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. Specifically,
we assume that the local spectrum of the standardized
random effect can be obtained by tempering a 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 the radiometric data. This allows the “smoothness”
of the random effect to vary locally. In addition the local
spatially correlated variance and “nugget” variance (apparently
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 improvement
in the precision of predictions of soil potassium
at validation sites. In addition, there was evidence that nonstationary
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 nonstationary
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 prediction
error variances at validation sites
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