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

This paper presents a numerical framework for the quantitative reconstruction of acoustic medium properties using passive imaging, where an iterative minimization scheme based on the expected value of cross-correlations from stochastic sources is developed and validated against active-source measurements in two and three dimensions.

Original authors: Jean Dutheil, Florian Faucher

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jean Dutheil, Florian Faucher

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out what a mysterious, foggy room looks like, but you aren't allowed to turn on the lights or throw a ball against the walls to hear the echo. Instead, you have to listen to the random, chaotic hum of the room itself—the creaking floorboards, the wind whistling through cracks, the distant traffic. This is the world of passive imaging.

For decades, scientists have tried to use these random "background noises" to map out the inside of things, like the Earth's crust or even the Sun. A popular trick has been to take two microphones, record the noise, and mix the signals together (a process called cross-correlation) to see if a clear pattern emerges. Usually, people have assumed that if you wait long enough, this random noise magically simplifies into a neat, predictable signal that looks exactly like a wave sent from a specific point.

But here is the twist: This paper argues that this "magical simplification" is often a lie. The authors, Jean Dutheil and Florian Faucher, say that in many real-world situations, that neat simplification doesn't hold up. Instead of pretending the noise is simple, they built a super-smart, complex computer model that treats the noise exactly as it is: a messy superposition of waves coming from random, uncorrelated sources.

The Big Idea: Listening to the Chaos

The team developed a method called Full Cross-Correlation Waveform Inversion (FCCWI). Think of it like this:

The Old Way (Active Imaging): You have a flashlight (a controlled source). You shine it on a wall, and you see the shadow. Easy.
The New Way (Passive Imaging): You are in the dark. You can't shine a light. You only have a room full of people whispering randomly.
The Paper's Innovation: Instead of just listening for the time it takes for a whisper to travel from one person to another (which is what most people do), this method listens to the entire shape of the sound wave that results from mixing all the whispers together.

The authors found that by mathematically modeling the expected value (the average pattern you'd see if you ran the experiment a million times) of these mixed signals, they could reconstruct the physical properties of the medium—specifically the wave speed and density—without ever needing a controlled flashlight.

The "Two-Step" Dance

To make this work, the authors had to solve a tricky math problem. In standard imaging, you only need to run a "reverse movie" of the sound once to figure out what went wrong. But because they are dealing with random noise correlations, they had to run two reverse movies (called "adjoint states") simultaneously.

Imagine trying to untangle a knot of headphones. Usually, you just pull one end. But here, the knot is so complex that you have to pull two ends at once, in a very specific dance, to see where the tangles are. The paper proves that this two-step dance is necessary to get the right answer when the sources of the noise are random and uncorrelated.

What the Simulations Showed

The team didn't just talk about this; they ran massive computer simulations to test it. Here is what they found:

It Works (Mostly): In a 2D test with a simple square and a disk hidden in a block, their method (FCCWI) did a great job finding the shapes, almost as well as the traditional "flashlight" method (FWI).
The "Unknown" Factor: The biggest hurdle is that they don't know exactly where the noise is coming from or how loud it is (the source covariance).
- When they tried to map the wave speed (how fast sound travels), the method worked well, even if they didn't know the noise source perfectly.
- However, when they tried to map the source covariance itself (the "loudness map" of the noise), the results were shaky. The simulations showed that while they could guess the general shape, the exact numbers were off. The paper suggests that the standard math they used (an $l_2$ -norm) is great for timing (phase) but not so great for measuring loudness (amplitude).
The 3D Challenge: They pushed the method into a 3D world with high-contrast objects (like a rock with a very different speed inside). The method successfully found the hidden rocks, even when starting with a blank, one-dimensional guess.
- The Catch: It was expensive. In their simulation, the new method took about 4 minutes per iteration, while the traditional method took only 1 minute. That's a fourfold increase in computer time because the math is so much heavier.

What They Explicitly Rule Out

The paper is very clear about what not to do. They explicitly reject the idea that you can always simplify the cross-correlation to just the "imaginary part of the Green's function" (a fancy way of saying "the neat, simple signal"). They argue that this "convenient source" assumption is often wrong in real applications and leads to bad results. They also rule out the idea that you can just use travel times; they insist on using the full wavefield.

How Sure Are They?

The authors are confident in their math and their simulations, but they are careful not to claim they have "solved" the problem for the real world yet.

Proven: The mathematical derivation of the two-adjoint-state method is solid.
Simulated: All the results (the 2D and 3D images) are based on synthetic data (computer-generated noise), not real-world recordings from the Earth or the Sun.
Suggested: They suggest that a two-stage process might be best: first, figure out the wave speed (which is easier), and then try to figure out the source noise. They admit that reconstructing the source noise itself is still "more difficult" and needs new strategies.

The Bottom Line

This paper is a blueprint for a new way to "see" in the dark. It shows that by embracing the full complexity of random noise rather than trying to simplify it, we can build accurate maps of the inside of objects. It's not a magic wand that makes the process instant or easy (it's actually four times slower on a computer), but it proves that we can get quantitative, detailed pictures of the world using only the background hum, provided we are willing to do the heavy mathematical lifting.

The authors are now looking forward to testing this on real data, like the vibrations of the Sun or the Earth, to see if the simulation magic holds up in the real, messy world.

Technical Summary: Full Cross-Correlation Inversion for Quantitative Passive Imaging

Problem Statement
This work addresses the quantitative inverse problem of reconstructing physical properties of a medium using passive imaging data. Unlike active-source imaging, which relies on controlled excitation, passive imaging utilizes ambient wavefields generated by stochastic sources (e.g., microseismic activity or turbulent convection). The central challenge is to relate these stochastic observations to deterministic medium properties. While traditional approaches often rely on extracting travel times from cross-correlations or assuming "convenient" source distributions (where the cross-correlation is proportional to the imaginary part of the Green's function), this paper focuses on a full-waveform approach. The goal is to reconstruct the medium's wave speed ( $c$ ) and density ( $\rho$ ), as well as the spatial covariance of the stochastic sources ( $S$ ), without reducing the data to travel times or assuming a uniform source distribution.

Methodology
The authors formulate the problem using time-harmonic acoustic wave propagation. The ambient signals are modeled as a superposition of waves from spatially uncorrelated, zero-mean stochastic sources. Under these assumptions, the expected value of the cross-correlation between signals at two locations is derived as a deterministic quantity related to the Green's function and the source covariance.

Forward Modeling: The wave equation is treated as a first-order system involving pressure and particle velocity. The expected cross-correlation is computed by solving two consecutive wave problems: first, a Green's function problem with a Dirac source to obtain the impulse response; second, a problem with a right-hand side constructed from the product of the source covariance and the Green's function. This formulation allows for the computation of cross-correlations between different field components (e.g., pressure and velocity).
Discretization: The numerical framework employs the Hybridizable Discontinuous Galerkin (HDG) method. This approach is chosen for its ability to handle first-order systems efficiently, providing direct access to both scalar and vector fields without enlarging the global linear system. It supports $hp$-adaptivity and parallelization. The HDG formulation leads to a global linear system involving only the degrees of freedom on the mesh faces, with local volume solutions recovered via static condensation.
Inverse Problem Formulation: The reconstruction is posed as a nonlinear minimization problem, analogous to Full Waveform Inversion (FWI), but using the expected cross-correlation as the data misfit. The objective function measures the $L^2$ distance between observed and simulated cross-correlations.
Gradient Computation: To solve the inverse problem iteratively, the gradient of the misfit functional is computed using the adjoint-state method. A key theoretical contribution is the derivation of the adjoint formulation for cross-correlation data. Unlike standard FWI, which requires a single adjoint state, this approach necessitates two adjoint states: one associated with the cross-correlation residual and a second one driven by the first adjoint state and the source covariance. This accounts for the dependency of the cross-correlation on both the medium parameters and the source distribution.
Optimization: The minimization is performed using a quasi-Newton L-BFGS algorithm with a line search, avoiding the explicit formation of the Hessian matrix.

Key Contributions

Full Cross-Correlation Waveform Inversion (FCCWI): The paper proposes a quantitative inversion framework that utilizes the full expected cross-correlation wavefield rather than just travel times or first arrivals.
Joint Reconstruction: The method simultaneously reconstructs the medium parameters ( $c, \rho$ ) and the source covariance ( $S$ ), treating the latter as an unknown rather than a fixed or simplified assumption.
Adjoint-State Derivation for Passive Imaging: The authors derive a specific adjoint-state formulation for cross-correlation data, demonstrating that two adjoint states are required to compute the gradient accurately.
HDG Implementation: The work details the implementation of this inverse problem using the HDG method, including the specific handling of the two-step forward solver and the two-step adjoint solver within the HDG static condensation framework.

Numerical Results
The methodology is validated through synthetic experiments in 2D and 3D:

Experiment 1 (2D, Full Data): In a unit square with contrasting inclusions, the FCCWI approach successfully recovers the wave speed, yielding results comparable to standard FWI, though with slightly more background oscillations. The reconstruction of the source covariance is shown to be less accurate than the wave speed, suggesting a two-stage inversion strategy (wave speed first, then source covariance) is beneficial.
Experiment 2 (2D, Seismic Configuration): Using the Overthrust model with surface-only data, FCCWI recovers the layered structure of the wave speed reasonably well, though with lower resolution than FWI. The reconstruction of the source covariance remains challenging, with the $L^2$ misfit functional proving more sensitive to phase (wave speed) than amplitude (source covariance).
Experiment 3 (3D, High Contrast): In a 3D domain with high-contrast inclusions, both FWI and FCCWI successfully recover the geometry of the inclusions. FCCWI shows improved resolution for inclusions near boundaries compared to direct wavefield data, though it slightly underestimates the amplitude within the inclusions.
Computational Cost: The 3D experiment highlights that FCCWI is computationally more expensive than FWI (approximately 4 times slower per iteration in the tested setup) due to the need to solve additional linear systems and assemble multiple right-hand sides.

Significance and Claims
The paper claims that the proposed framework enables the quantitative reconstruction of medium properties from passive ambient noise without relying on simplifying assumptions about the source distribution (such as uniformity or the "convenient source" assumption). While the authors acknowledge that controlled-source imaging (FWI) generally provides higher accuracy when available, they demonstrate that passive imaging using full cross-correlation waveforms can recover meaningful quantitative information about the medium.

The authors modestly note that the reconstruction of the source covariance is more difficult than that of the wave speed, particularly when using an $L^2$ misfit functional, and suggest that dedicated linear inversion strategies or alternative misfit functionals are necessary for future improvements. The work establishes a theoretical and numerical foundation for applying full-waveform inversion principles to passive imaging, paving the way for applications in fields like helioseismology and seismic monitoring where active sources are not feasible.