Research on microseismic localization algorithm with global search and local optimization

Abstract

A microseismic localization algorithm that combines global search and local optimization is proposed. The Fewer Conditions Trigger Difference (FCTD) objective function of global search and local optimization is constructed, the execution process of the algorithm is described by numerical simulation, and the global search and local optimization microseismic localization algorithm is verified and applied by field data analysis. The results show that: (1) the global search and local optimization methods have fast search speed in the global range, high convergence accuracy and stable localization results in the local range, and high localization accuracy and stability without relying on the velocity model and initial values in the process of search. (2) By comparing the localization results of different localization methods, the global search and local optimization algorithms have better localization results.

1. Introduction

Since the 1960s, microseismic monitoring technology has been widely used in mining engineering. Microseismic location is the core part of microseismic monitoring technology and has always been the focus of research on microseismic monitoring technology. At the present stage, with the in-depth research on the mechanism of occurrence and prevention technology of rock burst and mine seismic, higher requirements are put forward on the accuracy of microseismic positioning; therefore, it is vital to research microseismic positioning algorithm for the evolution of coal rock body rupture, monitoring, and prediction of surrounding rock stability and prevention of dynamic disasters in mining engineering [[[1], [2], [3], [4]]].

Scholars at home and abroad have researched microseismic localization algorithms and achieved many meaningful results. Currently, microseismic localization algorithms can be classified into the following basic types: First, the classical seismic source localization algorithm, such as the Geiger method [5], whose principle is to linearize the nonlinear equation system and solve it by the least square method, which is computationally intensive because of the need to solve partial derivatives and inverse matrices in the calculation, but with the development of computer technology, the positioning idea of Geiger method has been widely developed. Second, the simplex method. Li et al. [6] proposed a simplex localization method based on L1 norm statistics, and this method has high anti-interference for discrete data; Wang et al. [7]proposed a hybrid microseismic source location algorithm based on simplex and shortest path ray tracing; Third, heuristic algorithms, Chen et al. [8]proposed a hierarchical localization method for microseismic sources based on particle swarm algorithm with more stable localization results; Lv et al. [9] proposed a method based on steady simulated annealing-simplex hybrid algorithm; Wang et al. [10] proposed a differential evolutionary algorithm to improve the localization objective function, which can better improve the impact of erroneous traveling time on localization. Fourth, the seismic source scanning method, He et al. [11] proposed an improved seismic source scanning localization method, which improved the localization accuracy to some extent. Fifth, the method without pre-velocimetry, Dong et al. [12] proposed a microseismic positioning method without pre-velocimetry. The sixth is the direct search method, Wang [13] proposed a microseismic grid search localization algorithm based on the DIRECT algorithm. Jiang et al. [14] used a grid search-Newton iterative algorithm to obtain preliminary localization results. Soledad et al. [15] proposed an improved global search algorithm using a simulated annealing algorithm with a particle swarm algorithm, which improves localization speed. Jia et al. [16] proposed a microseismic localization method with variable-step acceleration search, which is effective for the localization of simplified velocity models. However, its application to non-uniform velocity models is yet to be verified.

The grid search method suffers from the problem that the computational effort will increase exponentially when the grid size decreases as the monitoring area increases when it is used for localization. Many scholars use the intelligent optimization algorithm in the global range of fast convergence characteristics to reduce the computational effort of grid search [ [[7], [8], [9], [10], [11]]], but the intelligent optimization algorithm is vulnerable to the influence of the parameters of the algorithm itself resulting in poor stability of the output results, which is not conducive to practical engineering applications. Therefore, we propose a global search and local optimization algorithm for microseismic localization, which uses an intelligent optimization algorithm to search quickly at a global scale to the vicinity of the earthquake source, the grid search method is used to solve the problem of unstable output results of the intelligent optimization algorithm in the local range, and the final output meets the positioning accuracy.

2. Principle and process of microseismic localization algorithm

2.1. Microseismic localization algorithm flow with global search and local optimization

The algorithms suitable for global search are mainly the intelligent optimization algorithm and Bayesian optimization algorithm [ [17,18]]. Considering that the localization problem of microseismic events is to find the extrema by constructing a quadratic function about the distance from the source point to the sensor, the propagation speed of seismic waves, and the time difference between different sensors, where the time difference, the range of the coordinates of the source point and the range of the propagation speed of seismic waves are known parameters, and the intelligent optimization algorithm can meet the localization requirements, so the intelligent optimization algorithm is set as a global search algorithm. The local search algorithm aims to solve the problem of unstable output results of the intelligent optimization algorithm, and the search range is small, and the grid search method can search the final results well. Global search is a method to initially determine the location of microseismic occurrence within the search area by global search algorithm; local optimization is a localization method to determine the location of microseismic occurrence by finely dividing the grid in the local area based on the global search output as the initial search value, and then conducting local optimization.

The process of the global search and local optimization microseismic localization algorithm is as follows: firstly, set the global search range, read the first arrival time of microseismic collected by each sensor when microseismic occurs, then execute the global search algorithm, search the source location globally, and get the initial source location after iterative optimization of the objective function, then divide the spatial grid near the initial source location, use the result of the global search algorithm as the initial search value of the local optimization algorithm, and obtain the optimal solution after iteration, which is the exact source location, and the algorithm process is shown in Fig. 1.

Fig. 1.

Global search and local Optimization algorithm flow chart.

2.2. Global search method - artificial fish swarm algorithm

The global search method requires a search algorithm that can converge quickly to the extreme value point in the global scope from a global perspective. The artificial fish swarm algorithm [19] satisfies the above conditions, and it is a new optimization algorithm that imitates the behavior of animal groups. In the process of global search, the algorithm adopts the idea of searching from the global to the individual, puts multiple individuals in the search range, and makes the group converge to the optimal objective function value by evaluating the objective function value of each individual, and solves the problem in the process of interaction between the individual and the group. The search speed is faster, taking the number of artificial fish individuals is 100 as an example, and its schematic diagram is shown in Fig. 2. At the beginning of the iteration, the artificial fish individuals are evenly distributed in the search space, and the color of the individuals closer to the source is darker and the value of the objective function is smaller, as shown in Fig. 2(a); as the iteration increases, the artificial fish individuals gradually approach near the source, but affected by the parameters of the algorithm itself, all the artificial and individuals are near the source, as shown in Fig. 2(b)(c)(d).

Fig. 2.

Iteration diagram of global search algorithm.

2.3. Local optimization methods - hill climbing algorithm

The local optimization method requires a search algorithm that converges quickly, with high convergence accuracy and stable output results in the local range from a local perspective. The hill-climbing algorithm [20] satisfies the above conditions, and in the local optimization process, a grid is divided near the output of the global search to generate several grid nodes, and each grid node generates an evaluation index by the objective function, and a new decision direction is generated by evaluating the objective function values of all grid nodes.

The computational steps of the local optimization search method are as follows.

• (1)the initial positioning point as the origin of the calculation, noted as (x₀, y₀, z₀), for the point in the x, y, z direction to take ± dx, ± dy, ± dz, that is, to get the spatial coordinates (x₀ ± dx, y₀ ± dy, z₀ ± dz), and then in the x, y, z direction, respectively, it will be divided into n copies, that is, to get n³ spatial coordinate points, through the objective function to calculate the evaluation index of each point, the evaluation index of the optimal point that is the next search direction, Fig. 3 is a schematic diagram of the local optimization method, where the initial search value is (3, 3, 3), dx, dy, dz are 2, n is 5.

• (2)Adjust the search direction, adjust the calculation origin to the point with the smallest evaluation index calculated in step (1), return to step (1), and iterate through the cycle until the optimal value point is found.

Fig. 3.

Schematic diagram of the local optimization algorithm.

2.4. Constructing the objective function

Most source localization methods rely on the wave velocity model. However, the stratigraphic structure is complex and changeable, and the wave velocity has excellent uncertainty, so its positioning accuracy will also be affected. Now an objective function that does not require wave velocity measurement is proposed, called FCTD (Fewer Conditions Trigger Difference). Assuming that the propagation velocity V of the P-wave in the stratum is unknown, the coordinates of the i-th sensor are (x_i, y_i, z_i), t₀ is the time of the earthquake occurs, (x, y, z) is the coordinate of the source point. Equation (1) is the calculated arrival time between the earthquake source and the sensors:

$$\begin{array}{c}{t}_i=t_0+\frac{\sqrt{{\left(x_i-x\right)}^2+{\left(y_i-y\right)}^2+{\left(z_i-z\right)}^2}}{V}\#\end{array}$$ (1)

Equation (2) is the calculated arrival time difference between two adjacent sensors:

$$\begin{array}{c}\Delta t_i=t_{i+1}-t_i=\frac{l_{i+1}-l_i}{V}=\frac{\Delta l_i}{V}\#\end{array}$$ (2)

where

$$\begin{array}{c}{l}_{i+1}=\sqrt{{\left(x_{i+1}-x\right)}^2+{\left(y_{i+1}-y\right)}^2+{\left(z_{i+1}-z\right)}^2}\#\end{array}$$ (3)

$$\begin{array}{c}{l}_i=\sqrt{{\left(x_i-x\right)}^2+{\left(y_i-y\right)}^2+{\left(z_i-z\right)}^2}\#\end{array}$$ (4)

Equations (3), (4) are used to calculate the distance from the source of the earthquake to the i-th and i+1-th sensors.

Equation (5) is the monitored arrival time difference between two adjacent sensors:

$$\begin{array}{c}\Delta T_i=T_{i+1}-T_i\#\end{array}$$ (5)

The sum of the squares of the difference between the monitored time difference ΔT_i and the calculated time difference Δt_i of all two adjacent sensors can describe the degree of deviation between the monitored value and the calculated value, that is

$$\begin{array}{c}Q\left(x,y,z,V\right)=\sum _{i=1}^n{\left(\Delta T_i-\frac{\Delta l_i}{V}\right)}^2\#\end{array}$$ (6)

Equation (6) is a non-negative quadratic function, so there is always a set of source search points (x, y, z, V) to obtain the minimum value when Q (x, y, z, V) takes the minimum value, the monitored time difference ΔT_i of the sensor fits best with the calculated time difference Δt_i, and the coordinates (x, y, z) at this time are the coordinates of the hypocenter point.

3. Numerical modeling

The search range of 100 m × 100 m × 100 m is used as an example to describe the global search and local optimization process for microseismic localization. It is assumed that the wave speed in the search range is 4200 m/s and by a 2% speed error. The source point coordinates are (39, 77, 80). Considering that the average number of sensors used in engineering monitoring is about 5, 5 sensors are set in the search range, and the coordinates and arrival time of the sensors are shown in Table 1.

Table 1.

The Sensor coordinates and the first arrival time.

Sensor number	Sensor coordinates/m			The first arrive time/ms
	X	Y	Z
1	8	13	4	24.681357
2	55	92	85	5.155820
3	75	2	92	20.012751
4	97	52	68	15.326765
5	16	66	36	12.107793

3.1. Global search algorithm localization process

The global search algorithm is a search method with faster convergence from a global perspective. In the global search, the global search range is set to (0, 100) in both x, y, and z directions, the number of artificial fish in the fish population is set to 100, the perceived distance of each artificial fish is 50 m, and the step length of each movement of an individual is 3 m. The global search algorithm is executed five times, and the localization results are shown in Table 2. From Table 2, we can see that the positioning results always hover around the true value.

Table 2.

Positioning results of the global search method.

Execution times	Positioning errors/m			Spatial errors/m
	X	Y	Z
1	0.5648	2.4141	0.9185	2.644
2	0.5422	2.4143	0.8795	2.6261
3	0.5304	2.4187	0.8788	2.6274
4	0.5351	2.4033	0.8829	2.6157
5	0.5425	2.4199	0.8831	2.6325

As can be seen from Table 2, the localization results output by the global search algorithm are unstable and always hover around the actual source location due to the influence of the artificial fish step length and the location of the artificial fish constantly changing with the change of food concentration during the global search process, which is not conducive to practical applications.

3.2. Local optimization algorithm localization process

The local optimization algorithm is a search method with fast convergence speed, high convergence accuracy, and stable convergence results in the local range. By setting the initial search value of the local optimization algorithm to the localization result of the global search algorithm, the localization speed and accuracy of microseismic events will be significantly improved, and the localization result will be stable.

The 0.1 m × 0.1 m × 0.1 m area near the localization result of the global search algorithm is divided into spatial grids, and the size of each grid is 0.01 m × 0.01 m × 0.01 m, and each grid node is a potential search value, and the localization result in Table 2 is set as the initial search value, and then the local optimization algorithm is executed, and the localization result is shown in Table 3.

Table 3.

Localization results of the local optimization method.

Execution times	Positioning errors/m			Spatial errors/m
	X	Y	Z
1	0.013	0.0245	0.028	0.0395
2	0.013	0.0245	0.028	0.0395
3	0.013	0.0245	0.028	0.0395
4	0.013	0.0245	0.028	0.0395
5	0.013	0.0245	0.028	0.0395

As can be seen from Table 3, compared with the global search algorithm, the local optimization algorithm has significantly improved the positioning accuracy, mainly because the local optimization algorithm divides the searched point into several spatial grids around the searched point in the search process, and then calculates the evaluation index of each grid node, and continuously adjusts the search direction according to the evaluation index until the global optimal value point is searched.

The search curve of the local optimization algorithm is shown in Fig. 4. As seen in Fig. 4, the search curve of the local optimization algorithm is a parabola with an upward opening. After the local optimization search starts, the objective function optimization value becomes a slowly decreasing trend and achieves the minimum value at the wave speed of 4200 m/s. As the iteration continues, the objective function optimization value becomes a slowly increasing trend.

Fig. 4.

Schematic diagram of the local optimization algorithm.

4. Field data analysis

Longjiapu coal mine is located in the Changchun New District Airport Economic Development Zone, with a wellfield area of 16.6 square kilometers and coal seams buried at depths mostly below 780 m. The Aramis M/E microseismic monitoring system is arranged at the mine site. The system has four components: underground seismic pickup, central surface station, data logging server, and data processing software, which can monitor vibration events with energy greater than 100 J and a frequency range of 0∼150 Hz.

Taking the blasting data from the 618 outer section workings of Longjiapu coal mine as an example, five manual blasting points from January 2022 to February 2022 were used for validation, and the blasting times and coordinates are shown in Table 4 and by the time of Table 5.

Table 4.

Blasting time and coordinates of blasting point.

Event Number	Time	Blasting point coordinates/m
		X	Y	Z
1	2022/1/29	69,710	73,609	1006
2	2022/2/7	69,698	73,640	1015
3	2022/2/11	69,694	73,660	1020
4	2022/2/15	69,722	73,615	1014
5	2022/2/20	69,665	73,669	1027

Table 5.

The coordinates of the first arrive time.

Sensor No.	Sensor coordinates/m			The first arrive time/s
	X	Y	Z	Event 1	Event 2	Event 3	Event 4	Event 5
1	69396.456	73659.927	1035.6	0.160	0.146	0.610	0.434	0.107
2	69245.908	73640.068	1051.4		0.178	0.642	0.468	0.141
5	70238.600	73671.700	911.20	0.198			0.468
7	69224.362	73815.770	1055.4			0.642		0.145
8	69379.560	73857.502	1026.4	0.172	0.160	0.624	0.448	0.123
9	69752.300	73617.200	1003.8	0.083	0.069	0.533	0.354	0.048
16	69950.469	73671.934	956.9	0.148	0.134	0.598	0.422	0.115

The global search and local optimization algorithm are executed, and the localization results are shown in Table 6 and Fig. 5.

Table 6.

Global search and local optimization algorithm positioning results.

Event Number	Positioning errors/m			Spatial errors/m
	X	Y	Z
1	2.9932	4.9673	7.7741	9.6990
2	2.6423	5.9853	2.9698	7.1851
3	1.9569	3.5598	2.7553	4.9085
4	4.6464	5.1519	3.4076	7.7294
5	4.7899	4.8768	5.0982	8.5275

Fig. 5.

Schematic diagram of the local optimization algorithm.

As can be seen from Fig. 3, the positioning errors of the field data are all under 10 m, and the positioning accuracy is good.

Comparative analysis of the accuracy of positioning algorithms.

The experimental data of the literature 1 [21] and 2 [12]were selected and compared with the localization results to verify the localization accuracy of the algorithm used in this paper. The blast coordinates of the literature are (8732.7, 6570.6, 511.3), and eight sensors received P-wave arrival times after the blast. The sensor coordinates and arrival times are shown in Table 7, and the comparison results are shown in Table 8.

Table 7.

Sensor coordinates and the first arrival times.

Sensor No.	Sensor coordinates/m			The first arrive time/ms
	X	Y	Z
9	8761	6614	522	34.9
21	8737	6609	565	36.6
5	8666	6600	520	39.3
17	8668	6599	565	41.1
4	8641	6515	520	42.3
8	8691	6684	520	44.5
2	8721	6449	520	47.8
26	8702	6604	647	50.0

Table 8.

Comparison of positioning results by method.

Positioning method	Positioning error/m			Spatial error/m
	X	Y	Z
Location of the epicenter	8732.7	6570.6	511.3	0
Least Squares	8720.6	6579.7	526.4	21.50
Simulated annealing - simplex method	8730.6	6580.7	504.5	12.40
Steady simulated annealing-simplex hybrid algorithm	8730.5	6580.9	503.9	12.90
Variable step size accelerated search	8731.4	6576.5	506.8	7.53
Global search and local optimization method	8732.3	6574.6	513.7	4.69

As shown in Table 8, the localization accuracy of the global search and local optimization algorithm improves by 78.19% over the Least squares method, 62.18% over the Simulated annealing-simplex method, 63.64% over the Steady simulated annealing-simplex hybrid algorithm, and 37.72% over the Variable step size accelerated search. In summary, the global search and local optimization algorithm is a feasible and high-accuracy localization algorithm.

5. Conclusion

• (1)The global search and local optimization algorithm combine the advantages of the artificial fish swarm algorithm for fast search in the global range and the hill climbing algorithm for high convergence accuracy and stable results in the local range, and high localization accuracy and stability without relying on the velocity model and initial values in the process of search.
• (2)By comparing the localization results of different localization methods, the global search and local optimization algorithms have better localization results.

Author contribution statement

Yimin Song: Conceived and designed the experiments; Analyzed and interpreted the data. Yazhou Zhao: Analyzed and interpreted the data; Wrote the paper. Yuanyu Zhang; Wenliang Deng: Performed the experiments. Chuang Lu: Contributed reagents, materials, analysis tools or data.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data included in article/supp. Material/referenced in article.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.