Field data of armour damage in mound breakwaters is scarce, and experimental testing methods usually neglect the influence of pre-existing damage on subsequent damage increments. This study proposes a probabilistic framework, based on a dataset of 44 cumulative damage experiments, to estimate long-term damage progression in a full-scale, non-overtopped, cube-armoured mound breakwater located in the depth-induced wave breaking zone. A Gaussian copula-based Bayesian network is constructed to model the multivariate relationships between existing armour damage (S_e), the increment of armour damage (ΔS_e), the dimensionless water depth at the toe of the structure (h_s/H_m0), wave steepness (H_m0/L_0p), and the stability number (N_s). Each variable is modelled with a univariate parametric distribution, enabling inference of probabilities for values not directly observed in the experimental dataset. Model validation includes testing the Gaussian copula assumption, which is deemed a reasonable model, and assessing the defined graph with satisfactory results. The model is conditionalized using a historical wave dataset to generate synthetic damage curves, which are subsequently used to quantify a gamma process to model the survivability of the structure along its design life. A case study of a hypothetical breakwater with D_n=2 m close to the port of Tarragona (Spain) illustrates the methodology. According to the results of the model, the probability of observing a dimensionless damage S_e>5 corresponding to Initiation of Destruction after 10 years is 0.29. Overall, the obtained results are deemed conservative; this could be caused by the use of data from 2D experiments which do not take into account oblique wave attack. However, the approach is adaptable to other datasets with additional variables, breakwaters in different conditions or with other configurations, and can also be used in combination with simulations of synthetic wave data, making it relevant under changing climate conditions.

Infrastructures are facing growing challenges due to their aging process while climate change and evolution of traffic and shipping fleets are increasing the uncertainty of loadings in the future. This study proposes a method to assess the survivability of structures with gradual deterioration under changing loading scenarios based on field data. The methodology is applied to the armor deterioration of a rock-armored groyne under ship-wave attack. First, we generate synthetic timeseries of damage by coupling a Poisson distribution to determine the number of passing ships per day, a vine-copula to quantify the multivariate joint distribution of the loading variables that define the primary wave height and a Bernoulli process and a bivariate copula to translate the primary wave height into the increment of damage. Afterwards, these damage curves are used to quantify a Gamma process. Thus, it is possible to conditionalize the joint distribution of the loading variables to generate the damage curves under different loading scenarios and evaluate the effects of these scenarios on the structure’s survivability. We exemplify the use of the methodology to assess the armor deterioration of a rock-armored groyne under ship-wave attack with and without a limitation in the speed velocity in the waterway.

Physical model experiments are conducted in a wave basin to investigate the influence of directional spreading on wave overtopping in shallow water. Offshore wave steepness, wave height, water depth, and directional spreading are systematically varied to assess their impact on the non-dimensional mean overtopping discharge (q∗). Additional tests with oblique wave attack are performed to examine the role of the wave direction. To better understand the underlying hydrodynamics, the dependence of low-frequency wave energy on directional spreading is analyzed. Results confirm that low-frequency wave energy strongly depends on directional spreading, consistent with previous studies. An empirical formulation is introduced to predict the ratio of low-frequency wave height to total incident wave height using the relative water depth, the offshore wave steepness, and offshore directional spreading, achieving an R ²=0.91. Excluding directional spreading from the formulation decreases the R ² to 0.38, highlighting its importance. Variations in low-frequency wave energy also affect q∗, as low-frequency waves temporarily raise the water level, leading to larger overtopping volumes and thus higher q∗. Consequently, directional spreading influences q∗ primarily through its effect on low-frequency energy, particularly in shallow water. To evaluate how existing prediction tools perform under these conditions, several formulations for q∗ are assessed. Their performance ranges from poor to reasonable, with the best results using the formulation in De Ridder et al. (2024) that were based on 2DV tests in shallow water rather than 3D tests including directional spreading. The tests with oblique waves show that the existing formulation captures the trends found in shallow water. Therefore, the existing formulation for the influence of oblique wave attack is also recommended for shallow water. To incorporate directional spreading effects into overtopping prediction, the relative crest height was adjusted by including the contribution of the low-frequency wave height as done in De Ridder et al. (2024). Due to reduced correlation between the short-wave steepness and the low-frequency height in this new dataset, coefficients could be estimated more reliably. The revised equations are validated against (long-crested) wave flume and new datasets with both short- and long-crested conditions and oblique attack. The expression including the low-frequency wave height results in the highest accuracy (R ²=0.87, Equation (32)) and is recommended, while a relatively simple expression with only the relative crest and short-wave steepness also performs well (R ²=0.83, Equation (28)).

Sea level rise can compromise the safety of coastal flood defences, as wave overtopping events are becoming more frequent and severe. This increasing threat emphasizes the need for accurate assessment of wave overtopping hydrodynamics over dikes, which is essential for evaluating flood safety. The currently available methods do not combine computational efficiency, detailed results and general applicability, which limits their use in modelling wave overtopping and the resulting dike erosion. To address these limitations, this study introduces the Wave Overtopping Surrogate Model (WOSM), a novel method for rapidly generating high-quality two-dimensional simulations of wave overtopping over the dike crest and landward slope. The foundation of the WOSM is the Vision Transformer Image to Image (ViTI2I), a new deep learning model that combines an adapted Vision Transformer with a convolutional decoder for next-frame prediction. Trained on CFD wave overtopping simulations, the WOSM accurately reproduces the overtopping hydrodynamics such as flow velocities, water depths, overtopping duration and vertical velocity profiles, including both spatial and temporal variations. The scope of the training data limits the applicability of the WOSM and its ability to consistently capture complex phenomena such as flow separation and reattachment, both of which could be improved by enriching the dataset. Its low computational demand makes it suitable for exploring additional applications, such as probabilistic design or simulating wave overtopping with evolving dike profiles for erosion assessment. Additionally, this study serves as a proof of concept that the WOSM framework could benefit other fields encountering comparable modelling constraints.

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.

This study presents a new explicit empirical formula to estimate wave transmission on Cubipod Homogeneous Low-Crested Structures (HLCS) under depth-limited breaking wave conditions. The formula was derived using Artificial Neural Networks (ANN) to identify and quantify the influence of nineteen candidate explanatory variables on the squared wave transmission coefficient, (Formula presented). A total of 210 two-dimensional physical model tests conducted at the Universitat Politècnica de València (Spain) were used to calibrate the formula. The dimensionless crest freeboard using the nominal diameter (R_c/D_n50) and the dimensionless incident wave height at the structure toe using the water depth (H_m0,I/h_s) were identified as the most relevant explanatory variables. A new two-variable formulation with 3 fitting-parameters was found to estimate the proportion of transmitted energy, (Formula presented), with a coefficient of determination R² = 0.89. The proposed formula was also applied to an external dataset of experimental tests on Cubipod HLCS previously reported in the literature. The results demonstrated a significantly better agreement than existing empirical formulas confirming the robustness and applicability of the new formula. The proposed formula is a reliable and easy-to-apply new tool for the preliminary design of emerged and submerged undamaged HLCS in depth-limited breaking wave conditions. The new explicit formula is particularly suited for low-crested structures aimed at combining coastal protection and ecosystem enhancement, such as artificial reefs in coral environments.

Individual overtopping events are important variables when designing a coastal structure as they can deviate significantly from the mean overtopping discharge. Thus, in this study, extreme overtopping events at rubble mound structures with a smooth crest in shallow water have been studied. Both the water layer thickness (flow depth), front velocity and individual overtopping volumes are measured in a wave flume for typical coastal structures with a smooth crest in shallow water for a large range of hydraulic conditions and three different foreshore slopes. An analysis of the individual overtopping volumes shows that the largest individual overtopping volumes arise from short waves that travel on the crest of a low-frequency wave in shallow water and short waves that travel on top of the trough in deep water. Due to the temporal water level variation caused by the low-frequency waves in shallow water, there are fewer overtopping events compared to deep water conditions with the same non-dimensional overtopping discharge. However, the individual overtopping volumes of these events are larger. To quantify the extreme overtopping variables, an empirical formulation based on the relative crest height and short-wave steepness is proposed for the non-dimensional 2 % exceedance water layer thickness, front velocity and individual overtopping volume in terms of incident waves with an R² of 0.84, R² of 0.55 and R² of 0.85 respectively. A further small improvement is found when the low-frequency wave height and 2% exceedance wave height are included, but the added value of this expression does not outweigh the additional wave variables needed for the expression. A log-normal distribution with a constant shape and an expression for the scale of the distribution is proposed to describe the distribution of the individual overtopping volumes in shallow water which accurately captures the distribution (R² of 0.90). Compared to most of the current design approach which is based on a cascade of empirical formulations, this is a significant improvement. In addition, the reasonable results for a distribution with a constant shape parameter show that the shape of the distribution does not change significantly for shallow water conditions.

Rock groins in the Elbe Estuary are constructed to maintain proper water levels for navigation and for embankment erosion protection. At certain localities, significant damages to rock groins have been observed due to the primary ship-generated waves. Primary waves are generated along the ship's hull and then propagate toward the river banks and groin fields, appearing in the interaction with the structures as a turbulent overflow phenomenon. Eventually, this overflowing may cause damages mainly to the crest and leeward side of the groins. Since this overflowing is the most pronounced with large primary waves at certain water levels, the estimation of the probabilities of extreme primary waves is a key element for a safe and reliable design of groins. For this goal, nonparametric Bayesian networks (NPBNs) are used here to infer the probability distribution function of the extreme primary wave heights at the tip of a groin in the Elbe Estuary. Results demonstrate the suitability of the NPBN in their prediction. The model framework allows the designer to predict the probabilities of primary ship-generated waves at groins when the information of ship dimensions, nautical parameters, and waterway geometry is available. These probabilities can later be used for design purposes for current and future conditions.

Aquaculture at sea is gaining increasing importance, not only as a (local) food source but also due to its potential of being combined with other offshore activities such as wind parks. Nevertheless, experience of offshore aquaculture is limited. This study aims to provide a framework to evaluate offshore aquaculture suitability accounting for the probabilistic dependence between relevant variables. This framework is applied to obtain suitability maps of aquaculture for the North Sea for the blue mussel Mytilus edulis and the sugar kelp Saccharina latissima. For each of these species, three ecological variables are selected and the optimal growth and critical survival limits are defined. Here, suitability is defined as the probability of meeting these conditions. Data on the selected variables is extracted from a large-scale 3D hydrodynamic and ecological model of the northwest European Shelf, of which daily extremes are sampled. The probabilistic model is developed using bivariate copula models, which are fitted to each variable pair to describe their joint distribution function at each studied location. Empirical distribution functions are used to describe the univariate distribution function of each variable and location. Using Monte-Carlo simulations, the probability of meeting the optimal and critical limits is estimated and suitability maps accounting for the probabilistic dependence between the variables are generated. In addition, suitability maps disregarding the dependence are generated and compared to those accounting for the probabilistic dependence. It was found that considering the dependence between variables significantly improves the accuracy of the results for optimal and critical growth conditions for both species. The presented method allows to identify potential areas where blue mussel and sugar kelp cultivation is the most suitable. For instance, in this study, a north-south elongated area west of the German and Danish coast appears to be most suitable for blue mussels, while estuaries and rivers are found the most suitable for the sugar kelp.

Rising sea levels caused by climate change are increasing the risk of overtopping on coastal structures. Moreover, there is a growing societal concern about the visual impact of these structures, which leads to the lowering of their crest freeboards. In previous studies, safety during overtopping events was assessed considering the overtopping layer thickness (h_c), the overtopping flow velocity (u_c) and the individual wave overtopping volume (V). Existing models in the literature to estimate h_c, u_c and V on mound breakwater crests are mainly deterministic, involve a chain of successive estimations leading to accumulated errors and/or do not account for the dependencies between h_c, u_c and V. This study proposes a model to describe the joint probability distribution of h_c, u_c and V based on bivariate copulas. Experimental data from small-scale 2D physical tests conducted on mound breakwaters with three armor layers (single-layer Cubipod®, and double-layer cubes and rocks) in depth-limited breaking wave conditions on two mild bottom slopes and dimensionless crest freeboards between 0.33 and 3.20 is used. Lognormal distribution functions are proposed for each variable and a multivariate dependence model is developed through a one-tree vine-copula. The parameters of this model are quantified directly using wave characteristics and the structure geometry minimizing the accumulated errors in the final predictions. The application of the model is illustrated by computing the probability of not fulfilling at least a tolerability limit for one of the studied variables (OR probability). The OR probability is computed both considering the dependence and assuming independence between the variables and a significant difference is obtained. It is concluded that by accounting for the multivariate dependence between the variables, it is possible to reduce the crest freeboard and, thus, achieve a more economic design within the required safety level.

The rapid changes in the shipping fleet during the last decades has increased the ship-induced loads and, thus, their impact on infrastructures, margin protections and ecosystems. Primary waves have been pointed out as the cause of those impacts, with heights that can exceed 2 m and periods around 2 minutes. Consequently, extensive literature can be found on their estimation mainly from a deterministic perspective with methods based on datasets limited to one location, making difficult their generalization. These studies propose either computationally expensive numerical models or empirical equations which often underestimate the extreme primary waves, hindering their use for design purposes. Moreover, a framework to allow the design of infrastructure under ship-wave attack based on probabilistic concepts such as return periods is still missing. In this study, a probabilistic model based on bivariate copulas is proposed to model the joint distribution of the primary wave height, the peak of the total energy flux, the ship length, the ship width, the relative velocity of the ship and the blockage factor. This model, a vine-copula, is developed and validated for four different deployments along the Savannah river (USA), with different locations and times. To do so, the model is quantified using part of the data in one deployment and validated using the rest of the data from this deployment and data of the other three. The vine-copula is validated from both a predictive performance point of view and with respect to the statistical properties. We prove that the probabilistic dependence of the data is preserved spatially and temporally in the Savannah river.