DISCUSSION

We have proved that the offset continuation equation correctly transforms common-offset seismic data modeled by the Kirchhoff integral approximation. The kinematic and dynamic equivalence of the OC equation has been proved previously by different methods Fomel (1995a,b). However, connecting this equation with Kirchhoff modeling opens new insights into the theoretical basis of DMO and offset continuation:

1.

The Kirchhoff integral can serve as a link between the wave-equation theory, conventionally used in seismic data processing, and the kinematically derived OC equation. Though the analysis in this paper follows the constant-velocity model, this link can be extended in principle to handle the case of a variable background velocity.

2.

The OC equation operates on the kernel of the Kirchhoff integral, which is independent of the local dip and curvature of the reflector. This proves that the true-amplitude OC and DMO operators can properly transform reflections from curved reflectors. Moreover, this result does not imply any special orientation of the reflector curvature matrix. Therefore, it does not require a commonly made 2.5-D assumption Bleistein and Cohen (1995); Fomel (1995a). Implicitly, this fact proves the amplitude preservation property of the three-dimensional azimuth moveout (AMO) operator Biondi and Chemingui (1994); Fomel and Biondi (1995), based on cascading the true-amplitude DMO and inverse DMO operators.