3D Bayesian Variational Full-Waveform Inversion

Seismic full-waveform inversion (FWI) produces high resolution images of the subsurface by exploiting information in full acoustic, seismic or electromagnetic waveforms, and has been applied at global, regional and industrial spatial scales. FWI inverse problems are traditionally solved by using optimization, in which one seeks a best model by minimizing the misfit between observed waveforms and model-predicted waveforms. Due to the nonlinearity of the physical relationship between model parameters and waveforms, a good starting model is often required to produce a reasonable result. In addition, the optimization methods cannot produce accurate uncertainty estimates, which are required to better interpret final model estimates.

In principle, nonlinear Bayesian methods can be deployed to solve both issues. Monte Carlo sampling is one such class of algorithms which are computationally expensive, and all Markov chain Monte Carlo-based methods are difficult to parallelise fully. Variational inference provides a fully parallelisable alternative methodology. This is a class of methods that optimize an approximation to a probability distribution describing post-inversion parameter uncertainties. Both Monte Carlo and variational full waveform inversion have been applied previously to solve 2D Bayesian FWI problems, but neither of them have been applied in 3D.

In this study we apply three variational methods to a 3D FWI problem and analyse their performance. Specifically we apply automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), and compare their results and computational costs. These tests show that ADVI is the least computationally demanding method, but its results are systematically biased as uncertainty is underestimated. The method might therefore be used to provide relatively rapid but approximate insights into the Bayesian solution. SVGD demands the highest computational cost, yet produces equally biased results. Adding a randomized term in the SVGD dynamics produces sSVGD, a Markov chain Monte Carlo method based on variational principles. This provides the most accurate results, at intermediate computational cost. We conclude that 3D variational full-waveform inversion is practically applicable, at least in small problems, and can be used to image the Earth’s interior and to provide reasonable uncertainty estimates on those images.