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Non-hyperbolic moveout

The hyperbolic model 1 is not accurate at large offsets in the case of non-hyperbolic moveouts, caused by vertical or lateral heterogeneity, reflector curvature, or anisotropy (Fomel and Grechka, 2001). One popular model for describing non-hyperbolic moveouts was developed by Malovichko (1978) and has the form of a shifted hyperbola (de Bazelaire, 1988; Castle, 1994; Siliqi and Bousquié, 2000)

\begin{displaymath}
t(l) = t_0\,\left(1-\frac{1}{S(t_0)}\right) +
\frac{1}{S(t_0)}\sqrt{t_0^2 + S(t_0)\,\frac{l^2}{v_n^2(t_0)}}\;,
\end{displaymath} (9)

Equation 9 contains an additional parameter $S$, which is related to heterogeneity and anisotropy of seismic velocities. We can eliminate this parameter by differentiating the equation twice and defining the second derivative $q = \partial p/\partial l = \partial^2
t/ \partial l^2$. Eliminating both $v_n$ and $S$ from equation 9 and equations
$\displaystyle p$ $\textstyle =$ $\displaystyle \frac{l}{\left[t_0 + S(t_0)\,\left(t-t_0\right)\right]\,v_n^2}\;,$ (10)
$\displaystyle q$ $\textstyle =$ $\displaystyle \frac{t_0^2}{\left[t_0 + S(t_0)\,\left(t-t_0\right)\right]^3\,v_n^2}\;$ (11)

leads to the velocity-independent non-hyperbolic moveout equation
\begin{displaymath}
t_0 = t - \frac{p\,l}{1 + \sqrt{\frac{q\,l}{p}}}\;.
\end{displaymath} (12)

If moveout parameters $v_n$ and $S$ are required for subsequent interpretation, one can easily extract them as special data attributes
$\displaystyle \frac{1}{v_n^2}$ $\textstyle =$ $\displaystyle t_0\,\sqrt{\frac{p^3}{q\,l^3}}\;,$ (13)
$\displaystyle S$ $\textstyle =$ $\displaystyle 1 + \frac{p\,(t - p\,l) - q\,l\,t}{\sqrt{q\,p^3\,l^3}}\;.$ (14)

One could estimate the function $q(t,l)$ in practice by numerically differentiating the local slope field $p(t,l)$.


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Next: Dix inversion Up: Oriented time-domain imaging Previous: - NMO

2013-07-26