  Uncertainty quantification is crucial to inverse problems, as it could
provide decision-makers with valuable information about the inversion results.
For example, seismic inversion is a notoriously ill-posed inverse problem due
to the band-limited and noisy nature of seismic data. It is therefore of
paramount importance to quantify the uncertainties associated to the inversion
process to ease the subsequent interpretation and decision making processes.
Within this framework of reference, sampling from a target posterior provides a
fundamental approach to quantifying the uncertainty in seismic inversion.
However, selecting appropriate prior information in a probabilistic inversion
is crucial, yet non-trivial, as it influences the ability of a sampling-based
inference in providing geological realism in the posterior samples. To overcome
such limitations, we present a regularized variational inference framework that
performs posterior inference by implicitly regularizing the Kullback-Leibler
divergence loss with a CNN-based denoiser by means of the Plug-and-Play
methods. We call this new algorithm Plug-and-Play Stein Variational Gradient
Descent (PnP-SVGD) and demonstrate its ability in producing high-resolution,
trustworthy samples representative of the subsurface structures, which we argue
could be used for post-inference tasks such as reservoir modelling and history
matching. To validate the proposed method, numerical tests are performed on
both synthetic and field post-stack seismic data.
