The strong smear function that you see in Figure 18
is Rocca's **DMO**+NMO operator,
the operator that converts a point on a constant-offset section
to a zero-offset section.
The important feature of this operator is that
the bulk of the energy is in a much narrower region
than the big ellipse of migration.
The narrowness of the Rocca operator is important
since it means that energies will not move far,
so the operator will not have a drastic effect
and be unduly affected by remote data.
(Being a small operator also makes it cheaper to apply).
The little signals you see away from the central burst
in Figure 18 result mainly from
my modulating the ellipse curve into into a sequence of dots.
However, noises from sampling and nearest-neighbor interpolation
also yield a figure much like Figure 18.
This warrants a more careful theoretical study
to see how to represent the Rocca operator
directly (rather than as a sequence of two nearly opposite operators).

frocca
Rocca's prestack partial-migration operator is
a superposition of hyperbolas, each with its top on an ellipse.
Figure 18 |

To get a sharper, more theoretical view of the Rocca operator, Figure 19 shows line drawings of the curves in a Rocca construction. It happens, and we will later show, that the Rocca operator lies along an ellipse that passes through (and hence is independent of velocity!) Curiously, we see something we could not see on Figure 18, that the Rocca curve ends part way up the ellipse and it does not reach the surface. The place where the Rocca operator ends and the velocity independent ellipse continues is, however, velocity dependent as we will see. The Rocca operator is along the curve of osculation in Figure 19, i.e., the smile-shaped curve where the hyperbolas reinforce one another.

Figure 19

12/26/2000