Seismic full-waveform inversion (FWI) can produce high resolution images of the Earth’s subsurface. Since full waveform modelling is significantly nonlinear with respect to velocities, Monte Carlo methods have been used to assess image uncertainties. However, because of the high computational cost of Monte Carlo sampling methods, uncertainty assessment remains intractable for larger data sets and 3D applications. In this study we propose a new method called variational full-waveform inversion (VFWI), which uses Stein variational gradient descent (SVGD) to solve FWI problems. We apply the method to a 2D synthetic example and demonstrate that the method produces accurate approximations to those obtained by Hamiltonian Monte Carlo (HMC). Since variational inference solves the problem using optimization, the method can be applied to larger datasets and 3D applications by using stochastic optimization and distributed optimization.

The ultimate goal of a scientific investigation is usually to find answers to specific, often low-dimensional questions: what is the size of a subsurface body? Does a hypothesised subsurface feature exist? Existing information is reviewed, an experiment is designed and performed to acquire new data, and the most likely answer is estimated. Typically the answer is interpreted from geological and geophysical data or models, but is biased because only one particular forward function is considered, one inversion method is applied, and because human interpretation is a biased process. Interrogation theory provides a systematic way to answer specific questions by combining forward, design, inverse and decision theories. The optimal answer is made more robust since it balances multiple possible forward models, inverse algorithms and model parametrizations, probabilistically. In a synthetic test, we evaluate the area of a low-velocity anomaly by interrogating Bayesian tomographic results. By combining the effect of four inversion algorithms, the optimal answer is very close to the true answer, even on a coarsely gridded parametrisation. In a field data test, we evaluate the volume of the East Irish Sea basins using 3D shear wave speed depth inversion results. This example shows that interrogation theory provides a useful way to answer realistic questions about the Earth. A key revelation is that while the majority of computation may be spent solving inverse problem, much of the skill and effort involved in answering questions may be spent defining and calculating those target function values in a clear and unbiased manner.