## 3 Source Localization by Means of Acoustic Emission  
Acoustic emission is often used in real-time nondestructive testing applications to monitor structures. Collected data can also be used to locate events at the origin of the detected AE hits. To locate an AE source, the triangulation method is often applied to signals detected by the different sensors at multiple reception points. This method revealed to be efficient when the wave group velocity \(\left(V_{g}\right)\) and the arrival time \(\left(t_{i}\right)\) of the AE hits are known. However, one should keep in mind that errors and uncertainties are inevitable in the process of measuring and identifying the location of AE sources [12; 26]. Figure 1: Flowchart of the proposed approach. Figure 2: Group velocity dispersion curves for Lamb waves propagating in an aluminum plate. The characteristics of the plate are: Young modulus E = 69 GPa, Poisson ratio \(\
u=0.34\) and density \(\rho=2900\) kg/m\({}^{3}\). Figure 3 illustrates a simple model to locate AE sources. In order to minimize errors related to the localization of AE sources, two main steps are performed. First, the continuous wavelet transform (CWT) is applied to determine the TOA of the AE signals detected by each sensor. Then, the EKF is used in order to iteratively estimate the location of the AE source. Errors related to the localization of the AE sources will be presented and discussed in the paper. Assuming a Cartesian coordinate system, where \((x_{s},\ y_{s})\) are the unknown coordinates of the AE source, \((x_{i},\ y_{i})\) are the known coordinates of the sensors and \(t_{i}\) is the arrival time of the elastic wave from the AE source to the \(i^{\text{th}}\) sensor, as shown in Figure 3. The distance \((d_{i})\) between the \(i^{\text{th}}\) sensor and the AE source can be calculated by multiplying the TOA and the propagation velocity of the involved elastic wave:  
\[d_{i}=\sqrt{\left(x_{i}-x_{s}\right)^{2}+\left(y_{i}-y_{s}\right)^{2}}=V_{g_{ i}}.t_{i} \tag{1}\]  
AE source location is then performed by calculating the differences between the TOA detected by the different sensors:  
\[(x_{i}-x_{s})^{2}+(y_{i}-y_{s})^{2}=\big{(}\ V_{g}\cdot t_{i}\big{)}^{2}= \big{[}(t_{m}+\Delta t_{mi})\cdot V_{g}\big{]}^{2} \tag{2}\]  
By combining Equations (1) and (2) we can determine \(\Delta t_{mi}\), the time difference between the master sensor and the \(i^{\text{th}}\) sensor [13; 37; 55]:  
\[\Delta t_{mi}=\frac{\sqrt{\big{(}x_{i}-x_{s}\big{)}^{2}+\left(y_{i}-y_{s} \right)^{2}}-\sqrt{\left(x_{m}-x_{s}\right)^{2}+\left(y_{m}-y_{s}\right)^{2}}} {V_{g}} \tag{3}\]  
As stated above, the accuracy of the arrival times is essential to well localize the AE sources. An imprecise knowledge of the properties of the propagation medium will have an impact on the estimation of the wave propagation velocity \(\big{(}V_{g}\big{)}\). For that reason, \(V_{g}\) is defined as an unknown parameter to be identified in addition to the coordinates of the AE source \((x_{s},\ y_{s})\). It should also be noticed that, since the propagation of Lamb waves is dispersive, the time difference of arrival (TDOA) corresponding to AE events will consequently depend on the frequency. The single frequency component in the output signal can be extracted, for each sensor, on the basis of a time-frequency analysis with the help of a continuous wavelet transform (CWT), whose effectiveness for studying dispersive waves was evaluated in different contributions such as in reference [56]. In order to take into account, the uncertainties, the unknowns \(\big{(}x_{s},\ y_{s},t_{i},V_{g}\big{)}\) are therefore treated as mutually independent gaussian random variables when applying the probabilistic approach. Based  
Figure 3: Two-dimensional localization principle of AE source.