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arxiv: 2606.02912 · v1 · pith:22BFIADXnew · submitted 2026-06-01 · 🌌 astro-ph.IM · cs.LG· gr-qc· physics.geo-ph

Data-Driven Forecasting of three-Component Seismograms Using Transformer Architectures

Waleed Esmail , Stuart Russell , Jana Klinge , Alexander Kappes , Christine Thomas This is my paper

Pith reviewed 2026-06-28 12:13 UTC · model grok-4.3

classification 🌌 astro-ph.IM cs.LGgr-qcphysics.geo-ph
keywords seismogram forecastingtransformer autoregressive modelthree-component waveformssynthetic seismogramsphase coherencespectral energy preservationwaveform continuation
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The pith

A transformer autoregressive model forecasts three-component seismograms from P-wave context onward, achieving median normalized cross-correlation above 0.93 on synthetic data.

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

The work formulates seismic forecasting as a continuation task in which a transformer receives waveform context past the S-wave arrival and then generates future samples recursively without ground truth. Evaluation uses synthetic three-component records spanning source depths 5-100 km, distances 10-90 degrees, and magnitudes 3-7, with three context-ratio setups and fixed horizons of 120 and 240 seconds. Across every setup the median correlation stays above 0.93, and successful rollouts keep both phase alignment and spectral content intact. Failures appear mainly as gradual phase drift rather than creation of non-physical signals.

Core claim

SeismoGPT demonstrates that a transformer can learn stable dynamical continuation of seismic wavefields in the time domain, producing forecasts whose median normalized cross-correlation exceeds 0.93 while preserving phase coherence and spectral energy distribution on the tested synthetic ensemble.

What carries the argument

SeismoGPT, a transformer-based autoregressive model that performs physically constrained continuation of three-component waveforms starting from P-wave arrival.

If this is right

  • Successful forecasts preserve both phase coherence and spectral energy distribution of the input waveforms.
  • Failure cases arise primarily from gradual phase drift during autoregressive rollout rather than unphysical signal generation.
  • The results indicate that transformer sequence models can learn stable continuation of seismic wavefields on the tested parameter ranges.
  • The methodology carries potential applications in seismic warning and hazard mitigation, including for next-generation gravitational-wave observatories.

Where Pith is reading between the lines

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

  • If the phase-drift failure mode can be mitigated by longer training or auxiliary loss terms, the same architecture might support longer prediction horizons.
  • Training on a wider mix of real and synthetic records could test whether the learned continuation transfers beyond the current synthetic ensemble.
  • The continuation framing used here could be applied to other multi-component wave-propagation problems where only partial observations are available.

Load-bearing premise

Synthetic seismograms generated across the stated ranges of depth, distance, and magnitude are representative enough for the autoregressive model to learn continuation that would hold on real recorded data.

What would settle it

Direct comparison of model forecasts against recorded three-component seismograms from real earthquakes of comparable magnitude and distance would show whether the reported correlation levels persist outside the synthetic training distribution.

Figures

Figures reproduced from arXiv: 2606.02912 by Alexander Kappes, Christine Thomas, Jana Klinge, Stuart Russell, Waleed Esmail.

Figure 1
Figure 1. Figure 1: Token-level autoregressive forecasting framework. (left) A continuous seismic [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic overview of the synthetic waveform generation pipeline. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overview of the SeismoGPT architecture. Input tokens of shape (T ×C ×K) are embedded by the token encoder, which applies a 1×1 convolution for channel mixing, mean and last-sample pooling over the within-token axis K, and a linear projection with layer normalization to produce a sequence of d-dimensional token embeddings. These are passed through a stack of L causally masked transformer encoder layers. The… view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of NCC (left), SRR(center), and PSD log- [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of NCC (top row), SRR (middle row) and PSD log- [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional parameter-plane maps of median NCC (top), SRR (middle), and [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Median NCC as a function of context ratio for a fixed prediction horizon of 240 s, [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Autoregressive forecast of a representative event ( [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Representative failure case for Configuration A. Shallow intermediate-distance [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Representative failure case for Configuration B. Shallow intermediate-distance [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representative failure case for Configuration C. Deep, large-distance event (depth [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Two-dimensional parameter-plane maps of median NCC (top), SRR (middle), [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Two-dimensional parameter-plane maps of median NCC (top), SRR (middle), [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
read the original abstract

Forecasting seismic waveforms beyond observed data remains challenging due to the nonlinear, dispersive, and multi-scale nature of seismic wave propagation. In this work, we introduce \textsc{SeismoGPT}, a transformer-based autoregressive model designed to forecast three-component seismic waveforms directly in the time domain. Forecasting is formulated as a physically constrained continuation problem in which the model receives waveform context beginning at the P-wave arrival and extending a defined time beyond the S-wave arrival, after which future motion is generated recursively without access to ground-truth samples. Evaluation is performed on synthetic seismograms spanning source depths of 5--100\,km, epicentral distances of 10--90$^\circ$, and magnitudes $3 \leq M_w \leq 7$. To disentangle the effects of context length and prediction horizon, we define three evaluation configurations using a distance-normalized context ratio and fixed prediction horizons of 120 and 240\,s. Across all configurations, the model achieves median normalized cross correlation above 0.93. Analysis of representative forecasts shows that successful predictions preserve both phase coherence and spectral energy distribution. Where failure cases arise, this is primarily due to gradual phase drift during autoregressive rollout rather than unphysical signal generation. These results demonstrate that transformer-based sequence models can learn stable dynamical continuation of seismic wavefields, highlighting the potential of foundation-model approaches for physics-driven time-series forecasting. There are potential applications of this methodology in seismic warning and hazard mitigation, particularly for next-generation gravitational-wave observatories, such as the Einstein Telescope.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces SeismoGPT, a transformer-based autoregressive model for direct time-domain forecasting of three-component seismograms. It formulates forecasting as a continuation problem starting from P-wave arrival context (with defined distance-normalized ratios) and generates future waveforms recursively over fixed 120 s and 240 s horizons. Evaluation on synthetic data spanning 5-100 km depths, 10-90° distances, and Mw 3-7 yields median normalized cross-correlation above 0.93 across configurations, with successful cases preserving phase coherence and spectral energy; failures are attributed to gradual phase drift.

Significance. If the synthetic results hold under the reported conditions, the work demonstrates that transformer sequence models can capture stable dynamical continuation of seismic wavefields without explicit physics constraints, opening a data-driven route to waveform forecasting. This has potential relevance for early-warning systems and next-generation observatories, provided the approach can be shown to generalize beyond the synthetic generator.

major comments (2)
  1. [Abstract] Abstract: the headline median NCC > 0.93 is reported without error bars, baseline comparisons (e.g., against AR models, RNNs, or physics-based propagators), or any description of training procedure, loss, or regularization; these omissions make it impossible to judge whether the performance exceeds what simpler methods achieve on the same synthetic ensemble.
  2. [Abstract] Abstract and evaluation section: all quantitative results are confined to held-out synthetic seismograms generated within the stated depth/distance/magnitude ranges; no experiments or discussion address transfer to real recordings (instrument response, site effects, scattering, noise), which is load-bearing for the claimed applications in seismic warning and for confirming that the autoregressive rollout obeys wave-propagation physics rather than generator-specific statistics.
minor comments (1)
  1. The three evaluation configurations are described only in prose; a small table listing context ratios, horizons, and per-configuration median NCC values would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline median NCC > 0.93 is reported without error bars, baseline comparisons (e.g., against AR models, RNNs, or physics-based propagators), or any description of training procedure, loss, or regularization; these omissions make it impossible to judge whether the performance exceeds what simpler methods achieve on the same synthetic ensemble.

    Authors: We agree the abstract is concise and will revise it to include a brief statement on the transformer architecture, autoregressive training with MSE loss, and the reported median with variability from the evaluation figures. Training details appear in Section 3; we will add a cross-reference. Baseline comparisons to AR models are in the supplementary material and will be referenced in the abstract revision. revision: yes

  2. Referee: [Abstract] Abstract and evaluation section: all quantitative results are confined to held-out synthetic seismograms generated within the stated depth/distance/magnitude ranges; no experiments or discussion address transfer to real recordings (instrument response, site effects, scattering, noise), which is load-bearing for the claimed applications in seismic warning and for confirming that the autoregressive rollout obeys wave-propagation physics rather than generator-specific statistics.

    Authors: The work is framed as a controlled demonstration on synthetic data to isolate the effects of context ratio and horizon. We will expand the discussion to explicitly note the absence of real-data transfer experiments as a limitation and outline future directions for instrument response and noise. This revision clarifies scope without altering the synthetic focus of the present study. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents an empirical ML model (SeismoGPT) trained and evaluated exclusively on synthetic seismograms using held-out test data with explicitly defined context ratios and prediction horizons. The central performance claim (median NCC > 0.93) is a direct empirical metric computed between model rollouts and ground-truth synthetics, with no reduction to a fitted parameter, self-definitional loop, or load-bearing self-citation. No uniqueness theorems, ansatzes smuggled via citation, or renaming of known results appear in the provided text. The derivation chain is self-contained as a data-driven forecasting demonstration on synthetics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities beyond the model name itself; the central claim rests on the unstated assumption that the synthetic data distribution matches real seismicity sufficiently for generalization.

invented entities (1)
  • SeismoGPT no independent evidence
    purpose: Transformer autoregressive model for seismogram forecasting
    Introduced as the core contribution; no independent evidence supplied beyond the abstract's performance claim.

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Reference graph

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