(17) |

(18) |

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

Setting and *v*=*v*_{v} in equation (16) leads to the
following kinematic equation for -continuation:

(19) |

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.

11/11/1997