Modeling 3-D anisotropic fractal media |

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

where , ; , and are the characteristic scales of the medium along the 3-dimensions and , and are the wavenumber components. is the modified Bessel function of order , where is the Hurst number (Mandelbrot, 1985,1983). The fractal dimension of a stochastic field characterized by a von Karman autocorrelation is given by:

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

whose three-dimensional Fourier transform is given by:

karman
Comparison of 1-dimensional isotropic von Karman autocorrelation functions for varying hurst number, .
Figure 1. |
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Figure 1 shows the one-dimensional isotropic von Karman correlation
function plotted
for different values of . The functions have exponential behavior
but different decay rates.
The higher the slope, the rougher the medium (i.e., the lower is ).
The exponential behavior is explained by the modified Bessel functions
which in the region behave as

and its 3-dimensional Fourier transform is given by:

Modeling 3-D anisotropic fractal media |

2013-03-03