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.