GA-PSO Algorithm for Microseismic Source Location

Abstract

Accurate source location is a critical component of microseismic monitoring and early warning systems. To improve the accuracy of microseismic source location, this manuscript proposes a GA-PSO algorithm that combines the Genetic Algorithm (GA) with Particle Swarm Optimization (PSO). The GA-PSO algorithm enhances the PSO algorithm by dynamically adjusting the balance between global exploration and local exploitation through a sinusoidal function for the nonlinear adjustment of both learning factors, and an adaptive inertia weight that decreases quadratically with iterations. Additionally, the precision of the solutions is further improved through the crossover and mutation operations of the GA. In the simulated location model, the GA-PSO algorithm demonstrated the smallest error value, outperforming both the GA and PSO algorithm in terms of accuracy. Furthermore, the GA-PSO algorithm exhibited minimal sensitivity to wave speed fluctuations of ±1%, ±3%, and ±5%, maintaining the error within 0.5 m. The validation through the blasting experiment at the Shizhuyuan mine further confirmed the enhanced accuracy of the GA-PSO algorithm, with a location error of 20.08 m, representing an improvement of 59% over the GA and 43% over the PSO algorithm.

1. Introduction

Mine rockbursts, air-mining zones, and mine gas leaks can cause serious disasters. To prevent these accidents, microseismic monitoring technology has been integrated into the mineral extraction process, enabling the prediction of small earthquake locations and magnitudes, thus mitigating serious injuries and fatalities. Accurate microseismic source location is crucial for microseismic monitoring, and many classic and widely used algorithms exist for locating microseismic events. Unlike traditional classification methods, source location algorithms are categorized into heuristic and non-heuristic algorithms. Non-heuristic algorithms include the classical location method proposed by Geiger and the simplex location algorithms introduced by A.F. Prugger et al. [1,2]. To reduce Geiger’s dependence on velocity models, Lin et al. [3] integrated a linear positioning approach with the Geiger method. This hybrid strategy addresses non-convergence issues caused by large initial value errors in time residual functions through iterative coordinate solving. Omran et al. [4] proposed a parameter-free, population-based simplex method (APS). This approach performs a local search by constructing a hypersphere around the best individual, where each population member selects a simplex and adjusts the diversification probability threshold according to the simplex’s volume. Ma et al. [5] applied the Gravitational Search Algorithm (GSA) to source localization.

In addition to these non-heuristic methods, several other innovations in seismic source localization have emerged. For instance, Bo Liu et al. proposed a novel data compression framework for seismic sensor networks using Distributed Principal Component Analysis, significantly improving the overall performance of seismic monitoring systems and the accuracy of seismic positioning [6]. WANG et al. introduced an interferometric travel time positioning method that iteratively solves the minimum residual function, eliminating certain errors and achieving accurate positioning [7]. Yong Hu et al. [8] proposed a combination of PA-PDEI and phase-amplitude-based full waveform inversion methods, enhancing low-frequency and phase information to reliably reconstruct velocity models and improve deeper region resolution. Ke Ma et al. [9] developed an intelligent microseismic localization model using Fully Convolutional Neural Networks (FCNNs).

Although non-heuristic algorithms exhibit theoretical simplicity, their strong dependence on initial values, computational complexity of derivatives, and reliance on objective function differentiability have driven researchers toward heuristic algorithms. The latter demonstrate enhanced search robustness by simulating natural phenomena.

Heuristic algorithms mainly include population intelligence optimization methods such as the GA, PSO algorithm, and the simulated annealing (SA) algorithm. Huang et al. improved the accuracy of microseismic location by optimizing the Ω penalty function using ant colony optimization [10]. Xiao, Y. et al. proposed a nonlinear modification to the PSO algorithm to prevent it from becoming trapped in local optima [11]. Yao et al. [12] integrated multiple strategies with the PSO algorithm to enhance PSO optimization performance. Yang and Deb introduced the Cuckoo Search Algorithm, optimizing the search by simulating the breeding behavior of cuckoos [13]. Dhiman and Kumar developed the Seagull Optimization Algorithm, which performs global searches during migration and local searches during attacks [14]. Novianty et al. [15] employed the GA for seismic data inversion of the Okada model in their experiments, optimizing convergence through an elitism strategy and an adaptive mutation rate. This method demonstrated reasonable accuracy in estimating earthquake magnitude and azimuthal direction. Despite various location schemes proposed by scholars, challenges remain, including difficulties in solving objective functions, local range locking, and slow convergence, which can lead to location errors. Improvements to single heuristic algorithms are constrained by inherent theoretical performance ceilings, which have propelled research into hybrid algorithms, particularly the synergistic combination of complementary heuristics.

Table 1 summarizes the characteristics of existing microseismic location methods: non-heuristic algorithms are theoretically simple but highly reliant on initial values and velocity models; single heuristic algorithms demonstrate enhanced robustness but face challenges in convergence and accuracy; hybrid heuristic algorithms enhance performance by leveraging collaborative mechanisms.

The PSO algorithm is inspired by the predatory behavior observed in bird flocks. This approach has gained widespread popularity due to its ease of implementation and its ability to be seamlessly integrated with other algorithms to achieve high precision [16]. Guo et al. [17] proposed a hybrid approach that combines Multi-Objective Particle Swarm Optimization (MOPSO) with SA. This method leverages MOPSO to obtain the optimal solution, while SA is employed to resolve the seismic source location. Liao Z. et al. [18] applied the NM-PSO algorithm to seismic source location, using PSO for global optimization to provide initial values for the Nelder–Mead (NM) algorithm. Furthermore, the integration of PSO with other optimization algorithms demonstrates significant potential in solving multimodal problems, such as Ant Colony Optimization (ACO) [19] and Differential Evolution (DE) [20].

The GA exhibits strong global search capabilities and does not rely on mathematical models, demonstrating significant robustness in solving nonlinear equations. Li et al. [21] utilized the GA to determine initial values, followed by a simplex search method for location determination. This hybrid localization approach demonstrated approximately a 10% improvement in accuracy over the traditional simplex algorithm. The Genetic Algorithm has shown tremendous potential in the field of seismic source localization.

The PSO algorithm has become an ideal foundation for the design of hybrid algorithms due to its ease of implementation and strong compatibility. Its integration with the GA is particularly noteworthy. Currently, the GA-PSO algorithm has demonstrated exceptional performance across multiple engineering fields. Examples include medical equipment development and disease diagnosis [22], antenna array design [23], workflow scheduling [24,25], power grid demand management [26], and slope stability prediction [27]. The cross-disciplinary success of the GA-PSO algorithm validates its robustness, highlighting its unique advantages in high-noise environments and multi-parameter coupled scenarios. These findings align well with the accuracy and stability requirements of microseismic localization.

This research innovatively introduces the GA-PSO algorithm into the field of microseismic localization by combining GA with the PSO algorithm, enhancing PSO by introducing dynamic learning factors and adaptively adjusting inertia weights. This approach enables global search in the early stages and local search in the later stages, increasing population diversity through GA integration, which significantly enhances the search capability and avoids local optima. The results demonstrate that this algorithm exhibits improved search capability, convergence speed, and location accuracy.

Table 1. Comparison of existing microseismic location methods.

Category	Algorithm	Key Innovation	Strengths	Limitations	Application
Non-Heuristic	Geiger [1]	Iterative time residual minimization	Theoretical simplicity, high efficiency	Initial value and velocity model sensitivity	Seismic detection in mining
	APS [4]	Hypersphere search, adaptive threshold	Parameter-free, efficient local search	High-dimensional performance drop	Small/medium sensor networks
Heuristic (Single)	GA [15]	Crossover and mutation	Strong global search capability	Slow convergence speed	Function optimization
	PSO [16]	Velocity–position update mechanism	Simple to implement, fast convergence	Parameter sensitivity, low diversity	Real-time localization
Heuristic (Hybrid)	NM-PSO [18]	NM local refinement, PSO global search	Robustness to initial values	Sensitive to initial conditions	Small-scale networks
	PSO-ACO [19]	Pheromone-guided particle trajectories	Strong multimodal problem-solving	High computational complexity	Path and network optimization

Table 1. Comparison of existing microseismic location methods.

Category	Algorithm	Key Innovation	Strengths	Limitations	Application
Non-Heuristic	Geiger [1]	Iterative time residual minimization	Theoretical simplicity, high efficiency	Initial value and velocity model sensitivity	Seismic detection in mining
	APS [4]	Hypersphere search, adaptive threshold	Parameter-free, efficient local search	High-dimensional performance drop	Small/medium sensor networks
Heuristic (Single)	GA [15]	Crossover and mutation	Strong global search capability	Slow convergence speed	Function optimization
	PSO [16]	Velocity–position update mechanism	Simple to implement, fast convergence	Parameter sensitivity, low diversity	Real-time localization
Heuristic (Hybrid)	NM-PSO [18]	NM local refinement, PSO global search	Robustness to initial values	Sensitive to initial conditions	Small-scale networks
	PSO-ACO [19]	Pheromone-guided particle trajectories	Strong multimodal problem-solving	High computational complexity	Path and network optimization

2. Materials and Methods

2.1. 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].

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:

$$\left(t_i-t_0\right)V_p=\sqrt{{\left(x_i-x_0\right)}^2+{\left(y_i-y_0\right)}^2+{\left(z_i-z_0\right)}^2}$$

(1)

where $(x_i,y_i,z_i)$ and $(x_j,y_j,z_j)$ 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:

$$\left\{\begin{array}{c}{V}_{pi}=\sqrt{{\left(x_i-x_0\right)}^2+{\left(y_i-y_0\right)}^2+{\left(z_i-z_0\right)}^2}/\left(t_i-t_0\right)\\ V_{pj}=\sqrt{{\left(x_j-x_0\right)}^2+{\left(y_j-y_0\right)}^2+{\left(z_j-z_0\right)}^2}/\left(t_j-t_0\right)\end{array}\right.$$

(2)

Since the wave speeds are the same, the velocity difference is

$${∆}_{\mathrm{i}\mathrm{j}}=\left|V_{pi}-V_{pj}\right|$$

(3)

Assuming there are N geophones in total, the objective function is formulated as

$$f={\sum }_{i=1}^{N,j=i+1}{∆}_{ij}$$

(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.

2.2. Implementation of GA-PSO Algorithm

The Genetic Algorithm is an algorithm proposed by John Holland to simulate the process of biological evolution based on the principles of genetics [29]. Starting with a population of chromosomally characterized individuals from the solution set, operations such as selection, crossover, and mutation are performed generation by generation from the initial population to produce a new population. Through the mechanisms of crossover and mutation, solution diversity is maintained, preventing the algorithm from becoming trapped in local optima. Mutation enables deeper exploration of local regions and enhances solution precision. Finally, the optimal chromosomes derived are decoded to obtain the best solution.

The PSO algorithm, a heuristic optimization method, was proposed by James Kennedy and Russ Eberhart [30]. Its main principle is to solve optimization problems by simulating the process of a flock of birds searching for food. By simulating the interactions among individual particles within the flock, the optimal solution is progressively identified through iterative optimization, with the fitness value serving as a metric for solution quality. This algorithm is straightforward to implement, provides high accuracy and rapid convergence, and effectively seeks the global optimum by leveraging collaboration among individuals within the population.

The algorithm steps of PSO are initialization, fitness evaluation, and then updating individual optimal positions and group optimal positions [31]. The formula for updating speed is

$$v_i\left(t+1\right)=w·v_i\left(t\right)+c_1·r_1·\left({Pbest}_i-x_i\left(t\right)\right)+c_2·r_2·({Gbest}_i-x_i\left(t\right))$$

(5)

The update formula for the position is

$$x_i\left(t+1\right)=x_i\left(t\right)+v_i(t+1)$$

(6)

where $w$ is the inertia weight. In general, a larger $w$ enhances the global search capability but weakens the local search capability, while a smaller $w$ strengthens the local search capability at the expense of global exploration. A dynamically adjusted $w$ allows for systematic variation during the PSO algorithm’s search process, improving the optimization performance and achieving optimal solutions. $c_1$, $c_2$ are the learning factors, where $i=\mathrm{1,2}$…N, and N is the total number of particles, usually $c_1$=$c_2$= 2. $r_1$, $r_2$ are random numbers in the range of 0–1, Pbest is the best position of an individual particle, and Gbest is the best position globally. $v_i$ is the velocity of the particle, and the value of $v_i$ is in between the values defining the velocity interval of the particle.

The PSO algorithm tends to experience diminished search capability in later stages and is susceptible to falling into local optima [32]. This manuscript enhances the PSO algorithm by incorporating a dynamic learning factor and adaptive inertia weight.

The dynamic learning factor can automatically adjust the particle behavior according to the iteration progress, without the need to manually adjust the parameters.

$c_1$ and $c_2$ denote two learning mechanisms: the individual learning factor and the social learning factor [33]. $c_1$ controls a particle’s tendency to follow its own best historical position, while $c_2$ governs its inclination to follow the global best position in the population. During different optimization phases, these two parameters need to be adjusted differently to maximize their effectiveness [34].

Research indicates that optimization strategies based on nonlinear variations using the sine function outperform those using the cosine function [35]. The use of a sine squared function, rather than linear decrement or increment, is preferred due to its smooth and continuous derivative, which mitigates oscillations caused by abrupt parameter changes and better accommodates the search requirements at different stages. Different magnitudes of adjustment are provided in each iteration stage through the nonlinear variation of the sinusoidal function, and this nonlinear dynamic variation is able to adapt to the different stages of the search process more efficiently than the traditional fixed-value learning factor. The dynamic learning factor is defined by the following formula:

$$c_1\left(t\right)=2·{\left(sin\right((\pi /2)·(1-t/T)\left)\right)}^2$$

(7)

$$c_2\left(t\right)=2·{\left(sin\right((\pi t/2)·T\left)\right)}^2$$

(8)

Here, $t$ represents the current iteration number and $T$ denotes the maximum iteration number.

The function $c_1\left(t\right)$ is approximately 2 in the early stages of iteration $(t\to 0)$, emphasizing the particle’s reliance on its own historical best, thereby promoting global exploration. As the iteration progresses, $c_1\left(t\right)$ gradually decreases, reducing the reliance on individual experience and preventing excessive dispersion. Initially $(t\to 0)$, $c_2\left(t\right)$ is close to 0, reducing the influence of the global best and preventing premature convergence; in later stages $(t\to T)$, $c_2\left(t\right)$ approaches 2, strengthening collective cooperation and facilitating local exploitation. Both $c_1\left(t\right)$ and $c_2\left(t\right)$ remain within the range of [0, 2], consistent with the empirical bounds of classical PSO learning factors (typically $c_1+c_2 \le 4$), ensuring the algorithm’s stability.

The inertia weight functions to adjust the update of particle velocity. When the inertia weight is relatively large, the particle retains more of its historical motion trend during the velocity update process, resulting in a wider search range. Conversely, when the inertia weight is relatively small, the particle tends to search around the current optimal position, thereby accelerating the convergence speed. The adaptive inertia weight is computed using the following equation [35]:

$$w=w_{max}-(w_{max}-w_{min})·{(i/T)}^2$$

(9)

where $i$ is the current iteration number and $T$ is the maximum iteration number.

This enhancement enables the algorithm to focus on extensive global search during the initial phase and local search in the latter phase. Additionally, integrating the improved PSO with the GA increases solution diversity through cross-mutations, thereby allowing the algorithm to adaptively adjust its search strategy across different stages.

The algorithm process is shown in Figure 1.

The steps for the implementation of the GA-PSO algorithm are as follows:

(1) Initialize GA and PSO parameters: Since real-valued encoding outperforms binary encoding in solving optimization problems [36], real-valued encoding is utilized for the chromosomes in the present study. Set the chromosome length to 3, denoting the three spatial coordinates of the microseismic source, the value range of the three coordinate axes is determined by the size of the model space. Set crossover probability to 0.8, mutation probability to 0.2, maximum iterations to 100, and population size to 20. Set $w_{max}=0.8$, $w_{min}=0.4$. Define limits for particle velocities and positions.

The number of iterations depends on the problem’s complexity, convergence speed, and optimization objectives. Under the experimental conditions of this study, the GA-PSO algorithm achieves stability within 100 iterations. Setting the maximum number of iterations to 100 ensures convergence while minimizing unnecessary computational overhead.

A typical population size ranges from 20 to 50 [37]. Given the moderate complexity of the objective function in this study, a population size of 20 is sufficient to ensure an effective search while minimizing computational waste associated with larger population sizes.

(2) Generate the initial particle population by creating a two-dimensional array of shape (popsize, lenchrom). The array values are randomly drawn from a uniform distribution within $[{pop}_{min}, {pop}_{max}]$, with velocity V randomly generated within $[V_{min}, V_{max}]$. Compute the fitness value according to the objective function, store it in the fitness array, and identify the current optimal individual to be stored in fitnesspbest.

(3) Initialize the individual best values and the global best values. Based on the fitness values, identify the index of the individual with the optimal fitness value in the current population. The corresponding position and fitness value of this individual are assigned as the global best position and the global best fitness value, respectively. At the same time, the positions and fitness values of all individuals in the population are initialized as their respective individual best positions and individual best fitness values.

(4) Iteratively update the algorithm: in each iteration, calculate the dynamic learning factors $c_1$ and $c_2$ using Equations (7) and (8), and calculate the inertia weights using Equation (9). The velocity update equation is

$$V[j+1]=w\cdot V\left[j\right]+c_1\cdot r_1\cdot \left(pbest\right[j]-pop[j\left]\right)+c_2\cdot r_2\cdot (\mathrm{g}best-pop[j\left]\right)$$

(10)

where $pbest\left[j\right]$ is the individual optimal solution in the $j$th dimension, $pop\left[j\right]$ represents the position of the $j$th particle, $gbest$ is the global best position, and $r_1$ and $r_2$ are random numbers within the range [0, 1]. Update the particle population positions using the following equation:

$$pop[j+1]=pop\left[j\right]+V[j+1]$$

(11)

Update the fitness values for the updated particle population positions.

(5) Compare the fitness value of each particle in the current swarm with its own historical individual best fitness value. The comparison is implicitly implemented using a Boolean mask, which identifies particles where the current fitness is better than the historical best. For these particles, their historical best positions and fitness values are updated simultaneously in a vectorized manner. Particles that do not meet the condition retain their previously recorded values.

(6) Evaluate whether the minimum fitness value in the current particle swarm is better than the recorded global optimal fitness value. If it is, update the global optimal solution and its corresponding fitness value accordingly. Otherwise, retain the previously recorded global optimal solution and fitness value.

(7) Check the termination condition: determine whether the final coordinates fall within the limited range of coordinates. If satisfied, output the global optimal value and the corresponding fitness value; otherwise, continue to step (8).

(8) Crossover operation: The best individuals are selected based on fitness. Select the fittest individuals based on the crossover probability $p_c$ and population size to participate in the crossover operation.

The offspring ${Child}_{x1}$ is generated by combining the genes of two parents:

$${Child}_{x1}=p_c\times {Pool}_x\left[{seed}_1\right]+(1-p_c)\times {Pool}_x\left[{seed}_2\right]$$

(12)

where $p_c$ is a random crossover probability factor between 0 and 1, seed denotes the index position of a parent individual in the population, and ${Pool}_x$ represents the population pool that contains the genes of all individuals. ${seed}_1$ and ${seed}_2$ are the indices of two parent individuals selected from the population.

Offspring velocity ${Child}_{v1}$ is a weighted average of the velocities of the two parents:

$${Child}_{v1}={\displaystyle \frac{‖{Pool}_v\left[{seed}_1\right]+{Pool}_v\left[{seed}_2\right]‖}{‖{Pool}_v\left[{seed}_1\right]‖}}\times ({Pool}_v\left[{seed}_1\right]+{Pool}_v\left[{seed}_2\right])$$

(13)

where ${Child}_{v1}$ represents the genetic data of the offspring individual generated. ${Pool}_v$ represents the set of velocities of all individuals in the population. After the crossover operation, the offspring particle ${Child}_{x1}$ replaces the corresponding particle in the original population with a better fitness.

(9) Mutation operation: An individual is selected according to the mutation probability $p_m$. The genes of the selected individual are then adjusted randomly:

$$mut=pop\left[{seed}_3\right]+(random.rand\left(lenchrom\right)-0.5)\times ({pop}_{max}-{pop}_{min})$$

(14)

where mut represents the mutated individual’s genes and $pop\left[{seed}_3\right]$ refers to the genes of the selected individual in the population. $random.rand\left(lenchrom\right)$ generates a random array of size lenchrom, where each value is in the range [0, 1). By subtracting 0.5, the random values are adjusted to the range [−0.5, 0.5], ensuring the disturbance is symmetric. ${pop}_{max} \mathrm{a}\mathrm{n}\mathrm{d} {pop}_{min}$ represent the maximum and minimum values of the genes, respectively.

(10) Calculate the fitness, update the population optimal solution, and record the optimal fitness, and return to step (5).

3. Experiments and Results

3.1. Simulation Validation of Microseismic Location Algorithms

3.1.1. Developing a Simulated Source Spatial Model

To validate the accuracy of the localization algorithm presented in this paper, 10 geophones and 6 microseismic events were deployed in a three-dimensional space. The model was constructed within a 500 m × 500 m × 500 m domain, with the microseismic event initiating at 0 s and a wave speed of 2570 m/s. The spatial distribution of the sensors and microseismic events is illustrated in Figure 2, and the geophone spatial coordinate are shown in

Table 2. Geophone spatial coordinate position.

Sensor Number	X/m	Y/m	Z/m
1	0	0	0
2	500	0	0
3	500	500	0
4	0	500	0
5	0	268	0
6	0	0	−500
7	500	64	−500
8	500	500	−500
9	0	500	−500
10	391	145	−247

.

The simulated microseismic event coordinates are shown in

Table 3. Location of microseismic source coordinates.

Location of Microseismic Source	X/m	Y/m	Z/m
A	55	192	−287
B	417	251	−96
C	285	72	−394
D	146	298	−163
E	88	384	−406
F	383	165	−230

, the principle of the positioning simulation verification is based on calculating the actual distance between each geophone and the simulated seismic source according to their three-dimensional spatial positions. The P-wave velocity and the initial time of seismicity ($t_0$) are assumed to determine the P-wave’s initial arrival time.

In this study, the arrival time of the P-wave is calculated using a geometric model based on the assumption of a uniform wave velocity, with the results representing theoretical predictions based on uniform propagation speed. The specific method is as follows: assuming the P-wave velocity, $V_p$, to be 2570 m/s and the origin time, $t_0$, to be 0 s, the locations of the seismic source and detectors are first defined, and the P-wave arrival time is then obtained using Equation (1).

Upon obtaining the P-wave arrival time, these values, along with the geophone locations, are input into the GA-PSO algorithm. The results generated by the algorithm are then compared with the actual microseismic source location, thus verifying the accuracy of the algorithm. The obtained P-wave first arrivals are shown in

Table 4. Detector and P-wave initial arrival.

Sensor Number	P-Wave Initial Arrival time/ms
	A	B	C	D	E	F
1	136.052	193.031	191.274	143.858	220.123	185.311
2	219.166	109.439	176.880	190.895	270.151	119.179
3	238.359	108.745	241.322	170.803	229.551	164.539
4	165.202	192.639	252.062	115.878	167.828	217.277
5	117.488	166.632	204.003	85.943	167.828	178.394
6	113.615	246.124	121.588	184.030	157.593	193.309
7	198.321	176.207	93.324	210.851	206.255	121.055
8	226.303	187.461	190.878	205.781	170.513	173.496
9	147.274	245.816	204.288	163.094	67.435	224.137
10	132.927	72.496	76.024	117.049	162.407	10.677

.

3.1.2. Analysis of the Simulation Results for Microseismic Source Location

To analyze the performance of the location algorithms, the PSO, GA, NM-PSO, PSO-ACO, and GA-PSO algorithms were compared. To ensure accuracy, the error and optimal function value for each seismic event were calculated, with each of the five algorithms being run 10 times. The number of particle swarms and populations was set to 20, and the number of iterations to 100. The maximum and minimum values of the final seismic source location results were excluded, and the average of the remaining results was taken as the convergence value, representing the final location result. Consequently,

Table 5. Comparison of location results of five algorithms.

Arithmetic	Event A
		X/m	Y/m	Z/m	Optimal Function Value	Search Time/s
PSO	Convergence value	54.7102	192.0557	−287.1071	5.097 × 10−3	0.33
	Real value	55	192	−287
	Errors	−0.2898	0.0557	−0.1071
GA	Convergence value	55.4110	191.7454	−286.8794	0.1310	1.18
	Real value	55	192	−287
	Errors	0.4110	−0.2546	0.1206
PSO-ACO	Convergence value	54.9964	192.0141	−286.9920
	Real value	55	192	−287	3.613 × 10−3	0.74
	Errors	−0.0036	0.0141	0.0080
NM-PSO	Convergence value	54.9923	191.9981	−287.0038
	Real value	55	192	−287	1.445 × 10−3	0.33
	Errors	−0.0077	−0.0019	−0.0038
GA-PSO	Convergence value	54.9987	191.9995	−286.9993	3.387 × 10−4	0.17
	Real value	55	192	−287
	Errors	−0.0013	0.0005	0.0007
Arithmetic	EventB
		X/m	Y/m	Z/m	Optimal Function Value	Search Time/s
PSO	Convergence value	417.4418	251.1682	−96.3564	5.219 × 10−3	0.43
	Real value	417	251	−96
	Errors	0.4418	0.1682	−0.3564
GA	Convergence value	417.5229	251.0520	−95.6258	0.1474	0.89
	Real value	417	251	−96
	Errors	0.5229	0.0520	0.3742
PSO-ACO	Convergence value	416.9981	251.0134	−95.9990	3.553 × 10−3	0.72
	Real value	417	251	−96
	Errors	−0.0019	0.0134	0.0010
NM-PSO	Convergence value	417.0008	250.9989	−95.9981
	Real value	417	251	−96	4.283 × 10−4	0.33
	Errors	0.0008	−0.0011	0.0019
GA-PSO	Convergence value	416.9988	251.0001	−95.9994	2.637 × 10−4	0.16
	Real value	417	251	−96
	Errors	−0.0012	0.0001	0.0006

presents the location results obtained from five algorithms for events A and event B.

The distance between the calculated and actual seismic source locations was considered as the absolute error. Figure 3a illustrates a comparison of the absolute errors in the localization results for six simulated microseismic source events across five algorithms. Among these, the errors of PSO-ACO, NM-PSO, and GA-PSO are minimal. To enhance clarity, Figure 3b provides an enlarged view of Figure 3a, focusing on the results of the three hybrid algorithms.

From Figure 3a, it can be seen that due to the tendency of the PSO algorithm to become trapped in local optima and the random search nature of the GA, both the PSO algorithm and GA show significant errors. In contrast, the GA-PSO algorithm, by optimizing the PSO algorithm and combining the strengths of both PSO and GA, demonstrates the smallest absolute error and improved precision.

Figure 3b presents a comparison among the GA-PSO, PSO-ACO, and NM-PSO algorithms. The results indicate that the errors of all three hybrid algorithms are confined within 0.03 m, demonstrating strong performance. Among these, the PSO-ACO algorithm performs slightly worse than the other two hybrid algorithms, while the NM-PSO and GA-PSO algorithms yield comparable results. However, the GA-PSO algorithm performs slightly better than the NM-PSO algorithm and exhibits a certain degree of robustness.

From Figure 3c, it can be seen that among the five algorithms, the search time of the GA-PSO algorithm is shorter and stable, ranging between 0.16 s and 0.17 s. In contrast, the GA requires breeding and selection in each generation of the population, which increases its operational complexity, usually leading to a greater need for computational resources and longer execution time. The PSO algorithm, on the other hand, typically finds the optimal solution in fewer iterations, thus reducing its computation time compared to the GA. The GA-PSO algorithm combines the advantages of both algorithms, enhancing local search efficiency while ensuring a global search, thereby reducing the complexity of each iteration and overall execution time. The PSO-ACO algorithm integrates the features of both PSO and ACO, including operations such as pheromone updates. Consequently, its higher computational complexity results in longer execution times. In contrast, the NM-PSO and GA-PSO algorithms demonstrate shorter runtimes. Notably, the GA-PSO algorithm reduces computation time by improving the PSO algorithm and integrating the advantages of the GA.

3.1.3. Comprehensive Comparative Experiments of Microseismic Location Algorithms

To verify the stability of the algorithm’s location accuracy, this manuscript re-evaluated detector arrivals under wave speed fluctuations of ±1%, ±3%, and ±5%. The P-wave velocity fluctuation is typically simulated by perturbing the wave velocity within a range of ±1% to ±5%, to reflect the natural variability commonly observed in geological environments [38]. Specifically, ±1% simulates minor environmental disturbances, ±3% simulates disturbances caused by moderate changes in the rock layer structure, and ±5% simulates disturbances under extreme conditions.

In the experiment, the base P-wave velocity is set to 2570 m/s, with fluctuations introduced within the ±1%, ±3%, and ±5% ranges of this velocity through random sampling. The fluctuated P-wave velocity is calculated using the formula below:

$$V_{pi}=V_b\times \left(1+{\delta }_i\right)$$

(15)

where $V_{pi}$ represents the P-wave velocity at the $i$-th detector after incorporating the velocity fluctuation; $V_b$ is the base P-wave velocity; $V_{ci}$ is a random variable; and ${\delta }_i$ represents the uniformly distributed perturbation coefficient, where ${\delta }_i~U(-r,r)$, with $r\in \left\{0.01\right., 0.03,\left. 0.05\right\}$. The geo-regime-specific bounds satisfy ${‖\delta ‖}_{\infty }\le 5\%$, preserving physical realizability in accordance with poroelasticity constraints.

Table 6. Sensor arrivals under different wave speed fluctuations.

Wave Velocity Fluctuate	Sensors	P-Wave Initial Arrival time/ms
		A	B	C	D	E	F
±1%	1	136.785	191.625	192.526	143.640	222.173	186.285
	2	220.346	108.641	178.038	190.606	272.667	119.806
	3	239.643	107.952	242.902	170.545	231.689	165.403
	4	166.091	191.235	253.713	115.703	169.391	218.419
	5	118.120	165.418	205.338	85.813	169.391	179.332
	6	114.226	244.331	122.384	183.752	159.060	194.325
	7	199.389	174.923	93.935	210.532	208.176	121.691
	8	227.522	186.095	192.128	205.469	172.101	174.408
	9	148.067	244.025	205.625	162.848	68.063	225.315
	10	133.642	71.968	76.522	116.872	163.919	10.733
±3%	1	138.601	192.003	195.194	144.552	223.367	187.829
	2	223.272	108.856	180.505	191.817	274.132	120.799
	3	242.825	108.166	246.267	171.628	232.933	166.774
	4	168.297	191.613	257.228	116.438	170.301	220.230
	5	119.689	165.745	208.183	86.358	170.301	180.818
	6	115.743	244.813	124.080	184.919	159.915	195.935
	7	202.036	175.268	95.237	211.869	209.294	122.700
	8	230.543	186.462	194.789	206.774	173.026	175.854
	9	150.033	244.507	208.474	163.882	68.429	227.183
	10	135.417	72.110	77.582	117.614	164.800	10.822
±5%	1	141.099	198.754	199.204	149.874	230.676	189.065
	2	227.295	112.683	184.213	198.879	283.102	121.593
	3	247.200	111.969	251.327	177.946	240.556	167.871
	4	171.329	198.350	262.512	120.724	175.874	221.678
	5	121.846	171.572	212.460	89.537	175.874	182.008
	6	117.829	253.420	126.629	191.727	165.148	197.224
	7	205.677	181.430	97.193	219.669	216.143	123.507
	8	234.697	193.018	198.791	214.387	178.688	177.010
	9	152.737	253.104	212.757	169.915	70.668	228.677
	10	137.857	74.645	79.176	121.944	170.193	10.894

presents the sensor arrival times under varying wave speed fluctuations.

As observed in Figure 4a, when the wave speed fluctuates by ±1%, the positioning error of both the PSO algorithm and the GA is minimal, with the GA-PSO algorithm demonstrating slightly better convergence results and maintaining a certain degree of accuracy. The NM-PSO algorithm performs slightly worse than the GA-PSO algorithm, demonstrating a certain level of robustness. The PSO-ACO algorithm also demonstrates good accuracy; however, its overall performance, particularly when compared to GA-PSO and NM-PSO, is slightly inferior. When the wave speed error is ±3% (Figure 4b), the impact on the GA-PSO algorithm remains minimal. It is evident that both the GA and the PSO algorithm display varying degrees of fluctuation in source location accuracy under different wave speed variations. Although GA has a strong global search capability to avoid falling into local optimal solutions, its search process is more random and prone to larger error fluctuations, and PSO’s updating mechanism makes its error fluctuations relatively small; the hybrid mechanism of GA-PSO algorithm is able to balance the advantages of both and thus maintains a small error fluctuation rate in most of the search process, which suggests that the GA-PSO algorithm achieves more accurate localization despite fluctuations in wave speed. Among the three hybrid algorithms, the PSO-ACO algorithm is more significantly affected by velocity fluctuations, mainly because its pheromone update mechanism does not quickly adapt to changes in velocity, resulting in inaccuracies during the search process. Meanwhile, the error rate of the NM-PSO algorithm increases more rapidly than that of the GA-PSO when subjected to velocity fluctuations. Figure 4c clearly demonstrates that in the NM-PSO algorithm, the limited adaptability of the Nelder–Mead method to global fluctuations results in error accumulation, which gradually increases due to velocity fluctuations. As each local solution may no longer represent the global optimum, the error eventually converges to a value similar to that of the PSO-ACO algorithm. In contrast, the localization effect of the GA-PSO algorithm is more pronounced.

As can be seen in Figure 5a, in the early stages of iteration, the GA-PSO algorithm leverages the crossover and mutation operations of GA to explore a larger search space. However, due to the random nature of GA’s initial exploration, while the positioning accuracy improves, it does not surpass the effectiveness of PSO’s early-stage local search. As the number of iterations increases, the velocity and position update mechanisms of PSO enable GA-PSO to converge rapidly within local regions, refine the solution towards a better optimum, and gradually reduce the positioning error. The PSO-ACO algorithm quickly identifies a promising initial solution through pheromone guidance, but its limited refinement capability results in slightly lower positioning accuracy compared to the other two hybrid algorithms. The NM algorithm excels in local optimization; however, when approaching the optimal solution, it makes excessive minor adjustments, leading to slower convergence speeds. The NM-PSO algorithm achieves higher accuracy than PSO-ACO, but its overall precision and iteration speed still lag behind those of the GA-PSO algorithm. In Figure 5b, after population initialization, GA performs a broad search across the entire solution space. However, it quickly converges to a relatively optimal solution, resulting in a larger optimal fitness value. In contrast, PSO directs all particles toward the global optimal solution through its particle update formula, which reduces population diversity in the later stages and further limits optimization accuracy. The convergence speed of the GA-PSO algorithm is between that of PSO-ACO and NM-PSO, but it can find the optimal solution within 50 to 70 iterations, with the optimal fitness value significantly outperforming the other two algorithms. GA-PSO introduces perturbations in the later stages through crossover and mutation operations, enabling refined local search optimization. This approach effectively avoids the “particle clustering” phenomenon commonly observed in standalone PSO optimization, resulting in fitness values that are closest to 0.

3.2. Validation Through Engineering Case Studies

3.2.1. Engineering Experiment for Microseismic Source Location

To further verify the accuracy of the GA-PSO algorithm, this study utilizes data from an artificial blasting positioning experiment conducted by Lin Feng et al. [3] at the Shizhuyuan mine to compare and analyze the positioning effects of the five algorithms. The blasting coordinates are recorded as 8732.7 m, 6570.6 m, and 511.3 m. Among the 30 stations, a total of eight stations detected vibration signals, with their location information and P-wave arrival times shown in

Table 7. Detector position and P-wave arrival.

Sensor Number	X/m	Y/m	Z/m	P-Wave Arrival/ms
1	8716	6614	522	34.9
2	8737	6609	565	36.6
3	8666	6600	520	39.3
4	8668	6599	565	41.1
5	8641	6515	520	42.3
6	8691	6684	520	44.5
7	8721	6449	520	47.8
8	8702	6604	647	50.0

.

3.2.2. Engineering Experiment Results

The positioning results of the blasting experiment were analyzed using five established methods: GA, PSO, PSO-ACO, NM-PSO, and the GA-PSO algorithm. The waveform propagation speed was set to 5000 m/s. The results are summarized in

Table 8. Comparison of positioning results using five location methods under engineering validation.

	X/m	Y/m	Z/m	X-Axis Error	Y-Axis Error	Z-Axis Error	Absolute Error
Actual source location	8732.70	6570.60	511.30
PSO	8720.56	6545.42	533.49	11.33	25.34	−22.16	35.52
GA	8719.87	6523.03	513.30	12.83	47.57	−2.00	49.31
PSO-ACO	8782.52	6555.73	500.02	−49.82	14.87	11.28	53.20
NM-PSO	8721.29	6544.60	532.51	11.41	26.00	−21.21	35.44
GA-PSO	8733.93	6550.59	510.12	−1.23	20.01	1.18	20.08

below.

In this engineering experiment, the PSO-ACO algorithm exhibited the largest error (53.20 m), indicating that it is not suitable for medium- to large-scale seismic events. Although the pheromone update mechanism of the ACO algorithm performs well in combinatorial optimization and path planning, its discrete nature is ill suited for continuous optimization problems in medium- to large-scale localization tasks. On the other hand, although the NM-PSO algorithm achieved a 33% improvement in accuracy compared to the PSO-ACO algorithm, it only improved by approximately 0.23% (from 35.52 m to 35.44 m) compared to the traditional PSO algorithm, indicating that the NM-PSO algorithm does not achieve satisfactory results in real data tests. As a result, when confronted with complex real seismic data, the NM-PSO algorithm’s localization performance cannot be substantially improved (with an error of 35.44 m).

The results indicate that the GA-PSO algorithm effectively integrates the strengths of both the GA and PSO algorithms. It achieves an improved balance between global exploration in the early stages and local refinement in the later stages. Under the experimental conditions, the GA-PSO algorithm resulted in an average positioning error of 20.08 m, representing an improvement of approximately 59% over the GA (49.31 m) and a 43% improvement over the PSO algorithm (35.52 m). Moreover, the GA-PSO algorithm outperforms the other two hybrid optimization algorithms, demonstrating superior accuracy and robustness.

4. Discussion and Conclusions

The microseismic source location method is a core component of microseismic monitoring and a key factor in improving the accuracy of disaster prediction. To address the limitations of traditional location algorithms, such as inaccurate location results or poor convergence, this study combines the improved PSO algorithm with the GA. The PSO algorithm is enhanced by introducing a dynamic learning factor and adaptive inertia weight updating, which prevents the algorithm from falling into local optima in the later stages. The integration of the GA’s strengths ensures faster convergence and higher location accuracy.

To verify the accuracy of the algorithm, simulated spatial experiments were conducted. The results demonstrate that the GA-PSO algorithm achieves the smallest error values, significantly outperforming both the GA and PSO algorithms, as well as the PSO-ACO and NM-PSO algorithms. It exhibits improved convergence under the same number of iterations and achieves the lowest fitness value.

In a comprehensive comparative experiment, this study simulated the positioning accuracy of the five algorithms under wave speed fluctuations of ±1%, ±3%, and ±5%. The results show that the GA-PSO algorithm is least affected by these fluctuations, maintaining a positioning error consistently below 0.5 m.

Finally, the algorithm was tested on a real engineering case. Compared with the GA and PSO algorithm, the GA-PSO algorithm demonstrates improved localization accuracy under the given experimental conditions, achieving an average localization error of approximately 20 m, which represents a 59% improvement over the GA and a 43% improvement over the PSO algorithm.

The GA-PSO algorithm currently provides precise and efficient seismic source localization and is capable of detecting minor vibrations. It is particularly well suited for monitoring surface or subsurface intrusions, such as real-time detection of illegal mining activities. However, in large-scale sensor networks (such as those with more than 100 nodes), high-frequency data acquisition may introduce computational delays, posing challenges to the algorithm’s computational efficiency. Furthermore, the current algorithm is designed for single-source localization and has not been tested for applicability in multi-source scenarios. Another limitation is the algorithm’s sensitivity to parameters, which may reduce its adaptability in complex environments.

Future research will focus on optimizing computational efficiency by integrating parallel computing and distributed processing frameworks to enhance real-time performance in large-scale applications. To improve the algorithm’s adaptability, future work will explore the integration of reinforcement learning strategies (such as Q-learning), thereby reducing the reliance on manual parameter tuning. The incorporation of deep learning-based clustering methods is expected to improve the algorithm’s localization capability in multi-source events, making it more suitable for complex monitoring environments. Additionally, future studies will involve field experiments and large-scale simulations to validate the effectiveness of the proposed improvements and assess its robustness in real-world applications.