arxiv: 2605.09564 · v1 · submitted 2026-05-10 · ⚛️ physics.geo-ph

Recognition: 2 theorem links

· Lean Theorem

Mitigating cycle skipping in full waveform inversion using max-pooling-based approximate envelope and shot patching

Xinru Mu , Omar M. Saad , Shaowen Wang , Tariq Alkhalifah

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:38 UTC · model grok-4.3

classification ⚛️ physics.geo-ph

keywords full waveform inversioncycle skippingmax-poolingapproximate envelopeshot patchingvelocity model buildingseismic inversion

0 comments

The pith

A max-pooling sequence yields a richer low-frequency envelope than the Hilbert transform, letting full-waveform inversion escape cycle skipping from poor initial models.

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

Full waveform inversion often fails when field data lack low frequencies, because the misfit landscape contains many local minima separated by cycle skips. The paper replaces the usual Hilbert-transform envelope with one built from repeated 2D max-pooling operations, claiming this produces stronger low-frequency content. The same envelope is then paired with a shot-patching scheme that normalizes the misfit inside each local patch, automatically balancing the adjoint-source energy. Experiments on both synthetic and field data show the combined method converges to more accurate velocity models than the Hilbert-transform baseline when the starting model is far from truth.

Core claim

The authors compute an approximate envelope by applying a sequence of 2D max-pooling operations to the seismic data; this envelope contains richer low-frequency components than the Hilbert-transform envelope and therefore reduces cycle skipping. They further formulate an MPBAEP loss by dividing each shot gather into patches and exploiting the normalization property of the Euclidean misfit inside each patch, which produces locally balanced adjoint sources and accelerates convergence.

What carries the argument

The max-pooling-based approximate envelope (MPBAE), obtained by successive 2D max-pooling operations that extract low-frequency information directly from the data without a Hilbert transform.

If this is right

MPBAE-FWI succeeds on initial models too inaccurate for HTE-FWI to converge.
The patching step in MPBAEP-FWI improves gradient balance and shortens the number of iterations needed.
Both variants remain effective on field data as well as synthetic data.
The method directly targets the low-frequency deficiency that currently limits many FWI applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If max-pooling preserves enough phase information, the same envelope could be inserted into other waveform-based imaging methods that currently rely on Hilbert transforms.
The local normalization effect of patching might generalize to 3D or to other misfit functions that suffer from uneven illumination.
Because the envelope is generated by cheap pooling rather than a transform, the approach may lower the cost of generating low-frequency data for large-scale inversions.

Load-bearing premise

The sequence of 2D max-pooling operations produces an envelope containing genuinely richer low-frequency information without introducing artifacts or losing phase information critical for the inversion.

What would settle it

On a controlled synthetic test with a deliberately poor initial model, MPBAE-FWI or MPBAEP-FWI should fail to reach a lower data misfit or a more accurate velocity model than standard HTE-FWI if the max-pooled envelope does not actually supply usable extra low-frequency content.

Figures

Figures reproduced from arXiv: 2605.09564 by Omar M. Saad, Shaowen Wang, Tariq Alkhalifah, Xinru Mu.

**Figure 1.** Figure 1: Ricker wavelet along with its envelopes and spectrum. (a) The Ricker wavelet and envelopes obtained using [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Schematic of the proposed MPBAEP-FWI. more balanced energy distribution, leading to more balanced gradients. A non-overlapping patching strategy is adopted in this study, and it does not introduce discontinuities into the inversion results. This is because when the inverted velocity model approaches the true model, the difference between the observed and simulated data becomes nearly zero. Consequently, ev… view at source ↗

**Figure 3.** Figure 3: Convexity analysis of the misfit functions (equations (2), (4), and (5)): (a) observed signal (blue) and simulated [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Convexity analysis of the misfit functions(equations (2), (4), and (5)): (a) observed signal (blue) and simulated [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Convexity analysis of the misfit functions(equations (7), (8), and (9)): (a) observed signal (blue) and simulated [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of inversion results for the Overthrust model: (a) true Overthrust model, (b) initial velocity model, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: (a) Convergence of normalized data residuals for different loss functions over iterations. (b) Evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Seismic data along with their envelopes and spectra. The first and second rows show the observed data and [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: Gradients of the Overthrust model after the first iteration using (a) Euclidean loss, (b) HTE loss, (c) MPBAE [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

**Figure 10.** Figure 10: Comparison of inversion results for the Marmousi model: (a) true Marmousi model, (b) initial velocity [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

**Figure 11.** Figure 11: Comparison of convergence performance for different FWI methods on the Marmousi model: (a) data [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗

**Figure 12.** Figure 12: Seismic data together with their envelopes and spectra. The first and second rows show the observed data and [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

**Figure 13.** Figure 13: Comparison of gradients for the Marmousi model after the first iteration using (a) Euclidean loss, (b) HTE [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗

**Figure 14.** Figure 14: Comparison of inversion results from different FWI methods applied to the Western Australia dataset: [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗

**Figure 15.** Figure 15: Comparison between the observed and synthetic data generated by (a) the initial model and the inversion [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗

**Figure 16.** Figure 16: Angle-domain common-image gathers at different locations computed using the initial model and the [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗

**Figure 17.** Figure 17: Composite Ricker wavelet and its corresponding MPBAEs and spectra. (a) Composite Ricker wavelet [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗

**Figure 18.** Figure 18: MPBAEP-FWI inversion results using patch sizes of (a) 1 [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗

**Figure 19.** Figure 19: Inversion results for the Marmousi model obtained using (a) MPBAEP-TV-FWI and (b) TV-FWI. [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗

read the original abstract

Full waveform inversion (FWI) can produce accurate subsurface velocity models. However, the lack of sufficiently low-frequency content in field data often causes cycle skipping and traps the inversion in local minima. The Hilbert-transform envelope (HTE) provides a low-frequency representation that helps mitigate cycle skipping, but it may be insufficient when the initial velocity model is highly inaccurate. To further enhance low-frequency information and reduce dependence on the initial model, we compute an approximate envelope using a sequence of 2D max-pooling operations. Compared with HTE, the resulting max-pooling-based approximate envelope (MPBAE) contains richer low-frequency components and better mitigates cycle skipping. We further combine the MPBAE loss with a shot patching strategy and exploit the inherent normalization property of the Euclidean loss to formulate the MPBAEP loss, in which each shot gather is divided into localized patches for misfit evaluation. This introduces local adjoint-source energy balancing, as the adjoint source associated with the Euclidean loss exhibits a normalization effect within each local region, thereby improving gradient balance and accelerating convergence. Numerical experiments on synthetic and field data demonstrate that MPBAE-FWI significantly outperforms HTE-FWI when the initial model is poor, while MPBAEP-FWI further improves inversion accuracy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Max-pooling envelope plus shot patching is a concrete tweak for cycle-skipping in FWI, but the abstract gives no numbers or analysis to show it actually works better than Hilbert envelopes.

read the letter

The main takeaway is that the authors swap the standard Hilbert-transform envelope for a sequence of 2D max-pooling operations to create a low-frequency proxy, then add local shot patching to the Euclidean loss so each patch normalizes its own adjoint source. They claim the result reduces cycle skipping when the starting model is poor and that the patched version improves accuracy further on both synthetic and field data. The specific pairing of max-pooling approximation with patch-wise loss is not in the earlier envelope FWI papers they cite, so that combination counts as new. The practical motivation is solid: cycle skipping remains a real bottleneck in seismic velocity building, and anything that makes the inversion less sensitive to bad initial models has direct value for exploration and monitoring work. The patching step also makes sense on its own as a way to balance energy across shots without extra weighting terms. The soft spots sit in the evidence and the missing justification for the envelope change. The abstract asserts clear outperformance but supplies no metrics, error bars, data details, or inversion parameters, so the size of any gain is impossible to judge. The stress-test point about max-pooling being nonlinear and non-invertible is not addressed; there is no Fourier comparison or phase-preservation test showing that the pooled result actually carries richer usable low frequencies without blocky artifacts or aliasing. It is therefore unclear whether the reported improvement comes from the envelope or simply from the patching normalization. The approach itself is algorithmic and carries low circularity risk, but the lack of implementation specifics on pooling and patch sizes leaves open questions about reproducibility. This is aimed at geophysicists who build or tune FWI codes and who already work with envelope or misfit-function variants. A reader looking for practical gradient-balancing tricks could extract the patching idea even if the envelope part needs more validation. I would send it to peer review because the underlying problem matters and the method is specific enough for referees to test the claims directly.

Referee Report

2 major / 1 minor

Summary. The paper proposes using a sequence of 2D max-pooling operations to compute an approximate envelope (MPBAE) for full-waveform inversion (FWI) that supplies richer low-frequency content than the conventional Hilbert-transform envelope (HTE), thereby reducing cycle skipping when the initial velocity model is poor. It further combines the MPBAE misfit with a shot-patching strategy (MPBAEP) that exploits local normalization of the Euclidean loss to improve gradient balance. Numerical experiments on synthetic and field data are reported to show that MPBAE-FWI outperforms HTE-FWI and that MPBAEP-FWI yields additional accuracy gains.

Significance. If the central claims hold, the work supplies a practical, parameter-light algorithmic improvement for a long-standing difficulty in seismic imaging. The numerical experiments on both synthetic and field data constitute a concrete strength, offering direct evidence of behavior under realistic conditions rather than purely theoretical arguments.

major comments (2)

[Abstract] Abstract: the statement that MPBAE-FWI 'significantly outperforms' HTE-FWI supplies no quantitative metrics (e.g., data misfit reduction, model error norms, or convergence curves), error bars, or details on data selection and inversion parameters, rendering the magnitude and reproducibility of the claimed improvement impossible to assess.
[Method (MPBAE)] Section describing the MPBAE construction: no Fourier analysis of the composite max-pooling operator, no comparison of its spectrum to the Hilbert envelope below the lowest data frequency, and no demonstration that phase/traveltime fidelity is preserved are provided. Without such support, any observed improvement could be attributable to the shot-patching normalization rather than to genuinely richer low-frequency content in the envelope.

minor comments (1)

[Numerical experiments] Ensure that the precise pooling kernel sizes, stride values, and patch dimensions are stated explicitly in the text and figure captions so that the experiments can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment point by point below and have revised the manuscript to strengthen the presentation of our results and methods.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that MPBAE-FWI 'significantly outperforms' HTE-FWI supplies no quantitative metrics (e.g., data misfit reduction, model error norms, or convergence curves), error bars, or details on data selection and inversion parameters, rendering the magnitude and reproducibility of the claimed improvement impossible to assess.

Authors: We agree that the abstract would be strengthened by including quantitative support for the performance claim. In the revised version we will update the abstract to report specific metrics drawn from the synthetic and field-data experiments, including model-error-norm reductions and data-misfit values for MPBAE-FWI versus HTE-FWI, together with a concise statement of the key inversion parameters and data characteristics used. revision: yes
Referee: [Method (MPBAE)] Section describing the MPBAE construction: no Fourier analysis of the composite max-pooling operator, no comparison of its spectrum to the Hilbert envelope below the lowest data frequency, and no demonstration that phase/traveltime fidelity is preserved are provided. Without such support, any observed improvement could be attributable to the shot-patching normalization rather than to genuinely richer low-frequency content in the envelope.

Authors: We thank the referee for identifying this analytical gap. While the manuscript already presents separate numerical results for MPBAE-FWI (prior to shot patching) that demonstrate improved cycle-skipping mitigation on both synthetic and field data, we acknowledge the value of explicit spectral support. In the revision we will add a dedicated subsection that (i) derives the frequency response of the composite 2-D max-pooling operator, (ii) compares its spectrum directly with that of the Hilbert-transform envelope below the lowest data frequency, and (iii) discusses preservation of traveltime and phase information. These additions will help isolate the contribution of the richer low-frequency content from any effects of the subsequent shot-patching normalization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic proposal validated empirically

full rationale

The paper presents a methodological contribution consisting of a sequence of 2D max-pooling operations to form an approximate envelope and a shot-patching strategy for the misfit function. These are described as algorithmic constructions without any derivation chain that reduces a claimed result to its own inputs by construction. No equations are shown that equate a prediction to a fitted parameter, no load-bearing self-citations appear in the abstract or described text, and the performance claims rest on numerical experiments on synthetic and field data rather than on a self-referential mathematical identity. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new physical axioms or entities; it relies on the standard acoustic or elastic wave equation and the assumption that low-frequency content mitigates cycle skipping.

axioms (1)

domain assumption Seismic wave propagation is governed by the wave equation
Standard assumption in all FWI literature

pith-pipeline@v0.9.0 · 5536 in / 1175 out tokens · 44995 ms · 2026-05-12T02:38:24.635652+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we compute an approximate envelope using a sequence of 2D max-pooling operations... LM P BAE = sqrt(sum (ζsim − ζobs)²)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Convexity analysis... MPBAE loss exhibits better convexity than the HTE loss

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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