In the 2-D Fourier transform we reviewed the phenomenon of spatial aliasing caused by undersampling of recorded data, and in further aspects of migration in practice we examined its adverse effects on migration. Prestack data may be made spatially uniform, for instance, by azimuth-moveout correction (AMO) , thus enabling use of finite-difference or frequency-wavenumber algorithms for prestack depth migration. When it becomes necessary to use Kirchhoff summation algorithm, operator aliasing is an issue that needs to be dealt with.
Shown in Figure 8.5-12 is a low-velocity hyperbolic summation path. Note that it intercepts more than one sample in a given input trace, each sample indicated by a shaded grid block. Thus, the summation along this path should include more than one sample per input trace. There are three different approaches to handle operator aliasing caused by mulitple samples per input trace included in the summation:
- The summation trajectory that represents the kinematics of the migration operator may be truncated to exclude the steep flanks that suffer from aliasing.
- Trace interpolation may be used to create additional traces so as to avoid multiple samples per input trace included in the summation (processing of 3-D seismic data).
- The frequency components that are aliased at a given dip along the summation trajectory are filtered out .
The first approach is undesirable since truncation is equivalent to limiting the migration aperture which can destroy steeply dipping events (kirchhoff migration in practice). The second approach can be taken but at a high cost in the case of prestack data. Trace interpolation also suffers in accuracy when applied to data with dipping events that conflict with one another. The third approach requires multiple copies of the input trace with different band-widths; this can be troublesome for very large input prestack data. Lumley  applied a triangular filter  to input traces in the time domain without creating multiple copies. An improved form of the triangular filtering method for 3-D Kirchhoff summation is given by Abma .
- Biondi et al., 1998, Biondi, B., Fomel, S., and Chemingui, N., 1998, Azimuth moveout for 3-D prestack imaging: Geophysics, 63, 574–588.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Gray, 1992, Gray, S., 1992, Frequency-selective design of the Kirchhoff migration operator: Geophys. Prosp., 40, 565–571.
- Lumley et al. (1994), Lumley, D. E., Claerbout, J. F., and Bevc, D., 1994, Antialiased Kirchhoff 3-D migration: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1282–1285.
- Claerbout, 1992, Claerbout, J.F., 1992, Earth soundings analysis: Blackwell Scientific Publications.
- Abma et al. (1999), Abma, R., Sun, J., and Bernitsas, N., 1999; Antialiasing methods in Kirchhoff migration: Geophysics, 64, 1783–1792.
- 3-D prestack depth migration
- Kirchhoff summation
- Calculation of traveltimes
- The eikonal equation
- Fermat’s principle
- Summation strategies
- Migration aperture
- 3-D common-offset depth migration