  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.
