Pith. sign in

REVIEW 3 major objections 4 minor 32 references

Solving All Seismic Tomographic Problems using Deep Learning

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that a graph mixture-density network, trained once on random 2D synthetic velocity models and their travel times, can estimate Bayesian posterior distributions for seismic tomography problems at other scales, station…

desk verdict A genuinely useful extension of graph MDNs to variable geometry and scale, with two real-data demos, but the 'comparable to McMC/SVGD' claim rests on visual comparison and the authors' own admitted discrepancies. read the letter →

arxiv 2504.14830 v1 pith:YK57DAXE submitted 2025-04-21 physics.geo-ph physics.data-an

Xin Zhang , Kaiwen Xia This is my paper · ORCID
classification physics.geo-phphysics.data-an
keywords seismictomographygraphneuralnetworksmixturedensityBayesianinversionsurfacewavetraveltimeuncertaintyquantificationdeeplearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a single neural network can solve seismic tomography problems it was never trained on, across different region sizes, station layouts, and data counts. The authors train a graph mixture density network on random 2D velocity models in a small 10 km by 10 km region with 30 randomly placed stations, using fast-marching travel times as data. Once trained, the same network is applied without retraining to larger synthetic problems, to Love-wave group-velocity data from 15 stations in South England, and to Rayleigh-wave phase-velocity data from 57 stations in Southwest China. In each case the network returns posterior mean and standard-deviation maps in about 0.6 seconds per period, and the paper reports these are broadly consistent with posteriors from Markov chain Monte Carlo and Stein variational gradient descent, which take hours. If correct, this would make fast, uncertainty-aware seismic imaging available for arbitrary new regions without bespoke Bayesian computation.

What carries the argument

The central object is the graph mixture density network (MDN): a graph neural network that outputs a mixture-of-Gaussians probability density instead of a point estimate. Seismic stations are graph nodes with normalized coordinates as node features; each station pair with a travel-time measurement defines an edge whose feature is the path-averaged velocity (travel time divided by inter-station distance). Graph transformer layers update node features using attention over neighbors, a global mean pooling layer aggregates nodes, and a linear layer outputs the mixture coefficients, means, and standard deviations of 10 Gaussian kernels, giving a diagonal-covariance approximation to the posterior over the 11 by 11 velocity grid. Training minimizes the negative log-likelihood of the mixture, with random dropping of stations and travel times so the network learns to handle variable data sizes. The scale-invariance property does the transfer work: proportional scaling of region and receivers preserves path-averaged velocity, so a network trained at one scale can be applied at another.

What would settle it

Generate a synthetic test set with a full 3D or anisotropic wave-equation forward model (or use real data known to contain strong off-path or depth-dependent sensitivity), then compare the trained network's posterior against a Monte Carlo inversion that uses the same correct forward model; if the network's mean or uncertainty falls outside the Monte Carlo posterior, the claim that one network solves all tomographic problems fails for that class.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that acquisition geometry and scale do not have to be fixed at training time for a neural-network tomographic solver. Because travel-time data are encoded as a graph whose node features are normalized station coordinates and whose edge features are path-averaged velocities, the representation is scale-invariant: for a given velocity structure, scaling the region and station layout proportionally leaves the path-averaged velocity unchanged. A graph MDN trained on random small-region synthetic models therefore transfers to larger regions, different station numbers, and real data at other sites. The paper applies the unchanged network to 20 x 20 km synthetic tests, South England group-velocity tomography at 4–14 s periods, and Southwest China phase-velocity tomography at 5–30 s periods, and reports that the resulting posterior means and standard deviations are broadly similar to those produced by McMC and SVGD, with differences attributed to insufficient training of the MDN or incomplete convergence of the samplers.

Load-bearing premise

Real surface-wave travel times must behave like the synthetic 2D path-averaged ray-model data used in training, so that scaling the region and receivers proportionally leaves the underlying velocity structure unchanged and no 3D, depth-dependent, or anisotropic sensitivity enters.

Editorial extensions

If this is right

  • A network trained once on random synthetic models can produce posterior mean and uncertainty maps for other 2D surface-wave tomography problems with different numbers of stations and different region sizes, without retraining.
  • Posterior inference becomes near-instant: about 0.6 seconds per period compared with roughly 16 hours for McMC and 3.9 hours for SVGD in the paper's examples.
  • Uncertainty maps can flag well-constrained anomalies and poorly covered regions in new datasets, just as full Bayesian inversions do.
  • The same approach could in principle be extended to 3D body-wave tomography or full waveform inversion by treating sources and receivers (or waveforms) as graph nodes and node features, though the authors note larger networks would be required.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is calibration: check whether the MDN's posterior standard deviations match the actual frequency of containing the true model when tested on many out-of-distribution synthetic models; the paper shows visual similarity to Bayesian baselines but does not report a quantitative calibration metric.
  • The transfer claim rests on period-by-period path-averaged velocity being scale-invariant; one could extend the same network to continuous dispersion curves by adding period as a node or edge feature, a step the paper does not take.
  • For real applications, prior choice matters: the posterior is conditioned on the training prior of 1.8–4.0 km/s on an 11x11 grid, so regions with velocities outside that range or needing finer spatial resolution may require retraining or a different architecture.
  • If the approach does generalize in practice, the largest payoff would be in rapid aftershock or volcano monitoring, where a pre-trained network could update probabilistic velocity images as data stream in, something the current hours-long Bayesian samplers cannot do.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a graph mixture density network (graph MDN) for 2D seismic travel-time tomography, trained on synthetic velocity models with random station distributions, and applies it to problems with different scales and acquisition geometries. The central claim is that a single trained network can provide posterior estimates 'comparable' to traditional Bayesian methods (McMC, SVGD) in seconds for unseen sites, as demonstrated on synthetic tests and two real datasets (South England Love-wave group velocities and Southwest China Rayleigh-wave phase velocities). The authors report computational costs and discuss limitations, including a fixed regular parameterization grid and possible insufficient training.

Significance. If the central claim holds, the work would provide a fast, amortized approximation to Bayesian seismic tomography that generalizes across acquisition geometries and scales, which is a practically valuable contribution. The method's strengths include a flexible graph representation of travel-time data, random training geometries that go beyond fixed-array networks, and two real-data demonstrations with reported runtime comparisons. However, the claim of 'comparable' posterior solutions is currently supported only by qualitative visual inspection, and the underlying domain assumptions about 2D ray theory and scale invariance are not tested against data that violate them. The paper would be strengthened substantially by quantitative accuracy/calibration metrics and a more carefully scoped statement of applicability.

major comments (3)
  1. [§3.3 and §3.4, Figs. 5–8] The central claim that graph MDNs provide 'comparable' solutions to McMC and SVGD is supported only by side-by-side visual comparison of mean and standard deviation maps. No quantitative discrepancy measures (e.g., mean absolute difference, correlation, structural similarity) or posterior calibration metrics (e.g., credible-interval coverage, probability integral transform histograms) are reported. The authors themselves state in §3.3 that McMC results show 'higher magnitude of velocity variations and lower uncertainty' and attribute the difference to 'insufficient training of the network'; this admitted, unquantified discrepancy concerns exactly the posterior mean and uncertainty that the paper claims to reproduce. I ask for quantitative comparisons on the synthetic examples (where the true model is known) and, if possible, calibration diagnostics on the real-data posteriors.
  2. [§3.1 and §3.4] The load-bearing assumption that a network trained on 2D fast-marching travel times on an 11×11 grid can be applied to real surface-wave data is not tested. The argument that 'path-averaged velocity remains the same for a given velocity structure when the scale of the study area and receiver distribution change proportionally' holds only under strict 2D ray theory and proportional scaling of the whole structure. Real surface-wave dispersion is sensitive to depth-dependent 3D structure, and the real-data comparisons against McMC and SVGD use the same 2D path-average forward model, so they cannot reveal a mismatch between the training distribution and real physics. The paper needs an explicit statement of the domain of validity, or a test on synthetic data generated from 3D/strongly 2.5D structures, to support the claim that the network output approximates the true Bayesian posterior for real data.
  3. [Title and Abstract; also §4 Discussion] The title and abstract claim that graph MDNs can solve 'all seismic tomographic problems,' but the method is restricted to 2D travel-time surface-wave tomography with a fixed regular 11×11 parameterization, a limitation acknowledged in §4. This overstatement is load-bearing because the 'all problems' claim is a central selling point. I recommend revising the title, abstract, and conclusion to state clearly the scope (2D path-averaged surface-wave travel-time tomography with a given parameterization), and to reframe the broader generalization to 3D or full-waveform problems as future work rather than as part of the demonstrated claim.
minor comments (4)
  1. [Figure 6 caption] The caption for Figure 6 lists periods of 5 s, 10 s, 15 s, 20 s, 25 s, and 30 s, but the text and Figure 5 describe the South England example at periods of 4 s, 6 s, 8 s, 10 s, 12 s, and 14 s. Please correct the mismatch so that the comparison panels correspond to the same periods.
  2. [§1 and §3.3 text] There is a typo in §3.3: 'sendimentary thickness' should be 'sedimentary thickness'. Also, the introduction's claim that Bayesian methods 'generally require expensive computational cost' is accurate but would benefit from a citation to a representative comparison; the later runtime table partly addresses this.
  3. [§2.3, Eq. (4)] The definition of the graph transformer layer would be easier to follow if the softmax argument were explicitly subscripted per layer and if the concatenation symbol were defined in the text; currently the notation 'C c=1' is formatted ambiguously.
  4. [General] The paper does not include a data availability or code availability statement. Given that the training data are synthetic and the architecture is described in detail, releasing the network implementation would materially strengthen reproducibility, which several journals now expect.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the graph MDN is trained on synthetic forward-modeled data and evaluated against independent Bayesian samplers, so the central claim is not forced by construction.

full rationale

The paper's derivation chain is a standard supervised conditional-density-estimation setup: 200,000 velocity models are drawn from a uniform prior, travel times are computed with a fast marching method, and a graph MDN is trained by minimizing the negative log-likelihood of the conditional mixture density (Section 3.1). The claimed results are then assessed on (i) held-out synthetic structures with different station numbers, geometries, and region sizes (Section 3.2), and (ii) real datasets in South England and Southwest China (Sections 3.3, 3.4). In both real-data cases the network outputs are compared against McMC or SVGD posterior estimates obtained by standard sampling algorithms, which are external benchmarks rather than quantities derived from the network. The only load-bearing modeling assumption is that path-averaged velocity is invariant under proportional rescaling and that real surface-wave data are adequately described by the same 2D fast-marching ray model used for training; this is an empirical domain assumption, not a circular reduction. The paper also cites the authors' earlier graph MDN work [32] for the architecture and prior variational/Bayesian work, but those citations are ordinary method references and do not supply the central claim or forbid alternatives. The admitted differences between MDN and McMC/SVGD results, and the absence of quantitative calibration metrics, are evidence-strength or correctness concerns rather than circularity. No equation or fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' own work.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a synthetic training distribution and on the transfer of a 2D path-averaged forward model to real surface-wave data at different scales. No physical constants are introduced; all listed quantities are choices made by the authors. No invented entities are proposed.

free parameters (6)
  • Gaussian noise standard deviation = 2% of travel times
    Chosen for training data; posterior uncertainty estimates scale with this value and it may not match real data noise.
  • Number of Gaussian mixture components = 10
    Set by hand in Section 3.1; affects expressiveness of predicted posterior.
  • Station and travel-time dropout ratios = 0.5 stations, 0.6 travel times
    Used at each training iteration to force variable input sizes; chosen by hand.
  • Velocity prior bounds = Uniform 1.8 to 4.0 km/s
    Defines training distribution and implicitly the prior for all applications; chosen by hand.
  • Network architecture hyperparameters = 6 transformer layers; channels 16,64,256,1024,512,512; 4 graph linear layers with 1000,600,600,1000 hidden units…
    Chosen without systematic search; may affect how well the posterior is approximated.
  • Output grid resolution = 11x11 regular grid (121 cells)
    Fixed in Section 3.1 and acknowledged in Section 4 as a limitation for other resolutions.
assumptions (7)
  • standard math Bayes' theorem and Gaussian likelihood define the target posterior (Eq. 1).
    Used in Section 2.1 as the definition of the inverse problem; not proved but standard.
  • standard math Mixture density networks can approximate arbitrary conditional densities to desired accuracy.
    Section 2.2 cites McLachlan and Basford [19] for universal approximation by mixtures.
  • domain assumption Fast marching method computes inter-receiver travel times accurately in 2D heterogeneous media.
    Section 3.1 uses fast marching [22] to generate all training labels; solver error is absorbed into the learned mapping.
  • domain assumption Real surface wave group and phase velocity measurements obey the same 2D path-averaged ray model as the synthetic training data.
    Sections 3.3 and 3.4 feed real edge features into a network trained on 2D fast-marching travel times; depth-dependent or 3D sensitivity would invalidate the posterior interpretation.
  • domain assumption Path-averaged velocity is invariant under proportional scaling of region and station coordinates.
    Section 3.1 states this to justify cross-scale transfer; no field-scale test of the assumption is provided.
  • domain assumption The synthetic training distribution (uniform 1.8-4.0 km/s pixel prior, 2% noise, random 30-station deployments) covers the relevant variability of real target regions.
    The network can only be as good as its training distribution; real geology may include structures outside this range.
  • ad hoc to paper A fixed 11x11 regular grid is an adequate parameterization for every target region, regardless of physical scale.
    Section 4 notes a finer or coarser grid would require retraining, contradicting the 'all problems' claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Solving All Seismic Tomographic Problems using Deep Learning." pith.science (2026). https://pith.science/paper/YK57DAXE

@misc{pith2026250414830,
  author       = {Pith},
  title        = {Pith review of: Solving All Seismic Tomographic Problems using Deep Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YK57DAXE}},
  note         = {Machine review of arXiv:2504.14830}
}
read the original abstract

In a variety of geoscientific applications scientists often need to image properties of the Earth's interior in order to understand the heterogeneity and processes taking place within the Earth. Seismic tomography is one such method which has been used widely to study properties of the subsurface. In order to solve tomographic problems efficiently, neural network-based methods have been introduced to geophysics. However, these methods can only be applied to certain types of problems with fixed acquisition geometry at a specific site. In this study we extend neural network-based methods to problems with various scales and acquisition geometries by using graph mixture density networks (MDNs). We train a graph MDN for 2D tomographic problems using simulated velocity models and travel time data, and apply the trained network to both synthetic and real data problems that have various scales and station distributions at different sites. The results demonstrate that graph MDNs can provide comparable solutions to those obtained using traditional Bayesian methods in seconds, and therefore provide the possibility to use graph MDNs to produce rapid solutions for all kinds of seismic tomographic problems over the world.

Figures

Figures reproduced from arXiv: 2504.14830 by the authors.

Figure 1
Figure 1. An example of (a) a velocity model and (b) associated travel times randomly selected from the training data. Black triangles denote seismometers whose locations are randomly assigned within the region. (c) The architecture of the graph MDN designed for seismic tomography. The network uses graph transformer layers and graph linear layers to update node features of the input graph, which are then fed into a global mea… view at source ↗
Figure 2
Figure 2. (a-c) The true velocity structure. (d-f) and (g-i) show the mean and standard deviation maps obtained using graph MDN respectively. Black triangles denote locations of seismometers. anomalies, which has also been observed in a range of previous studies [9, 28, 29, 11, 31] and reflects that these anomalies are well constrained by the data. In addition, the mean model obtained in the second set of tests matches the tr… view at source ↗
Figure 3
Figure 3. Results obtained using graph MDN for a larger region ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Topography maps and seismometer locations (red triangles) at two study areas in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The (a) mean and (b) standard deviation maps obtained using graph MDN at periods of 4 s, 6 s, 8 s, 10 s, 12 s and 14 s in south England. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The (a) mean and (b) standard deviation maps obtained using McMC at periods of 5 s, 10 s, 15 s, 20 s, 25 s and 30 s in south England. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The (a) mean and (b) standard deviation maps obtained using graph MDN at periods of 5 s, 10 s, 15 s, 20 s, 25 s and 30 s in southwest of China. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: The (a) mean and (b) standard deviation maps obtained using SVGD at periods of 5 s, 10 s, 15 s, 20 s, 25 s and 30 s in southwest of China. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 21 canonical work pages

  1. [32]

    Rapid Bayesian Seismic Tomography using Graph Mixture Density Networks

    Xin Zhang, Yan Wang, and Haijiang Zhang. Rapid bayesian seismic tomography using graph mixture density networks. arXiv preprint arXiv:2412.08080, 2024. 10 Solving All Seismic Tomographic Problems using Deep Learning A PREPRINT 3 2 1 0 51 52 53 54Latitude 4s (a) 3 2 1 0 51 52 53 54 6s 3 2 1 0 51 52 53 54 8s 3 2 1 0 Longitude 51 52 53 54Latitude 10s 3 2 1 0...

    work page Pith review arXiv 2024

  2. [1]

    Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1

    Keiiti Aki and WHK Lee. Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1. a homogeneous initial model. Journal of Geophysical research, 8 Solving All Seismic Tomographic Problems using Deep Learning A PREPRINT 81(23):4381–4399, 1976

    work page 1976

  3. [2]

    Deep-learning tomography.The Leading Edge, 37(1):58–66, 2018

    Mauricio Araya-Polo, Joseph Jennings, Amir Adler, and Taylor Dahlke. Deep-learning tomography.The Leading Edge, 37(1):58–66, 2018

    work page 2018

  4. [3]

    Parameter estimation and inverse problems

    Richard C Aster, Brian Borchers, and Clifford H Thurber. Parameter estimation and inverse problems. Elsevier, 2018

    2018

  5. [4]

    Two crustal low-velocity channels beneath se tibet revealed by joint inversion of rayleigh wave dispersion and receiver functions

    Xuewei Bao, Xiaoxiao Sun, Mingjie Xu, David W Eaton, Xiaodong Song, Liangshu Wang, Zhifeng Ding, Ning Mi, Hua Li, Dayong Yu, et al. Two crustal low-velocity channels beneath se tibet revealed by joint inversion of rayleigh wave dispersion and receiver functions. Earth and Planetary Science Letters, 415:16–24, 2015

    work page 2015

  6. [5]

    Pattern recognition and machine learning

    Christopher M Bishop. Pattern recognition and machine learning. springer, 2006

    2006

  7. [6]

    An efficient, probabilistic neural network approach to solving inverse problems: Inverting surface wave velocities for Eurasian crustal thickness

    RJR Devilee, A Curtis, and K Roy-Chowdhury. An efficient, probabilistic neural network approach to solving inverse problems: Inverting surface wave velocities for Eurasian crustal thickness. Journal of Geophysical Research: Solid Earth, 104(B12):28841–28857, 1999

    work page 1999

  8. [7]

    Probabilistic neural network-based 2d travel-time tomography

    Stephanie Earp and Andrew Curtis. Probabilistic neural network-based 2d travel-time tomography. NEURAL COMPUTING & APPLICATIONS, 2020

    work page 2020

Show all 32 references
  1. [8]

    Graph mixture density networks

    Federico Errica, Davide Bacciu, and Alessio Micheli. Graph mixture density networks. In International Confer- ence on Machine Learning, pages 3025–3035. PMLR, 2021

    2021

  2. [9]

    Uncertainty loops in travel-time to- mography from nonlinear wave physics

    Erica Galetti, Andrew Curtis, Giovanni Angelo Meles, and Brian Baptie. Uncertainty loops in travel-time to- mography from nonlinear wave physics. Physical review letters, 114(14):148501, 2015

    2015

  3. [10]

    Transdimensional love-wave tomography of the British Isles and shear-velocity structure of the east Irish Sea Basin from ambient-noise inter- ferometry

    Erica Galetti, Andrew Curtis, Brian Baptie, David Jenkins, and Heather Nicolson. Transdimensional love-wave tomography of the British Isles and shear-velocity structure of the east Irish Sea Basin from ambient-noise inter- ferometry. Geophysical Journal International, 208(1):3...

    2017

  4. [11]

    Bayesian elastic full-waveform inversion using Hamilto- nian Monte Carlo

    Lars Gebraad, Christian Boehm, and Andreas Fichtner. Bayesian elastic full-waveform inversion using Hamilto- nian Monte Carlo. Journal of Geophysical Research: Solid Earth, 125(3):e2019JB018428, 2020. doi: 10.1029/ 2019JB018428. URL https://agupubs.onlinelibrary.wiley.com/doi/...

    2020 doi

  5. [12]

    An adaptive metropolis algorithm

    Heikki Haario, Eero Saksman, Johanna Tamminen, et al. An adaptive metropolis algorithm. Bernoulli, 7(2): 223–242, 2001

    2001

  6. [13]

    Regularisation of mixture density networks

    Lars U Hjorth and Ian T Nabney. Regularisation of mixture density networks. In 1999 Ninth International Conference on Artificial Neural Networks ICANN 99.(Conf. Publ. No. 470) , volume 2, pages 521–526. IET, 1999

    1999

  7. [14]

    Adam: A method for stochastic optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    2014 arXiv

  8. [15]

    Stein variational gradient descent: A general purpose Byesian inference algorithm

    Qiang Liu and Dilin Wang. Stein variational gradient descent: A general purpose Byesian inference algorithm. In Advances In Neural Information Processing Systems, pages 2378–2386, 2016

    2016

  9. [16]

    The community velocity model v

    Ying Liu, Huajian Yao, Haijiang Zhang, and Hongjian Fang. The community velocity model v. 1.0 of southwest china, constructed from joint body-and surface-wave travel-time tomography.Seismological Society of America, 92(5):2972–2987, 2021

    2021

  10. [17]

    The high-resolution community velocity model v2

    Ying Liu, Ziye Yu, Zhiqi Zhang, Huajian Yao, Weitao Wang, Haijiang Zhang, Hongjian Fang, and Lihua Fang. The high-resolution community velocity model v2. 0 of southwest china, constructed by joint body and surface wave tomography of data recorded at temporary dense arrays.Scie...

    2023

  11. [18]

    Overcoming limitations of mixture density net- works: A sampling and fitting framework for multimodal future prediction

    Osama Makansi, Eddy Ilg, Ozgun Cicek, and Thomas Brox. Overcoming limitations of mixture density net- works: A sampling and fitting framework for multimodal future prediction. In Proceedings of the IEEE Confer- ence on Computer Vision and Pattern Recognition, pages 7144–7153, 2019

    2019

  12. [19]

    Mixture models: Inference and applications to clustering , vol- ume 38

    Geoffrey J McLachlan and Kaye E Basford. Mixture models: Inference and applications to clustering , vol- ume 38. M. Dekker New York, 1988

    1988

  13. [20]

    Global crustal thickness from neural network inversion of surface wave data

    Ueli Meier, Andrew Curtis, and Jeannot Trampert. Global crustal thickness from neural network inversion of surface wave data. Geophysical Journal International, 169(2):706–722, 2007

    2007

  14. [21]

    Monte Carlo sampling of solutions to inverse problems

    Klaus Mosegaard and Albert Tarantola. Monte Carlo sampling of solutions to inverse problems. Journal of Geophysical Research: Solid Earth, 100(B7):12431–12447, 1995

    1995

  15. [22]

    Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

    Nick Rawlinson and Malcolm Sambridge. Multiple reflection and transmission phases in complex layered media using a multistage fast marching method. Geophysics, 69(5):1338–1350, 2004. 9 Solving All Seismic Tomographic Problems using Deep Learning A PREPRINT

    2004

  16. [23]

    Neural networks and inversion of seismic data

    Gunter Röth and Albert Tarantola. Neural networks and inversion of seismic data. Journal of Geophysical Research: Solid Earth, 99(B4):6753–6768, 1994

    1994

  17. [24]

    Probabilistic programming in python using pymc3

    John Salvatier, Thomas V Wiecki, and Christopher Fonnesbeck. Probabilistic programming in python using pymc3. PeerJ Computer Science, 2:e55, 2016

    2016

  18. [25]

    Inverse problem theory and methods for model parameter estimation, volume 89

    Albert Tarantola. Inverse problem theory and methods for model parameter estimation, volume 89. SIAM, 2005

    2005

  19. [26]

    A new crustal shear-velocity model in southwest china from joint seismological inversion and its implications for regional crustal dynamics

    Yan Yang, Huajian Yao, Hanxiao Wu, Ping Zhang, and Maomao Wang. A new crustal shear-velocity model in southwest china from joint seismological inversion and its implications for regional crustal dynamics. Geophys- ical Journal International, 220(2):1379–1393, 2020

    2020

  20. [27]

    Deep learning for geophysics: Current and future trends

    Siwei Yu and Jianwei Ma. Deep learning for geophysics: Current and future trends. Reviews of Geophysics, 59 (3):e2021RG000742, 2021

    2021

  21. [28]

    Seismic tomography using variational inference methods.Journal of Geophysical Research: Solid Earth, 125(4):e2019JB018589, 2020

    Xin Zhang and Andrew Curtis. Seismic tomography using variational inference methods.Journal of Geophysical Research: Solid Earth, 125(4):e2019JB018589, 2020

    2020

  22. [29]

    Variational full-waveform inversion

    Xin Zhang and Andrew Curtis. Variational full-waveform inversion. Geophysical Journal International, 222(1): 406–411, 2020

    2020

  23. [30]

    Bayesian geophysical inversion using invertible neural networks

    Xin Zhang and Andrew Curtis. Bayesian geophysical inversion using invertible neural networks. Journal of Geophysical Research: Solid Earth, 126(7):e2021JB022320, 2021

    2021

  24. [31]

    3-d bayesian variational full wave- form inversion

    Xin Zhang, Angus Lomas, Muhong Zhou, York Zheng, and Andrew Curtis. 3-d bayesian variational full wave- form inversion. Geophysical Journal International, 234(1):546–561, 2023

    2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.