A new type of stochastic fractal for application in subsurface hydrology

Data suggest that many property distributions [P(x); p = property, x = length] are irregular [dP / dx discontinuous], exhibiting variations on all measurement scales. Logic then leads one to attempt to characterize these distributions through study of their increments or fluctuations [P(x + L) − P(x)], which results naturally in a statistical approach. Increment frequency distributions for a fixed L that fit the Levy/Gaussian family of probability density functions [PDFs] independent of position [statistically stationary in a spatial sense] have well-defined scaling properties characteristic of what are called self-affine stochastic fractals. Data show that measurements in natural systems display some of the scaling characteristics of this PDF family, but do not agree with theory in many important aspects. For example, increment ln(K) distributions show a consistently non-Gaussian shape for smaller values of L and become more Gaussian as L gets large. Particularly for smaller L, PDF tails display an approximate exponential decay and maintain a finite variance. In many cases they appear to be displaying a continuous change from exponential, or even sub-exponential, to Gaussian behavior with increasing L. Data supporting these observations are presented along with explanations of how the fractal-based concepts arise, both physically and mathematically. This is followed by the presentation and initial analysis of what appears to be a new type of stochastic fractal called fractional Laplace motion that displays more of the features observed in natural sedimentary systems.