Irregular acquisition of seismic data challenges processing. Frequency domain processes are difficult or impossible to apply. Kirchhoff methods adapt easily, but are likely to suffer from poor amplitude behavior and other problems; an acquisition 'footprint' is often visible on the final images Chemingui and Biondi (1996). If data can first be regularized, and large gaps interpolated, processing results will improve, and otherwise unavailable frequency domain processes will be made usable.
Sparsity of data, and the data aliasing which results, create a serious challenge to interpolation algorithms. Given a signal which is spatially aliased, it is difficult for an interpolation algorithm to determine the correct dip at which to model missing events. Transformation to velocity space is an attractive option, because the operator stacks and sprays data only along centered hyperbolas. Thus only reasonable dips are possible in the interpolated data. This type of interpolation does, however, have another pitfall. The optimum velocity spectrum, as estimated by least squares inversion, is likely to be too smooth to make an effective interpolation.
Least squares solutions to inverse problems are often smoother than desired Nichols (1994) Crawley (1995). Because squared errors are minimized, large noise bursts have much more impact on the final solution than smooth, low-amplitude, erroneous variations. However, in some applications these smooth variations are as serious a problem as large noise bursts. Numerous geophysical topics have been approached from the standpoint of finding parsimonious or 'spikey' solutions, including deconvolution Ulrych and Walker (1982), velocity analysis Vries and Berkhout (1984), and balancing Crawley (1995). A number of projects at SEP have sought to find spikey velocity space representations. Nichols 1994 showed results of Lp norm solutions to velocity space inversions, and presented methods for solving Lp problems using iteratively reweighted least squares (IRLS). Berryman Berryman (1996) discusses formulation of Lp norm problems, and gives some important mathematical basis for selection of p. In his PhD thesis, Ji 1994 showed that missing near offset traces in marine cmp gathers can be interpolated via application of a modified operator.
In this paper, I discuss and demonstrate the application of a preconditioned algorithm to interpolation beyond aliasing of 2D synthetic and real CMP gathers, and discuss the relationship between IRLS, Ji's approach, and preconditioning.