Physically Structured Variational Inference for Bayesian Full Waveform Inversion

Full waveform inversion (FWI) has become a commonly used tool to obtain high resolution subsurface images from seismic waveform data. Typically, FWI is solved using a local optimisation method which finds one model that best fits observed data. Due to the high non-linearity and non-uniqueness of FWI problems, finding globally best-fitting solutions is not necessarily desirable since they fit noise in the data, and quantifying uncertainties in the solution is challenging. In principle, Bayesian FWI calculates a posterior probability distribution function (pdf), which describes all possible model solutions and their probabilities. However, characterising the posterior pdf by sampling alone is often intractably expensive due to the high dimensionality of FWI problems and the computational expense of their forward functions. Alternatively, variational inference solves Bayesian FWI problems efficiently by minimising the difference between a predefined (variational) family of distributions and the true posterior distribution, requiring optimisation rather than random sampling. We propose a new variational methodology called physically structured variational inference (PSVI), in which a physics-based structure is imposed on the variational family. In a simple example motivated by prior information from past FWI solutions, we include parameter correlations between pairs of spatial locations within a dominant wavelength of each other, and set other correlations to zero. This makes the method far more efficient in terms of both memory requirements and computational cost. We demonstrate the proposed method with a 2D acoustic FWI scenario, and compare the results with those obtained using three other variational methods. This verifies that the method can produce accurate statistical information of the posterior distribution with significantly improved efficiency.