On a Model-Based Calibration Approach to Dynamic ...

ON A MODEL-BASED CALIBRATION APPROACH TO DYNAMIC BASELINE ESTIMATION FOR STRUCTURAL HEALTH MONITORING James S. Hall and Jennifer E. Michaels

School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250

ABSTRACT. Sparse arrays of permanently attached ultrasonic transducers are capable of quickly interrogating large areas, thereby reducing or eliminating the need for extensive bulk wave testing in plate-like structures. Current imaging methods often ignore dispersion, analyzing received signals by approximating wave propagation with a nominal group velocity. Dispersion compensation can significantly improve the resolution and accuracy of such methods by more accurately modeling the propagation environment and thereby leveraging a larger percentage of the energy and information contained in broadband signals. By adaptively estimating system parameters such as dispersion and relative transducer locations, algorithms will operate at the time of test with the most accurate model possible and will thus be robust to homogeneous environmental changes. This paper reviews features and limitations of current dispersion estimation algorithms and describes a new model-based approach that avoids a priori information and relies only on the physical nature of the dispersion curves. Performance of two contributing algorithms are evaluated using simulated signals obtained from known dispersion curves.

Keywords: Ultrasonic Guided Waves, Nondestructive Evaluation, Structural Health Monitoring, Sparse Arrays, Lamb Waves, Model-Based Calibration, Dispersion Estimation, Dynamic Baseline Estimation PACS: 43.20.Jr, 43.35.Cg, 43.35.Yb, 43.35.Zc, 43.60.Gk, 43.60.Jn, 43.60.Mn, 43.60.Pt, 43.60.Uv.

INTRODUCTION Current research efforts [1-3] are exploring the use of ultrasonic guided waves for damage detection and localization, which present some major advantages to traditional bulk wave NDE methods. Sparse arrays of permanently attached, lowǦcost transducers are capable of quickly interrogating large areas, including inaccessible or complex components, thereby reducing or eliminating the need for extensive testing. There are a significant number of challenges associated with obtaining comparable information from a sparse array of guided wave transducers as compared to a spatially dense bulk wave evaluation. Geometric dispersion and multiple reflections within the structure produce complex waveforms. Additionally, the propagation environment is multimodal and the dispersive nature of each mode can vary depending on the external

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environment. As such, in situ testing must address challenges associated with operation in uncontrolled environmental conditions. This paper documents the current status of efforts to develop an algorithm that adaptively estimates relative transducer distances, dispersion curves, propagation loss, transmitted signal function, and mode weighting coefficients in a multimode propagation environment using a sparse array of ultrasonic transducers and as little a priori information as possible. This information can then be used, along with additional system information, to dynamically estimate damage-free baselines for structural health monitoring. A model-based calibration approach is proposed for which a propagation model is assumed and model parameters (distances, dispersion curves, etc.) are obtained that allow the model to describe the received signals as closely as possible. Model-based calibration takes advantage of the inherent constraints of the system model in an effort to simplify the parameter search. By using as little a priori information as possible, errors in a priori assumptions, such as transducer distances and dispersion curves, are minimized and the resulting data-driven parameter estimates are robust to homogeneous environmental changes such as temperature. Alleyne and Cawley [4] successfully demonstrated a two-dimensional Fourier transform method that made all of the parameters being sought in this algorithm readily available. The experimental setup used a movable receiving transducer to obtain 64 equally spaced spatial records. However, the sparse nature of the fixed sensor arrangement in this application precludes the spatial sampling that is key to the twodimensional Fourier transform algorithm’s performance. Considering all of the parameters listed above, the estimation of dispersion curves and transducer distances is arguably the most important and has immediate applicability to current SHM systems. Current algorithms typically either ignore dispersion or anticipate dispersion effects using nominal dispersion curves. Because dispersion changes the shape of the signal over distance, back-propagation imaging techniques that leverage received signal correlation, such as conventional and MVDR beamforming [5], are degraded. This is because the correlation between signals received at two different propagation distances will be reduced as the difference in propagation distance is increased. By accounting for dispersion, a larger percentage of the signal energy and inherent information can be utilized, thereby improving the signal-to-noise ratio (SNR). Further, by adaptively estimating these parameters at the time of the test, algorithms that use the information for damage detection or localization can theoretically be made robust to homogeneous environmental factors. For example, Sicard et al. [6] and Wilcox [7] both proposed single-mode dispersion compensation techniques that take advantage of distance-time and frequency-wavenumber conversion techniques that leverage a priori dispersion curves to improve signal reconstruction. Understandably, inaccurate estimates of the dispersion curves result in degraded or biased performance. By adaptively estimating the dispersion curves at the time of test, these algorithms will theoretically sustain nominal performance across homogeneous environmental changes such as temperature, or even inaccuracies in estimation of plate thickness and material properties. With regard to dispersion estimation alone, current dispersion estimation efforts closely dovetail work by Sachse et al. [8], wherein group velocities and dispersion relations are obtained from the phase response of the received signal. The multichannel nature of the sparse array, however, provides a means of achieving similar results without knowledge or assumptions about the transmitted signal or transducer locations. Alternatively, time-frequency analysis representations [9, 10] require only a single signal and represent a dispersion estimation technique that may be a viable alternative to, or augmentation of, the proposed model-based calibration approach.

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The first part of this paper presents a description of the basic propagation model and discusses the inherent assumptions and goals. The second section details two select parts of the algorithm, which are somewhat mature and may be used independently. Finally the paper presents simulation results and conclusions. ASSUMPTIONS AND GOALS Model-based calibration is performed by transmitting an ultrasonic guided wave signal from one transducer and recording the response using all remaining transducers. The received signals are windowed to isolate the direct arrival of the signal, ignoring all reflections. Assuming a homogeneous, isotropic plate with N identical receivers and a common transmitted signal, a frequency domain model of the recorded signals is: RDAi Z T Z H DAi Z  Q M

T  Z ¦ D m  Z e J m Z d i  Q

for i 1...N

(1)

m 1

where m() = p() + jkm(). The additive noise term, , is a complex Gaussian random variable, ~N(0,2), that represents the incoherent, or electronic, noise associated with the receiving transducer. It is assumed that all receiver transfer functions are identical and are incorporated, along with the transmitter transfer function, into a combined isotropic transducer function, T(). A list of all model parameters is included in Table 1. It is important to note that the exponential term, m()di, in equation (1) cannot be separated mathematically from [Cm()][di /C], where C is an unknown constant, thus these model parameters can only be determined to within a common scaling factor. This scaling factor can be resolved after-the-fact by including a priori information about di. At this time, is assumed that mode isolation has been achieved and access to single-mode sub-signals is available. Although mode isolation is not a trivial requirement, several methods do exist to separate the sub-signals. The simplest technique is to position transducers sufficiently far apart from one another to achieve mode isolation in the time domain. For systems that cannot support such distances, another method uses sensors bonded at the same point of the propagating medium on opposite surfaces, which provides the ability to separate overlapping symmetric and asymmetric modes. With mode isolation achieved, the model for each sub-signal (mode) becomes: m RDAi Z T Z D m Z eJ m Z di Q

for i 1...N and m 1...M

TABLE 1. Description of model parameters. RDAi(): Measured direct arrival received at ith transducer M: Total number of propagating modes N: Total number of receiving transducers Solve for: T(): Combined isotropic transducer function HDAi(): Propagation transfer function for ith receiver di†: Distance between transmitter and ith receiver m(): Frequency dependent mth mode weighting coefficient p() *†: Frequency dependent and mode independent propagation loss km() *†: Frequency dependent mth dispersion curve *Due to sign convention and causality, these parameters will always be negative †These parameters can only be determined to within a common scaling factor Known:

898

(2)

Some critical assumptions are made about the noise term, , which should be noted here. First and foremost, the noise term is only intended to represent the electronic noise of the receiving system. Therefore, the noise terms are assumed to be complex, i.i.d. white Gaussian noise independent of the received signal and of the noise in other receivers. Coherent noise sources, such as undesired or unanticipated reflections, imperfect mode separation, and timing errors are assumed to be negligible for the purposes of this paper. Another important observation is that although the noise levels of each receiver are expected to be the same, signal power at each receiver will be different due to propagation loss. Therefore SNRs will not be identical between the receivers. Additionally, when considering frequency-dependent parameters, it is important to note that the SNR for a single frequency bin will vary over the bandwidth of the signal. Thus, frequency-dependent estimates will tend to have higher errors at frequencies for which the signal has low signal power. For this reason, weighted means are frequently used across receivers and frequencies to provide preference to higher SNR data. PARAMETER ESTIMATIONS Weighting Coefficient Estimation Weighting coefficients representing the relative power ratios between propagating modes are approximated using magnitude response ratios and a conservation of energy assumption. The magnitudes of each received single-mode sub-signal are modeled as: m ª RDA º 1 Z « » # « » « m » «¬ RDAN Z »¼

ª T Z D m Z e p Z d1  jkm  w d1  Q « « # « p Z d N  jkm  w d N « T Z D m  Z e Q ¬ p Z d1 ªe º ªP º « » T Z D m Z « # »  «« # »» « p Z d N » « P » ¬e ¼ ¬ ¼

º » » » » ¼ for m 1...M

(3)

The algebraic simplification is possible by noting that the complex random variable, ~N(0,2), has been replaced with a real random variable, , which represents the noise of the magnitude response. For sufficiently high SNR values in which |noise| < |signal| (SNR > 0dB) over all single frequencies,  will be another i.i.d. Gaussian random variable, ~N(0,2/2). If, however, SNR is not high enough,  will no longer be a Gaussian random variable and further analysis becomes much more complex. Assuming that the noise term, , is Gaussian, each of the received signal magnitudes can be described by a Gaussian random variable with a non-zero mean: m ª RDA º 1 Z « » # « » « m » R Z ¬« DAN  ¼»

ªR i 1m Z º 2 « » i im Z ~ N § T Z D Z e p Z di , VQ · « # » where R m ¨ 2 ¸¹ © « m » i R Z ¬« N  ¼»

899

(4)

Taking advantage of the mode independent propagation loss, p(), and transmitted signal term, T(), that are common to all modes, weighting coefficient ratios are obtained for each receiver with a simple division operation: m1 RDAi Z

|

m2 RDAi Z

D m1 Z

(5)

D m 2 Z

Due to the homogeneous and isotropic propagating environment combined with the assumed isotropic nature of the transmitted signal, weighting coefficient ratios are expected to be identical at each receiver and can therefore be combined across receivers to obtain a composite weighting coefficient ratio. Hinkley [11] showed that the ratio of uncorrelated, non-zero mean Gaussian random variables has a closed-form probability density function. This density function, however, can be heavy-tailed, causing a bias in the expected value. Two methods of compensation were explored for use when combining ratio estimates at each receiver: (1) use of the median value, and (2) a weighted mean based on SNR. While both methods showed improvement in SNR environments that produced a biased estimate, the weighted mean produced more consistent results. With a data-driven, zero bias, approximation of weighting coefficient ratios, a least-squares approach at each frequency can be used to produce the final frequencydependent estimate of m(). The governing equations are: D1  Z  "  D M  Z 2

D 2 Z

2

D 3 Z

2

 

D1 Z D 2 Z



2

1

2

D1  Z

2

2

D 2 Z

2

D 2 Z D3 Z

(6)

#

These equations can be rewritten and put into matrix form as: ª1 « «1  « « «0 « ¬#

"

1



D1 Z

D 2 Z



2

0 

1



D 2 Z

D 3 Z



2

1ºª 2 º » « D1  Z » »« » # »« » »« # » »« 2» » D Z %¼ ¬« M  ¼»

ª1 º «0» « » «# » « » ¬0¼

(7)

Note that this linear least-squares approach is only applicable if SNR levels are sufficiently high so that a zero-bias assumption can be made about the ratio distributions. In lieu of a zero-bias assumption, a non-linear regression would have to be performed that can accommodate non-Gaussian, non-zero mean noise. Group Velocity Estimation Group velocity and receiver distance estimation is based on measured differences in the phase response of each received signal. The phase response of a single-mode subsignal is modeled as: m (RDAi Z

 (T Z  d k Z  T mod 2S i m

900

for i

1...N and m 1...M

(8)

where  is an additive white Gaussian noise term with variance that is a function of SNR, ~N(0,2). Assuming that the sampling frequency is sufficiently high so that the phase response varies less than between adjacent frequencies, the complications inherent in dealing with a modulo 2 signal can be avoided by taking the derivative with respect to . This removes the modulo function and results in the following equation: ' m (RDAi Z 'Z

' ' (T Z  di km Z  - for i 1...N and m 1...M 'Z 'Z

(9)

where ~N(0, 22). Further, by using the difference of (9) between the ith and jth receivers, the T(w) phase contribution can be eliminated: ' ' m m (RDAi Z  (RDAj Z 'Z 'Z

d

i

dj

' km Z  I for i 1...N , j z i, and m 1...M 'Z

(10)

where I ~N(0, 42). Although this represents a 12 dB loss in phase response SNR, the resulting equation has now been simplified enough to estimate the remaining parameters. Placing all pair-wise combinations of receivers (10) into vector form yields:

JK m B Z

' ª ' º m m « 'Z (RDA1 Z  'Z (RDA 2 Z » « » # « » « ' » ' m m « (RDAN Z  (RDA1 Z » 'Z ¬ 'Z ¼ ª d1  d 2 º ªI º ' km Z «« # »»  «« # »» 'Z ¬« d N  d1 ¼» ¬«I ¼» JJK * JJK ' km Z D  ) 'Z

JJK*

for all m 1...M

(11)

JJK*

The normalized differential distance vector, D /|| D ||, can now be extracted from a JK weighted average of B m() across all frequencies, , and modes, m. One important note is that the noise term is i.i.d. white Gaussian noise, which ensures that the estimate will converge to the true vector. A dot-product between the normalized differential distance JK vector and B m() is then used to obtain a zero-bias estimate of the inverse scaled group velocity, km( )/. ª§ «¨ E «¨ «¨ ¬©

JJK* D JJK* D

T º · » ¸ JK m ¸ B Z » » ¸ ¹ ¼

ª § « JJK* ' ¨ E« D km Z  ¨ 'Z « ¨ © ¬ § JJK * ' ¨ D km Z  ¨ 'Z ¨ © JJK * D vgr Z

901

JJK * D JJK * D

JJK* D JJK* D

T · º ¸ JJK » ¸ )» ¸ » ¹ ¼

T

· JJK ¸ ¸ E ¬ª) ¼º ¸ ¹

(12)

RESULTS The results shown in Figures 1, 2, and 3 demonstrate the algorithms’ ability to extract mode weighting coefficients and group velocity. Simulated MATLAB test data was used to model an unloaded 3 mm thick aluminum plate with nominal S0 and A0 dispersion curves at 25°C. The transmitted signal, T(), is a 3-cycle, Hanning-windowed sinusoid with center frequency 250kHz. Signal to noise ratios for the received signals over a 300kHz bandwidth (100kHz – 400kHz) range from 12 to 45dB for the weaker S0 mode (S0() = 0.3) and 22 to 55dB for the stronger A0 mode (A0() = 0.95).

Distance (mm)

900

Rx 4

800 700

Rx 3

600

Rx 2

500

Rx 1

400 300 50

100

150

200

250

300

350

Time (s) FIGURE 1. A waterfall plot of the received signals showing clear mode separation in the time-domain. The signals have been normalized to have the same peak magnitude and demonstrate a decrease in SNR with distance due to propagation loss.

Mode Weighting Coefficient Ratio

1.5

Rx 1 Rx 2

1

Rx 3 Rx 4

0.5

0

150

200

250

300

350

400

Frequency (kHz)

Group Velocity (mm-s-1)

FIGURE 2. A plot of mode weighting coefficient ratio calculated from each receiver as a function of frequency. Note the increased noise from receivers that are a greater distance from the transmitter and the asymmetric variation from the constant true ratio of 0.34 at edge frequencies that have less power.

14

S0 Estimate

12

A0 Estimate

10

S0 Actual

8

A0 Actual

6 4 2 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency (MHz) x Thickness (mm) FIGURE 3. A plot of estimated group velocity as a function of frequency. Note the increased variance in the estimate for the weaker S0 mode. The additive white noise for the inverse group velocity estimate translates to a heavy-tailed noise distribution for the group velocity estimate as SNR decreases.

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CONCLUSIONS Although research is not complete, a model-based calibration approach to system parameter estimation is shown to adaptively estimate critical parameters of the propagation environment, such as relative transducer distances and dispersion curves. The ability to perform such estimations at the time of test will avoid system degradation due to inaccuracies in a priori information and assumptions as well as assist systems to perform dynamic baseline estimation by enabling them to operate effectively across homogeneous environmental factors, such as temperature. Two methods were presented in this paper that leveraged model assumptions to adaptively estimate mode weighting coefficients and group velocities. The paper discussed the impacts of SNR and provided simulation results that reflect satisfactory results, provided SNR is sufficiently high. ACKNOWLEDGEMENTS The authors would like to acknowledge the support of the National Science Foundation under Grant No. ECS-0401213 and NASA GSRP Grant No. NNX08AY93H. REFERENCES 1. J. Michaels, T. Michaels, “Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors,” Smart Mater. Struct., 17, 035035 (15pp) (2008). 2. X. Zhao, H. Gao, G. Zhang, B. Ayhan, F. Yah, C. Kwan, J. Rose, “Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. Defect detection, localization, and growth monitoring,” Smart Mater. Struct., 16, pp. 1208Ǧ1217 (2007). 3. T. Michaels, J. Michaels, B. Mi, M. Ruzzene, “Damage detection in plate structures using sparse ultrasonic transducer arrays and acoustic wavefield imaging,” in Review of Progress in QNDE, 24, pp. 938Ǧ945 (2005). 4. D. Alleyne, P. Cawley, “A twoǦdimensional Fourier transform method for the measurement of propagating multimode signals,” J. Acoust. Soc. Am., 89(3), pp. 1159Ǧ1168 (1991). 5. J. Michaels, J. Hall, G. Hickman, J. Krolik, “Sparse array imaging of change-detected ultrasonic signals by minimum variance processing,” to appear in Review of Progress in QNDE, 28, (2008). 6. R. Sicard, J. Goyette, D. Zellouf, “A numerical dispersion compensation technique for time recompression of Lamb wave signals,” Ultrasonics, 40, pp. 727-732, (2002). 7. P. Wilcox, “A rapid signal processing technique to remove the effect of dispersion from guided wave signals,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 50(4), pp. 419Ǧ427 (2003). 8. W. Sachse, Y.-H. Pao, “On the determination of phase and group velocities of dispersive waves in solids,” J. Appl. Phys., 49(8), pp. 4320-4327 (1978). 9. W. Prosser, M. Seale, B. Smith, “TimeǦfrequency analysis of Lamb modes,” J. Acoust. Soc. Am., 105 (5), pp. 2669Ǧ2676 (1999). 10. M. Neithammer, L. Jacobs, J. Qu, J. Jarzynski, “TimeǦfrequency representation of Lamb waves,” J. Acoust. Soc. Am., 109(5), pp. 97Ǧ102 (2001). 11. D. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika, 56, pp. 635-639 (1969).

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