Geologists often see the earth as homogeneous blocks separated by smoothly curving boundaries. In contrast, computer modeling algorithms based on finite‐difference schemes require elastic constants to be specified on the vertices of a regular rectangular grid. How can we convert a continuous geological model into a form suitable for a finite‐difference grid? One common way is to lay the finite‐difference grid down on the continuous geological model and use whatever elastic constants happen to lie beneath each of the grid points.

We compare various forms of single-arrival Kirchhoff prestack depth migration to a full-waveform, finitedifference migration image, using synthetic seismic data generated from the structurally complex 2-D Marmousi velocity model. First-arrival-traveltime Kirchhoff migration produces severe artifacts and image contamination in regions of the depth model where significant reflection energy propagates as late or multiple arrivals in the total reflection wavefield. Kirchhoff migrations using maximum-energy-arrival traveltime trajectories significantly improve the image in the complex zone of the Marmousi model, but are not as coherent as the finite-difference migration image. By carefully incorporating continuous phase estimates with the associated maximum-energy arrival traveltimes, we obtain single-arrival Kirchhoff images that are similar in quality to the finite-difference migration image. Furthermore, maximum-energy Green's function traveltime and phase values calculated within the seismic frequency band give a Kirchhoff image that is (1) far superior to a first-arrival-based image, (2) much better than the analogous high-frequency paraxial-ray Green's function image, and (3) closely matched in quality to the full-waveform finite-difference migration image.

Prestack Kirchhoff migration using first arrival traveltimes has been shown to fail in areas of complex structure. I propose a new method for calculating traveltimes that estimates the traveltime of the maximum energy arrival, rather than the first arrival. The method estimates a traveltime that is valid in the seismic frequency band, not the usual high frequency approximation. Instead of solving the eikonal equation for the traveltime, I solve the Helmholtz equation to estimate the wavefield for a few frequencies. I then perform a parametric fit to the wavefield to estimate a traveltime, amplitude, and phase. The images created by using these parameters are shown to be superior to those created by using first arrival traveltimes, or those created using maximum amplitude traveltimes calculated by paraxial ray tracing.

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