A stable method for linearized inversion of elastic parameters

Abstract

An inversion method that stabilizes the multiparameter inverse problem for a constant background elastic isotropic medium is developed. The inverse problem is formulated in the wavenumber domain and the operators acting on the individual elastic parameters are displayed and analysed. From here it is noticed that for certain scattering angles some parameters produce no scattering. This scattering angle dependence is combined with the frequency dependence of the parameters to reduce the multiparameter inverse problem to a single parameter inversion problem, which is known to yield stable results. The method developed here is an extension of an acoustical medical imaging method by Norton (1983) to the elastic seismic imaging problem which inherits limited view constraints. This method is compared to the elastic extension of a multiparameter acoustic inversion method developed by Devaney (1985) and the resulting improvements in stability are demonstrated on synthetic examples. From the theory it is observed that a fixed 90° scattering angle can be used with multiple frequencies to achieve highly stable inversion results of all elastic parameters provided specific elastic parameters are extracted from specific scattering modes. The attempt to extract all elastic parameters from the P-to-P scattering mode using multiple scattering angles is seen to be very ill-conditioned and not recommended. For the 90° scattering angle we find that surface reflection profiling (SRP) and cross-hole geometries have the same wavenumber domain coverage and the multi-offset vertical seismic profiling (MVSP) geometries give a better coverage than only a SRP or cross-hole geometry.