## 5 EKF-Based Approach for AE Source Localization during Quasi-Static Tensile Test
### EKF-Based Analysis of AE Data  
As previously described, the CWT of complex Morlet wavelet is applied to determine the arrival time, dominant frequency, and uncertainties corresponding to all AE events. Figure 11 illustrates the time and frequency characteristics of the AE signals detected by the four AE sensors at the end of the first stage of the crack growth (t = 201.84 s). Time domain signals show that for the same AE event, the AE signals do not have apparently the same characteristics. More details are given by the time-frequency representation of the same signals. The 2D scalogram of the CWT shows that the energy distribution of the same AE event can be different from one sensor to another. The energy density of the AE signal is not a uniformly distributed function as it can be seen on Figure 11. For instance, the AE signal has the maximum energy (the deepest red color) in the frequency range [180-350 kHz]. Under the abovementioned considerations, we can project on the time domain the average dominant frequency component in order to determine the arrival time of the involved Lamb wave. Figure 12 illustrates the applied proced  
Figure 11: Time domain signals and their CWT characteristics corresponding to the same AE event as detected by the four AE sensors at the end of the first stage (t = 201.84 s). Figure 10: Evolution of the cumulated AE energy and the crack length velocity as a function of time. and their average frequency \(\overline{f}=283.3\) kHz. We note that during the mechanical test, the scalogram maxima coefficients resulted slightly different depending on the transducer position. However, the associated frequencies were approximately the same. In order to locate the AE sources, the group velocity of the involved elastic guided waves must be known for each frequency. The frequency corresponding to AE hits emitted during the quasi-static tensile test of the aluminum plate were within [10-500 kHz]. According to the group velocity dispersion curve (see Figure 2), the involved Lamb modes within this frequency domain are A0 and S0. During the tensile test, it remains difficult to know a priori the mode generated by the cracks. The performed analyses showed that, depending on the detected AE hit, some of the sensors received A0 and S0 modes (as well as their reflections) and some other ones received only one mode, namely A0 mode (as well as its reflections). In order to locate the AE source position correctly, it remains important to use the same propagating mode which could be either extensional or flexural [65]. Recorded data revealed that the flexural mode A0 is always detected by all the sensors and contains at the same time the maximum energy of the detected AE hits. Depending on the involved frequency, group velocities corresponding to the detected A0 modes were between \(\sim\)2200 m/s and \(\sim\)3400 m/s. Here, it is important to note that under the multimodal condition, the EKF-based method should be adapted before carrying out the localization of the AE sources. The adapted EKF-based method is presented in the appendix. Once the group velocity and the time difference of arrival (TDOA) between the sensors and the master sensor are obtained, the adapted EKF-based method is applied to estimate the location and the propagation of the AE sources. Figure 13 shows the located AE sources using the proposed EKF-based method and the AE system. Both results show that the macro-crack is located inside the domain defined by the four sensors starting from the notch. However, when we superimpose these results to the crack path (obtained according to Figure 9c) we can see that the results obtained with the EKF method are closer to the crack than those obtained with the AE system. This is mainly related to the monitoring of the evolution of the group velocity (\(V_{g}\)), whose value changes during loading. The AE system does not update \(V_{g}\) and keeps the initial value to locate the different AE sources.