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Expected cross-correlations of ambient acoustic waves can be inverted for medium properties with the same quantitative fidelity as controlled-source full-waveform inversion.

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2026-07-13 03:20 UTC pith:BTCXWPYY

load-bearing objection Solid methods paper: full expected cross-correlation FWI with two-adjoint HDG, synthetic recovery comparable to active FWI under the uncorrelated-source model. the 3 major comments →

arxiv 2607.09392 v1 pith:BTCXWPYY submitted 2026-07-10 math.NA cs.NAmath.AP

Full cross-correlation inversion for quantitative passive imaging with time-harmonic acoustic waves

Jean Dutheil , Florian Faucher This is my paper
classification math.NA cs.NAmath.AP MSC 65N2135R3065M3286A22
keywords passive imagingcross-correlationfull-waveform inversionadjoint-state methodhybridizable discontinuous Galerkintime-harmonic acousticssource covarianceambient noise
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Passive imaging uses only ambient vibrations, not controlled sources, so the raw records are stochastic. This paper shows that if the sources are zero-mean and spatially uncorrelated, the expected value of the cross-correlation between any two receiver locations is a completely deterministic object: it is the solution of the same first-order acoustic wave equation driven by a right-hand side built from the Green’s function and the source covariance. That deterministic map is then inverted by iterative least-squares minimization whose gradient is obtained from two adjoint problems. Synthetic 2-D and 3-D experiments recover wave-speed structure of quality comparable to classical active-source full-waveform inversion, even when the source covariance itself is unknown. The result matters because it turns continuous, cheap ambient recordings into a quantitative imaging modality without requiring travel-time extraction or simplifying assumptions about source distribution.

Core claim

Under the assumption that ambient sources are zero-mean and spatially uncorrelated, the expected cross-correlation of time-harmonic acoustic fields is exactly the Green’s-function integral weighted by the source covariance; this quantity can be computed by two successive solves of the first-order wave equation and can be inverted for both medium parameters and source covariance by adjoint-state least-squares minimization.

What carries the argument

The expected-cross-correlation representation C[p](x1,x2)=∫ G(x1,y) S(y) G[p](y,x2) dy, which converts the stochastic passive data into a deterministic forward map that is then differentiated by a pair of adjoint states.

Load-bearing premise

The ambient sources must be spatially uncorrelated; if they possess significant spatial correlation the entire forward map used for inversion is misspecified.

What would settle it

On a laboratory or field data set whose ambient sources are known to be spatially correlated, check whether the recovered wave-speed model matches an independent active-source reconstruction; systematic bias would falsify the uncorrelated-source claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a quantitative passive-imaging framework for time-harmonic acoustic waves. Ambient signals are modeled as superpositions of waves from zero-mean, spatially uncorrelated stochastic sources. Under that assumption the expected cross-correlation of recorded fields is expressed via the deterministic Green’s tensor and the source covariance S (Eqs. 2.17–2.20). The first-order system is discretized by Hybridizable Discontinuous Galerkin (HDG); the expected cross-correlation is obtained by two consecutive solves (Green’s function then dense right-hand side). Medium parameters (wave speed, density) and S are recovered by iterative least-squares minimization of the misfit between simulated and observed expected cross-correlations. Gradients are obtained by an adjoint-state method that requires two adjoint problems (Eqs. 4.22 and 4.25). Synthetic 2-D and 3-D experiments compare the resulting Full Cross-Correlation Waveform Inversion (FCCWI) with classical active-source FWI and show that wave-speed structure can be recovered at a quality comparable to FWI, while recovery of S under pure L2 is weaker.

Significance. If the modeling assumptions hold, the work supplies a complete, discretization-consistent pipeline—from the representation of expected cross-correlations through HDG discretization and a two-adjoint gradient to quantitative reconstruction—that goes beyond travel-time or single-arrival approximations common in passive imaging. The explicit comparison with active-source FWI on the same synthetic models, the open-source HDG implementation (Hawen), and the careful treatment of first-order fields (pressure–velocity correlations) are concrete strengths. The main limitation is that all evidence is synthetic and generated under the same uncorrelated-source model used for inversion; real-data validation and the treatment of spatially correlated ambient sources remain open. Within its stated scope the contribution is a solid, usable numerical methodology for quantitative passive imaging.

major comments (3)
  1. Assumption 1 (Section 2.2.1) that Cov[f] is multiplication by a non-negative function S is load-bearing for the entire representation C=∫S(y)G(x1,y)⊗G(x2,y)dy and for the subsequent inverse problem. The paper correctly notes that the convenient-source simplification is unrealistic, yet it does not quantify how spatial correlation of real ambient sources would bias the recovered medium. A short numerical experiment with a non-diagonal covariance (or a clear statement that the method is restricted to the uncorrelated case) would make the domain of validity precise.
  2. All experiments (Section 5) use synthetic expected-value data generated under the same model that is inverted; the self-averaging property that justifies replacing long-time averages by E[C] is assumed rather than demonstrated. While this is standard for a methods paper, the claim of quantitative passive imaging would be stronger if at least one experiment replaced the exact expectation by a finite-sample average of stochastic realizations, so that the effect of residual variance on the L-BFGS iterates is visible.
  3. Source-covariance recovery under the pure L2 misfit (Eq. 5.1) is consistently weak (Figs. 3 and 7). The authors acknowledge that L2 is phase-oriented and therefore better suited to wave speed than to amplitude-controlled S. Because S is an unknown of the same inverse problem, the manuscript should either (i) demonstrate a more suitable amplitude-sensitive misfit or linear inversion strategy for S, or (ii) clearly relegate S-recovery to a secondary, optional step and focus the claims on medium parameters.
minor comments (5)
  1. Section 2.1: the free-surface / absorbing boundary conditions (2.3) are standard for Earth models; a one-sentence remark that the same derivation applies to other boundary operators would help readers from medical or laboratory acoustics.
  2. Algorithm 2: the two-stage loop over Σsrc is clear, but a brief complexity remark (factorization once, two multi-RHS solves) would make the computational advantage over a naïve double loop more explicit.
  3. Figure captions in Section 5 sometimes omit the frequency band or the number of iterations; adding these details would improve reproducibility.
  4. Typographical: “see Figure 1b” appears twice with slightly different wording; “convenient source” is sometimes hyphenated, sometimes not—standardize.
  5. References: the recent full-waveform ambient-noise inversion literature (e.g., Keating et al. 2025, already cited) could be contrasted more sharply with the present full-wave, non-travel-time approach in the introduction.

Circularity Check

0 steps flagged

No significant circularity: the expected-cross-correlation representation and the two-adjoint inversion are derived from the wave equation under stated assumptions; synthetic recoveries are not tautological.

full rationale

The paper starts from the first-order time-harmonic acoustic system (2.1), defines the Green’s kernel (Def. 1), invokes reciprocity (Prop. 1) and the representation formula (Prop. 2), then under Assumption 1 (spatially uncorrelated sources, Cov[f] = multiplication by S) obtains the deterministic integral representation C(x1,x2) = ∫ S(y) G(x1,y) ⊗ G(x2,y) dy (2.17) and the computable form via a second solve with RHS M = S G[p] (2.19–2.20, Alg. 1). The inverse problem is the ordinary least-squares misfit (4.1)/(4.4) between simulated and observed expected cross-correlations; the gradient is obtained from a Lagrangian that enforces the two HDG systems (3.22) and (3.25), yielding two adjoint problems (4.22) and (4.25) whose structure is the adjoint of the forward HDG operator. None of these steps is definitional of the target medium parameters: the observations are synthetic data generated from known (c,ρ,S) models and then inverted from incomplete initial guesses (Figs. 1–9). Self-citations to Hawen and the HDG-FWI adjoint machinery [22,23] supply discretization and gradient tools already validated for active-source FWI; they do not encode the passive-imaging claim. Source-covariance recovery is acknowledged to be weak under pure L2, which is an empirical limitation, not a circular reduction. The derivation is therefore self-contained against its own synthetic benchmarks; score 0.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The load-bearing content is a modeling assumption on ambient sources plus standard wave and optimization machinery. No new physical entity is postulated. Free parameters are ordinary inversion unknowns and numerical choices, not hidden constants fitted to claim a universal law. The central claim rests mainly on Assumption 1 and on treating expected cross-correlations as the data functional.

free parameters (4)
  • Source covariance field S(x)
    Unknown non-negative spatial field inverted or fixed; controls amplitude of expected cross-correlations. Not a universal constant, but a free model parameter the claim depends on when S is unknown.
  • Stabilization parameter τ in HDG
    Taken as τ=1/ρ0 on faces; a numerical modeling choice that can affect discrete solutions.
  • Frequency schedule and iteration counts
    e.g. 2–10 Hz with 30 L-BFGS iterations per frequency in 2D tests; hand-chosen inversion schedule affecting recovered models.
  • Cardinality and placement of Σsrc vs Σrcv
    Virtual-source subset chosen for cost (e.g. 16 of 149, or every second receiver); affects illumination and reconstruction quality.
axioms (6)
  • domain assumption Stochastic sources are zero-mean and spatially uncorrelated: Cov[f] is multiplication by S ∈ L∞(Ω) (Assumption 1).
    Section 2.2.1; enables C=∫ S(y) G⊗G dy and Algorithm 1.
  • domain assumption Time-harmonic first-order acoustic system (2.1) with free-surface and first-order absorbing BCs models the relevant wave physics.
    Section 2.1; viscoacoustic κ allowed but attenuation largely ignored in experiments.
  • domain assumption Long-time averages of ambient products converge to the expected cross-correlation (self-averaging), so E[u1 u2] is usable data.
    Introduction and Section 2.2; standard ambient-noise premise, not re-proved here.
  • standard math Green’s pressure component is reciprocal: G[p](x,y)=G[p](y,x).
    Proposition 1; used to rewrite C[p] as a forward solve with RHS M=S G[p].
  • ad hoc to paper Least-squares misfit on expected cross-correlations (or Green’s data for FWI) is a suitable objective for quantitative recovery.
    Section 4.1 and (5.1); authors later note L2 is weak for amplitude-dominated S.
  • standard math HDG discrete systems and Wirtinger calculus yield correct gradients of the discrete misfit.
    Sections 3–4; standard discrete adjoint reasoning with complex variables.

pith-pipeline@v1.1.0-grok45 · 29200 in / 3580 out tokens · 41165 ms · 2026-07-13T03:20:47.682104+00:00 · methodology

0 comments
read the original abstract

We consider the inverse problem for the quantitative reconstruction of physical properties in the context of passive imaging, where ambient wavefields are used to infer a medium. The data are modeled as a superposition of waves generated by stochastic sources. In this work, we focus on time-harmonic acoustic wave propagation and assume that the stochastic sources exciting the medium are zero-mean and spatially uncorrelated. Under these assumptions, the expected value of the cross-correlation between signals recorded at two locations can be related to the deterministic Green's function and the covariance of the source terms. We follow a first-order formulation of the wave equation, which enables the treatment of correlations between different types of wavefields. A numerical framework is developed for the resulting nonlinear inverse problem. The quantitative reconstruction is carried out using an iterative minimization scheme, in which the gradient of the misfit functional is computed via the adjoint-state method. Numerical experiments in two and three dimensions are performed using synthetic data, and inversions based on the expected value of cross-correlations are compared with those relying on direct wavefield measurements from active-source acquisitions.

Figures

Figures reproduced from arXiv: 2607.09392 by Florian Faucher, Jean Dutheil.

Figure 1
Figure 1. Figure 1: Experiment on the unit-square with the wave speed composed of two inclusions. The [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Wave speed reconstructions using either the direct wavefield data (FWI) or using cross [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Source covariance reconstruction depending on the (fixed) wave speed models used during [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The seismic model OT used for inversion. The domain has size 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Initial guesses for the reconstruction of the OT model [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Wave speed reconstructions of the OT model using either the direct wavefield data (FWI) [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Source covariance reconstruction for the OT model starting from the initial models shown [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Wave speed and source covariance for the 3D test case of size 2 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 3D wave speed reconstructions using either the direct wavefield data (FWI) or using [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗

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