Data-driven modeling of multivariate stochastic trajectories -- Application to water waves

TL;DR

The paper tackles the challenge of modeling joint distributions of multivariate stochastic water-wave trajectories in a data-driven manner. It combines Functional Principal Component Analysis to reduce trajectory dimensionality with a non-parametric vine copula for the distribution bulk and the Heffernan–Tawn conditional tail model for multivariate extremes, applied to DeRisk wave simulations with a wave-breaking filter. The approach demonstrates faithful replication of both the bulk behavior and extreme events, enabling generation of synthetic trajectories and assessment of response-variable distributions for reliability analyses. The method is scalable, interpretable via FPCA, and provides a pathway for probabilistic load predictions in marine-structure design across different sea states.

Abstract

A data-driven methodology is proposed to model the distribution of multivariate stochastic trajectories from an observed sample. As a first step, each trajectory in the sample is reduced to a vector of features by means of Functional Principal Component Analysis. Next, the joint distribution of features is modeled using (i) a non-parametric vine copula approach for the bulk of the distribution, and (ii) the conditional modeling framework of Heffernan and Tawn (2004) for the multivariate tail. The method is applied to the modeling of water waves. The dataset used is the DeRisk database, which consists of numerical simulations of water waves. The analysis is restricted to the portion of the wave period between the free-surface zero-upcrossing and the wave crest. The kinematic variables considered are the free-surface slope, the normal component of the fluid velocity at the free surface, and the vertical Lagrangian acceleration of the fluid at the free surface. The stochastic trajectories of these three variables are modeled jointly. The vertical Lagrangian acceleration of the fluid is employed to enforce a wave-breaking filter in the stochastic model. The capabilities of the model are illustrated by predicting the distributions of selected response variables and by generating synthetic trajectories.

Figures (5)

• Figure 1: Parameter space of sea states covered by the DeRisk simulations. Gray dots indicate individual simulation runs, while clusters of black dots, labeled with numbers, denote the sea states analyzed in this study (see Tab. \ref{['tab_Derisk_sims']}).
• Figure 2: Extraction of water-entry trajectories. The "clouds" of thin grey lines show the full dataset of stochastic trajectories extracted from the DeRisk database. The thick colored lines highlight the $10$ most extreme trajectories in terms of ${K_x}^{\rm max}$ value. Results are shown for sea states $\#1$ and $\#3$ in the first and second rows, respectively. Trajectories of $s$ and $u_n$ are shown in the first and second columns respectively. The corresponding wave profiles are shown in the third column. For the highlighted wave profiles, the solid portion (as opposed to the dotted portion) indicates the time interval corresponding to the water-entry phase. For a given water-entry event, the same color is used consistently across all three kinematic variables.
• Figure 3: Samples of stochastic trajectories for $u_n$ and $s$. The first two rows show samples for sea state $\# 1$, while the last row show samples for sea state $\#3$. The trajectory sample extracted from the DeRisk dataset is labeled as "true sample". The remaining samples (nine for sea state $\#1$ and four for sea state $\#3$) are independent synthetic samples generated by the stochastic model. Each synthetic sample contains the same number of trajectories as the corresponding true sample. For sea state $\#1$, the wave breaking filter is enabled in the stochastic model. In each plot, colors are used to help distinguish trajectories: for a given realization, the trajectories of $u_n$ and $s$ share the same color. The corresponding colormap was used from dark blue to dark red, resulting in the bulk of each sample being dominated by dark red.
• Figure 4: Marginal distributions of the response variables, $I_x$, $I_y$, ${K_x}^{\rm max}$ and ${K_y}^{\rm max}$, shown as exceedance probabilities. The first (resp. second) row shows results for sea state #1 (resp. #3). For sea state #1, two versions of the stochastic model are shown: one version with the breaking filter enabled (model WB), and one version with the breaking filter disabled (model NB). For sea state #3, enabling the breaking filter has effectively no effect. Therefore, only one curve is shown for the model.
• Figure 5: Bootstrap histograms of the $(1-1/n)$-quantiles of the response variables. The first and second rows show results for sea state $\#1$ (with the wave-breaking filter enabled) and sea state $\#3$, respectively. The histograms were generated from $400$ bootstrap samples.