Initial methods for recovering the angle-dependent reflectivity function were based on the assumption that the Earth can be considered as plane-layered within the range covered by a common midpoint gather (CMP), which is the basic assumption of standard CMP processing (Newman, 1973; Yu, 1985; Kolb and Chapel, 1989; Cunha, 1990). Although these methods properly compensate for the long wavelength losses in a reasonable number of practical problems, they cannot be applied to areas with structurally complex geology.
Recently, some more powerful methods for angle-dependent reflectivity recovery, based in migration algorithms, have been developed. De Bruin et al. (1990a) presented a method for estimation of the angle-dependent reflectivity using a scalar pre-stack phase-shift migration scheme. Tests with synthetic data showed that the method was able to retrieve the reflectivity function with considerable accuracy for horizontal layers, but when dip was present (de Bruin, 1990b) the retrieved function was incorrectly shifted, because of the assumption of a horizontally-layered earth, which is implicit in their imaging principle (integration along constant Snell parameter traces). Lumley and Beydoun (1991) introduced a different scheme for retrieving the reflectivity function, using an elastic pre-stack Kirchhoff migration scheme. Their imaging principle was not limited by the horizontal layer assumption and was successfully applied to synthetic and real data. Both approaches, however, are limited to the assumptions that the medium is isotropic medium, and the wavefields are decomposable.
This paper describes a method for retrieving the angle-dependent reflectivity that is based on anisotropic prestack reverse time migration. Reverse time migration was independently developed in the early 1980s by Whitmore (1983), Baysal et al. (1983), McMechan (1983), and Lowental and Mufty (1983). Levin (1984) gives a general discussion of the different formulations, clarifies the principle of reverse time migration and shows how it relates to depth extrapolation methods. Karrenbach (1991) developed an anisotropic reverse-time migration/inversion scheme using a tensorial imaging condition represented by the crosscorrelation between the different components of the upcoming and downgoing wavefields. Each of the resulting images is associated with a different component of the stiffness tensor. His approach is equivalent to the first step of a linearized inverse algorithm because the residuals between the modeled and recorded wavefields at the surface, rather than just the recorded wavefield, were propagated backward in time. As a result, the images were directly related to perturbations in the impedance, instead of to the reflectivity functions.
Similarly to the work developed by Karrenbach, the method described in this paper uses an anisotropic prestack, reverse-time migration algorithm. However, I use a different modeling scheme, which is based on the equivalent medium theory (Cunha, 1991), and an alternative approach for the surface boundary conditions. Instead of the tensorial imaging principle, this method uses the usual scalar imaging principle and an additional imaging condition to estimate the reflection angle associated with the imaged reflectivity.