  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.
