Introduction

Initial methods for recovering the angle-dependent reflectivity function were based on the assumption that the subsurface layers are nearly horizontal within the range covered by a common midpoint (CMP) gather 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 estimating the angle-dependent reflectivity, based on migration algorithms, have been developed. De Bruin et al. (1990a) presented a method for estimating the angle-dependent reflectivity using a scalar pre-stack $x-\omega$ migration scheme. First, the multi-component data were decomposed into P-P, P-S, S-P and S-S modes and each of them migrated separately. 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 (in angle) because of the assumption of a horizontally-layered earth which is implicit in their imaging principle (integration along constant Snell parameter traces). Another limitation in their method is that the estimated reflectivities do not correspond to the local, in depth, reflectivities because the decomposition is carried out before the depth extrapolation. As a result, a wavefield that leaves and arrives at the surface as a pure compressional mode (marine data) but is associated to a S-S reflection (PSSP mode) will not be imaged by their method. Lumley and Beydoun (1991) introduced a different method 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. As a consequence of the Kirchhoff approach, their method suffers the restrictions imposed by the ray approximation associated with the evaluation of the Green's function. Both methods, however, assume that the medium is isotropic and attempt to image only the converted modes associated with a single conversion at the reflection plane. Therefore, when applied to off-shore data, only PP reflectivities can be obtained with these approaches.

This paper discusses the imaging condition for retrieving the angle-dependent reflectivity using anisotropic prestack reverse time migration (RTM). With two different approaches to the imaging condition, Mora (1986), and Chang and McMechan (1987) introduced an elastic isotropic formulation for RTM. Etgen (1987) extended the extrapolation step for the general anisotropic case, but was able to migrate only S_H modes because of the restriction imposed by the scalar formulation of the imaging condition. 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 perturbation in one component of the stiffness tensor. His approach is equivalent to the first step of a linearized inverse algorithm because it backward propagates the residuals between the forward modeled and the recorded wavefields at the receivers, rather than just the recorded wavefield. As a result, the images were more closely related to perturbations in the impedance, instead of to the reflectivity functions.

The imaging conditions and imaging criteria discussed here are implemented in a reverse-time migration method described in Cunha (1992). The final result of the migration corresponds to four images, which are associated with the angle-dependent reflectivity functions for P-P, P-S, S-P, and S-S modes.