Optimizing microseismic sensor networks in underground space using Cramér–Rao Lower Bound and improved genetic encoding

Abstract

The layout of a sensor network is a critical determinant of the precision and reliability of microseismic source localization. Addressing the impact of sensor network configuration on positioning accuracy, this paper introduces an innovative approach to sensor network optimization in underground space. It utilizes the Cramér-Rao Lower Bound principle to formulate an optimization function for the sensor network layout, followed by the deployment of an enhanced genetic encoding to solve this function and determine the optimal layout. The efficacy of proposed method is rigorously tested through simulation experiments and pencil-lead break experiments, substantiating its superiority. Its practical utility is further demonstrated through its application in a mining process within underground spaces, where the optimized sensor network solved by the proposed method achieves remarkable localization accuracy of 15 m with an accuracy rate of 4.22% in on-site blasting experiments. Moreover, the study elucidates general principles for sensor network layout that can inform the strategic placement of sensors in standard monitoring systems.

Headnote

Received 28 June 2024; received in revised form 8 February 2025; accepted 17 February 2025

Available online 8 June 2025

Abstract

The layout of a sensor network is a critical determinant of the precision and reliability of microseismic source localization. Addressing the impact of sensor network configuration on positioning accuracy, this paper introduces an innovative approach to sensor network optimization in underground space. It utilizes the Cramér-Rao Lower Bound principle to formulate an optimization function for the sensor network layout, followed by the deployment of an enhanced genetic encoding to solve this function and determine the optimal layout. The efficacy of proposed method is rigorously tested through simulation experiments and pencil-lead break experiments, substantiating its superiority. Its practical utility is further demonstrated through its application in a mining process within underground spaces, where the optimized sensor network solved by the proposed method achieves remarkable localization accuracy of 15 m with an accuracy rate of 4.22% in on-site blasting experiments. Moreover, the study elucidates general principles for sensor network layout that can inform the strategic placement of sensors in standard monitoring systems.

Keywords: Underground space; Sensor network optimization; Microseismic monitoring; Cramér-Rao Lower Bound; Improved genetic encoding

(ProQuest: ... denotes formulae omitted.)

1 Introduction

As an important non-destructive monitoring technology, microseismic (MS) source location technique is widely used in the underground space such as mining engineering (Askaripour et al., 2022; Kan et al., 2022; Leake et al., 2017; Zhou et al., 2024) and tunnel engineering (Zhang et al., 2024, 2021). The main purpose of MS source location is to monitor rock damages and predict dynamic disasters in underground space (Ma et al., 2020; Rui et al., 2022b; Zhu et al., 2025). However, achieving high-precision MS source location in complex underground environments presents significant challenges (Chen ct al., 2024a, 2024b; Rui et al., 2024a, 2022a). While multiple factors affect position-ing accuracy, sensor network layout is particularly crucial as it fundamentally determines the long-term monitoring capabilities of the system (Dong et al., 2013; Huang et al., 2023; Li et al., 2017; Shang et al., 2022; Yang & Scheuing, 2006). Poor sensor geometry can lead to near-singular control equations, making the system highly susceptible to input errors (Rui et al., 2024b). For instance, sensors arranged on cube vertices of velocity-free systems can cause complete system failure (Zhou et al., 2021b). Therefore, optimal sensor network design is essential for reliable and accurate source localization.

The optimal layout of MS sensor networks is crucial for source location, yet research on geometric optimization remains limited. Currently, network deployment relies heavily on human judgment, with performance only becoming apparent after installation and operation. While minor adjustments can be made later by adding sites or relocating sensors (Ge, 2005), these changes are still largely based on subjective decisions. This subjective approach leads to high costs and potentially irreversible losses in observation data quality.

While some studies have explored theoretical approaches to optimize sensor network geometry (Bao & Li, 2010; Kijko & Sciocatti, 1995), these methods have limitations. The pioneering work (SATO, 1965) introduced Monte Carlo algorithms to evaluate monitoring capabilities and generate error contour maps for seismic parameters. However, these early theoretical frameworks, though valuable for seismic network design, have limited applicability to microseismic monitoring in underground spaces. Arora et al. (1978) conducted extensive studies on the localization capabilities of two-dimensional station networks with triangular, quadrilateral and six-station arrangements. Uhrhammer (1980, 1982) gave a quantitative evaluation method for the localization efficiency of specific network geometries. Souriau and Woodhouse (1985) developed a method to extend existing sensor networks, which improves the resolution of localization results by comparing different candidate points. However, the abovementioned methods only studied the sensor network arrangement in some special cases. Recently, Li et al. (2020) studied the error geometric amplification caused by the non-uniformity of the MS hyperbolic field, discussed the relationship between the location accuracy and typical paired sensors, and gave some rough suggestions on the layout of the sensor network. Based on the experimental statistical theory, Kijko (1977a, 1977b) proposed the Dplanning method, which determined the optimal layout of a regional seismic sensor network by minimizing the determinant of the covariance matrix of the ellipsoid errors. Based on the classical D-planning method, scholars also put forward some variations or extension methods (Gao et al., 2013; Steinberg & Rabinowitz, 2003; Tang et al., 2006). For example, Gong et al. (2010) proposed the integrated index method to determine the hazard monitoring area and sensor arrangement, and then the final scheme can be formed by using D-planning guidelines. Zhou et al. (2022) constructed a comprehensive evaluation index for sensor networks with the goal of providing energy efficiency, taking into account the coverage area, envelope volume, and effective coverage radius.

Current sensor network evaluation methods have several critical limitations: (1) existing optimization metrics inadequately address MS source parameter estimation and network localization capabilities; (2) fundamental principles governing optimal sensor arrangement remain undefined; (3) most approaches are incompatible with velocity-free monitoring systems (Zhou et ah, 2021b); (4) computational complexity poses significant challenges to optimization implementation. To address these limitations, this study (1) develops an optimization function based on the CramerRao Lower Bound (CRLB) framework; (2) implements an improved genetic encoding algorithm for solution optimization; (3) validates the proposed methodology through multiple experimental approaches, including simulation tests, pencil-lead break tests, and mine blasting experiments.

2 Method

2.1 Construction of optimization function

In the MS monitoring system, the target sensors cannot be arranged arbitrarily due to the limitations of the engineering conditions, material geometry, or experimental means, but there are often many candidate sensor locations to choose from. Therefore, how to determine the optimal sensor network layout from these candidate sensor locations is essential to the efficient operation of the monitoring system. Herein, the specific solutions are derived. First, let the vector u(x. y. z)v (it E Qf denote the MS source coordinates. Second, assume that all possible candidate locations of MS sensors together form a sensor domain Qs. Third, suppose that there are M target sensors to be identified in this sensor domain, denoted as sAx;, y^zA. i = 1,2, - The distance between the M target sensors Si and the MS source и can be expressed as

... (1)

Without loss of generality, the first sensor si is chosen as the reference sensor. Then the hyperbolic equation between the sensors and st is obtained as

... (2)

where d is an unknown propagation velocity of media; ¿ц expresses the time difference of arrival between the sensors s¡ and si, i.e., f\ = ti - where the symbols ti and t\ respectively denote the arrival times received by the zth and the first sensors. denotes the difference in distance from the source u to the sensors si and 5i, i.e., -Di-D\. Assume that the probability distribution of a MS event at any point in the Qu region is known, its density distribution function is pff.

In real scenarios, there is always inevitable measurement noise in the time difference of arrival, which can be modeled as

... (3)

where {*}· denotes the real value of {*}, and n^ expresses measurement noise. Moreover, the measurement noise n^ is assumed as Gaussian random process with zero mean, and is independent of arrival time difference and source coordinates. If the standard deviation of measurement noise of arrival time difference is a, its M x M dimensional covariance matrix can be expressed as (Chan & Ho, 1994) ... (4)

where ...

Thus, the probability density function of t^ conditional on a can be expressed as

... (5)

where ... is the vector form of

The CRLB provides a theoretical lower bound for source location uncertainty given any sensor network configuration. This property makes CRLB an effective tool for evaluating sensor array performance. The mathematical framework for calculating CRLB (ÆCrl) can be expressed through the following equation (Chan & Ho, 1994):

... (6)

CRLB establishes the theoretical lower limit on estimation precision in statistical theory. It defines the minimum variance achievable by any unbiased estimator, derived from the Fisher information matrix which quantifies the inherent parameter information content within a dataset. An estimator that attains the CRLB is considered efficient, having achieved optimal precision in parameter estimation. Conversely, if an estimator's variance exceeds the CRLB, it indicates sub-optimal performance and potential estimator bias.

By taking the natural logarithm of the Eq. (6) and differentiating it, the CRLB framework can be reduced to

... (7)

where is the M x 4 Jacobi matrix with the expression of

... (8)

and

... (9)

The CRLB obtained by Eq. (7) is a square matrix of the dimension 4x4, whose diagonal elements are the estimated lower bounds of parameters x ,y, z, and г in that order. Since the source localization problem focuses on the MS source coordinates x ,y, and z rather than the wave velocity v, only the first three elements of the matrix diagonal are considered in this paper. By minimizing the CRLB of source coordinates, the optimization function of the sensor network layout can be constructed as

... (10)

where the subscript (z, j) denotes the index of the z-th column and j-th row of the matrix. It is worth noting that since the constant g in Eqs. (10) and (7) does not affect the estimation results of the minimization process, c> is directly ignored to simplify the calculation.

The above optimization function in Eq. (10) can be used when the area of MS activity is relatively small or concentrated. For example, if a sensor network is planned to monitor a specific area such as a fault or an empty area of mining operation, the above optimization function can be used to evaluate the localization performance of the sensor network. However, the above optimization function needs to be improved when the area of MS activity is large. In this time, the MS coordinate vector и is random, and Eq. (11) is a function of parameter u. By integrating MS events in the whole monitoring region, the objective function can be given as

... (11)

where p(u) is the probability distribution function of the source u at different monitoring locations, by which the key monitoring areas can be considered and weighted effectively.

The above equation takes into account the probability that the source coordinate и occurs in the domain Qu. The best estimate of vector u over domain Qu needs to be guaranteed by the distribution of sensor s, i.e., the following conditions need to be satisfied,

... (12)

Minimizing the above optimization function, the best estimate of sensor network layout can be obtained. The above evaluation model does not give a specific localization accuracy, which only supports the solution of the optimal sensor network. However, the solution of the above objective function is never an easy task, and the optimal solution of trying to find by exhaustive search is hardly possible.

2.2 Estimation of optimal sensor network layout

This section introduces an improved genetic encoding approach to solve the complex objective function. Rather than directly encoding M sensor coordinates, which would result in an unwieldy gene segment, the proposed method represents candidate positions as binary chromosomes. Each gene in the chromosome uniquely contributes to the fitness function defined in Eq. (12), enabling a compact representation of complex sensor networks while enhancing computational efficiency and accuracy.

The encoding scheme preserves the core evolutionary characteristics of genetic algorithms. Starting with a population of feasible solutions, it iteratively evaluates individual fitness, performs probability-based selection of highfitness individuals, and applies crossover and mutation operations to generate successive populations. The process continues until reaching either a satisfactory fitness level or maximum iteration count, as illustrated in Fig. 1.

The genetic encoding algorithm comprises basic operations (selection, crossover, and mutation) and core components (fitness function construction, candidate solution encoding, initial population generation, and genetic operator design). These elements are detailed in the following section.

(1) Encoding

Binary encoding is adopted for its operational simplicity and efficiency among available methods including symbolic and floating-point encoding. The chromosome structure represents N candidate sensor positions as a binary string (sec Fig. 2), where each gene (0 or 1) indicates the absence or presence of a sensor at the corresponding location (Worden & Burrows, 2001). For instance, in a chromosome "1001110010", sensors are positioned at the 1st, 4th, 5th, 6th, and 9th candidate locations among ten possible positions.

If M sensors arc selected and bi is the gene value at the position i of the chromosome, then the gene values at all positions of the chromosome satisfy the following constraints:

... (13)

Therefore, the sensor network optimization problem solved using genetic encoding can be expressed as

... (14)

(2) Improved initializing

Population initialization begins with random generation of feasible solutions, with potential "seeding" near optimal regions. Each individual must satisfy the constraint in Eq. (13). As illustrated in Fig. 2, for a required sensor count M = 4, individuals that don't meet this constraint (such as individuals 3 and 4) undergo re-randomization until their chromosome's gene sum equals 4.

(3) Improved selection operator

The selection operator preserves superior solutions while eliminating inferior ones, either by direct inheritance or through crossover and mutation operations. Individual fitness is evaluated using Eq. (12), with higher values indicating better fitness. This study employs truncated selection, where individuals arc ranked by fitness and the top performers advance to the next generation. Using a 70% truncation ratio, the algorithm retains the top 70% of individuals and eliminates the bottom 30%, as shown in Fig. 3. This approach ensures rapid convergence by efficiently identifying superior solutions within large populations.

(4) Improved crossover operator

Crossover operation involves gene fragment exchange between paired individuals to produce new offspring, mimicking biological chromosome synapsis. This study implements uniform crossover, where multiple crossover points are selected based on probability thresholds. For instance, given a set of random numbers, positions with values exceeding 0.5 become crossover points, as illustrated in Fig. 4(a). Uniform crossover demonstrates superior efficiency compared to classical crossover methods for several reasons: (1) finer granularity through multiple segment divisions enables faster discovery of improved patterns; (2) enhanced ability to avoid local optima; (3) better recombination capabilities while maintaining effective information exchange; (4) improved design space exploration.

The improved method is to first generate a set of random numbers d, the length of which is equal to the coding length of the chromosomes. Record the positions where the random numbers are greater than 0.5 to generate the set I(l Ç {1 : M}). Assuming that the two parents are Pl and P2, their genetic positions need to satisfy (Guo et al., 2004)

... (15)

In other words, the sums of the gene values of two individuals at position set / are equal, and the random array is considered as a valid position identifier. Otherwise, a new random array will be created again until the constraint of Eq. (15) is satisfied. Positions with probability values greater than 0.5 are considered as identifiers. The gene fragments at the parent chromosome identifiers will be swapped to create two new strings. Figure 4(b) illustrates the exact process of this crossover scheme.

(5) Improved mutation operator

The crossover operation, while preserving beneficial genes, is limited to recombining existing genes and can only achieve local optimization. To overcome this limitation and pursue global optimality, mutation operators are introduced to randomly modify selected genes on chromosomes.

Traditional single-point mutation, which randomly flips one binary value (e.g., from 1 to 0), may violate problem constraints, as shown in Fig. 5(a). Therefore, an improved double-point mutation strategy is implemented: it first randomly selects and flips a 1 to 0, then compensates by flipping a 0 to 1 at another random position, as shown in Fig. 5(b). This balanced approach maintains the chromosome's validity while introducing genetic diversity.

By the crossover and mutation operations, if the parents satisfy the constraint, then their offspring should also satisfy the constraint.

(6) Elite retention strategy

Genetic encoding faces an inherent challenge: genes are often interdependent rather than independent, making them poor direct representations of problem solutions. Simple crossover and mutation operations can inadvertently disrupt beneficial gene combinations, counteracting the goal of preserving advantageous genetic patterns.

The elite retention strategy addresses this issue by directly preserving the fittest individuals into the next generation. These elite individuals, which possess optimal genetic structures and characteristics, are protected from potential degradation through selection, crossover, and mutation operations. This preservation mechanism is crucial for (1) ensuring algorithm convergence; (2) maintaining the best solutions discovered; (3) enhancing the global search capability of genetic encoding.

This strategy forms an essential component of effective genetic encoding, safeguarding valuable genetic combinations while allowing continued evolution.

The specific steps of elite retention strategy are: let the individual a(k) be the best individual in the population of the Æth generation, while the population of the next generation is denoted by 1); and if no individual in the population 1) is better than a(kf replace the worst individual in the population Л(k + 1) with the individual a(k).

(7) Convergence criterion

This paper gives the following two termination criteria:

(i) The optimization process can be terminated when the most excellent individuals remain unchanged across consecutive generations. While a minimum of 100 iterations is necessary to avoid premature convergence, this study employs 200 iterations to ensure solution accuracy based on extensive experimental validation.

(ii) The iteration is terminated if the difference between the minimum fitness value /min and the average fitness value /avg of individuals in a generation is less than the threshold e.

... (16)

where a is a very small value characterizing a small change, such as 10 3 or 10 5.

If either of the two convergence criteria is satisfied, the calculation terminates.

Figure 6 shows the flow chart of the improved genetic encoding for solving the optimal sensor network layout, and the specific steps are described as follows.

(i) Randomly generate an initial set of populations, with each individual in the population being binary coded.

(ii) Calculate the individual fitness value using Eq. (14) and determine whether the convergence criterion is satisfied. If so, output the best individual (solution) and terminate the iteration, otherwise go to step (iii).

(iii) Based on the individual fitness, the truncated selection is used to eliminate the individuals with lower fitness and select the individuals with higher fitness.

(iv) Generate new individuals by the improved crossover operator.

(v) Generate new individuals by the improved mutation operator.

(vi) Generate a new generation of populations from the improved crossover and mutation, returning to (ii).

2.3 Number of sensors and dynamic network adjustment

2.3.1 Determination of sensor numbers

Determining the optimal number of sensors is a crucial first step in sensor network optimization. This decision is primarily driven by positioning accuracy requirements and budget constraints. Given the dynamic nature of mining operations and the significant costs of underground system installation, careful assessment of both short- and long-term monitoring needs is essential to minimize future network modifications.

Engineering assessments not only determine the required sensor count but also identify suitable installation locations. Sites should be selected to avoid fault zones, large cavities, and areas with poor rock coupling conditions that could compromise signal quality (Dong et al., 2022).

While specific requirements vary, deploying a larger sensor array generally offers distinct advantages: it provides more constraint information for MS source localization and reduces the likelihood of geometric singularities in the monitoring network.

2.3.2 Strategy of dynamic network adjustment

Regarding the dynamic design of sensor networks, our method offers a flexible and adaptable approach. As mining operations progress, the monitoring requirements naturally evolve, requiring adjustments to the sensor network. Our proposed methodology can be readily adapted to these changing conditions through the following process:

(1) Re-evaluation of the monitoring region based on ongoing engineering progress.

(2) Identification of new candidate sensor positions according to the updated site conditions and previous sensor networks.

(3) Application of our proposed network layout optimization method to determine the new optimal sensor configuration.

This iterative optimization process ensures that the monitoring system maintains its effectiveness throughout the project lifecycle. The flexibility of our approach allows for

(1) continuous adaptation to changing geological conditions;

(2) optimization of coverage as the area of interest shifts;

(3) maintenance of monitoring accuracy despite evolving site conditions.

3 Simulation analysis and discussion

3.1 One-dimensional sensor network layout

To investigate how sensor network configurations affect positioning accuracy in one-dimensional (ID) monitoring systems, several four-sensor networks with different layouts were analyzed (Fig. 7(a)). The study evaluated the optimal positioning accuracy achievable under each configuration through simulation. MS sources were uniformly distributed across the monitoring area, with signals received by the four sensors. The simulation assumed an arrival time error with standard deviation of a = 0.3 fis and wave velocity of d = 5000 m/s.

The CRLB was employed to assess the theoretical positioning accuracy of each network configuration (Fig. 7(b)). Results revealed significant variations in positioning performance across different layouts. Network 1 achieved the highest accuracy with an error of 1.0 mm, while network 2 failed to provide reliable positioning. Cross marks in the figure indicate problematic layouts where source locations, particularly those outside the sensor array, could not be resolved effectively. Such configurations should be avoided in practical applications.

To determine the optimal sensor network layout for a ID positioning system, the following simulation tests are further conducted. It is assumed that there are 9 candidate sensor locations in a ID monitoring system with a length of 100 mm, as shown in Fig. 8(a). This experiment requires selecting M (M < 9) sensors from the 9 candidate sensor locations to form an optimal sensor network layout. Figure 8(b) shows the schematic diagram of the optimal sensor network layout for the number of sensors from 3 to 7 that are determined using the sensor network optimization method proposed in this paper. In order to verify the accuracy of the proposed method, all C9M possible sensor networks are traversed by using exhaustive enumeration. The results of the optimal sensor network obtained by the exhaustive enumeration are identical to that obtained by the new method, which verifies the accuracy of the proposed method. In addition, to explore the influence of different number of sensors on the positioning accuracy, the theoretical positioning accuracy for the optimal sensor network layout with different number of sensors is expressed by the CRLB in this section. Herein, the arrival data arc set to obey a normal distribution with the standard deviation of a = 0.3 ps, and the propagation speed of the medium is set as г = 5000 m/s. The positioning accuracy of the optimal sensor network layout for the number of sensors 3, 4, 5, 6 and 7 is given in Fig. 8(c). It can be seen that the theoretical localization accuracy denoted by the CRLB value tends to decrease negatively exponentially as the number of sensors increases; in other words, the localization accuracy keeps improving with increasing the number of sensors. This phenomenon is easily understood from an error analysis perspective, where more constraints can be provided to assist in determining the final source localization results when more sensors are used. In addition, a larger number of sensors often imply a more rational sensor network layout. According to the principle of great likelihood estimation, when the number of sensors is large enough, the localization result will approach the true value infinitely. Therefore, it can be concluded that when a greater number of sensors are available in the monitoring system, as many sensors as possible should be used (but the sensors should not be abnormal) to improve the location accuracy.

By summarizing the simulation results above, it can be found that the sensor network layout should be placed inside and at both ends of the monitoring system at the same time. It is not the best choice to place all sensors only inside or at the ends of the monitoring area. In addition, for the ID positioning system, the situation that all sensors are concentrated or approximately concentrated on one or two locations should be avoided, otherwise the positioning result will be completely invalid.

3.2 Two-dimensional sensor network layout

In two-dimensional (2D) monitoring systems, a minimum of four sensors is required for MS source localization due to the four unknown parameters: x, y, ty and t0. To evaluate how sensor network configurations affect positioning accuracy, several common four-sensor layouts were analyzed (Fig. 9(a)).

The study simulated uniformly distributed MS sources across the monitoring area, with signals detected by all four sensors in different network arrangements. Using the CRLB to assess positioning performance, theoretical accuracy was calculated assuming a = 0.3 ps arrival time error and v = 5000 m/s wave velocity. Results (Fig. 9(b)) demonstrated varying positioning accuracies across different layouts, with Network 5 achieving the highest accuracy at 2.8 mm. Notably, network 3, despite its symmetrical common layout, showed the poorest performance due to its sensors being positioned on a circle's circumference. This circular arrangement creates ambiguity in the governing equations during residual minimization, making layouts similar to network 4 unsuitable for practical applications.

To determine the optimal sensor network layout for varying numbers of sensors in a 2D monitoring system, a 100 mm x 100 mm plane with 25 candidate sensor positions was established (Fig. 10(a)). The study investigated optimal configurations for M sensors (where M = 4, 5, 6, 7, 8, 17) selected from these candidates, with results shown in Fig. 10(b).

The proposed optimization method's effectiveness was validated through exhaustive enumeration of all possible sensor combinations and their corresponding CRLB values. Both approaches identified identical optimal layouts, confirming the method's reliability. Analysis of positioning accuracy across different numbers of sensors (Fig. 10(c)) revealed that positioning error decreases exponentially as sensor count increases, demonstrating continuous improvement in system accuracy with additional sensors.

It is worth mentioning that many sensor networks are similar (or called identical), as shown in Fig. 11. The four sensor networks in the figure are referred to as similar networks and are either centrosymmetric or axisymmetric to each other. When the probability of MS events occurring in the monitoring area is equal, theoretically these similar sensor networks also have similar localization accuracy. The proposed method will give only one case of these networks, and other similar sensor networks can be determined and selected by users as needed.

To illustrate the feasibility of the proposed method when considering key monitoring areas, the paper is supplemented with the following simulation experiments. Figure 12(a) illustrates the distribution of candidate sensor locations and MS sources, with the red region being the key monitoring area, which can be given a higher weight in Eq. (12). Figure 12(b) gives the optimal network layout under this key monitoring region, and then traverses all kinds of sensor network combinations one by one to carry out the source localization; and the experimental results show that the network layout with the minimum localization accuracy is exactly the same as that of Fig. 12(b), and thus prove that the optimal network layout of the proposed method is completely correct when considering the key monitoring region. In addition, it can be seen that the optimal network layout with a key monitoring region is different from Fig. 12(c) without considering a key monitoring region. Therefore, the consideration of key monitoring regions also has a certain impact on the accuracy of the localization system.

Through the above simulation results, the general rules of sensor arrangement in 2D monitoring system can be summarized.

First, the sensor network layout should cover both dimensions of the monitoring system. Second, sensors should be placed both inside and at the edges of the monitoring system whenever possible. Third, placing all sensors only inside or at the edge of the monitoring area is not the best option. Finally, for the 2D monitoring system, the following two sensor arrays should be avoided, otherwise the localization will be completely invalid.

(1) All sensors cannot be collinear at the same time, or serious singularity and ill-condition will occur in the governing equation, resulting in unsolvable positioning results.

(2) All sensors cannot be on the circumference of a circle at once, or the localization result is invalid. The reason for this is that in this sensor array, the coordinates of the geometric center and the solution with a velocity value of 0 always satisfy the control equation, making the localization result invalid. Therefore, for 2D monitoring systems, it should be avoided to place or approximately place all sensors on the circumference of a circle. If most sensors need to be placed on the circumference of the same circle, another sensor can be placed at the circle center, at which point the positioning system will be effective.

3.3 Three-dimensional sensor network layout

Three-dimensional (3D) source localization problem is the most common scenario in real engineering projects. At least five sensors are required to achieve the source localization in 3D monitoring systems, because there are five unknowns x ,y, z, v and Zo in the localization system. To study the effect of different sensor network layouts on localization accuracy in 3D monitoring systems, six common 3D sensor network layouts (all with the number of target sensors being 5) are studied, as shown in Fig. 13(a). To quantitatively analyze the localization accuracy of these sensor networks, this paper assumes that the MS sources uniformly present in the monitoring area, and then the signals from the MS sources can all be received by five sensors with different network structures. Finally, CRLB (with a = 0.3 ps and г = 5000 m/s) is used to evaluate these sensor networks, as shown in Fig. 13(b). It can be seen that the localization accuracy of different sensor networks is different, and the network 1 has the highest positioning accuracy of 2.66 mm, while the sensor network 2 has the worst positioning accuracy of 88.5 mm. The reason why sensor network 2 is the worst is that all sensor arrays are located at the vertices of the cube (i.e., exactly on the spherical surface of the external sphere). In such sensor network, the center coordinates of the sphere always satisfy the control equation, which tends to cause the unrecognizability of the real source position. Therefore, such sensor network layout should be avoided in practical engineering applications.

To further study the optimal sensor network layout in 3D monitoring system, a monitoring system with the size of 100 mm x 100 mm x 100 mm is assumed in this section. There arc 27 candidate sensor positions in the system, as shown in Fig. 14(a). M (M < 27) sensors are used and selected from the 27 sensor positions to form the optimal sensor network layout. Figure 14(b) shows the optimal sensor network layouts with different number of sensors determined by the proposed sensor network optimization method. Meanwhile, the result of the optimal sensor network layout is obtained by traversing all possible sensor networks using the exhaustive enumeration, which is exactly the same as the optimal network layout obtained by the proposed method. Therefore, the accuracy of the proposed method is verified in determining the optimal 3D sensor network layout. Supposing that the error follows a normal distribution with a standard deviation of a = 0.5 ps and the propagation wave velocity of the medium is v = 5000 m/s, then the theoretical positioning accu- racy of the sensor network layouts of different number of sensors can be expressed by CRLB, as shown in Fig. 14 (c). As can be seen from the figure, with the continuous increase of the number of sensors, the CRLB value decreases linearly and the positioning accuracy improves rapidly.

From the above simulation results, it can be concluded that the sensor network layout should cover all three dimensions of the 3D monitoring system, and the sensors should try to be placed both inside and on the surface of the monitoring volume. Placing all the sensors only inside or on the surface of the monitoring area is not the optimal choice. In addition, the following 2 special sensor arrays should be avoided for 3D positioning systems, or they will make the positioning results completely invalid.

(1) All sensors cannot be coplanar at the same time, otherwise the governing equation of the source position will be seriously singular or ill-conditioned, resulting in the localization failure.

(2) The array formed by all sensors cannot be exactly on the surface of a sphere, otherwise the localization result is invalid. This is because the solution with the sphere geometric center coordinates and the wave velocity of 0 always satisfies the control equations in such sensor array, which makes the true localization result not easy to find. Therefore, for 3D monitoring systems, all sensors should be avoided to be placed on the same spherical surface. If most of the sensors need to be placed on the spherical surface, a good solution is to place another sensor at the center of the sphere, where the localization system will take effect.

Moreover, to evaluate the cost computation of the proposed method, the time-consuming calculation of the optimization process of sensor networks for proposed method is carried out and compared with the exhaustive method. Figure 15 illustrates the computational efficiency of the proposed method with the number of target sensors as 5 and 7. As can be seen from the figure, when the number of target sensors is 5, the running time of the proposed method is 5.1 s, while the exhaustive method takes 71.9 s, which is 14 times that of the new method. As the number of target sensors increases to 7, the time taken by the new method basically remains unchanged at 6.6 s, while the running time of the exhaustive method increases exponentially to 1405.0 s, which is 213 times that of the new method. Moreover, as the number of target sensors continues to increase, ordinary computers are unable to perform the calculations. Therefore, we conclude that the proposed method in this paper has considerable computational efficiency and significant computational advantages.

Notably, optimal sensor configurations vary depending on the positioning method employed. The present study specifically addresses sensor network layouts for velocityfree positioning methods, and all layout recommendations presented herein are tailored to these velocity-free approaches. For comparison, our previous work (Rui et al., 2024b) extensively examined sensor network configurations for localization methods that utilize predetermined wave velocities.

4 Pencil-lead breaks experiment

To verify the feasibility of the proposed method in the aspect of sensor network optimization, the planar granite sample is used as the propagation medium in this chapter to perform the source localization. In this experiment, a hard-black pencil lead with a diameter of 0.5 mm was broken at an angle of 30° from the plane. Sixteen rupture sources were generated uniformly in this planar system by breaking the pencil lead, and six piezoceramic resonant sensors were used to detect the signal emitted by the broken sources. The planar layout of 16 sources and 6 sensors is shown in Fig. 16, and the specific coordinates are shown in Tables 1 and 2. Figure 17(a) shows the waveform of the rupture signal received by the sensor, while Fig. 17(b) shows the frequency distribution of the rupture signal. After the rupture signal being received by the sensor, the original signal is amplified by a pre-gainer with a gain of 40 dB. Then the amplified signal was recorded by the DS5-16C holographic rupture signal analyzer. The sam- pling frequency of the device is set as 3 MHz, which can completely cover the frequency domain of the rupture signal without distortion. Finally, the received rupture signals are stored in a computer for arrival time picking. In this experiment, the moment when the rupture signal crosses the threshold for the first time is determined as the arrival time, which is a commonly used method to pick up the arrival time, as shown in Fig. 17(c). The picking threshold for arrival time was set at 10 mV, slightly above the amplitude of the ambient noise. The experimental process and equipment arc shown in Fig. 18.

Sensor coordinates listed in Table 2 are the candidate sensor locations, and 5 of them need to be selected to form the optimal sensor network layout. Selecting 5 from the 6 sensor locations, there are a total of = 6 sensor network combinations, as shown in Fig. 19. The optimal network layout of 5 sensors can be easily determined by the proposed method, and the calculation result is shown in network 1 in Fig. 19(a). Next, the localization results of the 16 uniformly distributed rupture sources mentioned above will be used to verify the sensor layout determined by the proposed method. Based on the sensor coordinates and arrival times, the conventional time-to-difference (TD) method (Dong et al., 2011) can sequentially localize the 16 rupture sources for different sensor networks 1 to 6 and obtain their localization errors. Figure 19(b) shows the average error obtained by the TD method for locating 16 rupture sources under different sensor networks. As can be seen from the figure, sensor network 1 has the smallest positioning error and the best location accuracy, which is completely consistent with the result obtained by the proposed optimization method. In addition, the sensor network 2 is the worst, because the network layout is approximately arranged on the circumference of a circle where the positioning process is prone to produce singularity, leading to the positioning failure.

5 Engineering application

The practical applicability of the proposed sensor optimization method was validated through a field experiment in a mine equipped with 10 existing sensors. These sensors were originally deployed based on careful consideration of geological conditions and roadway configuration. The study treated all 10 existing sensor locations as candidate positions and aimed to identify an optimal network layout consisting of 6 sensors.

Three blast vibration experiments were carried out to generate three MS sources. All 10 sensors received the signals emitted from the blast sources, and one of the blast vibration signals and its frequency distribution are shown in Fig. 20. Then, the p-wave arrival time is picked up by the build-in software. According to the p-wave arrival time and the corresponding sensor coordinates, the locations of the three blasting sources can be determined by using the existing method (Zhou et al., 2021a). The optimized sensor network is shown in Fig. 21(a), the sensor candidate points are the actual locations of the deployed sensors in the mine, and all 10 white circles indicate the candidate sensor locations. The blue circles are the selected target sensors which form the optimal 6-sensor network. The location errors of three sources in the sensor network arc shown in Fig. 21(b). It can be seen that the location errors of the three sources are 10.93, 19.32, and 14.81 m, respectively, and the accuracies relative to the monitoring scale (distance between the two farthest sensors) are 3.07%, 5.43%, and 4.16%, respectively. The average localization error of the three sources is 15 m, and has the relative accuracy of 4.22%. Therefore, the deployed sensor network has the high localization accuracy, and the proposed optimization method is accurate and effective in engineering practices.

6 Conclusions

Accurate location of MS sources is the essence to monitor accidents and hazards, and ensure the safe and efficient production. Considering the long-term influence of sensor network layout on the positioning accuracy, a sensor network optimization method is developed by using the CRLB principle and improved genetic encoding. The proposed method of sensor network optimization mainly highlights the following five advantages.

(1) An optimization function for the sensor network is meticulously crafted to take full account of the estimation accuracy of source coordinates.

(2) This method pioneers the optimal network layout for systems with unknown wave velocities and methods that do not require velocity measurement, thereby offering a more extensive application scenarios.

(3) The control equations incorporate weights for key monitoring areas, aligning more closely with real engineering practices where varying monitoring emphases are common.

(4) The improved genetic encoding has higher computational efficiency and can efficiently give the results of the optimal sensor network.

(5) The general laws and rules of sensor network layout are concluded by a large number of experiments and simulation studies.

The study employed a comprehensive validation approach encompassing simulation tests across ID, 2D, and 3D monitoring systems, which confirmed the optimization method's accuracy and established key layout principles. For example, for a 3D monitoring system, sensors must span all three dimensions, and combine interior and surface positions. In addition, all sensors cannot be arranged in a plane at the same time, nor can they all be located on the surface of a sphere, otherwise the positioning will be completely invalid. Pencil-lead breaks experiments are carried out to further verify the accuracy and effectiveness of the proposed sensor optimization method. Furthermore, blasting experiments are conducted to verify the ability of the proposed method on engineering application, and the results show that the sensor network optimized by the proposed method holds the high location accuracy of 15 m (accuracy rate of 4.22%).

While the method shows clear advantages, its reliance on an average velocity model may introduce uncertainties in complex underground environments. Future work will focus on incorporating more sophisticated velocity models and ray tracing principles to enhance accuracy in realworld applications.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

CRediT authorship contribution statement

Yichao Rui: Writing - review & editing, Writing - original draft, Resources, Methodology, Conceptualization. Jie Chen: Writing - review & editing, Supervision, Resources, Conceptualization. Junsheng Du: Writing - review & editing, Validation, Data curation. Xiang Peng: Funding acqui- sition. Zelin Zhou: Validation, Supervision, Resources, Data curation. Chun Zhu: Supervision, Formai analysis.

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.

Acknowledgement

All authors would like to acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 52304123), Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (Grant No. GZB20230914), the 10th Young Talent Lifting Project of the China Association for Science and Technology (No. 2024QNRC001), China Postdoctoral Science Foundation (Grant No. 2023M730412), Sichuan-Chongqing Science and Technology Innovation Cooperation Program Project (No. CSTB2024TIAD-CYKJCXX0016), and National Key Research and Development Program for Young Scientists (Grant No. 2021YFC2900400).

Sidebar

* Corresponding author at: State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China.

E-mail address: [email protected] (J. Chen).

Peer review under the responsibility of Tongji University

References

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