Estimating scour depth downstream of ski-jump spillways using the electric eel foraging optimization algorithm

Technical Summary: Estimating Scour Depth Downstream of Ski-Jump Spillways Using the Electric Eel Foraging Optimization Algorithm

Problem Statement
Ski-jump spillways are essential for energy dissipation in high dams, yet the impact of the water jet on the downstream riverbed induces significant local scour. This scouring threatens the stability of hydraulic structures and alters river morphology. While numerous empirical relationships have been proposed to predict scour depth (e.g., Veronese, 1937; Mason and Arumugam, 1985; Azmathullah et al., 2006), the paper notes that these methods often suffer from limited accuracy due to the nonlinear and stochastic nature of the scour process. Furthermore, traditional regression models lack sufficient precision, while "black-box" machine learning models often lack transparent mathematical formulations necessary for reliable extrapolation. The study aims to develop a predictive model that offers both high accuracy and a clear mathematical expression to address these limitations.

Methodology
The research utilizes a hybrid approach combining dimensional analysis with a novel metaheuristic optimization algorithm.

1. Data and Parameters: The model was calibrated using a dataset of 104 data points derived from experimental measurements by Azmathullah et al. (2005) and physical model tests of the Darian Reservoir Dam (Iranian Water Research Center, 2011). Key input variables include unit discharge ($q$), total drop height ($H$), bucket radius ($R$), ski-jump angle ($\Phi$), tailwater depth ($d_w$), and median bed particle size ($d_{50}$).
2. Dimensional Analysis: Using the Buckingham $\pi$ theorem, the authors derived a non-dimensional power-law relationship for scour depth ($d_s$). The relationship is expressed as a function of dimensionless groups: Froude number ($F$), relative drop height ($H/d_w$), relative bucket radius ($R/d_w$), relative grain size ($d_{50}/d_w$), and the bucket angle ($\Phi$).
3. Froude-Based Zoning: A novel zoning technique was implemented where the Froude number domain is divided into three distinct sub-regions. Within each zone, the model employs a unique set of calibrated coefficients for the power-law equation. The boundaries of these zones are not predefined but are determined as decision variables during the optimization process.
4. Optimization Algorithm: The Electric Eel Foraging Optimization (EEFO) algorithm was employed to calibrate the model. EEFO is a metaheuristic inspired by the collective hunting behaviors of electric eels (interaction, resting, hunting, and migration). The algorithm simultaneously optimizes:
   • The coefficients ($a_i, b_i, c_i, d_i, e_i, f_i$) for the power-law equation in each of the three zones.
   • The partitioning variables ($k_i$) that define the Froude number boundaries between zones.
   • The objective function minimizes the sum of squared errors (SSds) between observed and predicted scour depths, subject to constraints ensuring positive scour depth values.

Key Contributions

• Novel Algorithm Application: This study introduces the EEFO algorithm to the field of hydraulic engineering for the first time, specifically for scour depth prediction.
• Zoning Strategy: The implementation of a Froude-number-based zoning technique allows the model to adapt its mathematical behavior to different flow regimes, addressing the nonlinearity of the scour process more effectively than single-equation empirical formulas.
• Transparent Mathematical Formulation: Unlike black-box machine learning models (e.g., ANN, SVM), the resulting model provides an explicit, interpretable mathematical equation (Equation 42) that engineers can apply directly.

Results
The proposed EEFO-based model was evaluated using statistical indices: Ratio of Root Mean Square Error to Standard Deviation (RSR), Coefficient of Determination ($R^2$), Nash–Sutcliffe Efficiency (NSE), and Percent Bias (PBIAS).

• Performance Metrics: The model achieved an RSR of 0.167, $R^2$ of 0.972, NSE of 0.972, and PBIAS of 0.336. According to the rating scales cited in the paper (e.g., Haiati et al., 2025; Afaridegan et al., 2025), these values indicate "Very Good" performance.
• Comparison with Empirical Methods: The study compared the proposed model against 15 established empirical formulas (including those by Veronese, Mason and Arumugam, and Azmathullah et al.). The EEFO model significantly outperformed all empirical methods. For instance, the best-performing empirical method (Azmathullah et al., 2005) yielded an RSR of 0.390 and NSE of 0.847, whereas the proposed model achieved an RSR of 0.167 and NSE of 0.972.
• Calibrated Equation: The final output is a piecewise power-law equation (Equation 42) with specific coefficients for three Froude number ranges ($F \le 1.171$, $1.171 < F \le 2.774$, and $F > 2.774$).

Significance and Claims
The paper claims that the developed EEFO-based optimization method provides a highly accurate and reliable tool for predicting scour depth downstream of ski-jump spillways. By offering a transparent mathematical expression that outperforms traditional empirical formulas, the model addresses the dual need for precision and interpretability in hydraulic design. The authors state that these findings offer a practical solution for mitigating scour-related risks and optimizing the design of hydraulic structures, thereby enhancing their long-term safety and stability. The study concludes that the primary governing parameters are the Froude number, relative drop height, relative bucket radius, relative grain size, and bucket lip angle, with the model revealing specific direct and inverse relationships between these parameters and scour depth within the calibrated zones.