Introduction

Offset continuation (OC) by definition is an operator that transforms common-offset seismic gathers from one constant offset to another Bolondi et al. (1982). Bagaini et al. 1994 recently identified OC with a whole family of prestack continuation operators, such as shot continuation Bagaini and Spagnolini (1993); Schwab (1993), dip moveout as a particular case of OC to zero offset, and three-dimensional azimuth moveout Biondi and Chemingui (1994). Possible practical applications of OC operators include regularizing seismic data by partial stacking prior to prestack migration Chemingui and Biondi (1994) and interpolating missing data. Since dip moveout (DMO) represents a particular case of offset continuation to zero offset, the OC concept is also one of the possible approaches to DMO. Another prospective application of prestack continuation operators, pointed out recently by Fabio Rocca (personal communication), is prestack tomography-type velocity analysis.

In the theory of OC operators, two issues need to be addressed. The first is kinematic equivalence. We expect seismic sections obtained by OC to contain correctly positioned reflection traveltime curves. The second issue is amplitude equivalence. If the traveltimes are positioned correctly, it is wave amplitudes that deserve most of our attention. Since the final outputs of the seismic processing sequence are the migrated sections, the kinematic equivalence of OC concerns preserving the true geometry of seismic images, while the amplitude equivalence addresses preserving the desired brightness of the images. Apparently, there can be different definitions of amplitude-preserving or true-amplitude processing. The most commonly used one Black et al. (1993); Goldin (1992); Hubral et al. (1991); Tygel et al. (1992) refers to the reflectivity preservation. According to this definition, amplitude-preserving seismic data processing should make the image amplitudes proportional to the reflection coefficients that correspond to the initial constant-offset gathers. This point of view implies that an amplitude-preserving OC operator tends to transform offset-dependent amplitude factors, except for the reflection coefficient, in accordance with the geometric seismic laws.

In this paper I introduce a theoretical approach to constructing different types of OC operators with respect to both kinematic equivalence and amplitude preservation.

The first part presents the theory for a revised OC differential equation. As early as in 1982, Bolondi et al. came up with the idea of describing OC as a continuous process by means of a partial differential equation Bolondi et al. (1982). However, their approximate differential OC operator, built on the results of Deregowski and Rocca's classic paper 1981, turned out to fail in case of steep reflector dips or large offsets. In his famous Ph.D. thesis 1983 Dave Hale wrote:

The differences between this algorithm [DMO by Fourier transform] and previously published finite-difference DMO algorithms are analogous to the differences between frequency-wavenumber Gazdag (1978); Stolt (1978) and finite-difference Claerbout (1976) algorithms for migration. For example, just as finite-difference migration algorithms require approximations that break down at steep dips, finite-difference DMO algorithms are inaccurate for large offsets and steep dips, even for constant velocity.

Continuing this analogy, one can observe that both finite-difference and frequency-domain migration algorithms share a common origin: the wave equation. The new OC equation, presented in this paper and valid for all offsets and dips, can play an analogous role for offset continuation and dip moveout algorithms. The next section begins with a rigorous proof of the revised equation's kinematic validity. Since the OC process belongs to the wave type, it is appropriate to describe it by considering wavefronts (which in this case correspond to the traveltime curves) and ray trajectories (referred to in this paper as time rays). The laws of amplitude transport along the time rays illuminate the main dynamic properties of offset continuation and prove the OC equation's amplitude equivalence.