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 t2 stretching of the time axis as an alternative to Hampson's NMO correction. Kelamis and Chiburis (1992) used the t2 stretch, and by partially stacking the data, regularized the recording geometry to reduce the number of least-squares problems to be solved. Foster and Mosher (1992) showed that a hyperbolic moveout trajectory was more appropriate for NMO-corrected CMP gathers than a parabolic one.
In all of these cases, a time-invariant transform could be used because, after NMO correction or t2 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 t0 or ``pseudo depth'' axis. The migrated result is obtained by extracting the t=t0 slice from the cube.
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:
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.