Error Bounds for Extended Source Inversion applied to an Acoustic Transmission Inverse Problem

TL;DR

This paper develops error bounds for an extended source inversion method in acoustic wave problems, showing it reduces cycle-skipping issues and provides bounds on velocity estimates relative to noise and source support.

Contribution

It introduces an extended formulation with a weighted penalty that mitigates cycle-skipping and derives bounds on velocity errors at local minima.

Findings

Extended formulation reduces cycle-skipping in inverse problems.

Velocity error at local minima is bounded by source support and noise ratio.

Proper weighting ensures local minima are close to the global solution.

Abstract

A simple inverse problem for the wave equation requires determination of both the wave velocity in a homogenous acoustic material and the transient waveform of an isotropic point radiator, given the time history of the wavefield at a remote point in space. The duration (support) of the source waveform and the source-to-receiver distance are assumed known. A least squares formulation of this problem exhibits the "cycle-skipping" behaviour observed in field scale problems of this type, with many local minima differing greatly from the global minimizer. An extended formulation, dropping the support constraint on the source waveform in favor of a weighted quadratic penalty, eliminates this misbehaviour. With proper choice of the weight operator, the velocity component at any local minimizer of this extended objective function differs from the global minimizer of the least-squares formulation by less than a linear combination of the source waveform support radius and data noise-to-signal ratio.