Kriging a soil variable with a simple nonstationary variance model

Abstract

Kriging is a standard tool in the environmental sciences for spatial prediction from limited sample data, subject to the assumption of intrinsic stationarity, made about the underlying spatially correlated random function. It is generally well understood how the assumption of stationarity in the mean can be relaxed within the linear mixed model framework, using residual maximum likelihood to estimate variance parameters for the random effects. The Best Linear Unbiased Predictor (BLUP) is equivalent to the kriging predictor in these circumstances. However, nonstationarity in the variance is a harder problem to solve. Stationarity assumptions are necessary if the spatial covariance of a random process is to be estimated from the single realization which nature provides. However, they are not always plausible for variables arising from processes in complex landscapes across contrasting topography and geology. This article shows how a relatively simple extension of the random effects variance model in a Linear Mixed Model (LMM) for the slope of the soil surface across 10 km in western England allows nonstationarity in the variance to be modeled. Tests on the log-likelihood ratio provide evidence in favor of the nonstationary model, and the results of prediction at validation sites show that it characterizes the uncertainty of these predictions better than does a stationary equivalent. More complex models of nonstationary covariance may sometimes be needed, but the case study reported here shows that a relatively minor relaxation of the stationarity assumptions can lead to improved spatial modeling and prediction of a quite complex environmental variable.