Full Waveform Inversion using the Wasserstein metric for ultrasound transducer array based NDT

Daniel Rodrigues Pipa; Daniel Rossato; Gustavo Pinto Pires; Thiago Alberto Rigo Passarin

arxiv: 2602.22378 · v2 · submitted 2026-02-25 · 📡 eess.SP

Full Waveform Inversion using the Wasserstein metric for ultrasound transducer array based NDT

Daniel Rossato , Thiago Alberto Rigo Passarin , Gustavo Pinto Pires , Daniel Rodrigues Pipa This is my paper

Pith reviewed 2026-05-15 19:04 UTC · model grok-4.3

classification 📡 eess.SP

keywords full waveform inversionwasserstein metricultrasoundnondestructive testingphased arraysadjoint methodcycle skippingoptimal transport

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The pith

Wasserstein metric in FWI yields order-of-magnitude lower reconstruction errors for ultrasound NDT

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

The paper establishes that full waveform inversion using the squared Wasserstein distance as misfit functional reconstructs sound speed maps more accurately from ultrasonic phased array data than the conventional L2 approach. A key challenge in applying FWI to ultrasound is the absence of low frequencies, which intensifies cycle skipping and creates many local minima in the L2 misfit. The authors derive an analytical adjoint wavefield for the Wasserstein metric to compute gradients efficiently and implement it on GPU with low memory usage. Tests across six cases demonstrate that this substitution reduces pixel-wise squared error by at least a factor of ten in five instances while adding no more than two percent to gradient computation time.

Core claim

Full waveform inversion with the squared Wasserstein metric recovers sound speed distributions from ultrasound array measurements by minimizing the optimal transport distance between observed and simulated waveforms. An explicit time-domain expression for the corresponding adjoint field is derived and combined with a low-memory gradient strategy. When tested on six prototypical nondestructive evaluation scenarios, the Wasserstein-based reconstructions exhibit substantially smaller pixel-wise errors compared to those obtained with the least-squares misfit.

What carries the argument

Analytically derived adjoint acoustic field for the W2 misfit that enables back-propagation of the transport discrepancy to update the sound speed model

If this is right

Pixel-wise sum of squared errors in reconstructed sound speed maps drops by at least one order of magnitude in five out of six tested cases.
Gradient computation time increases by no more than two percent compared to the L2 implementation.
The direct model incorporates nonlinear effects including multiple reflections without linearizing assumptions.
Phased array ultrasound data can be used for high-resolution imaging even when low-frequency content is missing.

Where Pith is reading between the lines

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

The method may generalize to other high-frequency wave inversion problems in seismics or medical imaging where cycle skipping is prevalent.
Practical NDT workflows could adopt W2 to reduce dependence on initial model accuracy or additional regularization terms.
Further validation on real experimental datasets with noise and calibration uncertainties would strengthen applicability claims.

Load-bearing premise

The derived adjoint field for the Wasserstein misfit guides the inversion to the global minimum for the frequency content and transducer geometries in the ultrasound experiments.

What would settle it

Running the inversion on a synthetic dataset with known ground-truth sound speed and observing whether the final W2 reconstruction error stays below the L2 error by the reported margin, or whether both converge to similar local minima.

read the original abstract

Ultrasonic imaging methods often assume linear direct models, while in reality, many nonlinear phenomena are present, e.g. multiple reflections. A family of imaging methods called Full Waveform Inversion (FWI), which has been developed in the field of seismic imaging, uses full acoustic wave simulations as direct models, taking into account virtually all nonlinearities, which can ultimately enhance the accuracy of ultrasonic imaging. However, the problem of cycle skipping -- the existence of many local minima of the Least Squares (L2) misfit function due to the oscillatory nature of the signals -- is worsened when FWI is applied to ultrasound data because of a lack of low-frequency components. In this paper, we explore the use of the squared Wasserstein (W2) Optimal Transport Distance as the metric for the misfit between the acquired and the synthetic data, applying the method to Nondestructive Evaluation with ultrasonic phased arrays. An analytical continuous time-domain derivation of the adjoint acoustic field related to the W2 misfit is presented and used for the computation of the gradients. To cope with the computational burden of FWI, we apply a low-memory strategy that allows for the computation of the gradients without the storage of the full simulated fields. The GPU implementation of the method (in CUDA language) is detailed, and the source code is made available. Six prototypical cases are presented, and the corresponding sound speed maps are reconstructed with FWI using both the L2 and the W2 misfit functionals. In five of the six cases, the pixel-wise sum of squared errors obtained with W2 was at least one order of magnitude lower than that obtained with W2, with an increase in the gradient computation time not exceeding 2\%. The results highlight both the adequacy of the W2 misfit for ultrasonic FWI with phased arrays and its computational feasibility.

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.

Desk Editor's Note private letter to a colleague

The paper derives a continuous-time adjoint for Wasserstein-based FWI in ultrasound NDT and reports clear error reductions over L2 on most test cases, but the derivation itself is not shown.

read the letter

The core takeaway is that this work takes the Wasserstein metric, already used in seismic FWI, and works out a continuous-time adjoint derivation plus a low-memory GPU scheme specifically for phased-array ultrasound NDT. They release the CUDA code and show results on six simple cases where W2 cuts pixel-wise sum-of-squared-errors by at least 10x compared with L2 in five of them, with almost no extra runtime cost for the gradient step. That combination of application, derivation, and implementation is the actual new piece. The practical angle is useful: ultrasound NDT often lacks low frequencies, so cycle-skipping with L2 is a real issue, and the reported gains suggest the W2 misfit helps without blowing up the compute. Releasing the code is also a plus for anyone who wants to test it on their own arrays. The soft spot is straightforward. The abstract states an analytical derivation was done and used, yet supplies neither the steps nor the final expression for the adjoint field. Without that, it is impossible to confirm the gradient is the correct Fréchet derivative for the band-limited transducer signals in the experiments. Any error there would make the inversion results unreliable, and the headline claim rests entirely on those gradients. The experiments themselves are only summarized, so we cannot judge how representative the six cases are or whether the improvement holds under realistic noise or geometry variations. This is the kind of paper that belongs in a reading group on inverse problems or ultrasonic imaging. People working on NDT reconstruction or on optimal-transport misfits in wave problems would get value from the implementation details and the direct comparison. It is worth sending to peer review because the idea is grounded in existing literature, the code is public, and the empirical claim is specific enough that referees can check it once the derivation is supplied. A serious editor should ask for the adjoint equations and the full experimental setup in the first round.

Referee Report

1 major / 1 minor

Summary. The paper proposes replacing the conventional L2 misfit with the squared Wasserstein (W2) distance in full-waveform inversion (FWI) for ultrasonic phased-array nondestructive testing. It presents an analytical continuous-time derivation of the adjoint acoustic field associated with the W2 misfit, employs a low-memory strategy to compute gradients without storing full wavefields, details a CUDA GPU implementation with publicly released source code, and reports results on six prototypical cases in which W2 reduces pixel-wise sum-of-squared-error by at least one order of magnitude relative to L2 in five cases while increasing gradient computation time by no more than 2%.

Significance. If the adjoint derivation is correct, the reported error reductions would constitute a meaningful advance for ultrasound FWI, where cycle skipping is severe because of limited low-frequency content. The W2 metric supplies a transport-based alternative that is less prone to local minima, the low-memory approach addresses the memory bottleneck of time-domain FWI, and the open-source CUDA code supports reproducibility. These elements together could improve reconstruction accuracy for complex NDT geometries that exhibit multiple scattering.

major comments (1)

Abstract: the central empirical claim (≥10× lower pixel-wise SSE in 5/6 cases) rests entirely on gradients obtained from the analytically derived adjoint of the W2 misfit. No derivation steps or final expression for the adjoint field are supplied, preventing verification that the expression is the correct Fréchet derivative of the squared Wasserstein distance for the band-limited transducer signals employed. Any mismatch would render the reported inversion results unreliable.

minor comments (1)

Abstract, final sentence: the phrase 'lower than that obtained with W2' is evidently a typographical error and should read 'lower than that obtained with L2'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We agree that the adjoint derivation requires explicit presentation to allow independent verification and will revise the paper accordingly.

read point-by-point responses

Referee: [—] Abstract: the central empirical claim (≥10× lower pixel-wise SSE in 5/6 cases) rests entirely on gradients obtained from the analytically derived adjoint of the W2 misfit. No derivation steps or final expression for the adjoint field are supplied, preventing verification that the expression is the correct Fréchet derivative of the squared Wasserstein distance for the band-limited transducer signals employed. Any mismatch would render the reported inversion results unreliable.

Authors: We agree that the derivation must be supplied in full detail. The manuscript states that an analytical continuous-time derivation is presented, yet we acknowledge that the intermediate steps and final adjoint expression may not have been shown with sufficient explicitness. In the revised version we will add a complete, self-contained derivation: beginning from the definition of the squared Wasserstein distance between the observed and synthetic band-limited transducer signals, proceeding through the Kantorovich–Rubinstein dual formulation, and arriving at the explicit expression for the adjoint source term that is the Fréchet derivative with respect to the wavefield. This will confirm that the gradient used in our experiments is mathematically correct for the signals employed. The reported error reductions rest on this derivation; once the steps are visible, readers will be able to reproduce and verify the results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain self-contained

full rationale

The abstract applies the squared Wasserstein (W2) metric taken directly from the optimal transport literature and claims to present a new analytical continuous-time derivation of the adjoint acoustic field for use in gradient computation. No equations, fitted parameters, or self-citations are supplied that would reduce any claimed result (such as the reported SSE improvement) to an input by construction. The derivation is therefore independent of the present paper's data or prior author work and remains externally verifiable against the OT literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or non-standard axioms are stated. The work rests on the standard acoustic wave equation and the definition of the squared Wasserstein distance.

axioms (1)

domain assumption Acoustic wave propagation is governed by the standard second-order wave equation with spatially varying sound speed.
Implicit in all FWI formulations for ultrasound.

pith-pipeline@v0.9.0 · 5622 in / 1158 out tokens · 31426 ms · 2026-05-15T19:04:12.601125+00:00 · methodology

discussion (0)

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unclear
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An analytical continuous time-domain derivation of the adjoint acoustic field related to the W2 misfit is presented and used for the computation of the gradients.

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