Kirchhoff depth migration is a widely used algorithm for imaging seismic data in both two and three dimensions. To perform the summation at the heart of the algorithm, standard Kirchhoff migration requires a traveltime map for each source and receiver. True‐amplitude Kirchhoff migration in 2.5-D υ(x, z) media additionally requires maps of amplitudes, out‐of‐plane spreading factors, and takeoff angles; these quantities are necessary for calculating the true‐amplitude weight term in the summation. The increased input/output (I/O) and computational expense of including the true‐amplitude weight term is often not justified by significant improvement in the final muted and stacked image. For this reason, some authors advocate neglecting the weight term in the Kirchhoff summation entirely for most everyday imaging purposes. We demonstrate that for nearly the same expense as ignoring the weight term, a much better solution is possible. We first approximate the true‐amplitude weight term by the weight term for constant‐velocity media; this eliminates the need for additional source and receiver maps. With one small additional approximation, the weight term can then be moved entirely outside the innermost loop of the summation. The resulting Kirchhoff method produces images that are almost as good as for exact true‐amplitude Kirchhoff migration and at almost the same cost as standard methods that do not attempt to preserve amplitudes.

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Prestack depth migration needs a good velocity model to produce a good image; in fact, finding the velocity model is one of the goals of prestack depth migration. Migration velocity analysis uses information produced by the migration to update the current velocity model for use in the next migration iteration. Several techniques are currently used to estimate migration velocities, ranging from trial and error to automatic methods like reflection tomography. Here, we present a method that combines aspects of some of the more accurate methods into an interactive procedure for viewing the effects of residual normal moveout corrections on migrated common reflection point (CRP) gathers. The residual corrections are performed by computing traveltimes along raypaths through both the current velocity model and the velocity model plus suggested model perturbations. The differences between those sets of traveltimes are related to differences in depth, allowing the user to preview the approximate effects of a velocity change on the CRP gathers without remigrating the data. As with automatic tomography, the computed depth differences are essentially backprojected along raypaths through the model, yielding a velocity update that flattens the gathers. Unlike automatic tomography, in which an algebraic inverse problem is solved by the computer for all geologic layers simultaneously, our method estimates shallow velocities before proceeding deeper and requires substantial user intervention, both in flattening individual CRP gathers and in deciding the appropriateness of the suggested velocity updates in individual geologic units.

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