Modelling non-stationary spatial covariance structure from space-time monitoring data

Accurate interpolation of soil and climate variables at fine spatial scales is necessary for precise field management. Interpolation is needed to produce the input variables necessary for crop modelling. It is also important when deciding on regulations to limit environmental impacts from processes such as nitrate leaching. Non-stationarity may arise due to many factors, including differences in soil type, or heterogeneity in chemical concentrations. Many geostatistical methods make stationarity assumptions. Substantial improvements in interpolation or in the estimation of standard errors may be obtained by using non-stationary models of spatial covariances. This paper presents recent methodological developments for an approach to modelling non-stationary spatial covariance structure through deformations of the geographic coordinate system. This approach was first introduced by Sampson & Guttorp, although the estimation approach is updated in more recent papers. They compute a deformation of the geographic plane so that the spatial covariance structure can be considered stationary in terms of a new spatial coordinate system. This provides a non-stationary model for the spatial covariances between sampled locations and prediction locations. In this paper, we present a cross-validation procedure to avoid over-fitting of the sample dispersions. Results concerning the variability of the spatial covariance estimates are also presented. An example of the modelling of the spatial correlation field of rainfall at small regional scale is presented. Other directions in methodological development, including modelling temporally varying spatial correlation, and approaches to model temporal and spatial correlation are mentioned. Future directions for methodological development are indicated, including the modelling of multivariate processes and the use of external spatially dense covariables. Such covariates are frequently available in precision agriculture.