2-D random media with ellipsoidal autocorrelation functions

Abstract

The integration of surface seismic data with borehole seismic data and well‐log data requires a model of the earth which can explain all these measurements. We have chosen a model that consists of large and small scale inhomogeneities: the large scale inhomogeneities are the mean characteristics of the earth while the small scale inhomogeneities are fluctuations from these mean values. In this paper, we consider a two‐dimensional (2-D) model where the large scale inhomogeneities are represented by a homogeneous medium and small scale inhomogeneities are randomly distributed inside the homogeneous medium. The random distribution is characterized by an ellipsoidal autocorrelation function in the medium properties. The ellipsoidal autocorrelation function allows the parameterization of small scale inhomogeneities by two independent autocorrelation lengths a and b in the horizontal and the vertical Cartesian directions, respectively. Thus we can describe media in which the inhomogeneities are isotropic (a = b), or elongated in a direction parallel to either of the two Cartesian directions (a > b, a < b), or even taken to infinite extent in either dimension (e.g., a = infinity, b = finite: a 1-D medium) by the appropriate choice of the autocorrelation lengths. We also examine the response of seismic waves to this form of inhomogeneity. To do this in an accurate way, we used the finite‐difference technique to simulate seismic waves. Special care is taken to minimize errors due to grid dispersion and grid anisotropy. The source‐receiver configuration consists of receivers distributed along a quarter of a circle centered at the source point, so that the angle between the source‐receiver direction and the vertical Cartesian direction varies from 0 to 90 degrees. Pulse broadening, coda, and anisotropy (transverse isotropy) due to small scale inhomogeneities are clearly apparent in the synthetic seismograms. These properties can be recast as functions of the aspect ratio [Formula: see text] of the medium, especially the anisotropy and coda. For media with zero aspect ratio (1-D media), the coda energy is dominant at large angles. The coda energy gradually becomes uniformly distributed with respect to angle as the aspect ratio increases to unity. Our numerical results also suggest that, for small values of aspect ratio, the anisotropic behavior (i.e., the variations of pulse arrival times with angle) of the 2-D random media is similar to that of a 1-D random medium. The arrival times agree with the effective medium theory. As the aspect ratio increases to unity, the variations of pulse arrival times with angle gradually become isotropic. To retain the anisotropic behavior beyond the geometrical critical angle, we have used a low‐frequency pulse with a nonzero dc component.

Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

“…Frankel and Clayton (1986) investigated the influences of scattering on seismograms of local earthquakes using 2D finite-difference (FD) modeling. Ikelle et al (1993) used 2D FD modeling to investigate the influences of anisotropy in the heterogeneity of random media on pulse broadening, coda formation, and apparent anisotropy. They chose a scale appropriate for petroleum exploration and found that pulse broadening is relatively independent of the degree of anisotropy in the heterogeneity but that coda formation and apparent anisotropy were greatly influenced by the amount of anisotropy in the heterogeneity.…”

Numerical Simulations of the Lunar Penetrating Radar and Investigations of the Geological Structures of the Lunar Regolith Layer at the Chang’E 3 Landing Site

“…These small-scale media distributions can be considered as random processes to study the characteristics of their permittivity. The technique of random medium modeling is applied to the seismic numerical simulation [9][10][11].…”

Seismic Determination of Reservoir Heterogeneity: Application to the Characterization of Heavy Oil Reservoirs

“…Random realizations with a prescribed autocorrelation function (ACF ) are often computed by convolution of the zero-phase realization with a white-noise volume (Frankel and Clayton, 1986;Kerner, 1992;Ikelle et al, 1993). The autocorrelations described by the heterogeneity cubes, however, vary spatially.…”