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 of soil across a large region of eastern England in which the mean is modelled as a linear function of the passive gamma-ray emissions of the earth surface in the energy interval commonly associated with potassium decay. Non-stationary models are proposed for the random effect, which is the variation not captured by this regression. Specif-ically, we assume that the local spectrum of the standardized random effect can be obtained by tempering a common (sta-tionary) spectrum, that is to say raising its values to a power, the tempering parameter, which is itself modelled as a linear function of the radiometric data. This allows the "smooth-ness" of the random effect to vary locally. In addition the lo-cal spatially correlated variance and "nugget" variance (ap-parently uncorrelated given the resolution of the sampling) can also be modelled as a function of the radiometric data. Using the radiometric signal as a covariate gave some im-provement in the precision of predictions of soil potassium at validation sites. In addition, there was evidence that non-stationary models for the random effect fitted the data better than stationary models, and this difference was statistically significant. Non-stationary models also appeared to describe the error variance of predictions at the validation sites better. Further work is needed on selection among alternative non-stationary models, since simple procedures used here, based on comparing log-likelihood ratios of nested models and the Akaike information criterion for non-nested models, did not identify the model which gave the best account of the predic-tion error variances at validation sites.