Discovering the causal structure governing wave overtopping discharge using a combined approach of machine learning and physical constraints: Analysis of CLASH database data

Coastal structures such as breakwaters and coastal protection revetments are among the most important structures to protect the shoreline against waves and also to provide a calm condition for berthing and maneuvering ships. Wave overtopping over the crest of these coastal structures is one of the fundamental challenges of coastal engineering. It directly dictates the required freeboard and crest height. Its importance is even more highlighted under the intensifying impacts of climate change. Rising sea levels and increasing wave intensity can significantly increase overtopping rates, thereby affecting the safety and performance of coastal protection structures. Therefore, accurate prediction of wave overtopping is an essential issue to achieve an optimized structure with an acceptable performance. Although some empirical relationships and some machine learning models have been developed recently to predict and improve the prediction of the overtopping discharge over coastal structures, many of these approaches rely primarily on statistical correlations and are not able to identify the true causal relationships among the important variables governing the overtopping amount. The objective of this study is to identify the causal structure governing wave overtopping discharge by employing causal discovery methods together with other physical constraints derived from wave hydrodynamics principles. In this study, the well-known and approved CLASH database, which contains 10532 laboratory records, was analyzed first for use in the introduced approach. A combination of different causal discovery methods, including linear and nonlinear DAGMA, the PC algorithm, and Conditional Mutual Information, was applied alongside correlation-based approaches such as LASSO regression, Random Forest feature importance, and linear regression to improve the results. In addition, a weighted voting framework and a physical anomaly detection module were developed in this study to eliminate spurious relationships and proxy variables. The results of this research indicate that, after applying physical constraints and removing structural multicollinearity, the final causal parents of overtopping discharge are freeboard Rc, wave height Hm0, wave period Tm-10 toe, friction coefficient γ_f, and berm height hb. The effects of wave height, wave period, and water depth at the berm were found to be positive on the results, while the effects of freeboard and the roughness factor were negative,. These outcomes are consistent with physical principles and the EurOtop relationships. The result also shows that the presence of the berm can play an important role in controlling the overtopping rate not only by reducing wave energy, but also by changing the hydraulic conditions of incident wave propagation as a result of changing the effective water depth in the front of ​​the structure. Also, the removal of the variables h and Ac from the final parent set shows the importance of examining the confounding relationships and structural correlations in the data. Partial correlation analysis showed that the correlation between h and q almost reaches zero after controlling for the variable ht (partial correlation less than 0.05), which indicates that h effectively plays the role of a proxy variable for ht. The findings also reveal that the governing relationships of wave overtopping are inherently nonlinear, and that integrating physical knowledge with causal discovery methods can lead to the extraction of an interpretable and generalizable causal structure.