Dealing with Trend

Abstract

Where there is trend, i.e. smooth variation in space, also known as drift, the experimental variogram of the observations is no longer a function solely of a random variable. The assumption of stationarity no longer holds. An experimental variogram that increases with ever increasing gradient as the lag distance increases is usually symptomatic of trend. In these circumstances the process should be modelled as a combination of a deterministic trend plus spatially correlated random residuals from the trend. Estimation of the trend by ordinary least squares regression and a separate analysis of the residuals lead to bias in the variogram. Best practice is to estimate the trend and the parameters of the variogram by residual maximum likelihood (REML). Once this has been achieved, one can use Matheron’s universal kriging for prediction. The technique embodies simple functions of the coordinates that take the trend of given order into account. The methods are illustrated with an example of trend in the soil’s sand content of a field.