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REVIEW 3 major objections 4 minor 237 references

Towards Open Science: Monitoring Crustal Deformations in North America

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper claims that a wavelet-moment framework, WAMORE, can map crustal deformation across the entire North American GNSS network with calibrated per-station confidence intervals using standard laptop-class computing, running 100–1,000 ti

desk verdict Solid empirical scaling win for GNSS velocity estimation, but the paper overstates its theoretical guarantees and the key covariance shortcut has no error bound. read the letter →

arxiv 2607.16264 v1 pith:DYG7AZKY submitted 2026-06-29 physics.geo-ph stat.APstat.CO

Lionel Voirol , Haotian Xu , Yuming Zhang , Luca Insolia , Roberto Molinari , Stéphane Guerrier This is my paper · ORCID
classification physics.geo-phstat.APstat.CO MSC 62M1062F1262J05
keywords WaveletVarianceGeneralizedMethodofMomentsGNSStimeseriesmissingdatamodelselectionlong-memoryprocessescrustaldeformationcomputationalscalability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's aim is to make continent-scale GNSS analysis accessible: it introduces WAMORE, a method that estimates the standard trajectory-model parameters of GNSS position time series (velocity, seasonal terms, offsets, post-seismic transients) together with a full noise model, while explicitly accounting for missing observations. The authors claim WAMORE delivers inferential performance comparable to, or better than, maximum-likelihood-based software, while being 100 to 1,000 times faster. They demonstrate this by producing the first uncertainty-quantified velocity field for the full North American network of 4,793 stations (14,379 signals) in about 170 hours on ordinary hardware. If true, this removes a major practical barrier to open, reproducible monitoring of crustal deformation: individual researchers would no longer need high-performance computing clusters or weeks of computation.

What carries the argument

The engine is the Generalized Method of Wavelet Moments (GMWM): estimate β by least squares, then fit the noise parameters γ by minimizing a quadratic distance between the empirical wavelet variance of the residuals and the model-implied wavelet variance. Two explicit trace identities do the heavy lifting—ν_j = tr(A_j Σ) for the theoretical wavelet variance and v_jl = 2 tr(A_j Σ A_l Σ) for its covariance—allowing computations to run through the fast wavelet pyramid algorithm instead of forming n×n covariance matrices. The other load-bearing parts are the explicit missing-data covariance formula Φ_n and the O(log n)-lag interpolation scheme that makes the trace evaluation linear in n rather t

What would settle it

Simulate a 20-year daily GNSS series with a WN+MAT+RW noise model and 20% missingness under the paper's Markov mechanism, compute the theoretical wavelet variance ν_j exactly via tr(A_j Σ) on the full covariance matrix, and compare it with the paper's O(log n)-lag interpolated approximation: if the relative difference exceeds roughly 1% at typical scales, or if the empirical coverage of WAMORE's 95% confidence intervals falls below about 92% over 1,000 replications, the method's central claim of calibrated inference under the shortcut fails.

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Extended reading notes

Core claim

WAMORE's claim is that a two-step wavelet-moment estimator reproduces likelihood-level inference for GNSS trajectory models while running in linear time. The paper proves the theoretical wavelet variance of any finite-covariance process is ν_j = tr(A_j Σ) and the covariance of the empirical wavelet variance is v_jl = 2 tr(A_j Σ A_l Σ), and uses these to fit noise parameters by matching moments on the residuals, explicitly modeling the missing-data mechanism as a stationary Markov chain. To make the trace evaluation linear, it reads only O(log n) diagonal/superdiagonal sums and interpolates, and it replaces RΣR by RΣ. Simulations show nominal coverage, and on 300 real stations the trend concl

Load-bearing premise

The computational shortcut for the residual covariance—reading only O(log n) lags of the covariance and linearly interpolating, with the paper itself noting that 'a formal approximation error bound for this interpolation strategy is currently unavailable'—must leave the estimated noise parameters nearly unbiased; if it does not, the covariance matrix Φ_n and every confidence interval built from it is off.

Editorial extensions

If this is right

  • A full uncertainty-quantified velocity field for the 4,793-station North American network (14,379 signals, 90 million data points) is computed in about 170 hours; a 900-signal subset that takes over 49 days with a likelihood-based tool runs in about 12.7 hours.
  • A global deployment of 6,313 stations (18,939 signals) completes in under 254 hours, a scale the paper argues is infeasible for existing likelihood methods.
  • The proposed model-selection criterion reproduces AIC-like noisemodel selection, with median runtime of about 4 minutes versus 29 minutes on 40-year daily series.
  • The trace identities ν_j = tr(A_j Σ) and v_jl = 2 tr(A_j Σ A_l Σ) are new explicit finite-sample formulas for wavelet variance and its covariance, transferable to any wavelet-based inference beyond geodesy.
  • Simulations and emulation studies indicate WAMORE's trend confidence intervals stay better calibrated than the MLE's when a random-walk noise component is present.

Reading between the lines

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

  • The load-bearing point for the whole speedup is the unproved interpolation of diagonal/superdiagonal sums at O(log n) lags; a formal error bound for that step would let practitioners know which noise models and missingness patterns the speedup is safe for.
  • The same moment-matching machinery applies to any large dependent-data field where likelihoods are covariance-bound, such as climate series, financial high-frequency data, or arrays of inertial sensors.
  • If the approximation is as accurate as the simulations suggest, routine global reference-frame tracking and post-seismic monitoring could move from periodic cluster jobs to near-real-time desktop updates after major earthquakes.
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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

3 major / 4 minor

Summary. The paper proposes WAMORE, a scalable framework for estimating trend parameters and their uncertainties in GNSS position time series with missing data, based on the Generalized Method of Wavelet Moments. The method is applied to 4,793 North American GNSS stations (14,379 signals), producing a velocity field with CIs and a global 6,313-station analysis. The authors report computational gains of orders of magnitude relative to the MLE implemented in Hector, with comparable or better coverage properties in simulations, and they provide an open-source R package and reproducible code.

Significance. If the statistical guarantees hold, this is a valuable contribution to geodetic time series analysis: it would make continental-scale GNSS velocity estimation feasible on standard hardware, with calibrated uncertainties. The paper includes substantial simulation evidence, an external comparison against Hector's MLE on 300 stations, an emulation study, and openly available code and data products. The computational results—170 hours for the North American network and 254 hours globally—are notable. However, the central claims rest on an unvalidated covariance approximation and on theoretical statements that are broader than the proofs provided.

major comments (3)
  1. [Appendix 8.2.2, Eq. (33) and following] The finite-sample approximation replaces (I−P)Σ(I−P) by RΣ and evaluates diagonal/superdiagonal sums only at n* = 1+3log2(n) lags, linearly interpolating the rest. The text explicitly states that no formal error bound is available. The trace identity just before Eq. (34) only proves tr(RΣR)=tr(RΣ); it does not control the per-lag sums used in (30), which are what define the approximate WV. Since γ̂ in (8) is estimated from this approximate ν(γ,ϑ), any bias propagates into Φn in (5) and hence into every reported CI and the headline runtime results. The simulations use the same shortcut, so they cannot independently certify it. Please provide an error bound or, failing that, a direct validation against exact computation for representative n, missingness patterns, and models including random walk and long-memory cases.
  2. [Section 4.2, Appendix 8.1.7–8.1.8] The text after Eq. (8) states that Appendix 8.1.7 proves consistency of the WV estimator under 'these same regimes' (short- and long-memory), but Theorem 4 is proved only under Assumption F (strictly stationary α-mixing, i.e., short-memory) with summability conditions on the mixing coefficients. No long-memory analogue is provided, and the random-walk (non-stationary) case is not covered by Theorem 4. The case study includes WN+MAT+RW models, so the stated theoretical guarantee is broader than the proof. Please either prove consistency under the long-memory (Assumption E) and non-stationary settings used in the paper, or restrict the claims to the cases actually proved.
  3. [Section 4.3, Eq. (11)] The covariance formula v_{j,l}=2tr(AjΣAlΣ) is exact for Gaussian quadratic forms, but it is applied to ε⊙Z, which is non-Gaussian. The paper acknowledges this, but does not quantify the resulting error or justify the use of this V for the model-selection penalty in (10) and for the weighting matrix. A misspecified V could bias the penalty 2q and the selection criterion. The simulations cover only WN+FL and WN+MAT settings, and the missing-data mechanism itself generates non-Gaussianity. Please add a derivation of the fourth-order term or a sensitivity analysis with clearly non-Gaussian errors to establish that the criterion remains reliable.
minor comments (4)
  1. [Section 2, station selection thresholds] The thresholds (more than 10 years of observations, less than 40% missing data) are stated as a compromise but no sensitivity analysis is given. A short discussion of how results change under alternative thresholds would strengthen the case-study claims.
  2. [Appendix 8.2.1, Eq. (30)] The approximation formula in (30) appears specific to Haar wavelets (L_j=2^j and the explicit ±1 coefficients), while the paper earlier defines a generic wavelet filter h_j. Please state explicitly that the computational algorithms in 8.2.1–8.2.3 assume Haar filters, or generalize the notation.
  3. [Throughout] There are several typos and formatting issues, e.g., 'to denotes' in Section 4.1, and inconsistent use of μ versus μ(ϑ) in places. A careful proofread is needed.
  4. [Section 6, Hector comparison] The 85.11% agreement in significance of the trend parameter is reported, but the fraction of cases where the two methods disagree on the sign of the trend—not just on significance—would be more informative for tectonic interpretation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; central benchmarks are external and the main caveat is an unverified approximation, not a fitted-input prediction.

full rationale

The central derivation chain is self-contained relative to its inputs. The point estimator \hat{\beta} in Eq. (4) is ordinary least squares on masked data, and its covariance \Phi_n in Eq. (5) is derived in Appendix 8.1.3 rather than assumed. The theoretical wavelet variance in Eq. (7) and its covariance in Eq. (11) follow from quadratic-form derivations in Appendix 8.1.4. The noise parameter \gamma estimated in Eq. (8) affects only \Phi_n, not the point estimates of velocity, as made explicit in Algorithm 1, so the velocity field is not manufactured by a fitted constant. The case study is checked against an independent MLE implementation (Hector) on a 300-station subsample, with 85.11% significance agreement, and against established North American tectonic trends. The Appendix 8.2.2 shortcut (R\Sigma R \approx R\Sigma, evaluation at n* = 1 + 3 log2(n) lags, and linear interpolation) is an acknowledged approximation without a formal error bound; this is a correctness/robustness risk, not a circular reduction, because the approximation is not defined in terms of the outputs it is used to predict. Self-citations to prior GMWM and wavelet-variance work are present, but the specific results used are either re-proved in this manuscript (e.g., Theorem 7 for the O(n^-1) WV approximation) or externally benchmarked. Simulation parameters borrowed from the case study are a design choice, not a renamed prediction. Thus no step reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The framework loads on an independent Markov missingness model, standard fixed-design asymptotics, and two unproven computational shortcuts: the Gaussian WV-covariance formula for a non-Gaussian masked process and the interpolation approximation. No new physical entity is introduced. The counts are modest for a statistical-methods paper, but the unproven shortcuts directly support the advertised 'well-defined statistical properties'.

free parameters (2)
  • Interpolation lag count n* = 1 + 3 log2(n) = a=1, b=3 (empirically chosen)
    Appendix 8.2.2: the full vector of diagonal/superdiagonal sums S is evaluated only at n* lags and linearly interpolated; no formal error bound is given, and this approximation underlies every GMWM evaluation in the case study.
  • Station selection thresholds = >10 years of data, <40% missingness
    Section 2: these hand-set cutoffs define the 4,793-station network and therefore the coverage and results of the claimed full-network analysis.
assumptions (7)
  • domain assumption Missingness process Z is a stationary two-state Markov chain independent of Y (Eq. 13 / Assumption A)
    Central to Eq. (5); if missingness is informative, e.g. triggered by deformation or equipment failure correlated with the signal, the covariance formula and inference break.
  • domain assumption Error process ε is zero-mean and satisfies short-memory mixing (Assumptions B, F) or one-sided linear long-memory with d ∈ (0, 1/2) (Assumption E)
    Used for asymptotic normality of β̂ and consistency of the wavelet variance; Theorem 4 only proves the short-memory case, so the long-memory guarantee is an unproven extension.
  • standard math Design matrix X is deterministic with regular limits and bounded rows (Assumptions C, D, G)
    Standard fixed-design regularity conditions needed for the CLTs and for the residual-based wavelet variance proof.
  • ad hoc to paper The Gaussian quadratic-form covariance formula v_{j,l} = 2 tr(A_j Σ A_l Σ) applies to the non-Gaussian missing-data process ε∘Z
    Eq. (11) is derived for Gaussian processes, then asserted for ε∘Z without fourth-cumulant corrections or a proof; this affects the model-selection criterion and efficient weighting.
  • ad hoc to paper The interpolation-based approximation of diagonal/superdiagonal sums is accurate enough for estimation
    Appendix 8.2.2 admits 'a formal approximation error bound ... is currently unavailable', yet this shortcut is used throughout the case study.
  • standard math GMWM regularity conditions: compact parameter space, continuous and injective theoretical wavelet variance, consistent weighting matrix (Assumptions H–K)
    Standard GMM-type conditions needed for consistency of γ̂.
  • domain assumption Post-seismic relaxation times τ_k are fixed to 1 year (Section 3)
    User-provided in model (1); if the fixed relaxation time is misspecified, it can bias the deterministic fit and residuals, hence the estimated noise parameters.

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Cite this review

Pith. "Pith review of Towards Open Science: Monitoring Crustal Deformations in North America." pith.science (2026). https://pith.science/paper/DYG7AZKY

@misc{pith2026260716264,
  author       = {Pith},
  title        = {Pith review of: Towards Open Science: Monitoring Crustal Deformations in North America},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYG7AZKY}},
  note         = {Machine review of arXiv:2607.16264}
}
read the original abstract

The study of the Earth's behavior has greatly benefited from the widespread deployment of Global Navigation Satellite Systems (GNSS), enabling large-scale monitoring of crustal deformation and long-term geophysical trends. In this work, we focus on the North American region, where complex tectonic activity, particularly along the western margin, requires methods capable of processing and analyzing large collections of GNSS time series distributed across extensive spatial domains. Analyzing the full GNSS network provides a coherent view of deformation across multiple scales, allowing detection of long-wavelength signals, subtle intraplate strain and regionally consistent velocity fields that are difficult to capture through local or subsampled analyses. Despite the availability of such data, existing methodologies remain computationally prohibitive for large-scale analyses across this region (or others). To address this limitation, we introduce a highly scalable framework (implemented in open-source software) that enables inference on crustal deformation across the full North American GNSS network using standard computational resources. The proposed method achieves substantial computational gains while maintaining inferential performance comparable to existing approaches and confirms existing tectonic trends, thereby supporting efficient large-scale monitoring and contributing to ongoing efforts toward "Open Science".

Figures

Figures reproduced from arXiv: 2607.16264 by the authors.

Figure 1
Figure 1. Illustration of a subset of the GNSS station network provided by the Nevada Geodetic [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Average running time of the WAMORE (left panel) vs the MLE (right panel) for the [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Empirical type I error (in %) of the WAMORE (left panel) vs the MLE (right panel) for [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Estimated tectonic velocities and associated uncertainty using the WAMORE methodology. [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Median ratio of the length of confidence intervals for the setting with the WN + MAT + [PITH_FULL_IMAGE:figures/full_fig_p089_5.png]
Figure 6
Figure 6. Figure 6: Average running time of the WAMORE (left panel) vs the MLE (right panel) for the [PITH_FULL_IMAGE:figures/full_fig_p090_6.png]
Figure 7
Figure 7. Figure 7: Empirical type I error (in %) of the WAMORE (left panel) vs the MLE (right panel) for [PITH_FULL_IMAGE:figures/full_fig_p090_7.png]
Figure 8
Figure 8. Figure 8: Median ratio of the length of confidence intervals for the setting with the WN + FL [PITH_FULL_IMAGE:figures/full_fig_p091_8.png]
Figure 9
Figure 9. Figure 9: Average running time of the WAMORE (left panel) vs the MLE (right panel) for the [PITH_FULL_IMAGE:figures/full_fig_p091_9.png]
Figure 10
Figure 10. Figure 10: Empirical type I error of the WAMORE (left panel) vs the MLE (right panel) for the [PITH_FULL_IMAGE:figures/full_fig_p092_10.png]
Figure 11
Figure 11. Figure 11: Median ratio of the length of confidence intervals for the setting with the WN + PL [PITH_FULL_IMAGE:figures/full_fig_p092_11.png]
Figure 12
Figure 12. Figure 12: Selection rate of the AIC and the proposed WAMORE criterion for setting considering [PITH_FULL_IMAGE:figures/full_fig_p096_12.png]
Figure 13
Figure 13. Figure 13: Agreement proportion and case count by selected model combination for the AIC and [PITH_FULL_IMAGE:figures/full_fig_p097_13.png]
Figure 14
Figure 14. Figure 14: Ratio of median CI length for all combinations of selected models for the proposed [PITH_FULL_IMAGE:figures/full_fig_p098_14.png]
Figure 15
Figure 15. Figure 15: Empirical coverages of the WAMORE and MLE when the true model is WN + MAT + RW. We then run another emulation study by adding a RW to the true model, where the innovation variance of the RW is within range of those estimated when this model is selected in the signals …
Figure 16
Figure 16. Figure 16: Empirical coverages of the WAMORE and MLE at the selected model and empirical [PITH_FULL_IMAGE:figures/full_fig_p100_16.png]

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