Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability

[1] It has been shown by Neuman [1990], Di Federico and Neuman [1997], and Di Federico et al. [1999] that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or Gaussian variograms of overlapping, homogeneous random fields (modes) having mutually uncorrelated increments and variance proportional to a power 2H of the integral (spatial correlation) scale where H is the Hurst coefficient. Low- and high-frequency cutoffs are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy, created by a hiatus of modes associated with discrete ranges of scales. Alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed (mutually exclusive) materials or categories, each having its own architecture and attributes. The spatial distribution of categories is characterized by indicator random variables and their attributes by random fields. This paper focuses on expressions developed by Lu and Zhang [2002] in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories. Upon rewriting their expressions for statistically homogeneous and anisotropic media, it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space (overlap) in a way that preserves their local architecture, their attributes become mutually uncorrelated. Categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space; the idea is analogous to the well-known dual continuum concept in which two categories, most commonly fractures and porous blocks, are considered to overlap. In reality, the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space. The variogram of an attribute, sampled jointly over overlapping statistically homogeneous and anisotropic categories, is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories, weighted by their volumetric proportions. It is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or Gaussian variograms, such that the product of category probability density function (pdf) and attribute variance is proportional to a power 2H + 1 of the attribute integral scale, they represent a fractal field characterized by a power or truncated power variogram as in the work of Di Federico and Neuman. Any category pdf that satisfies this condition, and any attribute pdf having a variance that satisfies the same condition, are admissible; neither of them need to be Gaussian (to form fractional Brownian motion) or Levy-like (to approximate fractional Levy motion) for the attribute to form a random fractal. Intermediate cutoffs render the latter model discrete, as is that of Lu and Zhang. Hence the two multimodal models are mutually consistent and complementary.