Amplitude preservation for offset continuation:
Confirmation for Kirchhoff data

Sergey Fomel and Norman Bleistein

ABSTRACT

Offset continuation (OC) is the operator that transforms common-offset seismic reflection data from one offset to another. Earlier papers by the first author presented a partial differential equation in midpoint and offset to achieve this transformation. The equation was derived from the kinematics of the continuation process. This derivation is equivalent to proposing the wave equation from knowledge of the eikonal equation. While such a method will produce a PDE with the correct traveltimes, it does not guarantee that the amplitude will be correctly propagated by the resulting second-order partial differential equation. The second author (with J. K. Cohen) proposed a dip moveout (DMO) operator for which a verification of amplitude preservation was proven for Kirchhoff data. It was observed that the solution of the OC partial differential equation produced the same DMO solution when specialized to continue data to zero offset. Synthesizing these two approaches, we present here a proof that the solution of the OC partial differential equation does propagate amplitude properly at all offsets, at least to the same order of accuracy as the Kirchhoff approximation. That is, the OC equation provides a solution with the correct traveltime and correct leading-order amplitude. ``Correct amplitude'' in this case means that the transformed amplitude exhibits the right geometrical spreading and reflection-surface-curvature effects for the new offset. The reflection coefficient of the original offset is preserved in this transformation. This result is more general than the earlier results in that it does not rely on the two-and-one-half dimensional assumption.