# A prior regularized full waveform inversion using generative diffusion models

arXiv: https://arxiv.org/abs/2306.12776 · HTML: https://mlsanity.com/paper/2306.12776/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 22 Jun 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 7 of 7)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [Deep-Learning Inversion of Seismic Data](https://mlsanity.com/paper/1901.07733/) (2019)
- [MLReal: Bridging the gap between training on synthetic data and real data applications in machine learning](https://mlsanity.com/paper/2109.05294/) (2021)
- [Sparse-promoting Full Waveform Inversion based on Online Orthonormal Dictionary Learning](https://mlsanity.com/paper/1511.05194/) (2015)

## Built on by (most cited first, 30 of 31)

- [Controllable seismic velocity synthesis using generative diffusion models](https://mlsanity.com/paper/2402.06277/) (2024)
- [Paired Autoencoders for Likelihood-free Estimation in Inverse Problems](https://mlsanity.com/paper/2405.13220/) (2024)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Physics-informed waveform inversion using pretrained wavefield neural operators](https://mlsanity.com/paper/2509.08967/) (2025)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [Automatic Differentiation-based Full Waveform Inversion with Flexible Workflows](https://mlsanity.com/paper/2412.00486/) (2024)
- [A generative foundation model for an all-in-one seismic processing framework](https://mlsanity.com/paper/2502.01111/) (2025)
- [Stochastic full waveform inversion with deep generative prior for uncertainty quantification](https://mlsanity.com/paper/2406.04859/) (2024)
- [Diffusion prior as a direct regularization term for FWI](https://mlsanity.com/paper/2506.10141/) (2025)
- [Diffusion Model-Based Posterior Sampling in Full Waveform Inversion](https://mlsanity.com/paper/2512.12797/) (2025)
- [Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields](https://mlsanity.com/paper/2512.22421/) (2025)
- [Decoupled Latent Optimization of Diffusion Models for Full Waveform Inversion](https://mlsanity.com/paper/2606.14139/) (2026)
- [SG-DeepONet: Source-generalized deep operator learning for full waveform inversion](https://mlsanity.com/paper/2408.08005/) (2024)
- [Geological and Well prior assisted full waveform inversion using conditional diffusion models](https://mlsanity.com/paper/2412.06959/) (2024)
- [Diffusion models for multivariate subsurface generation and efficient probabilistic inversion](https://mlsanity.com/paper/2507.15809/) (2025)
- [Diffusion-based subsurface CO2 multiphysics monitoring and forecasting](https://mlsanity.com/paper/2407.18426/) (2024)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)
- [Full waveform inversion with CNN-based velocity representation extension](https://mlsanity.com/paper/2504.15826/) (2025)
- [Seismic Acoustic Impedance Inversion Framework Based on Conditional Latent Generative Diffusion Model](https://mlsanity.com/paper/2506.13529/) (2025)
- [Acoustic Waveform Inversion with Image-to-Image Schr"odinger Bridges](https://mlsanity.com/paper/2506.15346/) (2025)
- [Inspired by machine learning optimization: can gradient-based optimizers solve cycle skipping in full waveform inversion given sufficient iterations?](https://mlsanity.com/paper/2509.14919/) (2025)
- [Diffusion priors enhanced velocity model building from time-lag images using a neural operator](https://mlsanity.com/paper/2512.23375/) (2025)
- [Improving Full Waveform Inversion in Large Model Era](https://mlsanity.com/paper/2603.00377/) (2026)
- [Self-Flow-Matching assisted Full Waveform Inversion](https://mlsanity.com/paper/2603.13425/) (2026)
- [Deciphering Neural Reparameterized Full-Waveform Inversion with Neural Sensitivity Kernel and Wave Tangent Kernel](https://mlsanity.com/paper/2605.14370/) (2026)
- [SubsurfaceGen: Procedural Generation of Field-Scale Earth Models and Seismic Data](https://mlsanity.com/paper/2605.30541/) (2026)
- [Accelerating physics-informed neural networks for full waveform inversion using a hybrid quantum-classical finite-basis architecture](https://mlsanity.com/paper/2606.01110/) (2026)
- [Priors learned from legacy reconstructions inherit undetectable overconfidence](https://mlsanity.com/paper/2607.21721/) (2026)
- [Physical-State-Guided Diffusion Sampling for Full-Waveform Inversion](https://mlsanity.com/paper/2609.12899/) (2026)
- [Depth-progressive 3D seismic velocity model building via 2D generative diffusion models](https://mlsanity.com/paper/2610.14812/) (2026)

## Abstract (arXiv)

Full waveform inversion (FWI) has the potential to provide high-resolution subsurface model estimations. However, due to limitations in observation, e.g., regional noise, limited shots or receivers, and band-limited data, it is hard to obtain the desired high-resolution model with FWI. To address this challenge, we propose a new paradigm for FWI regularized by generative diffusion models. Specifically, we pre-train a diffusion model in a fully unsupervised manner on a prior velocity model distribution that represents our expectations of the subsurface and then adapt it to the seismic observations by incorporating the FWI into the sampling process of the generative diffusion models. What makes diffusion models uniquely appropriate for such an implementation is that the generative process retains the form and dimensions of the velocity model. Numerical examples demonstrate that our method can outperform the conventional FWI with only negligible additional computational cost. Even in cases of very sparse observations or observations with strong noise, the proposed method could still reconstruct a high-quality subsurface model. Thus, we can incorporate our prior expectations of the solutions in an efficient manner. We further test this approach on field data, which demonstrates the effectiveness of the proposed method.