A Generalized Convolution Model and Estimation for Non-stationary Random Fields
Résumé
Standard geostatistical models assume second order stationarity for the underlying random field. They rely on the variogram or covariogram to account for the spatial dependence of the observed data. In some instances, there is little reason to expect the spatial dependence structure to be stationary over the whole region of interest. We introduce a new model for non-stationary random fields as a convolution of an orthogonal random measure, with a spatially varying random weight function. This is a generalization of the common process-convolution approach, which use a non-random weight function. For a suitable choice of the random weight function, we derive a class of closed-form non-stationary spatial covariance functions that show locally a stationary behaviour. The parameters of these resulting non-stationary covariance functions are allowed to vary with location, yielding local variances, ranges, geometric anisotropies and smoothnesses. Under a single realization and local stationarity framework, we develop a weighted local variogram method-of-moments approach in combination with kernel smoothing technique to estimate efficiently local parameters that govern the spatial variability. A comparison scheme of ordinary kriging under stationary and non-stationary assumptions indicates that the proposed approach shows better prediction performances on a real dataset. Finally, we also show how carry out simulations of non-stationary random fields using a propagative version of the Gibbs sampler.