Estimation Of Non-Stationary Random Fields Through Space Deformation
Résumé
Stationary random functions have been successfully applied in geostatistical applications for several decades. In some instances, the assumption of a homogeneous spatial dependence structure across the entire domain of interest proves untenable. A useful approach for modelling and estimating non-stationary spatial dependence structure is considered. This consists of transforming a non-stationary random field to a stationary and isotropic one via a bijective bi-continuous deformation of the index space. So far, this approach has been sucessfully applied in the context of data from several independent realizations of a random field. In this communication, we propose an approach for non-stationary geostatistical modelling using space deformation in the context of a single realization with possibly irregularly spaced data. The estimation method is based on a non-stationary variogram estimator which serves as dissimilarity measure between two locations in the geographical space. The proposed procedure combines aspects of kernel smoothing, non-metric multi-dimensional scaling and thin-plate spline radial basis functions to transform the original non-stationary random field onto a new deformed space where it is stationary and isotropic. Standard techniques for prediction and simulation can be applied in the deformed space. The predicted and simulated results are then mapped back into the original space. On a simulated data, the method is able to retrieve the true deformation. A comparison scheme of ordinary kriging under stationary and non-stationary assumptions demonstrates that the proposed approach has better prediction performance on both simulated and real datasets. Finally, we also show that the method provides a tool for the exploratory analysis of non-stationarity.