Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

Let us consider the spatial fluctuations of seismic velocities to be small and to
constitute a second-order stochastic process.
We describe the fluctuations by using different realizations of the random function
with the expectation value
and with the spatial covariance
depending on the relative distance defined by

where is the standard deviation and is the spatial autocorrelation function with . The energy spectrum of the fluctuations in dimensions () is related to the autocorrelation by the Wiener-Khintchine theorem (Born and Wolf, 1964):

where is the spatial wave vector and is the Fourier transform of . The energy spectrum in equation 2 can be simplified, for an isotropic correlation function, to

where . The von Kármán autocorrelation function describes a self-affine medium relevant for geological structures (Goff and Jordan, 1988; Klimes, 2002; Dolan et al., 1998; Goff and Holliger, 2003; Sato and Fehler, 1998; Holliger and Levander, 1992). This function was initially derived by von Kármán (1948) while studying the velocity field in a turbulent fluid and has been used to describe heterogeneous media (Frankel and Clayton, 1986; Tatarski, 1961). The Fourier transform of was given by Lord (1954). The statistical autocorrelation and the energy spectrum in the Fourier domain are

where , is the modified Bessel function of the second kind with order , and is the Gamma function. Parameters describing the heterogeneities are characteristic distance , below which the distribution is fractal, and exponent , characterizing the roughness of the medium. We use the energy spectrum in equation 7 with to analyze sonic logs and with to predict 3D scattering attenuation.

- Fractal statistics
- Synthetic realizations
- Nonlinear parameter estimation on sonic well logs
- Fractal heterogeneities and cycles in sediments

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

2013-07-26