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Reconstructing effective ultrasound transducer models via distributed source inversion

This paper proposes and validates a distributed source inversion strategy to reconstruct effective spatio-temporal transducer models from experimental wavefields, demonstrating that such accurate source characterization significantly improves the performance of simulation-based ultrasound imaging and inversion workflows.

Original authors: Tim Bürchner, Simon Schmid, Ernst Rank, Stefan Kollmannsberger, Andreas Fichtner

Published 2026-03-26

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CC BY 4.0

Original authors: Tim Bürchner, Simon Schmid, Ernst Rank, Stefan Kollmannsberger, Andreas Fichtner

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 take a perfect photograph of a hidden object inside a block of aluminum using sound waves instead of light. This is what engineers do in Non-Destructive Testing (NDT): they use ultrasound to "see" cracks or holes without breaking the material.

For years, the computers doing the "seeing" have been using a very simple, lazy assumption: they think the ultrasound transducer (the speaker sending the sound) is just a single, tiny dot.

But in reality, a transducer is more like a large, flat drum. When you hit a drum, the sound doesn't come out perfectly evenly from the center. The edges vibrate differently, the middle might be slightly off-center, and the sound waves interfere with each other, creating a complex, wavy pattern.

If your computer thinks the sound comes from a tiny dot, but it actually comes from a complex drum, the computer's "photo" will be blurry, distorted, or completely wrong. This is especially true when looking at objects at weird angles or very close to the back wall.

The Problem: The "Lazy" Computer Model

The authors of this paper realized that advanced imaging techniques (like Reverse Time Migration and Full-Waveform Inversion) are like high-end cameras. They can take incredibly detailed pictures if the lens is perfect. But if the lens (the source model) is blurry, the picture is ruined.

Previously, scientists tried to guess what the sound looked like by measuring it directly or assuming it was uniform. But this often failed because:

The transducer isn't perfectly uniform.
The way it's glued to the metal changes the sound.
The sound waves bounce and interfere in complex ways across the surface of the transducer.

The Solution: "Reverse Engineering" the Sound

The team proposed a clever new method called Distributed Source Inversion (DSI).

Think of it like this:
Imagine you are in a dark room, and someone is playing a complex melody on a piano, but you can't see the piano. You only have microphones around the room recording the sound.

The Old Way: You guess, "Oh, they must be playing a single note in the middle of the piano."
The New Way (DSI): You say, "Let's assume there are 20 tiny speakers hidden all over the piano keys. I will adjust the volume and timing of each of those 20 speakers until the sound recorded by my microphones matches exactly what I hear in the room."

The computer does this mathematically. It doesn't need to know the physics of the piezoelectric crystals inside the transducer. It just asks: "What combination of sounds, coming from different spots on the transducer's surface, would create the exact wave pattern we measured?"

How They Tested It

They set up a real experiment with an aluminum half-cylinder (like a giant metal taco shell).

They hit the flat side with a real ultrasound transducer.
They placed 17 microphones around the curved side to catch the sound.
They ran their "Reverse Engineering" algorithm.

The Result:
The algorithm successfully reconstructed a "virtual transducer" that wasn't a single dot, but a complex map of sound. When they used this new map to simulate the sound, it matched the real-world recordings perfectly, even at sharp angles where the old "dot" model failed miserably.

Why Does This Matter? (The "Photo" Analogy)

To prove this wasn't just a neat trick, they used the new sound model to try and "photograph" three hidden holes in a block of metal.

Scenario A (The Old Dot Model): The computer tried to take a picture. It failed. The holes were invisible, or the image was full of ghostly artifacts. The "lens" was too blurry.
Scenario B (The New Distributed Model): The computer used the complex, reconstructed sound map. Suddenly, the image became sharp. It could clearly see all three holes, even the tiny one hiding right next to the back wall.

The Takeaway

This paper is about stop guessing and start measuring.

Instead of assuming a speaker is a simple point, we can mathematically "listen" to how it actually behaves and build a perfect digital twin of it. This allows engineers to see defects in materials with much higher precision, which is crucial for safety in things like airplanes, bridges, and medical imaging.

In short: They taught the computer to stop thinking of the ultrasound speaker as a tiny dot and start treating it like the complex, wavy drum it actually is. The result? Crystal clear pictures of the invisible.

1. Problem Statement

Accurate modeling of ultrasound wave propagation is critical for high-fidelity simulation-based imaging techniques like Reverse Time Migration (RTM) and Full-Waveform Inversion (FWI) in Non-Destructive Testing (NDT).

The Core Challenge: Conventional methods often assume a simple point source or a uniform excitation across the transducer aperture. However, for transducers with large apertures relative to the acoustic wavelength, spatial interference and non-uniform excitation (caused by piezoelectric eigenmodes and coupling variations) significantly alter the radiation pattern.
Consequence: Inaccurate source characterization leads to discrepancies between simulated and experimental data. This degrades the convergence and reliability of simulation-based inversion workflows, often causing imaging artifacts or failure to detect defects, particularly in complex scenarios involving diffraction and high-aperture angles.
Gap: Existing methods either rely on detailed knowledge of transducer structure (which is often unknown or complex) or use simplified point-source approximations that fail to capture spatial directivity.

2. Methodology

The authors propose an Adjoint-based Distributed Source Inversion (DSI) framework to reconstruct an effective spatio-temporal transducer model directly from experimental measurements.

Physical Model: The wave propagation is modeled using the second-order elastic wave equation in a 2D domain (representing a plane strain state), solved via the Spectral Element Method (SEM) using the parallel solver Salvus.
Optimization Problem: The goal is to find an optimal source term $f(x, t)$ $f (x, t)$ that minimizes the $L_2$ $L_{2}$ misfit between simulated wavefields and observed experimental data at receiver locations.
- Objective Function: Minimizes the difference between simulated and measured signals, weighted by time windows and receiver importance.
Source Representation:
- Instead of a single point, the source is parameterized as a collection of $N_s$ point sources distributed across the transducer aperture.
- Each point source has an independent spatio-temporal wavelet (separable basis of spatial and temporal functions).
- This allows the model to capture both amplitude variations and shape variations of the wavelet across the aperture.
Algorithm:
- Gradient Computation: Uses the Adjoint Method. A forward wavefield is simulated, followed by a backward-propagated adjoint wavefield driven by the data misfit. The gradient is computed by correlating the adjoint field with the source locations.
- Optimization: The L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shannon) algorithm is used to iteratively update the source coefficients.
- Step Length: An optimal step length is calculated analytically due to the linearity of the wave equation with respect to the source.
Experimental Validation Setup:
- Specimen: Aluminum half-cylinder (diameter 0.3 m).
- Source: 1 MHz P-wave transducer (12.7 mm active diameter).
- Receivers: 17 receivers arranged along the curved surface ( $\theta = -90^\circ$ to $90^\circ$ ).
- Weighting Schemes: Two strategies were tested:
  1. DSI 1: Uniform weighting (biased toward high-amplitude, low-angle signals).
  2. DSI 2: Amplitude-compensating weighting ( $1/\sqrt{A_r}$ ) to balance the contribution of high-aperture (low-amplitude) signals.

3. Key Contributions

General DSI Framework: A flexible inversion framework capable of estimating fully spatio-temporal transducer signatures without assuming spatial wavelet uniformity or relying on predefined excitation distributions.
Experimental Validation: Successful reconstruction of an effective source model using real-world data from an aluminum half-cylinder, demonstrating accurate reproduction of both waveform shapes and directivity patterns across all aperture angles.
Robustness Analysis: Systematic investigation showing that the method is robust to the number of source points (provided a minimum density is met) and requires a receiver angular spacing of at least $20^\circ$ to accurately capture directivity.
Impact Demonstration: Synthetic studies proving that accurate source modeling is a prerequisite for the success of RTM and FWI, significantly outperforming point-source and uniform-source approximations.

4. Key Results

Source Reconstruction:
- The DSI method successfully recovered a source model that accounted for a slight off-centering of the physical transducer and non-uniform excitation.
- DSI 2 (Amplitude-compensating) was superior, accurately reproducing waveforms at high-aperture angles ( $\pm 80^\circ$ ) where DSI 1 (uniform) failed.
- The reconstructed wavelets varied in both amplitude and shape across the aperture, capturing complex interference effects.
Directivity Patterns: The simulated directivity patterns using the reconstructed model matched experimental measurements closely, whereas a single point source produced an unrealistic omnidirectional pattern.
Imaging Performance (Synthetic Study):
- RTM (Reverse Time Migration):
  - Point Source (PS): Failed to identify any defects.
  - Uniform Source (US): Detected top defects but suffered from artifacts.
  - Heterogeneous Source (HS/DSI): Produced the cleanest image with fewer artifacts, correctly identifying the dominant scattering.
- FWI (Full-Waveform Inversion):
  - PS & US: Failed to converge to meaningful solutions; the mismatch in radiation patterns created unreliable gradients.
  - HS (DSI): Successfully reconstructed all three defects (including a small hole near the back wall) by exploiting subtle wavefield components (diffractions, high-aperture reflections) that were ignored by simpler models.

5. Significance and Conclusion

Practical Necessity: The paper establishes that for high-resolution waveform-based imaging (RTM/FWI), assuming a simple point source is insufficient when the transducer aperture is comparable to the wavelength. Accurate source modeling is not just an improvement but a prerequisite for defect detection in complex geometries.
Pragmatic Approach: The DSI framework bypasses the need for complex finite-element modeling of the piezoelectric crystal and coupling layers. Instead, it derives an "effective" source that implicitly accounts for experimental uncertainties (coupling, misalignment, manufacturing defects).
Future Applications: The method provides physically consistent training data for machine learning approaches in NDT and bridges the gap between numerical simulations and experimental reality.
Limitations: The reconstructed source is specific to the experimental setup (coupling, position) and may not transfer directly to significantly different configurations, though it serves as an excellent initial approximation. The current 2D formulation does not explicitly model receiver uncertainties.

In summary, this work demonstrates that distributed source inversion is a powerful tool for calibrating ultrasound transducers, enabling simulation-based imaging techniques to achieve their full potential in detecting small, closely spaced defects.