Bonakdari Hossein, Gholami Azadeh, Mosavi Amir, Kazemian-Kale-Kale Amin, Ebtehaj Isa, Azimi Amir Hossein
Department of Soils and Agri-Food Engineering, Université Laval, Québec, QC G1V0A6, Canada.
Environmental Research Centre, Department of Civil Engineering, Razi University, Kermanshah 6714414971, Iran.
Entropy (Basel). 2020 Oct 26;22(11):1218. doi: 10.3390/e22111218.
This paper presents an extensive and practical study of the estimation of stable channel bank shape and dimensions using the maximum entropy principle. The transverse slope (St) distribution of threshold channel bank cross-sections satisfies the properties of the probability space. The entropy of St is subject to two constraint conditions, and the principle of maximum entropy must be applied to find the least biased probability distribution. Accordingly, the Lagrange multiplier () as a critical parameter in the entropy equation is calculated numerically based on the maximum entropy principle. The main goal of the present paper is the investigation of the hydraulic parameters influence governing the mean transverse slope (St¯) value comprehensively using a Gene Expression Programming (GEP) by knowing the initial information (discharge () and mean sediment size ()) related to the intended problem. An explicit and simple equation of the St¯ of banks and the geometric and hydraulic parameters of flow is introduced based on the GEP in combination with the previous shape profile equation related to previous researchers. Therefore, a reliable numerical hybrid model is designed, namely Entropy-based Design Model of Threshold Channels (EDMTC) based on entropy theory combined with the evolutionary algorithm of the GEP model, for estimating the bank profile shape and also dimensions of threshold channels. A wide range of laboratory and field data are utilized to verify the proposed EDMTC. The results demonstrate that the used Shannon entropy model is accurate with a lower average value of Mean Absolute Relative Error (MARE) equal to 0.317 than a previous model proposed by Cao and Knight (1997) (MARE = 0.98) in estimating the bank profile shape of threshold channels based on entropy for the first time. Furthermore, the EDMTC proposed in this paper has acceptable accuracy in predicting the shape profile and consequently, the dimensions of threshold channel banks with a wide range of laboratory and field data when only the channel hydraulic characteristics (e.g., and ) are known. Thus, EDMTC can be used in threshold channel design and implementation applications in cases when the channel characteristics are unknown. Furthermore, the uncertainty analysis of the EDMTC supports the model's high reliability with a Width of Uncertainty Bound () of ±0.03 and standard deviation (Sd) of 0.24.
本文对基于最大熵原理估算稳定河道岸坡形状和尺寸进行了广泛而实用的研究。临界河道岸坡横断面的横向坡度（St）分布满足概率空间的性质。St的熵受两个约束条件限制，必须应用最大熵原理来找到偏差最小的概率分布。因此，基于最大熵原理通过数值计算得出熵方程中的关键参数拉格朗日乘数（）。本文的主要目标是通过基因表达式编程（GEP），在已知与目标问题相关的初始信息（流量（）和平均泥沙粒径（））的情况下，全面研究影响平均横向坡度（St¯）值的水力参数。结合前人的形状剖面方程，基于GEP引入了一个明确且简单的岸坡St¯与水流几何和水力参数的方程。因此，设计了一个可靠的数值混合模型，即基于熵理论结合GEP模型进化算法的临界河道熵基设计模型（EDMTC），用于估算临界河道的岸坡形状和尺寸。利用大量实验室和现场数据对所提出的EDMTC进行验证。结果表明，在首次基于熵估算临界河道岸坡形状时，所使用的香农熵模型比Cao和Knight（1997）提出的先前模型（MARE = 0.98）更准确，平均绝对相对误差（MARE）的平均值更低，为0.317。此外，本文提出的EDMTC在预测形状剖面以及临界河道岸坡尺寸方面具有可接受的精度，前提是仅知道河道水力特性（例如，和）。因此，当河道特性未知时，EDMTC可用于临界河道设计和实施应用。此外，EDMTC的不确定性分析支持该模型具有较高的可靠性，不确定性边界宽度（）为±0.03，标准差（Sd）为0.24。
