The time and space formulation of azimuth moveout

Introduction

Azimuth moveout (AMO) is by definition an operator that transforms common-azimuth common-offset seismic reflection data to different azimuths and offsets. A constructive approach to AMO was proposed by Biondi and Chemingui (1994). According to this approach, an AMO operator is built by cascading the dip moveout (DMO) operator that transforms the input common-azimuth data to zero offset, and the inverse DMO that transforms the zero-offset data to a new offset and azimuth. Evaluating the cascade of the frequency-domain DMO and inverse DMO operators by means of the stationary phase technique produces the integral (Kirchhoff-type) 3-D AMO operator in the time-space domain.

The first part of this paper applies an analogous idea to construct the AMO operator from the time-space domain DMO and achieves the same result in a simpler way. Cascading DMO and inverse DMO allows us to evaluate the AMO operator's summation path and the corresponding weighting function. However, it is not sufficient for evaluating the third major component of the integral operator, that is, its aperture (range of integration). To solve this problem, we apply an alternative approach, that defines AMO as a cascade of 3-D migration (inversion) for particular common-azimuth and common-offset data and 3-D modeling for a different azimuth and offset. This definition resembles the viewpoint on DMO developed by Deregowski and Rocca (1981). As with the DMO case, the migration and modeling approach reveals the physics of the AMO aperture and limits its boundaries. It is the aperture limitation that allows us to overcome the paradoxical inconsistency between 2-D and 3-D AMO operators discussed by Biondi and Chemingui (1994). If the aperture is chosen properly, the AMO operator converges to the 2-D offset continuation limit as the azimuth rotation approaches zero. This remarkable fact supports the proof of economical efficiency of AMO in comparison with the prestack migration operator, which is known to have an unlimited aperture.

The time and space formulation of azimuth moveout