Spectral tempering to model non-stationary variation of soil properties: Sensitivity to the initial stationary model

Spectral tempering is a method to model non-stationary variance structure in geostatistical models. It proceeds by spatially adapting an empirical spectrum that is computed from an initial stationary variance model applied to a set of basis locations. In this study we examined the sensitivity of the method to the choice of initial stationary variance model. We computed a profile residual log-likelihood function for simulated non-stationary data sets. This showed the residual log-likelihood for the best-fitting models, conditional on various initial stationary parameters of the spatial correlation. These profile likelihood surfaces were commonly, although not exclusively, somewhat ‘flat’ which shows that different parameters for the stationary model could provide a starting point from which to obtain non-stationary models that fitted the data well. However, the initial best-fitting stationary model was often not a good starting point for a non-stationary model. When we computed the profile residual log-likelihood for a real data set on the soil we obtained similar results. There were differences among non-stationary models with respect to both their residual log-likelihood and the success with which prediction error variances computed from them quantified the uncertainty of predictions at validation sites. However, these differences were small by comparison to the general advantage of non-stationary models over the stationary alternative. We recommend that the initial correlation parameters be estimated as part of the tempering process, possibly by screening a grid of trial combinations.

Research Highlights
► The best stationary model is not generally the best initial model for tempering.
► The profile likelihood on initial model parameters is often flat.
► Profiling the likelihood allows good initial models to be selected.
► The performance of a covariance model for prediction reflects its likelihood.
► The non-stationary models all outperformed the stationary alternative.