{"id":"d3333358-a52e-49b0-9ad4-3ad9011d123f","arxiv_id":"2507.16353","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A deterministic sampling scheme using sample extensions gives analytic Bayesian posteriors for travel time tomography from 51 forward simulations, but the results are biased because extensions cover limited space and parameter correlations are ignored.","lead":"This paper tests a way to speed up Bayesian seismic tomography by reusing each physics simulation to infer travel times over a whole region of possible Earth models. The method produces fast analytic uncertainty estimates, but the authors show it is biased and identify the two main causes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The on-ray extension, the cornerstone of the analytic posterior Eq. 13, replaces the true nonlinear first-arrival operator by an operator tied to a single pre-computed ray; Appendix B concedes this is not absolutely valid for finite cells, and the paper never directly quantifies its failure rate…","rationale":"I read the paper as an honestly-scoped methodological study. Its goal is to show that deterministic samples plus on-ray/off-ray extensions can produce a cheap analytic posterior approximation, while openly identifying bias and two major error sources. The reader's CONDITIONAL verdict is appropriate. My stress-test pass looked for the single most load-bearing assumption, and it is the on-ray extension, exactly as the reader identified. I did not find a fatal internal inconsistency: equations 1-6, 8-13 follow from the stated assumptions; the within-extension McMC checks in Figure 6 are credible internal validation; and the paper's own error analysis in Section 4.2 is unusually complete and honest. The identified concern is therefore not a rejection of the paper, but it is a genuine correctness risk: the analytic posterior in Eq. 13 is only the posterior of the stated problem if the on-ray assumption holds on the extension set, and the paper never verifies this directly. The proposed test would settle whether the assumption actually fails with non-negligible probability on the optimized extensions. If it fails, the method's central claim is materially overstated; if it passes, the CONDITIONAL verdict could later be upgraded. Either way, the appropriate verdict now is CONDITIONAL, unchanged from the reader.","tokens_in":21968,"tokens_out":2051,"duration_ms":23724,"concrete_test":"Implement the 5x5 single-datum example with an independent first-arrival solver (e.g., a fine-grid eikonal/FMM solver or a graph shortest-path solver on the same 5x5 cells). For each of the 51 optimal samples, draw many random test models inside the claimed extension box (on-ray slownesses decreased, off-ray slownesses increased, including the full volume shown in Figure 2b), and compare the claimed extension travel time (constant along off-ray directions, minus l_j*delta_m_j along on-ray directions) with the true first-arrival travel time computed by the independent solver. If a nonzero-volume subset has travel-time error exceeding the data uncertainty sigma=0.1 s, or if the violating fraction is more than a few percent across the 709 extensions, then Eq. 13 is not the posterior of the stated forward problem and the central claim is materially weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that one forward evaluation yields travel times, and hence an analytic posterior, over a continuous nonzero-volume extension subspace (Eq. 13). This rests on two assumptions made jointly in Section 2.1: off-ray slowness increases do not redirect the fastest ray, and on-ray slowness decreases leave the ray fixed. The off-ray rule is exactly true for first-arrival travel times. The on-ray rule is not exactly true for finite cells; Appendix B concedes it is 'not absolutely valid' and shows that if a competing ray has a longer segment in the modified cell, it can become faster. The paper's justification is heuristic: simultaneous decreases make a ray crossing less likely, and small cells make the error small. That is plausibly correct in some regimes, but it is not quantified. This assumption is load-bearing because Eq. 13 evaluates the Gaussian likelihood along a fixed ray vector l_k; any model inside the extension for which the argmin in f(m)=min(L^T m) shifts to another ray has a travel time that is not the claimed linear value. The optimized samples are extreme by construction: on-ray slownesses are at their upper bounds and off-ray slownesses at their lower bounds, so the samples sit on the boundary of the ray-valid region, and extension moves push directly toward the region where another ray may take over. The paper never tests the on-ray assumption directly: Figure 6 validates the analytic calculation against McMC run inside the extensions, and Figure 7 compares full-space results, but the full-space comparison conflates on-ray invalidity with the two acknowledged error sources (limited extension volume and mean-field combination). Thus the strongest claim, that the analytic posterior in Eq. 13 approximates the true posterior of the stated nonlinear forward problem, has an unquantified and potentially unbounded error source at its core.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes a semi-analytic Bayesian uncertainty assessment method for nonlinear travel time tomography based on the concept of sample extensions introduced by Curtis [16]. Two extension types are defined for a single forward evaluation: the off-ray extension, under which increasing slownesses in cells not on the fastest ray leaves the travel time unchanged, and the on-ray extension, under which decreasing slownesses in on-ray cells leaves the ray path unchanged and changes the travel time by the product of the slowness change and the ray segment length. Using a dictionary of 61 rays in a 5-by-5 grid, the authors solve an optimization problem to choose 51 slowness samples whose extensions maximize prior probability mass, obtaining 709 distinct extensions covering 3.3% of the prior space. Inside each extension the forward operator is linear (Eq. 1), so the single-datum posterior is an analytically tractable truncated Gaussian (Eq. 13). The analytic marginals are compared with McMC in the extension subspace and in the full space (Figures 6 and 7), and a multi-datum algorithm is built by combining single-datum marginals under a mean-field approximation (Eq. 17). Applied to a synthetic 7-by-7 tomography example with 28 travel times, the method recovers the low-slowness anomaly roughly but shows biases in several cells. An error decomposition (Table 1) attributes the largest errors to the limited extension hypervolume and to the mean-field combination of marginals.","tokens_in":22285,"tokens_out":4989,"duration_ms":60567,"significance":"If the extensions were physically valid over their stated hypervolumes, the method would be a notable efficiency gain: analytic posterior information from tens of deterministic forward evaluations instead of hundreds of thousands of McMC samples. The paper is unusually honest in its title and text about the method being biased and about its two fundamental inefficiency problems, and the analytic calculation inside a single extension is clean and internally self-consistent. The main scientific value is the demonstration that sample extensions for travel-time tomography provide a concrete, algorithmically usable subspace in which forward values are known analytically, together with a documented error budget for the subsequent approximations. However, the central physical assumption underlying the on-ray extension---that the fastest ray does not change when on-ray slownesses are decreased---is admitted in Appendix B to be 'not absolutely valid' for finite cells, and the manuscript never quantifies how often or by how much it fails. Because Eq.","major_comments":[{"comment":"The on-ray extension is load-bearing for the analytic posterior, but its validity is only asserted heuristically. Appendix B states that the extension is 'not absolutely valid' and that the competing-ray problem becomes less likely when cells are small or when many on-ray cells are decreased simultaneously. That reasoning is plausible but is not quantified. I ask for a direct numerical test: sample models uniformly from the extension of an optimized sample, compute the travel time using the extension formula l_k^T m, and compare it with a genuine nonlinear forward solver (e.g., FMM) for the same models. Report the fraction of the extension hypervolume in which the argmin in f(m) = min(L^T m) shifts to a different ray, or in which the travel-time error exceeds a meaningful threshold, across a range of cell sizes, contrast levels, and source-receiver geometries. Without this, the domain of validity of Eq. 13 is unknown.","section":"Section 2.1 and Appendix B, in relation to Eq. 13"},{"comment":"The paper's error decomposition omits the on-ray extension approximation as a source of bias. Table 1 lists five error sources, but the row 'Different forward modellers' compares the ray-dictionary forward operator (Eq. 2) with FMM in the full parameter space; it does not quantify the error caused by the on-ray extension itself inside an extension. This is a consequential omission because the optimized samples are constructed to lie at the boundary of ray validity: on-ray slownesses are at their upper bounds and off-ray slownesses at their lower bounds, so extension moves push directly into the region where a competing ray may become faster. The claim that the two dominant errors are 'limited extensions hypervolume' and 'construction of posterior pdf using marginals' can only be made after the on-ray approximation error has been measured and shown to be small, or explicitly acknowledged as a third major error source.","section":"Section 4.2 and Table 1"},{"comment":"The validation of the analytic posterior inside an extension is potentially circular. The text says the McMC inversions shown in Figures 6a and 6b are run 'inside this extension' and 'within extensions subspaces', but it does not state which forward operator was used for those McMC runs. If the McMC used the same linear formula l_k^T m that defines the extension, then the agreement between the orange analytic curves and the blue histograms only confirms the calculus of truncated Gaussian normalization; it does not test whether the extension travel-time prediction is physically correct. Please specify the forward solver used in the McMC runs of Figure 6, and if the current runs used the linear extension formula, add a comparison using FMM on a subsample of points inside the extension to directly test the on-ray assumption.","section":"Section 3.2 and Figures 6a/6b"}],"minor_comments":[{"comment":"The affiliation line contains a typo: 'Unite Kingdom' should be 'United Kingdom'.","section":"Author affiliation"},{"comment":"The formatting of the inequality constraints in Eq. 6 is inconsistent: 'm min < mj ≤ mmax, m min ≤ mk < mmax' uses an unspaced placeholder for m_min; please render the subscripts consistently.","section":"Equation 6"},{"comment":"The comparison between the analytic prior of Eq. 10 and numerical sampling inside extensions is a useful check, but the sentence 'the only explanation for deviations is due to any overlaps between hypervolumes of different extensions' is too strong; numerical sampling error and the non-normalized union in Eq. 10 could also contribute. Please soften or justify.","section":"Section 3.2 and Figure 5"},{"comment":"The panel labels (a)-(f) are described in the text, but the reader must map each panel to its method description manually; adding a short caption item per panel would improve clarity.","section":"Section 4.2, Figure 12"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely self-referential to Curtis [16], and the novelty with respect to that preprint should be made crisp or publication may be challenged. The main technical gap is the absence of a direct measurement of the on-ray extension's failure rate; adding such a test would substantially strengthen the paper and would align with the authors' own honest admission in Appendix B. I would not recommend rejection because the analytic steps are correct and the limitations are openly discussed, but the central physical approximation needs quantitative support."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this is a genuine methods contribution with an unusually honest error analysis, but the load-bearing on-ray extension is never validated directly, and that is the thing to push on in review.\n\nWhat's new: the deterministic sample-selection optimization (Eq. 6) is clean; the multi-datum marginal-combination algorithm in Section 3.3 is new; and Table 1's decomposition of error sources—with the isolating experiments in Figure 12—is more systematic than most geophysical inversion papers. The paper also earns credit for transparency: the title says \"biased,\" and the discussion names the two dominant error sources (limited extension volume and the mean-field combination of marginals). Within a single extension, the analytic posterior of Eq. 13 matches McMC run inside that extension (Figure 6), so the calculation is right conditional on the extension geometry.\n\nThe soft spot is the geometry itself. The on-ray extension assumes that decreasing slowness on the fastest ray does not change which ray is fastest. Appendix B concedes this is not absolutely valid for finite cells, and the paper never quantifies how often it fails in these test problems. The optimized samples are extreme by construction—on-ray slownesses at their upper bounds, off-ray at their lower bounds—so extension moves push straight into the regime where a competing ray can take over. If that happens, the travel time used in Eq. 13 is simply wrong for that part of the extension. The full-space comparisons in Figures 7 and 13 do not isolate this, because the limited-extension-volume and mean-field errors are also present. So the central claim has an unquantified error source at its core. That does not sink the paper, but it is a review-level issue, not a style point. Code is also not available, which makes it hard to check the ray dictionary construction.\n\nWho this is for: geophysicists working on cheap Bayesian inference for nonlinear problems. As a production method it is not there yet, but as a research direction it is worth engaging.\n\nRecommendation: send to peer review. The referee should ask for a direct test of the on-ray assumption—e.g., draw random models inside the extensions and compare the extension-predicted travel times against FMM—and quantification of how often and by how much the travel time is wrong. With that, the paper's claims would be on much firmer ground.\n\nYours,","headline":"Genuine methods contribution with an honest error analysis, but the on-ray extension at its core is never validated directly.","tokens_in":22865,"tokens_out":2944,"would_cite":true,"duration_ms":28831,"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":"This paper claims that Bayesian travel-time tomography can be made semi-analytic by extracting entire regions of known travel times from each forward simulation.","keywords":["sample extensions","travel time tomography","Bayesian inversion","uncertainty quantification","nonlinear inverse problems","deterministic sampling","mean field approximation","seismic tomography"],"falsifier":"A direct test is to take the 51 optimal models from the 5 by 5 example, sample many models uniformly inside their extension subspaces, and compare the extension-predicted travel time with the true first-arrival travel time computed by the fast marching method on the same model. If a non-negligible fraction of these models give differences larger than half the assumed data uncertainty, or if the actual fastest ray differs from the stored ray, the extension approximation is failing at the scale the method uses, exactly as the appendix predicts.","tokens_in":21766,"feed_emoji":"🧭","tokens_out":8267,"duration_ms":86291,"temperature":0.7,"pith_summary":"Travel time tomography usually needs hundreds of thousands of forward simulations to map out which subsurface slowness models fit the data. This paper claims that, for ray-based travel times, each simulation can be made to certify the travel time for an entire continuous region of model space, not just for one slowness model. The trick is to use two physical facts: changing slowness in cells the fastest ray does not touch leaves the travel time unchanged, and decreasing slowness in cells on the ray changes the travel time in a predictable linear way. The authors select models whose extensions cover as much prior probability as possible, then compute posterior marginal distributions analytically inside those regions. In a 25-parameter example, 51 optimally chosen simulations give analytic posteriors that approximate a 500,000-sample Monte Carlo reference at first-order level, with bias that the paper locates in limited extension volume and neglected parameter correlations.","feed_headline":"51 simulations replace 500,000 for tomography uncertainty","feed_subtitle":"Ray-physics 'extensions' let one forward run cover a volume of model space, yielding analytic posteriors with known bias.","key_machinery":"The central mechanism is the extension subspace formed by combining off-ray and on-ray extensions: within it, travel time is exactly linear in the on-ray slownesses, so the likelihood and posterior are analytically tractable. The deterministic sampling step solves an interior-point optimization problem whose objective is the negative logarithm of the extension hypervolume, subject to the constraint that the chosen ray remains fastest; this selects models whose extensions cover the most prior probability. The analytic posterior inside one extension is written as an integral over that subspace, and the multi-datum construction multiplies single-datum posterior marginals, an approximation the paper explicitly labels as a mean-field assumption.","core_discovery":"The central claim is that combining the off-ray extension (changing slowness in cells off the fastest ray leaves travel time unchanged) with the on-ray extension (changing slowness on the ray changes travel time linearly, exact in the small-cell limit) gives a continuous high-dimensional block of model space where travel times are known from a single forward evaluation. Within each such extension the travel-time function is exactly linear, so the posterior pdf can be written analytically. The authors define an optimization problem that picks slowness models maximizing the extension hypervolume while keeping the chosen ray fastest, apply it to a dictionary of 61 rays, and obtain 51 valid optimal samples that yield 709 extensions covering 3.3 percent of the 25-dimensional prior space. They compute single-datum posterior marginals analytically, correct some of the resulting bias using prior replacement, and combine single-datum marginals into multi-datum results using a mean-field product. In a 7 by 7 synthetic tomography example the method recovers the main anomaly, and the paper identifies two dominant error sources: the limited hypervolume spanned by the extensions and the neglect of parameter correlations in the mean-field construction.","pith_inferences":["The bias of extensions toward hypercorners of parameter space means the analytic prior over the covered region is not representative of the full prior; a natural test is to use these analytic extensions as proposals inside an informed-proposal Monte Carlo chain, which the paper itself flags as a possible remedy.","If the on-ray extension fails for finite cells as the paper's own appendix warns, the analytic posterior inside each extension will be overconfident; running a full-wave solver on models sampled inside the extensions would quantify how often a competing ray overtakes the stored fastest ray.","Because the extension hypervolume as a fraction of the prior shrinks exponentially with dimension, the semi-analytic advantage is likely to be confined to problems of modest dimensionality or with strong priors; a scaling study would show where the method stops beating direct Monte Carlo.","The mean-field error identified in the paper suggests a direct upgrade path: approximate each single-datum posterior by a mixture of Gaussians (as in boosting variational inference) and combine full distributions rather than marginals, which the paper proposes as a future direction."],"forward_implications":["A practitioner who trusts the extension approximations can obtain first-order posterior statistics—means, standard deviations, and marginals—with a small fixed number of forward evaluations, here 51, instead of hundreds of thousands or millions.","Inside each extension the Bayesian update is exactly linear, so the method yields reproducible analytic posteriors for those regions rather than sampling noise; the error appears as a known bias rather than random Monte Carlo error.","Because one solved template of rays and optimal samples can be scaled and rotated to different source-receiver pairs with the same local grid, the optimization cost is paid once and reused for many data.","The two dominant error sources identified in the paper—extension hypervolume and cross-parameter correlations—are separated, so each can be attacked independently in future work.","In the synthetic 7 by 7 example the method recovers the low-slowness anomaly with a mean model comparable to that from a full Markov chain Monte Carlo inversion, suggesting it can serve as a fast reconnaissance uncertainty estimate before more expensive sampling."],"supporting_citations":[{"why":"Introduces the sample-extension concept and the specific off-ray and on-ray extension physics that the whole method builds on.","marker":"[16]"},{"why":"Supplies the fast marching method used as the forward solver for reference Markov chain Monte Carlo inversions and comparisons.","marker":"[61, 62]"},{"why":"Provides the curse-of-dimensionality argument that motivates maximizing extension hypervolume and underlies the stated limitation.","marker":"[17]"},{"why":"Provides the prior replacement formulas used to correct bias in the posterior marginals computed on extension subspaces.","marker":"[82, 90]"},{"why":"Interior-point optimization algorithms used to solve the extension-maximization problem in equation 6.","marker":"[12, 11, 65]"}],"fun_headline_variants":["Sample extensions speed tomography uncertainty, but bias remains","51 rays map 3.3% of model space for tomography posterior","Analytic posterior from sample extensions, with known bias","Travel-time extensions cut simulations, yet limited and biased","Semi-analytic uncertainty assessment via sample extensions, biased"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the on-ray extension holds: lowering the slowness in cells along the current fastest ray never lets a different ray become faster, so every travel time inside the extension is simply the old time minus the slowness decrease times the path length, a condition the paper's own appendix says is exact only for vanishingly small cells and can fail for realistic finite cells.","fun_headline_variants_meta":{"raw":{"variants":["Sample extensions speed tomography uncertainty, but bias remains","51 rays map 3.3% of model space for tomography posterior","Analytic posterior from sample extensions, with known bias","Travel-time extensions cut simulations, yet limited and biased","Semi-analytic uncertainty assessment via sample extensions, biased"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1546,"prompt_tokens":1081,"completion_tokens":465,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":697,"tokens_out":465,"duration_ms":5279,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:11:48.687940+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test is to take the 51 optimal models from the 5 by 5 example, sample many models uniformly inside their extension subspaces, and compare the extension-predicted travel time with the true first-arrival travel time computed by the fast marching method on the same model. If a non-negligible fraction of these models give differences larger than half the assumed data uncertainty, or if the actual fastest ray differs from the stored ray, the extension approximation is failing at the scale the method uses, exactly as the appendix predicts.","supporting_citations":[{"cited_title":"Samples, symmetries and extensions: Exploring parameter space in nonlinear problems","cited_arxiv_id":null,"evidence_quote":"Introduces the sample-extension concept and the specific off-ray and on-ray extension physics that the whole method builds on."}],"review_version":1}