Multivariate Oceanographic Extremes in Time and Space

Tendijck, Stan and Tawn, Jonathan and Jonathan, Philip and Randell, David (2023) Multivariate Oceanographic Extremes in Time and Space. PhD thesis, Lancaster University.

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Abstract

This thesis contributes to the field of multivariate extremes. The work has been motivated by an application in oceanography to assess safety and reliability of offshore structures, vessels, and platforms, but we remark that the contents of the thesis are designed to be more generally applicable to other environmental or even financial applications. The model that forms the foundation for this work is the conditional extremes model Heffernan and Tawn (2004) in which the extremes of a multivariate random variable are modelled by conditioning on one of the variables being extreme. This model is called the Heffernan-Tawn model and it is one of the most flexible models for modelling extremes of multivariate random variables. We design a mixture model for significant wave height conditional on large wave periods in the North Sea by extending the Heffernan-Tawn model. Our extension helps with understanding the distribution of responses to offshore facilities or vessels that are dominated by resonance frequencies. A mixture model is necessary here because two types of waves are recorded in the North Sea: swell waves and wind sea waves; both of these can be associated to large wave periods. We calculate extremal properties of a model that is widely used by engineers in oceanographical applications. This model has a simple interpretation but is not motivated by extreme value theory. This led to the development of a mathematical toolset to calculate extremal characteristics for conditional models in general. This in turn allowed us to prove a new restriction on the space of the Heffernan-Tawn model parameters. Finally, we model the joint temporal evolution of oceanographic variables using an extension of the Heffernan-Tawn model to increase the understanding of what a $10,000$ year event would look like. This led to a generic formulation of a multivariate extremes temporal model.