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.

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