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Accelerating seismic inversion and uncertainty quantification with efficient high-rank Hessian approximations

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read High-rank Hessian approximations built from point-spread functions and pseudo-differential probing make seismic inversion and Bayesian uncertainty quantification practical, with MCMC variances that track the true posterior.

desk verdict A genuinely new PSF+ Hessian approximation with a convincing ideal-model test, but the Marmousi UQ validation leans on a low-rank benchmark that undercuts the paper's strongest claim. read the letter →

arxiv 2507.10804 v2 pith:RIF27DWR submitted 2025-07-14 math.NA cs.NA

Mathew Hu , Nick Alger , Rami Nammour , Omar Ghattas This is my paper · ORCID
classification math.NAcs.NA MSC 65N2186A1565C0562F15
keywords seismicfull-waveforminversionBayesianuncertaintyquantificationhigh-rankHessianapproximationpointspreadfunctionmethodpseudo-differentialoperatorprobingMCMC-gpCNMarmousimodelpreconditioning
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 make seismic full-waveform inversion and Bayesian uncertainty quantification tractable by approximating the misfit Hessian—the operator that encodes local curvature of the inversion objective as well as the covariance of the Laplace-approximated posterior—without forming it explicitly. Its central claim is that high-rank approximations built from point-spread functions and from pseudo-differential-operator probing, plus a new combined method called PSF+, reproduce the directional scalings of the posterior well enough that a generalized preconditioned Crank-Nicolson (gpCN) Markov chain Monte Carlo sampler explores the high-dimensional model space efficiently. On a synthetic quadratic problem with known uncertainty and on the Marmousi model, the authors find that samples from unapproximated or low-rank proposals concentrate in narrow regions, underestimate posterior variance, and overstate effective sample size, while the high-rank approximations yield variances that track the true posterior. If correct, this gives a practical route to trustworthy uncertainty estimates in seismic imaging at a fraction of the cost of exact Hessian construction.

What carries the argument

The central object is the Hessian $H = H_d + \Gamma_{pr}^{-1}$ of the negative log-posterior, accessed only through matrix-vector products that each cost two wave-equation solves per source. The argument treats $H_d$ as a pseudo-differential operator with symbol $s(x,\xi)$, a phase-space scaling function; PSF and PDO are dual ways of sampling that symbol, respectively along spatial rows via delta-function responses and along frequency columns via sinusoidal probing vectors. The load-bearing identities are $H_d\phi_\xi(x)=\phi_\xi(x)s(x,\xi)$ and $s(x_k,\xi)=(2\pi)^2\widehat{p_k}(\xi)$, which let a few operator applications recover enough of the symbol to reconstruct a low-rank separated approximation $s(x,\xi)\approx\sum_{k=1}^r a_k(x)b_k(\xi)$ that is applied in $O(rN\log N)$ time. For UQ, the symbol is square-rooted pointwise to keep the operator symmetric positive definite, and a high-pass filter plus low-rank correction separates the well-localized mid-frequency symbol from the smooth low-frequency part.

What would settle it

Run PSF+ on a synthetic model with a sharp high-contrast interface, compare the approximate Hessian's action on many random vectors against exact Hessian-vector products, and check whether gpCN samples still reproduce the variance of a converged reference sampler; growing relative error with contrast strength, or sample variance falling below the reference, would show the locality assumption has failed.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the seismic misfit Hessian $H_d$, although too large to store and too high-rank for low-rank compression, can be approximated as a pseudo-differential operator with a symbol $s(x,\xi)$ that is low-rank in a separated sense. The PSF method samples rows of this symbol by applying $H_d$ to delta functions; the PDO method samples columns by applying it to sums of sinusoids; the new PSF+ method combines both sets of samples by extracting a row basis with a singular value decomposition (SVD) and refining column coefficients through a small Tikhonov-regularized least-squares problem. After a high-pass filter removes nonlocal low-frequency content and a low-rank correction restores that content, the approximation is symmetric positive definite, so its inverse square root can seed Gaussian proposals for the gpCN sampler. The numerical evidence is that these proposals mix far faster than pCN or low-rank gpCN and, more importantly, that the spread of the samples matches the true posterior variance, whereas the baselines underestimate it and give misleadingly high effective sample sizes.

Load-bearing premise

The load-bearing premise is that the misfit Hessian behaves like a pseudo-differential operator with a symbol smooth and separable in space and frequency, so sampled point-spread functions stay localized and probed symbol columns can be interpolated; Section 5 states this can be violated in models with sharp parameter variations, and if it fails the approximations and the uncertainty gains break down.

Editorial extensions

If this is right

  • Full-waveform inversion can use the inverse Hessian approximation as an L-BFGS initial Hessian or Newton-Krylov preconditioner, cutting iteration counts and improving recovery in deeper, less-illuminated regions.
  • Bayesian UQ with gpCN becomes practical for high-rank seismic posteriors, since the proposal covariance captures directional scalings that low-rank and uninformative proposals miss.
  • Common MCMC diagnostics such as autocorrelation and effective sample size can report good mixing even when a chain is confined to a narrow region; the paper's histograms and variance maps expose that false confidence.
  • The PSF+ method inherits both the spatial locality of PSF sampling and the frequency locality of PDO probing, and in the benchmark experiments it is more accurate than either method alone.
  • The pseudo-differential row-column structure remains valid in three dimensions, so the approach is extensible to 3D seismic problems.

Reading between the lines

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

  • An implication the paper leaves implicit is that the row-column symbol completion strategy should transfer to other inverse problems governed by partial differential equations whose Hessians are pseudo-differential operators, such as elastic or electromagnetic inversion, though no such experiment appears here.
  • The paper's diagnostic lesson suggests a practical safeguard it does not itself implement: before trusting effective sample size for a Hessian-informed sampler, compare sample marginals against the Laplace approximation's marginals.
  • A natural testable extension, if sharp parameter contrasts violate symbol locality, is adaptive symbol sampling that places point-spread functions or probing frequencies where the symbol changes fastest, or a multiscale split that treats the nonlocal part separately.
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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

4 major / 5 minor

Summary. The paper adapts two established high-rank Hessian approximation methods (the point spread function method and the pseudo-differential operator probing method) and proposes a new PSF+ method that combines row and column symbol information to approximate the misfit Hessian of a seismic inverse problem. These approximations are used as preconditioners for L-BFGS in deterministic full-waveform inversion and as the proposal covariance in a generalized preconditioned Crank-Nicolson (gpCN) MCMC sampler for Bayesian uncertainty quantification. The methods are validated on an ideal quadratic model with analytically known posterior covariance and on the Marmousi model, where a low-rank Hessian-based gpCN run is used as the reference. The paper concludes that high-rank Hessian approximations are essential for accurate posterior variance estimation and can substantially improve MCMC efficiency.

Significance. If the central claim holds, the paper would make a meaningful contribution to large-scale Bayesian seismic UQ by showing that high-rank Hessian approximations can capture the directional scalings of the posterior in a regime where low-rank approximations are known to fail. The ideal quadratic model is a strong, falsifiable benchmark because the true posterior covariance is known analytically and the PSF+ method reproduces it, and the PSF/PDO duality derivation in Eqs. (39)-(42) is algebraically consistent. The paper also demonstrates concrete L-BFGS preconditioning benefits. However, the Marmousi validation currently lacks an independent reference, and the paper itself reports non-converged chains and acknowledges approximation limitations (Section 5, Appendix A3), so the strongest claims about posterior variance accuracy are not yet fully supported.

major comments (4)
  1. [Section 4.2.2, Table 4] The Marmousi benchmark is constructed as a 'brute-force low-rank approximation of the Hessian at the MAP model' followed by MCMC-gpCN sampling, but Sections 3.1 and 3.7 argue that low-rank approximations with insufficient rank underestimate posterior variance, and Table 4's own note states that the compared chains 'have not yet converged.' Therefore the statement that gpCN-PSF+ 'closely matches the benchmark' could simply reflect a shared low-rank truncation or non-convergence bias. The authors should provide a higher-rank or independently verified reference (for example, a Hessian approximation with explicit rank-convergence checks, or an independent sampling method with convergence diagnostics) before the Marmousi experiment can support the central claim about posterior variance accuracy.
  2. [Section 4.2.2, Table 4] The ESS values for gpCN-PSF+ on the Marmousi model (7.05, 6.14, 6.57, 4.88, 16.00) are not systematically higher than those of pCN or gpCN-LR, yet the text dismisses ESS and autocorrelation as 'heuristic' while relying on histograms that are compared only with the low-rank benchmark. Since no quantitative distance between the PSF+ marginal histograms and the benchmark is reported, and since non-convergence can affect all chains including the benchmark, the evidence for improved posterior variance accuracy in Marmousi is currently visual and indirect. Quantitative comparisons (for example, variance ratios against a trusted covariance, coverage of a reference interval, or a suitable distance between marginals) with convergence diagnostics are needed.
  3. [Section 3.7 and Appendix A3] The construction of UQ proposal samples via pointwise square roots of symbol rows and columns requires discarding negative symbol values, but the effect of this truncation on the resulting covariance is not analyzed. In addition, Appendix A3 states that the PDO method 'tends to overestimate high-frequency components' and that this 'poses challenges for UQ' because the proposal random vector is not band-limited; nevertheless, PDO is used as a UQ method in Sections 4.1.3 and 4.2.2. The paper should quantify the resulting bias in posterior marginal variances, for example by comparing PDO-based posterior variance against the analytic reference in the ideal quadratic model, and explain why the acknowledged high-frequency overestimate does not invalidate the UQ conclusions.
  4. [Section 5 and Figures 12-13] The methods rely on the assumption that the misfit Hessian has strong locality in both spatial and frequency domains, and Section 5 acknowledges that this 'may be violated in models with sharp parameter variations.' The Marmousi model contains strong velocity contrasts including the 400 m water layer, but no quantitative approximation error (for example, the relative operator norm error ||Hd - H_approx||_F / ||Hd||_F) is reported for the PSF, PDO, or PSF+ approximations on this model. A numerical assessment of the approximation error and its spatial distribution would be needed to rule out that the locality assumption is silently violated in the very setting used for the UQ validation.
minor comments (5)
  1. [Section 3.5, Eq. (39) and Figure 1] There are several typographical errors: 'self-adjont' should be 'self-adjoint', 'Too see' should be 'To see', and the Figure 1 caption 'An illustration of the duality ... this shown' should be 'this is shown.' These should be corrected.
  2. [Section 3, introductory paragraph] The sentence 'we introduce low-rank approximation as both a baseline and an auxiliary technique...' contains a duplicated word ('are are') in the preceding paragraph: 'two established Hessian approximation methods that are are widely used.' Please fix.
  3. [Sections 4.1.3 and 4.2.2] The MCMC settings are not fully specified: the step-size parameter beta in Eq. (14), the burn-in length, the number of chains, and the total number of samples after burn-in are not reported. Providing these details in the appendix would make the experiments reproducible and help interpret the reported ESS values.
  4. [Section 3.3, Eq. (27)] The definition of the symbol class is imprecise. The standard definition of S^1 requires derivative bounds of the form |∂_ξ^α ∂_x^β s(x,ξ)| ≤ C_{αβ}(1+|ξ|)^{1-|α|}; the text's wording 'bounded by polynomials of ξ of corresponding orders' is too vague. Similarly, Eq. (35) is an asymptotic expansion, not an exact polynomial expansion, and should be described as such.
  5. [Data Availability] The wave simulation code is proprietary, and the paper states it can be substituted with any standard code that supports gradient and Hessian-vector products. It would be helpful to specify at least the grid spacing, time-stepping scheme, source frequencies, and number of wave equation solves per Hessian-vector product, so that the reported numerical results can be meaningfully reproduced with an open-source solver.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central UQ claim is anchored in an exact analytical benchmark, and the Marmousi low-rank reference issue is a validation limitation, not a circular reduction.

full rationale

The derivation chain is self-contained. The PSF, PDO, and PSF+ Hessian approximations are built from Hessian-vector products applied to delta functions and sinusoids (Sections 3.2-3.6); no constant is fitted to the target posterior variance or to the MCMC output. The ideal quadratic model provides the load-bearing validation: the true posterior covariance is analytically known (Section 4.1), and the reported standard deviations are compared with the exact inverse Hessian (Fig. 10), not with the approximate Hessian used in the proposals. The low-rank correction is an additional approximation of the low-frequency subspace and is not used to define the evaluation metric. In the Marmousi experiment, the benchmark is a brute-force low-rank Hessian approximation (Section 4.2.2), which the paper itself argues underestimates variance (Sections 3.1, 4.1.3), and Table 4 explicitly concedes that the chains have not converged. This weakens the empirical support for the high-rank methods in the realistic setting, but it is a benchmarking and convergence weakness rather than a circular derivation: the paper does not define "posterior variance" as the benchmark variance, and no equation reduces the claimed result to its inputs. Section 5 and Appendix A3 honestly acknowledge the locality and high-frequency limitations. Self-citations to prior PSF/PDO work provide algorithms and theory; the present claims are tested against the exact quadratic solution, so the citations are not load-bearing in the circular sense.

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

The methods rest on several assumptions about the seismic Hessian that are standard in the pseudo-differential operator literature but not proven in this paper: locality in space and frequency, smoothness of the low-frequency part, and the pointwise-symbol rule for square roots. The UQ results additionally assume the posterior is close to Gaussian for gpCN to work. These are domain assumptions with partial empirical support in the synthetic experiments.

free parameters (6)
  • Low-rank correction rank = 100
    Number of leading eigenpairs added to the high-rank symbol approximation in all UQ experiments; chosen without sensitivity analysis (Sections 3.7, 4.1.3).
  • PDO angular sampling count = 8
    Number of symbol columns (8 equally spaced angles, 4 cosine probes) in PDO and PSF+; chosen as a 'good balance' with no sensitivity study (Appendix A1, Section 4.1.1).
  • Number of PSFs = 36 for ideal quadratic; not stated for Marmousi
    PSF sampling points chosen regularly; affects accuracy of the spatial interpolation in eq. (25); the Marmousi value is not reported (Section 4.1.1, Fig. 12).
  • gpCN step-size beta = not reported
    Tunable MCMC step-size in eq. (14) that strongly affects ESS and mixing; no value or tuning procedure reported (Sections 2.3, 4.1.3).
  • Tikhonov regularization coefficient alpha = alpha = ||B[:, Ic]||_F^2 / ||B||_F^2
    Heuristic choice in the PSF+ refinement (eq. 43) balancing fidelity to PDO columns against deviation from the PSF-based A0; no derivation or sensitivity analysis.
  • Prior and noise parameters (delta, gamma, Gamma_noise) = not reported
    The biharmonic prior (eq. 7) and noise covariance are essential inputs to the posteriors sampled; numerical values are not specified, limiting direct reproduction (Sections 2.1, 4.1, 4.2).
assumptions (6)
  • domain assumption The misfit Hessian Hd is (up to smooth error) a pseudo-differential operator of order 1, with a symbol that is low-rank in the sense of eq. (28).
    Invoked in Sections 3.3 and 3.4 to justify the PDO probing and symbol interpolation; grounded in prior work (Beylkin 1985; Bao & Symes 1996; Nammour & Symes 2011) but not proven in this paper.
  • domain assumption Hd has local point-spread functions: reflection effects are local, while transmission effects are shallow and low-frequency enough to be removed by a high-pass filter.
    Assumed in Section 3.2 so that multiple PSFs can be computed from a single vector; Section 5 admits this can fail for sharp parameter contrasts.
  • domain assumption The symbol columns of Hd are well-separated in frequency space, so applying Hd to a sum of sinusoids and Fourier-transforming the result separates the individual symbol columns.
    Used in Section 3.4 (eqs. 33-34) to extract r columns from one Hessian-vector product; requires spectral locality.
  • domain assumption The low-frequency part of Hd is smooth and low-rank, and the high-pass filtered part faithfully represents the pseudo-differential component.
    Stated in Section 3.7; justifies the high-pass filter plus rank-100 low-rank correction strategy for UQ.
  • standard math The symbol of the square root of a pseudo-differential operator is approximately the pointwise square root of the symbol.
    Used in Section 3.7 to build an SPD Hessian approximation for UQ; standard product-symbol asymptotics but only approximate, and discarding negative values is a further heuristic.
  • domain assumption The posterior is sufficiently close to Gaussian for the Laplace-approximation-based gpCN proposal to mix effectively when the Hessian is well approximated.
    The gpCN algorithm (Section 2.3) relies on a good Gaussian proposal; for multimodal or strongly non-Gaussian posteriors the method could fail, and this is not assessed.

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Pith. "Pith review of Accelerating seismic inversion and uncertainty quantification with efficient high-rank Hessian approximations." pith.science (2026). https://pith.science/paper/RIF27DWR

@misc{pith2026250710804,
  author       = {Pith},
  title        = {Pith review of: Accelerating seismic inversion and uncertainty quantification with efficient high-rank Hessian approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIF27DWR}},
  note         = {Machine review of arXiv:2507.10804}
}
read the original abstract

Efficient high-rank approximations of the Hessian can accelerate seismic full waveform inversion (FWI) and uncertainty quantification (UQ). In FWI, approximations of the inverse of the Hessian may be used as preconditioners for Newton-type or quasi-Newton algorithms, reducing computational costs and improving recovery in deeper subsurface regions. In Bayesian UQ, Hessian approximations enable the construction of Markov chain Monte Carlo (MCMC) proposals that capture the directional scalings of the posterior, enhancing the efficiency of MCMC. Computing the exact Hessian is intractable for large-scale problems because the Hessian is accessible only through matrix-vector products, and performing each matrix-vector product requires costly solution of wave equations. Moreover, the Hessian is high-rank, which means that low-rank methods, often employed in large-scale inverse problems, are inefficient. We adapt two existing high-rank Hessian approximations -- the point spread function method and the pseudo-differential operator probing method. Building on an observed duality between these approaches, we develop a novel method that unifies their complementary strengths. We validate these methods on a synthetic quadratic model and on the Marmousi model. Numerical experiments show that these high-rank Hessian approximations substantially reduce the computational costs in FWI. In UQ, MCMC samples computed using no Hessian approximation or a low-rank approximation explore the posterior slowly, providing little meaningful statistical information after tens of thousands of iterations and underestimating the variance. At the same time, the effective sample size is overestimated, providing false confidence. In contrast, MCMC samples generated using the high-rank Hessian approximations provide meaningful statistical information about the posterior and more accurately assess the posterior variance.

Figures

Figures reproduced from arXiv: 2507.10804 by the authors.

Figure 1
Figure 1. Illustration of the duality between the PSF and PDO methods. The PSF method approximates the misfit Hessian by interpolating sampled rows of the symbol. The PDO method approximates the misfit Hessian by interpolating sampled columns of the symbol. on sinusoids, and then constructs the symbol through interpolation in frequency space. An illustration of this intuition this shown in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 2
Figure 2. The components of the Hessian in the ideal quadratic model. To mitigate boundary artifacts, we apply a window function W that suppresses values near the domain edges, which is par￾ticularly suited to seismic applications. Near-surface structures are typically well-recovered by inexpensive methods, making inversion unnecessary in this region. Furthermore, the left, right, and bottom boundaries are poorly informed, yi… view at source ↗
Figure 3
Figure 3. The target solution and the right-hand-side in the ideal quadratic model [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Computed PSFs in the ideal quadratic model [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The PDO method in the ideal quadratic model: The probing vector and the result of applying misfit Hessian to the probing vector. cient techniques for computing symbol rows and columns continue to apply. ACKNOWLEDGMENTS We extend our sincere gratitude to TotalEnergies f…
Figure 6
Figure 6. Figure 6: L-BFGS convergence in the ideal quadratic model. Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B., 2017. Julia: A fresh approach to numerical computing, SIAM review, 59(1), 65–98. Brougois, A., Bourget, M., Lailly, P., Poulet, M., Ricarte, P., & Versteeg, R., 19…
Figure 7
Figure 7. Figure 7: L-BFGS solutions at the 5-th iteration in the ideal quadratic model. A2757. Rudolf, D. & Sprungk, B., 2018. On a generalization of the precondi￾tioned Crank–Nicolson Metropolis algorithm, Foundations of Computa￾tional Mathematics, 18, 309–343. Saad, Y., 2011. Numerical…
Figure 8
Figure 8. Figure 8: Sample positions for MCMC evaluations in the ideal quadratic model. (i) Choose frequency samples ξk = (ρ0, θk) based on the source frequency band. Verify that the frequency band condition fmax/fmin > cmax/cmin is satisfied, and set ρ0 ≈ fmax/cmax. Use r = 8 equally spa…
Figure 9
Figure 9. Figure 9: Performance of MCMC in the ideal quadratic model [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Standard deviations computed from the MCMC samples with different Hessian approximations in the ideal quadratic model [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Ground truth and the initial guess in the Marmousi model [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Computed PSFs for the initial misfit Hessian in the Marmousi model [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: PDO Method: The probing vector and the application of the misfit Hessian to the probing vector [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Comparison of preconditioners in L-BFGS solvers for the Marmousi model. The Hessian-based preconditioners lead to faster decay of data misfit, gradient norm, and solution error [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: Comparison of preconditioners in L-BFGS solvers for the Marmousi model. Each picture shows the difference between the current velocity model and the initial model. The Hessian-based preconditioners recover details faster [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Sample positions for MCMC evaluations in the Marmousi model [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 17
Figure 17. Figure 17: Performance of MCMC in the Marmousi model [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: Relative standard deviations computed from the MCMC samples with different Hessian approximations in the Marmousi model [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.