Geostatistical mapping of geomorphic variables in the presence of trend

Mapping geomorphic variables geostatistically, specifically by kriging, runs into difficulties when there is trend. The reason is that the variogram required for the kriging must be of residuals from any trend, which in turn cannot be estimated optimally by the usual method of trend surface analysis because the residuals are correlated. The difficulties can be overcome by the use of residual maximum likelihood (REML) to estimate both the trend and the variogram of the residuals simultaneously. We summarize the theory of REML as it applies to kriging in the presence of trend. We present the equations to show how estimates of the trend are combined with kriging of residuals to give empirical best linear unbiased predictions (E-BLUPs). We then apply the method to estimate the height of the sub-Upper-Chalk surface beneath the Chiltern Hills of southeast England from 238 borehole data. The variogram of the REML residuals is substantially different from that computed by ordinary least squares (OLS) analysis. The map of the predicted surface is similar to that made from kriging with the OLS variogram. The variances, however, are substantially larger because (a) they derive from a variogram with a much lager sill and (b) they include the uncertainty of the estimate of the trend. Copyright (c) 2006 John Wiley & Sons, Ltd.