Fabio Rocca developed a clear conceptual model for Sherwood's dip corrections. Figure 13 illustrates Rocca's concept of a prestack partial-migration operator.

Figure 13

Imagine a constant-offset section *P*(*t*,*y*,*h* = *h _{0}* ) containing
an impulse function at some particular (

The Rocca operator is the curve of osculation in Figure 13, i.e., the smile-shaped curve where the hyperbolas reinforce one another. If the hyperbolas in Figure 13 had been placed everywhere on the ellipse instead of at isolated points, then the osculation curve would be the only thing visible (and you wouldn't be able to see where it came from).

The analytic expression for the travel time on the Rocca smile is the end of a narrow ellipse, shown in Figure 14.

Figure 14

We will omit the derivation of the equation for this curve which turns out to be

(8) |

The Rocca operator transforms a constant-offset section
into a zero-offset section.
This transformation achieves two objectives:
first, it does normal-moveout correction;
second, it does Sherwood's dip corrections.
The operator of Figure 13 is convolved
across the midpoint axis of the constant-offset section,
giving as output a zero-offset section at just one time,
say, *t _{0}*.
For each

This operator is particularly attractive from a practical point of view. Instead of using a big, wide ellipse and doing the big job of migrating each constant-offset section, only the narrow, little Rocca operator is needed. From Figure 15 we see that the energy in the dip moveout operator concentrates narrowly near the bottom.

Figure 15

In the limiting case that *h* / *v t _{0}* is small,
the energy all goes to the bottom.
When all the energy is concentrated near the bottom point,
the Rocca operator is effectively a delta function.
After compensating each offset to zero offset,
velocity is determined by the normal-moveout residual;
then data is stacked and migrated.

The
*narrowness*
of the Rocca
ellipsoid is an advantage in two senses.
Practically, it means that
not many midpoints need to be brought into the computer
main memory before velocity estimation and stacking are done.
More fundamentally, since the operator is so compact, it does not
do a lot to the data.
This is important because the operation is done at an early stage,
before the velocity is well known.
So it may be satisfactory to choose the velocity for the Rocca
operator as a constant, regional value, say, 2.5 km/sec.

An expression for the travel-time curve of the dip moveout operator might be helpful for Kirchhoff-style implementations. More to the point is a Fourier representation for the operator itself, which we will find next.

10/31/1997