Bayesian full waveform inversion of surface waves with annealed stein variational gradient descent

Elastic full-waveform inversion has recently been utilized to estimate the physical properties of the upper tens of metres of the subsurface, leveraging its capability to exploit the complete information contained in recorded seismograms. However, due to the nonlinear and ill-posed nature of the problem, standard approaches typically require an optimal starting model to avoid producing non-physical solutions. Additionally, conventional optimization methods lack a robust uncertainty quantification, which is essential for subsequent informed decision-making. Bayesian inference offers a framework for estimating the posterior probability density function through the application of Bayes' theorem. Methods based on Markov Chain Monte Carlo processes use multiple sample chains to quantify and characterize the uncertainty of the solution. However, despite their ability to theoretically handle any form of distribution, these methods are computationally expensive, limiting their usage in large-scale problems with computationally expensive forward modellings, as in the case of full-waveform inversion. Variational inference provide an alternative approach to estimating the posterior distribution through a parametric or non-parametric proposal distribution. Among this class of methods, stein variational gradient descent stands out for its ability to iteratively refine a set of samples, usually referred to as particles, to approximate the target distribution through an optimization process. However, mode and variance-collapse issues affect this approach when applied to high-dimensional inverse problems. To address these challenges, in this work we propose to utilize an annealed variant of the stein variational gradient descent algorithm and apply this method to solve the elastic full-waveform inversion of surface waves. We validate our proposed approach with a synthetic test, where the velocity model is characterized by significant lateral and vertical velocity variations. Then, we invert a field data set from the InterPACIFIC project, proving that our method is robust against cycle-skipping issues and can provide reasonable uncertainty estimations with a limited computational cost.