On extracting coherent seismic wavefield using variational symmetric autoencoders

Pawan Bharadwaj

REVIEW 4 major objections 5 minor 1 cited by

On extracting coherent seismic wavefield using variational symmetric autoencoders

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A variational symmetric autoencoder disentangles shared earthquake source information from station-specific scattering, and reconstructs clean virtual seismograms without deconvolution.

desk verdict A genuinely useful new formulation for extracting source time functions without deconvolution, but the paper undersells its own contribution by leaning on qualitative validation and skipping baselines. read the letter →

arxiv 2411.15613 v1 pith:GN4UZUMC submitted 2024-11-23 physics.geo-ph

Pawan Bharadwaj This is my paper · ORCID

classification physics.geo-ph

keywords symmetricautoencodervariationaldisentangledrepresentationcoherentwavefieldextractionearthquakesourcetimefunctionnonlinearstackingvirtualseismogramtime-shifttransformer

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 sets out to show that a symmetric autoencoder (SymAE) can extract the coherent part of a seismic wavefield, namely the part truly shared across recordings, and separate it from per-receiver nuisance such as path scattering and noise. The model treats each group of seismograms as generated from a common source code $s$ and individual path codes $p_k$, with waveforms assumed independent given $s$. After training, a Kullback-Leibler divergence $H[x_j^k]$ between the accumulated source posterior and each individual posterior ranks how informative a seismogram is, and a latent-space optimization produces virtual seismograms with enhanced signal-to-noise ratio. Applied to teleseismic P-wave records of deep-focus earthquakes, the method recovers source time functions with multiple rupture episodes that standard deconvolution smooths out. A reader would care because the approach promises data-driven extraction of earthquake source information without assuming known path Green's functions.

What carries the argument

The central object is the Symmetric Autoencoder (SymAE), a variational autoencoder whose latent space is split into a coherent code $s$ shared by a whole group of waveforms and nuisance codes $p_k$ for each waveform. The load-bearing mechanism is the conditional independence assumption $P(x_j^k, x_j^l | s, p_k, p_l) = P(x_j^k | s, p_k) P(x_j^l | s, p_l)$ (eq. 9), which lets the encoder factorize the approximate posterior and accumulate source information across receivers as a product of Gaussians; the resulting precision-weighted mean and variance (eqs. 17-18) are a probabilistic generalization of nonlinear stacking. A time-shift transformer module, built on spatial-transformer-style localization networks, aligns waveforms before encoding so the model can handle timing variations without explicit cross-correlation.

What would settle it

Take the synthetic setup of Fig. 1 and add a common scattered coda, identical across all receivers in a group, on top of the per-receiver path effects; if the recovered source time function changes or contains that common coda, or if the eq. (19) ranking starts preferring contaminated traces, the conditional-independence assumption is broken and the extraction is not purely source information.

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Extended reading notes

Core claim

The central claim is that SymAE learns a disentangled latent representation in which one component, $s$, accumulates all information about the earthquake source shared by the seismograms in a group, while the remaining components $p_k$ carry waveform-specific nuisance. The accumulation is done by multiplying the per-seismogram approximate posteriors, which for Gaussians reduces to precision-weighted averages (eqs. 17-18). The paper defines an informativeness measure based on the KL divergence between $Q(s|X_j)$ and $Q(s|x_j^k)$ (eq. 19), shows that low-divergence seismograms carry reliable source content, and solves an optimization (eq. 21) for the nuisance code that minimizes this divergence while enforcing sparsity and normalization, thereby generating a virtual seismogram for each earthquake. On synthetic data the extracted source signatures match the ground-truth sources, and on real deep earthquakes the SymAE-derived source time functions are clearer and capture complex multi-event ruptures that traditional deconvolution obscures, all without an explicit deconvolution step.

Load-bearing premise

The method assumes that once the shared earthquake source is known, the recordings at different stations carry no additional shared information, so any leftover common path, noise, or site effect would be folded into the learned source and silently treated as source signal.

Editorial extensions

If this is right

Analysis of complex earthquakes with multiple rupture episodes becomes possible without deconvolution, because the source code $s$ is learned from the waveform population rather than inverted from a known path response.
The KL informativeness score (eq. 19) gives a principled way to rank, weight, or discard seismograms according to source content, potentially improving receiver-function and ambient-noise stacking workflows.
Virtual seismograms formed by swapping path codes between similar earthquakes (eq. 20) provide a data-driven redatuming tool, so a clear recording from one event can help visualize a smaller or more obscured event with a comparable source.
Source-similarity structure recovered from the information-loss heatmap (Fig. 6) aligns with known rupture complexity, suggesting the model can serve as an unsupervised grouping tool for earthquakes.

Reading between the lines

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

Editorial inference: If eq. (9) is violated, for example when receivers share near-source structure or common mantle heterogeneity, that correlated path energy will leak into $s$, and because the same encoder scores informativeness, the leakage may not be visible in the KL metric; an external comparison against back-projection or moment-tensor inversions would expose it.
Editorial inference: The same accumulation-by-conjunction recipe applies directly to receiver-based groups: treating a station's waveforms across many earthquakes as the group would let SymAE extract a coherent receiver function or site response without stacking individual teleseisms.
Editorial inference: A natural stress test is to add a synthetic common scattered coda to all receivers in a group; if the recovered source time function changes or the informativeness ranking shifts toward contaminated traces, the method's core independence assumption is falsified.
Editorial inference: The sparsity weight in eq. (21) controls a trade-off between temporal resolution and noise suppression; tuning it per target frequency band may let seismologists tailor virtual source functions to specific rupture processes.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague — quick take: this paper has a real idea. Bharadwaj extends the SymAE/ML-VAE machinery to earthquake source imaging, adding a KL-based informativeness metric (eq. 19) and an optimal virtual-seismogram objective (eq. 21) that together let you pull a coherent source time function from a collection of noisy teleseismic P waveforms without deconvolution. That's worth something: deconvolution-based source imaging struggles with multi-rupture events, and the real-data comparisons in Fig. 11 show the method producing sharper source functions for events like Bonin and Spain than a standard technique. The variational derivation is standard and internally consistent, and the time-shift transformer (Section 4) is a sensible fix for misalignment.

The soft spots are real, but they are mostly validation gaps, not fatal flaws. The synthetic experiments compare extracted sources to ground truth only by eye (Fig. 9). No correlation coefficient, no error bars, no seed variability. It would take an afternoon to add those. There's also no comparison against a simple non-linear stacking baseline, which undercuts the 'generalizes stacking' claim. And despite the Data Availability section saying all code and data are open access, there's no repository link or instruction—that's a practical problem for reproducibility.

The stress-test concern about eq. 9's conditional independence is legitimate but maybe slightly overplayed. Yes, if receiver waveforms share path structure beyond the source—near-source heterogeneity, common mantle attenuation—the model has nowhere to put it except the shared code. And yes, the informativeness metric uses the same encoder, so a contaminated source code could fool the self-evaluation. But that's a limitation of the framework, not a hidden contradiction. The synthetic test with independent path effects does ground the basic machinery; it just doesn't test the correlated-path scenario. The author should acknowledge this explicitly and, better, run a synthetic experiment with correlated path noise or a receiver-function-style grouping.

My recommendation: with the quantitative metrics, a baseline comparison, and the code released, this belongs in a good seismology or machine-learning-for-geophysics journal after revision. As is, it's a conditional accept rather than a desk reject. Send it to a referee who knows both the source-imaging literature and VAE practice—they'll be able to judge whether the absence of external validation is a dealbreaker for the specific claim of multi-rupture resolution. I'd bring it to our reading group, though I'd pair it with a classic ML-VAE paper to give context.

Referee Report

4 major / 5 minor

Summary. The paper presents a variational symmetric autoencoder (SymAE) that disentangles a set of seismograms into a shared source code and per-seismogram path codes. It introduces a KL-divergence-based informativeness metric (Eq. 19) and an optimization procedure (Eq. 21) to generate virtual seismograms enriched in coherent source information. The method is applied to synthetic data with known source time functions and to teleseismic P-wave records from deep-focus earthquakes, with comparisons to traditional deconvolution. The central claim is that SymAE extracts coherent source information without deconvolution, enabling analysis of complex earthquakes with multiple rupture episodes.

Significance. If the claims are substantiated, the paper offers a novel nonlinear stacking/representation framework for seismic source extraction, building on prior SymAE work and extending it with a practical informativeness criterion and a latent-space optimization for virtual seismograms. The variational formulation in Sections 2-3 is standard and internally consistent, and the synthetic experiment with known sources is a well-posed falsifiable test. The paper also states that all codes and data are open access, which is a strength for reproducibility, though no repository is provided. However, the current experimental validation is insufficient to establish the central claims, particularly because the synthetic results are only visually compared and the real-data comparison is qualitative.

major comments (4)

[§3, Eq. (9)] The conditional independence assumption P(x_k, x_l | s, p_k, p_l) = P(x_k | s, p_k) P(x_l | s, p_l) is load-bearing for the precision-weighted accumulation in Eqs. (17)-(18). For teleseismic P waves, receiver waveforms share path-related structure beyond the source (e.g., near-source heterogeneity, common mantle attenuation, source-side scattering) that has no dedicated latent variable in the model. When Eq. (9) is violated, shared path energy has nowhere to go except the shared code s, so the learned 'source' code is actually source plus common path. The synthetic experiment uses independent receiver-specific path effects and therefore cannot detect this failure. I recommend adding a synthetic experiment with correlated or partially shared path components across receivers and measuring whether the recovered s matches the true source under those conditions.
[§3.3-3.5, Eqs. (19)-(21)] Both the informativeness metric H and the optimal virtual seismogram objective in Eq. (21) are evaluated using the same trained encoder Q(s|x) that was fit under Eq. (9). If s contains shared path energy, these self-evaluations will not reveal the failure; H will be small even when the virtual seismogram is not source-dominated. The synthetic validation in Fig. 9 is only visual: no quantitative recovery metric (e.g., correlation or L2 error between the optimized virtual seismogram and the true band-limited source), no error bars, and no random-seed variability. Please add quantitative metrics with multiple seeds and, if possible, an out-of-sample decoder or a separate validation encoder to break the circularity.
[§5, Fig. 11] The real-data comparison against SCARDEC deconvolution is purely qualitative. The claim that SymAE 'captures complex features due to multiple events' is supported only by visual inspection. There is no external source time function, no quantitative similarity metric, and no uncertainty analysis. The statement that 'the half-durations of the source functions extracted using SymAE closely align with those determined from the raw seismograms' is not quantified. Please provide a quantitative comparison (e.g., cross-correlation, misfit, or at least a table of half-durations with uncertainties) and, if available, compare with independent source models for these events.
[§3.3-3.4 and Fig. 5] The text contradicts itself on the sign of H. Section 3.3 states that seismograms with lower H values are more informative (Eq. 19), while Section 3.4 states that 'seismograms that exhibit higher H values (refer to eq. 19) are considered informative' and Fig. 5's caption repeats this. This reversal is not cosmetic: it governs the interpretation of Figs. 4-6 and the redatuming results. Please reconcile the definition and apply it consistently throughout.

minor comments (5)

[§3.5, Eq. (21)] The display of the constraints in Eq. (21) is garbled; please write the zero-mean and unit-variance constraints clearly (e.g., sum_n (y_j[m])_n = 0 and (1/D) sum_n ((y_j[m])_n - mean)^2 = 1).
[§3.4, Fig. 6] The heatmap description is ambiguous: the text says 'higher values after normalization indicate greater similarity' while the caption says lighter colors indicate lower information loss; these are opposite unless the normalization inverts the KL values. Please clarify which quantity is plotted.
[§5] The paper states that SymAE is trained on 'roughly 5000 displacement seismograms (all components)' but does not specify the number of receivers, the time window, preprocessing details, network architecture (layer counts, latent dimensions), or the values of hyperparameters α and γ. Please add these details for reproducibility.
[§8] The data availability section says all data and codes are open access, but no repository or DOI is provided. Please include a link or accession code.
[References] There are several reference formatting errors, e.g., the Weaver 2001 entry is garbled ('RL Weaver. Lobkis 0 i. Ultrasonic without a sources Thermal fluctuation correlations at MHz frequencies...'). Please correct the reference list.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-referential informativeness metric, but the main source-extraction claim is independently grounded by the synthetic test and by comparison with deconvolution.

self definitional [Sec. 3.3, Eq. (19); used again in Sec. 3.5, Eq. (21)]
"After training SymAE, we can identify which specific seismograms contribute most significantly to the accumulated source information. This is accomplished by calculating the Kullback-Leibler (KL) divergence between Q(s | Xj), which represents the accumulated distribution, and Q(s | xk_j ), the distribution for an individual seismogram xk_j : H[xk_j ] = DKL(Q(s | Xj) ∥ Q(s | xk_j )). (19)"

The informativeness metric is defined entirely from the trained encoder's own outputs: Q(s|Xj) is the precision-weighted product of the individual posteriors (Eqs. 16-18), so H[xk_j] measures, by construction, how close one waveform's learned posterior is to the product of all learned posteriors. It is not an independent measure of source content. The same self-defined distance is then reused in Eq. (20) and as the primary objective in Eq. (21) to generate and evaluate virtual seismograms, so the model is its own judge at the metric level. The synthetic experiment (Fig. 9), which compares extracted waveforms with known sources, is an external check and prevents this self-referentiality from becoming load-bearing for the main source-extraction claim.

full rationale

The derivation of the SymAE objective (Eq. 14) follows from standard variational inference, and no step reduces a prediction to a fitted parameter. The conditional independence in Eq. 9 is an explicit modeling assumption rather than a hidden input, and the paper tests this framework in a synthetic setting with known sources and against a standard deconvolution method (Fig. 11). Citations to Bouchacourt et al. (2018) and Bharadwaj et al. (2022) are used for background and for the standard Gaussian-conjugate accumulation result, so they are not load-bearing self-citations. The one notable self-referential aspect is that the informativeness metric H in Eq. 19 and the Eq. 21 objective are evaluated with the trained encoder's own posteriors; this could conceal failures if correlated path energy leaks into the source code s. However, because the synthetic test provides an independent target and the real-data comparison is made against an external deconvolution method, the central claim retains independent content. Overall this is a minor self-referential evaluation step rather than a circular derivation, giving a score of 2.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the conditional independence assumption (eq. 9), the prior structure (eq. 8), and the faithfulness of the trained encoder as a source-evidence estimator, plus unreported hyperparameters alpha and gamma. The method's success therefore depends on model choices that are only partially validated.

free parameters (3)

alpha (sparsity weight, eq. 21) = not reported
Controls the trade-off between KL fidelity and L1 sparsity in the optimal virtual seismogram objective; value chosen by hand and not stated.
gamma (time-shift regularization, eq. 22) = not reported
Penalizes large time shifts; value not reported, affects alignment and thus source/path separation.
Latent dimensions and network architecture = not reported
Sizes of s and pk latents, number of convolutional layers, kernel sizes, learning rate, batch size, etc. are not specified; the preprint only states the ADAM optimizer and 50 epochs implied by figures.

assumptions (5)

domain assumption Conditional independence of waveforms given the shared source code (eq. 9)
Assumes that once the earthquake source s is known, receiver-specific waveforms carry no mutual information; this is the load-bearing structure that justifies combining posteriors. Correlated path effects shared between nearby stations would violate it.
domain assumption Prior independence of source and path codes (eq. 8)
Assumes the source and path latent factors are independent in the prior; if path effects modulate source radiation pattern systematically, this is violated.
standard math Factored approximate posterior (eq. 13)
Mean-field style factorization used to derive the tractable ELBO; standard in variational inference.
standard math Gaussian likelihood and diagonal-covariance Gaussian posteriors
Standard VAE assumptions; choose N(x | decoder output) and N(latent | encoder output).
domain assumption The learned Q(s|x_k) is a faithful estimator of source evidence
Underpins the informativeness metric and eq. 21; if the encoder maps noise into the source latent, the 'informative' ranking is an artifact.

invented entities (2)

Latent source code s
purpose: Captures coherent source information shared across all seismograms of an earthquake; used to accumulate evidence and generate virtual seismograms.
A latent variable postulated by the generative model; no direct physical measurement outside the model.
Per-seismogram latent path code p_k
purpose: Captures waveform-specific nuisance/path scattering so it can be separated from s and swapped in redatuming experiments.
Latent nuisance variable; not independently observable.

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Cite this review

Pith. "Pith review of On extracting coherent seismic wavefield using variational symmetric autoencoders." pith.science (2026). https://pith.science/paper/GN4UZUMC

@misc{pith2026241115613,
  author       = {Pith},
  title        = {Pith review of: On extracting coherent seismic wavefield using variational symmetric autoencoders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GN4UZUMC}},
  note         = {Machine review of arXiv:2411.15613}
}

read the original abstract

We discuss the variational formulation of the Symmetric Autoencoder (SymAE) and its role in achieving disentanglement within the latent space to extract coherent information from a collection of seismic waveforms. Disentanglement involves separating the latent space into components for coherent information shared by all waveforms and components for waveform-specific nuisance information. SymAE employs a generative model that independently generates waveforms based on coherent and nuisance components, and an inference model that estimates these components from observed wavefield. By assuming the independence of waveforms conditioned on coherent information, the model effectively accumulates this information across multiple waveforms. After training, a metric based on Kullback-Leibler divergence is used to evaluate the informativeness of individual waveforms, enabling latent-space optimization and the generation of synthetic seismograms with enhanced signal-to-noise ratios. To demonstrate the efficacy of our proposed method, we applied it to a data set of teleseismic displacement waveforms of the P wave from deep-focus earthquakes. By training the SymAE model on high-magnitude events, we successfully identified seismograms that contained robust source information. Furthermore, we generated high-resolution virtual seismograms enriched with relevant coherent source information and less influenced by scattering noise, allowing a deeper understanding of the characteristics of the earthquake source. Importantly, our method extracts coherent source information without relying on deconvolution, which is often used in traditional source imaging. This enables the analysis of complex earthquakes with multiple rupture episodes, a capability that is not easily achievable with conventional approaches.

Figures

Figures reproduced from arXiv: 2411.15613 by the authors.

**Figure 2.** Generative models. (a) SymAE: this model assumes each waveform (denoted by [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Disentangled representation learning for seismic waveforms using SymAE. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: SymAE, applied here to the wavefield data from the 2013-05-24 Mw 8.3 Sea of Okhotsk earthquake, identifies [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: In the framework of source imaging, seismograms that exhibit higher [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Heatmap illustrating the average information loss when generating virtual seismograms by combining source [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Architecture of the SymAE with time-shift transformer. The model incorporates a localization network [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: ELBO and KL divergence DKL (Q ∥ P) during SymAE training with and without time-shift transformer using noiseless seismograms (Figs. 1e and 1f). The time-shift transformer SymAE model exhibits lower reconstruction loss (mse). waveforms, we introduce the notation Wj = {w…

**Figure 9.** Figure 9: SymAE extracted coherent source information through latent space optimization for the synthetic experiment. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: Same as Fig. 8, but using noisy seismograms in Figs. 1g and 1h. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: Comparison of coherent source information extracted using SymAE (blue) for complex deep earthquakes [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Enhanced receiver function imaging of crustal structures using symmetric autoencoders
physics.geo-ph 2024-11 conditional novelty 6.0 of 10

A symmetric variational autoencoder separates coherent crustal signal from per-earthquake noise in receiver functions, producing cleaner virtual receiver functions than standard averaging.

Reference graph

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