  Uncertainty quantification provides quantitative measures on the reliability
of candidate solutions of ill-posed inverse problems. Due to their sequential
nature, Monte Carlo sampling methods require large numbers of sampling steps
for accurate Bayesian inference and are often computationally infeasible for
large-scale inverse problems, such as seismic imaging. Our main contribution is
a data-driven variational inference approach where we train a normalizing flow
(NF), a type of invertible neural net, capable of cheaply sampling the
posterior distribution given previously unseen seismic data from neighboring
surveys. To arrive at this result, we train the NF on pairs of low- and
high-fidelity migrated images. In our numerical example, we obtain
high-fidelity images from the Parihaka dataset and low-fidelity images are
derived from these images through the process of demigration, followed by
adding noise and migration. During inference, given shot records from a new
neighboring seismic survey, we first compute the reverse-time migration image.
Next, by feeding this low-fidelity migrated image to the NF we gain access to
samples from the posterior distribution virtually for free. We use these
samples to compute a high-fidelity image including a first assessment of the
image's reliability. To our knowledge, this is the first attempt to train a
conditional network on what we know from neighboring images to improve the
current image and assess its reliability.
