  Seismic tomography is a methodology to image the interior of solid or fluid
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 they are generally
computationally intractable for large datasets and high-dimensional parameter
spaces. To extend uncertainty analysis to larger systems we use 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 probabilistic results. In this study,
we applied two variational methods, automatic differential variational
inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic
tomography problems using both synthetic and real data and we compare the
results to those from two different Monte Carlo sampling methods. The results
show that variational inference methods can produce accurate approximations to
the results of Monte Carlo sampling methods at significantly lower
computational cost, provided that gradients of parameters with respect to data
can be calculated efficiently. We expect that the methods can be applied
fruitfully to many other types of geophysical inverse problems.
