Why Did Sudicky (1986) Find an Exponential-Like Spatial Correlation Structure for Hydraulic Conductivity at the Borden Research Site?

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The exponential-like spatial correlation structure for hydraulic conductivity that Sudicky (1986) found at the Borden research site arises from hierarchical sedimentary architecture. The sediments can be described in terms of a hierarchy of stratal unit types wherein larger-scale unit types (hierarchical level II) are composed of smaller-scale unit types (hierarchical level I). There is a hierarchy of permeability populations corresponding to this stratal hierarchy. The shape and effective range of the composite semivariogram is mostly defined by how the proportion of lag transitions crossing these unit types grows as a function of lag distance. Lag transitions across level I units but within the same level II unit type and transitions across level II unit types contribute about equally in defining the composite sample semivariogram in the vertical direction. These two types of cross transitions have shorter and longer length scales, respectively. The composite semivariogram reflects additive contributions from the two length scales. The proportion of cross-transition lags grows with an exponential-like curve because the unit types at either level reoccur with a relatively high variability in length. The shape and range of the semivariogram are modeled from knowing only the univariate statistics for length among the unit types, and the sill is modeled from knowing univariate statistics for hydraulic conductivity populations among the unit types. Understanding the relationship between the semivariogram structure and these quantifiable attributes of the hierarchical stratal architecture helps in reducing the equivocal aspects of choosing a semivariogram model. It also helps in identifying bias in sample semivariograms and in assessing stationarity of second-order spatial bivariate statistics.