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REVIEW 4 major objections 5 minor 86 references

Seismic tomography using variational inference methods

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that variational inference methods — which replace Bayesian sampling by optimization — can match Monte Carlo posterior uncertainty maps in seismic tomography at a small fraction of the computational cost, provided…

desk verdict Useful first demonstration of variational inference in travel-time tomography with real cost savings, but the abstract overstates accuracy; evidence is mostly visual. read the letter →

arxiv 1908.08356 v2 pith:7K5UEML4 submitted 2019-08-22 physics.geo-ph physics.comp-ph

Xin Zhang , Andrew Curtis This is my paper · ORCID
classification physics.geo-phphysics.comp-ph
keywords variationalinferenceseismictomographyBayesianinversionuncertaintyquantificationADVISVGDMarkovchainMonteCarloambientnoise
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

Seismic tomography normally needs Monte Carlo sampling to quantify uncertainty, and that sampling becomes computationally prohibitive for realistic data and model sizes. This paper tries to show that two variational inference methods, ADVI and SVGD, can instead produce accurate approximations to the Bayesian posterior distribution of subsurface velocity at far lower cost. It tests this on a 2D synthetic travel-time tomography problem and on real ambient-noise phase-velocity data from the Grane field, comparing against fixed-dimensional and trans-dimensional McMC. The central case is that if gradients of travel times with respect to velocity are available, variational inference turns the Bayesian problem into an optimization problem, needing thousands or hundreds of thousands of forward solves instead of millions. The paper reports that SVGD uncertainty maps closely match McMC while ADVI is faster but admits a Gaussian approximation that biases its posterior.

What carries the argument

The core mechanism is the replacement of Monte Carlo sampling by Kullback-Leibler divergence minimization: instead of generating posterior samples, one maximizes the ELBO over a chosen family of distributions. ADVI uses a Gaussian variational family in an unconstrained transformed space, with a Cholesky parameterization of the covariance and reparameterized Monte Carlo gradient estimates; a logit transform keeps velocities within their physical bounds. SVGD uses a set of particles moved deterministically along the kernelized Stein discrepancy direction, in which one term attracts particles toward high-probability regions and a second repulsive term prevents collapse, allowing approximation of arbitrary posterior shapes. Both methods depend on the same load-bearing object: gradients of the log posterior with respect to model parameters, which in this paper are obtained by ray-tracing the Eikonal equation with a fast marching method.

What would settle it

Run the paper's synthetic experiment with two well-separated low-velocity anomalies (or reduced ray coverage) so the posterior is genuinely multimodal, and compare ADVI and SVGD marginals at a point between the modes against a very long rj-McMC chain; if the VI marginals remain unimodal or their standard deviations do not bracket the two-mode spread, the general claim that VI produces accurate approximations to Monte Carlo posteriors fails in exactly the setting proposed.

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Extended reading notes

Core claim

The paper demonstrates that ADVI and SVGD can be applied to 2D seismic tomography and can approximate the posterior probability density functions that Monte Carlo methods deliver, at substantially lower computational cost. On the synthetic test, SVGD reproduces the characteristic double-loop uncertainty pattern seen in MH-McMC, whereas ADVI flattens that structure because its Gaussian approximation cannot represent non-Gaussian posterior shapes. On the Grane field data, the mean phase-velocity maps from ADVI and SVGD are nearly identical to each other and broadly consistent with rj-McMC, while the uncertainty maps differ, with SVGD showing more structure and rj-McMC showing much smaller uncertainties because of its lower-dimensional parameterization. The paper concludes that variational inference is an efficient alternative to McMC for tomography whenever gradients of parameters with respect to data can be computed efficiently.

Load-bearing premise

The load-bearing premise is that the posterior distribution of the velocity model has a shape the chosen variational family can represent; in particular ADVI's single Gaussian cannot represent a multimodal posterior, so in multimodal tomography problems its uncertainty maps are biased by construction.

Editorial extensions

If this is right

  • Seismic tomography can obtain Bayesian posterior mean and uncertainty maps by optimization rather than sampling, cutting compute from hundreds of CPU hours to single-digit or sub-hour CPU times in the synthetic test, and from over 1,800 CPU hours to under 150 CPU hours on the Grane field data.
  • SVGD is usable where the posterior shape is unknown, since it preserves the double-loop uncertainty pattern that MH-McMC shows and that ADVI's Gaussian approximation smooths away.
  • Because variational inference is an optimization problem, large data sets can be handled with stochastic minibatches and parallel gradient evaluation, a strategy McMC cannot use without breaking detailed balance.
  • The same recipe opens a route to Bayesian 3D tomography and full waveform inversion, where Monte Carlo cost is currently prohibitive, provided efficient gradients are available.
  • ADVI can serve as a very cheap screening tool for mean models and Gaussian uncertainty when a unimodal approximation is considered sufficient.

Reading between the lines

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

  • A testable extension the paper only gestures at: adjoint-based full waveform inversion meets the same gradient requirement, so SVGD-style posterior sampling should transfer to FWI problems that are currently out of reach for McMC; applying it there would reveal whether the cost advantage survives realistic waveform gradients.
  • Because rj-McMC and the fixed-grid VI inversions solve different parameter-estimation problems, the comparison conflates algorithm choice with parameterization choice; a trans-dimensional variational scheme, varying Voronoi-cell number and geometry, would narrow the gap in uncertainty magnitudes and test that distinction.
  • Running ADVI and SVGD side by side on the same data set functions as a practical multimodality diagnostic: where their uncertainty maps diverge, the posterior is likely non-Gaussian, and the ADVI map should not be interpreted as a faithful uncertainty estimate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper applies two variational inference methods, automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic travel-time tomography. In a synthetic test with a circular low-velocity anomaly, the authors compare mean and standard deviation maps and selected marginal posterior distributions against fixed-dimensional Metropolis-Hastings McMC (MH-McMC) and trans-dimensional reversible-jump McMC (rj-McMC), and they apply the methods to ambient-noise phase-velocity data from the Grane field. The paper reports that ADVI and SVGD recover mean models similar to the McMC benchmarks at considerably lower CPU cost, while ADVI's uncertainty estimates are biased by its Gaussian approximation. The central claim is that variational inference can produce accurate approximations to Monte Carlo sampling results at significantly lower computational cost, provided gradients with respect to data can be computed efficiently.

Significance. If fully substantiated, the paper would provide a practical route to Bayesian uncertainty quantification in seismic tomography at a fraction of Monte Carlo cost, which is a valuable contribution given the intractability of McMC for large 3D problems. The study is clearly written, the variational mathematics is standard and correctly transcribed, and the synthetic experiments include two independent Monte Carlo benchmarks as well as explicit convergence checks (doubled iterations and doubled particle count for ADVI and SVGD). The authors also honestly document ADVI's limitations and the parameterization mismatch with rj-McMC. However, because the central accuracy claim rests on visual comparison rather than quantitative metrics, and because ADVI is conceded to be biased, the paper's significance is presently smaller than its abstract claims.

major comments (4)
  1. [Abstract; §3.1; Fig. 10] The abstract's accuracy claim is not supported quantitatively: Section 3.1 and Figure 10 compare the ADVI, SVGD, MH-McMC and rj-McMC results only through visual inspection of mean and standard deviation maps and of marginal pdfs at three points, with no numerical discrepancy measure such as KL divergence, Wasserstein distance, coverage probability, or a calibration statistic. Since this is the load-bearing evidence for the paper's main claim, the authors should add a quantitative comparison of the VI posteriors against the fixed-dimensional MH-McMC posterior and report the corresponding metric.
  2. [§2.2; §5; Fig. 10f] The abstract's claim is also too broad for ADVI. Section 2.2 explicitly states that ADVI 'provides a unimodal approximation' and 'will not be effective for multimodal distributions,' and Figure 10f shows that the rj-McMC marginal at the anomaly boundary (1.8, 0) is clearly multimodal. Section 5 concedes that ADVI 'produces biased posterior pdfs.' The paper should therefore restrict the accuracy claim to SVGD, or condition it on unimodal posteriors, and the abstract and conclusion should be rewritten accordingly.
  3. [§3.1; §5] The comparison with rj-McMC cannot serve as a posterior-accuracy validation because the parameterizations are fundamentally different: the fixed grid inversions use 441 cells while the rj-McMC posterior lives in a trans-dimensional Voronoi space with roughly 10 cells, as stated in Section 3.1. Section 5 itself notes that the results are 'essentially not directly comparable.' Since the abstract claims accuracy relative to Monte Carlo sampling methods generically, the validation must rest on the MH-McMC comparison using the identical 21×21 grid; that comparison is currently only visual, which is the gap identified in the first major comment.
  4. [§3.2; Table 1] The computational-cost comparison is not fully quantitative because convergence is assessed subjectively. Section 3.2 acknowledges that 'the above comparison depends on the methods used to assess convergence for each method, which introduces some subjectivity in the comparison.' To substantiate the claim of 'significantly lower computational cost,' the authors should report convergence diagnostics or a cost-to-accuracy curve (e.g., CPU time required to reach a target posterior accuracy) rather than only raw CPU hours.
minor comments (5)
  1. [Author affiliation] The affiliation line contains a typo: 'Unite Kingdom' should be 'United Kingdom.'
  2. [Fig. 10 caption] Figure 10's caption refers to 'pluses in Figure 3,4,5,6' but the relevant figures are Figures 5–8.
  3. [§3.1] There is a typo in Section 3.1: 'reciever' should be 'receiver.'
  4. [§2.2; §3.2] The text alternates between 'constrains' and 'constraints'; the latter spelling should be used consistently.
  5. [§3.0] It would help to state explicitly whether the same initial particles or random seed were used for the SVGD convergence checks, so that the comparisons with doubled iterations and doubled particle count are cleanly interpretable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variational results are validated against independently run Monte Carlo benchmarks, and the accuracy claim does not reduce to the methods' inputs.

full rationale

The paper's central claim is that ADVI and SVGD can approximate Monte Carlo posterior results at lower cost. This claim is evaluated in a synthetic test by comparing the variational posteriors against MH-McMC and rj-McMC runs that use the same data, prior, and (for MH-McMC) the same 21x21 grid parameterization; the MC results are not used to fit or define the variational approximations. ADVI and SVGD are standard algorithms from the external literature (Kucukelbir et al. 2017; Liu and Wang 2016), and their objective functions are the ELBO and Stein discrepancy, respectively, not a fit to the Monte Carlo outputs. The self-citations that appear are used to set up the synthetic example (Galetti et al. 2015), supply a real-data noise level (Zhang et al. 2018), and provide benchmark context; these are empirical inputs and prior results, not definitions equivalent to the target conclusion. The paper explicitly concedes that ADVI gives a unimodal, biased approximation, which is a stated limitation rather than a disguised input. The comparison with rj-McMC is acknowledged to involve different parameterizations, and this is a validity caveat about interpretability, not a circular reduction. No equation in the paper is claimed as a prediction while actually being an input by construction. The central derivation chain is therefore self-contained against independently obtained Monte Carlo benchmarks, and the manuscript does not exhibit material circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard Bayesian and VI machinery; no new entities are introduced. The main user-chosen inputs are the noise level and the prior bounds; these are explicit and not fitted to force a particular conclusion.

free parameters (1)
  • Data noise level for VI inversions = 0.05 s
    Fixed at 0.05 s for both synthetic and real-data VI runs. For the real data this value is taken from the authors' prior hierarchical inversion (Zhang et al., 2019) rather than estimated within the VI inversion; the paper notes an improper noise level can bias tomographic results (§5).
assumptions (5)
  • domain assumption The likelihood p(d|m) is Gaussian with known variance
    Stated in §2.1: 'The likelihood is usually assumed to follow a Gaussian probability density function around the data predicted synthetically from model m.' This assumption is used in all four methods; if the true observational errors are non-Gaussian, the posterior comparison is affected.
  • domain assumption The fast marching Eikonal solver provides accurate forward travel times
    Used in §3 for the synthetic test and implicitly for the real data; the inversion quality and gradient computations depend on the forward operator being accurate.
  • domain assumption Gradients of log p with respect to model parameters are correctly computed by ray tracing
    Crucial for ADVI and SVGD updates (Eqs. 9, 10, 19). The paper computes gradients by tracing rays backwards using the travel time field (§3). If gradients are inaccurate, the optimization-based posterior approximation is invalid.
  • domain assumption The prior pdf for each cell is Uniform
    Stated in §3 for the synthetic test (0.5 to 3.0 km/s) and §4 for the real data (0.35 to 0.55 km/s). The posterior estimates depend on these choices.
  • standard math The RBF kernel with median heuristic is a valid Stein kernel
    Used in SVGD (Eq. 19); this is a standard choice from Liu and Wang (2016), but the approximation quality depends on it. Not an ad hoc invention; it is a standard method.

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Cite this review

Pith. "Pith review of Seismic tomography using variational inference methods." pith.science (2026). https://pith.science/paper/7K5UEML4

@misc{pith2026190808356,
  author       = {Pith},
  title        = {Pith review of: Seismic tomography using variational inference methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7K5UEML4}},
  note         = {Machine review of arXiv:1908.08356}
}
read the original abstract

Seismic tomography is a methodology to image the interior of solid or fluid media, and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large datasets and high-dimensional parameter spaces. To extend uncertainty analysis to larger systems we use variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem, yet still provide probabilistic results. In this study, we applied two variational methods, automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic tomography problems using both synthetic and real data and we compare the results to those from two different Monte Carlo sampling methods. The results show that variational inference methods can produce accurate approximations to the results of Monte Carlo sampling methods at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.

Figures

Figures reproduced from arXiv: 1908.08356 by the authors.

Figure 1
Figure 1. An illustration of the workflow of ADVI. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. An illustration of the transform in equation (11). The original variable is in a constrained space between 0.5 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. An illustration of the SVGD algorithm. The initial pdf is represented by the density of a set of particles (red [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The true velocity model and receivers (white triangle) used in the synthetic test. Sources are at the same [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The mean (left) and standard deviation (right) found using ADVI. The red pluses show locations which are [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The mean (left) and standard deviation (right) found using SVGD. The red pluses show locations which are [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: The mean (left) and standard deviation (right) found using MH-McMC. The red pluses show the point [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: The mean (left) and standard deviation (right) found using trans-dimensional rj-McMC. The red pluses show [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Two examples of the models sampled using rj-McMC algorithm. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: The marginal posterior pdfs of velocity at three points (pluses in Figure 3,4,5,6) derived using different [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: The distribution of receiver (blue and red triangles) across the Grane field used in this study. Red triangles [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: The mean (left) and standard deviation map (right) from ADVI. [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: The mean (left) and standard deviation map (right) from SVGD. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: The mean (left) and standard deviation map (right) from rj-McMC. [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

86 extracted references · 67 canonical work pages

  1. [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, 81(23):4381–4399, 1976

    1976

  2. [2]

    Interrogation theory

    Richard Arnold and Andrew Curtis. Interrogation theory. Geophysical Journal International, 214(3):1830–1846, 2018

    2018

  3. [3]

    A 3-D shear velocity model of the crust and uppermost mantle beneath the United States from ambient seismic noise

    GD Bensen, MH Ritzwoller, and Y Yang. A 3-D shear velocity model of the crust and uppermost mantle beneath the United States from ambient seismic noise. Geophysical Journal International, 177(3):1177–1196, 2009

    2009

  4. [4]

    Pattern recognition and machine learning

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

    2006

  5. [5]

    Variational inference: A review for statisticians

    David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518):859–877, 2017

    2017

  6. [6]

    Seismic tomography with the reversible jump algorithm

    Thomas Bodin and Malcolm Sambridge. Seismic tomography with the reversible jump algorithm. Geophysical Journal International, 178(3):1411–1436, 2009

    2009

  7. [7]

    Transdimensional inversion of receiver functions and surface wave dispersion.Journal of Geophysical Research: Solid Earth, 117(B2), 2012

    Thomas Bodin, Malcolm Sambridge, H Tkal ˇci´c, Pierre Arroucau, Kerry Gallagher, and Nicholas Rawlinson. Transdimensional inversion of receiver functions and surface wave dispersion.Journal of Geophysical Research: Solid Earth, 117(B2), 2012

    work page 2012

  8. [8]

    Velocity variations and uncertainty from transdimensional P-wave tomography of North America

    Scott Burdick and Vedran Leki ´c. Velocity variations and uncertainty from transdimensional P-wave tomography of North America. Geophysical Journal International, 209(2):1337–1351, 2017

    work page 2017

Show all 86 references
  1. [9]

    Long-range correlations in the diffuse seismic coda

    Michel Campillo and Anne Paul. Long-range correlations in the diffuse seismic coda. Science, 299(5606): 547–549, 2003

    2003

  2. [10]

    Probability and stochastics, volume 261

    Erhan Çınlar. Probability and stochastics, volume 261. Springer Science & Business Media, 2011

    2011

  3. [11]

    Transdimensional ambient noise tomography of Bass Strait, southeast Australia, reveals the sedimentary basin and deep crustal structure beneath a failed continental rift

    E Crowder, N Rawlinson, S Pilia, DG Cornwell, and AM Reading. Transdimensional ambient noise tomography of Bass Strait, southeast Australia, reveals the sedimentary basin and deep crustal structure beneath a failed continental rift. Geophysical Journal International, 217(2):97...

    2019

  4. [12]

    Prior information, sampling distributions, and the curse of dimensionality

    Andrew Curtis and Anthony Lomax. Prior information, sampling distributions, and the curse of dimensionality. Geophysics, 66(2):372–378, 2001

    2001

  5. [13]

    Reconditioning inverse problems using the genetic algorithm and revised parameterization

    Andrew Curtis and Roel Snieder. Reconditioning inverse problems using the genetic algorithm and revised parameterization. Geophysics, 62(5):1524–1532, 1997

    1997

  6. [14]

    Probing the earth’s interior with seismic tomography.International Geophysics Series, 81(A):861–874, 2002

    Andrew Curtis and Roel Snieder. Probing the earth’s interior with seismic tomography.International Geophysics Series, 81(A):861–874, 2002

    2002

  7. [15]

    Seismic interferometry – turning noise into signal

    Andrew Curtis, Peter Gerstoft, Haruo Sato, Roel Snieder, and Kees Wapenaar. Seismic interferometry – turning noise into signal. The Leading Edge, 25(9):1082–1092, 2006

    2006

  8. [16]

    Variational MCMC

    Nando De Freitas, Pedro Højen-Sørensen, Michael I Jordan, and Stuart Russell. Variational MCMC. InProceed- ings of the Seventeenth conference on Uncertainty in artificial intelligence , pages 120–127. Morgan Kaufmann Publishers Inc., 2001

    2001

  9. [17]

    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

    1999

  10. [18]

    Global images of the Earth’s interior

    Adam M Dziewonski and John H Woodhouse. Global images of the Earth’s interior. Science, 236(4797):37–48, 1987

    1987

  11. [19]

    Probabilistic neural-network based 2D travel time tomography

    Stephanie Earp and Andrew Curtis. Probabilistic neural-network based 2D travel time tomography. arXiv preprint arXiv:1907.00541, 2019

    1907 arXiv

  12. [20]

    Wavelet-based double-difference seismic tomography with sparsity regular- ization

    Hongjian Fang and Haijiang Zhang. Wavelet-based double-difference seismic tomography with sparsity regular- ization. Geophysical Journal International, 199(2):944–955, 2014

    2014

  13. [21]

    Transdimensional electrical resistivity tomography

    E Galetti and A Curtis. Transdimensional electrical resistivity tomography. Journal of Geophysical Research: Solid Earth, 123(8):6347–6377, 2018

    2018

  14. [22]

    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

  15. [23]

    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

  16. [24]

    Measuring sample quality with Stein’s method

    Jackson Gorham and Lester Mackey. Measuring sample quality with Stein’s method. In Advances in Neural Information Processing Systems, pages 226–234, 2015

    2015

  17. [25]

    Reversible jump Markov chain Monte Carlo computation and Byesian model determination

    Peter J Green. Reversible jump Markov chain Monte Carlo computation and Byesian model determination. Biometrika, pages 711–732, 1995

    1995

  18. [26]

    Reversible jump MCMC

    Peter J Green and David I Hastie. Reversible jump MCMC. Genetics, 155(3):1391–1403, 2009

    2009

  19. [27]

    Introduction to RKHS, and some simple kernel algorithms

    Arthur Gretton. Introduction to RKHS, and some simple kernel algorithms. 2013

    2013

  20. [28]

    Monte Carlo sampling methods using Markov chains and their applications

    W Keith Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57 (1):97–109, 1970

    1970

  21. [29]

    Geophysical imaging using trans-dimensional trees

    Rhys Hawkins and Malcolm Sambridge. Geophysical imaging using trans-dimensional trees. Geophysical Journal International, 203(2):972–1000, 2015

    2015

  22. [30]

    Structured stochastic variational inference

    Matthew D Hoffman and David M Blei. Structured stochastic variational inference. In Artificial Intelligence and Statistics, 2015

    2015

  23. [31]

    Seismic tomography: Theory and practice

    HM Iyer and Kazuro Hirahara. Seismic tomography: Theory and practice. Springer Science & Business Media, 1993

    1993

  24. [32]

    A first course in stochastic processes

    Samuel Karlin. A first course in stochastic processes. Academic press, 2014

    2014

  25. [33]

    A framework for fast probabilistic centroid-moment-tensor determination – inversion of regional static displacement measurements

    Paul Käufl, Andrew P Valentine, Thomas B O’Toole, and Jeannot Trampert. A framework for fast probabilistic centroid-moment-tensor determination – inversion of regional static displacement measurements. Geophysical Journal International, 196(3):1676–1693, 2013

    2013

  26. [34]

    Robust and fast probabilistic source param- eter estimation from near-field displacement waveforms using pattern recognition

    Paul Käufl, Andrew Valentine, Ralph de Wit, and Jeannot Trampert. Robust and fast probabilistic source param- eter estimation from near-field displacement waveforms using pattern recognition. Bulletin of the Seismological Society of America, 105(4):2299–2312, 2015

    2015

  27. [35]

    Auto-encoding variational Byes

    Diederik P Kingma and Max Welling. Auto-encoding variational Byes. arXiv preprint arXiv:1312.6114, 2013

    2013 arXiv

  28. [36]

    Stochastic approximation algorithms and applications

    CS Kubrusly and J Gravier. Stochastic approximation algorithms and applications. In 1973 IEEE conference on decision and control including the 12th symposium on adaptive processes, pages 763–766. IEEE, 1973

    1973

  29. [37]

    Automatic differentiation variational inference

    Alp Kucukelbir, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1):430–474, 2017

    2017

  30. [38]

    On information and sufficiency.The annals of mathematical statistics, 22(1):79–86, 1951

    Solomon Kullback and Richard A Leibler. On information and sufficiency.The annals of mathematical statistics, 22(1):79–86, 1951

    1951

  31. [39]

    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

  32. [40]

    A kernelized Stein discrepancy for goodness-of-fit tests

    Qiang Liu, Jason Lee, and Michael Jordan. A kernelized Stein discrepancy for goodness-of-fit tests. In Interna- tional Conference on Machine Learning, pages 276–284, 2016

    2016

  33. [41]

    Parsimonious Byesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem

    Alberto Malinverno. Parsimonious Byesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophysical Journal International, 151(3):675–688, 2002

    2002

  34. [42]

    Expanded uncertainty quantification in inverse problems: Hierarchi- cal Byes and empirical Byes

    Alberto Malinverno and Victoria A Briggs. Expanded uncertainty quantification in inverse problems: Hierarchi- cal Byes and empirical Byes. Geophysics, 69(4):1005–1016, 2004

    2004

  35. [43]

    A Monte Carlo method to quantify uncertainty in the inversion of zero-offset VSP data

    Alberto Malinverno, Scott Leaney, et al. A Monte Carlo method to quantify uncertainty in the inversion of zero-offset VSP data. In 2000 SEG Annual Meeting. Society of Exploration Geophysicists, 2000

    2000

  36. [44]

    An introduction to sampling via mea- sure transport

    Youssef Marzouk, Tarek Moselhy, Matthew Parno, and Alessio Spantini. An introduction to sampling via mea- sure transport. arXiv preprint arXiv:1602.05023, 2016

    2016 arXiv

  37. [45]

    A global crustal model constrained by nonlinearised inversion of fundamental mode surface waves

    U Meier, A Curtis, and J Trampert. A global crustal model constrained by nonlinearised inversion of fundamental mode surface waves. Geophysical Research Letters, 34:L16304, 2007

    2007

  38. [46]

    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

  39. [47]

    The Monte Carlo method

    Nicholas Metropolis and Stanislaw Ulam. The Monte Carlo method. Journal of the American statistical associ- ation, 44(247):335–341, 1949

    1949

  40. [48]

    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

  41. [49]

    Rapid discriminative variational Byesian inversion of geophysical data for the spatial distribution of geological properties

    MA Nawaz and A Curtis. Rapid discriminative variational Byesian inversion of geophysical data for the spatial distribution of geological properties. Journal of Geophysical Research: Solid Earth, 2019. 23 Seismic tomography using variational inference methods A PREPRINT

    2019

  42. [50]

    Variational Bayesian inversion (VBI) of quasi-localized seismic attributes for the spatial distribution of geological facies

    Muhammad Atif Nawaz and Andrew Curtis. Variational Bayesian inversion (VBI) of quasi-localized seismic attributes for the spatial distribution of geological facies. Geophysical Journal International, 214(2):845–875, 2018

    2018

  43. [51]

    Seismic interferometry and ambient noise tomography in the British Isles

    Heather Nicolson, Andrew Curtis, Brian Baptie, and Erica Galetti. Seismic interferometry and ambient noise tomography in the British Isles. Proceedings of the Geologists’ Association, 123(1):74–86, 2012

    2012

  44. [52]

    Rayleigh wave tomography of the British Isles from ambient seismic noise

    Heather Nicolson, Andrew Curtis, and Brian Baptie. Rayleigh wave tomography of the British Isles from ambient seismic noise. Geophysical Journal International, 198(2):637–655, 2014

    2014

  45. [53]

    Local three-dimensional earthquake tomography by trans-dimensional Monte Carlo sampling

    Nicola Piana Agostinetti, Genny Giacomuzzi, and Alberto Malinverno. Local three-dimensional earthquake tomography by trans-dimensional Monte Carlo sampling. Geophysical Journal International , 201(3):1598– 1617, 2015

    2015

  46. [54]

    Black box variational inference

    Rajesh Ranganath, Sean Gerrish, and David Blei. Black box variational inference. In Artificial Intelligence and Statistics, pages 814–822, 2014

    2014

  47. [55]

    Hierarchical variational models

    Rajesh Ranganath, Dustin Tran, and David Blei. Hierarchical variational models. In International Conference on Machine Learning, pages 324–333, 2016

    2016

  48. [56]

    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

    2004

  49. [57]

    Robust and accelerated Byesian inversion of marine controlled-source electromagnetic data using parallel tempering

    Anandaroop Ray, David L Alumbaugh, G Michael Hoversten, and Kerry Key. Robust and accelerated Byesian inversion of marine controlled-source electromagnetic data using parallel tempering. Geophysics, 78(6):E271– E280, 2013

    2013

  50. [58]

    Low frequency full waveform seismic inversion within a tree based Byesian framework

    Anandaroop Ray, Sam Kaplan, John Washbourne, and Uwe Albertin. Low frequency full waveform seismic inversion within a tree based Byesian framework. Geophysical Journal International, 212(1):522–542, 2017

    2017

  51. [59]

    Variational inference with normalizing flows

    Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015

    2015 arXiv

  52. [60]

    A stochastic approximation method

    Herbert Robbins and Sutton Monro. A stochastic approximation method. The annals of mathematical statistics, pages 400–407, 1951

    1951

  53. [61]

    Monte Carlo statistical methods

    Christian Robert and George Casella. Monte Carlo statistical methods . Springer Science & Business Media, 2013

    2013

  54. [62]

    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

  55. [63]

    Markov chain Monte Carlo and variational inference: Bridg- ing the gap

    Tim Salimans, Diederik Kingma, and Max Welling. Markov chain Monte Carlo and variational inference: Bridg- ing the gap. In International Conference on Machine Learning, pages 1218–1226, 2015

    2015

  56. [64]

    Geophysical inversion with a neighbourhood algorithm – i

    Malcolm Sambridge. Geophysical inversion with a neighbourhood algorithm – i. searching a parameter space. Geophysical journal international, 138(2):479–494, 1999

    1999

  57. [65]

    Exploiting tractable substructures in intractable networks

    Lawrence K Saul and Michael I Jordan. Exploiting tractable substructures in intractable networks. In Advances in neural information processing systems, pages 486–492, 1996

    1996

  58. [66]

    Fast probabilistic nonlinear petrophysical inversion

    Mohammad S Shahraeeni and Andrew Curtis. Fast probabilistic nonlinear petrophysical inversion. Geophysics, 76(2):E45–E58, 2011

    2011

  59. [67]

    Fast probabilistic petrophysical mapping of reser- voirs from 3D seismic data

    Mohammad S Shahraeeni, Andrew Curtis, and Gabriel Chao. Fast probabilistic petrophysical mapping of reser- voirs from 3D seismic data. Geophysics, 77(3):O1–O19, 2012

    2012

  60. [68]

    High-resolution surface-wave tomography from ambient seismic noise

    Nikolai M Shapiro, Michel Campillo, Laurent Stehly, and Michael H Ritzwoller. High-resolution surface-wave tomography from ambient seismic noise. Science, 307(5715):1615–1618, 2005

    2005

  61. [69]

    Joint inversion of surface wave dispersion and receiver functions: a Byesian Monte-Carlo approach

    Weisen Shen, Michael H Ritzwoller, Vera Schulte-Pelkum, and Fan-Chi Lin. Joint inversion of surface wave dispersion and receiver functions: a Byesian Monte-Carlo approach. Geophysical Journal International, 192(2): 807–836, 2012

    2012

  62. [70]

    Weisen Shen, Michael H Ritzwoller, and Vera Schulte-Pelkum. A 3-D model of the crust and uppermost mantle beneath the central and western US by joint inversion of receiver functions and surface wave dispersion.Journal of Geophysical Research: Solid Earth, 118(1):262–276, 2013

    2013

  63. [71]

    Data analysis: A Byesian tutorial (oxford science publications)

    DS Sivia. Data analysis: A Byesian tutorial (oxford science publications). 1996

    1996

  64. [72]

    Sequential Monte Carlo methods in practice

    Adrian Smith. Sequential Monte Carlo methods in practice. Springer Science & Business Media, 2013

    2013

  65. [73]

    A bound for the error in the normal approximation to the distribution of a sum of dependent random variables

    Charles Stein et al. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory. The Regents of the Univer...

    1972

  66. [74]

    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

  67. [75]

    Stan modeling language users guide and reference manual

    Stan Development Team et al. Stan modeling language users guide and reference manual. Technical report, 2016

    2016

  68. [76]

    The variational Gaussian process

    Dustin Tran, Rajesh Ranganath, and David M Blei. The variational Gaussian process. arXiv preprint arXiv:1511.06499, 2015

    2015 arXiv

  69. [77]

    Identification of parametric models from experimental data

    Eric Walter and Luc Pronzato. Identification of parametric models from experimental data . Springer Verlag, 1997

    1997

  70. [78]

    On the precision of noise correlation interferometry

    Richard L Weaver, Céline Hadziioannou, Eric Larose, and Michel Campillo. On the precision of noise correlation interferometry. Geophysical Journal International, 185(3):1384–1392, 2011

    2011

  71. [79]

    Analysis of ambient noise energy distribution and phase velocity bias in ambient noise tomography, with application to SE tibet

    Huajian Yao and Robert D Van Der Hilst. Analysis of ambient noise energy distribution and phase velocity bias in ambient noise tomography, with application to SE tibet. Geophysical Journal International, 179(2):1113–1132, 2009

    2009

  72. [80]

    Transdimensional inversion of ambient seismic noise for 3D shear velocity structure of the Tasmanian crust

    Mallory K Young, Nicholas Rawlinson, and Thomas Bodin. Transdimensional inversion of ambient seismic noise for 3D shear velocity structure of the Tasmanian crust. Geophysics, 78(3):WB49–WB62, 2013

    2013

  73. [81]

    Fully 3D Monte Carlo ambient noise tomography over Grane field

    X Zhang, F Hansteen, and A Curtis. Fully 3D Monte Carlo ambient noise tomography over Grane field. In 81st EAGE Conference and Exhibition 2019, 2019

    2019

  74. [82]

    Wavelet-based time-dependent travel time tomography method and its applica- tion in imaging the Etna volcano in Italy

    Xin Zhang and Haijiang Zhang. Wavelet-based time-dependent travel time tomography method and its applica- tion in imaging the Etna volcano in Italy. Journal of Geophysical Research: Solid Earth , 120(10):7068–7084, 2015

    2015

  75. [83]

    3-D Monte Carlo surface wave tomography

    Xin Zhang, Andrew Curtis, Erica Galetti, and Sjoerd de Ridder. 3-D Monte Carlo surface wave tomography. Geophysical Journal International, 215(3):1644–1658, 2018

    2018

  76. [84]

    Geophysical inverse theory and regularization problems, volume 36

    Michael S Zhdanov. Geophysical inverse theory and regularization problems, volume 36. Elsevier, 2002

    2002

  77. [85]

    Transdimensional Byesian seismic ambient noise tomography across SE tibet

    DingChang Zheng, Erdinc Saygin, Phil Cummins, Zengxi Ge, Zhaoxu Min, Athanasius Cipta, and Runhai Yang. Transdimensional Byesian seismic ambient noise tomography across SE tibet. Journal of Asian Earth Sciences, 134:86–93, 2017

    2017

  78. [86]

    Upper crustal structure of central Java, Indonesia, from transdimensional seismic ambient noise tomography

    Z Zulfakriza, Erdinc Saygin, PR Cummins, Sri Widiyantoro, Andri Dian Nugraha, B-G Lühr, and T Bodin. Upper crustal structure of central Java, Indonesia, from transdimensional seismic ambient noise tomography. Geophysical Journal International, 197(1):630–635, 2014. A The entro...

    2014

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