Prediction of depth‑averaged velocity in compound channels with vegetated floodplains using gene expression programming

Introduction:
Natural rivers are commonly characterized by a main channel for primary flow conveyance and a floodplain to carry extra flow during floods. Floodplains are usually partially or completely covered with vegetation such as shrubs or trees. Vegetation affects the depth of flow, velocity distribution as well as sediment transport(Yang et al.,2007). Predicting the lateral velocity distribution in compound channels is important for determining the stage-discharge curve and supporting the management of fluvial processes in vegetation condition in river(Tong and Knight, 2009). Previous research in the area of vegetated floodplains has primarily focused on the adaptation of theory driven resistance equations. Since the 1990’s, several Lateral Distribution Models (LDM) have been developed for obtaining lateral velocity distribution in compound channels. Among the velocity lateral distribution models, the Shiono and Knight Model (SKM) is more popular with widespread applications(Unal et al, 2010). Three calibrating coefficients need to be estimated for applying the SKM, namely eddy viscosity coefficient (λ), friction factor (f) and secondary flow coefficient (k). Determining these coefficients in natural channels is not always feasible and requires some experiences (Knight et al. 2010). Utilizing soft computing (SC) methods to solve different problems is another progressing concept. One of the newest and most powerful SC methods is gene expression programming (GEP), which is an extension of GP and GA, and was first introduced by Ferreira(2001). The GEP method mitigates the majority of problems of principal SC methods related to the absence of equations for practical engineering by presenting explicit equations. The aim of this study was to use GEP modeling to predict the depth averaged velocity distribution in compound channels with vegetated floodplains. The results of the best GEP model are presented as an equation and compared with the result of SKM model.

Methodology
In this study, by aid of nearly 508 depth-averaged velocity data reported in a study by Tavakoli(2019) and using gene expression programming (GEP), the depth-averaged velocity in compound channels with vegetated floodplains was modeled. 9 dimensionless input variables including, Relative flow depth 〖(D〗_r), Relative distance (χr), vegetation density (λ), shading factors , Dsa and Dsr, either aligned or randomly arranged, respectively, Dfp (the vegetation diameter over the width of the floodplain), y_n1,y_n2(the distance from the channel centerline to the measurement location in main channel and flooplain) and one output variable (depth averaged velocity) have been used in GEP. 70% of the experimental results are used for the training process and the remaining 30% for testing. After selecting the training set, the GEP learning environment should be defined. The five main steps in GEP training are as follows: First step: selecting fitness function. Second step: determining the function set(F) and terminal set (T) for chromosome generation. Third step: specifying the number of genes and the head length. Fourth step: defining the linking function for linking different sub-ETs. Fifth step: setting the values of different genetic operators, such as inversion, transposition and recombination(Fuladipanah, 2020). The values of these operators and other parameters used in GeneXpro program are presented in Table 2. Various statistical error analyses were performed to verify the reliability of the developed GEP model. An equation was derived from the best GEP model and its results were compared with the analytical method of Shiono and Knight. To obtain analytical solutions by SKM with an accepted accuracy, the drag coefficient, the shading factor, local friction factor, eddy viscosity and secondary flow term need to be determined, these parameters were discussed in the paper. Importance of the predictor variables for GEP models were also presented by using sensitivity analysis.

Results and Discussion
A number of expressions have been generated to predict depth averaged velocity distribution within a compound channel with vegetated floodplains using gene expression programming (GEP). For an initial attempt, the gene expression programming was run with all data in the non- dimensional form. The best produced formula was as given in Eq. 27. The amount of CC, RMSE, and MAE for GEP at the first scenario during training and testing phases were calculated as (0.919,0.13,0.093) and (0.874,0.156,0.096) respectively. This expression shows high positive correlation, however, this value may be misleading as correlation should only be used as a measure for normally distributed variables. Analysis of the Experimental data showed two distinct normalised distributions, for the lower velocities on the floodplain and the higher velocities on the main channel, respectively(Harris et al.,2003). So the data sets separated in to two data sets and the GeneXpro program was then applied to the two data sets separately, thus giving separate expressions for the two zones. Evaluation of model performance showed that the model presented for main channel, with CC of 0.902 and RMSE of 0.083 outperformed than the model presented for floodplain with CC of 0.843 and RMSE of 0.092. The velocity prediction on the main channel shows good correlation with R2=0.8536, see Fig. 2 but the floodplain results show a degree of scattering with R2=0.78 , this is due, in part, to data collection experimental error and the complexity of the flow around the vegetation. The sensitivity analysis results demonstrate that dimensionless shading factor of vegetation (Dsr), is the most influential parameter with regard to the depth averaged velocity distribution. As it is presented in Fig. 4. Dsr are the most important variable for GEP model, this conclusion is supported by the work of Naot et al. (1996). The results showed that the Shiono and Knight method (SKM) has shown satisfactory results for the prediction of depth-averaged velocity distribution in the lateral direction. The GEP model, with RMSE of 0.15, exhibits superior performance over the SKM model with RMSE of 0.24 for all data.

Conclusion
In this paper, two algorithms namely SKM and GEP have been applied to predict depth averaged velocity distribution in compound channels with vegetated floodplains. The results of these two mentioned algorithms were compared with experimental modeling. The paper highlights the advantages of using intelligent algorithm rather than the traditional approach to predict and extract the complicated and hidden relationship among dependent and independents variables.