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Zero-offset ray

The Taylor expansion of approximation 2 around the zero offset

\begin{displaymath}
t^2(x) = t_0^2 + \frac{x^2}{v^2} + \frac{A}{2}\,\frac{x^4}{v^4\,t_0^2} + O(x^6)
\end{displaymath} (18)

provides a convenient method for evaluating coefficients $t_0$, $v$, and $A$ by matching expansion 18 to the corresponding expansion of the exact traveltime. This is the method used previously for deriving approximations 11 and 12.

In the special case of an isotropic $V(z)$ medium, the coefficients are readily available and reduce to statistical averages of the velocity distribution (Bolshykh, 1956)

$\displaystyle t_0$ $\textstyle =$ $\displaystyle 2\,m_{-1}\;,$ (19)
$\displaystyle v^2$ $\textstyle =$ $\displaystyle \frac{m_1}{m_{-1}}\;,$ (20)
$\displaystyle A$ $\textstyle =$ $\displaystyle \frac{1}{2}\left(1-\frac{m_3\,m_{-1}}{m_1^2}\right)\;,$ (21)

where

\begin{displaymath}
m_k = \int\limits_{0}^{z} V^k(\zeta)\,d \zeta
\end{displaymath}

Equations 19-21 are easily extensible to the vertical transverse isotropy (VTI) case (Ursin and Stovas, 2006; Lyakhovitsky and Nevskiy, 1971; Alkhalifah, 1997b; Blias, 1983).


next up previous [pdf]

Next: Nonzero-offset ray Up: General method for parameter Previous: General method for parameter

2013-07-26