# Deformed SPDE models with an application to spatial modeling of significant wave height

## TL;DR

This paper introduces a novel non-stationary Gaussian random field model combining SPDE and deformation methods, enabling better spatial modeling of significant wave height for maritime risk assessment.

## Contribution

It develops a new non-stationary SPDE-based model that allows independent control of correlation range and variance, applied to ocean wave data.

## Key findings

- Model accurately fits significant wave height data
- Allows flexible non-stationary parameter control
- Provides reliable wave exceedance probability estimates

## Abstract

A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field becomes non-stationary. We show that if the stationary field is a Mat'ern field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method's more intuitive parameterisation of non-stationarity. In particular it allows for independent control over the non-stationary practical correlation range and the variance, which has not been possible with previously proposed non-stationary SPDE models. The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model parameters are estimated to data from the north Atlantic using a maximum likelihood approach. The fitted model is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicating that the model could be used for route optimization in naval logistics.