The multivariate Gaussian tail model: an application to oceanographic data Paola Bortot University of Padova, Italy and Stuart Coles and Jonathan Tawn Lancaster University, UK [Received May 1998. Final revision July 1999] Summary. Optimal design of sea-walls requires the extreme value analysis of a variety of oceanographic data. Asymptotic arguments suggest the use of multivariate extreme value models, but empirical studies based on data from several UK locations have revealed an inadequacy of this class for modelling the types of dependence that are often encountered in such data. This paper develops a speci®c model based on the marginal transformation of the tail of a multivariate Gaussian distribution and examines its utility in overcoming the limitations that are encountered with the current methodology. Diagnostics for the model are developed and the robustness of the model is demonstrated through a simulation study. Our analysis focuses on extreme sea-levels at Newlyn, a port in south-west England, for which previous studies had given con¯icting estimates of the probability of ¯ooding. The novel diagnostics suggest that this discrepancy may be due to the weak dependence at extreme levels between wave periods and both wave heights and still water levels. The multivariate Gaussian tail model is shown to resolve the con¯ict and to offer a convincing description of the extremal sea-state process at Newlyn. Keywords: Asymptotic independence; Extreme value theory; Gaussian distribution; Joint probabilities method; Multivariate extreme value distribution; Oceanography; Structure variable method; Threshold models 1. Introduction The sea-level is a complex dynamic process composed of a superposition of surface waves and the still water level, which itself is the composition of a tidal process that is deterministic in both time and space, and a surge process which, being meteorologically driven, is stochastic. Surface waves are also induced by meteorological conditions and, though complex, they can usually be summarized accurately by their height and period. The commonality of the driving mechanism induces a dependence between the surge, wave height and wave period processes. Moreover, the three processes are stochastically varying on dierent temporal and spatial scales, so individual analyses will require a consideration of the local sea-state behaviour. An extra complication is that variables are generally measured oshore, whereas applications, requiring the estimation of sea-wall overtopping rates for example, rely on estimates of the onshore climate. Deterministic hydrodynamical models are usually found to be suciently accurate to quantify this transfer process. Address for correspondence: Jonathan Tawn, Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK. E-mail: j.tawn@lancaster.ac.uk & 2000 Royal Statistical Society 0035±9254/00/49031 Appl. Statist. (2000) 49, Part 1, pp. 31±49