  Gaussian random fields (GRFs) constitute an important part of spatial
modelling, but can be computationally infeasible for general covariance
structures. An efficient approach is to specify GRFs via stochastic partial
differential equations (SPDEs) and derive Gaussian Markov random field (GMRF)
approximations of the solutions. We consider the construction of a class of
non-stationary GRFs with varying local anisotropy, where the local anisotropy
is introduced by allowing the coefficients in the SPDE to vary with position.
This is done by using a form of diffusion equation driven by Gaussian white
noise with a spatially varying diffusion matrix. This allows for the
introduction of parameters that control the GRF by parametrizing the diffusion
matrix. These parameters and the GRF may be considered to be part of a
hierarchical model and the parameters estimated in a Bayesian framework. The
results show that the use of an SPDE with non-constant coefficients is a
promising way of creating non-stationary spatial GMRFs that allow for physical
interpretability of the parameters, although there are several remaining
challenges that would need to be solved before these models can be put to
general practical use.
