Linearization

A useful approximation of equations (15) and (16) can be obtained by simply setting $\eta$ equal to zero in the right-hand side of the equations. Under this approximation, equation (15) leads to the kinematic velocity-continuation equation for elliptically anisotropic media, which has the following relatively simple form:  

$$\frac{\partial \tau}{\partial v} = {\frac{v\,\tau \,\left( 2... ...u }_x}}^2} \right) }{{{{v_v}}^2} + {v^4}\,{{{{\tau }_x}}^2}}}.$$ (17)

It is interesting to note that setting v=v_v, yields Fomel's expression for isotropic media Fomel (1996) given by  

$$\frac{\partial \tau}{\partial v} = v\,\tau \,{{{{\tau }_x}}^2}.$$ (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 $\tau=\frac{z}{v_v}$, where z is depth, we immediately obtain the equation

$$\frac{\partial \tau}{\partial v_v} = -\frac{\tau}{v_v},$$

which represents the vertical-velocity continuation.

Setting $\eta=0$ and v=v_v in equation (16) leads to the following kinematic equation for $\eta$-continuation:  

$$\frac{\partial \tau}{\partial \eta} = {\frac{\tau {v^4} \,{{{{\tau }_x}}^4}}{1 + {v^2}\,{{{{\tau }_x}}^2}}}.$$ (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.