On spatial prediction of soil properties in the presence of a spatial trend: the empirical best linear unbiased predictor (E-BLUP) with REML

Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed-effect model that is some linear function of spatial co-ordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical variograms estimated from the data by method-of-moments will be affected by both the random variation and that variation represented by the fixed effects. The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method-of-moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E-BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML-E-BLUP has not been widely used in soil science, and in most studies reported in the soils literature the variogram is estimated with methods that are seriously biased if the fixed-effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML-E-BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.