models are shown in Figures 13 and 14. The table of coefficients of the neurons involved in the GMDH network is also presented in Table 8. As can be seen from these figures, the statistical indices of GMDH and MLPNN models in the training stage are: GMDHTraining: R2 = 0.830, RMSE = 4.592 and MLPNNTraining: R2 = 0.982, RMSE = 1.595, respectively. In the testing stage are: GMDHTest: R2 = 0.69, RMSE = 5.412 and MLPNNTest: R2 = 0.933, RMSE = 2.183, respectively. A summary of the statistical indices of the models used is presented in Table 9. Evaluation of this table shows that the accuracy of the MARS model is higher than other models.

Table 8

The coefficients of developed GMDH network to estimate discharge in the main channel of the CCNPF

b0b1b2b3b4b5b6
layer-1 90.751 1.253 −14.305 −0.014 14.378 −2.292 90.751 
116.271 −91.802 −39.201 33.262 15.473 63.203 116.271 
101.198 0.168 −44.835 0.009 4.407 −0.881 101.198 
108.542 −7.859 −66.635 2.689 55.058 −2.445 108.542 
layer-2 −1,186.203 10.568 15.696 0.005 −0.023 −0.117 −1,186.203 
−985.800 6.668 15.201 0.014 −0.032 −0.092 −985.800 
−981.862 6.499 15.478 −0.005 −0.054 −0.053 −981.862 
layer-3 −90.627 7.223 −4.243 0.180 0.240 −0.431 −90.627 
b0b1b2b3b4b5b6
layer-1 90.751 1.253 −14.305 −0.014 14.378 −2.292 90.751 
116.271 −91.802 −39.201 33.262 15.473 63.203 116.271 
101.198 0.168 −44.835 0.009 4.407 −0.881 101.198 
108.542 −7.859 −66.635 2.689 55.058 −2.445 108.542 
layer-2 −1,186.203 10.568 15.696 0.005 −0.023 −0.117 −1,186.203 
−985.800 6.668 15.201 0.014 −0.032 −0.092 −985.800 
−981.862 6.499 15.478 −0.005 −0.054 −0.053 −981.862 
layer-3 −90.627 7.223 −4.243 0.180 0.240 −0.431 −90.627 
Table 9

Statistical indices of the models developed to estimate the discharge in the main channel

MethodsTraining Stage
Testing Stage
R2RMSER2RMSE
MARS 0.986 1.320 0.979 1.723 
GMDH 0.830 4.592 0.769 5.412 
MLP 0.982 1.595 0.933 2.183 
MethodsTraining Stage
Testing Stage
R2RMSER2RMSE
MARS 0.986 1.320 0.979 1.723 
GMDH 0.830 4.592 0.769 5.412 
MLP 0.982 1.595 0.933 2.183 
Figure 13

The structure developed the GMDH model to estimate flow rate in the main section.

Figure 13

The structure developed the GMDH model to estimate flow rate in the main section.

Close modal
Figure 14

The GMDH model structure developed to estimate discharge in main channel and floodplains.

Figure 14

The GMDH model structure developed to estimate discharge in main channel and floodplains.

Close modal
The performance of MARS, GMDH, and MLPNN models in the estimation of the contribution of floodplains from the total discharge passing (Qf) is investigated. The basis functions and related coefficients are presented in Table 10. The performance of the models in the training and testing stages is also shown in Figure 15.
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Table 10

Basis functions and coefficients of the MARS model developed for estimation of Qf

Basis functionsCoefficients
BF1 = max(0, 0.4 −Dr−130.933 
BF2 = max(0, xr −0.25) * max(0, θ −5.7) 6.336 
BF3 = max(0, 5.7 −θ−2.725 
BF4 = max(0, Dr −0.35) 142.337 
BF5 = BF1 * max(0, 0.25 −xr−317.615 
BF6 = max(0, 2 −fr2.555 
BF7 = max(0, 0.25 −xr) * max(0, θ +3.81) −5.756 
BF8 = max(0, 0.25 −xr) * max(0, θ −3.81) 6.965 
BF9 = max(0, 0.25 −xr) * max(0, 3.81 −θ−6.096 
BF10 = max(0, xr −0.25) * max(0, θ −3.81) −6.254 
BF11 = max(0, xr −0.25) * max(0, −3.81 −θ5.274 
BF12 = max(0, xr −0.25) * max(0, Dr −0.2) 1,218.284 
BF13 = max(0, xr −0.25) * max(0, 0.2 −Dr−2,068.526 
BF14 = max(0, xr −0.25) * max(0, Dr −0.25) −1,470.058 
BF15 = max(0, xr −0.25) * max(0, 0.25 −Dr1,880.897 
BF16 = BF3 * max(0, 0.3 −Dr−17.987 
BF17 = max(0, 0.35 −Dr) * max(0, xr −0.5) −318.956 
BF18 = max(0, 0.35 −Dr) * max(0, 0.5 −xr427.501 
BF19 = max(0, 0.35 −Dr) * max(0, 5.7 −θ12.529 
BF20 = BF3 * max(0, xr −0.75) −7.267 
BF21 = BF3 * max(0, 0.75 -xr5.216 
BF22 = max(0, xr −0.25) * max(0, fr −2) −5.471 
Basis functionsCoefficients
BF1 = max(0, 0.4 −Dr−130.933 
BF2 = max(0, xr −0.25) * max(0, θ −5.7) 6.336 
BF3 = max(0, 5.7 −θ−2.725 
BF4 = max(0, Dr −0.35) 142.337 
BF5 = BF1 * max(0, 0.25 −xr−317.615 
BF6 = max(0, 2 −fr2.555 
BF7 = max(0, 0.25 −xr) * max(0, θ +3.81) −5.756 
BF8 = max(0, 0.25 −xr) * max(0, θ −3.81) 6.965 
BF9 = max(0, 0.25 −xr) * max(0, 3.81 −θ−6.096 
BF10 = max(0, xr −0.25) * max(0, θ −3.81) −6.254 
BF11 = max(0, xr −0.25) * max(0, −3.81 −θ5.274 
BF12 = max(0, xr −0.25) * max(0, Dr −0.2) 1,218.284 
BF13 = max(0, xr −0.25) * max(0, 0.2 −Dr−2,068.526 
BF14 = max(0, xr −0.25) * max(0, Dr −0.25) −1,470.058 
BF15 = max(0, xr −0.25) * max(0, 0.25 −Dr1,880.897 
BF16 = BF3 * max(0, 0.3 −Dr−17.987 
BF17 = max(0, 0.35 −Dr) * max(0, xr −0.5) −318.956 
BF18 = max(0, 0.35 −Dr) * max(0, 0.5 −xr427.501 
BF19 = max(0, 0.35 −Dr) * max(0, 5.7 −θ12.529 
BF20 = BF3 * max(0, xr −0.75) −7.267 
BF21 = BF3 * max(0, 0.75 -xr5.216 
BF22 = max(0, xr −0.25) * max(0, fr −2) −5.471 
Figure 15

Performance of models developed to estimate discharge passing in floodplains.

Figure 15

Performance of models developed to estimate discharge passing in floodplains.

Close modal

The structure of the GMDH model developed to estimate the discharge passing in the floodplain is shown in Figure 16, and its associated coefficients are shown in Table 11. The structure of the neural network model is also shown in Figure 14. The error statistical indices of the developed models are generally presented in Table 12. This table shows that the MARS algorithm has the best accuracy.

Table 11

The coefficients of the developed GMDH model to estimate discharge passing in flood plains

b0b1b2b3b4b5b6
layer-1 −25.836 132.501 40.987 −44.750 −11.453 −80.880 −25.836 
13.044 −1.595 12.901 −0.005 −15.060 2.482 13.044 
−2.322 1.445 53.251 −0.145 13.106 −5.925 −2.322 
layer-2 −3.912 0.178 0.770 0.013 −0.009 0.021 −3.912 
−0.231 −0.763 1.666 0.036 −0.139 0.108 −0.231 
layer-3 −2.085 1.026 0.223 0.002 0.003 −0.010 −2.085 
b0b1b2b3b4b5b6
layer-1 −25.836 132.501 40.987 −44.750 −11.453 −80.880 −25.836 
13.044 −1.595 12.901 −0.005 −15.060 2.482 13.044 
−2.322 1.445 53.251 −0.145 13.106 −5.925 −2.322 
layer-2 −3.912 0.178 0.770 0.013 −0.009 0.021 −3.912 
−0.231 −0.763 1.666 0.036 −0.139 0.108 −0.231 
layer-3 −2.085 1.026 0.223 0.002 0.003 −0.010 −2.085 
Table 12

Statistical indices of models developed to estimate discharge passing in floodplains

MethodsTraining Stage
Testing Stage
R2RMSER2RMSE
MARS 0.991 1.094 0.956 2.274 
GMDH 0.862 4.394 0.864 3.373 
MLP 0.985 1.389 0.972 2.168 
MethodsTraining Stage
Testing Stage
R2RMSER2RMSE
MARS 0.991 1.094 0.956 2.274 
GMDH 0.862 4.394 0.864 3.373 
MLP 0.985 1.389 0.972 2.168 
Figure 16

GMDH model structure developed to estimate discharge passing in floodplains.

Figure 16

GMDH model structure developed to estimate discharge passing in floodplains.

Close modal

Comparison with previous findings

The performance of computational models developed in this study was compared with the ANFIS developed by Dos et al. [22]. Also, comparison and investigation of these findings showed that the structure of MARS and GMDH models is consistent with the results of the gamma test. In other words, both the structure of the GMDH and MARS models and the gamma test confirm that the most important parameters involved in estimating the discharge in CCNPF are δ*, So, Dr, Ar, fr, and Rr. Examination of the error indices of the model developed with these two models shows that the ANFIS models in the test stage are: R2 = 0.850, RMSE = 0.005. The comparison of the indices of the models developed in this research with ANFIS shows that the MARS and MLPNN models developed in this research can model and predict the discharge in CCNPF much more efficiently and accurately.

In the present study, the flow discharge (Q) in a CCNPF (convergent and divergent floodplains) was modeled and estimated using soft computational models including MARS, GMDH, and MLPNN models. For this purpose, the geometric and hydraulic characteristics of the flow include relative roughness (fr), relative area (Ar), relative hydraulic radius (Rr), flow aspect ratio (δ*), relative width (α), relative depth of the flow (Dr), relative longitudinal distance (xr), the angle of convergence or divergence (θ) of the floodplain relative to the main channel (positive for divergence angle and negative angle for convergence) and S0 as the bed slope were considered as inputs. Then, the performance of the developed models was compared with ANFIS (presented by previous researchers). The results of this study for global discharge in CCNPF showed that the statistical indices of MARS model error in the test stage are R2 = 0.962, RMSE = 0.003 and the statistical indices of the GMDH model in this stage are R2 = 0.930, RMSE = 0.004, and in the neural network model with a combination of the most important variables is R2 = 0.969, RMSE = 0.002. It should be noted that the error indices of the ANFIS model in the test phase are R2 = 0.850, RMSE = 0.005. Comparison of the performance of the mentioned models showed that the MARS model provides more accurate results than other models. Examination of the structure of GMDH and MARS models shows that the most important parameters influencing the discharge estimation are δ*, So, Dr, Ar, fr, and Rr. This result has been approved by previous researchers with the gamma test. In the following, the performance of utilized soft computing models used in modeling and estimating the share of each part of compound open channel including the floodplains and the main channel in the flood discharge was investigated. For this purpose, the parameters of Dr, xr, convergence or divergence angle (θ), and relative roughness (fr) were used as inputs. The results indicated that the MARS model with R2 = 0.979, RMSE = 1.723 for predictive modeling of discharge in the main channel in the testing stage, and R2 = 0.956, RMSE = 2.274 for discharge in floodplains has the best accuracy in comparison to other applied models.

If needed, the data can be made available to readers of the paper.

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