Dip moveout with v(z)

It is worth noticing that the concepts in this section are not limited to constant velocity but apply as well to v(z). However, the circle operator $\bold C$ presents some difficulties. Let us see why. Starting from the Dix moveout approximation, $t^2 = \tau^2 + x^2/v(\tau )^2$,we can directly solve for $t(\tau ,x)$but finding $\tau (t,x)$ is an iterative process at best. Even worse, at wide offsets, hyperbolas cross one another which means that $\tau (t,x)$ is multivalued. The spray (push) operators $\bold C$ and $\bold H$ loop over inputs and compute the location of their outputs. Thus $\vec {\bold z} = \bold C_h\ \vec {\bold t}$requires we compute $\tau$ from t so it is one of the troublesome cases. Likewise, the sum (pull) operators $\bold C'$ and $\bold H'$ loop over outputs. Thus $\vec {\bold t} = {\bold C'}_h\ \vec {\bold z}$causes us the same trouble. In both cases, the circle operator turns out to be the troublesome one. As a consequence, most practical work is done with the hyperbola operator.

A summary of the meaning of the Rocca smile and its adjoint is found in Figures 21 and 22, which were computed using subroutine flathyp() .

 

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Figure 21 Impulses on a zero-offset section migrate to semicircles. The corresponding constant-offset section contains the adjoint of the Rocca smile.

 

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Figure 22 Impulses on a constant-offset section become ellipses in depth and Rocca smiles on the zero-offset section.