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NONHYPERBOLIC MOVEOUT APPROXIMATION

Let represent the reflection traveltime as a function of the source-receiver offset . We propose the following general form of the moveout approximation:

$\begin{displaymath} t^2(x) \approx (1-\xi)\,(t_0^2+a\,x^2) + \xi\,\sqrt{t_0^4 + 2\,b\,t_0^2\,x^2 + c\,x^4}\;. \end{displaymath}$

(1)

The five parameters

, $\xi$ , and

describe the moveout behavior. By simple algebraic manipulations, one can also rewrite equation 1 as

$\begin{displaymath} t^2(x) \approx t_0^2+\frac{x^2}{v^2} + \frac{A\,x^4} {\displ... ... 2\,B\,t_0^2\,\frac{x^2}{v^2} + C\,\frac{x^4}{v^4}}\right)}\;, \end{displaymath}$

(2)

where the new set of parameters

, and

is related to the previous set by the equalities

$\displaystyle a$	$\textstyle =$	$\displaystyle \frac{A\,B+B^2-C}{v^2\,\left(A+B^2-C\right)}\;;$	(3)
$\displaystyle b$	$\textstyle =$	$\displaystyle \frac{B}{v^2}\;;$	(4)
$\displaystyle c$	$\textstyle =$	$\displaystyle \frac{C}{v^4}\;;$	(5)
$\displaystyle \xi$	$\textstyle =$	$\displaystyle \frac{A}{C-B^2}\;.$	(6)

The inverse transform is given by

$\displaystyle v^2$	$\textstyle =$	$\displaystyle \frac{1}{a\,(1-\xi) + b\,\xi}\;;$	(7)
$\displaystyle A$	$\textstyle =$	$\displaystyle \frac{\xi\,\left(c - b^2\right)} {\left[a\,(1-\xi) + b\,\xi\right]^2}\;;$	(8)
$\displaystyle B$	$\textstyle =$	$\displaystyle \frac{b}{a\,(1-\xi) + b\,\xi}\;;$	(9)
$\displaystyle C$	$\textstyle =$	$\displaystyle \frac{c}{\left[a\,(1-\xi) + b\,\xi\right]^2}\;.$	(10)