# Deep Bayesian inference for seismic imaging with tasks

arXiv: https://arxiv.org/abs/2110.04825 · HTML: https://mlsanity.com/paper/2110.04825/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 128 references that are in the map.
- First posted on arXiv 10 Oct 2021.

## Builds on (most cited first, 30 of 45)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Subspace Inference for Bayesian Deep Learning](https://mlsanity.com/paper/1907.07504/) (2019)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [A Bayesian Perspective on the Deep Image Prior](https://mlsanity.com/paper/1904.07457/) (2019)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [Loss Landscapes of Regularized Linear Autoencoders](https://mlsanity.com/paper/1901.08168/) (2019)
- [On the distance between two neural networks and the stability of learning](https://mlsanity.com/paper/2002.03432/) (2020)
- [Trumpets: Injective Flows for Inference and Inverse Problems](https://mlsanity.com/paper/2102.10461/) (2021)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Regularization by architecture: A deep prior approach for inverse problems](https://mlsanity.com/paper/1812.03889/) (2018)
- [A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions](https://mlsanity.com/paper/2010.09800/) (2020)
- [Deep Learning in Photoacoustic Tomography: Current approaches and future directions](https://mlsanity.com/paper/2009.07608/) (2020)

## Built on by (most cited first, 12 of 12)

- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)
- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Velocity continuation with Fourier neural operators for accelerated uncertainty quantification](https://mlsanity.com/paper/2203.14386/) (2022)
- [Deep Variational Inverse Scattering](https://mlsanity.com/paper/2212.04309/) (2022)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Dual-space posterior sampling for Bayesian inference in constrained inverse problems](https://mlsanity.com/paper/2603.00393/) (2026)
- [On the role of memorization in learned priors for geophysical inverse problems](https://mlsanity.com/paper/2603.19629/) (2026)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)
- [Priors learned from legacy reconstructions inherit undetectable overconfidence](https://mlsanity.com/paper/2607.21721/) (2026)

## Abstract (arXiv)

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.