Modelling non-stationary variance of soil properties by tempering an empirical spectrum

Geostatistical prediction of soil properties depends on the assumption of stationarity in the covariance, but this is often implausible, which implies that predictions may be suboptimal, and their computed variances will be unreliable. There have been attempts in the statistical literature to tackle this problem, and in this paper we discuss, develop and evaluate one approach. This is spectral tempering, in which we model non-stationary variances by spatially adapting the spectrum of an underlying stationary process. This paper describes spectral tempering from an empirical basis consisting of eigenvectors and eigenvalues of an initial covariance matrix, with modifications that allow us to adapt spatially the correlation properties, variance of the spatially-correlated component, and nugget variance; and to adapt these independently. The method can be applied to spatial data from any number of dimensions, from one or more realizations, and with locations in any arrangement, provided some initial covariance matrix (typically stationary) can be estimated. Spectral tempering is applied to a case study on nitrous oxide emissions and gravimetric water content, reserving half of the data for validation. Comparison with the optimal stationary model shows that kriged predictions are little affected, but prediction error variances (kriging variances) are more reliable under the non-stationary model.