Linearized Inverse Sca'itering Problems in Elasticity

Using the single-scattering approximation we invert for the material parameters of an acoustic two-parameter medium and then for a three-parameter isotropic elastic medium. Our procedure is related to various methods of depth migration in seismics, i.e. methods for locating major discontinuities in the subsurface material without specifying which quantities are discontinuous or by how much they jump. Our asymptotic multiparameter inversion makes use of amplitude information to reconstruct the size of the jumps in the parameters describing the medium. We allow spatially varying background parameters (both vertically and laterally) and an almost arbitrary source-receiver configuration. The computation is performed in the time domain and we use all available data even if it is redundant. This ability to incorporate the redundant information in a natural way is based upon a formula for double integrals over spheres. We solve for perturbations in different parameters treating separately P-to-P, P-to-S, S-to-P, and S-to-S data. It turns out that one may invert using subsets of the data, or all of it together. We also describe modifications to our scheme which allow us to use the Kirchhoff instead of the Born approximation for the forward problem when the scatterers are smooth surfaces of discontinuity.