Assessing and modelling extremal dependence in spatial extremes

Offshore structures, such as oil platforms and vessels, must be built such that they can withstand extreme environmental conditions (e.g., high waves and strong winds) that may occur during their lifetime. This means that it is essential to quantify probabilities of the occurrence of such extreme events. However, a difficulty arises in that there are very limited data available at these levels. The statistical field of extreme value theory provides asymptotically motivated models for extreme events, hence allowing extrapolation to very rare events. In addition to the risk to a single site, we are also interested in the joint risk of multiple offshore platforms being affected by the same extreme event. In order to understand joint extremal behaviour for two or more locations, the spatial dependence between the different locations must be considered. Extremal dependence between two locations can be of two types: asymptotic independence (AI) when the extremes at the two sites are unlikely to occur together, and asymptotic dependence (AD) when it is possible for both sites to be affected simultaneously. For finite samples it is often difficult to determine which type of dependence the data are more consistent with. In a large ocean basin it is reasonable to expect both of these features to be present, with some close by locations AD, with the dependence decreasing with distance, and some far apart locations AI. In this thesis we develop new diagnostic tools for distinguishing between AD and AI and illustrate these on North Sea wave height data. We also investigate how extremal dependence changes with direction and find evidence for spatial anisotropy in our data set. The most widely used spatial models assume asymptotic dependence or perfect independence between sites, which is often unrealistic in practice. Models that attempt to capture both AD and AI exist, but they are difficult to implement in practice due to their complexity and they are restricted in the forms of AD and AI they can model. In this thesis we introduce a family of bivariate distributions that exhibits all the required features of short, medium and long range extremal dependence required for pairwise dependence modelling in spatial applications

Assessing extremal dependence of North Sea storm severity

Extreme value theory provides asymptotically motivated methods for modelling the occurrences of extreme values of oceanographic variables, such as significant wave height at a single location. Problems are often multivariate or spatial in nature, where interest lies in the risk associated with joint occurrences of rare events, e.g. the risk to multiple offshore structures from a storm event. There are two different classes of joint tail behaviour that have very different implications: asymptotic independence suggesting that extreme events are unlikely to occur together, and asymptotic dependence implying that extreme events can occur simultaneously. It is vital to have good diagnostics to identify the appropriate dependence class. If variables are asymptotically independent, incorrectly assuming an asymptotically dependent model can lead to overestimation of the joint risk of extreme events, and hence to higher than necessary design costs of offshore structures. We develop improved diagnostics for differentiating between these two classes, which leads to increased confidence in model selection. Application to samples of North Sea sea-state and storm-peak significant wave height suggests that tail dependence changes with wave direction and distance between spatial locations, and that in most cases asymptotic independence seems to be the more appropriate assumption

On the spatial dependence of extreme ocean storm seas

Contemporaneous occurrences of extreme seas at multiple locations in a neighbourhood can cause greater structural reliability and human safety concerns than extremes at a single location. Understanding spatial dependence of extreme seas is important therefore in metocean design, yet has received little rigorous attention in the offshore engineering literature. We characterise the spatial dependence of storm peak significant wave height using three models motivated by max-stable processes for locations in the northern North Sea. Models for marginal extremes per location, and dependence of extremes between locations, are estimated using Bayesian inference with composite spatial likelihoods. We show that, in addition to marginal directional non-stationarity of extreme seas per location, all three models indicate spatial anisotropy in extremal dependence quantified by the spatial covariance matrix of the corresponding max-stable process. Estimates suggest that extreme seas show greater extremal dependence from West to East than from North to South