A Generalized Convolution Model and Estimation for Non-stationary Random Fields
Résumé
Standard geostatistical models assume second order stationarity for the underlying random function. They rely on the variogram or covariance 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. In this work, we introduce a new model for non-stationary random functions as a convolution of an orthogonal random measure, with a spatially varying random weighting function. This is a generalization of the common process-convolution approach, which use a non-random weighting function. For a suitable choice of the random weighting function, we derive a class of closed-form non-stationary spatial covariance functions that show locally a stationary behaviour. The parameters of these latter 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 dependence. Performances are assessed on a soil dataset. It is shown in particular that our approach outperforms the stationary approach. It takes into account certain local characteristics of the regionalization that the stationary approach is unable to retrieve. The proposed approach provides a tool for the exploratory analysis of the non-stationarity.