Probabilistic estimation of the mean wave overtopping discharge on mound breakwaters

Abstract

This study develops a probabilistic model, a Gaussian copula-based Bayesian Network (BN), to explain the joint probability distribution of the dimensionless mean wave overtopping discharge (Q=q/gH_m0³, being q the mean wave overtopping discharge, g the gravity acceleration and H_m0 the spectral significant wave height) and a set of explanatory variables on mound breakwaters. This model estimates the distribution of Q conditional to the values of (all or some of) the explanatory variables. The goal of this model is to allow the incorporation of the uncertainties of the structural response and the overtopping phenomenon to probabilistic frameworks. Given a tolerable Q value, a probability of failure can be directly computed from the distribution of Q estimated by the developed BN, differently to current methods in the literature which are deterministic. To develop the BN, a subset of CLASH database focused on mound breakwaters is used (3,179 tests), using 80% of those tests for training and 20% for statistical and performance testing. Ten dimensionless explanatory variables are selected with the following experimental ranges: bottom slope, 7.6≤m≤1000; wave attack angle, 0≤β≤80° roughness factor, 0.38≤γ_f≤1.00; dimensionless crest freeboard, 0≤R_c/H_m0≤4.37; wave steepness, 1.31⋅10⁻³≤s_−1,0≤0.069; dimensionless width of the crest berm, 0≤G_c/H_m0≤6.67; dimensionless height of the crest berm, 0≤A_c/H_m0≤4.2; dimensionless width of the crest of the toe berm, 0≤B_t/H_m0≤15.9; dimensionless water depth at the toe of the structure, 1.03≤h/H_m0≤17.6; and armor slope, 1.19≤cotα≤4. Empirical cumulative distribution functions are used to quantify the nodes of the BN. The Gaussian copula assumption is successfully validated using the training subset. The proposed model is evaluated using the testing subset in both statistical and performance terms. In statistical terms, the proposed model seems to satisfactorily capture the dependence structure between the studied variables. In performance terms, the predicted mean of the distribution of Q is a reasonable estimator of Q (R²=0.78) and the percentage of the observations that lay within the predicted 90% confidence intervals is close to the expected 90%. Finally, the use of the model for the probabilistic design of the crest elevation of mound breakwaters is also illustrated through one example. It should be noted that the less information provided to the model, the wider the estimated distribution of Q as the uncertainty is higher.