Advanced Methods for Time-Of-Flight Estimation with ... - CiteSeerX

Title: Advanced Methods for Time-Of-Flight Estimation with Application to Lamb Wave Structural Health Monitoring

Authors:

Buli Xu Lingyu Yu Victor Giurgiutiu

Paper to be presented at:

The 7th International Workshop on Structural Health Monitoring – 2009, Stanford University, Palo Alto, CA

ABSTRACT For single-mode nondispersive bulk wave, gating and peak-detection techniques are usually adequate for time of flight (TOF) estimation. However, for multimodal and dispersive Lamb waves, TOF estimation is more complicated and requires special methods. In this paper, several TOF estimation methods, including cross-correlation, envelop moment, matching pursuit decomposition and dispersion compensation, are studied and compared using a dispersive Lamb wave mode, both in simulation and in experiment. The performance of each method is evaluated by comparing the extracted TOF with the theoretical TOF value. We found that, the correlation method is much less efficient in processing experimental wave signals that simulated ones. The envelop method exhibits high accuracy when applied to the simulated S0 Lamb wave mode. However, its accuracy deteriorated when evaluating the experimental data due to the presence of noise. Among all the methods presented, dispersion compensation gives the best TOF estimation though it only works well for single mode waves. Further comparison is under investigation for piezoelectric wafer active sensor (PWAS) phased array implementation. 1

INTRODUCTION

Time of flight (TOF) describes the time that it takes for a particle, object or stream to reach a detector while traveling over a certain distance. Many ultrasonic testing applications, such as thickness gauge, tomography imaging and phased array, are based on the estimation of the TOF of ultrasonic echoes. Lamb waves are ultrasonic guided waves that can propagate long distance without much attenuation in thinwall structures; they have been widely used in nondestructive evaluation/testing (NDE/NDT) and structural health monitoring (SHM). For single-mode nondispersive bulk wave, gating and peak-detection techniques are usually adequate for TOF estimation. However, for multimodal and dispersive Lamb waves, TOF estimation is more complicated and requires special methods. TOF can be estimated non-deterministically using parameter estimation and optimization methods based on theoretical models of the waveform under analysis. Heijden (2003, 2004) presented statistical method for TOF estimation based on covariance models. Alternatively, TOF can be extracted deterministically by methods such as 2

magnitude thresholding (Tong et al, 2001), matched filter (cross-correlation) (Couch, 2001), envelop moment (Demirli, 2001), time-frequency methods (Hou et al 2004), etc. In this chapter, several TOF estimation methods, including cross-correlation, envelop moment, matching pursuit decomposition and dispersion compensation, are studied and compared using a dispersive Lamb wave signal theoretically and experimentally. The performance of each method is evaluated by comparing the extracted TOF with the theoretical TOF value. 2 2.1

TIME-OF-FLIGHT ESTIMATION Signal under Analysis

To demonstrate the performance of these methods, we consider TOF estimation of a 350 kHz S0 mode waves excited by 3.5-count Hanning windowed tone burst after propagation distance x = 300 mm on a 3-mm aluminum plate. Figure 1a shows the simulated results while Figure 1b shows the experimental data collected on real specimen. TOF is defined as the group delay at the tone burst central frequency 350 kHz and measures the average TOF for this frequency. At 350 kHz, the wave group velocity IS cgrs0 = 4989 m/s, hence the theoretical TOF is equal to 60.132 µs. In the following sections, this theoretical TOF will be compared with the TOF values determined through different methods as discussed in each section. 2.2

Cross-Correlation Method

The cross-correlation function is a quantitative operation in the time domain to describe the relationship between data measured at a point and data obtained at another observation point. The cross correlation function is given as 1 T ψ xy (τ ) = lim ∫ f x (t ) f y (t + τ )dt (1) T →∞ T 0 where fx(t) is the signal at point x , at time t , and f y (t + τ ) is the signal at a point y

at time t + τ . The TOF is the value of the time instant τ for which the cross correlation integral reaches a maximum. Figure 2 shows the TOF estimation of the simulated 350 kHz S0 mode using the original excitation signal as reference. The simulated S0 wave packet is slightly dispersed but is still similar in shape to its excitation. The estimated TOF is 60.143 µs, which is very close to the theoretical TOF of 60.132 µs. For comparison, Figure 3 shows the cross-correlation method applied to experimental data; the estimated TOF is found to be 62.320 µs, which is slightly off the theoretical TOF = 60.132 µs. These results may indicate that the correlation method would work well for slightly dispersive waves. However, when the waveform under analysis is very dispersive, cross-correlation will cease to work becausethe cross-correlation method lies on the assumption that the response signal is only a shifted, scaled version of reference signal buried in additive Gaussian white noise. The reference signal can be the impulse response or be a more exact signal obtained prior to actual experiment. However, the received signal has undergone shape distortion, such as the dispersion of propagating Lamb wave, then the cross-correlation method may become inefficient.

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Waveforms of S0 mode at 350 kHz on a 3mm aluminum plate after propagation distance x = 300 mm: (a) simulated wave; (c) experimental wave

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TOF estimation of simulated S0 mode wave by cross-correlation method: (a) 3.5-count tone burst excitation centered at 350 kHz; (b) simulated S0 wave packet at x = 300 mm; (c) cross-correlation of waves in (a) and (b) 0.01

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TOF estimation of experimental S0 mode wave by cross-correlation method: (a) experimental S0 wave at x = 300 mm; (c) cross-correlation

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2.3

Envelope Moment Method

2.3.1 Envelop Extraction by Hilbert Transform Hilbert transform is widely used in communications to map a real signal into an analytical signal (complex envelope) such that a single sideband (SSB) signal is obtained. The analytical signal s(t) of a real signal x(t) is defined as s (t ) = x(t ) + jh(t ) (2)

where h(t ) is Hilbert transform of x(t ) and j = −1 . The magnitude of the analytical signal, which is identical to the magnitude of the real signal, is called the envelope.

2.3.2 TOF Estimation Using Envelop To demonstrate the performance of TOF estimation using the envelope, the waveforms in Figure 1 were considered in Figure 4. The estimated TOF was found to be 60.476 μs for the simulated wave (Figure 4a) and 62.206 μs for the experimental wave (Figure 4b). A rectangular window was applied to the experimental wave to extract the wave packet of interest before applying the Hilbert transform. 2.4

Dispersion Compensation Method

Figure 5 shows the TOF estimation of the simulated S0 mode wave using dispersion compensation method (Xu et al., 2009a). The estimated TOF is equal to 60.363 μs. For the experimental S0 mode wave, the TOF is found to be 60.936 μs as shown in Figure 6. Both of the estimated TOF are relatively closed to the theoretical value at 60.132 µs.

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TOF estimation by envelop method. (s) envelop of simulated S0 mode; (b) envelop of experimental S0 mode.

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TOF estimation by dispersion compensation method of the simulated S0 wave: (a) simulated S0 wave at 300 mm; (b) dispersion compensated wave

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TOF estimation by dispersion compensation method of the experimental S0 wave: (a) experimental S0 wave packet at 300 mm; (b) dispersion compensated wave

Chirplet Matching Pursuit Decomposition

Chirplet matching pursuit decomposition (Xu et al 2009b) was applied to the simulated and experimental S0 mode waves to extract their TOF. Figure 7 shows the decomposed first two atoms of the simulated waves. Because the first atoms contain most of the energy of the wave, TOF estimation is based on the first atoms. Figure 8 shows the decomposed first two atoms of the experimental waves. The peak location of the S0 wave is found at 64 µs for the simulated wave and 66.56 µs for the experimental waveform. Since the total TOF is determined by peak to peak and the location of the peak of the 3.5-count toneburst excitation of 350 kHz is at 5 µs, the TOF is determined to be 59 µs for the simulated S0 wave, and 61.56 µs for the experimental wave.

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(s, u, f0 ,c)= (128, 64.00μs, 331.5 kHz, 2.28e+10)

1st atom

(s, u, f0 ,c)= (128, 68.57μs, 436.4 kHz, 7.68e+9)

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Figure 7

First two decomposed chirplet atoms and their parameters of simulated S0 mode wave

(s, u, f0 ,c)= (512, 66.560μs, 356.4 kHz, 2.203e+10)

1st atom

(s, u, f0 ,c)= (512, 71.680μs, 482.2 kHz, -6.015e+9)

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Figure 8

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Experimental S0 mode wave first two decomposed chirplet atoms and their parameters

COMPARISON AND CONCLUSIONS

Four TOF estimation methods are presented in this paper to extract TOF of a simulated and an experimental S0 mode waves centered at 350 kHz. Table 1 shows the performance of these methods. Accuracy of each method is evaluated by compared the extracted TOF to its theoretical value. Figure 9 shows the accuracy of 7

Table 1

Comparison of TOF estimation by various methods (TOFTh = 60.132 µs)

Correlation TOF (µs) Error (%) Sim. 60.143 0.033 Exp. 62.320 3.654

(a)

Envelop TOF (µs) Error (%) 60.476 0.587 62.206 3.465

Error (%) 4 3.5 3 2.5 2 1.5 1 0.5 Correlation 0

MP TOF (µs) 59.000 61.560

Disp. Comp. Error (%) 1.87 2.391

TOF (µs) 60.363 60.936

Error (%) 0.399 1.352

MP

Envelop

Disp. Comp.

Error (%)

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Figure 9

4 Correlation 3.5 3 2.5 2 1.5 1 0.5 0

Envelop MP Disp. Comp.

Comparison of accuracy of various methods. (a) TOF estimation of the simulated S0 mode wave; (b) TOF estimation of the experimental S0 mode wave

these methods for TOF estimation of the simulated and experimental S0 mode waves, respectively. The correlation method is very efficient in simulation but quite inefficient when processing the experimental wave. This may be due to the fact that the experimental wave is more dispersive than its simulation, The envelop method shows high accuracy when extracting TOF of the simulated S0 mode wave. However, its accuracy deteriorated when evaluating the experimental wave. This may due to the effect of noise in the experimental wave. Moreover, envelop method only works well to extract TOF of a single wave packet. To extract TOF of the experimental S0 mode wave, a rectangular window has to be applied to extract the wave packet of interest. This makes the method hard to be automated for SHM application. Among these methods, dispersion compensation gives the best TOF estimation for experimental data. However, dispersion compensation only works well for single-mode waves.

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Accuracy of matching pursuit method is stable and acceptable, around 2% error in both cases. In addition, this method outputs parameters of decomposed atoms and can be easily automated for TOF estimation. Also, considering dispersion and multi modes in Lamb waves, matching pursuit may be the preferred method for TOF estimation. 4

ACKNOWLEDGMENTS

The financial supports from National Science Foundation under Grant #CMS0408578 is thankfully acknowledged. 5

REFERENCE

Couch, L. W. (2001) Digital and Analog Communication Systems, 6th edition, Prentice Hall, pp. 445-452, 2001 Demirli, R. (2001) Model Based Estimation of Ultrasonic Echoes: Analysis, Algorithms, and Applications, Ph.D. dissertation, Illinois Institute of Technology, pp. 39- 51, 2001 Heijden, F.; Tuquerres, G. ; Regtien, P. (2003) “Time-of-flight Estimation Based on Covariance Models”, Meas. Sci. Technol., Vol. 12, pp. 1295-1304, 2003 Heijden, F.; Duin, R.P.W., Ridder, D.; Tax, D.M.J. (2004) Classification, Parameter Estimation and State Estimation – An Engineering Approach using MATLAB, John Wiley & Sons, Ltd, pp. 319-338, 2004 Hou, J.; Leonard, K.R.; Hinder, M.K. (2004) “Automatic Multi-mode Lamb Wave Arrival Time Extraction for Improved Tomographic Reconstruction”, Inverse Problem, Vol. 20, pp. 1873-1888, 2004 Giurgiutiu, V.; Zagrai, A. N. (2000) “Characterization of Piezoelectric Wafer Active Sensors”, Journal of Intelligent Material Systems and Structures, Vol. 11, pp. 959-976, 2000 Tong, C.; Figueroa, J.F. (2001) “A Method for Short or Long Range Time-of-Flight Measurements Using Phase-Detection with an Analog Circuit”, IEEE Trans. on Inst. and Meas., vol. 50, pp. 1324-1328, Oct., 2001 Xu, B.; Yu, L.; Giurgiutiu, V. (2009a) “Lamb wave dispersion compensation in piezoelectric wafer active sensor phased-array applications”, SPIE Health Monitoring of Structural and Biological Systems III conference, March 2009, San Diego, CA, Paper #7295-74 Xu, B.; Giurgiutiu, V.; Yu, L. (2009b) “Lamb waves decomposition using matching pursuit method”, SPIE Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems Conference, March 2009, San Diego, CA, Paper #7292-18

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