Modeling 3-D anisotropic fractal media

von Karman correlation functions

The three-dimensional anisotropic von Karman function is given by (Goff and Jordan, 1988):

$$C(r)=\frac{4\pi{\nu}H^2r^{\nu}K_{\nu}(r)}{K_{\nu}(0)} \$$ (4)

and its three-dimensional Fourier transform is:

$$P(k)=\frac{4\pi{\nu}H^2}{K_{\nu}(0)}\frac{a_x^2+a_y^2+a_z^2}{{(1+k^2)}^{\nu+\frac {3} {2}}} \$$ (5)

where $r=\sqrt{\frac{x^2}{a_x^2}+\frac{y^2}{a_y^2}+\frac{z^2}{a_z^2}}$, $k=\sqrt{k_x^2a_x^2+k_y^2a_y^2+k_z^2a_z^2}$; $a_x$, $a_y$ and $a_z$ are the characteristic scales of the medium along the 3-dimensions and $k_x$, $k_y$ and $k_z$ are the wavenumber components. $K_\nu$ is the modified Bessel function of order $\nu$, where $0.0<\nu<1.0$ is the Hurst number (Mandelbrot, 1985,1983). The fractal dimension of a stochastic field characterized by a von Karman autocorrelation is given by:

$$D=E+1-\nu \$$ (6)

where $E$ is the Euclidean dimension i.e., $E=3$ for the three-dimensional problem. The special case of $\nu=0.5$ yields to the exponential covariance function that corresponds to a Markov process (Feller, 1971).

$$C(r)=H^2e^{-r} \$$ (7)

whose three-dimensional Fourier transform is given by:

$$P(k)=H^2\frac{a_x^2+a_y^2+a_z^2}{{(1+k^2)}^{2}} \$$ (8)

karman
Figure 1. Comparison of 1-dimensional isotropic von Karman autocorrelation functions for varying hurst number, $\nu$.

Decreasing the Hurst number, $\nu$, increases the roughness of the medium. The limiting cases of unity and zero correspond to a smooth Euclidean random field and a space-filling field respectively.

Figure 1 shows the one-dimensional isotropic von Karman correlation function plotted for different values of $\nu$. The functions have exponential behavior but different decay rates. The higher the slope, the rougher the medium (i.e., the lower is $\nu$). The exponential behavior is explained by the modified Bessel functions $K_\nu(x)$ which in the region $x \gg \nu$ behave as

$$K_\nu(x) \approx \frac {1} {\sqrt{2\pi x}} \exp{(x)} \$$ (9)

For comparison of the results, I also include the anisotropic Gaussian autocovariance function, which in 3-D has the familiar form:

$$C(r)=H^2e^{-r^2} \$$ (10)

and its 3-dimensional Fourier transform is given by:

$$P(k)=\frac{a_xa_ya_z}{2}H^2e^{\frac{k^2}{4}} \$$ (11)

Modeling 3-D anisotropic fractal media