Although the numerical solutions are sufficient and efficient enough to perform prestack migration, analytical solutions are important for obtaining insights into the problem. Analytical solutions are also important for inversion purposes where we need to calculate the solution and its derivatives for different medium parameters in an iterative fashion.

For homogeneous isotropic media, the stationary point is given by

(19) |

The 2-point solution, shown in Appendix B,
is the same as equation (19) without the term in square brackets. As a result, this
equation fits the exact solution at *p*_{x}=0 and , and, therefore,
I will refer to it as the 2-point solution.
The low curvature of *T* around
its maximum (see Figure 1)
makes any error attributed to equation (19) or to the 2-point solution
to be insignificant, even for large offsets.

The stationary point solutions for homogeneous VTI media are much more complicated, as shown
in Appendix B, than their isotropic counterparts.
Nevertheless, they have the same form (2- and 3-point fitting solutions)
as that for isotropic media. The solutions
for *p*_{x}=0 and *p*_{x}=*p*_{h}, however, are no longer exact because we dropped some terms (related
to strong anisotropy and large offsets) in their
derivation. Nevertheless, the approximations, as shown in
Appendix B and in the following subsection, are relatively accurate. In fact, using the perturbation
theory along with Shanks transforms, one can obtain even better approximations of *p*_{h}
(see Appendix C).

11/11/1997