On the geostatistical characterization of hierarchical media

The subsurface exhibits hierarchical structures with multiscale hydraulic and transport properties. The structure and properties are often well characterized by stationary variograms. We propose that the latter is an artifact of sampling over finite windows, which disappears when one uses truncated power variograms (TPVs). We cite evidence that parameters of traditional stationary variograms vary with support and window scales; demonstrate the ability of TPVs to capture these scale variations; show that stationary variograms are often difficult to distinguish from TPVs; note that TPVs are unique in their ability to represent multiscale random fields having either Gaussian or heavy-tailed symmetric Levy stable probability distributions; detail the way in which TPVs allow conditioning on multiscale measurements via co-kriging; and illustrate these capabilities on multiscale hydraulic data from Tübingen, Germany.