# An Algorithm of Acoustic Emission Location for Complex Composite Structure
## 2 Algorithms  
### Location Algorithm  
As a kind of elastic wave, AE has the same wave equation as seismic wave. However, AE is an ultrasonic wave that is much higher in frequency than seismic wave. The first arrival phase is usually used to locate AE source since the first arrival phase is easy to identify while other seismic phases are often polluted by earlier seismic phases. For isotropic and homogeneous materials with known wave velocity, the equation for the location using the first arrival time can be written as:  
\[\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}+\left(z-z_{i}\right)^ {2}}=V(t_{i}-t) \tag{1}\]  
where \((x_{i}\),\(y_{i}\),\(z_{i})\) is the 3D coordinate of the \(i\)th sensor, \(t_{i}\) is the first arrival time detected by the ith sensor, \((x\),\(y\),\(z)\) is the 3D coordinate of the source, \(V\) is the wave velocity, and \(t\) is the onset of the source.  
Equation (1) has four unknowns. Therefore, at least four sensors are required to achieve the 3D location of the source.  
For 3D wave velocity structure, Equation (1) can be rewritten as Equation (2) when the travel time of each node in the structure is known.  
\[T(x_{i},\ y_{i},\ z_{i},x,y,z)=t_{i}-t \tag{2}\]  
where, \(T(x_{i},\ y_{i},\ z_{i},x,y,z)\) is the travel time from \((x_{i},\ y_{i},\ z_{i})\) to \((x,y,z)\).  
Equation (2) still has 4 unknowns and requires at least 4 arrival times to calculate the source location. However, the location error will be large when only four equations are used because of the uncertainties in arrival time and wave velocity. Thus, more than four sensors are usually used to obtain more travel time data and build overdetermined equations. By doing this, the optimal solution of the overdetermined equations can be solved with the least square method and subsequently the location error is reduced. The least square method is the optimal method when the errors of the arrival time obey the normal distribution. In addition, the location accuracy will increase with an increasing amount of arrival times. However, the least square method will produce large location errors under the existence of individual data (outliers) with large errors.  
The time-reversal imaging method is another location algorithm commonly used in seismology [37; 38; 39]. The method assumes that the rays emit from the detectors to the source in the reverse time axis. In this case, the common intersection of more than four rays in the reverse time axis can be determined as the location and onset of the source. It is not necessary to determine which rays belong to the same event by using the method. Instead, the method focuses on how many rays intersect at one point in the reverse time axis, which means that individual outliers have little impact on the location. As a result, the time-reversal imaging method is more robust. Moreover, not only the arrival times but also the waveforms can be used in the method. Thus, the method is used in this paper to locate AE sources. However, this does not mean that other location algorithms cannot be used. In fact, all the seismic location algorithms based on travel time can be used to locate the sources of 3D complex structures after the 3D travel time table is obtained. Therefore, the key to AE location in 3D complex structure is how to obtain the 3D travel time table. In this paper, we focus on the algorithm of 3D travel time.