# Microsseismic Source Location Method and Application Based on NM-PSO Algorithm
## 2 Time Difference of Arrival (TDOA) Source Location Theory
\[D_{i}=\sqrt{\left(x_{i}-x_{0}\right)^{2}+\left(y_{i}-y_{0}\right)^{2}+\left(z _{i}-z_{0}\right)^{2}} \tag{4}\]  
The actual travel time \(t_{a}^{i}\) from the source \(S\) to the \(i\)-th sensor as follows:  
\[t_{i}^{a}=t_{i}-t_{0} \tag{5}\]  
Then the residua \(\gamma_{i}\) between the actual travel time from the source \(S\) to the \(i\)-th sensor and the calculated travel time is:  
\[\gamma_{i}=t_{i}^{a}-t_{i}^{c}=t_{i}-t_{0}-t_{i}^{c} \tag{6}\]  
After the microseismic event occurs, the microseismic wave mainly propagates to the outside world as two spherical waves, P-wave and S-wave. Because P-wave is a longitudinal wave and S-wave is a shear wave, the wave velocity of the P-wave is greater than that of the S-wave. In the research of microseismic underground, S-wave will arrive at the sensor later than P-wave; however, due to the small monitoring range, S-wave is often superimposed at the tail of the P-wave, and it is not easy to separate P-wave and S-wave. Because P-wave arrives first, it is relatively easy to pick up at the first arrival. Therefore,  
Figure 2: Principle of TDOA source location. when a microseismic location is performed underground, the first arrival of the P-wave is used to participate in the source location calculation in most cases. Considering the applicability of microseismic location and the simplified calculation, when microseismic location is performed underground, it is generally assumed that the coal and rock mass is a homogeneous and isotropic medium, and the wave velocity of the vibration wave adopts a fixed value during the propagation process. The residual between the actual travel time and the calculated travel time is as follows:  
\[\gamma_{i}=t_{i}-t_{0}-\frac{\sqrt{\left(x_{i}-x_{0}\right)^{2}+\left(y_{i}-y_ {0}\right)^{2}+\left(z_{i}-z_{0}\right)^{2}}}{v_{p}} \tag{7}\]  
where \(x_{0}\), \(y_{0}\), \(z_{0}\), and \(t_{0}\) represent the microseismic source parameters, which are unknown parameters to be determined; \(x_{i}\), \(y_{i}\), \(z_{i}\), and t\({}_{i}\) represent the three-dimensional coordinates of the \(i\)-th sensor involved in the calculation and the first arrival of the P-wave, respectively, belonging to known parameters; \(i\) = 1, 2, 3,..., \(n\), where \(n\) is the number of effective sensors. When \(\gamma_{i}\) = 0, the above equation transforms into:  
\[F(m)=D \tag{1}\]0  
Since there are four unknown parameters to be solved, in order to have a solution to the equation, the number of effective sensors must be \(n\)\(\geq\) 4. When \(n\) = 4, the equation can theoretically solve each unknown parameter exactly. When \(n\) is greater than 4, the equation solution is not unique, and it is generally necessary to use an optimization method to solve the source parameters. Calculate the error sum of squares of the adequate sensor residuals \(\gamma_{i}\) to get  
\[F(m)=D \tag{1}\]1  
By iteratively solving the minimum value of this objective function in the range of the definition domain, the hypocenter parameters can be obtained when \(n\) is greater than 4.