Bayesian Geophysical Inversion Using Invertible Neural Networks

Constraining geophysical models with observed data usually involves solving nonlinear and nonunique inverse problems. Neural mixture density networks (MDNs) provide an efficient way to estimate Bayesian posterior marginal probability density functions (pdf's) that represent the nonunique solution. However, it is difficult to infer correlations between parameters using MDNs, and in turn to draw samples from the posterior pdf. We introduce an alternative to resolve these issues: invertible neural networks (INNs). These are simultaneously trained to represent uncertain forward functions and to solve Bayesian inverse problems. In its usual form, the method does not account for uncertainty caused by data noise and becomes less effective in high dimensionality. To overcome these issues, in this study, we include data uncertainties as additional model parameters, and train the network by maximizing the likelihood of the data used for training. We apply the method to two types of imaging problems: One-dimensional surface wave dispersion inversion and two-dimensional travel time tomography, and we compare the results to those obtained using Monte Carlo and MDNs. Results show that INNs provide comparable posterior pdfs to those obtained using Monte Carlo, including correlations between parameters, and provide more accurate marginal distributions than MDNs. After training, INNs estimate posterior pdfs in seconds on a typical desktop computer. Hence they can be used to provide efficient solutions for repeated inverse problems using different data sets. Also even accounting for training time, our results show that INNs can be more efficient than Monte Carlo methods for solving inverse problems only once.