A frequency-domain implementation is also possible for non-linear moveout trajectories, as long as the moveout correction is time-invariant. Kostov used a time-invariant hyperbolic transform to detect the signal from a drill bit. A time-invariant transform was appropriate in this case because the drill bit operates continuously over time, so a given moveout trajectory may be observed at any time.

For impulsive-source data, however, even in the constant-velocity
case, a time-invariant transform is not appropriate.
Several authors have found ways around this problem,
for application to the problem of multiple suppression.
Hampson (1986) noted that after NMO correction, residual moveout
of multiples in a CMP gather is approximately time-invariant and
parabolic. He used a time-invariant parabolic transform
to separate multiples and reflections.
Yilmaz (1989) proposed a *t ^{2}* stretching of the
time axis as an alternative to Hampson's NMO correction.
Kelamis and Chiburis (1992) used the

In all of these cases, a time-invariant transform could
be used because, after NMO correction or *t ^{2}* stretching,
multiples exhibit moveout that is time-invariant.

Kirchhoff migration, also a summation along hyperbolic
trajectories, suffers from similar aperture effects.
Some work has been done on limited aperture migration;
Carrion *et al.* (1991) present an overview, and offer
a wavefront set method that uses deterministic and
stochastic criteria to compensate for the limited aperture.

In this paper, we propose applying the least-squares transform approach described above to the Kirchhoff migration case. As in the slant stack case, a least-squares Kirchoff migration algorithm should offer improved resolution.

Using a time-invariant transform is important, as it was in the slant stack case, in reducing the computational cost of the algorithm. Unfortunately, even in the constant velocity case, the summation trajectory for Kirchhoff migration is time-variant. There are several possible alternatives at this point. We could decide to work in the time domain, where we can use time-variant transform. This approach is taken by another paper in this report (Ji, 1992). Although the system of equations to be solved is larger in the time domain, this may be the best approach when accurate velocity information is available. A second approach, which we have not tried, would be to perform NMO correction, and then assume that any residual moveout of events is approximately parabolic or hyperbolic and time-invariant. A third approach, which is the subject of this paper, is to use time-invariant transforms, despite their drawbacks, and find a way to relate them to the time-variant transforms of a conventional migration.

We describe a version of Kirchhoff migration where time-invariant
transforms are used. Input 2D data, a function of position *x*
and time *t*, is transformed to a three-dimensional space where
two of the axes are *x* and *t*. The various panels of the
3D cube correspond to different time-invariant moveout corrections.
One can refer to this as the ``hyperbola shape'' axis.
Imaging involves selecting, for each time sample, the result
of summing along the hyperbola with the correct shape.
For a constant velocity, we might use *nt* different hyperbolas,
where *nt* is the number of points on the time axis, each
appropriate for the given velocity and a different time zero.
In this case, the
third axis is the *t _{0}* or ``pseudo depth'' axis.
The migrated result is obtained
by extracting the

This is a costly procedure. The cost of computing the 3D migration cube is typically 40 times greater for the least-squares case than for a conventional Kirchhoff approach. We believe that it is worth considering, however, for three reasons:

- The time-invariant transform allows the various frequencies to be processed in parallel, reducing the effective cost on parallel computers.
- While straightforward imaging involves throwing away all but one slice of the 3D cube computed by the program, the rest of the cube actually contains information, similar to a migration velocity analysis, that allows us to determine whether the velocity used to migrate the data was correct.
- The added cost may be justified if this scheme allows us to reduce the aperture in migration of field data.

We begin our explanation of the least-squares scheme by a review of Kostov's least-squares slant-stack transforms, and show how a simple modification leads to a Kirchhoff migration scheme.

11/17/1997