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arxiv: 2605.11763 · v1 · submitted 2026-05-12 · 📡 eess.SP

Recognition: 2 theorem links

· Lean Theorem

Assessment of Time-of-Arrival Estimation Methods for Impact Detection in Isotropic Plates using Piezoceramic Sensors

Alexander Humer, Ayech Benjeddou, Lukas Grasboeck

Pith reviewed 2026-05-13 05:49 UTC · model grok-4.3

classification 📡 eess.SP
keywords time-of-arrival estimationLamb wavesimpact detectionisotropic platespiezoceramic sensorscontinuous wavelet transformAkaike information criterionwave mode detection
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The pith

Time-of-arrival methods detect both symmetric and anti-symmetric Lamb wave modes from impacts on isotropic plates, with noise mainly disrupting the earliest arrivals.

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

This paper evaluates five techniques for estimating the arrival times of waves created by impacts on flat isotropic plates. The techniques are tested on synthetic signals generated by transient finite element simulations that have been tuned to match experimental data from piezoceramic sensors on the plate surface. The assessment shows that nearly all methods identify both the faster symmetric and slower anti-symmetric fundamental modes when signals are free of noise. Noise makes the symmetric-mode arrivals harder to detect, yet the anti-symmetric arrivals can still be captured reliably through preprocessing or combined time-frequency analysis. The authors introduce two practical enhancements: a frequency-domain threshold crossing step inside the continuous wavelet transform and the use of local minima within the Akaike information criterion to improve detection of the symmetric mode.

Core claim

The paper establishes that threshold crossing, continuous wavelet transform, short/long term average, modified energy ratio, and Akaike information criterion methods can detect the fundamental Lamb wave modes from impact-induced waves monitored by piezoceramic sensors on isotropic plates. Under noise-free conditions nearly all methods capture both symmetric and anti-symmetric arrivals. Noise affects symmetric mode detection most, but anti-symmetric TOA can be estimated meaningfully with preprocessing or time-frequency methods. Novel elements are a frequency-domain threshold crossing inside the CWT framework for better robustness and accuracy, and using local minima of the AIC for the TOA of

What carries the argument

Time-of-arrival estimation algorithms applied to piezoceramic sensor signals recording impact-generated Lamb waves, including a frequency-domain threshold crossing extension to the continuous wavelet transform and local-minima consideration in the Akaike information criterion.

If this is right

  • Nearly all assessed methods capture both symmetric and anti-symmetric fundamental Lamb wave mode arrivals under noise-free conditions.
  • Noise primarily impairs detection of the earliest symmetric-mode arrivals.
  • Meaningful anti-symmetric-mode TOA estimates remain obtainable through suitable preprocessing or time-frequency analysis even with noise.
  • The new frequency-domain threshold crossing within the CWT framework improves both robustness and accuracy of TOA estimation.
  • Considering local minima in the AIC proves effective for detecting the TOA of the fundamental symmetric mode.

Where Pith is reading between the lines

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

  • These TOA techniques could support hybrid detection systems that switch between methods based on measured noise levels to maintain localization accuracy across varying impact conditions.
  • Extending the assessment to anisotropic or layered plates would require incorporating direction-dependent group velocities into the signal models.
  • The practical guidelines on method selection and preprocessing could guide sensor array design to minimize the impact of noise on early arrivals.

Load-bearing premise

The transient finite element simulations, after experimental calibration for excitation and dispersion, sufficiently represent real sensor signals for impacts at varying positions and force profiles, including under added noise.

What would settle it

Direct comparison of the methods' TOA outputs against known arrival times measured in controlled physical impact experiments on the same plate geometry, especially for signals with added noise levels matching the simulations.

Figures

Figures reproduced from arXiv: 2605.11763 by Alexander Humer, Ayech Benjeddou, Lukas Grasboeck.

Figure 1
Figure 1. Figure 1: (a) Schematic illustration of the fundamental mode shapes 𝑆0 and 𝐴0 ; (b) solutions of Rayleigh-Lamb equation (expressed in terms of phase velocities 𝑐ph) for symmetric (𝑆0 and 𝑆1 ) and antisymmetric (𝐴0 and 𝐴1 ) cases. Additionally, the phase velocities of flexural (𝑐f l) and axial (𝑐ax) plate waves according to Kirchhoff-Love plate theory are illustrated. (c) Corresponding group velocities 𝑐g for the res… view at source ↗
Figure 2
Figure 2. Figure 2: Positions of sensors (S𝑖 ; 𝑖 = 1-4) and impacts (𝐼𝑖 ; 𝑖 = 1, 2) on aluminum plate [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Driving voltage of piezo to obtain the dispersion characteristics of the aluminum plate (a) time-history and (b) frequency-spectrum. Subfigure (c) shows schematically how B-scans are performed for different orientations upon excitation with a piezoelectric transducer. by the piezoelectric actuator, see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dispersion characteristics of the aluminum plate: The contour plot shows the experimental results, the orange dashed line shows the results obtained from the simulation (line of local maxima in the wavenumber-domain as a function of the frequency) and the solid white line is the dispersion curve of the 𝐴0 mode from Rayleigh-Lamb equations (22). In what follows, we discuss the properties of the impacts whic… view at source ↗
Figure 5
Figure 5. Figure 5: Recorded force during an impact on the aluminum plate: (a) time-signal and (b) the corresponding frequency￾spectrum. As a second impact, we chose to generate an idealized force signal by impacting a massive aluminum block with an impact hammer. In this case, the time-signal of the force sensor truly resembles an impulse-like excitation, see [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Recorded force during an impact on a massive aluminum block: (a) time-signal and (b) the corresponding frequency-spectrum. can be calculated by determining the mechanical power on the structure through FE simulation. To achieve this, the velocities of the nodes in the impact area are multiplied by the applied force and then summed over the respective area. Integrating the mechanical power over the entire t… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic overview of selected TOA-estimation methods based on [22]. The methods in bold are investigated hereafter. Remark 2. Some of the methods assessed in what follows originate in the field of seismology, which is characterized not only by a different frequency-range, but also by different ‘modes’ of wave propagation. In seismology, longitudinal 𝑃 -waves and transverse 𝑆-waves propagate in the interio… view at source ↗
Figure 8
Figure 8. Figure 8: Characteristics of resolution in time and frequency domains: (a) Time-frequency boxes of two wavelets 𝜓𝑎1 ,𝑏1 and 𝜓𝑎2 ,𝑏2 , adapted from [31, p. 18]. Sub-figures (b) and (c) show the tiling of the time-frequency plane in time-frequency boxes when applying TFA on a signal. In case of STFT (b) the dimensions of boxes are constant, which correspond to a constant time-frequency resolution in the entire time-fr… view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the two-step AIC algorithm applied to a generic tone-burst signal. (a) In the first step, the sensor signal 𝑠(𝑡 𝑖 ) (blue) and its characteristic function 𝛽(𝑡 𝑖 ) (green) are shown. The AIC function 𝜒 ( 𝑡 𝑖 ) (black) is evaluated in the first AIC window [𝑡 lb1, 𝑡ub1]. The minimum of 𝜒 ( 𝑡 𝑖 ) in this window yields the first TOA-estimate 𝑡 fe (orange). (b) In the second AIC step, 𝜒 ( 𝑡 𝑖 ) i… view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the sensor signals for 𝐼  1 . Time-history of the sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. 0 500 1000 1500 2000 2500 3000 t in µs −1000 −500 0 500 1000 signal in mV (a) S1 S2 S3 S4 0 50 100 150 200 250 t in µs −0.5 0.0 0.5 signal in mV (b) S1 S2 S3 S4 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Illustration of the sensor signals for 𝐼  2 . Time-history of the sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. trace. To avoid crowding, we skip labeling the markers as they can be identified by the condition 𝑡𝑆0 < 𝑡𝐴0 . Note that 𝑡𝑆0 and 𝑡𝐴0 are different for each sensor. The corresponding values are listed in Tab. 2. 𝑡𝑆0 in µs 𝑡𝐴0 in… view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of the sensor signals at 𝐼  1 . Time-history of the sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. 0 500 1000 1500 2000 2500 3000 t in µs −3000 −1500 0 1500 3000 signal in mV (a) S1 S2 S3 S4 0 50 100 150 200 250 t in µs −1.50 −0.75 0.00 0.75 1.50 signal in mV (b) S1 S2 S3 S4 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Illustration of the sensor signals at 𝐼  2 . Time-history of the sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. To assess the robustness of the TOA-estimation methods under more realistic conditions, artificial noise is added to the ideal sensor signals obtained from the FE-simulations. The noise level is derived from experimental measur… view at source ↗
Figure 14
Figure 14. Figure 14: Illustration of the sensor signals at 𝐼  1 . Time-history of the noise-contaminated sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. 0 500 1000 1500 2000 2500 3000 t in µs −1000 −500 0 500 1000 signal in mV (a) S1 S2 S3 S4 0 100 200 300 400 500 t in µs −5.0 −2.5 0.0 2.5 5.0 signal in mV (b) S1 S2 S3 S4 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 15
Figure 15. Figure 15: Illustration of the sensor signals at 𝐼  2 . Time-history of the noise-contaminated sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. 0 500 1000 1500 2000 2500 3000 t in µs −1500 −750 0 750 1500 signal in mV (a) S1 S2 S3 S4 0 100 200 300 400 500 t in µs −10 −5 0 5 10 signal in mV (b) S1 S2 S3 S4 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Illustration of the sensor signals at 𝐼  1 . Time-history of the noise-contaminated sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. To investigate how sensitive the TOA-estimation for our sensor signals is with respect to the threshold level, we perform a parametric study in which the threshold is varied. In an application scenario, rathe… view at source ↗
Figure 17
Figure 17. Figure 17: Illustration of the sensor signals at 𝐼  2 . Time-history of the noise-contaminated sensor signals over (a) the entire time-range considered and (b) a detailed view at the onset of the first events. where 𝑟 is the number of sensors. The smallest maximum value of all sensors is used as the base value for calculating the threshold. This ensures that every sensor signal exceeds the threshold. For the parame… view at source ↗
Figure 18
Figure 18. Figure 18: Results of TOA-estimation with TC method applied on noise-free signals: Evolution of 𝑡̂ for each sensor signal for (a) 𝐼  1 and (b) 𝐼  2 , when a common threshold is used for all sensor signals. The base value for the threshold determination, i.e., the smallest maximum of the sensor signals is 135.4 mV for impact 𝐼  1 and 40.6 mV for impact 𝐼  2 . The dotted lines indicate the first arrivals of the fu… view at source ↗
Figure 19
Figure 19. Figure 19: Same representation as [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Same representation as [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Same representation as [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Scalograms of noise-free sensor signals S1–S4 for the idealized impact 𝐼  1 : The normalized squares of the CWTs’ absolute values are illustrated for the entire time-span (3 ms) in the frequency-range of 5 kHz to 100 kHz. The COI is indicated by white dash-dotted lines [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Scalograms of noise-free sensor signals S1–S4 for the idealized impact 𝐼  1 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 0.15 ms and a frequency-range of 30 kHz to 100 kHz. Dashed red lines illustrate TOA-estimates as a function of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1×10−2 is reached. Dash-dotted white lines represent the COI.… view at source ↗
Figure 24
Figure 24. Figure 24: Sections of scalograms for noise-free sensor signals at 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz and 100 kHz for impact 𝐼  1 . The threshold 𝑠th = 1 × 10−2 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. To gain further insight, we consider sections along the temporal axis through the (normalized) scalograms for s… view at source ↗
Figure 25
Figure 25. Figure 25: Scalograms of noise-free sensor signals S1–S4 for the experiment-based impact 𝐼  1 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 0.15 ms and a frequency-range of 30 kHz to 100 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1 × 10−2 is reached. Dash-dotted white lines represent … view at source ↗
Figure 26
Figure 26. Figure 26: Sections of scalograms for noise-free sensor signals at 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz and 100 kHz for impact 𝐼  1 . The threshold 𝑠th = 1 × 10−2 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. to sensor S1 . As for the first impact, we first consider the idealized impact 𝐼  2 . The scalograms obtained … view at source ↗
Figure 27
Figure 27. Figure 27: Scalograms of noise-free sensor signals S1–S4 for idealized impact 𝐼  2 : The normalized square of the CWT’s absolute value is illustrated for a frequency-range of 30 kHz to 100 kHz and time-windows of 0.2 ms each. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1 × 10−2 is reached. Dash-dotted white lines represent the … view at source ↗
Figure 28
Figure 28. Figure 28: Sections of scalograms for noise-free sensor signals at 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz and 100 kHz for impact 𝐼  2 . The threshold 𝑠th = 1 × 10−2 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. L. Grasboeck et al.: Preprint submitted to Elsevier Page 29 of 60 [PITH_FULL_IMAGE:figures/full_fig_p029_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Scalograms of noise-free sensor signals S1–S4 for the experiment-based 𝐼  2 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 0.2 ms and a frequency-range of 30 kHz to 100 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1 × 10−2 is reached. Dash-dotted white lines represent the COI.… view at source ↗
Figure 30
Figure 30. Figure 30: Sections of scalograms for noise-free sensor signals at 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz and 100 kHz for impact 𝐼  2 : The threshold 𝑠th = 1 × 10−2 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. Frequency in kHz Relative times in ms – 𝐼  2 Relative times in ms – 𝐼  2 𝑡̂ 2 − 𝑡̂ 1 𝑡̂ 3 − 𝑡̂ 1 𝑡̂ 4 − 𝑡̂ 1 … view at source ↗
Figure 31
Figure 31. Figure 31: Scalograms of noise-contaminated sensor signals S1–S4 for the idealized impact 𝐼  1 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 0.15 ms and a frequency-range of 30 kHz to 100 kHz. Dashed red lines illustrate TOA-estimates as a function of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 2 × 10−1 is reached. Dash-dotted white lines represen… view at source ↗
Figure 32
Figure 32. Figure 32: Sections of scalograms for noise-contaminated sensor signals at 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz and 100 kHz for impact 𝐼  1 . The threshold 𝑠th = 2 × 10−1 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. L. Grasboeck et al.: Preprint submitted to Elsevier Page 32 of 60 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 33
Figure 33. Figure 33: Scalograms of noise-contaminated sensor signals S1–S4 for the idealized impact 𝐼  1 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 1.5 ms and a frequency-range of 2 kHz to 15 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1 × 10−3 is reached. Dash-dotted white lines represent th… view at source ↗
Figure 34
Figure 34. Figure 34: Sections of scalograms for noise-contaminated sensor signals at 4 kHz, 6 kHz, 8 kHz, 10 kHz, 12 kHz and 14 kHz for impact 𝐼  1 . The threshold 𝑠th = 1 × 10−3 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. Frequency in kHz Threshold crossing in ms – 𝐼  1 Threshold crossing in ms – 𝐼  1 S1 S2 S3 S4 S1 S2 S3 S4 … view at source ↗
Figure 35
Figure 35. Figure 35: Scalograms of noise-contaminated sensor signals S1–S4 for the experiment-based impact 𝐼  1 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 1.5 ms and a frequency-range of 2 kHz to 15 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 1 × 10−3 is reached. Dash-dotted white lines repre… view at source ↗
Figure 36
Figure 36. Figure 36: Sections of scalograms for noise-contaminated sensor signals at 4 kHz, 6 kHz, 8 kHz, 10 kHz, 12 kHz and 14 kHz for impact 𝐼  1 . The threshold 𝑠th = 1 × 10−3 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. L. Grasboeck et al.: Preprint submitted to Elsevier Page 35 of 60 [PITH_FULL_IMAGE:figures/full_fig_p035_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Scalograms of noise-contaminated sensor signals S1–S4 for the idealized impact 𝐼  2 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 1.5 ms and a frequency-range of 2 kHz to 15 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 5 × 10−3 is reached. Dash-dotted white lines represent th… view at source ↗
Figure 38
Figure 38. Figure 38: Scalograms of noise-contaminated sensor signals S1–S4 for the experiment-based impact 𝐼  2 : The normalized square of the CWT’s absolute value is illustrated in time-spans of 1.5 ms and a frequency-range of 2 kHz to 15 kHz. Dashed red lines illustrate TOA-estimates as functions of the frequency, i.e., times at which a (non-dimensional) threshold of 𝑠th = 5 × 10−3 is reached. Dash-dotted white lines repre… view at source ↗
Figure 39
Figure 39. Figure 39: Sections of scalograms for noise-contaminated sensor signals at 4 kHz, 6 kHz, 8 kHz, 10 kHz, 12 kHz and 14 kHz for impact 𝐼  2 . The threshold 𝑠th = 5 × 10−3 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. 0.0 0.5 1.0 1.5 t in ms 10−6 10−4 10−2 100 102 |Wψ| 2/ ¯W2 (a) 0.0 0.5 1.0 1.5 t in ms 10−6 10−4 10−2 100 1… view at source ↗
Figure 40
Figure 40. Figure 40: Sections of scalograms for noise-contaminated sensor signals at 4 kHz, 6 kHz, 8 kHz, 10 kHz, 12 kHz and 14 kHz for impact 𝐼  2 . The threshold 𝑠th = 5 × 10−3 is indicated by dashed red lines; their (first) intersections with the scalograms’ sections determine the TOA for the respective frequencies. L. Grasboeck et al.: Preprint submitted to Elsevier Page 38 of 60 [PITH_FULL_IMAGE:figures/full_fig_p038_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: Parametric study of the SLA algorithm for signal of sensor S3 (noise-free) upon impact 𝐼  1 : (a) Contour plot of TOA-estimates as a function of 𝛼 and 𝛽. The white dashed line at 𝛼𝛽 = 270 corresponds to 𝑡𝑆0 = 115.4 µs. (b) Histogram of TOA-estimates, including a zoomed view centered on 𝑡𝑆0 . Most estimates cluster near 𝑡𝑆0 , while only few parameter combinations capture 𝑡𝐴0 . indicating a high robustness… view at source ↗
Figure 42
Figure 42. Figure 42: Result of TOA-estimation using the SLA method on noise-free sensor signals for impact position 𝐼1 . The solid line represents the mean TOA-estimates, while the color-matched shaded regions indicate the corresponding standard deviations over all (𝛼, 𝛽)-combinations producing that product. The red and green dashed lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes 𝑡𝑆0 and 𝑡𝐴0 , respectively. The g… view at source ↗
Figure 43
Figure 43. Figure 43: Result of TOA-estimation using the SLA method on noise-free sensor signals for impact position 𝐼2 . The solid line represents the mean TOA-estimates, while the color-matched shaded regions indicate the corresponding standard deviations over all (𝛼, 𝛽)-combinations producing that product. The red and green dashed lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes 𝑡𝑆0 and 𝑡𝐴0 , respectively. The g… view at source ↗
Figure 44
Figure 44. Figure 44: Parametric study of the SLA algorithm for signal of sensor S3 (noise-contaminated) upon impact 𝐼  1 : (a) Contour plot of TOA-estimates as a function of 𝛼 and 𝛽. (b) Histogram of TOA-estimates. The orange bars in (b) correspond to the TOA-intervals highlighted as discrete colored bands in the contour plot in (a), i.e. [465, 470] µs, [510, 515] µs, [565, 570] µs, [595, 600] µs, and [665, 670] µs. The grey… view at source ↗
Figure 45
Figure 45. Figure 45: Result of TOA-estimation using the SLA method on noise-contaminated sensor signals for impact position 𝐼1 . The solid line represents the mean TOA-estimates, while the color-matched shaded regions indicate the corresponding standard deviations over all (𝛼, 𝛽)-combinations producing that product. The red and green dashed lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes 𝑡𝑆0 and 𝑡𝐴0 , respectivel… view at source ↗
Figure 46
Figure 46. Figure 46: Result of TOA-estimation using the SLA method on noise-contaminated sensor signals for impact position 𝐼2 . The solid line represents the mean TOA-estimates, while the color-matched shaded regions indicate the corresponding standard deviations over all (𝛼, 𝛽)-combinations producing that product. The red and green dashed lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes 𝑡𝑆0 and 𝑡𝐴0 , respectivel… view at source ↗
Figure 47
Figure 47. Figure 47: Results of the parametric study of MER algorithm applied to noise-free signals from sensors S1–S4 in subfigures (a)–(d), for the idealized and experiment-based impacts at location 𝐼1 . The dashed reference lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes, 𝑡𝑆0 and 𝑡𝐴0 . The grey shaded area marks the earliest possible arrival window of the 𝐴0 mode between 10 kHz to 20 kHz. The situation changes… view at source ↗
Figure 48
Figure 48. Figure 48: Results of parametric study of MER algorithm applied to noise-free signals of sensors S1–S4 in subfigures (a)–(d) for idealized and experiment-based impact at location 𝐼2 . The dashed reference lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes, 𝑡𝑆0 and 𝑡𝐴0 . The grey shaded area marks the earliest possible arrival window of the 𝐴0 mode between 10 kHz to 20 kHz. 20 40 60 80 100 α 102 103 ˆt1 in … view at source ↗
Figure 49
Figure 49. Figure 49: Results of the parametric study of the MER algorithm applied to noise-contaminated signals of sensors S1–S4 , shown in subfigures (a)–(d), for the idealized and experiment-based impacts at location 𝐼1 . The dashed reference lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes, 𝑡𝑆0 and 𝑡𝐴0 . The grey shaded area marks the earliest possible 𝐴0 -arrival window for frequencies between 10 kHz to 20 kHz… view at source ↗
Figure 50
Figure 50. Figure 50: Results of parametric study of MER algorithm applied to noise-contaminated signals of sensors S1–S4 in subfigures (a)–(d) for idealized and experiment-based impact at location 𝐼2 . The dashed reference lines indicate the first arrival time of the 𝑆0 and 𝐴0 modes, 𝑡𝑆0 and 𝑡𝐴0 . The grey shaded area marks the earliest possible arrival window of the 𝐴0 mode between 10 kHz to 20 kHz. These velocities indicate… view at source ↗
Figure 51
Figure 51. Figure 51: TOA-estimation results obtained by MER with a window size of 𝑛e = 1250 (𝛼 = 25) after low-pass filtering of the noise-free sensor signals with varying cutoff frequency 𝑓𝑐 , shown for all impacts under consideration: (a) 𝐼  1 , (b) 𝐼  1 , (c) 𝐼  2 and (d) 𝐼  2 . ̂𝑐𝑔 in m s−1: idealized impact ̂𝑐𝑔 in m s−1: experiment-based impact S1 S2 S3 S4 S1 S2 S3 S4 𝐼1 991 1036 1074 992 1133 1065 1092 1140 𝐼2 3014 … view at source ↗
Figure 52
Figure 52. Figure 52: TOA-estimation results obtained by MER with a window size of 𝑛e = 1250 (𝛼 = 25) after low-pass filtering of the noise-contaminated sensor signals with varying cutoff frequency 𝑓𝑐 , shown for all impacts under consideration: (a) 𝐼  1 , (b) 𝐼  1 , (c) 𝐼  2 and (d) 𝐼  2 . wave speeds that can be associated with 𝐴0 . For impacts at location 𝐼2 , on the other hand, the resulting TOA-estimates show substant… view at source ↗
Figure 53
Figure 53. Figure 53: Demonstration of the two-step AIC procedure on a representative noise-free sensor signal (S2 for 𝐼  1 ). The grey shaded area indicates the window, where the AIC, i.e., 𝜒(𝑡 𝑖 ), is applied to the signal. In the first AIC step (a), the AIC window is determined by the AIC picker via the characteristic function 𝛽(𝑡 𝑖 ) (Allen’s formula (51)) and in the second AIC step (b) the TOA-estimation is repeated in a… view at source ↗
Figure 54
Figure 54. Figure 54: Influence of varying the upper bound 𝑡 ub1 of the first AIC window on the estimated TOA for sensors (a) S1 , (b) S2 , (c) S3 and (d) S4 at impact location 𝐼1 . Shown in each case is the comparison between AIC-GM and AIC-LM for idealized and experiment-based impacts. The green and red dashed lines indicate the reference events 𝑡𝑆0 and 𝑡𝐴0 , respectively. The grey shaded area marks the earliest possible arr… view at source ↗
Figure 55
Figure 55. Figure 55: Same analysis as in [PITH_FULL_IMAGE:figures/full_fig_p051_55.png] view at source ↗
Figure 56
Figure 56. Figure 56: Influence of varying the upper bound 𝑡 ub1 of the first AIC window on the estimated TOA for sensors (a) S1 , (b) S2 , (c) S3 and (d) S4 at impact location 𝐼1 . Shown in each case is the comparison between AIC-GM and AIC-LM for idealized and experiment-based impacts. The green and red dashed lines indicate the reference events 𝑡𝑆0 and 𝑡𝐴0 , respectively. The grey shaded area marks the earliest possible arr… view at source ↗
Figure 57
Figure 57. Figure 57: Same analysis as in [PITH_FULL_IMAGE:figures/full_fig_p053_57.png] view at source ↗
Figure 58
Figure 58. Figure 58: TOA-estimation results (obtained by AIC) by filtering the sensor signals with different cutoff frequencies for all impacts under consideration: (a) 𝐼  1 , (b) 𝐼  1 , (c) 𝐼  2 and (d) 𝐼  2 . L. Grasboeck et al.: Preprint submitted to Elsevier Page 53 of 60 [PITH_FULL_IMAGE:figures/full_fig_p053_58.png] view at source ↗
Figure 59
Figure 59. Figure 59: TOA-estimation results (obtained by AIC) by filtering the noise-contaminated sensor signals with different cutoff frequencies for all impacts under consideration: (a) 𝐼  1 , (b) 𝐼  1 , (c) 𝐼  2 and (d) 𝐼  2 . 6. Summary and Conclusions The main objective of this work is to provide a good understanding of the behavior of different TOA-estimation methods and to derive guidelines for obtaining the most a… view at source ↗
Figure 60
Figure 60. Figure 60: Experimental setup of impacting the aluminum plate with the shaker tip. (a) Plate on foam mat with wooden pallet underneath and (b) shaker tip with the force sensor through a hole in the foam mat. L. Grasboeck et al.: Preprint submitted to Elsevier Page 56 of 60 [PITH_FULL_IMAGE:figures/full_fig_p056_60.png] view at source ↗
read the original abstract

This work describes and assesses different methods for estimating the time-of-arrival (TOA) of impact-induced waves in isotropic plate-like structures. The methods considered include threshold crossing (TC), continuous wavelet transform (CWT), short/long term average (SLA), modified energy ratio (MER), and the Akaike information criterion (AIC). Their advantages, limitations, and sensitivities to method-specific parameters are systematically investigated. The assessment is based on synthetic data from transient finite element simulations that are experimentally calibrated with respect to excitation and dispersion characteristics. Wave propagation is monitored using piezoceramic patch sensors bonded to the plate surface, and robustness is evaluated for impacts of varying positions and force profiles, including noise-contaminated sensor signals in order to account for practically relevant measurement conditions. The results show that the methods are capable of detecting the fundamental Lamb wave modes, with nearly all capturing both the symmetric and anti-symmetric mode arrivals under noise-free conditions. In particular, noise primarily impairs the detection of the earliest symmetric-mode arrivals, while meaningful anti-symmetric-mode TOA-estimates can still be obtained by suitable preprocessing or time-frequency analysis. Besides, new contributions to the assessed TOA-estimation methods include a frequency-domain threshold crossing within the CWT framework that improves both robustness and accuracy of TOA-estimation, and the consideration of local minima in the AIC that proves effective for detecting the TOA of the fundamental symmetric mode. Beyond these findings, the research provides practical guidelines and insights into the specific characteristics of each assessed method, supporting accurate and reliable TOA-estimation for applications such as impact localization.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript assesses multiple time-of-arrival (TOA) estimation methods—threshold crossing (TC), continuous wavelet transform (CWT), short/long-term average (SLA), modified energy ratio (MER), and Akaike information criterion (AIC)—for detecting impact-induced Lamb waves in isotropic plates using piezoceramic sensors. Evaluation uses synthetic data from transient finite-element simulations experimentally calibrated for excitation amplitude and dispersion curves. Performance is examined across impact positions, force profiles, and additive noise. New contributions include a frequency-domain threshold-crossing variant inside the CWT framework and selection of local minima in the AIC for S0-mode detection. Results indicate that nearly all methods capture both S0 and A0 arrivals under noise-free conditions, noise primarily degrades earliest S0 detections while A0 estimates remain recoverable via preprocessing or time-frequency analysis, and the work supplies practical guidelines for method selection and tuning.

Significance. If the synthetic waveforms faithfully reproduce real piezoceramic sensor statistics under noise, the study supplies actionable guidelines for TOA estimation in structural-health-monitoring applications. The systematic empirical comparison on calibrated simulations, together with the two proposed algorithmic tweaks, could help practitioners improve robustness of impact localization. The low circularity (performance metrics are not derived from the same fitted parameters used to tune the methods) is a positive feature.

major comments (3)
  1. [Methods (synthetic data generation) and Results (noise-contaminated cases)] The central claims on noise robustness—that noise impairs S0 arrivals while meaningful A0 TOA estimates remain obtainable via preprocessing or CWT—are derived exclusively from synthetic noise added to FE traces calibrated only for excitation amplitude and dispersion curves. The manuscript does not demonstrate that the resulting time-frequency content, modal attenuation, sensor ringing, or non-stationary noise statistics match experimental piezoceramic outputs for the same impact locations and force histories (see abstract and the description of the synthetic-data pipeline). This is load-bearing for the reported detection rates and accuracy rankings.
  2. [Results and Discussion (proposed method variants)] The two new contributions—a frequency-domain threshold crossing inside the CWT and local-minima selection in the AIC—are asserted to improve robustness and accuracy, yet the manuscript provides neither quantitative error bars, statistical significance tests, nor cross-validation against held-out experimental traces to substantiate the magnitude of these gains over baseline implementations.
  3. [Methods] Parameter-sensitivity analysis, data-exclusion rules, and the precise definition of “meaningful” TOA estimates are not fully specified, preventing independent verification of the claimed systematic assessment and practical guidelines.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the noise level (SNR) and impact parameters for each panel to improve readability.
  2. [Methods] Notation for the new CWT threshold-crossing variant should be introduced with a clear equation or pseudocode block.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and indicate the changes we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Methods (synthetic data generation) and Results (noise-contaminated cases)] The central claims on noise robustness—that noise impairs S0 arrivals while meaningful A0 TOA estimates remain obtainable via preprocessing or CWT—are derived exclusively from synthetic noise added to FE traces calibrated only for excitation amplitude and dispersion curves. The manuscript does not demonstrate that the resulting time-frequency content, modal attenuation, sensor ringing, or non-stationary noise statistics match experimental piezoceramic outputs for the same impact locations and force histories (see abstract and the description of the synthetic-data pipeline). This is load-bearing for the reported detection rates and accuracy rankings.

    Authors: We agree that the noise-robustness claims rest on additive Gaussian noise applied to FE traces calibrated solely for amplitude and dispersion. The study does not match experimental sensor ringing or non-stationary noise statistics. In revision we will (i) explicitly state these modeling assumptions in the Methods section, (ii) add a dedicated limitations paragraph in the Discussion, and (iii) rephrase the abstract and conclusions to present the results as indicative under the modeled conditions rather than experimentally validated. revision: yes

  2. Referee: [Results and Discussion (proposed method variants)] The two new contributions—a frequency-domain threshold crossing inside the CWT and local-minima selection in the AIC—are asserted to improve robustness and accuracy, yet the manuscript provides neither quantitative error bars, statistical significance tests, nor cross-validation against held-out experimental traces to substantiate the magnitude of these gains over baseline implementations.

    Authors: We will add error bars (standard deviation across noise realizations and impact positions) to all performance plots and tables. Where sample sizes permit, we will include simple statistical comparisons. Because the work uses only synthetic data, cross-validation on held-out experimental traces is not possible; we will note this limitation and identify experimental validation as future work. revision: partial

  3. Referee: [Methods] Parameter-sensitivity analysis, data-exclusion rules, and the precise definition of “meaningful” TOA estimates are not fully specified, preventing independent verification of the claimed systematic assessment and practical guidelines.

    Authors: We will expand the Methods section with a new subsection on parameter sensitivity, provide explicit data-exclusion criteria (e.g., TOA detections rejected when the estimated arrival falls outside the physically plausible window), and define “meaningful” TOA as an estimate whose absolute error is below a stated threshold relative to the known simulation arrival time. These additions will enable independent reproduction of the reported rankings and guidelines. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical assessment on independent synthetic benchmarks

full rationale

The paper performs a comparative evaluation of TOA estimators (TC, CWT, SLA, MER, AIC) plus two proposed tweaks on transient FE-generated waveforms that were calibrated only for excitation amplitude and dispersion curves. All reported detection rates, accuracy rankings, and robustness statements are obtained by direct application of the methods to these fixed synthetic traces (with and without added noise). No performance metric is obtained by fitting a parameter to a subset of the assessment data and then re-using that parameter to compute the same or a closely related quantity. No self-citation chain is invoked to justify uniqueness or to close a derivation loop. The central claims therefore remain falsifiable against external experimental data and do not reduce to the input data by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The assessment rests on the assumption that calibrated finite-element wave-propagation models faithfully reproduce experimental sensor signals across impact locations, force profiles, and added noise. No new free parameters, ad-hoc axioms, or invented physical entities are introduced beyond standard Lamb-wave and signal-processing assumptions.

axioms (1)
  • domain assumption Transient finite-element simulations, once calibrated to experimental excitation and dispersion data, produce sensor signals representative of real piezoceramic measurements on isotropic plates.
    The entire assessment is performed on synthetic data generated under this premise.

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