{"id":"e979957c-116c-4799-8e74-9c09daaa8a6a","arxiv_id":"1908.08356","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Variational inference can approximate the results of Monte Carlo seismic tomography at much lower computational cost, but the approximate posterior uncertainties, especially from ADVI, are biased.","lead":"This paper applies two variational inference methods, ADVI and SVGD, to 2D seismic tomography and compares them with Monte Carlo sampling. On synthetic and real data, the variational methods produce similar mean velocity maps at far lower computational cost, though their uncertainty estimates are less reliable than the text suggests.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own Section 5 concedes ADVI produces biased posterior pdfs, and the claimed 'accurate approximations' to Monte Carlo posteriors rest on visual comparison with no quantitative accuracy metric; the missing validation is the load-bearing step.","rationale":"The reader's weakest assumption focuses on ADVI's unimodal family and the multimodal marginal in Figure 10f, which is a real and central limitation. My concern is broader: even setting aside multimodality, the paper never defines or measures 'accurate approximation' quantitatively, so the central claim is not fully testable from the reported results. This reinforces the CONDITIONAL verdict rather than changing it. The paper is an honest demonstration with explicit limitations (Section 5, and the discussion of parameterization differences), so a REJECT would be too strong; the missing quantitative validation and absent code/data mean ACCEPT is not warranted. I agree partially with the reader because the unimodality issue is a specific instance of the broader missing-accuracy-validation concern, but the broader issue also applies to SVGD and to the rj-McMC comparison, so it is not identical to the reader's stated weakest assumption.","tokens_in":52,"tokens_out":4435,"duration_ms":110461,"concrete_test":"Re-run the synthetic inversion and compute a quantitative posterior-accuracy check: for the three points in Figure 10, compute the 1-Wasserstein distance between each variational marginal (ADVI and SVGD) and the corresponding MH-McMC marginal, and also compute 90% credible-interval coverage of the true velocity model over all grid cells for ADVI, SVGD, and MH-McMC. If ADVI's anomaly-boundary Wasserstein distance is substantially larger than SVGD's, or if either variational method's coverage departs from 0.90 by more than the Monte Carlo error of the MH-McMC reference, then the abstract's 'accurate approximations' claim must be restricted to mean-model estimation and the uncertainty maps should be labeled biased.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in the abstract is that variational inference methods 'can produce accurate approximations to the results of Monte Carlo sampling methods at significantly lower computational cost.' The evidence reported does not establish that claim as stated. Section 2.2 says ADVI provides a 'unimodal approximation' and 'will not be effective for multimodal distributions'; Section 5 concedes ADVI 'produces biased posterior pdfs.' In the synthetic experiment, Figure 10f shows the rj-McMC marginal at the anomaly boundary is clearly multimodal, so ADVI's uncertainty at that point is biased by construction. For SVGD, the claimed accuracy is supported only by visual similarity of mean and standard-deviation maps and a few marginals relative to MH-McMC; no KL divergence, Wasserstein distance, coverage, or calibration statistic is reported. The comparison with rj-McMC is also not a like-for-like posterior check because the parameterizations differ substantially (441 regular grid cells versus roughly 10 Voronoi cells), as the authors themselves acknowledge in Sections 3.1 and 5. The experiments therefore support the weaker conclusion that variational methods can recover similar mean models far more cheaply, but not the stronger claim that they produce accurate posterior approximations to Monte Carlo sampling results in general. The weakest load-bearing step is the unquantified, visually assessed accuracy assertion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21427,"tokens_out":4640,"duration_ms":44630,"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":[{"comment":"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.","section":"Abstract; §3.1; Fig. 10"},{"comment":"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.","section":"§2.2; §5; Fig. 10f"},{"comment":"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.","section":"§3.1; §5"},{"comment":"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.","section":"§3.2; Table 1"}],"minor_comments":[{"comment":"The affiliation line contains a typo: 'Unite Kingdom' should be 'United Kingdom.'","section":"Author affiliation"},{"comment":"Figure 10's caption refers to 'pluses in Figure 3,4,5,6' but the relevant figures are Figures 5–8.","section":"Fig. 10 caption"},{"comment":"There is a typo in Section 3.1: 'reciever' should be 'receiver.'","section":"§3.1"},{"comment":"The text alternates between 'constrains' and 'constraints'; the latter spelling should be used consistently.","section":"§2.2; §3.2"},{"comment":"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.","section":"§3.0"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent application of standard variational inference algorithms to a geophysical inverse problem. Its main weakness is the mismatch between the strong abstract claim and the visual-only evidence; this is fixable with quantitative comparisons and a moderated claim. The novelty is incremental but appropriate for a methods-application venue. I recommend major revision rather than rejection because the underlying experiments are sound and the caveats are mostly present in the body."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis is a useful first demonstration of ADVI and SVGD for 2D travel-time tomography, with an honest body text that in places undercuts its own abstract. Worth engaging, but the central claim needs qualification.\n\nThe new piece is straightforward: they take two off-the-shelf variational methods, apply them to a 2D ambient-noise-style travel-time problem, and compare against two McMC benchmarks. The cost savings are real and substantial—ADVI is about three orders of magnitude cheaper than fixed-dimensional McMC in their table, SVGD about two orders, even after parallelization. The synthetic tests are clearly described, and on the mean-model level the variational results match MH-McMC well. The real-data example is a sensible demonstration. The paper is also honest about the big caveats: ADVI's Gaussian approximation is explicitly acknowledged to produce biased posteriors, and the rj-McMC comparison is confounded by parameterization (441 fixed cells vs ~10 Voronoi cells), which they discuss.\n\nThe soft spot is the accuracy claim. The abstract says VI 'can produce accurate approximations' to McMC posteriors, but the evidence is visual: a few marginal distributions and standard-deviation maps, no KL divergence, no Wasserstein distance, no coverage or calibration statistic. Where they do show marginals, ADVI is visibly more concentrated than both SVGD and MH-McMC at the anomaly boundary, which is exactly the kind of place a Gaussian approximation fails. SVGD's marginals do look close to MH-McMC, which is genuinely encouraging, but 'close' is doing a lot of work without a metric. The speed comparison is also sensitive to how convergence is judged; they flag this too. Minor: no code or data, which would materially help others adopt these methods.\n\nNone of this sinks the paper. It is a legitimate application paper, and the authors appear to know their limitations better than their abstract does. It deserves peer review, with a request to either add quantitative posterior accuracy metrics or soften the abstract's claim. This is for geophysicists doing Bayesian inversion who want to scale up to larger problems.","headline":"Useful first demonstration of variational inference in travel-time tomography with real cost savings, but the abstract overstates accuracy; evidence is mostly visual.","tokens_in":21922,"tokens_out":3711,"would_cite":true,"duration_ms":31521,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["variational inference","seismic tomography","Bayesian inversion","uncertainty quantification","ADVI","SVGD","Markov chain Monte Carlo","ambient noise tomography"],"falsifier":"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.","tokens_in":20976,"feed_emoji":"📉","tokens_out":7862,"duration_ms":83796,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bayesian tomography maps at a fraction of Monte Carlo cost","feed_subtitle":"Two gradient-based methods turn the inference into optimization, cutting CPU time from days to hours.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the ADVI algorithm: Gaussian variational family in a transformed space, Cholesky covariance parameterization, and reparameterized ELBO gradients.","marker":"[37]"},{"why":"Supplies SVGD, including the particle update based on the kernelized Stein discrepancy and the RBF kernel used in both experiments.","marker":"[39]"},{"why":"Defines the kernelized Stein discrepancy that provides the optimal perturbation direction for the SVGD particle update.","marker":"[40]"},{"why":"Provides the synthetic circular low-velocity anomaly test problem and the double uncertainty-loop benchmark that SVGD reproduces.","marker":"[22]"},{"why":"Introduces reversible-jump McMC, the trans-dimensional sampling framework used as the comparison reference.","marker":"[25]"},{"why":"Introduces the trans-dimensional Voronoi parameterization and its application to rj-McMC tomography, used to construct the comparison models.","marker":"[6]"},{"why":"Provides the fast-marching Eikonal solver used for forward modeling and for computing travel-time gradients on which both variational methods depend.","marker":"[56]"},{"why":"Provides the Grane-field ambient-noise data processing and the phase-velocity picks at 0.9 s used in the real-data test.","marker":"[81]"},{"why":"Provides the 3D Monte Carlo surface-wave tomography framework, the noise-level estimate, and proposal choices used in the rj-McMC comparison.","marker":"[83]"}],"fun_headline_variants":["Variational inference speeds seismic tomography","Bayesian tomography without the Monte Carlo wait","ADVI and SVGD slash tomography computing time","Seismic imaging uncertainty at lower cost","Gradient-based Bayesian tomography runs faster"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Variational inference speeds seismic tomography","Bayesian tomography without the Monte Carlo wait","ADVI and SVGD slash tomography computing time","Seismic imaging uncertainty at lower cost","Gradient-based Bayesian tomography runs faster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000138,"raw_usage":{"total_tokens":1134,"prompt_tokens":908,"completion_tokens":226,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":163}},"tokens_in":524,"tokens_out":226,"duration_ms":3089,"temperature":1.0,"reasoning_tokens":163,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:40:23.417796+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Stein variational gradient descent: A general purpose Byesian inference algorithm","cited_arxiv_id":null,"evidence_quote":"Supplies SVGD, including the particle update based on the kernelized Stein discrepancy and the RBF kernel used in both experiments."},{"cited_title":"A kernelized Stein discrepancy for goodness-of-fit tests","cited_arxiv_id":null,"evidence_quote":"Defines the kernelized Stein discrepancy that provides the optimal perturbation direction for the SVGD particle update."},{"cited_title":"Uncertainty loops in travel-time to- mography from nonlinear wave physics","cited_arxiv_id":null,"evidence_quote":"Provides the synthetic circular low-velocity anomaly test problem and the double uncertainty-loop benchmark that SVGD reproduces."},{"cited_title":"Reversible jump Markov chain Monte Carlo computation and Byesian model determination","cited_arxiv_id":null,"evidence_quote":"Introduces reversible-jump McMC, the trans-dimensional sampling framework used as the comparison reference."},{"cited_title":"Multiple reflection and transmission phases in complex layered media using a multistage fast marching method","cited_arxiv_id":null,"evidence_quote":"Provides the fast-marching Eikonal solver used for forward modeling and for computing travel-time gradients on which both variational methods depend."},{"cited_title":"Fully 3D Monte Carlo ambient noise tomography over Grane field","cited_arxiv_id":null,"evidence_quote":"Provides the Grane-field ambient-noise data processing and the phase-velocity picks at 0.9 s used in the real-data test."},{"cited_title":"3-D Monte Carlo surface wave tomography","cited_arxiv_id":null,"evidence_quote":"Provides the 3D Monte Carlo surface-wave tomography framework, the noise-level estimate, and proposal choices used in the rj-McMC comparison."}],"review_version":1}