Inverse problems for torsional modes

Summary. We consider a spherically symmétric, non-rotating earth consisting of an isotropic, perfect elastic material where the density and the S -wave velocity may have one or two discontinuities in the upper mantle. We show that given the velocity throughout the mantle and the crust and given the density in the lower mantle, then the frequencies of the torsional oscillations of one angular order (one torsional spectrum), determine the density in the upper mantle and in the crust uniquely. If the velocity is known only in the lower mantle, then the frequencies of the torsional oscillations of two angular orders uniquely determine both the density and the velocity in the upper mantle and in the crust. In particular, the position and size of the discontinuities in the density and velocity are uniquely determined by two torsional spectra.