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NONHYPERBOLIC MOVEOUT APPROXIMATION

Let $t(x)$ represent the reflection traveltime as a function of the source-receiver offset $x$. 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 $a$, $b$, $c$, $\xi$, and $t_0$ 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 $A$, $B$, $C$, $v$, and $t_0$ 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)

The existence of the nonhyperbolic part in the traveltime approximation 1 and 2 is controlled by parameter $A$. When $A$ is zero (which implies that $\xi=0$ or $c=b^2$), approximation 1 is hyperbolic. When both $B$ and $C$ are very large, approximation 2 also reduces to the hyperbolic form.



Subsections
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Next: Connection with other approximations Up: Fomel & Stovas: Generalized Previous: INTRODUCTION

2013-03-02