Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion

TL;DR

This paper investigates the stability of Gibbs posteriors derived from the Wasserstein loss in Bayesian full waveform inversion, demonstrating their robustness to noise and comparing them with other loss functions.

Contribution

It establishes the existence and stability of Wasserstein-based Gibbs posteriors in Bayesian FWI and compares their performance with other loss functions.

Findings

Wasserstein loss leads to stable Gibbs posteriors under high-frequency noise

Numerical comparisons show differences in uncertainty estimates

Wasserstein-based posteriors are robust to data noise

Abstract

Recently, the Wasserstein loss function has been proven to be effective when applied to deterministic full-waveform inversion (FWI) problems. We consider the application of this loss function in Bayesian FWI so that the uncertainty can be captured in the solution. Other loss functions that are commonly used in practice are also considered for comparison. Existence and stability of the resulting Gibbs posteriors are shown on function space under weak assumptions on the prior and model. In particular, the distribution arising from the Wasserstein loss is shown to be quite stable with respect to high-frequency noise in the data. We then illustrate the difference between the resulting distributions numerically, using Laplace approximations to estimate the unknown velocity field and uncertainty associated with the estimates.