Seismic Tomography Using Variational Inference Methods

Abstract:

In a variety of geoscientific applications we require maps of subsurface properties together with the corresponding maps of uncertainties to assess their reliability. Seismic tomography is a method that is widely used to generate those maps. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large data sets and high-dimensionality parameter spaces. To extend uncertainty analysis to larger systems, we introduce variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem yet still provide fully nonlinear, probabilistic results. This is achieved by minimizing the Kullback-Leibler (KL) divergence between approximate and target probability distributions within a predefined family of probability distributions.

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, but nominally at increased cost. However, this increase is partly because 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 included in the quoted cost estimates. 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 which is impossible for Monte Carlo methods without incurring probabilistic errors. 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.