Imaging beyond aliasing is therefore a very active area of research. A promising direction aims to exploit the redundancy of multi-offset data sets to generate alias-free and high-resolution stacks Brink et al. (1997) or partial stacks Chemingui and Biondi (1998). However, these methods are based on the inversion of prestack imaging operators (DMO or AMO) that are computationally demanding. Another approach based on inversion, but less computationally demanding, is the interpolation of aliased data by prediction-error filters Crawley (1998). Wisecup 1998 presented an interesting method to exploit data redundancy over offset to overcome temporal aliasing. The application of his idea to overcome spatial aliasing certainly has potential, but has not been developed yet. Huygen-migration methods are also promising Neidell (1997), although their theoretical development is still not mature.
The goal of the anti-aliasing method presented in this paper is to improve the image resolution by properly imaging some of the aliased components of the recorded data, that would not be imaged by applying standard anti-aliasing methods of Kirchhoff operators Abma (1998); Bevc and Claerbout (1992); Gray (1992); Lumley et al. (1994). Ours is thus a more modest goal than the ones of the research discussed above. But the proposed method has the advantage of not requiring complex, and sometime unreliable, inversion techniques. In addition, it does not require data redundancy, and thus can be applied to the imaging of minimal data sets Padhi and Holley (1997), such as zero-offset data. I show that by a simple modification of the anti-aliasing conditions of the Kirchhoff migration operator, some of the aliased components in the data are nicely imaged, without generating aliasing noise. The proposed ``imaging beyond aliasing'' is possible by exploiting a priori assumptions on the dip bandwidth of the data. These assumptions are realistic in many important cases, such as the example that I show of the imaging of a the steep flanks of a salt dome in the Gulf of Mexico. Pica 1996 presented a model-based anti-aliasing method that is related to the one presented in this paper. However, his analysis is based on a priori information on the image dips instead of data dips, and it yields different anti-aliasing conditions.
To derive the proposed anti-aliasing method, I analyze in detail the aliasing of Kirchhoff imaging operators. I distinguish the image-space aliasing from the data-space aliasing, often referred to as operator aliasing. Image aliasing occurs when the imaging operator, represented by its spreading formulation (e.g. ellipsoid for migration) is too coarsely sampled in image space. Dually, operator aliasing occurs when the gathering formulation of the imaging operator (e.g. hyperboloid for migration) is too sparsely sampled. I show that image aliasing is closely related to operator aliasing when the data sampling and the image sampling are the same and the data are not aliased. Therefore, the two concepts are seldom distinguished and often only the conditions necessary to avoid operator aliasing are used to avoid aliasing artifacts Abma (1998); Bevc and Claerbout (1992); Gray (1992); Lumley et al. (1994), or more rarely, only the conditions to avoid image aliasing Hale (1991). However, the distinction becomes important when exploiting a priori knowledge on the dip spectrum of the data, because in this case the image needs to be more densely sampled than the data to avoid image aliasing. In addition, the distinction is important for prestack data sets that have larger dimensionality, and consequently a different sampling, than the image.