Correlation Functions of Random Media

Summary— 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 and reviewed in this paper, together with an 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 Kármán, self-affine, Kummer), are then derived and briefly discussed. The corresponding elliptically anisotropic correlation functions can simply be obtained by linear coordinate transforms.

[1]  Robert W. Clayton,et al.  Finite difference simulations of seismic scattering: Implications for the propagation of short‐period seismic waves in the crust and models of crustal heterogeneity , 1986 .

[2]  Paul S. Addison,et al.  Fractals and Chaos: An Illustrated Course , 1997 .

[3]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[4]  George Papanicolaou,et al.  Transport equations for elastic and other waves in random media , 1996 .

[5]  S. Shapiro,et al.  The effect of random isotropic inhomogeneities on the phase velocity of seismic waves , 1996 .

[6]  H. Maurer,et al.  Stochastic regularization: Smoothness or similarity? , 1998 .

[7]  Mj Martin Bastiaans.  Signal Description By Means Of A Local Frequency Spectrum , 1984, Other Conferences.

[8]  D. Turcotte.  Fractals in geology and geophysics , 2009, Encyclopedia of Complexity and Systems Science.

[9]  Walter Schottky,et al.  Small-Shot Effect and Flicker Effect , 1926 .

[10]  M. Korn.  Seismic waves in random media , 1993 .

[11]  Richard A. Silverman,et al.  Wave Propagation in a Random Medium , 1960 .

[12]  Mark Pilkington,et al.  Stochastic inversion for scaling geology , 1990 .

[13]  Thomas H. Jordan,et al.  Stochastic Modeling of Seafloor Morphology: Inversion of Sea Beam Data for Second-Order Statistics , 1988 .

[14]  S. Flatté,et al.  Small-scale structure in the lithosphere and asthenosphere deduced from arrival time and amplitude fluctuations at NORSAR , 1988 .

[15]  K. Aki,et al.  Elastic wave scattering by a random medium and the small‐scale inhomogeneities in the lithosphere , 1985 .

[16]  Application of the Medium Covariance Functions to Travel-time Tomography , 2002 .

[17]  Wu Ru-Shan.  Attenuation of short period seismic waves due to scattering , 1982 .

[18]  S. Shapiro,et al.  Most probable seismic pulses in single realizations of two‐ and three‐dimensional random media , 2001 .

[19]  Ru-Shan Wu,et al.  The perturbation method in elastic wave scattering , 1989 .

[20]  Transformations in optical signal processing , 1984 .

[21]  S. Shapiro,et al.  Seismic Attenuation By Scattering: Theory and Numerical Results , 1993 .

[22]  David Crossley,et al.  Fractal Velocity Models in Refraction Seismology , 1989 .

[23]  Michael Fehler,et al.  Seismic Wave Propagation and Scattering in the Heterogeneous Earth , 2012 .