Bayesian Full-waveform Inversion with Weak Priors

Seismic full waveform inversion (FWI) produces high resolution images of the subsurface directly from seismic waveforms, and has been applied at global, regional and industrial spatial scales. FWI is traditionally solved using optimization methods, by iteratively updating a model of the subsurface so as to minimize the misfit between observed waveforms and those predicted by linearised (approximate) physics. Due to the nonlinearity of the physical relation between model parameters and seismic waveforms, a good starting model (derived from other methods) and hence strong prior information is required in order that the linearised physical relationships are reasonable in the vicinity of the true solution. Such linearised methods cannot provide accurate estimates of uncertainties, which are required if we are to understand and interpret the results appropriately.

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 FWI problems, but only with strong prior information about the velocity structure to limit the space of possible models. Unfortunately such strong information is almost never available in practice. In addition, VFWI has only been applied to wavefield transmission problems in which seismic data are recorded on a receiver array that lies above the structure to be imaged, given known, double-couple (earthquake-like) sources located underneath the same structure. In practice, knowledge of such sources is never definitive, and usually depends circularly on the unknown structure itself. In this study, we present the first application of VFWI to seismic reflection data acquired from known (deliberately fired) near-surface sources. We also apply variational inference (specifically, Stein variational gradient descent) to solve FWI problems with realistic prior information. We perform multiple inversions using data from different frequency ranges, and show that VFWI produces high resolution mean and uncertainty models using both low and high frequency data, and given only weak prior information. This is usually impossible using traditional linearised methods. We conclude that VFWI may be a useful method to produce high resolution images and reliable uncertainties, at least in smaller FWI problems.