## 2 Improvement of Post-Processing and Localization
### AE Source Localization  
[PERSON]'s method was applied for AE source localization in which the best approximation of the source location is computed iteratively based on a least-squares approach. For a detailed description of the algorithm, the reader is referred to Ge [25]. In this algorithm, the wave velocity is considered to be homogeneous, and a straight propagation path is assumed between the source and the sensor. A trial hypocenter (\(x_{0}\), \(y_{0}\), \(z_{0}\), \(t_{0}\)) is needed as a first guess of the source location. In this research, the middle of the rebar was chosen as a first hypocenter. The arrival times for each sensor can be calculated as:  
\[t_{\text{a,i}}=t_{0}+\frac{1}{v_{\text{p}}}\sqrt{(x_{i}-x_{0})^{2}+(y_{i}-y_{0 })^{2}+(z_{i}-z_{0})^{2}} \tag{4}\]  
with \(x_{0}\), \(y_{0}\) and \(z_{0}\) being the coordinates of an AE source, i.e., trial hypocenter, \(t_{0}\) the onset time of the source, \(x_{i}\), \(y_{i}\) and \(z_{i}\) the coordinates of the \(i\)th sensor, and \(v_{\text{p}}\) the wave velocity. The wave velocity is determined during calibration before the start of a test. As the trial hypocenter is a first guess, the calculated arrival times are different from the observed ones. The arrival time residuals (\(\gamma_{i}\)) can then be calculated for each sensor and should be minimized by applying correction factors (\(\delta x\), \(\delta y\), \(\delta z\), and \(\delta t\)). If the corrections are relatively small, the arrival time function can be linearized and can be approximated as a first-order Taylor expansion:  
\[ \begin{bmatrix}\gamma_{1}\\ \gamma_{2}\\ \vdots\\ \gamma_{m}\end{bmatrix} = \begin{bmatrix}\frac{\partial f_{1}}{\partial x}& \frac{\partial f_{1}}{\partial y}&\frac{\partial f_{1}}{\partial z}&\frac{ \partial f_{1}}{\partial t}\\ \frac{\partial f_{2}}{\partial x}&\frac{\partial f_{2}}{\partial y}&\frac{ \partial f_{2}}{\partial z}&\frac{\partial f_{2}}{\partial t}\\ \vdots&\vdots&\vdots&\vdots\\ \frac{\partial f_{m}}{\partial x}&\frac{\partial f_{m}}{\partial y}&\frac{ \partial f_{m}}{\partial z}&\frac{\partial f_{m}}{\partial t}\\ \end{bmatrix} \cdot \begin{bmatrix}\delta x\\ \delta y\\ \delta z\\ \delta t\end{bmatrix} \]  
or short:  
\[\mathbf{\gamma}=\mathbf{AB} \tag{5}\]  
The Moore-Penrose inverse is applied to determine the correction factors:  
\[\mathbf{B}=(\mathbf{A^{T}A})^{-1}\mathbf{A^{T}\mathbf{\gamma}} \tag{6}\]  
At the beginning of each iteration step, the trial solution is updated by adding the correction factor from the previous iteration step. The process is repeated until two stopping criteria are reached. In this research, the first criterion is based on the size of the correction vector. When this correction vector goes to zero, it is a sign of convergency. In this research, a tolerance of 0.1 mm was chosen. The second stopping criterion is that the maximum number of iterations is set to 100. Error ellipsoids can be used to visualize an estimation of the localization accuracy. Therefore, the covariance matrix can be calculated [29]:  
\[\mathbf{C}=\sigma_{d}^{2}(\mathbf{A^{T}A})^{-1} \tag{7}\]  
where \(\sigma_{d}^{2}\) is the data variance. As only spatial errors are of interest, \(\mathbf{C}\) is a 3 \(\times\) 3 matrix with the variances of the source parameters in direction of the three coordinates on its diagonal. The length of the semi-axis and the orientation of the ellipsoid can be determined from the eigenvalues \(\lambda_{i}\) and the eigenvectors \(v_{i}\) respectively. The length of the semiaxes, \(l_{i}\), for a 68%-ellipsoid is determined as:  
\[l_{i}=\sqrt{3.53\lambda_{i}} \tag{8}\]The data variance \(\sigma_{d}\) is usually unknown. In this research, following expression is used as presented by [PERSON] [32]:  
\[\sigma_{d}^{2}=\frac{\sum\limits_{i=1}^{m}\gamma_{i}^{\mathrm{T}}\gamma_{i}}{m} \tag{9}\]  
with \(m\) the amount of observed arrival times. Signals can be filtered based on a desired accuracy, meaning that a threshold could be set on the error (i.e., radius of the error ellipsoid).