ADCIGs are attractive because they provide straightforward information for amplitude analysis, that is, amplitude versus angle (AVA) instead of the more common amplitude versus offset (AVO) analysis. The main focus of this paper is to evaluate how accurate is the the amplitude response of wave-equation migration as a function of reflection angle.
Angle-domain common-image gathers are representations of the seismic images sorted by the incidence angle at the reflection point. Angle-gathers can be obtained using wave-equation techniques either for shot-profile migration, as described by de Bruin et al. (1990), or for shot-geophone migration, as described by Prucha et al. (1999). In either case, angle-gathers are evaluated using slant-stacks on the downward continued wavefield, prior to imaging. However, decomposing the downward continued wavefield before imaging produces angle-gathers as a function of the less intuitive offset ray-parameter instead of the true reflection angle. Angle-domain gathers can also be computed by slant-stacking the image, instead of the downward continued wavefield. This alternative procedure directly produces angle-gathers as a function of reflection angle. In both cases, the slant-stack transformation can be more conveniently performed by a radial-trace transform (RTT) in the frequency-wavenumber domain Ottolini (1982).
AVA analysis requires that the procedures used to compute ADCIGs preserve the amplitude of the reflections as a function of angle. It is thus puzzling that straightforward implementations of both ADCIG methods produce contradictory amplitudes, and that downward-continuation migration is not a good approximation of the corresponding upward-continuation modeling for either method.
To solve the puzzle, we must take into account the weighting function that is introduced in the migration process by the imaging step. We show that this weighting is well approximated by a diagonal operator in the frequency-wavenumber domain. Since the two methods for computing ADCIGs perform a slant-stack at different stages, one before imaging and the other one after imaging, the corresponding weighting functions are different. Once the weights are taken into account, the AVA responses produced by the two methods are consistent, and migration is an approximate inverse of forward modeling. We find that, when restricted to flat reflectors, the weight we obtain for amplitude compensation is identical to those derived by Wapenaar et al. (1999).
According to the physical model for reflection data, the weights can be set to make migration a good approximation of a linearized inversion. We adopt the physical model proposed by Stolt and Benson (1986) and define the appropriate weights for both methods used to compute ADCIGs. Modeling and migration can also be easily made pseudo-unitary, if needed for an iterative estimation procedure. We achieve this by splitting in half the weighting factor between modeling and migration.