In geophysics, the correlation functions of random media are of principal importance for understanding and inverting the properties of seismic waves propagating in geological structures. Unfortunately, the kinds of correlation functions inappropriate for the description of geological structures are often assumed and applied. The most frequently used types of correlation functions are thus summarized in this paper, together with explanation of the physical meaning of their parameters.
A stationary random medium is assumed to be realized in terms of a white noise filtered by a spectral filter. The spectral filter is considered isotropic, in a simple general form enabling the random media used in geophysics to be specified. The medium correlation functions, corresponding to the individual special cases of the general random medium (Gaussian, exponential, von Karman, self-affine, Kummer), are then derived and briefly discussed. The corresponding elliptically anisotropic correlation functions can simply be obtained by linear coordinate transforms.
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Klimes, L.: Correlation functions of random media. In: Seismic Waves in Complex 3-D Structures, Report 6, pp. 25-40, Dep. Geophys., Charles Univ., Prague, 1997.
Klimes, L.: Correlation functions of random media. In: Seismic Waves in Complex 3-D Structures, Report 11, pp. 91-109, Dep. Geophys., Charles Univ., Prague, 2001 [to appear in Pure and Applied Geophysics, 159 (2002)].