Connection with other approximations

Equations 1-2 reduce to some well-known approximations with special choices of parameters.

, the proposed approximation reduces to the classic hyperbolic form

$\begin{displaymath} t^2(x) \approx t_0^2 + \frac{x^2}{v^2}\;, \end{displaymath}$

(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:

$\begin{displaymath} t(x) \approx t_0\,\left(1-\frac{1}{s}\right) + \frac{1}{s}\,\sqrt{t_0^2+s\,\frac{x^2}{v^2}}\;. \end{displaymath}$

(12)

The choice of parameters $A = - 4\,\eta$ ; $B = 1 + 2\,\eta$ ; $C=(1 + 2\,\eta)^2$ reduces approximation 2 to the form proposed by Alkhalifah and Tsvankin (1995) for VTI media, which is the following three-parameter approximation:

$\begin{displaymath} t^2(x) \approx t_0^2 + \frac{x^2}{v^2} - \frac{2\,\eta\... ...yle v^4\,\left[t_0^2 + (1+2\,\eta)\,\frac{x^2}{v^2}\right]}\;. \end{displaymath}$

(13)

The choice of parameters $A = - 2\,\gamma\,t_0^2\,v^2$ ;

;

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):

$\begin{displaymath} t^2(x) \approx t_0^2 + \frac{x^2}{v^2\,(1+\gamma\,x^2)}\;. \end{displaymath}$

(14)

The choice of parameters

;

reduces the proposed approximation to the following three-parameter approximation suggested by Blias (2009):

$\begin{displaymath} t(x) \approx \frac{1}{2}\,\sqrt{t_0^2+\left(1-\sqrt{s-1... ...\,\sqrt{t_0^2+\left(1+\sqrt{s-1}\right)\,\frac{x^2}{v^2}} \;. \end{displaymath}$

(15)

The choice of parameters

; $C=2\,A$ reduces the proposed approximation to the following three-parameter approximation also suggested by Blias (2009):

$\begin{displaymath} t^2(x) \approx \frac{t_0^2}{2} + \frac{x^2}{v^2} + \frac{1}{2}\,\sqrt{t_0^4 + \frac{2\,A\,x^4}{v^4}}\;. \end{displaymath}$

(16)

The choice of parameters $A = 2\,\tan^2{\theta}$ , $B=1-\tan^2{\theta}$ , $C=1/\cos^4{\theta}$ reduces the proposed approximation to the double-square-root expression

$\displaystyle t(x)$	$\textstyle \approx$	$\displaystyle \frac{1}{2}\, \sqrt{t_0^2 + \frac{x\,(x+t_0\,v\,\sin{2\theta})}{... ...}{2}\, \sqrt{t_0^2 + \frac{x\,(x-t_0\,v\,\sin{2\theta})}{v^2\,\cos^2{\theta}}}$
	$\textstyle =$	$\displaystyle \frac{\sqrt{z^2 + (y+x/2)^2}}{V} + \frac{\sqrt{z^2 + (y-x/2)^2}}{V}\;,$	(17)

where $V = v\,\cos{\theta}$ , $z=(t_0\,V/2)\,\cos{\theta}$ , and $y=(t_0\,V/2)\,\sin{\theta}$ . 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.

General method for parameter

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NONHYPERBOLIC MOVEOUT APPROXIMATION

2013-03-02