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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 presents some difficulties.
Let us see why.
Starting from the Dix moveout approximation,
,we can directly solve for but finding is an iterative process at best.
Even worse, at wide offsets, hyperbolas cross one another
which means that is multivalued.
The spray (push)
operators and loop over inputs
and compute the location of their outputs.
Thus
requires we compute from *t* so it is
one of the troublesome cases.
Likewise, the sum (pull)
operators and loop over outputs.
Thus
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()` .

**yalei2
**

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

**yalei1
**

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

** Next:** GARDNER'S SMEAR OPERATOR
** Up:** ROCCA'S SMEAR OPERATOR
** Previous:** Push and pull
Stanford Exploration Project

12/26/2000