As early as 20 years ago, researchers at SEP considered data interpolation as one of the most important problems of seismic data processing Claerbout (1980, 1981); Thorson (1981). In 2-D exploration, the interpolation problem arises because of missing near and far offsets, spatial aliasing and occasional bad traces. In 3-D exploration, the importance of this problem increases dramatically because 3-D acquisition almost never provides a complete regular coverage in both midpoint and offset coordinates Biondi (1999). Data regularization in 3-D can solve the problem of Kirchoff migration artifacts Gardner and Canning (1994), prepare the data for common-azimuth imaging Biondi and Palacharla (1996), or provide the spatial coverage required for 3-D multiple elimination van Dedem and Verschuur (1998).
Claerbout (1992, 1999) formulates the following general principle of missing data interpolation:
A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy.How can one specify an appropriate filtering for a given interpolation problem? Smooth surfaces are conveniently interpolated with Laplacian filtering Briggs (1974); Fomel (2000). Steering filters help us interpolate data with predefined dip fields Clapp et al. (1997); Fomel et al. (1997); Fomel (1999). Prediction-error filters in time-space or frequency-space domain successfully interpolate data composed of distinctive plane waves Claerbout (1999); Spitz (1991). Because prestack seismic data is not stationary in the offset direction, non-stationary prediction-error filters need to be estimated, which leads to an accurate but relatively expensive method with many adjustable parameters Clapp et al. (1999); Crawley (1999).
A simple model for reflection seismic data is a set of hyperbolic events on a common midpoint gather. The simplest filter for this model is the first derivative in the offset direction applied after the normal moveout correction. Going one step beyond this simple approximation requires taking the dip moveout (DMO) effect into account Deregowski (1986). The DMO effect is fully incorporated in the offset continuation differential equation Fomel (1994, 1995a).
Offset continuation is a process of seismic data transformation between different offsets Bolondi et al. (1982); Deregowski and Rocca (1981); Salvador and Savelli (1982). Different types of DMO operators Hale (1995) can be regarded as continuation to zero offset and derived as solutions of an initial-value problem with the revised offset continuation equation Fomel (1995b). Within a constant-velocity assumption, this equation not only provides correct traveltimes on the continued sections, but also correctly transforms the corresponding wave amplitudes Fomel and Bleistein (1996). Integral offset continuation operators have been derived independently by Stovas and Fomel (1996), Bagaini and Spagnolini (1996), and Chemingui and Biondi (1994). The 3-D analog is known as azimuth moveout (AMO) Biondi et al. (1998). In the shot-record domain, integral offset continuation transforms to shot continuation Bagaini and Spagnolini (1993); Schwab (1993); Spagnolini and Opreni (1996). Integral continuation operators can be applied directly for missing data interpolation and regularization Bagaini et al. (1994); Mazzucchelli and Rocca (1999). However, they don't behave well for continuation at small distances in the offset space because of limited integration apertures and, therefore, are not well suited for interpolating neighboring records. Additionally, as all integral (Kirchoff-type) operators they suffer from irregularities in the input geometry. The latter problem is addressed by accurate but expensive inversion to common offset Chemingui (1999).
In this paper, I propose an application of offset continuation in the form of a finite-difference filter for Claerbout's method of missing data interpolation. The filter is designed in the log-stretch frequency domain, where each frequency slice can be interpolated independently. Small filter size and easy parallelization among different frequencies assure a high efficiency of the proposed approach. Although the offset continuation filter lacks the predictive power of non-stationary prediction-error filters, it is much simpler to handle and serves as a good a priori guess of an interpolative filter for seismic reflection data. I test the proposed method by interpolating randomly missing traces in a constant-velocity synthetic and by restoring near offsets and intermediate shot gathers in the Marmousi synthetic dataset. These early tests produce encouraging results. In the final section of the paper, I discuss possible strategies for improving the method.