2-D RANDOM-MEDIA WITH ELLIPSOIDAL AUTOCORRELATION FUNCTIONS

The integration of surface seismic data with borehole seismic data and well-log data requires a model of the earth which can explain all the se measurements. We have chosen a model that consists of large and sma ll scale inhomogeneities: the large scale inhomogeneities are the mean characteristics of the earth while the small scale inhomogeneities ar e fluctuations from these mean values. In this paper, we consider a tw o-dimensional (2-D) model where the large scale inhomogeneities are re presented by a homogeneous medium and small scale inhomogeneities are randomly distributed inside the homogeneous medium. The random distrib ution is characterized by an ellipsoidal autocorrelation function in t he 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 Ca rtesian 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 autocorre lation lengths. We also examine the response of seismic waves to this form of inhomogeneity. To do this in an accurate way, we used the fini te-difference technique to simulate seismic waves. Special care is tak en 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 direc tion varies from 0 to 90 degrees. Pulse broadening, coda, and anisotro py (transverse isotropy) due to small scale inhomogeneities are clearl y apparent in the synthetic seismograms. These properties can be recas t as functions of the aspect ratio (r0 = b/a) of the medium, especiall y the anisotropy and coda. For media with zero aspect ratio (1-D media ), the coda energy is dominant at large angles. The coda energy gradua lly 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 var iations 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 t he effective medium theory. As the aspect ratio increases to unity, th e variations of pulse arrival times with angle gradually become isotro pic. To retain the anisotropic behavior beyond the geometrical critica l angle, we have used a low-frequency pulse with a nonzero dc componen t.