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Perturbation theory is based on expressing the solution in terms of power-series expansions
of parameters that are expected to be small. Thus, higher power terms have
smaller contributions, and
as a result, they are usually dropped. The degree of truncation depends on the convergence
behavior of the series.
I will apply
the perturbation theory to evaluate the stationary phase solutions around in
VTI media.

Analytical solutions for the quartic equation (24) in *p*_{s}^{2} can be evaluated. They are, however,
complicated, and some of them actually do not exist () for =0.
Recognizing that can be small,
we develop a perturbation series, that
is apply a power-series expansion in terms of . Unlike
weak anisotropy approximations, the resultant solution based on perturbation theory
yields good results even for strong anisotropy (). The
key here is to recognize the behavior of the series for large powers
of using Shanks transforms. According to perturbation
theory Buchanan and Turner (1978), the solution of equation (24)
can be represented in a power-series expansion in terms of as follows

| |
(34) |

where *y*_{i} are coefficients of this power series.
For practical applications, the power series of equation (34) is truncated
to *n* terms as follows
| |
(35) |

The coefficients, *y*_{i}, are determined by inserting the truncated form
of equation (34)
(three terms of the series are enough here)
into equation (24) and then solving for *y*_{i}, recursively. Because is
a variable, we can set the coefficients of each power of separately to equal zero.
This gives a sequence of equations for the *y*_{i} expansion coefficients. For example, *y*_{0} is
obtained directly from setting =0, and the result corresponds
to the solution for isotropic media.
For large , *A*_{n} converges slowly to the exact solution, and, therefore, yields
sub-accurate results when used, even if we go up to
*A*_{10}. Truncating after the second term (linear in , *A*_{1}) is
referred to as the weak anisotropy approximation. Using Shank transforms
Buchanan and Turner (1978), one can
predict the behavior of the series for large *n*, and, therefore, eliminate the most pronounced
transient behavior of the series
(to eliminate the term that has the slowest decay). Following Shanks
transform, the solution is evaluated
using the following relation

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Stanford Exploration Project

7/5/1998