Algorithms in the -k domain, such as Stolt's and phase-shift, lend themselves perfectly to anisotropic extension, since such algorithms use the dispersion relation explicitly for downward continuation and imaging. In general the agreement between the exact transverse isotropic (TI) dispersion relation and the double elliptic approximation, equation 1, is very good. The approximate algorithm is even able to model triplications. Figures and compare modeling and migration impulse responses. The right column plots the difference between the results in the left and middle columns using the same scale. Only the P and SV wave types show some small differences, while the SH wave is exact. The approximation fits the dispersion relation exactly on the vertical and horizontal axes. The largest discrepancy is in the area of triplication. The crucial test, however, is to model a TI media using the exact dispersion relation and then migrate it with the double elliptic approximation. Figure shows that, more or less, the correct spikes are obtained. The middle column shows migration results, where both, modeling and migration, used the exact dispersion relation. The right column shows migration results where the exact form was used for modeling and the approximation was used for imaging. The algorithm, which uses the exact dispersion relation, reproduces the spike, which is convolved with a ricker wavelet, very well. The double elliptic approximation shows the spikes slightly blurred. The results suggest that phase domain algorithms can be expected to perform reasonably well, when used to image an anisotropic scalar eigenfield.

12/18/1997