The chosen elastic parameters are P-impedance, S-impedance and density. The medium is considered to be composed of a known constant background plus an unknown small variation of the elastic parameters. The hypothesis of small contrasts enables us to use the Born approximation and to linearize the scattering properties of point diffractors (Wu and Aki, 1985). The formulation is similar to that developed in Clayton and Stolt (1981) and Miller et al. (1987) for the acoustic case. The extension to the elastic case is done by expressing the reflectivity term as a linear function of elastic perturbations (Stolt and Weglein, 1985, Beydoun and Mendes, 1989, De Nicolao et al., 1993).
We plan to insert the algorithm in a target-oriented inversion scheme, so limiting its application to a restricted zone of the subsurface, where the hypothesis of small variations of the parameters holds. For a realistic earth model, we must assume that the target is embedded in a complex stratified or inhomogeneous overburden, which has propagation effects such as geometrical spreading, transmission coefficients across elastic interfaces, anisotropy, wavefront distortion, multiples. If the overburden parameters are known, its effects can be removed from the inversion problem by a downward continuation (datuming) of sources and receivers to the target location. Our current work, not included in this paper, involves the development of a datuming scheme in the k- domain (Bernasconi et al., 1995). Moreover, a target-oriented approach is well suited for 3D geometries, as we can concentrate the computing resources on a small part of the model. In this paper we consider 2D media, but the extension to 3D media is straightforward.
Operating in the k- domain, we gain the additional advantage of the simplicity of the theory: perturbations of a uniform background are decomposed in sinusoidal components. The Bragg resonance condition (Born and Wolf, 1959) links plane monochromatic incident and reflected waves to the medium wavenumber, so we obtain a simple relation between data and parameters. The inversion can be done separately for each point of the model spectrum (k domain), so we deal with small matrices. In practice we collect the contributions to the reflected field of one sinusoidal component at a time.
Inversion and analysis of the information is achieved by means of singular value decomposition of the data-model relation and least-squares estimation along the eigenvectors. In this way, we obtain a robust, fast and efficient algorithm. The recovered model is a ``filtered'' but reliable version of the original model, as it contains only and all the information that we can extract from the data.
The ill conditioning of the inverse problem (De Nicolao et al. 1993) does not allow a good estimation of all three elastic parameters. Even small numerical errors can cause the quality of inversion to deteriorate and produce interference between parameter estimates. Similar effects result from the practical aspects of data collection and processing. Truncations due to the finite length of the cable, the finite number of sources and receivers, and the limited bandwidth of the wavelet produce aliasing and a distorted reconstruction of the original model. The main consequence is that P-P reflections allow an accurate estimation of P-impedance perturbation only. To recover S-impedance perturbation we must use multicomponent data, namely P-SV reflections. We demonstrate this with synthetic data. Particular care is taken in the choice of the processing sequence that minimizes the disturbances.
Although not investigated in this paper, we think that an improvement of the results can be achieved adding some a priori information (interpretative inversion). If we impose boundaries to the variations of the elastic parameters, we reduce the null space and the indetermination of the inversion.