Enabling Uncertainty Quantification in a standard Full Waveform Inversion method using Normalizing Flows

In order to maximize the utility of seismic imaging and inversion results, we need to compute not only a final image but also quantify the uncertainty in that image. While the most thorough approach to quantify the uncertainty is to use a method such as Markov chain Monte Carlo (MCMC), which systematically samples the entire posterior distribution, this is often inefficient and not all applications require a full representation of the posterior. We use normalizing flows (NF), a machine learning technique to perform uncertainty quantification (UQ) in full waveform inversion (FWI), specifically for time-lapse data. As with any machine learning algorithm, the NF learns only the mapping from the part of the prior spanned by the training data to the distribution of final models spanned by the training data. Here we make use of this property to perform UQ efficiently by learning a mapping from the prior to the distribution that really characterizes the model perturbations within a specific range. Our approach involves using a range of starting models, paired with final models from a standard FWI as training data. While this does not capture the full posterior of the FWI problem, it enables us to quantify the uncertainties associated with updating from an initial to a final model. Since our target is to perform UQ for time-lapse imaging, we use a local wave-equation solver that allows us to solve the wave equation in a small subset of our entire model, thereby keeping computational costs low. Numerical examples demonstrate that incorporating the training step for NF provides a distribution of model perturbations, which is dependent on a designated prior, to quantify the uncertainty of FWI results.