A GENERAL SPATIAL COVARIANCE MODEL WITH HIERARCHICAL ORGANIZATION, AND APPLICATION TO PERMEABILITY IN CROSS-BEDDED SEDIMENT

We present a general form for any of the common measures of spatial covariance, written to correspond to the hierarchical organization in sedimentary deposits. Here we apply the model to study cross-bedded sedimentary deposits that have antipersistent variation in permeability arising across unit types defined at all hierarchical levels. In such deposits, the correlation (which is negative) across different unit types is predominant in the scaling. At each higher level (larger scale) the cross-transition probabilities for units with greatest relative difference in permeability dictate the shape and range of the spatial covariance. The shape lies somewhere in the evolution between a perfectly periodic piece-wise linear structure and an a-periodic exponential structure, dependant largely on the coefficient of variation in length for the unit types and their pattern. An exponential semivariogram, commonly assumed for permeability in cross-bedded sediments, can be explained by antipersistence arising across unit types having a relatively large coefficient of variation in length.