Amplitude preservation for offset continuation: Confirmation for Kirchhoff data

Introduction

Offset continuation (OC) is the operator that transforms common-offset seismic reflection data to data with a different offset. Following the classic results of Deregowski and Rocca (1981), Bolondi et al. (1984,1982) described OC as a continuous process of gradual change of the offset by means of a partial differential equation. Because it is based on the small-offset small-dip approximation, Bolondi's equation failed at large offsets or steep reflector dips. Nevertheless, the OC concept inspired a flood of research on dip moveout (DMO) correction (Hale, 1991). Since one can view DMO as a particular case of OC (continuation to zero offset), the offset continuation theory can serve as a natural basis for DMO theory. Its immediate application is in interpolating data undersampled in the offset dimension.

Fomel (2003,1994) introduced a revised version of the OC differential equation and proved that it provides the correct kinematics of the continued wavefield for any offset and reflector dip under the assumption of constant effective velocity. The equation is interpreted as an ``image wave equation'' by Hubral et al. (1996). Studying the laws of amplitude transformation shows that in 2.5-D media the amplitudes of continued seismic gathers transform according to the rules of geometric seismics, except for the reflection coefficient, which remains unchanged (Fomel, 2003; Goldin and Fomel, 1995). The solution of the boundary problem on the OC equation for the DMO case (Fomel, 2003) coincides in high-frequency asymptotics with the amplitude-preserving DMO, also known as Born DMO (Bleistein, 1990; Liner, 1991). However, for the purposes of verifying that the amplitude is correct for any offset, this derivation is incomplete.

In this paper, we perform a direct test on the amplitude properties of the OC equation. We describe the input common-offset data by the Kirchhoff modeling integral, which represents the high-frequency approximation of a reflected (scattered) wavefield, recorded at the surface at nonzero offset (Bleistein, 1984). For reflected waves, the Kirchhoff approximation is accurate up to the two orders in the high-frequency series (the ray series) for the differential operator applied to the solution, with the first order describing the phase function alone and the second order describing the amplitude. We prove that both orders of accuracy are satisfied when the offset continuation equation is applied to Kirchhoff data. Thus, this differential equation is the ``right'' equation to two orders, producing the correct amplitude as well as the correct phase for offset continuation. That is, the geometric spreading effects and curvature effects of the reflected data are properly transformed. The angularly dependent reflection coefficient of the original offset is preserved.

This proof relates the OC equation with ``wave-equation'' processing. It also provides additional confirmation of the fact that the true-amplitude OC and DMO operators (Tygel et al., 1998; Goldin and Fomel, 1995; Santos et al., 1997; Black et al., 1993) do not depend on the reflector curvature and can properly transform reflections from arbitrarily shaped reflectors (Bleistein et al., 2001; Goldin, 1990; Tygel et al., 1996). The latter result was specifically a 2.5-D result, whereas the result of this paper does not depend on the 2.5-D assumption. That is, the result presented here remains valid when the reflector has out-of-plane variation.

Our method of proof is indirect. We first write the Kirchhoff representation for the reflected wave in a form that can be easily matched to the solution of the OC differential equation. We then present the analogues of the eikonal and transport equations for the OC equation and show that the amplitude and phase of the Kirchhoff representation satisfy those two equations.

Amplitude preservation for offset continuation: Confirmation for Kirchhoff data