Prediction of downstream hydraulic geometry in alluvial channels using the hybrid intelligent algorithm (GMDH-HS)

Alluvial rivers always change their hydraulic geometry to achieve a balance between water discharge, input and output sediment. Hydraulic geometry focuses specifically on the evolution of the river form and how the bed and channel influence this evolution. The morphology of alluvial rivers has led to the creation of two concepts: (1) at-a-station hydraulic geometry and (2) downstream hydraulic geometry (Julien, 2015). Downstream hydraulic geometry is defined by the top channel width (W), average flow depth (h), mean flow velocity (V), and slope of the flow energy (S) under bankfull conditions. Downstream hydraulic geometry as a function of hydraulic parameters and bed conditions, including flow rate, median size of bed particles and the Shields parameter is paramount to determine the state of a river. Therefore, various relationships have been derived based on various methods to estimate the channel hydraulic geometry, include: empirical relationships based on collected fields observation and theoretical relationships based on governing equations such as flow rate, resistance to flow, secondary flow and sediment transport in alluvial river. Result of theoretical derivations indicated reasonable agreement with field observations and regime equations. In recent years, intelligent data driven methods is used as new methods for predicting and estimating the parameters of complex hydraulic models. One of the common methods is the Group Method of Data Handling (GMDH) with self-organization approach, which has the ability to solve complex non-linear problems with higher accuracy and simpler structure. In GMDH method the coefficients of the polynomial are found by a Least Square Estimation (LSE) method. It is possible that combined the GMDH methods and optimization algorithms. By doing this, a hybrid method will be created. In this method an optimization algorithm used to calibrate the weights of each neuron in GMDH rather than LSE method and so the hybrid methods may have better performance. The GMDH method is combined with artificial intelligence and optimization techniques such as harmony search (HS) optimization method. Harmony search algorithm (HS) is one of the optimization methods that used to solve nonlinear problems, which was introduced in 2001 by Geem et al. based on a metaheuristic technique. The advantages of this algorithm are less computations to find a solution, fast convergence and significant ability to achieve the optimal solution due to the appropriate structure. HS algorithm has become one of the most used optimization algorithms because can be used for both continuous and discrete problems. Since the GMDH algorithm has a self-organizing approach and its structure is initially unclear, the Harmony search algorithm is used to train and optimize the weights in the structure of each neuron in the GMDH network. In fact, the objective of HS sub model is to determine the optimal weights in short time to achieve the optimal GMDH structure and minimize the cumulative error between the measured and computed data sets.

In this study, to predict and improve the accuracy of the relations of the downstream hydraulic geometry in alluvial channels the GMDH model and a hybrid intelligent model based on the combination of GMDH and HS optimization algorithm that called GMDH-HS developed in the MATLAB software. Before using the developed models to predict and improve the accuracy of downstream hydraulic geometry in alluvial channels, it is necessary to check the accuracy of their results. Typically, researchers use the Mackey and Glass standard time series to verify the accuracy of developed models. Therefore, the validation of the developed models was done using the Mackey-Glass time series and calculated the verification results of the models for the two prediction methods by the CE, RMSE, MSRE, MPAE and RB criteria. To evaluate the developed models, 880 data series were collected from previous research. This data set include a wide range of field and laboratory measurement data. From this collection, 498 series were used to train and 382 remaining data series employed to test models. The results of GMDH and GMDH-HS are compared with observed data. Also, the intelligent models results are compared with theoretical equations proposed by Lee and Julian (2006).

Evaluate the performance of developed models using statistical indices CE, MSRE, MAPE, RMSE, RB and R2 indicate satisfactory performance of both models in predicting the downstream hydraulic geometry alluvial channels. A closer examination and comparison of the results of the two models showed that the hybrid GMDH-HS model had a much higher performance in predicting hydraulic geometry. Also, the comparison of the statistical indices obtained from the results of both developed models with the theoretical relationships presented by Lee and Julein (2006) suggests the very satisfactory performance of the hybrid intelligent model GMDH-HS in predicting the downstream hydraulic channel geometry in alluvial channels.