Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so-called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimized. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non-linear, high-dimensional Bayesian full waveform inversion problems. This paper introduces an efficient method to find possible images of the Earth's interior structures and estimate the uncertainties of these images using seismic data observed at the near surface. The method represents the imaging results with a flexible expression. We demonstrate the method by imaging the subsurface using two different types of seismic data, and by comparing the imaging results with other methods that have been introduced previously. We further show that the obtained imaging results can be used to estimate the size of a subsurface structure of interest with minimum bias. We apply boosting variational inference to Bayesian inversion, which uses a mixture of Gaussians to approximate the posterior distributionThe method is shown to be efficient and accurate, and constructs a fully analytic expression for a high-dimensional posterior distributionThe analytic solution allows extremely efficient methods to be used to answer scientific questions with minimum bias