Automatic Identification of Rock Discontinuity Sets by a Fuzzy C-Means Clustering Method Based on Artificial Bee Colony Algorithm

Abstract

The identification and classification of rock discontinuities are crucial for studying rock mechanical properties and rock engineering optimization design and safety assessment. An improved artificial bee colony (ABC) algorithm is proposed and combined with the fuzzy C-means (FCM) clustering method to develop an FCM clustering method for automatically identifying rock discontinuity sets based on the ABC algorithm (FCM-ABC method). All the equations of the method are fully developed, and the methodology is presented in its entirety. Moreover, the rock structural planes are investigated in a gold mine in China using a ShapeMetriX 3D system. Based on the measured structural plane data, the specific calculation process, selection of parameters, effectiveness of grouping, and the dominant orientation of the proposed method for structural plane occurrence classification are analyzed and discussed, and satisfactory clustering results are achieved. This validates the validity and reliability of the method. Furthermore, multiple aspects of the excellent performance of this method for the identification of structural plane sets compared to traditional clustering methods are demonstrated. In addition, the significance of structural plane identification in the prevention and control of rock engineering disasters is discussed. This new method theoretically expands the technology of rock mass structural plane identification and has important application value in practical engineering.

1. Introduction

The grouping and orientation characterization of discontinuity (structural plane) sets is crucially important for rock mass characterization, geological disaster prevention and control, rock engineering applications (such as geothermal reservoirs, mines, tunnels, slopes, and water conservancy and hydropower), and earth science research [1,2]. Over a long period of geological evolution, the crust has undergone complex geological processes of construction and transformation, and various types and scales of structural planes, such as joints, fractures, and faults, are widely developed in the natural rock mass [3,4,5,6]. Because of the presence of discontinuity sets, the mechanical properties of rock masses exhibit significant anisotropic characteristics and are predominantly controlled by the structural plane. The occurrence, scale, density, morphology, and their combination of structural planes not only govern the stability of underground rock engineering but also dominate the shape, scale, and trend of the induced sliding body [7,8], because essentially, the structural plane changes the stress distribution mode in the rock mass. Also, the accurate identification of structural planes contributes significantly to the classification of the rock mass and the subsequent prevention and control of engineering geological disasters. As a consequence, on-site measurement and statistics of structural plane parameters in rock masses to detect structural plane characteristics and their combination distribution patterns are necessary prerequisites for quantitative analysis and the calculation of rock mass engineering stability.

In recent years, considerable efforts have been devoted to investigating effective methods for rock mass structural plane identification. Because of the strong randomness and uncertainty of the parameters of the structural plane, deterministic research methods are difficult to directly adopt. To analyze the regularity of developed structural planes, structural planes with some common characteristics are often classified, among which the most common is to group the structural planes and determine the dominant orientation according to their occurrence [9]. The traditional approaches for grouping structural planes generally adopt rose diagrams, pole plots, and isodensity plots. The representation of structural plane direction data and the identification of structural planes are usually conducted using techniques for the hemispherical projection of discontinuity poles (i.e., unit vectors with directions normal to structural planes) [10]. The advantage of this type of method lies in the ease of making an intuitive judgment on the distribution of the main structural planes, but the grouping results primarily depend on experience; in particular, the differences in results are even more pronounced when the boundaries between clusters are not clear [11,12]. Moreover, the visual clustering method of calculating density contour lines by calculating the number of poles falling within the reference circle is a commonly used method, but due to sampling bias, this method has some problems that need to be corrected [13,14]. In summary, these approaches may not be entirely satisfactory in some conditions, leading to the updating of alternative technologies for the automatic identification of structural planes.

Numerous attempts to address subjective issues have already been made using cluster analyses. For instance, Shanley and Mahtab [15] first proposed a clustering algorithm for structural plane orientation in 1976, which was then upgraded by Mahtab and Yegulalp [16] and Hammah and Curran [17] to develop an FCM clustering algorithm for the automatic identification of structural planes. Dershowitz et al. [18] described a random algorithm for clustering structural planes, which can handle the additional information provided; this algorithm is based on the probability distribution that defines each crack feature and involves the integration of these probability distribution functions. The above clustering algorithms are all dynamic clustering algorithms, which generally require a predetermined number of target sets to be divided and the determination of initial clustering centers for each type. Due to the significant impact of initial cluster centers on classification results, different initial cluster centers result in differences in the classification results. To this end, Cai et al. [7] proposed an FCM clustering method for rock mass discontinuity sets using a genetic algorithm, which avoided the subjectivity of artificially delineating classification boundaries. Zhou et al. [19] adopted a combination of fuzzy equivalence clustering and fuzzy soft clustering, using the optimal grouping results obtained from the fuzzy equivalence clustering method as the initial division of the fuzzy soft clustering approach, and using the clustering center obtained from the soft clustering method as the central orientation of each classification. Jimenez-Rodriguez and Sitar [20] used a spectral clustering algorithm to construct an affinity matrix for similarity measurement between discontinuities, used the eigenvalue of the affinity matrix to construct space vectors corresponding to the projection poles of discontinuities, and then employed the K-means method for classification. Behzad et al. [21] adopted the K-means clustering algorithm to group the multi-property data of discontinuity sets and applied the principal component analysis approach to calculate the eigenvalue of the covariance matrix of the measured data.

Additionally, statistical tools such as discriminant analysis, regressions, and decision analysis are used for exploratory analyses of structural plane data, and clustering methods that can combine other information besides structural plane orientation, such as planarity, weathering, spacing, or roughness, are proposed. Furthermore, methods of classifying structural plane sets using artificial intelligence technologies such as neural networks and machine learning have also been developed [22]. However, without prior information on the groups, data must be classified into homogeneous object groups, and cluster analysis is the most suitable tool for applications [17]. The clustering method is essentially a local search optimization approach, which is prone to getting stuck in local minima, thus dramatically affecting the clustering results of structural planes. Moreover, clustering analysis requires the number of clusters and fuzzy weighting index to be given in advance; how to scientifically and reasonably select these two parameters is currently an unresolved problem.

Against this background, new technical means need to be introduced to solve the above problems. In the present study, an improved ABC algorithm is proposed and combined with the FCM clustering method, and a new FCM clustering method for automatically identifying dominant rock mass structural planes based on the ABC algorithm (named the FCM-ABC method) is developed. All the equations of this approach are fully developed, and the methodology is presented in its entirety. Moreover, the structural planes of rock masses are investigated at multiple depth levels in a gold mine in China by using a ShapeMetriX 3D system (3GSM GmbH, Graz, Austria). Based on the measured data of structural planes, the specific calculation process, the selection of parameters, the effectiveness of grouping, and the determination of the dominant orientation of the proposed method for structural plane occurrence classification are analyzed and discussed, and multiple aspects including the higher recognition accuracy, faster convergence speed, and excellent performance of this method for the identification of structural plane sets compared to traditional clustering methods used in rock engineering applications are demonstrated. This method theoretically expands the technology of rock mass structural plane identification and has important application value in practical engineering.

2. FCM Clustering Algorithm

2.1. Measurement of Similarity

The FCM clustering algorithm was first proposed by Dunn [23] in 1974 and then extended by Bezdek [24]. In the FCM algorithm, the membership degree of each sample point to all class centers is obtained by optimizing the objective function, thereby determining the class membership of the sample points and achieving the automatic classification of sample data. Each sample is assigned a membership function belonging to each cluster, and the samples are classified according to their membership values. Among numerous fuzzy clustering algorithms, the FCM clustering algorithm is the most widely and successfully applied algorithm, which has been used in fields such as data mining and image processing.

A sample is a point in a feature space, and the distance between points reflects whether the corresponding samples belong to different types [11]. According to the distribution of main discontinuity sets in rock masses, the dip direction and dip angle parameters describing the occurrence of the structural planes are usually used as statistical indicators for cluster analysis. To facilitate analyses and comparisons, it is necessary to standardize the statistical indicators of each representative point. A good standardization method, while dimensionless, should also maintain the resolution of the original statistical indicators, that is, the magnitude of variability. For the clustering results and the proximity between the clustering center and the samples, the sine value of the included angle between normal vectors, similarity coefficient, or Euclidean distance are often used to represent them. Due to the characteristics of rotation invariance, sample features being unit vectors, and there being no need for the linear transformation of coordinate units, the Euclidean distance is adopted to measure the similarity between discontinuity sets in this study.

2.2. Characteristic Function

FCM clustering is to obtain the clustering center by minimizing the objective function, which is essentially the sum of the Euclidean distances (the sum of the squares of errors) from each point to each class. The objective function can be expressed as follows:

$$J_m={\displaystyle \sum _{i=1}^c{\displaystyle \sum _{k=1}^n{u}_{ik}^m{‖X_k-V_i‖}^2}}$$

(1)

where $X=\left\{X_1,X_2,\cdots ,X_n\right\}\subset {ℝ}^2$ is a finite sample in the two-dimensional real number space ${ℝ}^2$, which contains n sample data subsets; $X_k=\left(x_{k1},x_{k2}\right)\in {ℝ}^2$ is the sample feature vector, that is, the dip direction and dip angle parameter vector of the k-th sample; $x_{k1}$ represents the dip direction; $x_{k2}$ represents the dip angle; $V_i=\left(v_{i1},v_{i2}\right)$ is the i-th cluster center; c is the number of cluster centers; and $u_{ik}$ represents the degree to which the k-th sample belongs to the i-th cluster center and satisfies $u_{ik}\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^c{u}_{ik}=1}$.

When $X_k\ne V_i$, the following can be written:

$$u_{ik}=\left[{\left({\displaystyle \frac{1}{{‖X_k-V_i‖}^2}}\right)}^{{\displaystyle \frac{1}{m-1}}}\right]\cdot {\left[{\displaystyle \sum _{j=1}^c{\left(\frac{1}{{‖X_k-V_j‖}^2}\right)}^{\frac{1}{m-1}}}\right]}^{-1}$$

(2)

$$V_i={\displaystyle \frac{{\displaystyle \sum _{k=1}^n{u}_{ik}^m{X}_k}}{{\displaystyle \sum _{k=1}^n{u}_{ik}^m}}},\left(1\le i\le c,1\le k\le n\right)$$

(3)

where m is the fuzzy weighted index and $m\in \left[1,\infty \right)$, and j is a randomly selected subscript. The index m controls the allocation of membership degree and the fuzzy degree of clustering. The selection of m has been discussed in previous studies [17,25], and in general, taking m = 2 can meet the requirements.

By using the FCM clustering algorithm mentioned above, a fuzzy C-partition of $X$ can be determined. This fuzzy processing method can precisely characterize the actual distribution of data, which is particularly suitable for situations in which various data points have overlapping distributions. Nevertheless, because the FCM clustering algorithm is realized through the iterative optimization of the objective function, the objective function J_m decreases in the iterative process, and the option of the initial clustering centers and the input order of samples have a remarkable impact on the final clustering results. If the initial clustering centers are imbalanced in the entire sample space, the FCM clustering algorithm will probably fall into local minima, which is particularly prominent when the data volume is large, especially in high-dimensional cases. Thus, it is crucially important to give the initial cluster center as accurately as possible. That is, using clustering methods to divide the occurrence data of structural planes requires a rough judgment of the grouping beforehand, including the number of clusters and the orientation of cluster centers. For this problem, the common countermeasure is to cluster with several different initial clustering centers and then choose the most satisfactory one as the final clustering result. When this method is used for data analyses of the structural plane orientation, it not only requires a large amount of work but also cannot guarantee the optimality of clustering results. In view of this, an attempt is made to use the ABC algorithm to solve the global optimization problem in the FCM clustering algorithm of rock mass structural planes.

3. FCM-ABC Method for Rock Mass Structural Plane Identification

3.1. Principle of the ABC Algorithm

The ABC algorithm, developed by Derviş Karaboğa [26] in 2005, is a meta-heuristic optimization algorithm that imitates the behavior of bee colonies. It seeks the optimal solution by simulating the search and food collection processes of bee colonies, aiming at solving complex optimization problems, especially those that are difficult to solve by traditional methods. The algorithm exhibits the characteristics of simple individual behavior, distributed control, and strong robustness and scalability and is not constrained by domain knowledge. Compared to other methods, such as genetic algorithms, differential evolution algorithms, and particle swarm optimization algorithms, the ABC algorithm has greatly improved its convergence speed and algorithm performance [27,28] and has been applied in related fields such as cluster analysis [29], large-scale problems and project design optimization [30], and constrained optimization problems [31]. Combining the ABC algorithm with the FCM clustering method has advantages in terms of its global search ability, diversity maintenance, flexible parameter adjustment, efficient convergence, and adaptability to large-scale problems.

According to the different divisions of labor, bee colonies are classified into hiring bees, following bees, and reconnaissance bees [32]. The main task of hiring bees is to explore potential solutions in the search space. They choose a honey source (candidate solution), collect information on the honey source, and choose the next honey source to explore according to the quality of the current solution. The following bees observe the actions of the hiring bees and decide which honey source is worthy of further study based on the quality of the solutions discovered by the hiring bees. They explore this honey source with the hiring bees to obtain more information. If a honey source contains a better solution, the following bees will inform other bees. Three strategies for finding new honey sources are provided, as illustrated in Figure 1. The search space of reconnaissance bees is not limited to certain areas. If reconnaissance bees discover a potential honey source, they will transmit the information to other bees so that other bees can further explore the area. Therefore, the kernel of the ABC algorithm involves the following three parts: the hiring bees search for honey sources; the following bees select honey sources with a certain probability of searching according to the honey source information shared by the hiring bees; and the reconnaissance bees randomly search in the search space. In the search and optimization process of the ABC algorithm, the entire process is repeated over and over again until the end condition is met (i.e., the optimal solution is obtained).

3.2. Improved ABC Algorithm

When using the ABC algorithm to solve optimization problems, the location of the honey source is abstracted as a point in the solution space, which represents the potential solution of the problem. The mass of the honey source g (g = 1, 2, …, SN) corresponds to the fitness value fit_g of the solution, where SN is the quantity of the honey sources. Let D be the dimension of the problem to be solved, and the position of the honey source g at t iterations can be expressed as follows:

$$L_g^t=\left[l_{g1}^t, l_{g2}^t, l_{g3}^t, \cdots , l_{gD}^t\right]$$

(4)

The initial location of the honey source g is stochastically produced in the search space based on Equation (5):

$$x_{gh}=Q_h+rand\left(0, 1\right)\left(G_h-Q_h\right)$$

(5)

where x_gh∈(G_h, Q_h); G_h and Q_h are the lower and upper limits of the search space, respectively; h = 1, 2, …, D; and rand stands for a random number between (0, 1).

In the ABC algorithm, the richness of the nectar contained in each honey source, that is, the fitness fit_g of the solution it represents, is as follows:

$$fi{t}_g={\displaystyle \frac{1}{J_m+1}}$$

(6)

In which

$$J_m={\displaystyle \sum _{l=1}^R{\displaystyle \sum _{g=1}^S{m}_{gl}{d}^2}}\left(P_g,w_l\right)$$

(7)

where J_m is the sum of the squares of the deviations; S is the number of structural planes; R is the number of groups divided by S structural planes; P_g is the g-th grouped structural plane; w_l is the center vector (i.e., group center) of the l-th grouped structural plane W_l; d(P_g, w_l) is the distance between P_g and w_l; and m_gl represents the attribution relationship between P_g and W_l as follows:

$$m_{gl}=\left\{\begin{array}{l}1\hspace{1em}\left(P_g\in W_l\right)\\ 0\hspace{1em}\left(P_g\notin W_l\right)\end{array}\right.$$

(8)

At the beginning of the search, the hiring bees search around the honey source g according to Equation (9) to generate a new honey source, that is, update the solution:

$$v_{gh}=x_{gh}+o_{gh}\left(x_{gh}-x_{bh}\right)$$

(9)

where v_gh is a new honey source (solution) obtained from the x_gh neighborhood search; b is a honey source different from g (b ≠ g) randomly selected from SN honey sources; b∈{1, 2, …, SN}; o_gh represents a random number in the range of [0, 1], which controls the generation range of the x_gh neighborhood; and h represents a random integer in [1, D], which implies that the hiring bees stochastically select one dimension to search.

The newly produced possible solution is compared with the original solution:

New: v_gh = {v_g1, v_g2, …, v_gD}

(10)

Old: x_gh = {x_g1, x_g2, …, x_gD}

(11)

The greedy selection strategy is adopted to preserve the better solution.

The choice of the solution by the following bees is to judge the fitness value of the solution by observing the swing dance of the leading bees and selecting which hiring bees to follow according to the selection probability. The fitness value fit_g is computed using the following equation:

$$fi{t}_g=\left\{\begin{array}{l}{\displaystyle \frac{1}{1+f_g}},f_g>0\\ 1+\left|f_g\right|,f_g\le 0\end{array}\right.$$

(12)

where f_g represents the objective function value of the g-th solution.

The following bees follow the others according to the probability calculated by Equation (13) based on the honey source information shared by the hiring bees:

$$P_g={\displaystyle \frac{fi{t}_g}{{\displaystyle \sum _{e=1}^{SN}fi{t}_n}}}$$

(13)

where P_g represents the probability that the g-th honey source is chosen; and e denotes the maximum number of cycles.

In addition to the two control parameters of the number of honey sources and the maximum number of cycles, there is also another crucial control parameter in the ABC algorithm, i.e., the maximum number of times the same honey source is collected (limit), which is employed to record the number of times a solution is updated. If a solution has not been improved after limit consecutive cycles, it indicates that this solution is trapped into a local optimum and should be discarded, and then a new solution will be stochastically generated using Equation (5) to replace the original solution. The ABC algorithm defines two main behavior patterns that are essential for self-organization and swarm intelligence, namely, the positive feedback mechanism of recruiting more following bees to exploit richer honey sources and the negative feedback mechanism of abandoning honey source exhaustion [12]. The ABC algorithm combines global search and local search methods, achieving a good balance between honey source exploration and extraction for the bees, thereby greatly improving the performance of the algorithm.

In this study, the detailed calculation process of the FCM-ABC method is presented in Figure 2, which is automatically completed using a self-designed MATLAB program. The above approach also introduces the idea of a roulette wheel algorithm, which can provide better optimization direction and higher-quality candidate solutions in the iterative process of the ABC algorithm, strengthen the robustness and convergence speed of this algorithm, and balance its local search and global search capabilities.

3.3. Clustering Validity Verification

The clustering results obtained needed to be verified. At present, a variety of validity test indicators have been proposed. From the effectiveness of clustering, a good clustering result should be as clear of a division as possible [17,25]. Here, two test indicators, i.e., the classification entropy coefficient H_m and the fuzzy classification coefficient F_m, are employed to test the effectiveness of clustering, and the formulas for both are as follows [24]:

$$H_m=-{\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{i=1}^c{u}_{ik}}{\log }_a}\left(u_{ik}\right)$$

(14)

$$F_m={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^n{\displaystyle \sum _{i=1}^c{u}_{ik}^2}}$$

(15)

where the base number of logarithm a∈(1, ∞), and it is agreed that when u_ik = 0, u_iklog_a(u_ik) = 0 (the natural logarithm is taken in this study). For the two test indicators H_m and F_m, when H_m→0 and F_m→1, it indicates that the smaller the fuzziness value of the classification, the better the clustering effect.

In addition, to make an objective judgment on the optimal clustering results, it is necessary to use multiple indicators to test the clustering effect because fewer evaluation indicators may lead to one-sided evaluation results. Thus, another evaluation indicator, i.e., the fuzzy hypervolume F_h, is also used to evaluate the clustering effect. A minimum F_h value indicates the optimal fuzzy partition. The calculation formula for F_h is as follows [33]:

$$F_h={{\displaystyle \sum _{i=1}^c\left(\det \left(T_i\right)\right)}}^{0.5}$$

(16)

In which, T_i is a fuzzy covariance matrix and the following can be written:

$$T_i={\displaystyle \frac{{\displaystyle \sum _{j=1}^n{\left(u_{ij}\right)}^m}\left(X_j-V_i\right){\left(X_j-V_i\right)}^{\mathrm{T}}}{{\displaystyle \sum _{j=1}^n{\left(u_{ij}\right)}^m}}}$$

(17)

According to the proposed FCM-ABC method, after the clustering effect test, the appropriate clustering number c obtained is the structural plane grouping number. Each group represents the dominant structural plane at the measurement point, and the clustering center stands for the mean normal vector of this group of the dominant structural planes in that group, which can be converted into the average occurrence according to Equation (18). Meanwhile, the sample size in each group can also reflect the different levels of importance of the dominant structural plane.

$$X_l=\left(\cos {\alpha }_l\sin {\beta }_l, \sin {\alpha }_l\sin {\beta }_l, \sin {\alpha }_l\cos {\beta }_l\right)$$

(18)

where α_l and β_l represent the dip direction and dip angle of the l-th structural plane, respectively, and l∈[1, n].

4. Engineering Application

4.1. Discontinuity Set Survey in a Gold Mine

A gold mine is located in Shandong province, China, with a mining depth exceeding one kilometer underground. The structural planes such as joints and fissures in this mine area are quite developed (Figure 3), which seriously threatens the stability of roadways and stopes. Hence, investigating the scale, occurrence, and distribution of the structural planes in this mine is crucial for the failure mode analysis and stability evaluation of the surrounding rock. A 3D non-contact measuring system (ShapeMetriX 3D) for the geometric parameters of the rock mass is used to investigate the rock mass structural planes of the mine, which primarily contains a high-resolution standard DSLR zoom camera (Nikon Imaging Japan Inc., Tokyo, Japan), corresponding marker rod components, and a 3D visual analysis software system, as shown in Figure 4. This system can provide real 3D images of the rock mass surface and determine the geometric measurement data of the rock discontinuity sets by being combined with the matching image analysis software, thus recording the tunnel outline and the spatial position of the rock mass discontinuity. The system essentially relies on the digital image correlation technique, which organizes the digital information of the rock mass to obtain information on the rock mass structure plane occurrence.

Figure 5 shows the basic principle of the 3D non-contact photogrammetry. Firstly, the investigation area is determined, and the left and right views of the exposure surface are obtained through the calibrated high-pixel camera, such as the points P₁(u, v) and P₂(u, v). Afterwards, the two views in the picture are imported into a software analysis system, and the system automatically identifies the points P₁(u, v) and P₂(u, v) and reconstructs them in three dimensions to obtain 3D points P(x, y, z) based on a series of techniques such as pixel point matching and image distortion deviation correction. Finally, the 3D model is synthesized and the direction and distance are realistic to obtain a true 3D digital model of the rock mass surface. According to the reconstructed 3D surface, the discontinuity sets of rock masses can be marked, such as the joint trace length, spacing, and fault distance, and thus, the geometric occurrence information of structural planes can be statistically analyzed.

The structural planes of the rock masses are investigated at multiple depth levels in the gold mine by using the ShapeMetriX 3D system. The following is an example of measuring the structural planes at a measuring point at a depth of 765 m underground. The left and right views of the survey location are identified (Figure 6a). These two views are imported into the ShapeMetriX 3D software analysis system, the key measurement areas are delineated, the 3D models are synthesized, and the direction and distance are realistically transformed to obtain the 3D view of the rock mass surface (Figure 6b). On the synthesized 3D map, rock joints are grouped according to the distribution of the main joints and fractures, and different colors represent different groups. There are two groups of the main joint distribution, namely, the red group and the blue group in Figure 6c. Based on the spatial distribution and grouping of discontinuity sets, the stereographic projection map (Figure 6d) and the structural plane trace distribution map (Figure 6e), which can reflect the joint spacing, are drawn.

The spatial distribution of rock discontinuity sets can be described by spatial geometric parameters such as occurrence, shape, scale, and spacing, and these basic geometrical features can be used to determine the spatial shape characteristics of each group of rock mass discontinuities. The collected data are counted and calculated using the software of the ShapeMetriX 3D system, and the geometric parameters such as the trace length, dip direction, dip angle, and spacing of each group of joints are counted, respectively, and the statistical distribution law of each set of discontinuities is obtained, as listed in

Table 1. Geometric parameter values of structural planes and their distribution models.

Group	Trace Length (m)			Dip Direction (°)			Dip Angle (°)			Spacing (m)
	Distribution Model	Mean Value	Standard Deviation	Distribution Model	Mean Value	Standard Deviation	Distribution Model	Mean Value	Standard Deviation	Distribution Model	Mean Value	Standard Deviation
No. 1	Normal	0.44	0.103	Normal	251.88	39.4	Normal	56.65	16.06	Negative exponential	0.34	0.35
No. 2	Normal	0.25	0.06	Normal	49.21	19.75	Normal	42.70	16.92	Negative exponential	0.11	0.07

.

4.2. Verification of Discontinuity Sets’ Clustering Effect

The measured data of the structural planes at a depth of 765 m in this gold mine are used to validate the effectiveness and reliability of the automatic identification method for the structural planes proposed in this study. The FCM-ABC method is employed to cluster the joint parameters of each group together. The measured group size is 81, and the relevant parameters are set as follows: m = 2, c = 2–7, limit = 100, and e = 200. The two-dimensional distribution of joints on the surface of the original rock mass is plotted in Figure 7, showing two significant groups of dominant structural planes, which are used for comparison. The clustering results of the field discontinuity data automatically computed by the self-designed MATLAB code using different c values are presented in Figure 8, and the obtained values of the three cluster validity test indicators under different c values are provided in

Table 2. Obtained values of the three cluster validity test indicators under different c values.

c	2	3	4	5	6	7
Hm	0.10	0.21	0.19	0.19	0.19	0.22
Fm	0.58	0.25	0.18	0.13	0.10	0.09
Fh	998.47	1288.00	1340.69	1297.49	1322.03	1459.50

. Apparently, for the tested sample data, from c = 2 to c = 7, when c = 2, the classification entropy coefficient H_m is the closest to 0, the fuzzy classification coefficient F_m is the closest to 1, and the fuzzy hypervolume F_h is the smallest, indicating that the clustering effect is the best in this case and the grouping result is very satisfactory. According to the calculation results of the cluster center when c = 2, the average occurrence of the two sets of predominant discontinuity sets is determined, i.e., the dip direction of the first set of discontinuities is 248.16°, and the dip angle is 58.17°; the second set of discontinuities has a dip direction of 52.33° and a dip angle of 44.86°. The clustering calculation results are quite comparable to the actual measured results (Table 1 and Figure 7), which implies that the FCM-ABC method has a high solving accuracy and also verifies the correctness of the new algorithm.

Moreover, the correlation between the values of the cluster validity test indicators and c values is intuitively presented in Figure 9. Evidently, with the increase in the c value, H_m and F_h show an overall increasing trend, while F_m shows a decreasing trend. This indicates that the clustering effect is best when c = 2, which is identical to the real situation of the two dominant structural planes. The selection of the c value has a significant influence on the clustering results. Hence, choosing an appropriate c value is crucial. The proposed FCM-ABC method can accurately obtain the clustering center and membership degree through extensive training, which can effectively solve the problem of conventional clustering methods not being able to effectively select the appropriate initial clustering center. Accordingly, the new clustering approach can be applied to solve the problem of grouping the dominant structural planes of rock masses and reveal the geometric characteristics of the structural planes. In addition, the quality of clustering results will affect the convergence process to some extent. Figure 10 shows the process of searching for optimal solutions during the structural plane clustering calculation. The optimal solution is reached after around 125 cycles, and the actual time to find the optimum is approximately 0.02 s. The important feature of the ABC algorithm converging to the global optimal solution ensures that the proposed FCM-ABC method can quickly obtain the global optimal solution.

To further validate the performance and advantages of the proposed FCM-ABC method, this method was compared with other clustering algorithms (the FCM algorithm, genetic algorithm, and neural network) based on the same data. The clustering results that achieve the optimal solutions obtained by different methods are listed in

Table 3. Clustering results obtained by different clustering algorithms.

Algorithm	Hm	Fm	Fh	Iterations	Convergence Time/s
FCM-ABC	0.10	0.58	998.47	125	0.02
FCM	0.15	0.27	1195.91	308	8.99
Genetic algorithm	0.18	0.21	1453.83	562	306.69
Neural network	0.20	0.13	1501.55	40037	509.56

. Compared to the traditional clustering algorithms, the FCM-ABC method significantly reduces the convergence times, demonstrating a better algorithm performance and faster convergence speed. Consequently, the calculation time of the program can be greatly reduced in engineering applications, which is convenient for repeated calculations. Moreover, this new method recruits following bees through a roulette wheel algorithm according to the probability determined by the calculated Euclidean distance of the clustering and constantly updates the Euclidean distance, inducing the optimization process to approach the optimal solution more closely. While improving the convergence speed, it can also converge to better results, greatly improving the reliability and objectivity of the clustering results. Therefore, the approach that we developed has outstanding advantages in grouping analyses of structural planes, effectively avoids the shortcoming of conventional clustering algorithms easily falling into local optima, and has broad application prospects. This study provides an effective approach and method for the automatic identification and grouping of structural planes.

In the grouping calculation of structural planes, for different geological conditions, a clustering analysis can be performed as long as there are structural plane parameters of the dip direction and dip angle. Of course, other parameters can also be selected, but they have not been verified in this paper. For low-quality or incomplete structural plane data, they can be eliminated in advance manually or by compiling a program that meets certain rules to avoid affecting the accuracy of the clustering calculation results. Moreover, based on the clustering calculation via the FCM-ABC method using the dip direction and dip angle of the engineering structural plane in this study, the empirical thresholds of the three clustering effectiveness indicators H_m, F_m, and F_h could be recommended as H_m < 0.15, F_m > 0.5, and F_h < 1000, respectively, to evaluate whether the clustering calculation has reached the optimal solution. This may need to be adjusted according to different data types, specific engineering conditions, and actual requirements. In addition, the selection scheme for an appropriate number of clusters is generally predetermined with different numbers of partition groups (usually between two and ten) [34], and the corresponding grouping result is the best grouping result when the clustering effectiveness indicator values are optimal.

Notably, the above method is also used to cluster analyze the structural plane data (all of which are two groups of dominant structural planes) investigated at other depth levels in this gold mine, and good recognition results are also reached. Due to the space limitation, the clustering calculation results of the dominant grouping for these structural planes are not provided. Nevertheless, it should be emphasized that only the sample data containing two groups of dominant structural planes are used to analyze and verify the proposed FCM-ABC method in this study. To further verify the applicability of the proposed method, the sample data containing at least three groups of dominant structural planes will be used for the clustering analysis in the next step, which is also conducive to optimizing and perfecting the method. In addition, only two parameters, i.e., dip direction and dip angle, are employed to group the dominant structural planes of rock masses, and the proposed method should be used to identify the dominant structural planes based on more parameters in the future because more parameters can more accurately characterize the distribution and development laws of structural planes.

5. Discussion

Numerous studies [35,36,37] have indicated that the occurrence of rock engineering disasters (such as spalling, falling roofs, collapses, large deformations of surrounding rock, rockburst, and water inrush) is mostly due to insufficient attention given to the influence of structural planes in the rock mass. In practice, generally, rock engineering is not suitable for being built in rock masses with completely continuous structural planes but is usually built in rock masses with discontinuous structural planes. Discontinuous structural planes usually crisscross in rock masses, forming non-continuous bodies with complex structural characteristics. The influence of the structural plane on the mechanical response behavior of engineering rock masses is multifaceted. The morphology, size, spatial distribution, and combination characteristics of structural planes are key factors determining the mechanical and engineering properties of rock masses and also controlling the stability and failure mode of engineering rock masses.

Specifically, structural planes can cause additional deformations of the surrounding rock, such as the closure and sliding of the structural planes. The existence of structural planes can seriously deteriorate the integrity and bearing capacity of the rock mass, and the existing structural planes are the source of new cracks, making the surrounding rock more prone to deformation and damage in engineering. Furthermore, structural planes can also alter the stress distribution pattern in engineering rock masses (such as changes in the magnitude and direction of the circumferential stress) [3,38,39], resulting in an abnormal stress distribution in local areas and thus causing different failure modes of the surrounding rock. In particular, the failure mode is directly related to the occurrence of structural planes. In addition, the mechanical properties of rock masses exhibit remarkable anisotropic characteristics, and the interaction between structural planes and between structural planes and engineering structures (such as tunnels) increases the complexity and unpredictability of the mechanical response and disaster behavior of engineering rock masses.

The stability analysis of rock masses often relies on a thorough understanding of the characteristics of structural planes. In practical engineering, the structural planes of rock masses may have complex geometric features (such as different shapes, orientations, dip angles, and distribution densities), which have a decisive impact on the strength, deformation, and failure modes of the rock mass. By using automated methods to identify and analyze these structural planes, the spatial distribution information of the planes can be accurately obtained, providing reliable data for rock mass stability assessment. Moreover, the characteristics of rock mass structural planes directly influence the design and construction of rock engineering projects (such as underground tunnels, mining, and underground storage facilities). Engineering design needs to comprehensively consider the geometric characteristics, mechanical properties, and distribution of structural planes within the rock mass. Thus, accurately mastering this information is crucial for optimizing design and improving construction safety. Additionally, during underground engineering and mining operations, the structural planes of rock masses are often the source of disasters such as rock bursts, spalling, and collapses. Changes in the distribution and characteristics of structural planes directly affect the failure behavior of the rock mass under external loading. Therefore, the automatic recognition technology of structural planes provides new ideas and methods for disaster warning and risk management in rock engineering. More accurate data on structural planes could enhance the scientific and effective prevention of disasters, ensuring the safety of the project. In particular, as rock engineering moves towards intelligence and digitization, the application of automatic recognition technology provides fundamental data for the intelligent management of rock engineering. Combined with big data analysis and machine learning, the technology of automatically recognizing structural planes in rock masses not only provides real-time monitoring information but also allows for long-term dynamic analysis, offering real-time support for engineering decision making. In summary, automatically identifying structural planes and accurately obtaining their geometric information can provide a data foundation for engineering rock mass quality grading, rock mass stability assessment and analysis, 3D geomechanical model construction, rock support design, and disaster monitoring and early warning involved in rock engineering disaster prevention and control. This is conducive to solving the dual scientific challenges of the prevention and precise control of rock engineering disasters and intelligent emergency decision making.

It should be noted that although considerable progress has been made in the theory and technology of the automatic identification of rock discontinuities, due to the complexity and uncertainty of geological bodies and the urgent need for accurate prevention and control of rock engineering disasters, relevant research is still needed in the future to strengthen the refined and intelligent identification, extraction, analysis, and characterization of discontinuities, so as to obtain complete and true information, such as the occurrence and spatial distribution of structural planes, to improve the theoretical and technical level of disaster prevention and control.

6. Conclusions

The traditional classification methods for rock mass structural planes have many limitations when dealing with complex rock mass structures and large-scale data. To overcome the drawback of the FCM clustering algorithm easily falling into local optima, in this paper, an improved ABC algorithm is proposed and combined with FCM clustering, and a new FCM-ABC method is developed. Using the measured structural plane data, the specific calculation process, the selection of parameters, the effectiveness of grouping, and the dominant orientation of the proposed method for structural plane occurrence classification are analyzed and discussed, and satisfactory clustering results are achieved, verifying the effectiveness of the method. Also, multiple aspects of the excellent performance of this method for the identification of structural plane sets compared to traditional clustering methods are demonstrated. The approach can not only solve the problem of the sensitivity of traditional clustering algorithms to the initial clustering center but also accelerate the convergence speed of this algorithm and improve the clustering accuracy and stability of the algorithm, showing great potential in solving complex optimization problems. In particular, this method can avoid the subjectivity of artificially defining classification boundaries when the grouping boundary of dominant structural planes is not clear and prior knowledge is insufficient. Additionally, it can give objective and reliable results without relying on engineering experience, providing a reliable basis for the identification and classification of rock discontinuity sets, the optimization design and stability analysis of rock engineering, and the accurate prevention and control of engineering disasters. The method theoretically expands the technology of rock mass structural plane identification and has important application value in practical engineering. Note that to further verify the applicability and reliability of this method, the sample data containing at least three groups of dominant structural planes should be used for clustering analysis based on more parameters in the future. Also, corresponding software platforms to achieve the intelligent and refined grouping of structural planes should be developed to guide engineering practice.