# A Review of Acoustic Emission Source Localization Techniques in Different Dimensions
## 4 Acoustic Emission Source Localization Techniques for Three-Dimensional (3D) Structures
### Extend the Two-Dimensional Source Localization  
The most classic two-dimensional positioning method is the time difference method. The time difference localization method in three-dimensional structures first assumes that the sound velocity is known and constant. By calculating the distance difference  
Figure 9: Method for 3D sound source localization. between the sound source and each sensor, the spatial coordinates of the sound source are determined (Figure 10). Four sensors are located in the same plane (z = 0), and S is the position of the sound source. Assuming the coordinates of \(T_{2}\) are (0, 0, 0), \(T_{0}\) is (x0, y0, z0), \(T_{1}\) is (x1, y1, z1), \(T_{3}\) is (x3, y3, z3) and S is (x, y, z), the distance difference is:  
\[ \begin{array}{l}|ST_{0}|-|ST_{2}|=d_{02}\\ |ST_{1}|-|ST_{2}|=d_{12}\\ |ST_{3}|-|ST_{2}|=d_{32}\end{array} \]  
Substitute the coordinates into the equation to obtain the system of Equation (1)  
\[ \begin{array}{l}\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+ \left(z-z_{0}\right)^{2}}-\sqrt{x^{2}+y^{2}+z^{2}}=d_{02}\\ \sqrt{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}+\left(z-z_{1}\right) ^{2}}-\sqrt{x^{2}+y^{2}+z^{2}}=d_{12}\\ \sqrt{\left(x-x_{3}\right)^{2}+\left(y-y_{3}\right)^{2}+\left(z-z_{3}\right) ^{2}}-\sqrt{x^{2}+y^{2}+z^{2}}=d_{32}\end{array} \]  
Simplify to obtain Equation (2)  
\[\begin{array}{l}2(x_{0}x+y_{0}y+z_{0}z)+2d_{02}\sqrt{x^{2}+y^{2}+z^{2}}=x_{0 }{}^{2}+y_{0}{}^{2}+z_{0}{}^{2}-d_{02}{}^{2}\\ 2(x_{1}x+y_{1}y+z_{1}z)+2d_{12  
Substitute the initial condition z_0 = z1 = z_3 = 0 to obtain the coordinates of the sound source  
\[ \begin{array}{l}x=\frac{(d_{0}-c_{0}d_{1})(y_{0}-c_{0}y_{3})-(d_{0}-c_{0}c_{0}d _{3})(y_{0}-c_{0}c_{0}y_{1})}{(x_{0}-c_{0}c_{1}y_{1})(y_{0}-c_{0}c_{0}y_{3})-(x_ {0}-c_{0}c_{0}x_{3})(y_{0}-c_{0}c_{1}y_{1})}\\ y=\frac{(d_{0}-c_{0}d_{1})(x_{0}-c_{0}x_{3})-(d_{0}-c_{0}d_{3})(x_{0}-c_{0}c_{1} x_{1})}{(y_{0}-c_{0}c_{1}y_{1})(x_{0}-c_{0}c_{0}x_{3})-(y_{0}-c_{0}c_{0}x_{3})(x_ {0}-c_{0}c_{1}x_{1})}\\ z=\sqrt{\left[\frac{d_{0}-(x_{0}+y_{0}y)}{d_{0}}\right]^{2}-(x^{2}+y^{2})} \end{array} \]  
[PERSON] et al. [68] used multiple sensors to localize the damage location in reinforced concrete beams was based on the time difference method, where the time difference recorded by multiple sensors was used to localize the damage location (Figure 11). Subsequently, an improved AIC method was proposed by [PERSON] et al. [69]. The improved AIC (Akaike Information Criterion) method was successfully applied to AE detection during a three-point bending test of a fiber-reinforced concrete beam to analyze the crack pattern. Experiments have shown that the deviations in the results obtained by the improved method ranged from 3% to 4% from the correct results (Figure 12). Figure 11: In situ retrofitted RC beam test. (**a**) Scheme of the beam cross-section. (**b**) Photo of a flexural crack in between transducers 3 and 4 (using an optical microscopy with magnification 100\(\times\)). (**c**) Scheme of the beam indicating localized AE sources. (**d**) AE counting number for each sensor Si during the loading test. (**e**) The b-value during the loading test. Figure 12: 3D structural sound source localization based on AIC. [PERSON], [PERSON] et al. [70] proposed a new method for locating artificial acoustic emission sources (named the H-N source [71]) based on the time difference method, called \"Delta T\" source location. This method does not require knowledge of the sensor location or wave spread. Compared with traditional TOA source location methods, this method had significant improvements, reducing the positioning error from 4.81% using traditional time of arrival (TOA) technology to 1.77% using \"Delta T\" location. [PERSON] et al. [72] localized acoustic sources in cylindrical coal samples in this manner. [PERSON] and [PERSON] [73] simplified the source localization in three-dimensional bodies by placing sensors at selected corner points of a cube and then deriving an analytical expression for source localization (Figure 13). [PERSON] et al.