Conditioning bounds for traveltime tomography in layered media (Q2888756)

The authors revisit the problem of recovering a smooth, isotropic, layered wave speed profile from surface travel-time information. While it is classic knowledge that the diving (refracted) rays classically determine the wave speed in a weakly well-posed fashion via the Abel transform, they show in this paper that travel-times of reflected rays do not contain enough information to recover the medium in a well-posed manner, regardless of the discretization. The counterpart of the Abel transform in the case of reflected rays is a Fredholm kernel of the first kind, which is shown to have singular values that decay at least root exponentially. Kinematically equivalent media are characterized in terms of a sequence of matching moments. This severe conditioning issue comes on top of the well-known rearrangement ambiguity due to low-velocity zones. Numerical experiments in an ideal scenario show that a waveform-based model inversion code fits the data accurately while converging to the wrong wave speed profile.