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) analyzed theoretically in Chapter .
Offset continuation is a process of seismic data transformation across different offsets Bolondi et al. (1982); Deregowski and Rocca (1981); Salvador and Savelli (1982). As I show in Chapter 6, different types of DMO operators Hale (1995) can be regarded as a 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); Fomel (1995a). Integral offset continuation operators have been derived independently by Stovas and Fomel (1993, 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 do not 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, like all integral (Kirchhoff-type) operators, they suffer from irregularities in the input geometry. The latter problem is addressed by the accurate but expensive method of inversion to common offset Chemingui (1999).
In this section, I propose an application of offset continuation in the form of a finite-difference filter for seismic data regularization. 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 the 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 shot gathers in a constant-velocity synthetic.