Overall Resilient Evaluation of Surrounding Rock of In-Service High-Speed Railway Tunnel Based on Information Fusion-Improved Fuzzy Matter-Element

Abstract

Once the high-speed railway tunnel is put into use, its resilience will determine the possibility of permanent safety of the tunnel due to the closure of the structural space of the high-speed railway tunnel in service. Resilience theory is introduced into a risk analysis of operating high-speed rail tunnels to improve the ability to respond to risks in operating high-speed rail tunnels and to relieve the aging phenomenon caused by changes in the tunnel with time. First, an evaluation framework for the safety resilience of existing high-speed railway tunnels is constructed. Starting from the attributes of resilience such as resistance, adaptability, and resilience, and considering the characteristics of high-speed railway tunnels, protective measures, emergency management measures, and other factors, we fit the risk factors and probability of accident type of the high-speed railway tunnel and establish a tunnel safety resilience evaluation index system with 10 indexes. Secondly, the method of information fusion is used to combine subjective weighting and objective weighting. Then, the comprehensive weight of the evaluation index is obtained based on the principle of minimum discriminant information. Thirdly, the system resilience evaluation model based on the TOPSIS improved fuzzy matter-element is constructed to determine the classification criteria of resilience. On this basis, based on the temporal and spatial variability of the ductile tunnel, the concepts of ductile transition and ductile attenuation are introduced and the tunnel toughness optimization model is established to suppress the attenuation situation, enhance the transition ability, and then improve the system resilience level. On this basis, an optimal lifting scheme is obtained. Finally, taking Ai-Min tunnel of Ha-Mu high-speed railway as the engineering background, the flexibility of the resilience system is calculated, and the resilience grade (3) of the rock system surrounding the tunnel is obtained. Combined with the numerical model, improvement measures for specific tunnel facilities are proposed. The results show that the Ai-Min tunnel system has a general ability to resist external intrusion and prevent disasters, and the resilience level is general. It should focus on improving the resilience level of the transition index. The resilience evaluation results of the evaluation model are consistent with the actual situation of the project.

1. Introduction

The rapid construction of a modern comprehensive transportation system has led to the comprehensive development of tunnels and underground engineering. Railway tunnels, especially high-speed rail tunnels, often have inevitable defects after they are built and put into use. The normal operation and service life of the tunnel will be affected and threatened because of the existence of such defects. The surrounding rock is the skeleton and the foundation of the tunnel. Surrounding rock and its protection measures do not exist alone. They are systems composed of many factors. The safety of the system affects the safety of high-speed rail operations [1]. To ensure the permanent safety of the built high-speed railway tunnel and reduce the accidents caused by the failure of the surrounding rock system, it is necessary to have a further understanding of the characteristics of the surrounding rock and its protection measures and emergency management measures.

In recent years, domestic and foreign scholars have studied the safety risk assessment methods for tunnel construction. For example, Chang Xiangyu et al. [2] proposed a hybrid method, using BN and RVM to perform a probabilistic risk assessment of tunnel-related risks and probabilistic risk assessment of the damage caused by existing property; PADTUN was introduced and used by Vania Dimitrova et al. [3]. It can help to make a pathological diagnosis and evaluation of the tunnel. PADTUN can understand the disorder of the tunnel and the influencing factors of diagnosis. Some scholars use the index weighting method to assign the weight of each index and establish the risk assessment model of the tunnel-surrounding rock [4,5,6,7]. However, these studies mostly evaluate the vulnerability of the inherent attributes of the tunnel, ignoring the correlation and dynamics of risk factors, and lack research on the attributes of the tunnel-surrounding rock system such as resistance and resilience.

At present, in the field of safety engineering, safety risk identification and evaluation methods are relatively perfect from qualitative to quantitative, but not all fields are applicable, and some bottlenecks are encountered in risk research. The concept of resilience provides a new and important way to study the complex social technology system. To improve the ability of tunnel operations to deal with risks, resilience theory is introduced into the risk assessment of high-speed railway tunnels. Since the 1990s, resilience theory has attracted more and more scholars’ attention. Huang et al. [8] applied resilience theory to the field of safety science and proposed the definition of system safety resilience. In recent years, the theory of safety resilience has been gradually integrated into tunnel engineering practice. Aiming at the DGL tunnel of the Sichuan-Tibet railway, WEI Qiang et al. [9] put forward the concept of tunnel construction safety resilience based on the concept and characteristics of resilience theory and constructed the evaluation model of construction resilience. Zhang J. et al. [10,11,12] believe that the tunnel has great risk factors in the construction stage, and the construction resilience of shield tunnels or other types of tunnels is evaluated. However, after the tunnel is completed and put into use, its ability to maintain integrity is often not taken seriously. Some studies have found that the time and economic cost of tunnel safety construction are several times that of the design and construction [13]. On this basis, NI Xin [14] proposed a quantitative evaluation method for the resilience of shield tunnel structures. Using ABAQUS numerical analysis software, the influence of different buried depths, different water content, and different seismic grade on the surrounding rock of a loess tunnel was evaluated. However, the influence surface involved is narrow, ignoring the results under the combined action of various physical fields and not involving the protection measures and emergency management measures of the tunnel-surrounding rock. This method cannot fully reflect the ability of the tunnel to maintain permanent operation. Based on the concept of resilience theory, WEI Qiang et al. [9] put forward the concept of safety resilience of tunnel construction and constructed the evaluation model of construction resilience. However, these studies only constructed the evaluation model of system resilience. They only stayed at the theoretical level, lacked practical application, could not solve practical problems, and ignored the variability of resilience in time scale. To make it convenient for decision makers to improve the resilience grade of tunnel-surrounding rock through reasonable methods, we studied the principle of space–time variation of the resilience of the surrounding rock system. We put forward the concept of resilience transition and resilience attenuation, and established an optimization model to suppress the attenuation trend and enhance the transition ability, to improve the resilience level of the surrounding rock of a high-speed railway tunnel, and, finally, to obtain the optimal order of index transition.

Evaluation methods play an important role in resilience assessment. Xiao S. et al. [15,16,17] try to find a more suitable resilience assessment method for fuzzy indicators to quantify qualitative indicators as much as possible. In the existing literature, the establishment of an evaluation model is often divorced from decision makers and fails to fully consider the coupling relationship between indicators and the lack of practical application value. Therefore, according to the load evolution mechanism of coupling characteristics, we establish a weighting method based on information fusion. This method adopts a combination of subjective weighting and objective weighting and integrates case information and EWM to assign weights. Considering the influence of the system’s information on the weight distribution and the practical significance of the data, the subjectivity of the weighting is reduced and the defects of the traditional method are overcome. The resilience model contains some qualitative indicators. Considering that the relationships between the indicators overlap and affect each other, we use the optimal solution and the worst solution as the basis, combined with the concept of fuzzy matter-element, and use TOPSIS to improve the fuzzy matter-element. This method can quantify some factors that are not easy to quantify, avoid unreliable evaluation results due to insufficient quantitative information, better identify the pros and cons of the scheme, and enhance the practical engineering application value.

Given the above arguments and descriptions, we introduce the resilience theory to study the impact of various risk factors on the safety of in-service high-speed railway tunnels. We determine the comprehensive weight through the method of information fusion, and use TOPSIS to improve the fuzzy matter-element to realize the whole process of evaluating the resilience of the surrounding rock of a high-speed railway tunnel in service. Based on the principle of time–space variation of system resilience, we study the strategy of improving resilience, propose a method to enhance the resilience of tunnel-surrounding rock systems and obtain the optimal index improvement scheme. These studies aim to reduce the risk of tunnel-surrounding rock systems, improve the ability of high-speed railway tunnels to resist risks, restore ability and adaptability, and then improve the overall resilience level and provide new ideas for maintaining the permanent operation capacity of a high-speed railway tunnel.

2. Resilience of Surrounding Rock of In-Service High-Speed Railway Tunnel

2.1. Basic Characteristics of Surrounding Rock Resilience

Although resilience is defined differently in various disciplines, it essentially describes the ability of a research system to adapt to normal conditions through autonomous adjustment after external shocks and disturbances, and to respond to shocks and disturbances sustainably, thereby achieving sustainable development of system security [18]. Based on the concept and characteristics of resilience, combined with the systematicness, environmental complexity, diversity of equipment and facilities, and timeliness of emergency management of high-speed railway tunnels in operation, the safety resilience of high-speed railway tunnels in service is defined as In the process of operation of high-speed railway tunnels, in the face of a variety of uncertain risk factors, the ability of the tunnel-surrounding rock system to resist intrusion (tunnel-surrounding rock—equipment and facilities to protect surrounding rock environment inside and outside the tunnel—emergency management system), ensure structural safety, and maintain normal operation of the system can be guaranteed without external help. Based on the characteristics of resilient cities and resilient transportation networks, several of the literature, data, and accident types are summarized. According to the characteristics of high-speed railway tunnels and the characteristics of tunnel safety resilience, the characteristics of resilient high-speed railway tunnels are analyzed:

• Robustness: The high-speed railway tunnel resists disasters and reduces the economic, social, personnel, material, and other losses of the high-speed railway operation system caused by disasters.
• Rapidity: Ability to recover quickly after a certain degree of injury. The high-speed railway tunnel can recover to a certain functional level within a certain time after damage. The high-speed railway tunnel with strong resilience can restore certain functions in a short time.
• Redundancy: The key functions or equipment and facilities in the surrounding rock system of the high-speed railway tunnel can maintain the normal operation of the system through emergency management measures or their own resilience after bearing certain damage. The spare function or module can make the whole system still play a certain level of function, and the system will not be completely paralyzed.
• Adaptability: The tunnel surrounding the rock system can learn from past damage accidents and improve the adaptability of the system to disasters.

Space–time variability: After a long period of damage and disturbance, the tunnel surrounding the rock system with a low resilience level leads to the decline of the overall resilience level and the attenuation of resilience ability due to its low resilience level. However, after the decision-making rectification, the system can still restore the original resilience level or even jump to the next level.

2.2. Framework for Evaluating System Resilience

At present, the research on system resilience evaluation mainly involves quantitative analysis and qualitative analysis. According to the resilience evaluation framework given by Francis and Bekera [19], starting from the resilience, adaptability, and resistance of the studied system, combined with the four main influencing aspects of the rock mass properties of the tunnel surrounding the rock system, the equipment and facilities of the protection system, the environment inside and outside the tunnel, and the emergency management measures, 10 representative evaluation indexes that can reflect the resilience characteristics of the tunnel-surrounding rock system are selected. Combined with the relevant technical specifications and many scholars’ research on the surrounding rock, the resilience evaluation framework is established, as shown in Figure 1.

Construct a system composed of evaluation indicators. The more indicators we select, the more complex the index system we build, and we may cover up the impact of important factors; if we select fewer evaluation indicators, the risk assessment process is simple, and the evaluation results are difficult to fully and systematically reflect the objective situation of the evaluation object. Therefore, we need to follow scientific, systematic, stable, and quantifiable principles [20].

Combined with the relevant technical specifications and many scholars on the surrounding rock research and exploration, we divide the evaluation criteria into the following 10 indicators (Figure 2).

The model intuitively reflects the whole dynamic process of the surrounding rock of the high-speed railway tunnel being disturbed and takes the most representative influencing factors in each influencing aspect as the resilience evaluation index. Starting from the three resilience characteristics of absorption capacity, adaptability, and recovery ability of the surrounding rock, the evaluation model has high feasibility and operability.

3. Assessing Safety Resilience

3.1. Evaluation Grade

3.1.1. Establish Evaluation Grade

The focus of the establishment and numerical determination of the resilience evaluation indicators is to find out one or more risk factors. These factors not only have the ability to erode the system, but also can be resisted and adapted by the system to restore the system itself. A preliminary evaluation index system is constructed, and the quantitative values of some indicators are given according to a series of references and actual data. The process of establishing the index system is shown in Figure 3.

The 10 evaluation indexes given in the framework of resilience evaluation in Figure 3 are graded, representing the resilience of the surrounding rock in turn: Bad effect (Grade 1), Less effective (Grade 2), General effective (Grade 3), Good effect (Grade 4), and More effective (Grade 5). The specific division is shown in

Table 1. Division of evaluation indicators.

Evaluation Indicators	Basis for Classification
	Grade 1	Grade 2	Grade 3	Grade 4	Grade 5
Rock lithology C1	I	II	III	IV	V-VI
Buried depth C2/m	C2 < 20	20 ≤ C2 < 50	50 ≤ C2 < 200	200 ≤ C2 < 500	C2 ≥ 500
Annual rainfall C3/mm	≥2000	[1500, 2000)	[1000, 1500)	[500, 1000)	[0, 500)
Seepage C4/ [L·(min·10 m)−1]	C4 ≥ 125	100 ≤ C4 < 125	50 ≤ C4 < 100	25 ≤ C4 < 50	C4 < 25
Seismic disturbance C5	Damage disturbance	Effective disturbance	Disturbance	Tiny disturbance	Minimal disturbance
Mean temperature in the coldest month C6/°C	C6 < −20	−20 ≤ C6 < −15	−15 ≤ C6 < −10	−10 ≤ C6 < 0	0 ≤ C6
Cold and antifreeze measures C7 1	Bad effect	Less effective	Generally effective	Good effect	More effective
Drainage facilities C8	Below proof	Eligible	Middle	Good	Excellent
Train track dynamic load effects C9	Damage disturbance	Effective disturbance	Disturbance	Tiny disturbance	Minimal disturbance
Emergency Management Measures C10	Bad effect	Less effective	Generally effective	Good effect	More effective

¹ Concerning the corresponding specification [21] and the literature [22,23,24,25], the above indexes are graded. According to the different ambient temperatures, the demand of tunnel-surrounding rock for cold and frost resistance measures is different. Among them, the cold and antifreeze measures C7, based on the classification criteria given in the literature, if the coldest month temperature is greater than 0 °C, the cold and antifreeze measure C7 is regarded as Grade 5 (More effective).

.

3.1.2. Fuzzy Description of the Qualitative Index

The influence factors of the tunnel-surrounding rock are diverse and complex. Because of this characteristic, some evaluation indexes are difficult to give specific values. Therefore, the evaluation indexes given in Table 1 are mostly fuzzy descriptions. To ensure the preciseness of the classification of evaluation indexes and the scientificity of the data, it is necessary to analyze the classification basis of the fuzzy qualitative indexes in Table 1:

• Rock lithology C1

Rock lithology C1 is the basis for evaluating the resilience of the surrounding rock. The risk assessment of the surrounding rock lithology is often based on the I~VI grade of the surrounding rock. Therefore, by consulting the literature [22] and combining expert opinions, we establish a bridge between the grade of the surrounding rock and its resilience grade. The surrounding rock of grade VI is hard rock with a saturated compressive ultimate strength R > 60 MP, which is slightly affected by the geological structure. The joints are not developed, there is no weak surface or interlayer, and the layered rock layer is a thick layer with good interlayer bonding. The grade V surrounding rock is hard rock, and the saturated compressive ultimate strength R > 30 MP. It is seriously affected by the geological structure, and the joints are more developed. There are a small number of weak surfaces or interlayers and penetrating micro-tensile joints, but its occurrence and combination relationship do not produce sliding. The surrounding rock structure of grade V and VI has good resilience and will not slide. To ensure the simplicity of the assignment, we regard both grade V and VI as Grade 5.

2.

The buried depth C5

We qualitatively express C5 as the following five levels: Minimal disturbance (the frequency of micro-inductive earthquakes with a source level of 3 or below is less than 20 times, and there are no earthquakes above 3); Tiny disturbance (the frequency of micro-inductive earthquakes with a source level of 3 or below is less than 70 times, and there is no earthquake above 3); Disturbance (the frequency of micro-inductive earthquakes with a source level of 3 or below is more than 70 times, and there is no earthquake with a source level of 3 or above; the total frequency of earthquakes is more than 70 times, and the ratio of the frequency of micro-inductive earthquakes of magnitude 3 and below to the frequency of earthquakes of magnitude 6 and below is greater than 1/1.5); Effective disturbance (the average occurrence of more than one earthquake of magnitude 6~7 in a year); Damage disturbance (an average of at least one earthquake above 7 occurred within one year).

3.

Cold and antifreeze measures C7

According to the application and effect of cold and frost-resistant equipment and facilities in high-speed railway tunnels in current engineering practice [23], C7 is classified into the following five grades: Bad effect (does not have any cold antifreeze facilities); Less effective (only consider setting insulation or active heating, and the insulation or active heating is less reliable); Generally effective (taking into account the combination of active heating and passive insulation, but only three of the following cold and antifreeze facilities); Good effect (taking into account the combination of active heating and passive insulation, and set up a variety of cold and frost-resistant facilities, can resist local large temperature difference freezing damage); More effective (according to the characteristics of the tunnel itself and the environmental characteristics, set up a set of cold and frost-resistant equipment and facilities suitable for the specific tunnel, it can guarantee the integrity of the tunnel to resist the influence of freezing damage). In the tunnel environment where the minimum temperature is greater than 0 °C all year round, C7 cannot be reflected in the evaluation of resilience.

4.

Drainage facilities C8

Based on the application and effectiveness of drainage facilities in high-speed rail tunnels [24], C8 is classified into the following five grades: Below proof (no drainage equipment is set); Eligible (only one or two of the cold-proof drainage holes, central deep-buried ditches, and thermal insulation ditches are set, and the drainage measures have been broken or do not meet the requirements); Middle (only one or two of the cold-proof drainage holes, central deep-buried ditches, and thermal insulation ditches are set, and the facilities can be used normally); Good (basic drainage equipment and mechanical drainage equipment are set well); Excellent (according to the tunnel’s characteristics and environmental factors, set up a set of drainage and flood control facilities suitable for the specific tunnel, which can ensure the integrity of the tunnel to resist the impact of waterlogging disasters). According to Table 1, C8 cannot be reflected in the resilience evaluation when the annual rainfall and water seepage are Grade 5.

5.

Dynamic load generated by train-track system C9

The C9 value is more suitable for qualitative analysis. According to the application and effect of the train-track system in the high-speed railway tunnel in the current engineering practice [24], we classify C9 into the following five grades: Minimal disturbance (only one track is set in the tunnel, and the train speed v < 200 km/h); Tiny disturbance (only one track is set in the tunnel, and 200 km/h ≤ v < 350 km/h); Disturbance (only one track is set in the tunnel, and 350 km/h ≤ v, or more than one track is set in the tunnel, v < 200 km/h); Effective disturbance (more than one track is set in the tunnel, and v > 200 km/h); Damage disturbance (more than one track is set in the tunnel, and 350 km/h ≤ v).

6.

Emergency management measures C10

According to the application and effect of emergency management measures in high-speed railway tunnels in current engineering practice [25], C10 is characterized by the following five levels: More effective (good emergency drill effect, complete emergency plan system, emergency management organization equipment sound); Good effect (good emergency drill effect, complete emergency plan system, emergency management organization equipment sound degree is good); Generally effective (general emergency drill effect, lack of emergency plan system, emergency management organization equipment sound degree is general); Less effective (poor emergency drill effect, lack of emergency plan system, emergency management organization equipment sound degree is poor); Bad effect (no emergency drill, no emergency plan system, no emergency management organization equipment). Emergency management measures in a timely and effective high-speed rail tunnel can more quickly adapt to the erosion of risk factors, thereby improving the resilience of the tunnel-surrounding rock system. C10 should be based on the actual situation of the tunnel operation fuzzy evaluation.

The fuzzy index should be quantified as 1~5 according to the pros and cons of its impact on the resilience, corresponding to the five grades of the surrounding rock resilience of the high-speed railway tunnel in service. The quantitative results of the specific evaluation index are shown in

Table 2. Evaluation index quantitative score.

Evaluation Indicators	Basis for Classification
	Grade 1	Grade 2	Grade 3	Grade 4	Grade 5
C1	1	2	3	4	5
C2/m	0	20	50	200	500
C3/mm	2000	1500	1000	500	0
C4/ [L·(min·10 m)−1]	125	100	50	25	0
C5	5	4	3	2	1
C6/°C	0	10	15	20	25
C7	1	2	3	4	5
C8	1	2	3	4	5
C9	5	4	3	2	1
C10	1	2	3	4	5

.

3.2. Combination Weighting of Surrounding Rock Resilience System Based on Information Fusion

Based on the influence of 10 index factors on the classification given in Table 1, we propose a combination weighting method of resilience system based on information fusion. Through information fusion determine the local weight and reduce the subjectivity of weight, using a combination of subjective and objective ways to improve the accuracy of weighing.

3.2.1. Subjective Weighting Based on DEMATEL

This paper uses the DEMATEL method to calculate the cause degree and centrality of each index through the influence degree and the influence degree between the indexes and to determine the subjective weight value of each evaluation index. The relevant formulas of the DEMATEL weighting method are as follows:

$$K=\frac{1}{\underset{1\le j\le m}{\max {\displaystyle \sum _{i=1}^m{z}_{ji}}}}{\left[I-\frac{1}{\underset{1\le j\le m}{\max {\displaystyle \sum _{i=1}^m{z}_{ji}}}}\right]}^{-1}$$

(1)

$$D={\displaystyle \sum _{i=1}^m{k}_{ij}}+{\displaystyle \sum _{j=1}^m{k}_{ji}}$$

(2)

$${\gamma }_j=\frac{D_j}{{\displaystyle \sum _{j=1}^m{D}_j}}$$

(3)

In the formula, z_ji is the element in the direct influence matrix (i = 1, 2… n; j = 1, 2… m), I is the unit matrix, K is the comprehensive influence matrix, D is the centrality between the indexes, and γ_j is the subjective weight of each evaluation index.

3.2.2. Objective Weighting Based on the Entropy Weight Method (EWM)

Calculate the entropy of the index i:

$$\left\{\begin{array}{l}{p}_{ij}=\frac{{\xi }_{ij}}{{\displaystyle \sum _{i=1}^m{\xi }_{ij}}}\\ k=\frac{1}{\ln m}\end{array}\right.$$

(4)

In the formula, p_ij represents the proportion of the eigenvalue of the i-th description object of the j-th indicator; k denotes an excessive variable.

$$\left\{\begin{array}{l}{e}_j=-k{\displaystyle \sum _{i=1}^m{p}_{ij}\ln p_{ij}}\\ {\eta }_j=\frac{1-e_j}{{\displaystyle \sum _{j=1}^{\mathrm{n}}\left(1-e_j\right)}}\end{array}\right.$$

(5)

In the formula, e_j represents the entropy value of the index j, and η_j represents the weight of the index j.

3.2.3. Combination Weighting Based on Information Fusion

We combine the influence degree distribution of 10 index factors on the resilience system with subjective weighting to determine the local weight distribution. We also reduce the subjectivity of weights by integrating information at the decision level [26]. From the perspective of influencing factors of 10 groups of indicators, the data to be evaluated and the evaluation indicators are subjectively weighted and scored, and two sets of subjective weight distribution are obtained. According to the influence degree distribution of 10 index factors, the local weight distribution is obtained. The calculation formula is as follows:

$${\gamma }_j{}^{*}={\displaystyle \sum _{\begin{array}{l}t=1\\ j=1\end{array}}^m{k}_t·{\gamma }_j}$$

(6)

In the formula, k_t represents the influence degree distribution of each evaluation index (t = 1, 2… m), and γ_j^* represents the local subjective weight value of the index.

To avoid the “multiplier effect” of linear weighting and multiplicative weighting [27], and to ensure that the combined weight is not biased toward subjective or objective weight, we use the principle of minimum discriminant information [28]. The optimization model is established by using the optimization method with the minimum consistency between the comprehensive weight and the local subjective weight and the objective weight as the optimization objective. The optimization model is as follows:

$$\left\{\begin{array}{l}\min J\left(\omega \right)={\displaystyle \sum _{j=1}^m\left[{\omega }_j\ln \frac{{\omega }_j}{{\gamma }^{*}{}_j}+{\omega }_j\ln \frac{{\omega }_j}{{\tau }_j}\right]}\\ \mathrm{s}·\mathrm{t}·{\displaystyle \sum _{j=1}^m{\omega }_j=1,{\omega }_j\ge 0}\end{array}\right.$$

(7)

In the formula, ω_j represents the comprehensive weight of index _j.

Using the Lagrange multiplier method to solve the above optimization model, the final comprehensive weight is shown in the following formula:

$${\omega }_j=\frac{\sqrt{{\gamma }^{*}{}_j{\tau }_j}}{{\displaystyle \sum _{j=1}^m\sqrt{{\gamma }^{*}{}_j{\tau }_j}}}$$

(8)

Then, determine the optimal combination of weights. The final weight values are shown in

Table 3. Combination weight value.

j	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
ωj	0.20	0.114	0.056	0.088	0.143	0.107	0.099	0.103	0.012	0.078

.

3.3. Improved Fuzzy Matter Element Evaluation Method Based on TOPSIS

The evaluation model of the surrounding rock resilience of the high-speed railway tunnel in service contains some qualitative indexes, and the relationships between the indexes overlap and influence each other. The two kinds of evaluation methods are combined, and the comprehensive weight obtained by the combination weighting method based on information fusion is used to replace the weight obtained by the fuzzy evaluation method, and then the closeness degree between the fuzzy matter-element to be evaluated and the ideal fuzzy matter-element is calculated to determine the quality of the object to be evaluated [9].

3.3.1. Constructing the Composite Fuzzy Matter-Element

The so-called fuzzy matter-element is an orderly combination of three elements (things, characteristics, and fuzzy value) as the basic elements to describe things [29]. The fuzzy matter-element includes three elements: describing things M, feature C, and value v, among which the value v is fuzzy. If M has n characteristics C₁, C₂, …, C_n, its fuzzy value is v₁, v₂, …, v_n. According to the characteristics of the evaluation object and the characteristics of the evaluation index of the resilience system, we regard M as a discrete resilience level, and the 10 evaluation indexes corresponding to M are regarded as the characteristics C of the resilience system. The fuzzy index score is the fuzzy value v, and the description object can be called an n-dimensional fuzzy matter-element. It can be abbreviated as R = (M, C, v), as shown in Formula (10).

$$\mathrm{Fuzzy} \mathrm{matter}\text{-}\mathrm{element}=\left[\begin{array}{cc}& \mathrm{Level} \mathrm{after} \mathrm{discretization}\\ \mathrm{Resilience} \mathrm{of} \mathrm{indicators}& \mathrm{Fuzzy} \mathrm{quantity} \mathrm{value}\end{array}\right]$$

(9)

By combining the n-dimensional fuzzy matter-elements of m description objects, the fuzzy complex element R_mn is formed, as shown in Formula (10).

$$R_{mn}=\left[\begin{array}{ccccc}& M_1& M_2& \cdots & M_m\\ C_1& v_{11}& v_{21}& \cdots & v_{m1}\\ C_2& v_{12}& v_{22}& \dots & v_{m2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ C_n& v_{1n}& v_{2n}& \cdots & v_{mn}\end{array}\right]$$

(10)

In the formula, R_mn is an n-dimensional composite fuzzy matter-element with m resilience levels; M_i is the i-th resilience evaluation object, where i = 1, 2, …, m; C_j is the characteristic of evaluation index j, where j = 1, 2, …, n; v_ij is the fuzzy value of the j-th index of the i-th resilience evaluation object.

3.3.2. Determining Priority Membership

The membership degree of the fuzzy value corresponding to each single evaluation index is called the optimal membership degree [29]. In economics, because of the characteristic value of each index for the evaluation results, the greater the efficiency indicators, the better the cost-based indicators are. Therefore, different calculation formulas are used for different subordinate degrees, as follows:

$${\xi }_{ij}=\left\{\begin{array}{l}{\xi }^{+}{}_{ij}\iff \mathrm{Benefit}\\ {\xi }^{-}{}_{ij}\iff \mathrm{Cos}\mathrm{t}\end{array}\right.$$

(11)

$${\xi }^{+}{}_{ij}=\frac{v_{ij}-\min v_{ij}}{\max v_{ij}-\min v_{ij}}$$

(12)

$${\xi }^{-}{}_{ij}=\frac{\max v_{ij}-v_{ij}}{\max v_{ij}-\min v_{ij}}$$

(13)

In the formula, ζ_ij represents the optimal membership of an index. When the index is a benefit index, ζ⁺_ij represents the subordinate degree of the index; when the index is cost-based, ζ⁻_ij represents the subordinate degree of the index; v_ij is the fuzzy value corresponding to the j-th index of the i-th description object; max-v_ij is the maximum fuzzy value corresponding to the j-th index in all objects; min-v_ij is the minimum of the fuzzy values corresponding to the j-th indicator in all objects.

The larger superior index should be calculated according to Formula (12), and the smaller superior index should be calculated according to Formula (13).

3.3.3. Construct the Fuzzy Matter-Element

According to the definition of optimal value in TOPSIS, if all attribute index values of a fuzzy matter-element are optimal values, we define the fuzzy matter-element as an ideal fuzzy matter-element and use R^* to represent an ideal fuzzy matter-element.

$$R^{*}=\left[\begin{array}{cc}& M^{*}\\ C^1& {\xi }^{*}{}_1\\ C^2& {\xi }^{*}{}_2\\ ⋮& ⋮\\ C^n& {\xi }^{*}{}_n\end{array}\right]$$

(14)

In the formula, ξ*j is the ideal fuzzy matter-element value, ${\xi }^{*}{}_j=\underset{1\le i\le m}{\max }{\xi }_{ij}$.

In order to improve the reliability of the evaluation, the square value Δ_ij of the difference between the ideal fuzzy matter-element R^* and the composite fuzzy matter-element R_mn is used to calculate the distance from each value to the optimal solution, and then the difference square composite fuzzy matter-element R_Δ is obtained.

$${\Delta }_{ij}={\left({\xi }^{*}{}_j-{\xi }_{ij}\right)}^2$$

(15)

$$R_{mn}=\left[\begin{array}{ccccc}& M_1& M_2& \cdots & M_m\\ C_1& {\Delta }_{11}& {\Delta }_{21}& \cdots & {\Delta }_{m1}\\ C_2& {\Delta }_{12}& {\Delta }_{22}& \dots & {\Delta }_{m2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ C_n& {\Delta }_{1n}& {\Delta }_{2n}& \cdots & {\Delta }_{mn}\end{array}\right]$$

(16)

3.3.4. Proximity Calculation

Euclidean proximity is introduced to measure the proximity between the matter-element to be evaluated and the ideal fuzzy matter-element. The greater the value of proximity is, the greater the proximity is, and vice versa. Therefore, the safety resilience level of the tunnel construction can be divided by calculating the Euclidean closeness degree. The calculation of Euclidean closeness degree adopts the M(, +) algorithm.

$$\rho H_i=1-\sqrt{{\displaystyle \sum _{j=1}^n{\omega }_j{\Delta }_{ij}}}$$

(17)

In the formula, ω_j is the weight of the index j, and ρH_i represents the proximity between the matter-element to be evaluated and the ideal matter-element. The larger the value is, the closer the two are, and vice versa.

Based on this, the composite fuzzy matter-element R_ρH of the Euclidean closeness degree is constructed as follows:

$$R_{\rho H}=\left[\begin{array}{ccccc}& M_1& M_2& \cdots & M_m\\ \rho H_i& \rho H_1^{}& \rho H_2& \cdots & \rho H_m\end{array}\right]$$

(18)

3.4. Standard for Classification of Resilience

The compound fuzzy matter-element R_ρH containing the Euclidean closeness degree is obtained as follows:

$$R_{\rho H}=\left[\begin{array}{cccccc}& M_1& M_2& M_3& M_4& M_5\\ \rho H_i& 0& 0.2329& 0.4557& 0.6967& 1\end{array}\right]$$

(19)

In the formula, ρH_i represents the flexibility of the tunnel-surrounding rock system.

According to the flexibility in Formula (19), the resilience grade is divided. The results are shown in

Table 4. Grade standard of division.

Resilience Grade of Surrounding Rock System 1	The Level of Resilience	The Range of Grades	Rank Difference
Grade 1	Easy to damage; Low adaptability; Impossible to recover	[0, 0.2329)	0.2329
Grade 2	Easier to damage; Low adaptability; Hard to recover	[0.2329, 0.4557)	0.2228
Grade 3	General injury; Has certain adaptability; Can be restored	[0.4557, 0.6967)	0.2410
Grade 4	More difficult to damage; Good adaptability; Recovery	[0.6967, 1)	0.3033
Grade 5	It is difficult to damage; Excellent adaptability; Fast recovery	1

¹ The resilience level in Table 4 is based on the three capabilities of the resilience system. The simplified criteria can be recorded as Bad effect (Grade 1), Less effective (Grade 2), Generally effective (Grade 3), Good effect (Grade 4), and More effective (Grade 5).

:

4. Research on Resilience Improvement Strategy Based on Spatio-Temporal Variability

System resilience is not a fixed value but a variable. It is found that a tunnel-surrounding rock system with low resilience levels will experience long-term damage and disturbance. Due to its low resilience level, the overall resilience level of the system will decline, and the system’s resilience ability will continue to decay, but the decision-making rectification can still restore the original resilience level or even jump to the next level. Therefore, to improve the ability of the tunnel to resist risks, we start from the perspective of the temporal and spatial variability of resilience (resilience transition and resilience attenuation) to alleviate the attenuation of system resilience, to enhance the ability of resilience transition, to maintain the resilience level of the system, and even to improve it.

4.1. Resilience Transition

To enable decision makers to make better decisions and improve the resilience level of the surrounding rock of high-speed railway tunnels, we propose the concept of resilience transition. It refers to the improvement of one or more variable indicators (surrounding rock protection equipment and facilities, etc.), which can improve the resilience level of the tunnel-surrounding rock (for example, from the first-level resilience transition to the third-level resilience). The range represents the allowable size in the tunnel resilience grade interval. However, a larger grade difference does not mean that the possibility of a tunnel-surrounding rock system appearing in this interval is greater, but the grade difference can indicate the difficulty of surrounding rock resilience transition and that the grade difference between each resilience grade is not increasing, as shown in Figure 4.

According to Figure 4 and Figure 5, the transition from Grade 2 resilience to Grade 3 resilience is relatively easy. For an unused high-speed rail tunnel with a resilience assessment level of level 2, we can improve its resilience by improving the quality of its indicators. In order to describe the difficulty and possibility of ductile transition more quantitatively, we construct a transition model such as Equation (20):

$${\displaystyle \sum _{t=1}^q{k}_t{\omega }_t{\Delta }_t}={\left(1-\rho H_{i+1}\right)}^2-{\displaystyle \sum _{s=1}^{m-q}{\omega }_s{\Delta }_s}$$

(20)

In the formula, t represents the index in the evaluation index that can be artificially controlled for transition (t = 1, 2, …, q); k_t denotes the coefficient of expansion; Δ_t represents the square of the difference between indicators; ω_t represents the weight of the t-th variable; ρH_{i + 1} represents the flexibility of the Grade i + 1 tunnel-surrounding rock system; s indicates non-transitional indicators in the assessment indicators (s = 1, 2, …, m − q); ω_s represents the weight of the s-th immutable indicator.

According to Formula (22), when the decision maker pushes the tunnel’s surrounding rock resilience to the next level, there are always s immutable indicators that limit its transition, such as lithology, rainfall, and seepage. To increase the resilience levels, decision makers must change the variable based on the constant factor, which is the transition indicator. Therefore, we introduce an expansion coefficient k_t, which will represent the expansion of t different levels of transitional indicators. However, considering the economic problem of decision making, we must not make a certain coefficient k too large alone. Although the increase of k may improve the resilience of the tunnel-surrounding rock system, we must consider the practical significance and economic problems of the index. Therefore, to improve the resilience of the system, decision makers should improve the comprehensive level of each index. The optimal solution should first consider the variable index with the largest weight and the lowest difficulty. Based on the results of the division of weights in Figure 5, we believe that the order of index improvement should be: C2 > C8 > C7 > C10 > C9, which is to improve system resilience while saving costs.

4.2. Attenuation of Resilience

The biggest difference between the resilience of the built high-speed railway tunnel and the high-speed railway tunnel under construction is that the former has temporal and spatial variability, which indicates that the resilience of the system is continuously attenuated. The study found that on the time scale, considering the accumulation of tunnel-surrounding rock damage, the overall resilience of the high-speed rail tunnel-surrounding rock system is from high to low, gradually decaying, showing a downward trend.

The mathematical model of resilience degradation is shown in Formula (21):

$$\left\{\begin{array}{l}{\omega }_{j^{\prime }}=\frac{\sqrt{{\gamma }^{*}{}_{j^{\prime }}{\tau }_{j^{\prime }}}}{{\displaystyle \sum _{j^{\prime }=1}^k\sqrt{{\gamma }^{*}{}_{j^{\prime }}{\tau }_{j^{\prime }}}}}\\ j^{\prime }=j_1+j_2\\ \rho {H^{\prime }}_i=1-\sqrt{{\displaystyle \sum _{j^{\prime }=1}^n{\omega }_{j^{\prime }}{\Delta }_{i{j}^{\prime }}}}\end{array}\right.$$

(21)

In the formula, j indicates the evaluation index of the damaged tunnel (j = 1, 2…k; k ≥ 10); j₁ indicates the original 10 resilience indicators; and j₂ represents the additional indicator that occurs after the risk causes damage to the surrounding rock resilience system. The risks referred to here are often those that tunnels cannot withstand; ρH_i′ represents the flexibility of the damaged tunnel-surrounding rock system.

According to Formula (21), it can be seen that the high-speed railway tunnel with a lower resilience level will produce j₂ additional indicators after the risk of erosion of the surrounding rock resilience system that cannot be resisted. The additional indicators represent the resilience of the system itself after the damage to the surrounding rock of the tunnel. Despite the loss of ability, the study found that the tunnel still has strong resilience unless it is subjected to irreversible damage, but the influence of additional indicators will inevitably lead to ρH_i′ ≤ ρH_i; except for the non-transition s indicators, the resilience level of the transition t indicators has decreased, as shown in Figure 6.

According to Figure 6, the resilience level of the surrounding rock system of the high-speed railway tunnel shows an attenuation trend. If no influence is exerted, this attenuation trend will gradually flatten out. What causes this attenuation trend is an irresistible risk or a long-term, multi-frequency risk erosion and risk interference; the tunnel system with a good resilience level can slow down this attenuation trend and resist most of the risk erosion. The tunnel system with a low resilience level is not enough to resist erosion because of its poor resilience. Currently, the system is difficult to adapt to changes in a short time, and it is impossible to restore health, which ultimately leads to a decline in its resilience level. Therefore, decision makers can only suppress the attenuation trend of the system’s resilience, improve the resilience of the system, and resist risk erosion by increasing the level of one or several indicators that can be increased. In practical applications, the difficulty of index improvement should also be considered.

5. Analysis of Application Examples

5.1. Specifics of the Tunnel

The Ai-Min tunnel of the Ha-Mu high-speed railway is a tunnel with an extremely high risk level in the alpine region of China. The extreme temperature is −35 °C and the maximum buried depth is 60.3 m. At the same time, the tunnel passes through a number of loose sedimentary layers, tertiary sediment rock, and other strata; V or VI rock lengths account for 95%; the line design speed is 250 km; the tunnel is a single tunnel, set up with two tracks, using CRTS III slab ballastless track; the average annual precipitation is 541.3 mm, and the average water seepage is 142.36 mm; to prevent the Ai-Min tunnel from freezing after a large leakage of water during operation, the surrounding rock of the tunnel is provided with an insulation layer, and there is a 6 km resistance heating system on both sides of the tunnel line and in the upper and lower side ditches. The constant temperature is set at 8 °C−15 °C in winter. When the temperature in the tunnel side ditch reaches the lower limit, the heating system in the tunnel will automatically heat to prevent the tunnel drainage ditch from freezing. According to the seismic data released by China Seismic Network and the distribution of seismic zones, the seismic disturbance phenomenon of the tunnel is classified as a 4-level micro disturbance [30].

5.2. Analysis of Resilience Ability

According to the method of Formulas (12)~(14), the index data of the Ai-Min tunnel are normalized, and the normalized data values are shown in

Table 5. Resilience evaluation index.

Evaluation Indicators	Practical Measurement Data	Standardized Data
Rock lithology C1	V-VI	1
Buried depth C2/m	60.3	0.1
Annual rainfall C3/mm	541.3	0.75
Seepage C4/[L·(min·10 m)−1]	142.36	0
Seismic disturbance C5	Tiny disturbance	0.75
Mean temperature in the coldest month C6/°C	−21	0
Cold and antifreeze measures C7	More effective	1
Drainage facilities C8	Good	0.75
Train track dynamic load effects C9	Effective disturbance	0.5
Emergency management measures C10	More effective	1

.

According to the method of Formulas (9)~(19), then, combined with the weight distribution of Table 3, the R_Δ of the optimal fuzzy matter-element and the composite fuzzy matter-element of the surrounding rock resilience of the Ai-Min high-speed railway tunnel is calculated, and the European paste progress is solved.

$$\left\{\begin{array}{l}{R}_{\Delta }=\left[\begin{array}{ccccccccccc}& C_1& C_2& C_3& C_4& C_5& C_6& C_7& C_8& C_9& C_{10}\\ M& 0& 0.81& 0.0625& 1& 0.0625& 1& 0& 0.0625& 0.25& 0\end{array}\right]\\ \rho H_i=0.5561\end{array}\right.$$

(22)

According to Formula (22), the flexibility of the tunnel-surrounding rock system is calculated to be 0.5561, and the resilience level of the Ai-Min tunnel-surrounding rock system is grade 3. Its resilience level is general, indicating that the resilience level of the system needs to be improved, but it can still resist certain risks. According to the division of the resilience ability of the indicators in Figure 3, only four of the nine adaptability indicators have the best ability, and the adaptability of the system is general; among the six resistance indexes, four indexes have low ability, and the system has low bearing capacity. One of the three recovery indicators is not up to standard, and the recovery ability of the system needs to be improved. Therefore, the Ai-Min tunnel will be affected by risk and disaster factors during operation. The resilience of the tunnel to maintain a normal state is general, but it has certain adaptability. Some components in the system can be recovered, but it takes a certain time.

5.3. Analysis of Strategies to Improve Resilience

To further analyze the impact of each resilience index on the evaluation results, we use the product of the resilience index value and the weight to represent the impact value, and then compare the actual impact value with the impact value under the extremely high resilience state, as shown in Figure 7.

From Figure 7, it can be seen that the tunnel is deeply affected by many factors, which leads to poor results. In addition, rainfall and underground water seepage have a greater impact on the resilience system of the surrounding rock of the tunnel. The selected address of the tunnel has rich groundwater. The amount of rainfall in the rainy season and the amount of snow in winter are large. In addition, the drainage equipment and facilities do not fully meet the optimal setting conditions, and the water (atmospheric rainfall and summer snowmelt) can easily penetrate the ground, which leads to the imbalance of groundwater force, which may lead to sudden water gushing, tunnel collapse, and other safety accidents. At the same time, the temperature of the coldest month in the region reaches −35 °C; although the effect of thermal insulation measures is better, it still may lead to a local freeze–thaw phenomenon, and then damage the fine structure of the surrounding rock; there are many geological disasters in the area where the tunnel is located. The earthquakes will disturb the surrounding rock of the tunnel, resulting in a low safety factor.

Based on the results of the above analysis, the two major challenges facing the safe operation of the Ai-Min tunnel are extreme geological disasters and extreme environmental disasters. According to the influence rate and improvement difficulty of each index, the appropriate index should be selected for another decision, so that the tunnel resilience can be transitioned. At the same time, the ability of the tunnel to resist risks should be improved to avoid the tunnel suffering from irresistible risks during operation. Decision makers should optimize the improvement plan to optimize the system’s ability to resist risks, increase the resilience level of the system, mitigate the system’s resilience degradation, and achieve long-term tunnel operations.

6. Phronesis

In solving practical engineering problems, we usually need to consider the influence of multiple factors, which may have different importance and priority. When weighing these factors, we can use the method of weight to quantify and measure different factors and sort and make decisions according to their importance. This method can help us to better consider the influence of different factors and balance them to obtain the optimal decision.

However, some scholars [31,32] believe that the determination of weights should pay more attention to the balance of weighting factors rather than just representing importance. Scholars believe that in practical applications, there may be mutual constraints and mutual influences between different factors. If only the importance of factors is considered, it is easy to cause the decision-making results to be biased in a certain direction and ignore the influence of other important factors. Therefore, scholars believe that the relationship and balance between different factors need to be considered more comprehensively when determining the weight. However, in the tunnel engineering problem, we must set sufficient constraints, which can make the decision problem of improving the toughness level better applied.

At the same time, the determination of weight is also closely related to mathematical problems. In mathematics, we can use multi-objective programming and vector optimization methods to solve multiple objectives with weights and obtain the optimal decision-making scheme. These methods can not only help us to determine the weight, but also, after the weight is determined, the weight can be used again to weight different objectives and solve them to obtain the optimal decision-making scheme.

Therefore, the determination of weight not only represents the importance, but also needs to comprehensively consider the relationship and balance between different factors. In solving practical problems, we need to use the weight method flexibly according to the specific background and needs, and combine mathematical methods and tools to obtain effective solutions. At the same time, we also need to fully understand the subjectivity and uncertainty of weight determination and adopt corresponding techniques and methods to reduce these effects and improve the accuracy and practicability of research results.

7. Conclusions

Based on specific in-service high-speed railway tunnels, we introduce resilience theory to study the impact of risk factors on the safety of in-service high-speed railway tunnel systems. The specific research results are as follows:

• Based on the concept and characteristics of resilience, we put forward the concept of safety resilience of in-service high-speed railway tunnels, and then give the characteristics of resilient high-speed railway tunnels: Robustness, rapidity, redundancy, adaptability, and time–space variability. Accordingly, we should explore a sustainable way to deal with the impact and disturbance, to realize the sustainable development of a high-speed railway tunnel and its surrounding rock system.
• We construct a valuation framework for the safety resilience of in-service high-speed railway tunnels. From the three aspects of resistance, adaptability, and recovery ability, the resilience evaluation index system of the tunnel-surrounding rock system is constructed, including 10 indexes, and then the quantitative standard of index grade is given. Some fuzzy indexes are quantified according to the measured data and research status, and the specific index assignment method is explained to ensure the preciseness of the classification of the evaluation index and the scientificity of the value. Corresponding to the evaluation index, the three resilience characteristics of surrounding rock absorption capacity, adaptability, and recovery ability, make the evaluation model have high feasibility and strong operability.
• We propose a combination weighting method for a resilient system based on information fusion. We determine the local weight, reduce the subjectivity of weight, and combine subjective weighting and objective weighting to improve the accuracy of the weight through information fusion. Based on the TOPSIS improved fuzzy matter-element, a safety resilience evaluation model of an in-service high-speed railway tunnel is proposed, and the resilience level of the system is analyzed by combining Euclidean proximity quantification, and then the resilience grading standard is obtained, which more intuitively reflects the resilience degree of the surrounding rock system of a high-speed railway tunnel.
• To enable decision makers to make better decisions and improve the resilience level of the surrounding rock of the high-speed railway tunnel, we study the strategy of improving the resilience of the system based on the temporal and spatial variability of the ductile tunnel. Then, we put forward the concept of resilience transition and resilience attenuation, and establish the transition model to evaluate the ability and difficulty of improving the resilience of the surrounding rock system in the initial state. Finally, we gain the optimal transition order of the index: C2 > C8 > C7 > C10 > C9. Based on the evolution model of high-speed railway tunnel-surrounding rock, we found that on the time scale, considering the accumulation of tunnel-surrounding rock damage, the overall resilience change of the high-speed railway tunnel-surrounding rock system is from high to low, gradually decaying, showing a downward trend, and finally tending to be gentle. To suppress this attenuation trend, we introduce an attenuation model, which is used for studying the mechanism of resilience attenuation, and then we propose a method to enhance resilience. Improving the resilience level of one or more transition indexes that are easily damaged and restraining the system resilience attenuation trend is beneficial to improving the system resilience and the ability to resist risk erosion.
• It is calculated that the resilience level of the surrounding rock system of the Ai-Min tunnel is grade 3, and the resilience level is general, indicating that the resilience level of the system needs to be improved, but it can still resist certain risks. According to the comprehensive analysis results, extreme geological disasters and environmental disasters are the two major challenges faced in the safe operation of the Ai-Min tunnel. Based on the influence rate of each index and the difficulty of improvement, we should select the appropriate index for decision making, so that the tunnel resilience can be increased. The evaluation results are consistent with the actual engineering situation, which verifies the effectiveness of the model.