Alkhalifah and Tsvankin (1995) have shown that time-domain processing algorithms for elliptically anisotropic media should be the same as those for isotropic media. However, in anisotropic continuation, elliptical anisotropy and isotropy differ by a vertical scaling factor that is related to the difference between the vertical and NMO velocities. In isotropic media, when velocity is continued, both the vertical and NMO velocities (which are the same) are continued together, whereas in anisotropic media (including elliptically anisotropic) the NMO-velocity continuation is separated from the vertical-velocity one, and equation (19) corresponds to continuation only in the NMO velocity. This also implies that equation (19) is more flexible than equation (18), in that we can isolate the vertical-velocity continuation (a parameter that is usually ambiguous in surface processing) from the rest of the continuation process. Using , where z is depth, we immediately obtain the equation
which represents the vertical-velocity continuation.
Setting and v=vv in equation (16) leads to the following kinematic equation for -continuation:
We include more discussion about different aspects of linearization in Appendix B. The next section presents the analytic solution of equation (19). Later in this paper, we compare the analytic solution with a numerical synthetic example.