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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}](img8.png) |
(1) |
The five parameters
,
,
,
, 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}](img14.png) |
(2) |
where the new set of parameters
,
,
,
, and
is
related to the previous set by the equalities
The inverse transform is given by
The existence of the nonhyperbolic part in the traveltime
approximation 1 and 2 is controlled by parameter
. When
is zero (which implies that
or
),
approximation 1 is hyperbolic. When both
and
are
very large, approximation 2 also reduces to the hyperbolic
form.
Subsections
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Previous: INTRODUCTION
2013-03-02