3D Variational Full-Waveform Inversion

Seismic full-waveform inversion (FWI) produces high resolution images of the subsurface by exploiting information in full seismic waveforms, and has been applied at global, regional and industrial spatial scales. FWI is 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 model. In addition, the optimization methods cannot produce accurate uncertainty estimates, which are required to better interpret the results.

To estimate uncertainties more accurately, nonlinear Bayesian methods have been deployed to solve the FWI problem. Monte Carlo sampling is one such algorithm but it is computationally expensive, and all Markov chain Monte Carlo-based methods are difficult to parallelise fully. Variational inference provides an efficient, 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 (VFWI) have been applied previously to solve 2D FWI problems, but neither of them have been applied to 3D FWI. In this study we apply the VFWI method to a 3D FWI problem. Specifically we use Stein variational gradient descent (SVGD) method to solve the 3D Bayesian FWI problem and to obtain an optimised set of samples of the full posterior probability distribution. The aim of this study is to explore performance of the method in 3D, to assess the computational requirements and to provide useful information for practitioners. Our results demonstrate that the 3D VFWI is practical, at least for small problems, and can be applied to image the subsurface in reality.