Linearized Elastic inversion in the k-w domain

Elastic inversion aims to give a detailed description of earth parameters. We explore the implementation aspects of the elastic linearized inversion in the k-ω domain introduced by Rocca and De Nicolao (1993). The hypothesis of small contrasts enables to use the Born approximation and to linearize the scattering properties of point diffractors. Operating in the k-ω 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 links plane monochromatic incident and reflected waves to the medium wavenumber, so we obtain a simple relation between data and parameters. Inversion and analysis of the information is achieved by means of singular value decomposition of the data-parameters relation. 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 deteriorate the quality of inversion and produce interferences between parameter estimates. Similar effects come 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 initial model. Particular care is applied in choosing the processing sequence that minimizes these disturbances.