Modeling 3-D anisotropic fractal media

RANDOM FIELDS

A stochastic model is constructed for the properties of the random medium. We first construct a distribution function P(x) for the properties of the medium h(x). From such a probability function, we can recover the statistical properties of the distribution (i.e., mean, variance , etc.) through its N-point statistical moments (Goff and Jordan, 1988).

$$C_N(x_1,x_2,...,x_N) = <h(x_1)...h(x_N)>$$

$$= \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}h_1...h_NP( h_1, ...,h_N)dh_1...dh_N$$ (1)

where $h_N=h(x_N)$. The key assumption of spatial homogeneity (stationarity) means that the N-point moments are taken to depend only on the vector joining these points and not on their absolute positions. These moments describe the magnitude and smoothness of the fluctuations of h(x).