Modeling 3-D anisotropic fractal media

Numerical Implementation

The generation of synthetic random media is done in the wave number domain. First, we compute the power spectrum of the field, i.e, the Fourier spectrum of the autocorrelation function. Then we compute the Fourier spectrum by multiplying the square root of the power spectrum by a random phase factor $e^{2\pi\eta}$ where $\eta$ is a uniform deviate that lies in the interval [0,1). In a final step we apply an inverse fast Fourier transform to obtain the spatial domain representation of the random medium. The numerical implementation of the method is very straightforward, although special care is required to handle D.C. and Nyquist wavenumbers.
Algorithms are similar for the one-, two- and three-dimensional problem although if we do not care about computer expenses, 1- and 2-D random sequences can be simply extracted as arrays or sections from 3-D simulations.

gauss
Figure 2. Synthetic random field with anisotropic Gaussian autocorrelation function; $a_x=15$, $a_y=25$, $a_z=35$.

expo
Figure 3. Synthetic random field with anisotropic exponential autocorrelation function; $a_x=15$, $a_y=25$, $a_z=35$.

fractal
Figure 4. Synthetic random field with anisotropic von Karman autocorrelation function; $a_x=15$, $a_y=25$, $a_z=35$.

Modeling 3-D anisotropic fractal media