# Learning by example: fast reliability-aware seismic imaging with normalizing flows

arXiv: https://arxiv.org/abs/2104.06255 · HTML: https://mlsanity.com/paper/2104.06255/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 13 Apr 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [Invert to Learn to Invert](https://mlsanity.com/paper/1911.10914/) (2019)
- [Fully Hyperbolic Convolutional Neural Networks](https://mlsanity.com/paper/1905.10484/) (2019)
- [Parameterizing uncertainty by deep invertible networks, an application to reservoir characterization](https://mlsanity.com/paper/2004.07871/) (2020)
- [A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification](https://mlsanity.com/paper/2001.04567/) (2020)
- [Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows](https://mlsanity.com/paper/2007.07985/) (2020)
- [Learned imaging with constraints and uncertainty quantification](https://mlsanity.com/paper/1909.06473/) (2019)
- [Uncertainty quantification in imaging and automatic horizon tracking: a Bayesian deep-prior based approach](https://mlsanity.com/paper/2004.00227/) (2020)
- [Fully reversible neural networks for large-scale 3D seismic horizon tracking](https://mlsanity.com/paper/2003.08466/) (2020)

## Built on by (most cited first, 10 of 10)

- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [Solving multiphysics-based inverse problems with learned surrogates and constraints](https://mlsanity.com/paper/2307.11099/) (2023)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [Uncertainty quantification for ptychography using normalizing flows](https://mlsanity.com/paper/2111.00745/) (2021)
- [Velocity continuation with Fourier neural operators for accelerated uncertainty quantification](https://mlsanity.com/paper/2203.14386/) (2022)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Priors learned from legacy reconstructions inherit undetectable overconfidence](https://mlsanity.com/paper/2607.21721/) (2026)

## Abstract (arXiv)

Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low- and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the image's reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.