The geostatistical literature uses many different terms for what are essentially the same or at least very similar techniques. This confuses the users and distracts them from using the right technique for their mapping projects. In fact, both universal kriging, kriging with external drift, and regression-kriging are basically the same technique. Matheron (1969) originally termed the technique
Le krigeage universel, however, the technique was intended as a generalized case of kriging where the trend is modelled as a function of coordinates. Thus, many authors reserve the term
universal kriging (UK) for the case when only the coordinates are used as predictors. If the deterministic part of variation (
drift) is defined externally as a linear function of some auxiliary variables, rather than the coordinates, the term
kriging with external drift (KED) is preferred (according to Hengl 2007, "About regression-kriging: From equations to case studies"). In the case of UK or KED, the predictions are made as with kriging, with the difference that the covariance matrix of residuals is extended with the auxiliary predictors. However, the drift and residuals can also be estimated separately and then summed. This procedure was suggested by Ahmed et al. (1987) and Odeh et al. (1995) later named it
regression-kriging, while Goovaerts (1997) uses the term
kriging with a trend model to refer to a family of interpolators, and refers to RK as
simple kriging with varying local means. Minasny and McBratney (2007) simply call this technique Empirical Best Linear Unbiased Predictor i.e.
E-BLUP. : \mathbf{C}^\mathtt{KED} = \left[ \begin{array}{ccccccc} C(\mathbf{s}_1 , \mathbf{s}_1) & \cdots & C(\mathbf{s}_1, \mathbf{s}_n ) & 1 & q_1 (\mathbf{s}_1 ) & \cdots & q_p (\mathbf{s}_1 ) \\ \vdots & & \vdots & \vdots & \vdots & & \vdots \\ C(\mathbf{s}_n, \mathbf{s}_1 ) & \cdots & C(\mathbf{s}_n ,\mathbf{s}_n ) & 1 & q_1 (\mathbf{s}_n ) & \cdots & q_p (\mathbf{s}_n ) \\ 1 & \cdots & 1 & 0 & 0 & \cdots & 0 \\ q_1 (\mathbf{s}_1 ) & \cdots & q_1 (\mathbf{s}_n ) & 0 & 0 & \cdots & 0 \\ \vdots & & \vdots & \vdots & \vdots & & \vdots \\ q_p (\mathbf{s}_1 ) & \cdots & q_p (\mathbf{s}_n ) & 0 & 0 & \cdots & 0 \end{array} \right] and \mathbf{c}_{\mathbf{0}}^{\mathtt{KED}} like this: : \mathbf{c}_\mathbf{0}^\mathtt{KED} = \left\{ C(\mathbf{s}_0, \mathbf{s}_1 ), \ldots , C(\mathbf{s}_0, \mathbf{s}_n ), q_0 (\mathbf{s}_0 ), q_1 (\mathbf{s}_0 ), \ldots ,q_p (\mathbf{s}_0 ) \right\}^\mathbf{T}; q_0 (\mathbf{s}_0 ) = 1 Hence, KED looks exactly as ordinary kriging, except the covariance matrix/vector are extended with values of auxiliary predictors. Although the KED seems, at first glance, to be computationally more straightforward than RK, the parameters of the
variogram for KED must also be estimated from regression residuals, thus requiring a separate regression modelling step. This regression should be GLS because of the likely spatial correlation between residuals. Note that many analyst use instead the OLS residuals, which may not be too different from the GLS residuals. However, they are not optimal if there is any spatial correlation, and indeed they may be quite different for clustered sample points or if the number of samples is relatively small (\ll 200). A limitation of KED is the instability of the extended matrix in the case that the covariate does not vary smoothly in space. RK has the advantage that it explicitly separates trend estimation from spatial prediction of residuals, allowing the use of arbitrarily-complex forms of regression, rather than the simple linear techniques that can be used with KED. In addition, it allows the separate interpretation of the two interpolated components. The emphasis on regression is important also because fitting of the deterministic part of variation (regression) is often more beneficial for the quality of final maps than fitting of the stochastic part (residuals). == Software to run regression-kriging ==