Kirchhoff migration is generally accepted to be the most efficient method of imaging 2-D and 3-D prestack seismic data. However, many researchers have discovered that Kirchhoff algorithms that use first-arrival traveltimes do a poor job of imaging complex structures. (Audebert et al., 1994b; Gray and May, 1993; Geoltrain and Brac, 1993). Even methods which calculate multiple arrivals and most energetic arrivals along with estimates of amplitude and phase do not always result in satisfactory images. It is generally accepted that algorithms which use recursive wavefield continuation to backwards propagate the received wavefield produce the best images. Unfortunately, these methods often require regular spatial sampling and are computationally intensive. That is why nonrecursive methods based on the Kirchhoff integral are attractive, especially for 3-D prestack imaging objectives. Kirchhoff algorithms can easily accommodate irregular sampling and they can be applied in a target-oriented fashion. The use of first-arrival traveltimes is popular because they are efficiently computed and they have the attractive property of filling the entire computational grid.

In their 1993 *Geophysics* article, Geoltrain and Brac ask the
question ``Can we image complex structures with first-arrival
traveltime?'' They conclude that they cannot, and that they should
either ray trace to find the most energetic arrivals, or calculate
multiple-arrival Green's functions. Nichols (1994) calculates
band-limited Green's functions to estimate the most energetic
arrivals. He estimates not only traveltime, but also amplitude and
phase. Both these approaches are computationally complex
and much more costly than first-arrival traveltime computation methods.
My approach is simpler: by breaking up the complex
velocity structure, I am able
to calculate traveltimes in velocity models where
finite-differencing the eikonal equation is valid.
This results in images comparable to those obtained by
Nichols' method and by shot-profile migration at a reduced
computational cost.

2/12/2001