Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.