Bayesian inference for non-stationary spatial covariance structure via spatial deformations

In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, i.e. the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin. However, in environmental problems, such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data. The paper proposes a Bayesian model to address the anisotropy problem. Following Sampson and Guttorp, we define the correlation function of the spatial process by reference to a latent space, denoted by D, where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by G. We adopt a Bayesian approach in which the mapping between G and D is represented by an unknown function d(.). A Gaussian process prior distribution is defined for d(.). Unlike the Sampson-Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Markov chain Monte Carlo methods are used to obtain samples from the posterior distributions. Two examples are discussed: a simulated data set and the solar radiation data set that also was analysed by Sampson and Guttorp.