  Uncertainty quantification for full-waveform inversion provides a
probabilistic characterization of the ill-conditioning of the problem,
comprising the sensitivity of the solution with respect to the starting model
and data noise. This analysis allows to assess the confidence in the candidate
solution and how it is reflected in the tasks that are typically performed
after imaging (e.g., stratigraphic segmentation following reservoir
characterization). Classically, uncertainty comes in the form of a probability
distribution formulated from Bayesian principles, from which we seek to obtain
samples. A popular solution involves Monte Carlo sampling. Here, we propose
instead an approach characterized by training a deep network that "pushes
forward" Gaussian random inputs into the model space (representing, for
example, density or velocity) as if they were sampled from the actual posterior
distribution. Such network is designed to solve a variational optimization
problem based on the Kullback-Leibler divergence between the posterior and the
network output distributions. This work is fundamentally rooted in recent
developments for invertible networks. Special invertible architectures, besides
being computational advantageous with respect to traditional networks, do also
enable analytic computation of the output density function. Therefore, after
training, these networks can be readily used as a new prior for a related
inversion problem. This stands in stark contrast with Monte-Carlo methods,
which only produce samples. We validate these ideas with an application to
angle-versus-ray parameter analysis for reservoir characterization.

Geophysics;Machine Learning;Machine Learning