To apply prestack VTI migration in the ()-domain, we will need the NMO velocity field as well as . Using a smoothed version of both fields, shown above, I prestack migrate the anisotropic Marmousi dataset in the same way I did the isotropic one. However, we must first compute the traveltime map using the ray tracing equations of Alkhalifah et al. (1997), which hold for VTI media as well.
Figure 8 shows rays emanating from a source at surface position of 4000 m. The traveltime information along the rays are interpolated to a regular grid in the same way as the isotropic case. The contours in Figure 8 show traveltimes corresponding to the interpolated Field. The rays and traveltimes are different from the isotropic ones, shown in Figure 3, especially for large ray angles.
Figure 9 (top) shows, again, a stacked section after prestack time-migrated, but for the anisotropic Marmousi dataset. Overall, the migrated section seems well focused, better focused than the isotropic result. This is because the anisotropic dataset, which is new, has slightly higher peak frequencies and a better, less vibrating, source. Figure 10 shows common CMP gathers after prestack anisotropic time migration. Despite the large nonhyperbolic moveout that often accompany reflections in VTI media (see Alkhalifah (1997b)), the moveout is well aligned here. These outstanding alignments hold for the complex, as well as the smooth, regions of the model.
Finally, Figure 6 (bottom) shows the stacked section after prestack anisotropic time migration in depth. Here, I use the same velocity model used in the isotropic case for the conversion from time to depth, since this velocity also represents the vertical velocity. In practical applications, we would need to use a vertical velocity model built from information extracted from the available wells in the area, since the surface P-wave seismic data do not hold any explicit information about the vertical velocity. The time migration, in its original intension, allowed us to delay the depth representation of the seismic image to whenever such depth information becomes available. Yet, using this time migration, we managed to image and focus data as complex as the Marmousi model. This feature is particularly important in prestack-based parameter estimation.