Scattering techniques for the geophysical inverse problem

The inverse problem for the propagation of elastic waves in stratified media. The one dimensional inverse problem for the propagation of shear waves is studied in the context of spectral theory. Certain inverse scattering methods in quantum mechanics are generalized to determine the properties of the stratification from experimental knowledge gathered at the origin of the halfline. This method is applied with some mathematical manipulations to arrive at a procedure to solve the inverse problem for a three dimensional layered half space. This formulation of the problem lead to the complete and unique reconstruction of the depth dependences of the Lame parameters and the density from surface data. A discussion on the applicability of the proposed profile inversion method is given, and it is demonstrated with a numerical example.