A linear mixed model, with non-stationary mean and covariance, for soil potassium based on gamma radiometry

摘要

In this paper we present a linear mixed model for the potassium content ofsoil across a large region of eastern England in which the mean is modelledas a linear function of the passive gamma-ray emissions of the earth surfacein the energy interval commonly associated with potassium decay.Non-stationary models are proposed for the random effect, which is thevariation not captured by this regression. Specifically, we assume that thelocal spectrum of the standardized random effect can be obtained by temperinga common (stationary) spectrum, that is to say raising its values to a power,the tempering parameter, which is itself modelled as a linear function of theradiometric data. This allows the "smoothness" of the random effect to varylocally. In addition the local spatially correlated variance and "nugget"variance (apparently uncorrelated given the resolution of the sampling) canalso be modelled as a function of the radiometric data. Using the radiometricsignal as a covariate gave some improvement in the precision of predictionsof soil potassium at validation sites. In addition, there was evidence thatnon-stationary models for the random effect fitted the data better thanstationary models, and this difference was statistically significant.Non-stationary models also appeared to describe the error variance ofpredictions at the validation sites better. Further work is needed onselection among alternative non-stationary models, since simple proceduresused here, based on comparing log-likelihood ratios of nested models and theAkaike information criterion for non-nested models, did not identify themodel which gave the best account of the prediction error variances atvalidation sites.