On spatial conditional extremes for ocean storm severity

Abstract

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.

Keywords: conditional extremes; nonstationary; significant wave height; spatial dependence.

Figure 1

Northern North Sea and central North Sea locations considered

Figure 2

Illustration of notation used to describe the disposition of points on the line, enabling pooling of data from pairs of locations by distance. The (c,j) notation is shown below the line, and distance h _c0j is given above each point. Points with the same value of h _c0j are shown in the same colour; Δ is the interlocation spacing

Figure 3

NNS:N‐S transect, free model: parameter estimates for (a) α, (b) β, (c) μ, and (d) σ with distance h, summarised using posterior means (disc) and 95% pseudolikelihood credible intervals (with end points shown as solid triangles). NNS = northern North Sea; N‐S = north–south

Figure 4

Scatterplots illustrating dependence between values of Laplace‐scale storm peak H _S at different relative distances for NNS:N‐S transects, from (a) original sample and (b) simulation under the fitted free model. Black points are the original data on Laplace scale; red points are data simulated under the fitted model. NNS = northern North Sea; N‐S = north–south

Figure 5

Observed spatial processes from the NNS:N‐S transect with Laplace‐scale values at the left‐hand location in the interval [3.5,4.5], together with posterior predictive estimates from simulation under the fitted free model, represented using the median (black) and upper and lower limits of a 95% pseudolikelihood credible interval. Red lines are simulated spatial processes from the fitted model

Figure 6

NNS:N‐S transect, parametric‐α model: parameter estimates for (a) α, (b) β, (c) μ, and (d) σ with distance h, summarised using posterior means (disc) and 95% pseudolikelihood credible intervals (with end points shown as solid triangles). NNS = northern North Sea; N‐S = north–south

Figure 7

NNS:E‐W transect, parametric α(h) model: estimates for (a) α(h), (b) β(h), (c) μ(h), and (d) σ(h) with distance h, summarised using posterior means (disc) and 95% pseudolikelihood credible intervals (with end points shown as solid triangles). NNS = northern North Sea; E‐W = east–west

Figure 8

CNS:N‐S transect, parametric‐α model: parameter estimates for (a) α, (b) β, (c) μ, and (d) σ with distance h, summarised using posterior means (disc) and 95% pseudolikelihood credible intervals (with end points shown as solid triangles). CNS = central North Sea; N‐S = north–south

Figure 9

CNS:E‐W transect, parametric‐α model: parameter estimates for (a) α, (b) β, (c) μ, and (d) σ with distance h, summarised using posterior means (disc) and 95% pseudolikelihood credible intervals (with end points shown as solid triangles). CNS = central North Sea; E‐W = east–west

Figure 10

Pseudolikelihood credible intervals for (a) the conditional mean and (b) the conditional standard deviation of the fitted dependence model as a function of distance in kilometres, for conditioning Laplace‐scale value of 5, and the different transects, NNS:N‐W (red), NNS:E‐W (magenta), CNS:N‐S (blue), and CNS:E‐W (cyan). NNS = northern North Sea; N‐S = north–south; E‐W = east–west; CNS = central North Sea