  To obtain high-resolution images of subsurface structures from seismic data,
seismic imaging techniques such as Full Waveform Inversion (FWI) serve as
crucial tools. However, FWI involves solving a nonlinear and often non-unique
inverse problem, presenting challenges such as local minima trapping and
inadequate handling of inherent uncertainties. In addressing these challenges,
we propose leveraging deep generative models as the prior distribution of
geophysical parameters for stochastic Bayesian inversion. This approach
integrates the adjoint state gradient for efficient back-propagation from the
numerical solution of partial differential equations. Additionally, we
introduce explicit and implicit variational Bayesian inference methods. The
explicit method computes variational distribution density using a normalizing
flow-based neural network, enabling computation of the Bayesian posterior of
parameters. Conversely, the implicit method employs an inference network
attached to a pretrained generative model to estimate density, incorporating an
entropy estimator. Furthermore, we also experimented with the Stein Variational
Gradient Descent (SVGD) method as another variational inference technique,
using particles. We compare these variational Bayesian inference methods with
conventional Markov chain Monte Carlo (McMC) sampling. Each method is able to
quantify uncertainties and to generate seismic data-conditioned realizations of
subsurface geophysical parameters. This framework provides insights into
subsurface structures while accounting for inherent uncertainties.
