Xuebin Zhao, Andrew Curtis, Xin Zhang

We test a fully non-linear method to solve seismic tomographic problems using data consisting of observed travel times of first-arriving waves. We use variational inference to calculate the posterior probability distribution which describes the solution to the Bayesian tomographic inverse problem. The variational method is an efficient alternate to Monte Carlo methods, which seeks the best approximation to the posterior distribution. This approximation is found using an optimization framework, and the method provides fully probabilistic results. We apply a new variational method for geophysics -- normalizing flows. The method models the posterior distribution by employing a series of invertible and differentiable transforms -- the flows. By optimizing the parameters of these transforms the flows are designed to convert a simple and analytically known distribution into a good approximation of the posterior distribution. Numerical examples show that normalizing flows can provide an accurate tomographic result including full uncertainty information while significantly decreasing the computational cost compared to Monte Carlo and other variational methods. In addition, this method provides analytic solutions for the posterior distribution rather than an ensemble of posterior samples. This opens the possibility that subsequent calculations about the posterior distribution might be performed analytically.