Migration of seismic data may give rise to aliasing problems in four distinct situations. Two of these aliasing situations we consider to be well understood and have effective solutions. These are: 1) improper discretization of the wavefield recorded at the surface can lead to aliasing of the features in the raw data (i.e. poorly sampled hyperbolas), and 2) not establishing an image space with twice finer sampling to accomodate the sum of spatial frequencies due to the multiplication of the wavefields in the imaging condition. Proper planning and layout of the acquisition mitigate the first problem, while Zhang et al. (2003) points out that interpolation of source and receiver wavefields by a factor of two eliminates the aliasing from multiplication. For the duration of this discussion we will assume both of these issues have been effectively controlled.
This paper presents an evaluation of two additional aliasing situations that have yet to be examined by the geophysical community. First, and the focus of this presentation, aliasing phenomenon can arise during the adoption of a wavefield coordinate system on which both source and receiver wavefields can be propagated and combined to create the image. Zero-offset migration, by definition, has common source and receiver locations. However, prestack migration requires choosing from two probably unequal source and receiver sampling intervals (doubly compounded for 3D surveys). This paper explores the ramifications of this problem and defines appropriate bandlimits. Finally, the extrapolation operator can introduce aliasing by moving energy to too high wavenumbers as it convolves the wavefields with the earth velocity model. During later propagation steps, that same energy could move back within appropriate limits again. We have not identified how to capitalize on this effect.
The discretized representation of seismic wavefields and wavefield continuation operators requires a strategy to eliminate contamination from aliasing. Fourier sampling theory allows for the development of the rigorous Nyquist limits for arbitrary sampling of data and image axes. Using these requirements to restrict the discretized wavefield continuation process, we present criteria for determining appropriate image space Nyquist limits for arbitrary sampling choices.
As an example, we show a simple numerical case where aliased energy is introduced into the image space during migration after subsampling the shot axis. We then present three ways in which operator aliasing problems may be resolved in shot-profile migration strategies, and discuss the implications of operator aliasing on source-receiver migration formulations. Finally, an Appendix is included to provide, a rigorous development of the appropriate energy wavenumber limits as a function of data axes sampling, the extension of that development to explain the equivalence of shot-profile migration and source-receiver migration, and the introduction of imaging condition aliasing in the source-receiver migration algorithm.
This analysis has several important ramifications. Shot axis subsampling is a common practice before migration of large data sets to save time or cost. Narrow azimuth acquisition strategies, common to marine surveys, have inherent trade-offs between strike and dip resolution that are sometimes difficult to quantify. Wide azimuth land surveys are also often constrained by unequal in-line and cross-line sampling. Migrations of ocean bottom cable data also suffer from this problem due to their acquisition idiosyncrasies (though the reciprocal of our example presented later). Thus for any situation, be it acquisition design or processing choices, where one is forced to migrate data without equal numbers of shots and receivers at the same locations in both surface directions, the aliasing criteria explained herein can easily be implemented in standard wavefield continuation migration programs to enhance the quality of the image.