Shanks Transform

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 $\eta=0$ in VTI media.

Analytical solutions for the quartic equation (24) in p_s² can be evaluated. They are, however, complicated, and some of them actually do not exist ($\rightarrow \infty$) for $\eta$=0. Recognizing that $\eta$ can be small, we develop a perturbation series, that is apply a power-series expansion in terms of $\eta$. Unlike weak anisotropy approximations, the resultant solution based on perturbation theory yields good results even for strong anisotropy ($\eta\gt.5$). The key here is to recognize the behavior of the series for large powers of $\eta$ 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 $\eta$as follows  

$$y = \sum_{i=0}^{\infty} {y_i \eta^i},$$ (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  

$$A_n = \sum_{i=0}^{n} {y_i \eta^i}.$$ (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 $\eta$ is a variable, we can set the coefficients of each power of $\eta$ separately to equal zero. This gives a sequence of equations for the y_i expansion coefficients. For example, y₀ is obtained directly from setting $\eta$=0, and the result corresponds to the solution for isotropic media. For large $\eta$, A_n converges slowly to the exact solution, and, therefore, yields sub-accurate results when used, even if we go up to A₁₀. Truncating after the second term (linear in $\eta$, A₁) 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

$$y_s = \frac{A_2 A_0 - A_1^2}{A_2+A_0-2 A_1}.$$