# Microsseismic Source Location Method and Application Based on NM-PSO Algorithm
## 2 Time Difference of Arrival (TDOA) Source Location Theory  
Microseismic source location is the core of microseismic monitoring, and it is also an important realization goal of microseismic monitoring. Accurate microseismic location results can effectively identify the time and location of microseismic occurrences, accurately determine the fractured coal and rock mass area, and then study the roof fracture mechanism. The positioning principle is shown in Figure 1. Vibration wave produced by underground rock mass fracture [43; 44], and the geophone receives the stress wave. According to the received waveform and arrival time difference, the three-dimensional coordinates of the source are calculated by algorithm. As an inversion problem based on stress wave propagation, microseismic source location can be expressed as:  
\[F(m)=D \tag{1}\]  
where \(D\) is the waveform received by the geophone, the arrival time, the three-dimensional coordinates of the geophone, etc. ; \(F\) is the stress wave propagation function used, and m is the relevant model parameters, including the location of the source, the propagation medium, and the wave speed. When mining activities are carried out underground, due to frequent stress changes, the amount of signals received by the geophone is large, the frequency of microseismic events is high, the location method based on waveform information is expensive, and the amount of data to be processed is cumbersome, which is not conducive to the quick positioning of underground microseismic events. The source localization method based on arrival time difference is currently the leading research direction in rock engineering. The arrival time difference positioning method can be expressed as:  
\[\Delta t=f(r,v,s) \tag{2}\]  
In the formula: \(\Delta t\) is the time difference data between sensors obtained according to the observed waveform; \(f\) is the forward modeling function of stress wave propagation, and the required parameters are the sensor position \(r\), the model velocity information \(v\), and the hypocenter position \(s\). Based on the above formula, the main factors affecting the location of the source can be summarized as follows:  
1. The accuracy of the time difference information \(\Delta t\) between the initial arrival of the stress wave in the observation data;
2. The accuracy of the model parameters such as the velocity model information v and the performance of the related inversion strategy;  
Figure 1: Schematic diagram of microseismic positioning principle. 3. The influence of the sensor location on the completeness of observational information and the multiplicity of inversion results. As shown in Figure 2, the source \(S(x_{0},y_{0},z_{0},t_{0})\) is the unknown parameter that needs to be solved, including the three-dimensional coordinates of the source \(S(x_{0},y_{0},z_{0})\) and the seismic time \(t_{0}\), \(\mathrm{T_{i}}(x_{i},y_{i},z_{i},t_{i})\) is the \(i\)-th sensor in the microseismic monitoring network, Its three-dimensional coordinates \(\mathrm{T_{i}}(x_{i},y_{i},z_{i})\) have been measured in advance, and the first arrival time \(\mathrm{t_{i}}\) of the microseismic wave can be obtained by the first arrival picking method. Then the calculated travel time \(t_{i}^{c}\) from the source \(S\) to the \(i\)-th sensor is:  
\[t_{i}^{c}=\frac{D_{i}}{v_{i}(x_{i},y_{i},z_{i})} \tag{3}\]  
where \(v_{i}(x_{i},y_{i},z_{i})\) is the propagation velocity of the microseismic wave from the source \(S\) to the \(i\)-th sensor, and \(D_{i}\) is the distance from the source \(S\) to the \(i\)-th sensor.