## 3 Principle of the AE Source Location Technique  
### Time Difference of Arrival (TDOA)  
The AE source location technique employed in this study has been documented by [PERSON] et al. [10], [PERSON] et al. [11], and [PERSON] et al. [12] in detail. It is analogous to the back analysis of an earthquake epicenter. Based on the principle of TDOA, the location and the generation time of the AE source can be computed. According to [PERSON] [13], the travel distance between the AE source and the \(i\)-th sensor can be obtained by Equation (1) given below:  
\[d_{i}=(t_{i}-t)\times v=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{ 2}+\left(z-z_{i}\right)^{2}} \tag{1}\]  
where \(d_{i}\) is the distance between AE signal source and the \(i\)-th sensor;  
\(t\) is the generation time;  
\(t_{i}\) is detected arrival time at the \(i\)-th AE sensor;  
\(v\) is the AE wave velocity;  
\((x,y,\text{and }z)\) are the location of AE source;  
\((x_{i},y_{i},\text{ and }z_{i})\) are the location of \(i\)-th AE sensor. If one AE source is obtained by four AE sensors (e.g., No. S1 to No. S4), Equation (1) can be rewritten as Equation (2) given below:  
\[\left\{ \begin{array}{l}f_{1}=\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{ 2}+\left(z-z_{1}\right)^{2}-\left(t-t_{1}\right)^{2}v^{2}=0\\ f_{2}=\left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}+\left(z-z_{2}\right)^{ 2}-\left(t-t_{2}\right)^{2}v^{2}=0\\ f_{3}=\left(x-x_{3}\right)^{2}+\left(y-y_{3}\right)^{2}+\left(z-z_{3}\right)^{ 2}-\left(t-t_{3}\right)^{2}v^{2}=0\\ f_{4}=\left(x-x_{4}\right)^{2}+\left(y-y_{4}\right)^{2}+\left(z-z_{4}\right)^{ 2}-\left(t-t_{4}\right)^{2}v^{2}=0\end{array} \right\}\right. \tag{2}\]  
The location of the sensors can be obtained from Table 1, and the wave velocity is assumed to be a constant value for a fully saturated sand specimen, as suggested by [PERSON] et al. [11]. To verify the accuracy, a steel metal specimen was used in this study because of its lower attenuation property. Thus, the unknown generation time (\(t\)) and the location of the AE source (\(x\), \(y\), and \(z\)) can be calculated by solving Equation (2). Theoretically, it is possible to locate an AE source by using four AE sensors. However, in practice, more sensors are used to reduce the effects of potential errors, such as those of the positions of the attached sensors. Regarding the detected arrival time, an automatic arrival time determination algorithm was developed by [PERSON] et al. [10] based on the autoregressive-Akaike information criteria (AR-AIC, Akaike [14]). Further details of this algorithm are presented in [PERSON] et al. [10] and [PERSON] et al. [11]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tests** & **Location** & \(x_{0}\)**/m** & \(y_{0}\)**/m** & \(z_{0}\)**/m** & \(d_{0}\)**/m** \\ \hline \multirow{6}{*}{PLB} & A & \(0.000\) & \(0.025\) & \(0.080\) & \(0.084\) \\  & B & \(0.000\) & \(0.025\) & \(0.050\) & \(0.056\) \\  & C & \(0.000\) & \(0.025\) & \(0.020\) & \(0.032\) \\ \cline{2-6}  & A & \(0.000\) & \(-0.025\) & \(0.080\) & \(0.084\) \\  & B & \(0.000\) & \(-0.025\) & \(0.050\) & \(0.056\) \\  & C & \(0.000\) & \(-0.025\) & \(0.020\) & \(0.032\) \\ \cline{2-6}  & A & \(-0.025\) & \(0.000\) & \(0.080\) & \(0.084\) \\  & B & \(-0.025\) & \(0.000\) & \(0.050\) & \(0.056\) \\  & C & \(-0.025\) & \(0.000\) & \(0.020\) & \(0.032\) \\ \cline{2-6}  & A & \(0.025\) & \(0.000\) & \(0.080\) & \(0.084\) \\  & B & \(0.025\) & \(0.000\) & \(0.050\) & \(0.056\) \\  & C & \(0.025\) & \(0.000\) & \(0.020\) & \(0.032\) \\ \hline Noise check & Front & B & \(0.000\) & \(0.025\) & \(0.050\) & \(0.056\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The location of all pencil lead break sources. Four channels were selected randomly to solve Equation (2) by the Newton-Raphson method [15].