Geoscientists use observed data to estimate properties of the Earth’s interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so-called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease to optimise. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate solutions to fully non-linear, high-dimensional Bayesian full waveform inversion problems.

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.