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Gaussian Processes for Probabilistic Estimates of Earthquake Ground Shaking: A 1-D Proof-of-Concept

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Fusing two seismic velocity models with Gaussian processes and simulating waves through the sampled profiles shows that peak ground displacement spreads much wider than any single model predicts.

desk verdict Useful proof-of-concept for propagating velocity-model disagreement through wave simulations, but the synthetic setup doesn't validate the quantitative uncertainty width. read the letter →

arxiv 2412.03299 v1 pith:LLUKWS47 submitted 2024-12-04 physics.geo-ph cs.LGstat.AP

Sam A. Scivier , Tarje Nissen-Meyer , Paula Koelemeijer , At{i}l{i}m Güneş Baydin This is my paper · ORCID
classification physics.geo-phcs.LGstat.AP
keywords Gaussianprocessregressionseismicvelocitymodelsearthquakegroundmotionpeakdisplacementprobabilisticfusionuncertaintyquantificationacousticwaveequationphysics-basedhazard
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

Seismic hazard estimates depend on which velocity model is chosen, and current physics-based workflows ignore disagreement among models. This paper proposes treating two or more velocity models as noisy observations of one true profile, fusing them with scalable Gaussian-process regression, and then sampling velocity profiles to drive acoustic wave-propagation simulations. The central finding is that the distribution of peak ground displacement across these samples is much wider than the spread implied by the two original models alone. The paper argues that probabilistic fusion is therefore necessary to capture the uncertainty that model disagreement adds to physics-based hazard analysis.

What carries the argument

The central object is the parametric predictive Gaussian process regressor (PPGPR), a sparse variational GP variant whose objective directly targets the predictive distribution $q(y_\ast)$ rather than the marginal likelihood. This yields an input-dependent variance $\sigma_f(x_\ast)^2$ over the underlying velocity function, which the paper interprets as uncertainty due to disagreement between velocity models. The covariance structure of the GP then allows spatially coherent velocity samples to be drawn from the predictive distribution. Each sample is passed through a finite-difference solver for the 1-D acoustic wave equation with a free-surface boundary condition at the surface and a perfectly matched layer at depth, and the peak surface displacement is recorded.

What would settle it

Replace the second synthetic model with a copy of the first; if the workflow is calibrated, the predictive variance should collapse and the PGD distribution should concentrate on a single value. If substantial predictive variance or a wide PGD spread remains, the uncertainty attributed to model disagreement is not actually measuring disagreement.

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Extended reading notes

Core claim

The paper demonstrates that fitting a parametric predictive Gaussian process regressor (PPGPR) to two synthetic 1-D velocity profiles simultaneously yields a predictive distribution whose uncertainty tracks the local differences between the models. This contrasts with standard sparse variational Gaussian processes, which absorb disagreement into a single observational noise term and produce noisy, spatially incoherent samples. Drawing 200 samples from the PPGPR predictive distribution and simulating the acoustic wave equation with a Ricker source produces a histogram of peak ground displacement whose median and middle 70% span far more than the PGD values obtained from the two input models alone. The paper concludes that using only the input models under-represents possible ground motions, and that the workflow is a proof-of-concept for probabilistic physics-based hazard analysis.

Load-bearing premise

The load-bearing premise is that the two input velocity models can be treated as conditionally independent, zero-mean noisy observations of a single true velocity profile; the synthetic construction makes them correlated, and real velocity models likely share systematic biases, so if this premise fails the predictive variance can no longer be read as uncertainty due to model disagreement.

Editorial extensions

If this is right

  • Physics-based hazard analyses that rely on a single velocity model will systematically understate the range of possible peak ground displacements, because the two input models' PGD values fall well inside the middle 70% of the distribution produced from GP samples.
  • The GP fusion step accepts any number of overlapping velocity models, so adding more input models does not change the workflow, although unequal data density would require weighting.
  • Because the predictive samples carry spatial covariance, the velocity profiles are plausible structures rather than pointwise noise, which is what makes the downstream wave simulations physically meaningful.
  • For real hazard assessment the workflow must be extended from 1-D synthetic profiles to 2-D and 3-D elastic media, as the paper itself notes, so the direct consequences here are limited to the proof-of-concept setting.

Reading between the lines

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

  • A natural calibration test would run the same fusion on many synthetic model pairs and check whether the GP's predictive intervals match the true spread of PGD, rather than relying on a single realization.
  • If real velocity models carry correlated errors, the diagonal Gaussian likelihood used here will likely understate uncertainty; encoding cross-model correlation in the process covariance would be the next step.
  • Because the input models' own uncertainties are currently ignored, incorporating them as heteroscedastic observation noise would presumably widen the final PGD distribution further.
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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 / 5 minor

Summary. The manuscript proposes a proof-of-concept workflow for propagating uncertainty in seismic velocity models into probabilistic earthquake ground-motion predictions. Two synthetic 1-D velocity profiles, m1 and m2, are fused with a sparse variational Gaussian process (PPGPR) so that disagreements between the models appear as predictive variance. Two hundred velocity samples are drawn from the GP predictive distribution and propagated through a 1-D acoustic finite-difference wave solver; the resulting peak ground displacement (PGD) histogram is compared with simulations using only m1 or m2. The paper concludes that the probabilistic fusion yields a much wider PGD distribution than the two individual velocity models, motivating probabilistic treatment of velocity-model uncertainty in physics-based hazard analysis.

Significance. If the result is validated, the workflow addresses a real gap in physics-based seismic hazard analysis: the arbitrary choice of velocity model is currently not treated as an uncertainty source, and this paper shows a practical way to turn model disagreement into a distribution of ground-motion outcomes. The paper's strengths are its controlled synthetic setup, use of a standard wave-propagation solver, a clear two-model benchmark, and released code, which makes the experiment reproducible. The significance is at the proof-of-concept level: the method is not yet calibrated or compared with existing fusion approaches, and the synthetic example is too small to demonstrate scalability in practice.

major comments (3)
  1. [Section 3] The synthetic construction violates the conditional-independence assumption of the likelihood used in the fusion. With m1 = s1 and m2 = (2/3)s1 + (1/3)s2, no latent function f and independent noises ε1, ε2 can represent both profiles as y_i = f + ε_i: if f = s1, then ε2 = (s2 - s1)/3 is correlated with f; if f is any other function, then ε1 and ε2 are correlated through their shared dependence on s1 and s2. Since the diagonal Gaussian likelihood in Section 3 is exactly what converts the two input profiles into predictive variance, the width of the posterior, and hence of the PGD histogram in Fig. 2g, is not established as the correct model-disagreement uncertainty. This issue is not flagged in the Limitations section. Please re-run the synthetic experiment with profiles generated as independent noisy observations of one common latent profile, or explicitly model the correlation between the two models.
  2. [Section 3] The reported validation against N((m1+m2)/2, ((m1-m2)/2)^2) is a per-location two-point heuristic, not a test of probabilistic calibration. Low RMSE of the PPGPR mean and variance relative to this heuristic shows that the predictive distribution tracks the local difference between the two profiles, but it does not establish that the predictive intervals have correct coverage for the true velocity or for the resulting PGD. A coverage or probability-integral-transform check on the synthetic data would directly support the 'probabilistic' claim, which is central to the paper's message.
  3. [Section 4] The headline comparison in Fig. 2g contrasts a continuous predictive distribution with two deterministic endpoint simulations. Because the predictive distribution is constructed to have nonzero width, the conclusion that the PGD spread is 'much wider' is partly by construction; what is missing is a quantitative statement of how often the sampled PGD values fall outside the range defined by the two endpoint simulations, and whether the m1 and m2 PGD values are plausible under the predictive distribution. Reporting quantiles, the fraction of samples outside the endpoint range, or a proper scoring rule would make the central claim more informative and less visual.
minor comments (5)
  1. [Section 2 and Fig. 1] The abbreviation is introduced as SVGP, but Fig. 1 and some text use 'SVGPR'; please unify the notation throughout.
  2. [Section 2] The observational noise variance is denoted σ2_y and then later σ2_obs; please use one symbol consistently.
  3. [Section 3] The paper states m = 20 inducing points with n = 25 data points per profile, so the total dataset n = 50 and the condition m ≪ n is not satisfied; the scalability argument should be stated as a motivation for future large-scale applications rather than a demonstrated property of this experiment.
  4. [Section 4] The Ricker wavelet parameters (central frequency and amplitude) are not specified in the text; since PGD scales with source amplitude, these values should be given in the paper for reproducibility, even if the code is available.
  5. [Software] There is a typo: 'libaries' should be 'libraries'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PGD distribution is a forward Monte Carlo propagation of a GP posterior and is not used to refit the model.

full rationale

The central claimed result is the distribution of peak ground displacements obtained by drawing 200 velocity samples from the PPGPR predictive distribution and simulating acoustic wave propagation for each sample (Section 4). This is a genuine forward propagation: the wave equation solver and the PGD extraction are independent of the GP fit, and the resulting histogram is not fed back into the regression or used to adjust any parameter. The GP itself is fit to the two synthetic velocity profiles, so the predictive variance is, by design, a probabilistic account of their differences; but calling this a 'fit' rather than an independent prediction is not a circularity because the paper's contribution is the downstream probabilistic hazard workflow, not a claim that the velocity posterior was validated out-of-sample. The validation metric comparing the predictive variance to N((m1+m2)/2, ((m1-m2)/2)^2) is a descriptive sanity check of the GP fit, not an independent benchmark, but it is not used to manufacture the PGD result. The limitations section explicitly acknowledges missing input uncertainties and the synthetic nature of the study. Self-citations (e.g., wave-propagation codes [27,29] and the code repository [15]) are not load-bearing for the central derivation. The conditional-independence assumption on m1 and m2, which is indeed violated by the construction m2 = (2/3)s1 + (1/3)s2, is a potential correctness/calibration concern, not a circularity: the argument does not reduce to assuming its conclusion.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central demonstration introduces no new physical entities. It relies on standard GP machinery plus several domain assumptions, the most fragile being that mismatches between velocity models are zero-mean noise and that the 1-D acoustic simulation is representative. Several hand-set numbers (20 inducing points, 200 samples, Ricker source parameters) affect the quantitative output but not the qualitative claim of widening the PGD spread.

free parameters (6)
  • GP kernel hyperparameters (RBF length scale and output scale) = learned via Adam; values in Zenodo code
    Control the smoothness and variance of the fused velocity profiles, and therefore the width of the PGD histogram.
  • Gaussian likelihood noise variance (sigma_y^2) = learned via Adam; then ignored in favor of latent predictive samples
    PPGPR by design drives sigma_y^2 low and sigma_f^2 high; the choice to ignore sigma_y when drawing samples is a modeling decision that affects the smoothness of the samples.
  • Number of inducing points m = 20
    Chosen by hand for both SVGP and PPGPR; affects approximation quality and computational cost.
  • Number of sampled velocity profiles = 200
    Chosen as a round number for the Monte Carlo ground-motion study; no convergence analysis is provided.
  • Source wavelet parameters (Ricker frequency and amplitude) = not specified in text; set in code
    PGD values are in arbitrary units, but the relative spread of PGD may depend on source frequency relative to structure.
  • Synthetic model weighting coefficient for m2 = m2 = (2/3)s1 + (1/3)s2
    Ad hoc choice to create larger-scale similarities and smaller-scale differences; it also makes m1 and m2 statistically dependent, which the GP model ignores.
assumptions (6)
  • standard math Sparse variational GP and PPGPR predictive equations from Titsias (2009), Matthews et al. (2016), and Jankowiak et al. (2020).
    Used as the statistical engine for fusion in Section 2 and taken as background.
  • domain assumption The true velocity profile is a GP with a stationary RBF kernel, and mismatches between velocity models are zero-mean Gaussian noise.
    Section 3 generates the synthetic models from GP samples and fits an RBF-based GP; this in-sample assumption makes the demonstration favorable.
  • domain assumption The two input models are conditionally independent observations given the latent function.
    Section 3 concatenates the datasets and uses a diagonal Gaussian likelihood; the construction m2 = (2/3)s1 + (1/3)s2 violates this independence.
  • domain assumption The 1-D acoustic wave equation with a free-surface boundary condition and Chern PML approximates seismic ground motion for the proof-of-concept.
    Section 4 uses finite-difference acoustic simulations; real seismology requires elastic waves in 2-D and 3-D.
  • domain assumption Predictive samples from q(f*) can be treated as deterministic input velocity models for the wave solver.
    Section 4 draws 200 samples and runs one simulation per sample; this ignores uncertainty in the GP hyperparameters.
  • domain assumption Input velocity models carry no uncertainty of their own.
    Stated as a limitation in Section 5; the workflow treats the input models as exact observations.

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Cite this review

Pith. "Pith review of Gaussian Processes for Probabilistic Estimates of Earthquake Ground Shaking: A 1-D Proof-of-Concept." pith.science (2026). https://pith.science/paper/LLUKWS47

@misc{pith2026241203299,
  author       = {Pith},
  title        = {Pith review of: Gaussian Processes for Probabilistic Estimates of Earthquake Ground Shaking: A 1-D Proof-of-Concept},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLUKWS47}},
  note         = {Machine review of arXiv:2412.03299}
}
read the original abstract

Estimates of seismic wave speeds in the Earth (seismic velocity models) are key input parameters to earthquake simulations for ground motion prediction. Owing to the non-uniqueness of the seismic inverse problem, typically many velocity models exist for any given region. The arbitrary choice of which velocity model to use in earthquake simulations impacts ground motion predictions. However, current hazard analysis methods do not account for this source of uncertainty. We present a proof-of-concept ground motion prediction workflow for incorporating uncertainties arising from inconsistencies between existing seismic velocity models. Our analysis is based on the probabilistic fusion of overlapping seismic velocity models using scalable Gaussian process (GP) regression. Specifically, we fit a GP to two synthetic 1-D velocity profiles simultaneously, and show that the predictive uncertainty accounts for the differences between the models. We subsequently draw velocity model samples from the predictive distribution and estimate peak ground displacement using acoustic wave propagation through the velocity models. The resulting distribution of possible ground motion amplitudes is much wider than would be predicted by simulating shaking using only the two input velocity models. This proof-of-concept illustrates the importance of probabilistic methods for physics-based seismic hazard analysis.

Figures

Figures reproduced from arXiv: 2412.03299 by the authors.

Figure 1
Figure 1. Comparison of SVGPR and PPGPR for the probabilistic fusion of seismic velocity [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Wavefield snapshots and probabilistic ground motion prediction. (a)–(f) shows wavefield [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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    doi: 10.1109/TNNLS.2019.2957109

    2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.