Extreme environmental events endanger human life and cause serious damage to property and infrastructure. For example, Storm Desmond (2015) caused approximately £500m of damage in Lancashire and Cumbria, UK from high winds and flooding, while Storm Britta (2006) damaged shipping vessels and offshore structures in the southern North Sea, and led to coastal flooding. Estimating the probability of the occurrence of such events is key in designing structures and infrastructure that are able to withstand their impacts.

Due to the rarity of these events, extreme value theory techniques are used for
inference. This thesis focusses on developing novel spatial extreme value methods motivated by applications to significant wave height in the North Sea and north Atlantic, and extreme precipitation for the Netherlands.

We develop methodology for analysing the dependence structure of significant
wave height by utilising spatial conditional extreme value methods. Since the dependence structure of extremes between locations is likely to be complicated, with contributing factors including distance and covariates, we model dependence flexibly; otherwise, the incorrect assumption on the dependence between sites may lead to inaccurate estimation of the probabilities of spatial extreme events occurring. Existing methods for spatial extremes typically assume a particular form of extremal
dependence termed asymptotic dependence, and often have intractable forms for describing the dependence of joint events over large numbers of locations. The model developed here overcomes these deficiencies. Moreover, the estimation of joint probabilities across sites under both asymptotic independence and asymptotic dependence, the two limiting extremal dependence classes, is possible with our model; this is not the case with other methods.

We propose a method for the estimation of marginal extreme precipitation quantiles, utilising a Bayesian spatio-temporal hierarchical model. Our model parameters incorporate an autoregressive prior distribution, and use spatial interpolation to pool information on model parameters across neighbouring sites.