Beydoun and Mendes (1989) greatly reduced the magnitude of complexity of the full nonlinear inverse problem by assuming a linear relation between elastic parameter perturbations and seismic reflection data under the Born approximation. Although their method is much more tractable than that of Tarantola et al., it is not clear that the Born approximation is valid for seismic reflection data, in which impedance contrasts may be large and nonlinear point scatterer-to-scatterer interaction (i.e., reflection) may be significant.
However, since elastic reflectivity, and not elastic parameters themselves, is linearly related to seismic reflection data amplitudes, it may be natural to pose the inverse problem as first inverting for elastic angle-dependent reflection coefficients, and then subsequently performing a much-simplified nonlinear inversion of Zoeppritz-like reflectivity, in order to estimate elastic parameters (e.g., Lumley and Beydoun, 1991).
Cunha (1992) developed an elastic reverse-time migration method based on finite differences to forward model and continue elastic displacement wavefields. Estimates of elastic angle-dependent reflectivity were attempted by performing numerical correlations of attributes of the continued wavefields. Although theoretically appealing, Cunha's preliminary results were inconclusive. Possible disadvantages to Cunha's method include its prohibitive computational expense, and significant potential for instability in the presence of noise due to its fundamental shot-domain implementation.
In a recent work, de Bruin (1992) uses an w-x finite difference depth migration method to make migrated midpoint gathers as a function of depth and ray parameter. His migrated gathers show excellent recovery of angle-dependent reflectivity amplitudes in synthetic tests. Unfortunately, de Bruin's method interprets the ray parameter values in terms of reflection angles, which is appropriate only in a 1-D medium. In a 2-D (or 3-D) medium, de Bruin must raytrace the reflection angles external to the migration/inversion procedure, using a posteriori information about the depth model structure and dip.
In previous works, I formulate the inverse problem in the space-time constant offset domain, which offers the potential for enhanced numerical inversion stability (Lumley and Beydoun, 1991; Lumley, 1992a; and Lumley 1993). The least-squares integral solutions are compressed by the use of ray-valid WKBJ Green's tensors (Aki and Richards, 1980), thus reducing the number of floating point operations ultimately required to perform wavefield continuation and correlation. Both the angle-dependent reflectivity, and the associated reflection angles, are estimated directly from the data. This approach is related to the Kirchhoff integral formulation of Parsons (1986), which is based on the work of Beylkin (1985) and Bleistein (1987), except that I consider elastodynamic vector wave propagation as opposed to acoustics, and find a potentially robust least-squares solution as opposed to a potentially unstable ``direct'' integral inverse. The theoretical results of Lumley and Beydoun (1991) and Lumley (1993) form the basic foundation for the algorithms and results presented in this real data study.