## 3 Source Localization by Means of Acoustic Emission
### Continuous Wavelet Transform (CWT)  
The CWT of the time signal \(x(t)\) is defined as [13; 37]:  
\[X(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}x(t)\psi^{*}\bigg{(}\frac{t- b}{a}\bigg{)}dt \tag{5}\]  
where \(\psi^{*}(t)\) denotes the complex conjugate of the mother wavelet \(\psi(t)\). \(a\) is the dilation or scale parameter defining the support width of the wavelet and \(b\) is the translation parameter locating the wavelet in the time domain. The CWT is based upon a family of functions defined as:  
\[\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\bigg{(}\frac{t-b}{a}\bigg{)} \tag{6}\]  
The time and frequency resolution of the wavelet transform can be expressed as a function of the scale parameter \(a\) through the following equation [13; 37]:  
\[\sigma_{t}=a\sigma_{t_{g}},\;\sigma_{\omega}=\frac{\sigma_{\omega_{\psi}}}{a} \tag{7}\]  
where \(\sigma_{t_{g}}\) and \(\sigma_{\omega_{\psi}}\) are the duration and bandwidth of the mother wavelet, respectively.  
In literature, various studies based on the use of the wavelet transform have succeeded in locating AE sources in isotropic and anisotropic structures [13; 14; 15; 21; 39]. The present study relies on the use of a complex Morlet wavelet thanks to its ability to separate amplitude and phase information. The complex Morlet wavelet, which can also measure instantaneous frequencies and their time evolution, is defined by the following equation [13; 37]:  
\[\psi(t)=\frac{1}{\sqrt{\pi\,f_{b}}}e^{-\frac{t^{2}}{f_{b}}}[\cos(\omega_{c}t)+ j\sin(\omega_{c}t)] \tag{8}\]  
\(f_{c}=\omega_{c}/2\pi\) is the central frequency. \(f_{b}\) represents the envelope factor and consequently controls the bandwidth of the wavelet spectrum. Values of \(f_{c}\) and \(f_{b}\) are fixed by considering the experimental conditions as it will be explained below [57].  
When using Morlet as a mother wavelet, the function \(\psi_{a,b}(t)\) presented above (see Equation (6)) is determined around \(t=b\) and its Fourier transform is centred around \(\omega=\omega_{c}/\alpha\)[13; 37].  
The squared modulus of the CWT coefficients (scalogram) represents the energy density of a signal in the time-frequency domain. The maximum value of the coefficients of the scalogram is achieved at the dominant frequency. The arrival time of the wave packet is the time which corresponds to the peak magnitude in the time-frequency domain, reached at the dominant frequency [13; 37]. The relationship between the scale parameter \(a\) and the frequency component \(f_{i}\) can be expressed as:  
\[f_{i}=\frac{f_{c}}{a\;T_{e}} \tag{9}\]  
where \(f_{i}\) is the frequency of interest, \(f_{c}\) is the center frequency of the wavelet and \(T_{e}\) is the sampling period. Since the dominant frequency can be different depending on the position of each sensor, an average frequency \(\overline{f}\) is used to calculate the TOA:  
\[\overline{f}=\frac{\sum_{1}^{n}f_{i}}{n} \tag{10}\]Following these considerations, the time and frequency resolutions \(\sigma_{t}\) and \(\sigma_{\omega}\) of the complex Morlet wavelet are given by the following equations [13]:  
\[\sigma_{t}=\frac{f_{c}\sqrt{f_{b}}}{2\overline{f}} \tag{11}\]  
\[\sigma_{\omega}=\frac{\overline{f}}{f_{c}\sqrt{f_{b}}} \tag{12}\]  
Once the TOA is known, the time difference \(\Delta t_{mi}\) can be calculated using Equation (2). By substituting Equation (11) into Equation (3), the variance of the TDOA is given by the following equation [13]:  
\[\sigma_{\Delta t_{mi}}^{2}=2\left(\frac{f_{c}\sqrt{f_{b}}}{2\overline{f}} \right)^{2} \tag{13}\]  
This value is used to model the uncertainty of the measurement vector when applying the probabilistic approach. Hence, based on the algorithm discussed in the next section, the coordinates of the AE source as well as the wave group velocity can be identified.