The method alternates steps of datuming and imaging. Because traveltimes are computed for each step, the adverse effects of caustics and headwaves do not develop. The method is semi-recursive because the wavefield is downward continued and re-synthesized at multiple depth levels. When the imaging problem is divided in this way, the traveltimes are better behaved and some multiple arrivals are accounted for.

The great appeal of this approach is that it combines the efficiency of first-arrival Kirchhoff imaging with excellent image quality. In principle, this method only requires the same number of traveltime calculations as a standard migration. The Kirchhoff datuming steps can be implemented so that they are much more efficient than Kirchhoff migration. Therefore, the total cost of the algorithm is only a few times more than that of standard first-arrival Kirchhoff migration.

The Marmousi synthetic data set has been a popular testbed for seismic velocity estimation and imaging algorithms and it has been demonstrated that Kirchhoff algorithms using first arrival traveltimes fail even when the correct velocity model is used. This failure is illustrated in the top panel of Figure . This type of failure is disappointing because Kirchhoff migration is the most popular 2-D and 3-D prestack imaging method. This failure has stimulated the development of more accurate traveltime calculation methods that aim to handle complex propagation effects. It has led to methods that perform local averages of the velocity model in order to impose a frequency dependence on the eikonal equation (Biondi, 1992; Lomax, 1992), methods that track the most energetic arrivals, and methods which calculate traveltimes in the seismic frequency band (Nichols, 1994). Some of these methods also estimate amplitude and phase. Complicated velocity models can result in multi-valued traveltime tables, so these capabilities have also been incorporated into Kirchhoff algorithms.

My approach is based on the observation that most of the adverse effects of wavefield propagation that are difficult to parameterize with first-arrival traveltimes do not occur until the wavefield has evolved for some time. By limiting the first-arrival traveltime calculation, I am able to parameterize the asymptotic Green's functions accurately. As shown in the bottom panel of Figure , this approach leads to an excellent imaging result.

Figure 2

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