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Equations 1-2 reduce to some well-known
approximations with special choices of parameters.
- If
, the proposed approximation reduces to
the classic hyperbolic form
![\begin{displaymath}
t^2(x) \approx t_0^2 + \frac{x^2}{v^2}\;,
\end{displaymath}](img39.png) |
(11) |
which is a two-parameter approximation.
- The choice of parameters
;
;
reduces the proposed
approximation to the shifted hyperbola (Malovichko, 1978; de Bazelaire, 1988; Castle, 1994), which is the following three-parameter
approximation:
![\begin{displaymath}
t(x) \approx t_0\,\left(1-\frac{1}{s}\right) +
\frac{1}{s}\,\sqrt{t_0^2+s\,\frac{x^2}{v^2}}\;.
\end{displaymath}](img43.png) |
(12) |
- The choice of parameters
;
;
reduces approximation 2 to the form
proposed by Alkhalifah and Tsvankin (1995) for VTI media, which is the following
three-parameter approximation:
![\begin{displaymath}
t^2(x) \approx t_0^2 + \frac{x^2}{v^2} -
\frac{2\,\eta\...
...yle v^4\,\left[t_0^2 + (1+2\,\eta)\,\frac{x^2}{v^2}\right]}\;.
\end{displaymath}](img47.png) |
(13) |
- The choice of parameters
;
;
reduces approximation 2 to the following
three-parameter approximation suggested by Blias (2007) and
reminiscent of the ``velocity acceleration'' equation proposed by
Taner et al. (2005,2007):
![\begin{displaymath}
t^2(x) \approx t_0^2 + \frac{x^2}{v^2\,(1+\gamma\,x^2)}\;.
\end{displaymath}](img51.png) |
(14) |
- The choice of parameters
;
;
reduces the proposed approximation to the following
three-parameter approximation suggested by Blias (2009):
![\begin{displaymath}
t(x) \approx
\frac{1}{2}\,\sqrt{t_0^2+\left(1-\sqrt{s-1...
...\,\sqrt{t_0^2+\left(1+\sqrt{s-1}\right)\,\frac{x^2}{v^2}} \;.
\end{displaymath}](img54.png) |
(15) |
- The choice of parameters
;
reduces the proposed approximation to the following
three-parameter approximation also suggested by Blias (2009):
![\begin{displaymath}
t^2(x) \approx \frac{t_0^2}{2} + \frac{x^2}{v^2} + \frac{1}{2}\,\sqrt{t_0^4 + \frac{2\,A\,x^4}{v^4}}\;.
\end{displaymath}](img57.png) |
(16) |
- The choice of parameters
,
,
reduces the proposed
approximation to the double-square-root expression
where
,
, and
. Equation 17 describes
moveout precisely for the case of a diffraction point in a constant
velocity medium.
Thus, the proposed approximation encompasses some other known forms but
introduces more degrees of freedom for optimal fitting.
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2013-07-26