  Most geophysical inverse problems are nonlinear and rely upon numerical
forward solvers involving discretization and simplified representations of the
underlying physics. As a result, forward modeling errors are inevitable. In
practice, such model errors tend to be either completely ignored, which leads
to biased and over-confident inversion results, or only partly taken into
account using restrictive Gaussian assumptions. Here, we rely on deep
generative neural networks to learn problem-specific low-dimensional
probabilistic representations of the discrepancy between high-fidelity and
low-fidelity forward solvers. These representations are then used to
probabilistically invert for the model error jointly with the target
geophysical property field, using the computationally-cheap, low-fidelity
forward solver. To this end, we combine a Markov-chain-Monte-Carlo (MCMC)
inversion algorithm with a trained convolutional neural network of the spatial
generative adversarial network (SGAN) type, whereby at each MCMC step, the
simulated low-fidelity forward response is corrected using a proposed
model-error realization. Considering the crosshole ground-penetrating radar
traveltime tomography inverse problem, we train SGAN networks on traveltime
discrepancy images between: (1) curved-ray (high fidelity) and straight-ray
(low fidelity) forward solvers; and (2) finite-difference-time-domain (high
fidelity) and straight-ray (low fidelity) forward solvers. We demonstrate that
the SGAN is able to learn the spatial statistics of the model error and that
suitable representations of both the subsurface model and model error can be
recovered by MCMC. In comparison with inversion results obtained when model
errors are either ignored or approximated by a Gaussian distribution, we find
that our method has lower posterior parameter bias and better explains the
observed traveltime data.[...]
