On the geostatistical characterization of hierarchical media

The subsurface consists of porous and fractured materials exhibiting a hierarchical geologic structure, which gives rise to systematic and random spatial and directional variations in hydraulic and transport properties on a multiplicity of scales. Traditional geostatistical moment analysis allows one to infer the spatial covariance structure of such hierarchical, multiscale geologic materials on the basis of numerous measurements on a given support scale across a domain or window'' of a given length scale. The resultant sample variogram often appears to fit a stationary variogram model with constant variance (sill) and integral (spatial correlation) scale. In fact, some authors, who recognize that hierarchical sedimentary architecture and associated log hydraulic conductivity fields tend to be nonstationary, nevertheless associate them with stationary exponential-like'' transition probabilities and variograms, respectively, the latter being a consequence of the former. We propose that (1) the apparent ability of stationary spatial statistics to characterize the covariance structure of nonstationary hierarchical media is an artifact stemming from the finite size of the windows within which geologic and hydrologic variables are ubiquitously sampled, and (2) the artifact is eliminated upon characterizing the covariance structure of such media with the aid of truncated power variograms, which represent stationary random fields obtained upon sampling a nonstationary fractal over finite windows. To support our opinion, we note that truncated power variograms arise formally when a hierarchical medium is sampled jointly across all geologic categories and scales within a window; cite direct evidence that geostatistical parameters (variance and integral scale) inferred on the basis of traditional variograms vary systematically with support and window scales; demonstrate the ability of truncated power models to capture these variations in terms of a few scaling parameters; show that exponential and truncated power variograms are often difficult to distinguish from each other, which helps explain why hierarchical data may appear to fit the former; note that truncated power models 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 these models allow conditioning the spatial statistics of such fields on multiscale measurements via cokriging; and illustrate these capabilities on multiscale hydraulic data from an unconfined aquifer near Tubingen, Germany.