Obtaining multiparameter elastic subsurface images from seismic reflection data is a crucial step toward seismic reservoir characterization. High resolution maps of elastic properties may be indicative of lithology, subsurface physical states (e.g., pressure and temperature), and pore-fluid content. Information about such physical material properties may be useful in hydrocarbon exploration (including direct detection), reservoir management (infill drilling and field extension), and monitoring time-variable reservoir production and EOR processes (Nur, 1989).
Seismic reflectivity is intrinsically related to subsurface physical material properties by their elastic impedance contrasts. Estimation of elastic specular reflection coefficients can therefore be viewed as a preprocessing step prior to elastic parameter inversion. I derive elastic vector integral solutions for the nine-component elastic reflectivity matrix. This approach is somewhat similar in spirit to that 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 least-squares solution as opposed to a ``direct'' integral inversion.
In a related work, Tarantola (1986) proposed a nonlinear strategy to perform full waveform inversion of seismic reflection data directly for elastic parameters, essentially bypassing the reflectivity estimation step. This approach was implemented by Crase et al. (1990) on a marine data set, and was extremely computer intensive. Furthermore, because this one-step approach offers no intermediate processed steps to check for quality control, their results may be somewhat ambiguous in terms of the reliability of the elastic impedance results, robustness of multiple suppression, and sensitivity to the initial background velocity model.
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 scatterer-to-scatterer interaction (i.e., reflections) may be significant.
Since elastic reflectivity, and not elastic parameters themselves, is linearly related to seismic reflection data amplitudes, it is natural to pose the inverse problem as first inverting for elastic specular reflection coefficients, and then subsequently performing a much-simplified nonlinear inversion of Zoeppritz-like reflectivity, in order to estimate elastic parameters. 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 obtained by performing the appropriate numerical correlations of attributes of the continued wavefields. Although potentially promising, possible disadvantages to Cunha's method include its computational expense, and significant potential for instability in the presence of noise due to its mandatory shot gather domain implementation.
As a result, I seek to formulate the inverse problem in the space-time constant offset domain, which offers the potential for enhanced numerical inversion stability. The integral solutions are compressed by the use of ray-valid WKBJ Green's tensors, thus reducing the number of floating point operations ultimately required to perform wavefield continuation and correlation. Finally, since the Green's function integral kernels have a physical ray-based interpretation, it may be easier to ``tweak'' the inversion process based on physical intuition, as opposed to black box processing by finite differences or iterative gradient techniques.
This paper proceeds as follows. I first pose the reflectivity estimation as a formal least-squares inverse problem, which I then solve by stationary point analysis. Next, the elastodynamic wave equation is introduced, and a representation integral solution is presented by use of Betti's Theorem. The representation integrals are evaluated by assuming WKBJ approximate Green's tensors, which are in turn evaluated by the ray theoretic eikonal and transport equations. The representation integrals are substituted back into the generalized least-squares solution to obtain expressions for elastic specular reflectivity. I give explicit integral expressions for the and reflectivities as surface integrals over the recorded vector displacement wavefields.