A methodological approach of predicting threshold channel bank profile by multi-objective evolutionary optimization of ANFIS
Introduction
Natural channels and streams tend to reach stability at the banks (Huang et al., 2014), while there is ongoing sediment transfer in the central region of the channel bed. Most researchers define a stable channel as a channel whose bank profile is in a state where the particles at the wetted perimeter are in incipient motion (Dey, 2001). Hence, the term ‘threshold channel’ is used in conditions where channel widening is the ultimate state (Yu and Knight, 1998). Stable channel also refers to a state when the lateral diffusion of momentum due to turbulence from the channel centre towards the banks leads to shear stress redistribution to the channel boundaries (Vigilar and Diplas, 1997). The above definition is one of the most descriptive of a channel that will achieve equilibrium state (Diplas, 1990). The cross-sectional shape of a channel in stable state is very important for the bank profile's geometric formation. Fig. 1 represents the characteristics of a bank profile. In the figure, x is the transverse distance from the channel centre, y is the vertical boundary point level in stable channel form, h is the flow depth at the channel centre in stable state and T is the water surface width.
A few researchers have noted the importance of channel bank profile shape in stable state and suggested various optimized shapes. Glover and Florey (1951) presented the primary assumptions for stable channels. They defined equilibrium as the lack of particle motion at all channel points, in which case the bank profile is referred to as cosine. The movement of particles on the channel bed is incompatible with the notion of a stable channel and the stable channel paradox is derived (Ikeda et al., 1988). Parker (1978) resolved particle movement on the channel bed by using lateral momentum transfer from areas with high momentum towards areas with low momentum and non-uniform distribution of shear stress, where the stress on the channel bed and banks is more and less than the critical stress allowed, respectively. Therefore, particle movement on the channel bed is justified. Ikeda (1981) was the first to suggest an exponential channel shape. Pizzuto (1990) modeled the final widening process in a channel and introduced an exponential equation for stable channel shape. According to Pizzuto (1990), stable channel shape changes over time to a cosine bank profile shape. Diplas and Vigilar (1992) suggested the fifth-degree polynomial function for a bank profile with a numerical solution to the momentum equilibrium and force equilibrium equations for sediment particles in impending motion (threshold channel). Vigilar and Diplas (1997) provided a numerical model and considered the lateral diffusion of momentum induced by turbulence to investigate a channel with "stable bank and mobile bed" that consists of a flat bed and two curving bank regions. Then Vigilar and Diplas (1998) verified their numerical model using experimental data and suggested a three-degree polynomial bank profile. Yu and Knight (1998) presented a numerical model by combining the depth averaged momentum equation with the resistance laws and threshold conditions. This combination yielded a geometric model based on physics for threshold channel design. Cao and Knight (1998) provided a numerical model to solve the stable channel paradox in terms of the secondary flow portion in the lateral diffusion of momentum. Moreover, combining the bank profile equations based on Cao and Knight's (1997) entropy concept, flow continuity condition and frictional resistance with sediment transport relations produces a geometric model for stable alluvial channels. Babaeyan-Koopaei and Valentine (1998) used an experimental model and studied the bank profile of a straight stable channel. They suggested hyperbolic functions for the bank profile shape of stable channels. Dey (2001) proposed a polynomial shape using the power law and the model is highly compliant with Parker (1978), Diplas and Vigilar (1992) and Yu and Knight's (1998) models. Khodashenas (2016) compared the performance of 13 previous models using experimental sets. Their comparison results indicate that all prior models produced significant error in predicting the bank profile of a threshold channel and thus required further study. Preceding studies therefore denote the importance of predicting stable channel bank profile shape. Most studies reported above are based on either experimental and numerical models or mathematical principles. Today, hydrology and hydraulics scientists widely use artificial intelligence methods as powerful time- and cost-saving tools (Emiroglu et al., 2010, Emiroglu et al., 2011; Kisi et al., 2013; Zhang et al., 2015; Ebtehaj and Bonakdari, 2016; Gholami et al., 2017a; Huang et al., 2017). Madvar et al. (2011), Taher-Shamsi et al. (2013) and Bonakdari and Gholami (2016) used artificial neural networks (ANN), while Gholami et al. (2017b) and Shaghaghi et al. (2017) used a group method of data handling (GMDH) neural model to predict stable channel geometry. All models estimate the geometry and hydraulic parameters, such water surface width (w), depth at the centreline (h) and longitudinal water surface slope (s) for stable channel design. However, the mentioned researchers did not report any bank profile results nor proposed shapes. More recently, combining fuzzy logic with neural networks has led to a new neuro-fuzzy system called the Adaptive Neural Fuzzy Inference System (ANFIS) (Khanlari et al., 2012; Mishra and Basu, 2013; Basarir et al., 2014; Kayabasi et al., 2015; Gholami et al., 2017c). The ANFIS model is one of the most common and powerful techniques for modeling complicated, nonlinear problems (Kulatilake et al., 2010; Azimi et al., 2017; Öge, 2017). However, the number of IF-THEN fuzzy rules in the ANFIS network structure becomes very large with the increment in membership functions (MF) in the system's input apace. In this case, a large number of IF-THEN fuzzy rules is not desirable to map inputs to output parameters due to the overfitting problem that reduces the ANFIS model's generalization ability for predicting unforeseen data. To achieve highly precise models for solving nonlinear problems, all inconsistent objective functions should be considered, which have a remarkable effect on modeling (Coello et al., 2007). In multiobjective optimization the existence of different objective functions leads to a set of Pareto optimal solutions (Shaghaghi et al., 2017). Thus, a program was coded in MATLAB to design a hybrid ANFIS model associated with two evolutionary algorithms. The authors found no study related to the use of these algorithms to predict stable channel bank profile shape.
Therefore, the main objective of this study is to estimate the cross-sectional profile shape of a threshold channel bank profiles using a new hybrid multiobjective method. In this method, the multiobjective Differential Evolution (DE) algorithm and Singular Value Decomposition (SVD) are applied in combination with the ANFIS network (ANFIS-DE/SVD). DE is employed to optimally select the variables of the Gaussian membership function's nonlinear coefficients in the antecedent part of ANFIS. Moreover, the SVD technique is utilized to compute the linear coefficients of the consequent part of the ANFIS network. Training error (TE) and prediction error (PE) are considered significant conflicting objectives in the optimal multiobjective ANFIS network design. Experiments were conducted to gain a wide range of data for different discharge rates. The bank profile shape result is compared with existing models (based on analytical and numerical investigations) and a simple ANFIS model.
Section snippets
Overview of ANFIS
An adaptive neuro-fuzzy inference system (ANFIS) includes a set of Takagi-Sugeno-Kang (TSK) fuzzy rules to map the input onto the output space (Sugeno and Kang, 1988). The main purpose of an ANFIS model is to find a function (f) using n inputs, one output and a number of M different observational samples:where l = {1, 2, …, n}, ji∈{1, 2, …, r}, r is the number of fuzzy sets in each rule, Wl = {w0l, w1l, w2l, …, wnl} is the parameter set of the
Experimental model
For this study, the authors obtained four datasets from the hydraulic laboratory of the Department of Civil and Geological Engineering, University of Saskatchewan, Canada. Two parameters, namely the depth or vertical alignment of points located at the channel boundary (y) and the transverse distance of points on the channel axis (x) were measured in balanced channel mode. The flume used for the channel was 20 × 1.22 × 0.6 m (length, width and height) with a longitudinal channel slope (S) of
Parameter definitions and overview of existing models
The various parameters used in this study are defined in this section. Fig. 1 displays the bank profile shape. In this figure, the parameters are made dimensionless as follows:where x⁎ = non-dimensional transverse distance, y⁎ = non-dimensional balance-border and T⁎ is the non-dimensional water surface width.
Vigilar and Diplas (1997) presented the following relationships to determine h, T⁎and δcr∗, respectively:
Model performance evaluation using statistical indices
In the present study, artificial intelligence models are compared with earlier models and experimental values from different statistical equations based on regression. The deviation of the predicted values from the experimental data is measured with the error indices Mean Absolute Relative Error (MARE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Error (ME), determination coefficient (R2), BIAS and ρ. Error indices RMSE, MARE, MAPE, MAE
Threshold channel bank profile modeling using ANFIS-DE/SVD
This section presents and explains threshold channel bank profile modeling using the hybrid ANFIS-DE/SVD. In the current study, two variables, i.e. transverse distance from the center line (x) and the discharge (Q) were employed as input parameters to estimate the vertical level (y) as the output parameter. There were 241 samples in total, of which 50% were used for training (121) and the rest for model validation in the testing stage. Other parameters to be set were related to the DE
Model performance evaluation
In this section, the performance of the ANFIS and ANFIS-DE/SVD models in predicting stable channel shape is evaluated and the results are compared. The authors' experimental results for predicting stable channel section shape at four discharge rates were used to train and test the models. The numbers of measured points for 1.157, 2.18, 2.57 and 6.2 l/s were 39, 127, 32 and 43 data, respectively. Therefore, there were a total of 241 x data for 4 discharge rates and a corresponding 241 y data.
Conclusion
In this study, a new hybrid model based on ANFIS combined with the DE algorithm and SVD (ANFIS-DE/SVD) was employed to predict stable channel profile shape. To provide a flexible algorithm for multiple datasets, two different objective functions were defined and the Pareto curve was used to select the optimal point, which is the tradeoff between the two objectives. The bank profile shape proposed by ANFIS-DE/SVD was compared with the top seven existing models in terms of prediction accuracy.
Acknowledgement
The experiments for this study were performed at the Hydraulic Laboratory of Civil and Geological Engineering Department, University of Saskatchewan, Saskatoon, Canada, in a period of sabbatical leave of the fifth author.
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