Fred Molz

Clemson University

Fractal Models for Heterogeneous Property Distributions: Some Mathematical Problems

Fractal models of heterogeneous subsurface property distributions, such as hydraulic conductivity (K), may be developed based on the frequency distributions of K increments or log(K) increments. The most common practice in subsurface hydrology has been to use Log(K) increments and a Gaussian distribution as a starting point for developing fractional Gaussian noise (fGn)- or fractional Brownian motion (fBm)-type models for Log(K) distributions. Generalizing to the Levy-stable family of PDF's yields the much more irregular fractals known as fractional Levy noise (fLn) and fractional Levy Motion (fLm), as introduced to subsurface hydrology by Painter and colleagues. All of the above functions are true fractals in that scaling is exhibited for increment distributions over all lags from zero to infinity. However, a recent study published in Science (Avnir et al., 1998) indicates that with few exceptions fractal-like scaling is exhibited in nature over lag-size ranges of approximately two or three orders of magnitude. In the present study, we examine the development of new fractals and empirical scaling models based on PDF'ss that are not members of the Levy-stable family. Most applications of the Levy model require truncation of the tails of the distribution which ultimately become too fat because of the power law die off. Truncation also causes the distribution that is actually used to have a finite variance. Gaussian distributions have tails that decrease too rapidly. Empirical evidence is presented that the Laplace PDF, which has tails that decrease exponentially and a finite variance, is a good model for several log(K) data sets. While an empirical scaling model may be developed based on the Laplace PDF, the distribution is not closed under convolution and so does not produce theoretically a fractal with consistent scaling over an infinite lag range. Nevertheless, we argue that such a model may still be useful since the property distributions themselves don't appear to be true fractals. What seems ubiquitous in nature is scaling over a finite range, and this appears to be reproducible with what we are calling a Laplace scaling model. The Laplace PDF is a limiting member of the Gamma family of PDF's, and this family is closed under convolutions and so can serve as the basis for the development of a set of infinite scaling models. An interesting observation concerning the Gamma family is that its use leads to the development of a class of multifractals rather than monofractals of the Gaussian or Levy type. Examples are presented and the implications of the various scaling models are discussed. In particular, it is noted that the Levy family serves as the basis for development of universal multifractals, and here the infinite moments may have a more natural interpretation.

Back to Colloquium Series