The inverse eigenvalue problems of free oscillation data inversion: the Backus—Gilbert conjecture for angular-order spectra

Summary. In their fundamental paper of 1967, Backus & Gilbert made the following conjecture (p. 265): '… that if we seek uniquely to determine v of the functions ρ, k, μ (with ν = 1, 2, 3) from the mass, moment, and eigen-frequencies of the normal modes, we need 2ν different infinite sequences of eigenfrequencies, each sequence consisting either of all toroìdal modes of a given angular order or all spheroidal modes of a given angular order…' . We show that, at least if attention is not restricted to geophysically realistic solutions, the conjecture is false. The question of its validity remains open when attention is restricted to geophysically realistic solutions. However, as the present result indicates, the fact cannot now be ignored that, even in a quite appropriate geophysical context, an examination of angular-order spectra is not sufficient to determine ρ, k and μ.