Variational Probabilistic Tomography

Seismic Tomography is a method to image the interior of solid media, and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose but the ‘curse of dimensionality’ generally makes them computationally intractable for large data sets and high-dimensional parameter spaces. To extend uncertainty analysis to larger systems we introduce variational inference methods to conduct seismic tomography. In contrast to Monte Carlo stochastic sampling, variational methods solve the Bayesian inference problem as an optimization problem, yet still provide probabilistic results.

We apply variational inference to solve two types of tomographic problems using synthetic and real data: travel time tomography and full waveform inversion. We test two different variational methods: automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD). In each case we compare the results to solutions given by a variety of Monte Carlo methods.

In the travel time tomography example we show that ADVI provides a robust mean velocity model but biased uncertainties due to an implicit Gaussian approximation, and that it cannot be used to find multi-modal Bayesian posterior probability distributions. SVGD produces an accurate match to the fully probabilistic results of Markov chain Monte Carlo analysis, but at significantly reduced computational cost – provided that gradients of model parameters with respect to data can be calculated efficiently.

In our waveform inversion example, the SVGD method produces results of similar quality to published results from an efficient Hamiltonian Monte Carlo analysis, at around the same cost. However, that particular Monte Carlo method has significant ‘hidden’ costs: these are necessarily incurred by running a substantial number of pre-run tests to determine suitable settings of run-time parameters, and are not generally included in quoted cost estimates. By contrast, SVGD has relatively low pre-run costs. In addition, SVGD is significantly easier to parallelize, and for very large problems can be run in minibatch mode; this is impossible for Monte Carlo methods without incurring probabilistic errors as so-called ‘detailed balance’ can not be maintained in minibatch Hamiltonian Monte Carlo. We therefore contend that the variational method may have greater potential to extend probabilistic analysis to higher dimensional tomographic systems than current Monte Carlo methods.