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arxiv: 2606.21789 · v1 · pith:YF5MU2Y6new · submitted 2026-06-19 · ⚛️ physics.geo-ph · cs.LG

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

Ryoichiro Agata , Kazuya Shiraishi , Gou Fujie , Dan Bassett This is my paper

Pith reviewed 2026-06-26 12:11 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.LG
keywords seismic tomographyBayesian inferencephysics-informed neural networkstravel-time tomographyuncertainty quantificationactive-source datapassive-source dataneural velocity representation
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The pith

A neural representation of velocity structure enables tractable Bayesian 3D seismic travel-time tomography with active and passive sources.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a meshless 3D Bayesian travel-time tomography method that pairs physics-informed neural networks with a neural representation of the velocity field. The approach performs inference via function-space particle-based variational inference and analytically marginalizes uncertain passive-source parameters as nuisance variables. Traditional grid-based Bayesian methods encounter prohibitive computational costs in three dimensions, leaving rigorous uncertainty quantification largely out of reach for margin-scale problems. The new method therefore targets the practical need for probabilistic velocity models that support seismicity monitoring and hazard assessment.

Core claim

The central claim is that the neural velocity representation combined with function-space particle-based variational inference makes full Bayesian estimation tractable and data-efficient for three-dimensional travel-time tomography, while analytical marginalization of passive-source parameters allows joint use of active- and passive-source data without explicit joint sampling. Synthetic tests confirm recovery of known structures, and application to marine active-source and earthquake data off the Kii Peninsula yields an ensemble that resolves geological features, supplies spatially varying uncertainty, and reduces storage for the full posterior.

What carries the argument

Meshless neural representation of the velocity structure together with function-space particle-based variational inference and analytical marginalization of passive-source parameters.

If this is right

  • The method recovers key geological features from a real marine dataset off the Kii Peninsula.
  • It produces spatially varying, data-consistent uncertainty maps across the velocity volume.
  • Posterior hypocenters shift 10-15 km mainly in the vertical direction, consistent with prior relocation studies.
  • Storage cost for the entire ensemble of velocity models drops dramatically compared with grid-based storage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The storage reduction from the neural representation could make ensemble modeling feasible at regional or global scales where grid storage is prohibitive.
  • The analytical marginalization step might transfer to other geophysical inverse problems that treat source or instrument parameters as nuisance variables.
  • Joint inversion with additional data types such as gravity could be tested by extending the same neural representation and inference scheme.

Load-bearing premise

The neural network must be expressive enough to capture the true velocity structure and the variational approximation must be close enough to the true posterior for the reported uncertainties to be reliable.

What would settle it

A side-by-side comparison, on a known synthetic 3D velocity model, of the posterior mean and credible intervals obtained by this method versus those obtained by a conventional grid-based Bayesian tomography code.

Figures

Figures reproduced from arXiv: 2606.21789 by Dan Bassett, Gou Fujie, Kazuya Shiraishi, Ryoichiro Agata.

Figure 1
Figure 1. Figure 1: Schematic flow of the proposed PINN-based Bayesian travel-time tomography using fParVI. (a) (b) 2(a) 2(b) 3(a) 3(b) −70 −60 −50 −40 −30 −20 −10 0 10 z (km) Mean Mean±3sigma 2.5 5.0 7.5 Vp (km/s) (c) [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Analysis setting for the synthetic numerical experiment. (a) True 3D P-wave velocity structure used to generate the synthetic travel-time data and locations of active sources and their receivers. (b) Distribution of passive sources, their receivers, and incorrect source-location guesses adopted in Case 2. (c) The mean and ±3 − σ range of the prior velocity distribution [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 3
Figure 3. Figure 3: Cross-sections of the synthetic numerical experiment Case 1 in the x-direction at x = 10 and 70 km. From top to bottom, the panels show (a) the true model, (b) the ensemble mean of the proposed method, and (c) the ensemble standard deviation of the proposed method [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cross-sections of the synthetic numerical experiment Case 1 in the y-direction at y = 30 and 130 km. The panel arrangement is the same as in [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Pointwise priors and histograms of the posterior probability density functions of velocity at the three representative points marked in [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cross-sections of the synthetic numerical experiment Case 2 in the x-direction at x = 10 and 70 km. From top to bottom, the panels show (a) the true model, (b) the ensemble mean of the proposed method, (c) the ensemble standard deviation of the proposed method, (d) the mean of the reference inversion results with fixed incorrect passive-source locations, and (e) the corresponding standard deviation [PITH_… view at source ↗
Figure 7
Figure 7. Figure 7: Cross-sections of the synthetic numerical experiment Case 2 in the y-direction at y = 30 and 130 km. The panel arrangement is the same as in [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Study area and dataset geometry for the application off the Kii Peninsula. (a) The target model domain with the distributions of the active-source shots and receivers. (b) Distribution of passive earthquake sources and their receivers [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cross-sections of the estimated P-wave velocity structure off the Kii Peninsula. Panels (a)–(d) show the sections along x = 20, 62.5, and 100 km and along y = 140 km, respectively. For each section, the upper, middle, and lower rows show the ensemble mean of our result, the corresponding standard deviation, and the model of Arnulf et al. (2022), respectively. SE, NW, SW, NE indicate the directions of the s… view at source ↗
Figure 10
Figure 10. Figure 10: Pointwise prior and histogram of the posterior probability density functions of velocity at the three representative points marked in [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representative posterior sample corresponding to the cross-section of [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relocation of JMA-catalog hypocenters for passive sources. (a) Map view comparing prior and pos￾terior mean source locations. (b)-(c) Cross-sectional views comparing the posterior mean and standard deviation of source locations. SE and NW indicate the directions of the sections [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Examples of 3D velocity models sampled from the 255-member posterior ensemble for the Kii Peninsula application. The solid black lines denote the coastlines, below which lies the terrestrial region [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to introduce a meshless 3D Bayesian travel-time tomography method that represents velocity structure via a neural network within a PINN framework, performs tractable Bayesian inference using function-space particle-based variational inference, and analytically marginalizes over uncertain passive-source parameters (with post-processing relocation). Synthetic experiments validate the approach for 3D problems, and application to marine active-source and earthquake data from the Nankai Trough yields an ensemble that resolves key geological features, supplies data-consistent uncertainty maps, and produces hypocenter shifts of 10-15 km vertically that match prior results; the neural representation is also noted to reduce ensemble storage requirements.

Significance. If the central claims hold, the work is significant because it provides a scalable route to rigorous Bayesian UQ for margin-scale 3D tomography, directly addressing the curse of dimensionality that has limited such analyses. Credit is due for the synthetic experiments that test the full pipeline and the real-data application to the Nankai Trough that demonstrates consistency with independent relocation results; the neural representation's storage reduction is a practical strength for ensemble dissemination.

minor comments (3)
  1. [§4.2] §4.2: the convergence diagnostics and sensitivity tests for the particle-based VI (e.g., number of particles, learning-rate schedules) are only summarized; explicit reporting of these choices and their effect on posterior spread would strengthen reproducibility.
  2. [Figure 7] Figure 7 and associated text: the vertical hypocenter shifts are stated as 10-15 km but the figure panels do not include error bars or the full posterior marginals for the relocated events; adding these would clarify the uncertainty quantification.
  3. [§3] The notation for the neural velocity field (e.g., the precise form of the PINN loss and the parameterization of the velocity network) is introduced in §3 but the explicit functional form is not restated when the marginalization step is derived; a short equation block linking the two would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive and positive review, which recognizes the significance of the meshless Bayesian PINN framework for 3D travel-time tomography and its application to the Nankai Trough dataset. The recommendation for minor revision is appreciated. No specific major comments are listed in the report, so we have no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proposes a meshless Bayesian tomography method using PINNs for neural velocity representation and function-space particle-based variational inference, with analytical marginalization for passive sources. All load-bearing steps rely on standard PINN physics constraints and VI approximations that are validated directly against synthetic experiments and independent real-world Nankai Trough data; no derivation reduces by construction to fitted inputs, self-citations, or renamed ansatzes within the manuscript. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are stated beyond standard assumptions of PINN solvability and variational approximation quality.

pith-pipeline@v0.9.1-grok · 5834 in / 1083 out tokens · 26015 ms · 2026-06-26T12:11:56.057431+00:00 · methodology

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Reference graph

Works this paper leans on

56 extracted references · 3 canonical work pages · 1 internal anchor

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