# Bayesian three-dimensional seismic travel-time tomography for active- and passive-source seismic data using physics-informed neural network

arXiv: https://arxiv.org/abs/2606.21789 · HTML: https://mlsanity.com/paper/2606.21789/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 43 references that are in the map.
- First posted on arXiv 19 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Mish: A Self Regularized Non-Monotonic Activation Function](https://mlsanity.com/paper/1908.08681/) (2019)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Function Space Particle Optimization for Bayesian Neural Networks](https://mlsanity.com/paper/1902.09754/) (2019)
- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [DPVI: A Dynamic-Weight Particle-Based Variational Inference Framework](https://mlsanity.com/paper/2112.00945/) (2021)

## Abstract (arXiv)

Accurate 3D seismic velocity modeling through seismic travel-time tomography using both active- and passive-source data provides critical underpinning models for seismicity monitoring and hazard assessment. Because travel-time tomography is an inherently ill-posed inverse problem, UQ of the estimated models using Bayesian methods is also important for reliable downstream interpretations and analyses. However, Bayesian inference for 3D tomography based on conventional grid-based representations faces the ``curse of dimensionality'' and severe computational bottlenecks. Consequently, rigorous Bayesian UQ for margin-wide 3D travel-time tomography has remained largely unexplored. In this study, we propose a meshless 3D Bayesian travel-time tomography method that combines PINNs with a neural representation of the velocity structure, enabling tractable and data-efficient Bayesian inference through function-space particle-based variational inference. To efficiently integrate passive-source data into the Bayesian estimation of the velocity structure, we conduct analytical marginalization treating uncertain source parameters as nuisance parameters, with passive-source relocation carried out in post-processing. We validated the capability of our approach for 3D problems through synthetic experiments. Furthermore, we applied the method to a real-world dataset from marine active-source surveys and natural earthquakes off the Kii Peninsula, Nankai Trough. Our probabilistic 3D ensemble successfully resolves key geological features and provides data-consistent uncertainty maps. The posterior mean hypocenters shifted mainly in the vertical direction by 10-15 km, consistent with a previous relocation result. Finally, the neural representation drastically reduces storage requirements for the entire ensemble velocity model, highlighting the scalability and data efficiency of the proposed framework.