We describe a model for the conditional dependence of a spatial process measured at one or more remote locations given extreme values of the process at a conditioning location, motivated by the conditional extremes methodology of Heffernan and Tawn. Compared to alternative descriptions in terms of max‐stable spatial processes, the model is advantageous because it is conceptually straightforward and admits different forms of extremal dependence (including asymptotic dependence and asymptotic independence). We use the model within a Bayesian framework to estimate the extremal dependence of ocean storm severity (quantified using significant wave height, H S) for locations on spatial transects with approximate east–west (E‐W) and north–south (N‐S) orientations in the northern North Sea (NNS) and central North Sea (CNS). For H S on the standard Laplace marginal scale, the conditional extremes “linear slope” parameter α decays approximately exponentially with distance for all transects. Furthermore, the decay of mean dependence with distance is found to be faster in CNS than NNS. The persistence of mean dependence is greatest for the E‐W transect in NNS, potentially because this transect is approximately aligned with the direction of propagation of the most severe storms in the region.
Environmental contours are used in structural reliability analysis of marine and coastal structures as an approximate means to locate the boundary of the distribution of environmental variables, and hence sets of environmental conditions giving rise to extreme structural loads and responses. Outline guidance concerning the application of environmental contour methods is given in recent design guidelines from many organisations.
Physical considerations and previous studies suggest that extremal dependence between ocean storm severity at two locations exhibits near asymptotic dependence at short inter-location distances, leading to asymptotic independence and perfect independence with increasing distance. We present a spatial conditional extremes (SCE) model for storm severity, characterising extremal spatial dependence of severe storms by distance and direction. The model is an extension of Shooter et al. 2019 (Environmetrics 30, e2562, 2019) and Wadsworth and Tawn (2019), incorporating piecewise linear representations for SCE model parameters with distance and direction; model variants including parametric representations of some SCE model parameters are also considered. The SCE residual process is assumed to follow the delta-Laplace form marginally, with distance-dependent parameter. Residual dependence of remote locations given conditioning location is characterised by a conditional Gaussian covariance dependent on the distances between remote locations, and distances of remote locations to the conditioning location. We apply the model using Bayesian inference to estimates extremal spatial dependence of storm peak significant wave height on a neighbourhood of 150 locations covering over 200,000 km2 in the North Sea.
Motivated by recent work on Markov extremal models, we develop a nonstationary extension and use it to characterize the time evolution of extreme sea state significant wave height (HS) and storm direction in the vicinity of the storm peak sea state. The approach first requires transformation of HS from a physical to a standard Laplace scale achieved using a nonstationary directional marginal extreme value model. The evolution of Laplace‐scale HS is subsequently characterized using a Markov extremal model and that of the rate of change of storm direction described by an autoregressive model, the evolution variance of which is HS‐dependent. Simulations on the physical scale under the estimated model give realistic realizations of storm trajectories consistent with historical data for storm trajectories at a northern North Sea location.
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