Deep Bayesian inference for seismic imaging with tasks

Abstract

We 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 are 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 developed to translate uncertainty due to noise in the data to the confidence intervals of automatically tracked horizons in the image. The uncertainty in the seismic image is characterized by a convolutional neural network (CNN) that is used to reparameterize the image. To assess these uncertainties, samples are drawn from the posterior distribution of the CNN weights. Compared with 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, including medical imaging, compressive sensing, and diffraction tomography. The method of stochastic gradient Langevin dynamics is used 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 the maximum a posteriori estimate that is prone to overfitting, access to these samples allows 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.