## 4 EKF-Based Approach for AE Source Localization during Pencil Lead Breaks Tests  
In order to validate the performance of the proposed EKF-based AE source localization algorithm, we consider an aluminum plate having the dimensions (\(230\times 250\times 2\)) mm\({}^{3}\). The Young modulus, Poisson ratio and density of the aluminum plate are \(=69\) GPa, \(\
u=0.34\), and \(\rho=2700\) kg/m\({}^{3}\), respectively. To simulate broadband AE sources, Hsu-Nielsen pencil lead breaks (PLB) tests were performed on the specimen at 18 different positions [58]. A square mesh of 10 mm was made on the plate in order to ensure a good repeatability of the different PLB. Four identical broadband (0.1-1 MHz) PZT sensors (model PAC 15a) were used to detect the elastic waves propagating within the plate. Acoustic signals were 40 dB preamplified and then acquired using a data acquisition (DAQ) system (Mistras Micro Express) with 2 MHz sampling frequency. The initial location was calculated based on Equations (A8) and (A9) defined in Appendix A.2. The use of the PLB method to generate elastic waves makes the amplitude of the extensional mode negligible compared to that of the flexural mode. Therefore, the flexural mode A0 was then easily distinguishable and was therefore used to locate the different AE sources. Acquired waveforms were processed following the EKF-based localization procedure as described in Section 3. With the help of the MatlabTM (version 2019b, MathWorks, Natick, MA, USA) code implemented by the authors, waveforms were analyzed based on the wavelet transform in order to find the arrival times and the dominant frequencies at the different sensors' positions. The number of iterations during the EKF stage was fixed at 5 and the complex Morlet parameters were taken as \(f_{c}=0.7\) Hz and \(f_{b}=1.2\)\(s^{2}\). \(f_{b}\) and \(f_{c}\) values take into account the broadband pulse excitation and the frequency bandwidth of the AE sensors which is from \(\sim\)100 kHz to \(\sim\)1 MHz. Their values were fixed by performing several localization tests. On the basis of the time of arrival determined with the help of the CWT and the propagation distance, the calculated group velocities were found to be in agreement with the ones given by the dispersion curves. The EKF-based localization procedure is then applied to iteratively estimate the location of the acoustic sources. The peak of the squared modulus of the CWT corresponding to each AE signal was used to indicate the arrival time of the generated guided waves. Figure 4 shows the detected AE signals as well as their time-frequency characteristics when the artificial source position is at \(PLB(15,30)\). For the latter position, for instance, the time-frequency analysis of the different AE signals showed that the frequency for which the energy is maximum is between 227.6 kHz and 298.3 kHz for all the sensors. On the other hand, Figure 5 shows the line profile of the abovementioned signals at the average frequency \(\overline{f}=282.63\) kHz, where the time delay is taken at the maximum energy of each AE hit clearly appears. Besides, for each AE event, the group velocity was calculated with the help of the Monte Carlo simulation based on the randomly generated 150,000 samples of the variable \(\overline{fd}\) as shown in Figure 6. Note that the starting values \(\widehat{V}_{g_{0}}\) may not exactly coincide with the values that would be found in the dispersion curve of the group velocity. However, the algorithm based on EKF allows the convergence between the initial values \((\hat{x}_{g_{0}},\ \hat{y}_{g_{0}})\) and \(\widehat{V}_{g_{0}}\) as well as the differences in time of arrival of the AE signals detected by the different sensors. The injection of the variance is used to approach the true value. The interest of applying such an approach appears when the properties of the material are not known in a precise way or when the material is subjected to one or more constraints in which case the velocity of the propagating wave changes according to the state of stress.