  We propose to use techniques from Bayesian inference and deep neural networks
to translate uncertainty in seismic imaging to uncertainty in tasks performed
on the image, such as horizon tracking. Seismic imaging is an ill-posed inverse
problem because of bandwidth and aperture limitations, which is hampered by the
presence of noise and linearization errors. Many regularization methods, such
as transform-domain sparsity promotion, have been designed to deal with the
adverse effects of these errors, however, these methods run the risk of biasing
the solution and do not provide information on uncertainty in the image space
and how this uncertainty impacts certain tasks on the image. A systematic
approach is proposed to translate uncertainty due to noise in the data to
confidence intervals of automatically tracked horizons in the image. The
uncertainty is characterized by a convolutional neural network (CNN) and to
assess these uncertainties, samples are drawn from the posterior distribution
of the CNN weights, used to parameterize the image. Compared to traditional
priors, it is argued in the literature that these CNNs introduce a flexible
inductive bias that is a surprisingly good fit for a diverse set of problems.
The method of stochastic gradient Langevin dynamics is employed to sample from
the posterior distribution. This method is designed to handle large scale
Bayesian inference problems with computationally expensive forward operators as
in seismic imaging. Aside from offering a robust alternative to maximum a
posteriori estimate that is prone to overfitting, access to these samples allow
us to translate uncertainty in the image, due to noise in the data, to
uncertainty on the tracked horizons. For instance, it admits estimates for the
pointwise standard deviation on the image and for confidence intervals on its
automatically tracked horizons.
