Conclusions

We conclude that operator aliasing artifacts are indeed introduced during prestack wavefield continuation migration when s- or r-axes are allowed to deviate from a regular grid. This problem is thus manifest both when we choose to subsample the number of shots available from the data or design a survey without shots at every receiver location (or in reciprocal cases like OBC surveys).

Source-receiver migration strategies conveniently side-step the bulk of the operator aliasing problem by effectively performing all propagation and imaging on the coarsest available grid. Thus during every wavefield continuation step, we are only propagating energy that lies below the Nyquist wavenumber for the final image space. However, this constraint can be much too stringent when compared to the sampling criteria set out, and presumably met, during the acquisition effort. This migration style will propagate energy along fewer traces than shot-profile methodologies, although it lacks some ability to control the quality of the image. The fact that the model output of source-receiver migration is on a twice finer grid than acquisition or the model produced with shot-profile migration does not exclude this migration strategy from the imaging condition aliasing described by Zhang et al. (2003). The division in the mapping of equation (1) is responsible for this, but does not perform the necessary interpolation to avoid imaging condition aliasing. Performing the imaging condition in the Fourier domain does not entail additional cost for this split-step Fourier migration kernel, and conveniently eliminates the need to interpolate the wavefields by a factor of two to address the aliasing due to multiplication as the output of the convolution is inherently twice the size of the two inputs.

Given some knowledge of the dip content of the data, it is possible to extend the boundaries of the rigorous anti-aliasing criteria presented. Using both positive and negative one-sided band-limits can include high wavenumber energy that can improve the image in areas of steep dips and the shallow section of the model. Therefore, when challenged with imaging important steeply dipping targets, decisions concerning acquisition design or the level of decimation along different directions for migration can be made with a better understanding of the consequences to the final product. Finally, it has been noticed that the rigorous Nyquist limits are substantially too restrictive in practice with real data. In reality, the benefits of some level of anti-aliasing are realized when inspecting the dipping canyon features of the Marmousi data, though the limitation imposed during the imaging condition was much more relaxed than indicated by the theory and needed to be found through experimentation.

Further, if a subsample of a data set is imaged with a shot profile migration strategy, full bandwidth source and receiver wavefields, $W_s(\bold x,z,\omega)$ and $W_r(\bold x,z,\omega)$, could be saved for future migration efforts. An imaging condition with the appropriate band-limitation across the spatial axes can be applied with partial or complete sets of these migrated volumes. Thus incremental increases in image quality can be achieved while avoiding re-migration of data by augmenting a library of wavefields (if adequate storage capacity is available).