Introduction

Rockburst is a relatively common underground hard rock engineering in high-stress areas with considerable risk of engineering geological hazards. As the depth of rock engineering increases, the elastic energy storage in the rock grows with it. When a rockburst occurs, the elastic energy stored in the rock is released dramatically, causing rock fragments to separate and eject from the rock mass [1], as shown in Fig. 1. Strong rockburst often causes casualties and equipment damage and even triggers local earthquakes and other catastrophic consequences, seriously hindering the progress of production. And as the mining depth increases, the rockburst problem becomes more and more serious and urgent. Before the actual construction of underground works, if we can accurately predict the level of propensity to rockburst disasters in the future, we can take appropriate preventive measures in advance to reduce casualties and property damage, so it is necessary to study the prediction of rockburst propensity. Therefore, how to accurately and effectively predict the rockburst has become the focus of many scholars [3]. In this regard, a lot of research has been carried out. Initially, to explore the laws and mechanisms of the occurrence of rockburst, people have carried out a large number of theoretical and experimental studies on different aspects of strength, stiffness, energy, etc., and proposed a series of rockburst criteria and classification methods, such as Hoek criterion [4], Russenes criterion [5], Turchaninov criterion [6], Barton criterion [7], Tao Zhenyu criterion [8], etc.

Fig. 1
figure 1

Actual situation of rockbursts [2]

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Although the evaluation method using a single evaluation index has the advantages of simplicity in form and convenience in engineering application, the evaluation results of different evaluation indexes often vary greatly, which shows that the single index method has obvious limitations [3]. As a result, scholars at home and abroad have proposed different multi-factor comprehensive prediction methods, which is the focus of current research on rockburst prediction. Some of these studies are shown in Table 1. These prediction methods can be divided into two categories. One is the prediction model based on a large amount of a priori rockburst engineering case data, using machine learning algorithms such as support vector machine, neural network, etc. For example, Zhou et al. [9] comprehensively considered eight impact indicators and predicted the rockburst grade with ten different supervised learning machine learning algorithms. Liu et al. [18] used stacking integration algorithm to integrate eight different machine learning algorithms to predict rockburst. Although many research results show that the rockburst prediction methods based on machine learning are acceptable, most machine learning algorithms ignore the actual physical meaning of the data, which is just like an opaque “black box” [11].

Table 1 Summary of existing rockburst prediction studies
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Another rockburst prediction method is based on the rockburst criterion and the fuzzy mathematical theory of rockburst prediction. Because the rock is not a homogeneous material, and there are many factors that affect the occurrence of a rock burst, so it is not possible to form a definitive rockburst rating criteria, that is, based on the judgment of the occurrence of rockburst level, there is a certain degree of uncertainty and randomness. The application of fuzzy mathematical theory can be a good solution to the problem. Rockburst prediction can be considered as a multi-indicator classification decision problem. Among them, using fuzzy evaluation methods can effectively overcome the associated uncertainties [30]. In the use of fuzzy mathematical theory to solve the classification problem of rockburst level, normal cloud theory (NC) and unascertained measure theory (UM) have been proven to have certain advantages and reliability in dealing with the randomness and fuzziness of rockburst [28, 31]. Aiming at the complexity and uncertainty of rockburst prediction, Liu et al. [32] combined with rough set and normal cloud theory researched rockburst grade prediction. Guo and Liu et al. [21, 33] combined the objective weighting method of Criteria Importance Through Intercriteria Correlation (CRITIC) with a multi-dimensional normal cloud model and used it to predict the grade of rockburst tendency. To effectively deal with the fuzzy information in the process of rockburst intensity prediction, Jia et al. [25] used the unascertained measure theory to determine the fuzzy membership of each evaluation index and establish the rockburst classification prediction model. Wu et al. [34] combined unascertained measure theory with intuitionistic fuzzy set theory and multi-source weight to establish a rockburst prediction model. Zhou et al. [35] comprehensively considered the subjective and objective factors of the index weight, established the combined weighting method, and established the rockburst risk evaluation model in combination with the unascertained measurement theory.

In essence, rockburst prediction is the process of converting the quantitative values of indicators to the qualitative concept of grade. However, when using the unascertained measure theory to establish the rockburst prediction model, most of the existing studies adopt segmented linear functions to determine the contribution of indicators to the results, which leads to the incomplete consideration of the randomness and fuzziness of rockburst. In many areas, the normal distribution is better suited to most cases of reality [36]. The normal cloud model applies this property of normal distribution. It treats the contribution of indicator values to the results within a certain rank interval as a normal distribution. Therefore, it is a good attempt to use the normal cloud model to determine the contribution of indicator values to the results.

This paper proposes NC-UM rockburst prediction model, combines the advantages of normal cloud theory and unascertained measure theory, and uses normal cloud theory to construct the single index measurement matrix. The game theory is used to synthesize the three different weighting methods, CRITIC, Entropy Weight (EW), and Analytic Hierarchy Process (AHP), to calculate the optimal comprehensive weight of the index, and a rockburst prediction model based on the improved unascertained measure theory with normal cloud model (NC-UM) is proposed. The model is tested with the data of rockburst cases, and compared with the prediction results based on the traditional unconfirmed measure theory model, the validity of this paper's model is demonstrated.

D depth, σθ maximum tangential stress, σc uniaxial compressive strength, σt tensile strength, σθ/σc stress coefficient, σc/σt brittleness coefficient, Wet elastic deformation energy index, Kv integrity coefficient of rock mass, RQD rock quality designation, σL axial geostress

Methodology of prediction model

The advantage of using unascertained measure theory in rockburst prediction model is that it takes into account the fuzzy and random characteristics of the evaluation factors. However, it does not specify a uniform affiliation function to characterize the affiliation of the indicator values to the rockburst level. In the previous studies, people usually use segmented linear affiliation function for triangles [28, 37, 38]. Other types of fuzzy evaluation often used are trapezoidal affiliation function [39] and normal affiliation function (bell-shaped) [40], as shown in Fig. 2. In this paper, we propose a rockburst prediction model that optimizes the unascertained measure using a normal cloud. The related fundamental theory will be carefully elaborated in this section.

Fig. 2
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Different types of affiliation functions

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Normal cloud theory

Cloud modeling is a tool to elaborate the problem of fuzziness and randomness of concepts in natural language, which was first proposed by Li et al. [41]. The normal cloud model is a basic cloud model, which that can characterize the uncertainty in the process of transforming qualitative concepts and quantitative representations [32]. The normal distribution is widely used in intelligent control and system evaluation, because it conforms to the laws of distribution in most social and natural sciences. Assuming that U is a domain and C is a qualitative concept on domain U. If the quantitative value x ∈ U, and x is a random realization of the qualitative concept C and satisfies the normal distribution x ~ N(Ex, En'2), En' ~ (En, He2), where Ex, En, and He is called expectation, entropy, and hyperentropy, respectively, then the membership of the quantitative value x to the qualitative concept C satisfies the following formula:

$$ \mu = {\text{exp}}\left[ { - \left( {x - Ex} \right)^{2} /2En^{{{^{\prime}}2}} } \right]. $$
(1)

Cloud drops characterizing the target properties in the cloud model are generated by the cloud generator. It is divided into two types: positive cloud generator and reversed cloud generator. The former can realize the transformation from qualitative concept to quantitative value, while the latter can realize the transformation from quantitative value to qualitative concept [21]. In this paper, the positive cloud generator is used to quantify the membership of each rockburst sample to each rockburst grade, as shown in Fig. 3. The main algorithm steps are as follows:

Fig. 3
figure 3

Positive cloud generator

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Step 1: According to the rating criteria of different indexes, the corresponding digital features of the cloud model are calculated Ex(Ex1, Ex2,…,Exm), En(En1, En2,…,Enm), He(He1, He2,…,Hem). The calculation method is mentioned in Section “Digital features of cloud model”.

Step 2: Generate normal random numbers En'(En1', En2',…,Enm') with En and He.

Step 3: Generate n normal random numbers xi with Ex as expectation and En', as standard deviation.

Step 4: Calculate the membership degree μ(xi) of quantitative value xi to a qualitative concept C using formula (1).

Unascertained measurement theory

Unascertained measurement theory is mainly used to solve the mathematical problems of fuzziness and uncertainty. Because of its significant advantages in the classification of ordered fuzzy problems, it is gradually applied to rockburst prediction [28]. Its core is to calculate the membership degree of each rockburst risk level, which is used to represent the uncertainty relationship between the evaluation index and the evaluation grade. Finally, introduce the confidence theory to identify the final prediction level. The relevant definitions and calculation steps are as follows [34, 35].

Assuming that there are n evaluation samples within the evaluation object, that is, the evaluation sample set R = {R1, R2,…,Rn}. Each evaluation object contains m evaluation indexes, that is, the index space of samples I = {I1, I2,…,Im}. Then, each evaluation index contains η risk evaluation grades, that is, the evaluation grade space C = {C1, C2,…,Cη}, and Cn is more dangerous than Cn-1, which is recorded as Cn > Cn–1, If Cη > Cη-1 > ··· > C2 > C1, {C1, C2,…,Cη} is called the ordered classification of evaluation grade space C. The main steps of unascertained measurement theory are as follows:

Step 1: Define the single index measure matrix

According to the single index measure function, the measure value μijk = u(Iij ∈ Ck) is calculated, where μijk represents the membership of index Iij to grade k, and μ meets the following conditions:

Nonnegativity and boundedness:

$$ 0 \le \mu_{ijk} \left( {I_{ij} \in C_{k} } \right) \le 1. $$
(2)

Uniformity:

$$ \mu_{ij} \left( {I_{ij} \in C} \right) = 1. $$
(3)

Additivity:

$$ \mu_{ij} \left( {I_{ij} \in \bigcup\limits_{k = 1}^{p} {C_{k} } } \right) = \mathop \sum \limits_{k = 1}^{p} \mu_{ij} \left( {I_{ij} \in C_{k} } \right){ }\left( {k = 1,2, \ldots ,p} \right). $$
(4)

If the above conditions are met at the same time, the single index measurement matrix can be constructed as follows:

$$ \left( {\mu_{ijk} } \right)_{m \times \eta } = \left[ {\begin{array}{*{20}c} {\mu_{i11} } & {\mu_{i12} } & {\begin{array}{*{20}c} \cdots & {\mu_{i1\eta } } \\ \end{array} } \\ {\mu_{i21} } & {\mu_{i22} } & {\begin{array}{*{20}c} \cdots & {\mu_{i2\eta } } \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots \\ {\mu_{im1} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {\mu_{im2} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} \ddots & \vdots \\ \end{array} } \\ {\begin{array}{*{20}c} \cdots & {\mu_{im\eta } } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]. $$
(5)

Step 2: Calculating multi-index comprehensive measure vector vi

$$ v_{ik} = \mathop \sum \limits_{j = 1}^{m} w_{j} \mu_{ijk} { }\left( {i = 1,2, \ldots ,n;k = 1,2, \ldots ,\eta } \right) $$
(6)
$$ v_{i} = \left[ {v_{1} ,v_{2} , \ldots ,v_{\eta } } \right]. $$
(7)

Step 3: Analysis of confidence recognition criteria

When the evaluation grade space C is orderly, the maximum affiliation identification criterion does not characterize the true evaluation level well [28]. For example, if the comprehensive measure vector of rockburst evaluation sample v = {0.4, 0.3, 0.3, 0}, it is judged to belong to no rockburst (I) according to the maximum affiliation identification criterion, but it is easy to find that the total affiliation degree of weak rockburst (II) and medium rockburst (III) reaches 0.6, more than half of the total membership, so it is unreasonable to classify rockburst grade as no rockburst (I). The model of confidence recognition criteria is as follows:

$$ P_{k} = {\text{min}}\left\{ {k:{ }\mathop \sum \limits_{i = 1}^{k} v_{i} \ge \lambda ,{ }k = 1,2, \ldots ,\eta } \right\}, $$
(8)

where λ is the confidence (usually 0.6 or 0.7). When the value of k is taken from small to large until the above formula is met, it is considered that identified results belong to the k-th evaluation grade Ck.

Comprehensive weighting of game theory

When using fuzzy mathematical theory to solve multi-indicator evaluation problems, it is inevitable to consider the indicator weighting problem. In the existing studies, it has been confirmed that the evaluation index weights play a crucial role in the prediction accuracy of the model [35]. It is difficult to ensure the uniformity of weights of indicators by a single method of assigning weights, and there are great differences between the results obtained by different methods. Therefore, to ensure the stability and uniformity of index weights, this paper uses the game-integrated weight theory to obtain the optimal weights, which integrates three different weighting results of CRITIC, EW, and AHP. The basic idea of game theory is to find a balanced optimal weight among the weights obtained by different methods. The optimal weight is a linear combination of different basic weights such that it minimizes the deviation from the basic weights. The calculation method is as follows [42]:

Step 1: Assuming the existing index weight wi (i = 1, 2, 3,…,ξ) calculated by ξ different methods

$$ w\left( l \right) = \left[ {w_{l1} ,w_{l2} ,w_{l3} , \ldots ,w_{lm} } \right]{ }l = 1,2, \ldots ,L. $$
(9)

Suppose any linear combination of ξ weight vectors is w

$$ w = \mathop \sum \limits_{l = 1}^{L} \alpha_{l} w_{l}^{T} . $$
(10)

Step 2: To minimize the deviation between w and each basic weight, the linear combination coefficient α in the above formula is optimized, and finally, the optimal combination weight w* is calculated. That is, the following conditions are satisfied:

$$ \min \mathop \sum \limits_{l = 1}^{L} \alpha_{l} w_{l}^{T} - w_{p2} \left( {p = 1,2, \ldots ,\xi } \right). $$
(11)

Its optimal solution is as follows:

$$ \mathop \sum \limits_{l = 1}^{L} \alpha_{l} w_{p} w_{l}^{T} = w_{p} w_{p}^{T} . $$
(12)

Calculating the linear combination coefficient α1, α2,…,αξ and normalize it according to the following formula:

$$ \alpha_{l}^{*} = \frac{{\left| {\alpha_{l} } \right|}}{{\left| {\alpha_{1} } \right| + \left| {\alpha_{2} } \right| + \cdots + \left| {\alpha_{\xi } } \right|}}{ }\left( {l = 1,2, \ldots ,\xi } \right). $$
(13)

Calculating the comprehensive weight vector of the index

$$ w^{*} = \mathop \sum \limits_{l = 1}^{L} \alpha_{l}^{*} w_{l}^{T} . $$
(14)

Model building and calculation

The core of the rockburst prediction is the numerical analysis and calculation of multiple indicators to achieve the quantification of the propensity level. This section shows the details of the model calculation steps. And determine cloud model feature parameters and calculate index weights.

Framework of prediction model

Due to the randomness and fuzziness of rockburst, the evaluation indexes have the randomness of distribution and the variability of rating in different grading intervals. This paper proposes to the rockburst prediction model of normal cloud-unascertained measure theory. The output of the cloud model represents the affiliation degree of indicators to different rockburst levels, resulting in an improved single index measurement matrix. Then, the rockburst intensity grade is identified based on the unascertained measure theory. The framework of the prediction model is shown in Fig. 4, and the main process is as follows:

  1. 1.

    Determine the rock explosion prediction index system. Factors affecting the occurrence of rockburst are multifaceted. The selection of appropriate evaluation indicators is significant for prediction results.

  2. 2.

    Based on previous studies, combined with the rockburst evaluation index system, establishing the rockburst grade classification criteria.

  3. 3.

    Calculate the digital features of the cloud model based on the classification criteria (Exi, Eni, Hei).

  4. 4.

    Review relevant literature and obtain data on rockburst cases in underground projects. Divide all data into weight set and test set. The former is used to calculate the objective weights of the indicators, and the latter is used to verify the prediction results.

  5. 5.

    Combined with the data from test set, generate normal clouds using the positive cloud generator. Each cloud represents the affiliation between the evaluation index values and the different rockburst level properties.

  6. 6.

    Construct an improved single index measurement matrix based on the information from cloud drops.

  7. 7.

    Combined with the data from weight set, three different weighting methods, CRITIC, EW, and AHP, are integrated through game theory to calculate the optimal combination weight.

  8. 8.

    Calculate a comprehensive measurement evaluation vector of multiple indexes. Identify the rockburst intensity level by the confidence criterion and output the model prediction results.

Fig. 4
figure 4

Schematic diagram of prediction model

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Data screening and selection of evaluation indicators

There are 267 groups of data collected in this paper, all occurring in tunnels, hydraulic and hydroelectric projects, or mines in the world [9, 26, 43,44,45,46,47]. Referring to the research on the existing rockburst evaluation system [28, 33], four influencing factors are selected as the evaluation index set for rockburst grade prediction, including uniaxial compressive strength σc, stress coefficient σθ/σc, elastic deformation energy index Wet, and brittleness index σc/σt. The rockburst intensity grade classification standards are shown in Table 2. To obtain better model stability, 18 rockburst samples containing the instability measured values are removed after data screening. Including 11 samples with unstable stress coefficient index and 7 samples with unstable brittleness index, as shown in Fig. 5. A total of 249 rockburst samples are left as the weighting set to calculate the comprehensive weight of each index. Thirty cases were randomly selected as the model test set to test the reliability and accuracy of the prediction model, as shown in Table 3.

Table 2 Rockburst evaluation index system and classification criteria [28, 29]
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Fig. 5
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Box diagram of data set

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Table 3 Samples of model test set
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Digital features of cloud model

Expectation (Ex), entropy (En), and hyperentropy (He) are the three digital features that characterize the qualitative and quantitative concepts of the normal cloud model. As shown in Fig. 6, Ex indicates the center of the cloud, that is, the center point of the evaluation interval of the index. En denotes the size of the range of the cloud droplet group. According to statistical knowledge, the range of Ex around 3En should cover 99% of conceptually acceptable elements, that is, 3En criterion. He indicates the thickness of the cloud; the larger the He, the greater the dispersion of the cloud drops.

Fig. 6
figure 6

Digital characteristics of cloud model

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Referring to the research methods of Liu and Guo et al. [32, 33], this paper uses the following methods to calculate the digital features of the normal cloud model, and the calculation results are shown in Table 4:

$$ Ex_{ij} = \left( {B_{\min } + B_{\max } } \right)/2 $$
(15)
$$ En_{ij} = \left( {B_{max} - B_{min} } \right)/6 $$
(16)
$$ He = k, $$
(17)

where i = 1, 2,…,m; j = 1, 2,…,η. Bmax and Bmin are the upper and lower limits of the ith index corresponding to the jth rockburst grade standard respectively. For the case of single boundary, the value limit of data is used to replace the missing boundary value. He is expressed as cloud thickness, which is an uncertainty measure of entropy. It can be taken according to the size of k. generally, the greater k is, the greater He is.

Table 4 The results of digital features
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Comprehensive index weighting based on game theory

The index weight has a great impact on the results of rockburst prediction. The existing weighting methods are mainly divided into subjective weighting methods and objective weighting methods. Each method has its limitations. Therefore, in most cases, it is difficult to truly characterize the index weights using a single method. Thus, this paper uses game theory to combine three classical weighting methods: AHP, EW, and CRITIC. Taking into account the information contained in the indexes and the correlation between the indexes, the optimal comprehensive weight is calculated according to the steps described in Section “Uncertainty measurement theory”.

Based on the 249 groups of rockburst data in the weighing set, the weights of the four evaluation indexes were calculated using different weighting methods. The specific calculation method can be referred to Lu, Shi, Guo et al. [27, 28, 33]. This paper does not elaborate overly. The weight values calculated by CRITIC, EW, and AHP are as follows, respectively: w1 = [0.2471, 0.1867, 0.3022, 0.2639]: w2 = [0.3150, 0.1568, 0.3520, 0.1763]: w3 = [0.2, 0.2, 0.4, 0.2]. Finally, the game comprehensive weighting method is used to obtain the optimal combination weight coefficient \(\alpha\) = [0.2972, 0.3404, 0.3624], and the optimal comprehensive weight is calculated \(w\) = [0.2531, 0.1813, 0.3546, 0.2109]. The weighting results of each method are shown in Fig. 7.

Fig. 7
figure 7

Index weights under different weighting methods

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Model validation

To verify the validity and feasibility of this proposed model, 30 groups of data from the test set were used to conduct an analysis. Measured values of indicators are listed in Table 3.

Single index measurement matrix

The construction of an acceptable single index measurement matrix is the core of the prediction model in this paper. Because each element of the matrix should be able to fully express the classified attributes of the evaluation object. The traditional construction method is to characterize the randomness of index values in different rating intervals using a segmented linear function [28]. In this paper, the normal cloud model is used to improve the traditional linear measure function, and generates the improved measure function based on the positive cloud generator, as shown in Fig. 8. Combined with the digital characteristics in Table 4, the rockburst data in the test set were used as model inputs and substituted into Eq. (1). The membership degree uijk of each index to different rockburst grades is obtained, and the single index measurement matrix μijk(m×n) is obtained after normalization.

Fig. 8
figure 8

Single index measurement function improved by the normal cloud model

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The distribution diagram of the single index measurement of samples 12 and 27 is as shown in Fig. 9. From the diagram, we can see the affiliation of each evaluation index on the rank attribute. For example, for the No. 12 rockburst sample, the uniaxial compressive strength index value calculated by the cloud model has membership degrees of 0.9175 and 0.0825 for grade II and III, respectively; The membership degree of stress coefficient index values to grade II and III is 0.9751 and 0.0249 respectively; The membership degree of elastic energy index value to grade III and IV is 0.7602 and 0.2397, respectively; the membership degree of brittleness index value to grade III reaches 0.9992.

Fig. 9
figure 9

Distribution of single index measures of samples 12 and 27. a Single index measurement of sample 12; b single index measurement of sample 27

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Prediction results and analysis

In Section “Comprehensive index weighting based on game theory”, based on the game theory and combining the three weighting methods of CRITIC, EW, and AHP, the importance ranking of the four indexes has been obtained as elastic deformation energy index > uniaxial compressive strength > brittleness index > stress coefficient, and the comprehensive weight values of the index are 0.3546, 0.2531, 0.2109, and 0.1813, respectively. Substitute the single index measure matrix into Eq. (12) to obtain the multi-index comprehensive measure vector (C1, C2, C3, C4) of each sample of the test set, as shown in Table 6. Then identify the prediction result according to Eq. (11) (λ = 0.6), and compare it with the actual grades and the prediction result of the traditional unascertained measure model method [28]. The results are shown in Fig. 10.

Table 5 Calculated results of multi-index comprehensive measure vectors
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Fig. 10
figure 10

Predicted results using the improved model and traditional model

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In Fig. 10, the left side of the histogram in the same group represents the model of this paper, and the right side represents the traditional model. It can be easily seen that, except for samples No.16 and No.23, the prediction results of the remaining 28 samples are consistent with the actual grade. Compared with the traditional unascertained measure method, the prediction accuracy is improved from 76.7% to 93.3%, which shows that using the cloud model as a transformation method between the quantitative index and qualitative concept is reliable and has certain advantages. It has a higher accuracy rate than traditional prediction methods.

Engineering application

For verifying the feasibility and applicability of the prediction model in this paper, ten rockburst cases of large underground projects in the world are selected in this chapter to test the model. Meanwhile, introduce more comparison of the results of other prediction method, ideal solution (TOPSIS), and multi-dimensional normal cloud (MNC). To ensure the reasonability of the index weights, the comprehensive weights derived from the previous section based on 249 groups of case data are still adopted. After model calculation, the grade prediction results and real rockburst grades of each rockburst example are shown in Table 7.

Table 6 Rockburst cases and prediction results of some projects at home and abroad
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It can be seen that the prediction accuracy of the model proposed in this paper reaches 80%. The deviation between the predicted grade and the actual grade is no more than one grade. MNC, NC-UM has a higher prediction accuracy, compared with the UM, TOPSIS. The application results show that the NC-UM rockburst grade prediction model has higher reliability and universality. Compared with the traditional segmental linear measure function, the normal cloud model can better characterize the affiliation between the index values and the rank attributes when constructing the single index measure matrix. Among these, the prediction accuracy of the MNC model is the lowest, which is partly because the cloud model numerical features are taken on the total interval, which leads to a large difference between the calculated affiliation and the actual affiliation; on the other hand, the method takes the maximum affiliation criterion to identify the rockburst level, and its performance on the ordered classification space problem is not as good as the confidence identification criterion.

Conclusion

In this paper, a rockburst prediction method is proposed. It is based on the unascertained measure model which is improved by the normal cloud theory. Using normal cloud theory to construct the single index measurement matrix, which optimized the uncertainty representation method. And comprehensively considering the subjective and objective weights of the index, getting the comprehensive weight with the game theory. After verifying the accuracy of the model, the following conclusions are obtained:

  1. (1)

    Uniaxial compressive strength σc, rock stress coefficient σθ/σc, elastic deformation energy index Wet, and rock brittleness coefficient σc/σt are selected as the main evaluation indexes of rockburst prediction. After data screening, 249 groups of rockburst case data are obtained. Considering the index weights calculated by CRITIC, EW, and AHP, the optimal combination weight is calculated by game theory. The results show that the order of importance of the four indexes is Wet > σc > σc/σt > σθ/σc, and the comprehensive weight values of the index are 0.3546, 0.2531, 0.2109, and 0.1813, respectively.

  2. (2)

    Based on the normal cloud model, the calculation method of the single index measure is improved. After verification of the test set, the prediction accuracy of the improved method reaches 93.3%. And the prediction grade is increased by 16.6% compared with the traditional prediction method. It shows that the established prediction model is effective and feasible.

  3. (3)

    Combine the data from ten typical rockburst cases around the world to verify the universality of the model. The results show that the prediction accuracy of the NC-UM model proposed in this paper reaches 80%. Compared with UM, TOPSIS, and MNC methods, it has the highest accuracy. It shows that the method proposed in this paper can better predict the rockburst tendency.

  4. (4)

    The NC-UM model was applied to rockburst grade prediction, and enables to evaluate the rockburst grade of the sample. Compared with other methods such as traditional UM, TOPSIS, the method improves the evaluation accuracy. However, the method is only a preliminary attempt, and the influence of factors such as the calculation of comprehensive weights of indicators and the selection of numerical features of the cloud model on the performance of the model has yet to be explored in more depth.