Modeling 3-D anisotropic fractal media

Second-order Statistics

I restricted this research to the study of second-order statistics of random fields. This means the study of random media characterized by Gaussian distributions, where a Gaussian process is completely specified by its first- and second-order moments. Furthermore if I define the field $h(x)$ to be a zero mean process:

$$<h(x)>=\int_{-\infty}^{+\infty}h(x)P(h(x))dh=0 \$$ (2)

then $h(x)$ is fully described by its two-point moment that is its autocovariance function which we write as a function of the correlation function:

$$C_{hh}(r)=E[h(x)h(x+r)]=H^2\rho_{hh}(r) \$$ (3)

where $P(h(x))$ is the probability density function of $h(x)$, $r$ is the lag vector, $E$ is the expected value, $H^2$ is the variance (i.e. $C_{hh}(0)$) and $\rho_{hh}$ is the three-dimensional correlation function. Equation (3) shows that the random medium can be adequately specified by its autocorrelation function. More generally, an anisotropic random field can be described by a monotonically decaying autocorrelation function whose rate of decay depends on direction. The roughness of the medium is function of the decay rate of the correlation. The Fourier transform of the autocorrelation is the power spectrum of the field (Bracewell, 1978). Three types of correlation functions were commonly used in the field of seismic modeling: the Gaussian, exponential and von Karman functions. These special functions are described by analytic expressions of their autocorrelations and Fourier transforms.

Two-dimensional cases have been studied for some time(Wu and Aki, 1985; Holliger et al., 1993; Goff and Jordan, 1988; Frankel and Clayton, 1986; Holliger and Levander, 1992). Within the last several years, computer capacity and speed have grown rapidly. It is now feasible to extend our models and simulations to the three-dimensional case.

Modeling 3-D anisotropic fractal media