A Bayesian Approach for Localization of Acoustic Emission Source in Plate-Like Structures

This paper presents a Bayesian approach for localizing acoustic emission (AE) source in plate-like structures with consideration of uncertainties from modeling error and measurement noise. A PZT sensor network is deployed to monitor and acquire AE wave signals released by possible damage. By using continuous wavelet transform (CWT), the time-of-flight (TOF) information of the AE wave signals is extracted and measured. With a theoretical TOF model, a Bayesian parameter identification procedure is developed to obtain the AE source location and the wave velocity at a specific frequency simultaneously and meanwhile quantify their uncertainties. It is based on Bayes’ theorem that the posterior distributions of the parameters about the AE source location and the wave velocity are obtained by relating their priors and the likelihood of the measured time difference data. A Markov chain Monte Carlo (MCMC) algorithm is employed to draw samples to approximate the posteriors. Also, a data fusion scheme is performed to fuse results identified at multiple frequencies to increase accuracy and reduce uncertainty of the final localization results. Experimental studies on a stiffened aluminum panel with simulated AE events by pensile lead breaks (PLBs) are conducted to validate the proposed Bayesian AE source localization approach.


Introduction
The increasing emphasis on integrity of critical structures such as aircrafts urges the needs to monitor structures in situ and real-time to detect damages at an early stage to prevent catastrophic failure.In general, an effective online structural health monitoring system should consist of two major components: active monitoring and passive monitoring [1,2].Active monitoring needs both actuators and sensors to evaluate the damage of a structure, while passive monitoring uses only sensors to "listen" to the acoustic emission (AE) wave signals emitted by internal or external sources such as propagating cracks or impacts.In the present study, we deal with passive monitoring to determine the AE source location in plate-like structures where possible damage may exist.
Up to date, a lot of AE source localization approaches have been developed, including several relatively new introduced ones, such as genetic algorithms [3], neural networks [4], support vector machines [5], time reversal [6], beamforming [7], and Voronoi construction [8].Among these approaches, the triangulation-based approaches utilizing the time-of-flight (TOF) information of the AE wave signals are most widely used [9][10][11][12][13][14].With the known positions of the sensors and the velocities of the AE wave signals, the location of the AE source can be determined by solving a set of nonlinear equations directly or by iterative optimization algorithms.These triangulation approaches are originally developed for isotropic structures and then extended to anisotropic composite structures, for which damage caused by low velocity impact is a major concern [13,14].In these approaches, the time arrivals are usually obtained by threshold crossing [15], cross-correlation [16], and statistical Akaike information criterion (AIC) [17].With the development of modal acoustic emission (MAE) technique in which the concept of guided waves is introduced to improve the interpretation of the AE wave signals, the continuous wavelet transform (CWT) has become a useful tool for the time-frequency representation of transient AE waves in dispersive medium with its advantage of good resolution in both time and frequency domains.It makes the determination of the time arrivals of the dispersive AE waves at each local frequency more accurate [10,11,[18][19][20].Other advanced signal processing techniques, such as Mathematical Problems in Engineering Hilbert transform (HT) [21], short time Fourier transform (STFT) [22], and warped Fourier transform (WFT) [23] have been also employed for processing the AE wave signals in recent years.
Generally, most of the triangulation approaches developed can be categorized as deterministic methods, in which uncertainties are not considered and only single solution is given as the final result.However, in reality errors and uncertainties are always unavoidable in the measurement and identification process.For example, the time arrival determined by threshold crossing is usually vulnerable to the measurement noise.Even by employing CWT, systematic uncertainty could exist due to the Heisenberg uncertainty principle [20].In addition, uncertainties may arise from the theoretical wave velocities, since they depend on the material property values of a specific structure, which may have variations from those nominal values and be influenced by several factors, such as environmental temperature changes [24].Under such a circumstance, probabilistic approaches may be more appropriate than deterministic approaches for AE source localization in that probability distributions can be used to quantify the various uncertainties.This leads to the research of probability-based approaches, to give a more reasonable source characterization.Niri and Salamone proposed a probabilistic approach for AE source localization in plate-like structures based on extended Kalman filter (EKF).The main advantage of EKF algorithm over traditional methods is that it can take into account uncertainties in TOF measurement and wave velocity [20].They also proposed an adaptive multisensor fusion framework to combine with EKF to locate the AE sources in noisy operation conditions [21].Schumacher et al. proposed a Bayesian analysis approach for passive AE source location in concrete bridge column to account for uncertainties and errors that exist within the measurement and calculation process [25].
The aim of this work is to develop a Bayesian approach for localizing AE source in plate-like structures, while considering uncertainties from modeling error and measurement noise.It is an extension of an early work dealing with active monitoring of damage for plate-like structures using Lamb waves [26].The rest of this paper is structured as follows.Section 2 describes the proposed Bayesian approach for AE source localization and outlines the main idea of Markov chain Monte Carlo (MCMC) method for sampling the posterior distributions of the unknown parameters, while a data fusion scheme to increase accuracy and reduce uncertainty of the localization results is also presented.Experimental studies on a stiffened aluminum panel with simulated AE events are conducted in Section 3 to validate the proposed probabilistic localization approach.Final conclusions are given in Section 4.  sensor (or master sensor).Assume that the sensor network works in such a mode: once the voltage level of the data acquisition (DAQ) channel connected to the trigger sensor exceeds a predefined threshold value (usually higher than the noise level), an AE event is regarded to have occurred; all of the channels of the sensor network start to acquire the AE wave signals simultaneously and save them to the computer for further analysis.As illustrated in Figure 1, using the TOF of the AE waves is a straightforward way to triangulate the AE source; however, the absolute time of the initiation of an AE event is usually unknown, only the time differences between the AE wave signals received by the rest of the sensors and that received by the trigger sensor can be measured after the time arrivals in each channel are extracted.Theoretically, the time difference between the th sensor and the trigger sensor (assume it is the th sensor) Δ   can be calculated as

Bayesian Approach for AE Source Localization
where (  ,   ), (  ,   ), and (  ,   ) are the coordinates of the AE source location and the th and th sensors, respectively.  is the wave velocity of the AE waves at a specific frequency.Usually, in plate-like structures, the AE waves contain symmetric and antisymmetric modes of guided waves governed by Rayleigh-Lamb equation [27], and   is dependent of the selected frequency due to the dispersive nature of guide waves, resulting in Δ   becoming dependent of frequency as well.In the present study, only isotropic plate is considered;   is independent of propagation direction.Since uncertainties from material properties could influence the accuracy of the theoretical wave velocity value, in the present study, the wave velocity   in (1) is set as an additional unknown parameter to be identified besides the coordinates of the AE source location, rather than calculated using the nominal material property values beforehand by solving Rayleigh-Lamb equation or by using approximate plate theory solution [28].Thus, the unknown parameters that should be identified in our AE source localization approach are the center location of the AE source (  ,   ) and the wave velocity   at a specific frequency.They can be written in a vector style as  = [  ,   ,   ]  .
For localizing the AE source using TOF-based method, the time arrivals of the AE wave signals in each channel at specific frequencies should be obtained first.As aforementioned in the introduction, CWT has good resolution in both time and frequency domains; it is popular in analyzing dispersive guided waves.In the present study, a complex Morlet wavelet is employed to determine the time arrivals of the AE wave signals in the interested frequencies.The details of CWT of complex Morlet wavelet can be found in [11,20] and are not repeated here for conciseness.Theoretical and experimental results have demonstrated that the first peak of the magnitude of CWT coefficients represents the arrival time of the dispersive wave with the group velocity at local frequencies [10,11,[18][19][20].This important property is used in the present study to extract the time arrivals of the AE waves in each channel.After that, the measured time difference between the th sensor and the trigger sensor Δ   can be measured simply as in which    and    are the measured time arrivals of the th sensor and the trigger sensor (the th sensor), respectively.

Bayesian Approach for Parameter Identification.
After the unknown parameters are defined, the AE source localization is formulated as a parameter identification problem with the measured time difference data.In the present study, it is solved from a probabilistic point of view by employing Bayesian identification.Rather than pinpointing a single solution by deterministic approaches, the Bayesian approach can provide probabilistic distributions of the unknown parameters, giving both point and interval estimates [29][30][31][32][33][34].The basic idea of Bayesian identification is that it treats the parameters, currently denoted by a vector , as random variables with joint distribution ().It aims to update the posterior distributions of the uncertain parameters for a given set of measured data.The final parameter estimates can be taken as the mean values of the posteriors or use the values that maximize the posteriors.
Assume that the uncertainties from modeling error and measurement noise can be described by a Gaussian type random variable ; the probabilistic description of the measured time difference between the th sensor and the triggersensor can be expressed as where Δ   is the theoretically calculated time difference for the th sensor using (1), with the coordinates of AE source location and the wave velocity contained in the parameter vector ,  is usually assumed as a variable with mean of zero and variance of  2  , and assumption is made that the uncertainties from different sensors have the same variance and they are uncorrelated.In this case, the likelihood function can be written as The likelihood function (D | ) is a probabilistic statement about the distribution of the measured time difference data given the predicted time difference values, determined by (1) and parameter vector .
Usually, the variance  2  is unknown, and a conjugate Gamma prior with parameters  and  is adopted for the precision  = 1/ 2  in the present study [24].With this choice, the variance can be analytically integrated out from (4), leading to a new likelihood expression as where Γ() = ∫ +∞ 0  −1  −  is the Gamma function.This new likelihood function instead of ( 4) is used in the following calculation.
By using Bayes' theorem, the posterior distribution of parameter vector  for a given set of measured time difference data D is constructed by relating the prior and the likelihood function as where ( | D) is the posterior distribution of ,   () is the prior distribution of , and (D) is a normalizing constant that ensure the integration of the distribution over the predefined domain is equal to unity.For each parameter   ( = 1, 2, 3), the marginal posterior distribution can be obtained by integrating (6) with respect to the rest parameters over the domain of interest as where the notation ∫  − denotes the multidimensional integral over all of the rest parameters other than   in .

Markov Chain Monte Carlo
Method.Equation (7) gives a general expression of the updated marginal distribution for each parameter in  using measured time difference data.
After that, the parameters can be estimated from the posterior distributions, and the AE source location and the wave velocity can be identified.However, analytical evaluation of ( 7) is usually difficult since it involves integration over multidimensional parameters.In the present study, MCMC method is resorted to obtain samples to approximate the posterior distributions and make inference.The pioneer work of MCMC was done by Metropolis and his colleagues in the 1950s.They recognized that Markov chains could be applied to some difficult sampling problems.After about two decades, Hastings expanded and formalized MCMC into the procedure now known as Metropolis-Hastings (M-H) algorithm for performing such sampling [35].Let () be the target distribution from which random samples are desired.The M-H algorithm generates a sequence of   for the target distribution () by a two-step rejection sampling procedure.At stage , a candidate value  * is sampled based on the current value  −1 ; it is sampled from a chosen proposal or transition distribution function ℎ( * |  −1 ).A Bernoulli trial is performed with success probability If the result of the trial is success,   is set to  * ; otherwise   is kept to  * .This process is repeated for a sufficient number of iterations, until the resulting Markov chain is stationary and can be used to represent the target distribution.Usually, when starting from an arbitrary state, there is a nonstationary period of iterations before the chain gets converged and stationary.It is called "burn-in" period and should be discarded in the final approximation of the posterior distribution.The proposal distribution could be an arbitrary distribution specified by the user, but proper ones can lead to better convergence in the resulting Markov chain.Often symmetrical proposal distribution, which means ℎ( * |  −1 ) is equal to ℎ( −1 | * ), is used to simplify the acceptance probability.
In the present study, a modified M-H algorithm is employed to sample the posterior distributions of the unknown parameters in  defined in Section 2.1.It is similar to the algorithm developed in [26,33].This algorithm combines Gibbs sampling concept that the full conditional distribution of each parameter can be thought of as its posterior distribution if other parameters' values are known.It updates each parameter sequentially by using the most recent sampled values.Uniform transition distribution is employed to propose candidate value via a random walk scheme.For each parameter, the interval of random walk has been continuously tuned during the "burn-in" period to achieve an appropriate acceptance rate and improve the performance of MCMC with scaling factors suggested in [33].Unlike the algorithm in [33], since the variance is already integrated out from the likelihood function, sampling of the parameters is independent of the variance; it is not necessary to sample the variance within the algorithm.

Data Fusion at Multiple
Frequencies.One of the advantages of extracting TOF information by CWT in timefrequency domain over other traditional methods is that we can obtain TOF data at multiple frequencies.All of these data at different frequencies can be used to localize the AE source individually with the proposed Bayesian approach.A proper data fusion scheme can be employed to increase accuracy and reduce uncertainty of the AE source locations.As suggested by Su et al. [36], usually three fusion architectures, that is, independent, centralized, and decentralized, can be chosen to perform data fusion.In the present study, an independent architecture is applied to fuse the results; namely, individually obtained AE source locations by time difference data at multiple frequencies are fused to get the final results.
After the MCMC algorithm obtains samples from the posterior distributions, it is reasonable to use Gaussian distribution to approximate these posteriors.For plate-like structures, an AE source location can be represented by a two-dimensional Gaussian distribution in a canonical form as where  = [    ]  and Σ = [ ] are the mean and covariance matrix of the distribution, respectively;   and   are means of the and -coordinates of the identified AE source location;   and   are standard deviations of the and -coordinates;  is the correlation coefficient, respectively.Since individual localization results with Gaussian distributions can be obtained by time difference data at multiple frequencies, a fusion method is required to determine the combined parameters from those of individual distributions.In the present study, formulations for merging Gaussian distributions in [37,38] are adopted.For Gaussian distributions, the mean, standard deviations, and orientation of the major axis are independent of scaling; they can be extracted from the resulting merged covariance matrices without considering absolute probability values [38].Take two Gaussian distributions as an example; the covariance matrix of the resulting merged distribution can be combined by individual covariance matrices as where Σ * is the covariance matrix of the merged distribution and Σ 1 and Σ 2 are the covariance matrices of the two individual distributions.The mean of the resulting merged distribution is computed from the means and covariance matrices of the individual distributions as where  * is the mean of the merged distribution and  1 and  2 are the means of the two individual distributions.More Gaussian distributions can be merged one by one with the above formulations.In the present study, six sensors are sufficient to collect time difference data to identify the three unknown parameters; however, a later study on the fail-safe distribution of sensors could help to increase the reliability of the monitoring system [39].An illustration of the stiffened panel and sensor placement is shown in Figure 2. The origin of the coordinate system is set at the center of the panel, and the coordinates of the functioning sensors are listed in Table 1.In the experiments, pensile lead breaks (PLBs) are used to simulate sources of  AE events.Five PLBs at different locations are performed.These simulated AE sources are labeled in Figure 2 and their coordinates are listed in Table 2.During the experiments,  4 is selected as the trigger sensor for the sensor network, and the threshold voltage value for triggering  4 is set as 0.1 V.After  4 is triggered, the response wave signals are received by the sensors and acquired by the digitizer, whose sampling rate is set at 10 MHz.   3.

Results for Single-Frequency Data.
After the time difference data is obtained, MCMC procedure is then performed to obtain samples for the unknown parameters to approximate their posterior distributions.For the initial values, the -coordinate of the AE source location is uniformly sampled from [−150, 150] mm, the coordinate of the AE source location is uniformly sampled from [−200, 200] mm, and the wave velocity at 20 kHz are uniformly sampled from [0, 3000] m/s.Figures 5(a) and 5(b) show the samples of x-y coordinates of the AE source location by MCMC, and Figure 5(c) shows the simultaneously sampled wave velocity of the AE wave at 20 kHz for the simulated AE event PLB 1 .For each parameter, 100000 samples in total are obtained and the first 40000 samples are set as the "burn-in" period.Gaussian distributions are used to fit these samples.Figure 6 illustrates the fitted joint distribution of - coordinates of the AE source location in 2D contour view for the simulated AE event PLB 1 .The actual PLB 1 source location is also labeled in Figure 6 for comparison.The means of the fitted distributions of the unknown parameters are output as their estimates.For PLB 1 , the identified AE source location is (0.4, −2.0) mm; the distance between the identified AE source location and the actual one is less than 3.0 mm with a relative error of 1.4% with respect to the minimum sensor spacing (150 mm), demonstrating the effectiveness of the proposed Bayesian AE source localization approach.For the wave velocity, the identified value at 20 kHz is 1187.8m/s, which is close to the theoretical value 1200 m/s calculated by freeware "Vallen Dispersion" contained in "AGU-Vallen Wavelet" package provided by the AE company Vallen System [40], with nominal material properties for the stiffened aluminum panel as Young's modulus 73 GPa, Poisson's ratio 0.32, and density 2790 kg/m 3 .It should be noted that, due to the various uncertainties involved, these nominal material properties are not exactly the same as those for this specific stiffened aluminum panel; thus the theoretical wave velocity value is only used for reference.The results also give the standard deviations and correlation coefficient for the AE source location and the wave velocity as listed in Table 1, which can be used for uncertainty analysis.
Similar identification results are obtained for the other four PLB events.Figures 7(a 3. The same MCMC procedure as presented for PLB 1 case is then performed for these four PLB cases.The identification results are listed in Table 4 together with those for PLB 1 .Figures 9(a)-9(d) illustrate the fitted joint distributions of x-y coordinates of the identified AE source locations for these four cases, respectively.The actual PLB locations are also labeled in Figure 9 for comparison.Figure 10(a) shows the identified wave velocities compared with the theoretical one at 20 kHz for the five PLB events; error bars with three times of standard deviations are plotted with the mean values.From Table 4 and Figures 6 and  9, it can be seen that the identified mean values of the simulated AE source locations are quite close to the actual ones.In addition to the point estimates, the Bayesian approach also provides interval estimates for uncertainty analysis.As aforementioned, one appealing property of CWT for time-frequency analysis of AE wave signals is that it can obtain TOF information of the AE waves at different frequencies without any difficulties.To further demonstrate the effectiveness of the proposed AE source localization approach, we take time difference data at other two frequencies, that is, 30 kHz and 40 kHz, to localize the AE sources.The CWT measured time differences of AE waves at 30 kHz and 40 kHz in these five PLB cases are listed in Tables 5  and 6, respectively.Tables 7 and 8 present the identified AE    9. From Figure 12 and Table 9, it can be seen that, compared with the individually obtained localization results, for most of the cases, the fused results have mean values closer to the actual ones and narrower standard deviations, demonstrating that the data fusion scheme indeed can increase the accuracy of the AE source localization results and reduce their associated uncertainties.

Figure 1 :
Figure 1: Illustration of AE source localization and sensor network.

1 Figure 2 :
Figure 2: Test configuration and sensor arrangement for the stiffened aluminum panel.

Figure 6 :
Figure 6: Identified source location for simulated AE event PLB 1 by time difference data at 20 kHz.

Figure 3
shows the AE wave signals emitted by the simulated AE event PLB 1 and received by the six sensors.To obtain the time arrivals of these AE wave signals, CWT of complex Morlet wavelet is applied.

Figure 4
shows the magnitudes of CWT coefficients of the AE wave signals at frequency of 20 kHz.From the figure, the time arrivals of the AE wave signals represented by the first peaks can be clearly identified.The corresponding time arrivals for the AE wave signals at 20 kHz are determined and labeled by triangles in Figure 4. Then the time difference data between the rest five sensors and the trigger sensor  4 can be easily calculated by subtraction as listed in Table
)-7(d) show the AE wave signals emitted by the simulated AE events PLB 2 -PLB 5 .Figures 8(a)-8(d) show the magnitudes of CWT coefficients of the AE wave signals at frequency of 20 kHz.The corresponding time arrivals of the AE wave signals at 20 kHz are determined and labeled by triangles in Figure 8.The CWT measured time difference data at 20 kHz for these four cases are listed in Table

Figure 11 :
Figure 11: Merged distribution for and -coordinates of source location for simulated AE event PLB 1 .
Figure 11  shows the merging of three individual distributions obtained by time difference data at three frequencies into one distribution for x-and y-coordinates of the source location for simulated AE event PLB 1 .Figure12(a)illustrates the fused joint distribution of x-y coordinates of the AE source location in 2D contour view corresponding to Figure11.The actual PLB 1 source location is also labeled in Figure12(a) for comparison.Figures12(b)-12(e) illustrate the fused joint distributions of x-y coordinates of the identified AE source locations for the rest of the four cases, respectively.Also these fuse AE source localization results are listed in Table

Table 1 :
Coordinates of the sensors on the stiffened aluminum panel.

Table 2 :
Coordinates of the pensile lead breaks on the stiffened aluminum panel.
The dimensions of the stiffened aluminum panel are 800 mm × 600 mm × 2 mm.A sensor network containing eight PZT transducers, denoted by  1 to  8 , with diameter of 10 mm and thickness of 1 mm are surface mounted on an area of 300 mm × 400 mm between two riveted stiffeners.These sensors are connected to the DAQ channels of the digitizer for monitoring the AE event and collecting AE wave signals.Unfortunately,  1 and  5 malfunction in the following test; only the remaining six sensors are employed for AE source localization.

Table 3 :
CWT measured time difference data at 20 kHz.

Table 4 :
Identified results by CWT measured time difference data at 20 kHz.

Table 5 :
CWT measured time difference data at 30 kHz.

Table 6 :
CWT measured time difference data at 40 kHz.sourcelocations and corresponding wave velocities at 30 kHz and 40 kHz, respectively.Figures10(b) and 10(c) show the identified wave velocities at 30 kHz and 40 kHz compared with the theoretical ones, respectively.3.2.2.Fused Results for Multiple-Frequency Data.After AE source localization results from time difference data at different frequencies, that is, 20 kHz, 30 kHz, and 40 kHz in the present study, are obtained and fitted by two-dimensional Gaussian distributions; the proposed data fusion scheme described in Section 2.4 can be applied to get the final fused results.