We show that a scheme to solve the 2-D eikonal equation by a finite‐difference method can violate causality for moderate to large velocity contrasts [Formula: see text]. As an alternative, we present a finite‐difference scheme in which the solution region progresses outward from an “expanding wavefront” rather than an “expanding square,” and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.

Variable-size (dynamic) smoothing operator constraints are applied in crosswell traveltime tomography to reconstruct both the smooth-and fine-scale details of the tomogram. In mixed and underdetermined problems a large number of iterations may be necessary to introduce the slowly varying slowness features into the tomogram. To speed up convergence, the dynamic smoothing operator applies adaptive regularization to the traveltime prediction error function with the help of the model covariance matrix. By so doing, the regularization term has a larger weight at initial iterations and the prediction error term dominates the final iterations with a small regularization term weight. In addition, it is shown that adaptive regularization acts by reweighting the adjoint modeling operator (preconditioning) and by providing additional damping.Comparisons of two dynamic smoothing operators, the low-pass filter smoothing and the multigrid technique, with the fixed-size (static) smoothing operators show that the dynamic smoothing operator yields more accurate velocity distributions with greater stability for larger velocity contrasts. Consequently, it is a preferred choice for regularization.

Primary reflections, like multiples, can generate false images when reverse time migration (RTM) algorithms are used. The false images are formed by the zero-lag correlation of the source wavefields and primary reflections, which are propagated by the migration algorithm along nonphysical paths. These paths are generated by strong velocity gradients or reflection interfaces when the two-way wave equation is used. Conceptually, this type of artifact can be removed by separating up- and downgoing waves, but such separation may be impractical because it often requires storing the entire wavefields at all time steps. We have developed a de-primary RTM method in which such separation can be accomplished without saving the wavefields. The computational cost of the proposed method was only approximately 33% higher than that of conventional RTM algorithms. Using field and synthetic data sets, we have demonstrated the existence of this endemic RTM problem and verified the effectiveness of the de-primary RTM technique for removing the false events.

A beam implementation is presented for efficient full‐volume 3-D prestack Kirchhoff depth migration of seismic data. Unlike conventional Kirchhoff migration in which the input seismic traces in time are migrated one trace at a time into the 3-D image volume for the earth’s subsurface, the beam migration processes a group of input traces (a supergather) together. The requirement for a supergather is that the source and receiver coordinates of the traces fall into two small surface patches. The patches are small enough that a single set of time maps pertaining to the centers of the patches can be used to migrate all the traces within the supergather by Taylor expansion or interpolation. The migration of a supergather consists of two major steps: stacking the traces into a τ-P beam volume, and mapping the beams into the image volume. Since the beam volume is much smaller than the image volume, the beam migration cost is roughly proportional to the number of input supergathers. The computational speedup of beam migration over conventional Kirchhoff migration is roughly proportional to [Formula: see text], the average number of traces per supergather, resulting a theoretical speedup up to two orders of magnitudes. The beam migration was successfully implemented and has been in production use for several years. A factor of 5–25 speedup has been achieved in our in‐house depth migrations. The implementation made 3-D prestack full‐volume depth imaging feasible in a parallel distributed environment.

Huygens’s principle is used to derive a method for calculating first arrival traveltimes in anisotropic media. Numerical results for several transverse isotropic (TI) models show that calculated Huygens traveltimes are in good agreement with traveltimes computed by a finite‐difference solution to the anisotropic wave equation. This Huygens method is stable and accurate for the test models and so it may be useful in anisotropic data inversion and wave propagation visualization.

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