# GA-PSO Algorithm for Microseismic Source Location
## 2 Materials and Methods  
### Construction of the Objective Function  
Since the P-wave is easier to capture and recognize during microseismic signal propagation, it is typically used for detection and location. Given that P-waves are affected by rock noise and other interferences during propagation, it is typically assumed that P-waves maintain a constant wave speed [28].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Category** & **Algorithm** & **Key Innovation** & **Strengths** & **Limitations** & **Application** \\ \hline \multirow{4}{*}{Non-Heuristic} & \multirow{2}{*}{Geiger [1]} & Iterative time & Theoretical & Initial value and & Seismic detection \\  & & residual & simplicity, & velocity model & in mining \\  & & minimization & high efficiency & sensitivity & \\ \cline{2-6}  & \multirow{2}{*}{APS [4]} & Hypersphere search, & Parameter-free, & High-dimensional & Small/medium \\  & & adaptive threshold & efficient local & performance drop & sensor networks \\  & & & search & & \\ \hline \multirow{4}{*}{Heuristic (Single)} & GA [15] & Crossover and & Strong global & Slow convergence & Function \\  & & mutation & search capability & speed & optimization \\ \cline{2-6}  & \multirow{2}{*}{PSO [16]} & Velocity–position & Simple to & Parameter sensitivity, & Real-time \\  & & update mechanism & implement, & low diversity & localization \\  & & & fast convergence & & \\ \hline \multirow{4}{*}{Heuristic (Hybrid)} & \multirow{2}{*}{NM-PSO [18]} & NM local refinement, & Robustness to & Sensitive to initial & Small-scale \\  & & PSO global search & initial values & conditions & networks \\ \cline{2-6}  & \multirow{2}{*}{PSO-ACO [19]} & Pheromone-guided & Strong multimodal & High computational & Path and network \\ \cline{1-1}  & & particle trajectories & problem-solving & complexity & optimization \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of existing microseismic location methods.  
The average propagation velocity \(V_{p}\), for each geophone signal arriving at the seismic source \((x_{0},y_{0},z_{0})\) is defined, and the travel time equation is established as follows:  
\[(t_{i}-t_{0})V_{p}=\sqrt{\left(x_{i}-x_{0}\right)^{2}+\left(y_{i}-y_{0}\right)^{ 2}+\left(z_{i}-z_{0}\right)^{2}} \tag{1}\]  
where \((x_{i},y_{i},z_{i})\) and \(\left(x_{j},y_{j},z_{j}\right)\) represent the coordinates of the geophones, \(t_{i}\) is the arrival time of the P-wave, and \(t_{0}\) is the origin time of the microseismic event. Given that the wave speeds are identical, the velocity difference is expressed as follows:  
\[\left\{\begin{array}{l}V_{pi}=\sqrt{\left(x_{i}-x_{0}\right)^{2}+\left(y_{i }-y_{0}\right)^{2}+\left(z_{i}-z_{0}\right)^{2}}/\left(t_{i}-t_{0}\right)\\ V_{pj}=\sqrt{\left(x_{j}-x_{0}\right)^{2}+\left(y_{j}-y_{0}\right)^{2}+\left( z_{j}-z_{0}\right)^{2}}/\left(t_{j}-t_{0}\right)\end{array}\right. \tag{2}\]  
Since the wave speeds are the same, the velocity difference is  
\[\Delta_{\mathrm{ij}}=\left|V_{pi}-V_{pj}\right| \tag{3}\]  
Assuming there are \(N\) geophones in total, the objective function is formulated as  
\[f=\sum_{i=1}^{N_{i}j=i+1}\Delta_{ij} \tag{4}\]  
The objective function requires at least two geophones to complete the location process. When the objective function approaches or equals zero, the corresponding seismic source coordinates \((x_{0},y_{0},z_{0})\) are closest to the true source, leading to more accurate location results.