  Travel time tomography for the velocity structure of a medium is a highly
non-linear and non-unique inverse problem. Monte Carlo methods are becoming
increasingly common choices to provide probabilistic solutions to tomographic
problems but those methods are computationally expensive. Neural networks can
often be used to solve highly non-linear problems at a much lower computational
cost when multiple inversions are needed from similar data types. We present
the first method to perform fully non-linear, rapid and probabilistic Bayesian
inversion of travel time data for 2D velocity maps using a mixture density
network. We compare multiple methods to estimate probability density functions
that represent the tomographic solution, using different sets of prior
information and different training methodologies. We demonstrate the importance
of prior information in such high dimensional inverse problems due to the curse
of dimensionality: unrealistically informative prior probability distributions
may result in better estimates of the mean velocity structure, however the
uncertainties represented in the posterior probability density functions then
contain less information than is obtained when using a less informative prior.
This is illustrated by the emergence of uncertainty loops in posterior standard
deviation maps when inverting travel time data using a less informative prior,
which are not observed when using networks trained on prior information that
includes (unrealistic) a priori smoothness constraints in the velocity models.
We show that after an expensive program of training the networks, repeated
high-dimensional, probabilistic tomography is possible on timescales of the
order of a second on a standard desktop computer.
