Conditioning bounds for traveltime tomography in layered media

TL;DR

This paper analyzes the limits of traveltime tomography in layered media, showing reflected rays lack sufficient information for well-posed recovery and highlighting severe conditioning issues in waveform inversion.

Contribution

It demonstrates that reflected rays do not provide enough information for well-posed inversion and characterizes the severe ill-conditioning of the problem.

Findings

Reflected ray traveltimes are insufficient for unique medium recovery.

The Fredholm kernel associated with reflected rays has exponentially decaying singular values.

Numerical experiments show waveform inversion can fit data but converge to incorrect profiles.

Abstract

This paper revisits the problem of recovering a smooth, isotropic, layered wave speed profile from surface traveltime 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, we show in this paper that traveltimes 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 data accurately while converging to the wrong wave speed profile.