An Improved Unascertained Measure-Set Pair Analysis Model Based on Fuzzy AHP and Entropy for Landslide Susceptibility Zonation Mapping

Abstract

Landslides are one of the most destructive and common geological disasters in the Tonglvshan mining area, which seriously threatens the safety of surrounding residents and the Tonglvshan ancient copper mine site. Therefore, to effectively reduce the landslide risk and protect the safety of the Tonglvshan ancient copper mine site, it is necessary to carry out a systematic assessment of the landslide susceptibility in the study area. Combining the unascertained measure (UM) theory, the dynamic comprehensive weighting (DCW) method based on the fuzzy analytic hierarchy process (AHP)-entropy weight method and the set pair analysis (SPA) theory, an improved UM-SPA coupling model for landslide susceptibility assessment is proposed in this study. First, a hierarchical evaluation index system including 10 landslide conditioning factors is constructed. Then, the dynamic comprehensive weighting method based on the fuzzy AHP-entropy weight method is used to assign independent comprehensive weights to each evaluation unit. Finally, we optimize the credible degree recognition criteria of UM theory by introducing SPA theory to quantitatively determine the landslide susceptibility level. The results show that the improved UM-SPA model can produce landslide susceptibility zoning maps with high reliability. The whole study area is divided into five susceptibility levels. 5.8% and 10.16% of the Tonglvshan mining area are divided into extremely high susceptibility areas and high susceptibility areas, respectively. The low and extremely low susceptibility areas account for 30.87% and 34.14% of the total area of the study area, respectively. Comparison with the AHP and Entropy-FAHP models indicates that the improved UM-SPA model (AUC = 0.777) shows a better performance than the Entropy-FAHP models (AUC = 0.764) and the conventional AHP (AUC = 0.698). Therefore, these results can provide reference for emergency planning, disaster reduction and prevention decision-making in the Tonglvshan mining area.

1. Introduction

Landslides, as one of the major geological disasters in nature [1,2,3], have caused great economic losses and potential safety hazards to mining, water conservancy, transportation, national defense, protection of cultural relics and other fields, posing a serious threat to people’s work and life safety [4,5,6,7]. Due to complex engineering geological conditions, large mining scale, frequent activities and many construction disturbance factors, the stability of open-pit slopes affects the safety of the whole mining process. Therefore, in order to effectively assess the potential landslide risk and reduce the impact of landslide geological hazards, it is necessary to conduct landslide susceptibility analysis on the regions prone to landslide disasters [8,9,10]. However, it is still a challenging task to reliably evaluate and predict LSZ mapping due to the combined influence of landform, climatic conditions, rainfall, stratigraphic lithology and structure, as well as human engineering activities [11,12,13].

Landslide susceptibility is the spatial probability of the possibility of slope failure under specific geological conditions [14]. Over the last decades, researchers have developed a variety of GIS-based landslide susceptibility assessment methods. These methods can be broadly divided into three categories: qualitative, semi-quantitative and quantitative methods [8,15]. Qualitative methods, which depend on the expert’s experience, are based on the field geological survey and the expert’s understanding of the landslide to assess landslide susceptibility [9,15,16]. Semi-quantitative research is based on qualitative research, combined with original field data, and uses subjective methods to rank the importance of each influencing factor [17,18], such as analytic hierarchy process (AHP) [1] and weight linear combination (WLC) [19]. As a comparison, quantitative methods employ mathematical models to evaluate the relationship between conditioning factors and landslide instability and then define hazard zones on a continuous scale [9,16,17]. Hence, it is considered to be objective. Quantitative methods usually refer to deterministic analysis and statistical methods [9,17]. In the deterministic method, the limit equilibrium method or numerical analysis is used to determine the possibility of potential landslides. However, it is difficult to obtain the spatial variability of geotechnical parameters, which limits its application in large areas [9,20]. Statistical methods indirectly depend on the inherent relationship of occurred landslides with conditioning factors, and based on that, the spatial probability of landslides is established. Typical statistical methods include frequency ratio (FR) [17,21], logistic regression (LR) [12], information value (IV) [22], artificial neural network (ANN) [15,17], support vector machines (SVM) [23,24], weights-of-evidence (WOE) [25], etc. Recently, some new hybrid methods have been developed for landslide susceptibility analysis, such as the adaptive neuro-fuzzy inference system and frequency ratio (ANFIS-FR) [26], the AHP-fuzzy [19,24], AHP and artificial neural network (AHP-ANN) [27] and the wavelet packet-statistical model (WP-SM) [28].

Although many different LSZ methods have been proven effective, there is no agreement on which technology and method are the most suitable. This is because each LSZ method and technology has its own characteristics and limitations. For example, ANN and SVM models are very good at dealing with nonlinear problems in landslide susceptibility assessment, but their prediction accuracy depends on the integrity of the landslide inventory. AHP is not affected by any prior statistical relationship between historical landslides and causative factors, but its decision-making process is greatly influenced by subjective factors. The fuzzy comprehensive evaluation method is suitable for solving the fuzzy multi-criteria decision problem, but it overemphasizes the role of extreme values, which makes it easy to lose intermediate information. Therefore, finding a single prediction model to meet the needs of landslide susceptibility assessment accuracy has been difficult. Research shows that the accuracy of LSZ depends largely on the scale of the study area, data availability and the selection of appropriate modeling methods [14,29]. Landslide instability is the result of multiple factors and has fuzziness, complexity, and uncertainty [19,30]. Therefore, the evaluation of landslide susceptibility is an uncertain problem. UM theory is a new method that can deal with a lot of uncertain information and can be quantitative analysis [31]. Compared with other mathematical theories, UM theory not only satisfies the principles of “non-negative boundedness”, “normalization” and “additivity”, but also has significant advantages in solving the problem of ordered segmentation. Therefore, we try to introduce UM theory into the study of LSZ. In addition, another noteworthy problem is the impractical determination of the weight of the controlling factors. Previous LSZ studies assigned fixed weights to the entire study area, which did not reflect the spatial variability of the controlling factors in different locations.

Aiming at the above problems, combining UM theory, the DCW method based on fuzzy AHP-entropy weight method and SPA theory [32], an improved UM-SPA coupling model for landslide susceptibility assessment is proposed in this study. Among them, the DCW method based on fuzzy AHP-entropy weight method is a kind of weighting method combining subjective and objective weights. However, in contrast to the previous research, this research takes a single index unascertained measure matrix as the research object to determine the objective weight, so that the final comprehensive weight changes along with the evaluation unit. The Tonglvshan Ancient Copper Mine is by far the oldest, largest and highest mining and smelting level, and the most well-preserved ancient copper mine site in China. The ancient mining and smelting site of Tonglvshan has an important position and significance in the history of human civilization. Therefore, this paper takes the Tonglvshan mining area in Daye as an example to study the landslide susceptibility. First, a hierarchical evaluation index system including 10 landslide conditioning factors such as lithology, aspect, slope, relief amplitude, distance of draft, elevation, distance of fault, topographic wetness index (TWI), land cover/land use (LULC) and their subclasses is constructed. To reduce the bias of factor selection, multicollinearity analysis was used to examine the independence of selected factors. Then, the UM model of landslide susceptibility evaluation based on the comprehensive evaluation index system is constructed, and the dynamic comprehensive weighting method is used to assign independent comprehensive weights to each evaluation unit. Finally, the identity-discrepancy-contrary connection degree in SPA theory is introduced to optimize the credible degree recognition criteria of UM theory and quantitatively determine the vulnerability grade of a landslide. The application results show the potential of the proposed method in this paper and these results can provide reference for emergency planning, disaster reduction and prevention decision-making in the Tonglvshan mining area.

2. Proposed Methodology

Drawing on the basic ideas of UM theory and SPA theory, and taking into account the actual situation of the Tonglvshan mining area in Daye, China, an improved UM-SPA coupling model for landslide susceptibility assessment is proposed in this study. The construction steps of the improved UM-SPA model are described as follows.

2.1. Unascertained Measure Theory

The concept of unascertained information was first put forward by Wang Guangyuan [31], and then expanded and studied, and the mathematical theory of unascertained information was established. This method is mainly used to solve the mathematical problems of fuzziness, randomness and uncertainty in a system, and has been widely used in the field of natural science [33,34]. The core step is to construct the evaluation system of each evaluation index, divide the evaluation grade standard, establish the single index measure evaluation matrix of the evaluation object based on the measure function, and then combine the weight value of the evaluation index to determine the multi-index comprehensive measure evaluation matrix.

Suppose there are $n$ objects $R$ to be evaluated in the system; then the evaluation object space is $R=\left\{R_1,R_2,\cdots ,R_n\right\}$, and for each evaluation object $R_i(i=1,2,\cdots ,n)$, there are $m$ single evaluation index space, that is $X=\left\{X_1,X_2,\cdots ,X_m\right\}$. $R_i$ can be expressed as m-dimension vector $R_i=\left\{x_{i1},x_{i2},\cdots ,x_{im}\right\}$, where $x_{ij}$ represents the measured value of the object $R_i$ to be evaluated regarding the evaluation index. Suppose each sub-item $x_{ij}$ has $p$ evaluation levels; then the evaluation level space is $U=\left\{C_1,C_2,\cdots C_p\right\}$. Set $C_k(k=1,2,\cdots ,p)$ as the $k-\mathrm{th}$ grade evaluation level, and the stability of the $k-\mathrm{th}$ grade is stronger than that of the $(k+1)-\mathrm{th}$ level, and is denoted as $C_k>C_{k+1}$; then set $\left\{C_1,C_2,\cdots C_p\right\}$ is an ordered segmentation class on the evaluation space $U$.

If $z_{ijk}=z(x_{ij}\in C_k)$ represents the degree that the measured value of the sample $x_{ij}$ belongs to the $k-\mathrm{th}$ grade $C_k$ and meets the requirements of Equations (1)–(3), $z$ is called uncertainty measure.

$$0\le z(x_{ij}\in C_k)\le 1$$

(1)

$$z(x_{ij}\in U)=1$$

(2)

$$z\left(x_{ij}\in {\displaystyle \underset{l=1}{\overset{k}{\cup }}{C}_l}\right)={\displaystyle \sum _{l=1}^{k}z\left(x_{ij}\in C_l\right)}$$

(3)

where, Equation (1) is called the non-bounded property of $z$ in the evaluation space $U$, Equation (2) is called the uniformity of $z$ to the evaluation space $U$ and Equation (3) is called the additivity of $z$ to the evaluation space $U$.

According to the definition, to determine the measure value $z_{ijk}$ of each index of the evaluation object $R_i$, it is necessary to construct the measure function $z(x_{ij}\in C_k)$ of each evaluation index. The matrix ${(z_{ijk})}_{m\times p}$ composed of each index measure value is called the single index measure evaluation matrix. The measure matrix is shown as follows:

$$z={(z_{ijk})}_{m\times p}=\left[\begin{array}{cccc}{z}_{i11}& z_{i12}& \cdots & z_{i1p}\\ z_{i21}& z_{i22}& \cdots & z_{i2p}\\ ⋮& ⋮& \ddots & ⋮\\ z_{im1}& z_{im2}& \cdots & z_{imp}\end{array}\right]$$

(4)

where the $j-\mathrm{th}$ row vector $(z_{ij1},z_{ij2},\cdots ,z_{ijp})$ of the measure matrix is the single index measure evaluation vector of the evaluation index $x_{ij}$.

2.2. Set Pair Analysis Theory

SPA theory, proposed by Zhao Keqin [32], is a systematic analysis method used to deal with the “determination-uncertainty problem”. The core idea is to regard the certainty and uncertainty of the problem studied as a whole system, using the identity degree and contrary degree to describe the certainty of the system, using the discrepancy degree to describe the uncertainty of the system and to quantitatively describe the relationship between the certainty and uncertainty of the system from the three aspects of similarities and differences [35,36].

Set pair refers to the pair formed by two sets with a certain relation in the system. Suppose there are two mutually related sets $A$ and $B$, which have $N$ indexes to represent their characteristics, and the set pair formed is $H(A,B)$; then the relation degree $\mu$ of the set pair $H(A,B)$ can be expressed as:

$$\mu =\frac{S}{N}+\frac{F}{N}p+\frac{P}{N}q=a+bp+cq$$

(5)

where, $\mu$ is the connection degree of the set pair $H(A,B)$, $N$ is the total number of characteristics of the set pair, $S,F,P$ are the identity characteristic number, the anisotropy characteristic number and the opposition characteristic number of set pair, respectively. $a,b,c$, respectively, represent the identity degree, discrepancy degree and contrary degree of set pairs, and satisfy $a+b+c=1$; $p$ is the coefficient of discrepancy degree, $p\in [-1,1]$; $q$ is the coefficient of opposition. In general, $q=-1$.

The conventional SPA theory roughly divides the characteristics of the set pair $H(A,B)$ into three categories, which is too rough to accurately describe the intricate uncertainty relationship between the risk level and various influencing factors, affecting the accuracy of the final evaluation results. Thus, it is necessary to extend the ternary connection degree $\mu$ to the multivariate connection degree ${\mu }_n$ to reflect different degrees of difference [37]. The expression of multivariate connection degree is as follows:

$${\mu }_n=\frac{S}{N}+\frac{F_1}{N}{p}_1+\frac{F_2}{N}{p}_2+\cdots +\frac{F_{n-2}}{N}{p}_{n-2}+\frac{P}{N}q$$

(6)

Equation (6) can be represented as:

$${\mu }_n=a+b_1{p}_1+b_2{p}_2+\cdots +b_{n-2}{p}_{n-2}+cq=\Gamma \cdot E$$

(7)

where $p_1,p_2,\cdots ,p_{n-2}$ is the coefficient of discrepancy degree, $b_1,b_2,\cdots ,b_{n-2}$ is the component of discrepancy degree, representing different levels of the discrepancy degree; $a\in [0,1], b_{n-2}\in [0,1], c\in [0,1]$, and $a+b_1+b_2+\cdots +b_{n-2}+c=1$; $\Gamma$ is the IDC membership matrix stated as $\Gamma =[a,b_1,b_2,\cdots ,b_{n-2},c]$; $E$ is the component matrix of connection degree represented as $E={[1,p_1,p_2,\cdots ,p_{n-2},q]}^T$.

2.3. Improved UM-SPA Model

The evaluation method of geological hazard zoning of landslides in mining area mainly includes four parts: the selection of landslide influencing factors, the multicollinearity analysis of evaluation indexes, the calculation of dynamic comprehensive weight of each evaluation index and the construction of the evaluation model. The dynamic comprehensive weight of the evaluation index is determined based on the fuzzy AHP and entropy weight method, and the flow chart of landslide susceptibility zonation in the Tonglvshan mining area is shown in Figure 1.

In step 1, build a comprehensive evaluation index system. The landslide geological hazard in the Tonglvshan mining area is a complex dynamic phenomenon of hydrogeology and mining. The construction of a comprehensive evaluation index system is the basis of the evaluation model. Accordingly, in step 2, calculate the dynamic comprehensive weight of each indicator. By analyzing the combination of various control factors in detail, the partition state variable weight vector can be constructed. In step 3, the multi-attribute comprehensive unascertained evaluation quantity of each evaluation index can be determined by combining the dynamic comprehensive weight with the single index unascertained degree matrix. Finally, based on SPA theory, the landslide susceptibility grade of the whole mining area is divided.

2.3.1. Construct Single Index Unascertained Measure Function

According to the definition of unascertained degree, the key to determine the evaluation matrix ${(z_{ijk})}_{m\times p}$ of the single index measure of the evaluation object is to construct a reasonable unascertained measure function. Currently, the construction methods of unascertained measure function mainly include the straight-line method (SLM), the exponential curve method (ECM), the sine curve method (SCM) and the quadratic curve method (QCM) [33,34]. In practice, decision-makers should choose the appropriate unascertained measure function according to the change characteristics of specific evaluation index. However, irrespective of the type of measure function adopted, the principles of “non-negative, additivity, normalization” must be met. Among them, the linear unascertained measure function is widely used in various risk assessment models because of its simple calculation and strong adaptability [34]. Therefore, the straight-line type (SLT) unascertained measure function is adopted for analysis and calculation in this paper, as shown in Figure 2. The corresponding expression of interval $[d_k,d_{k+1}]$ is shown in Equation (8).

$$\begin{array}{l}\{\begin{array}{r}{z}_{ijk}=z_{jk}\left(x_{ij}\right)=\{\begin{array}{l}\frac{-x_{ij}}{d_{k+1}-d_k}+\frac{d_{k+1}}{d_{k+1}-d_k}, d_k<x_{ij}⩽d_{k+1}\hfill \\ 0,x_{ij}>d_{k+1}\hfill \end{array}\\ z_{ij(k+1)}=z_{j(k+1)}\left(x_{ij}\right)=\{\begin{array}{l}0,x_{ij}⩽d_k\hfill \\ \frac{x_{ij}}{d_{k+1}-d_k}-\frac{d_k}{d_{k+1}-d_k}, d_k<x_{ij}⩽d_{k+1}\hfill \end{array}\end{array}\\ \end{array}$$

(8)

where $x_{ij}$ is the measured value of the evaluation index and $d_{k+1}$ and $d_k$ are the upper and lower boundary values of $k-\mathrm{th}$ grade, which can be determined based on the grading standard value of evaluation indices.

2.3.2. Determine the Subjective Weight Using Fuzzy AHP

AHP is a semi-quantitative decision-making analysis method [1,38,39], which has been widely used in the multi-criteria decision-making (MCDM) process to obtain the subjective weights for different criteria. However, since the conventional AHP cannot properly depict the decision maker’s options based on quantitative articulation of preference, AHP based on fuzzy theory extension was introduced to solve the fuzzy hierarchical problems [14,19]. The fuzzy AHP model adopted by this research is to achieve fuzzy hierarchical analysis by assigning fuzzy numbers to the pairwise comparison results. There are various types of fuzzy numbers [14,40], but for this study, the triangular fuzzy number (TFN) was used for the subjective weights calculation process. it is necessary to briefly introduce the basic concepts and operation rules of TFN.

Let a fuzzy number $\tilde{M}$ be a triangular fuzzy number on $R$, then its member function ${\mu }_{\tilde{M}}(x)\mathsf{:} R\to [0,1]$ can be defined as:

$${\mu }_{\tilde{M}}(x)=\{\begin{array}{ll}\frac{x}{m-l}-\frac{l}{m-l},& l\le x\le m\\ \frac{x}{m-u}-\frac{u}{m-u},& m\le x\le u \\ 0,& \mathrm{otherwise}\end{array}$$

(9)

where $l\le m\le u$, $l, u$ represent the lower and upper value of the interval of $l, u$ respectively, and $m$ represent the modal value. The triangular fuzzy number $\tilde{M}$ can be expressed as: $\tilde{M}=(l, m, u)$.

Let two triangular fuzzy numbers ${\tilde{M}}_1=(l_1, m_1, u_1)$ and ${\tilde{M}}_2=(l_2, m_2, u_2)$, then their operation rules are as follows:

$${\tilde{M}}_1\oplus {\tilde{M}}_2=(l_1+l_2,m_1+m_2,u_1+u_2)$$

(10)

$${\tilde{M}}_1\otimes {\tilde{M}}_2(l_1\times l_2,m_1\times m_2,u_1\times u_2)$$

(11)

$$\lambda \otimes {\tilde{M}}_1=(\lambda \times l_1,\lambda \times m_1,\lambda \times u_1)$$

(12)

$${\tilde{M}}_1^{-1}={(l_1,m_1,u_1)}^{-1}=(\frac{1}{u_1},\frac{1}{m_1},\frac{1}{l_1})$$

(13)

The fuzzy AHP based on TFN has been discussed and studied by many experts and scholars. Among them, Chang’s [41] study is phenomenal. Chang developed a new method, which uses TFN to determine the pairwise comparison scale of fuzzy AHP, and adopts the extent analysis method to obtain the comprehensive extent value of pairwise comparison. In this study, Chang’s extent analysis method is employed to determine the subjective weight of landslide conditioning factors. The detailed steps can be summarized as follows:

Step 1: Constructing fuzzy judgment matrix based on TFN. First, experts are invited to compare each pair of factors in the index layer in pairs to obtain the comparative results of the relative importance of each index. Then, the triangular fuzzy number is used to construct the fuzzy judgment matrix. The corresponding relationship between linguistic variables and fuzzy numbers is shown in

Table 1. Triangular fuzzy number scale and linguistic variables [8,14,40].

Linguistic Variables	Triangular Fuzzy Numbers	Reciprocal of Triangular Fuzzy Numbers
Equally Important	(1,1,1)	(1,1,1)
Slightly Important	(2,3,4)	(1/4,1/3,1/2)
Moderately Important	(4,5,6)	(1/6,1/5,1/4)
Very Important	(6,7,8)	(1/8,1/7,1/6)
Extremely Important	(9,9,9)	(1/9,1/9,1/9)
Intermediate Value	(1,2,3), (3,4,5), (5,6,7), (7,8,9)	(1/3,1/2,1), (1/5,1/4,1/3), (1/7,1/6,1/5), (1/9,1/8,1/7)

$$\tilde{K}=\left[\begin{array}{cccc}\tilde{1}& {\tilde{k}}_{12}& \cdots & {\tilde{k}}_{1n}\\ {\tilde{k}}_{21}& \tilde{1}& \cdots & {\tilde{k}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{k}}_{n1}& {\tilde{k}}_{n2}& \cdots & \tilde{1}\end{array}\right]=\left[\begin{array}{cccc}\tilde{1}& {\tilde{k}}_{12}& \dots & {\tilde{k}}_{1n}\\ {\tilde{k}}_{12}{}^{-1}& \tilde{1}& \dots & {\tilde{k}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{k}}_{1n}{}^{-1}& {\tilde{k}}_{2n}{}^{-1}& \dots & \tilde{1}\end{array}\right]$$

(14)

where ${\tilde{k}}_{ij}=(l_{ij}, m_{ij}, u_{ij}), (i,j=1,2,\cdots ,n \mathrm{and} i\ne j)$ denotes the relative importance of comparison value of criterion $i$ and criterion $j$; let $\tilde{1}$ be $(1,1,1)$, when $i$ equal $j(\mathrm{i}.\mathrm{e}. i=j)$; whereas ${\tilde{k}}_{ji}={\tilde{k}}_{ij}^{-1}=(1/u_{ij}, 1/m_{ij}, 1/l_{ij})$ denotes the reciprocal fuzzy comparison value. Note that $i$ and $j$ here only represent the sequence number of variables.

If $H$ experts evaluate each pair of evaluation index in pairs, we can get a series of triangular fuzzy numbers: $(l_1, m_1, u_1)$, $(l_2, m_2, u_2),\cdots ,(l_h, m_h, u_h)$. Then, ${\tilde{k}}_{ij}$ is called the synthetic triangular fuzzy number.

$${\tilde{k}}_{ij}=\frac{1}{H}\otimes \left({\tilde{k}}_{ij}^1+{\tilde{k}}_{ij}^2+\cdots +{\tilde{k}}_{ij}^h\right)$$

(15)

where ${\tilde{k}}_{ij}^h=(l_{ij}^h, m_{ij}^h, u_{ij}^h), (h=1,2,\cdots H)$ is the triangular fuzzy number given by the $h-\mathrm{th}$ expert.

Step 2: Determine the fuzzy comprehensive value of each evaluation index. Firstly, each row of fuzzy judgment matrix is summed. Then, the rows of the fuzzy judgment matrix are normalized and the fuzzy comprehensive value of the $i-\mathrm{th}$ evaluation index is calculated.

$${\tilde{S}}_i={\displaystyle \sum _{j=1}^m{a}_{ij}}\otimes {\left[{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{a}_{ij}}}\right]}^{-1}=\left(\frac{{\displaystyle {\sum }_{j-1}^n{l}_{ij}}}{{\displaystyle {\sum }_{k=1}^n{\displaystyle {\sum }_{j=1}^n{u}_{kj}}}},\frac{{\displaystyle {\sum }_{j-1}^n{m}_{ij}}}{{\displaystyle {\sum }_{k=1}^n{\displaystyle {\sum }_{j=1}^n{m}_{kj}}}},\frac{{\displaystyle {\sum }_{j-1}^n{u}_{ij}}}{{\displaystyle {\sum }_{k=1}^n{\displaystyle {\sum }_{j=1}^n{l}_{kj}}}}\right)$$

(16)

Step 3: The final weight of evaluation index is obtained by defuzzification. In order to obtain the estimated value of the weight vector of each evaluation index, it is necessary to defuzzify the fuzzy three-value weight. Usually, adopt the fuzzy number comparison principle for processing, using the following equation to calculate the possibility of fuzzy number ${\tilde{S}}_i\ge {\tilde{S}}_j$:

$$V\left({\tilde{S}}_i\ge {\tilde{S}}_j{}_i\right)=\underset{x\ge y}{\sup }\left\{\min \left[{\mu }_{{\tilde{M}}_i}(x),{\mu }_{{\tilde{M}}_j}(y)\right]\right\}$$

(17)

The equation can also be expressed as:

$$V\left({\tilde{S}}_i\ge {\tilde{S}}_j\right)=\{\begin{array}{cc}1& m_i\ge m_j\\ \frac{u_i-l_j}{\left(u_i-m_i\right)+\left(m_j-l_j\right)}& l_j\ge u_i\\ 0& \mathrm{otherwise} \end{array}$$

(18)

If the probability of a fuzzy number ${\tilde{S}}_i$ is greater than that of other $n-1$ fuzzy numbers, it can be defined as:

$$V\left({\tilde{S}}_i\ge {\tilde{S}}_j{}_i|j=1,2,\cdots ,n-1;j\ne i\right)=\min \left({\tilde{S}}_i\ge {\tilde{S}}_j{}_i\right)$$

(19)

According to Chang’s proposed method, the minimum possible degree obtained by the above equation is the initial weight of the fuzzy number, and it is denoted as: $d(A_i)$

$$d(A_i)=\min V(S_i\ge S_j\ne 0)$$

(20)

Finally, the initial weight is normalized to get the final subjective weight.

$$W_i=\frac{V\left({\tilde{S}}_i\ge {\tilde{S}}_j{}_i|j=1,2,\cdots ,n-1;j\ne i\right)}{{\displaystyle \sum _{i=1}^{n}V\left({\tilde{S}}_i\ge {\tilde{S}}_j{}_i|j=1,2,\cdots ,n-1;j\ne i\right)}}$$

(21)

2.3.3. Determine the Objective Weight Using Entropy

Entropy, originally a thermodynamic concept used to quantitatively characterize the degree of system disorder, was first introduced into information theory by Shannon [42]. In information theory, Shannon entropy reflects the disordered degree of information and can be used to measure the amount of information. The lower the entropy value of an evaluation index, the more information it carries, indicating that the index plays a greater role in the comprehensive decision-making process. Therefore, Shannon entropy can be used to evaluate the order degree and utility of the information obtained, namely, the judgment matrix composed of evaluation index values to determine its weight.

Taking single index unascertained measure matrix as the research object, according to the conventional concept of information entropy, the entropy of the $j-\mathrm{th}$ evaluation index can be defined as [29,33]:

$$H_j=-\frac{1}{\ln p}{\displaystyle \sum _{k=1}^p{z}_{jk}}\ln z_{jk} ,(j=1,2,\cdots ,m;k=1,2,\cdots ,p)$$

(22)

where $p$ is the number of evaluation levels of the system; $z_{jk}$ is satisfied such that $j-\mathrm{th}$. We stipulate that $H=0$ when $z_{jk}=0$.

According to the calculation principle of the entropy weight method, the entropy weight of the $j-\mathrm{th}$ evaluation factor can be defined as:

$$V_j=\frac{1-H_j}{m-{\displaystyle \sum _{j=1}^m{H}_j}}$$

(23)

where $V_j$ is the entropy weight of the $j-\mathrm{th}$ evaluation index. Similarly, the corresponding weights of other indexes can be obtained. Then, the weight vector of the evaluation index can be expressed as: $V_j=(V_1,V_2,\cdots ,V_m)$.

2.3.4. Calculation of Dynamic Comprehensive Weights

As mentioned above, the fuzzy analytic hierarchy process builds a fuzzy judgment matrix according to subjective experience of experts, while the evaluation process of the entropy weight method depends on the law of field-measured data. In order to not only reflect the experts’ intuitive understanding of landslide disaster in the field, but also reflect the objective law and authenticity of field-measured data, this paper introduces the principle of minimum information entropy [43] and combines the fuzzy analytic hierarchy process and entropy weight method to establish a dynamic comprehensive weight that changes with the specific situation of the object to be evaluated.

According to the principle of minimum information entropy, the deviation function of dynamic comprehensive weight can be constructed as:

$$\min F={\displaystyle \sum _{j=1}^m{\omega }_z(\ln {\omega }_j-\ln W_I)+}{\displaystyle \sum _{j=1}^m{\omega }_z(\ln {\omega }_j-\ln V_j)}$$

(24)

$$\mathrm{s}.\mathrm{t}. {\displaystyle \sum _j^m{\omega }_j}=1 ,({\omega }_j>0,j=1,2,\cdots ,m)$$

(25)

The Lagrange multiplier algorithm is used to solve Equations (25) and (26); then, the dynamic comprehensive weight of the $j-\mathrm{th}$ evaluation index is:

$${\omega }_j=\frac{{(W_j{V}_j)}^{0.5}}{{\displaystyle \sum _i^n{(W_j{V}_j)}^{0.5}}}$$

(26)

2.3.5. Multi-Index Comprehensive Measure Evaluation Vector

The single-index unascertained measure aims to quantify the degree of uncertainty of a single index, but cannot reflect the influence of these indexes on the whole evaluation system. Therefore, by introducing the weight coefficient to carry on the comprehensive consideration and by combining the unascertained measure matrix of single index with the dynamic synthetic weight, the multi-index synthetic measure evaluation vector of the evaluation object can be obtained [33,34]. Let $z_{ik}=z(R_i\in C_k)$ be the degree to which the sample $R_i$ to be evaluated belongs to the $k-\mathrm{th}$ evaluation grade $C_k$; then,

$$z_{ik}={\displaystyle \sum _{j=1}^m{w}_j{z}_{ijk}} (i=1,2,\cdots ,n; k=1,2,\cdots ,p)$$

(27)

where in, $0\le z_{ik}\le 1,{\displaystyle \sum _{k=1}^p{z}_{ik}=1}$; then, $z_{ik}$ is called an unascertained measure; $z_{ik}=(z_{i1},z_{i2},\cdots ,z_{ip})$ is called the evaluation vector of the multi-index comprehensive measure of the evaluation object $R_i$.

2.3.6. Classification of Landslide Susceptibility Zonation

In order to improve the confidence $\lambda$ arbitrary value $\lambda >0.5$ in the confidence criterion of unascertained measure theory, this study adopts SPA theory to construct the corresponding relationship between the correlation number and the landslide susceptibility evaluation level, and directly calculate the susceptibility level of the evaluation object quantitatively [33,44].

In this study, the comprehensive evaluation vector $z_{ik}$ of the unascertained measure of multiple indexes is regarded as the same and different anti-membership matrix of the evaluation object $R_i$ at the levels of landslide risk regionalization is substituted into Equation (7) to obtain the $T-\mathrm{element}$ connection number ${\mu }_t$ of the overall index comprehensive evaluation of the evaluation object $R_i$:

$${\mu }_t=z_{ik}\cdot E=z_{ik}\cdot \left[1,p_{i1},p_{i2},\cdots ,p_{i(k-1)},\cdots ,p_{i(t-2)},q\right]$$

(28)

Since ${\mu }_t$ is a $T-\mathrm{element}$ connection number and ${\mu }_t\in [-1,1]$, the value range of ${\mu }_t$ is divided into $t-1$ equivalents based on the principle of equipartition, and $p_{i(t-2)},p_{i(t-1)},\cdots ,p_2,p_1$ are assigned to $t-1$ average fractions in the range [–1, 1] from left to right, $q=-1$; then, the component matrix $E$ of the $T-\mathrm{element}$ connection number is shown as follows:

$$E={\left[1,\frac{t-3}{t-1},\frac{t-5}{t-1},\cdots ,\frac{t+1-2k}{t-1},\cdots ,\frac{5-2t}{t-1},-1\right]}^{\mathrm{T}}$$

(29)

With regard to the classification of landslide susceptibility zoning, this study is based on the principle of average division [36,44] and divides the judgment interval of the susceptibility level corresponding to the comprehensive connection degree. The interval $[-1,1]$ was divided into $t$ equal fractions to obtain $t$ value intervals, each of which corresponds to a susceptibility assessment level.

3. Study Area and Materials

3.1. Study Area

The Tonglvshan mining area is located in Daye City, Hubei Province, approximately 3 km southwest of Daye City and 99 km southeast of Wuhan City. The administrative region is under the jurisdiction of the Jinhu Sub-District Office of Daye City, and its geographical coordinates are as follows: with east longitude 114°55′26″–114°57′19″ and north latitude 30°04′21″–30°05′46″, the mining area is approximately 5.61 km². The location of the research area is shown in Figure 3. The mining area is low mountain and hilly landform, and the surrounding terrain is high in the south and low in the north. Before mining, the terrain has little fluctuation, and the elevation is 14.5~58.15 m. At present, the topography and geomorphology in the study area have changed greatly under the influence of mining. After the shallow mining resources are exposed, two adjacent open mining pits with a length of approximately 1500 m from north to south and a width of approximately 500 m from east to west have been formed. The slope top elevation of the stope is 30–50 m, the lowest elevation of the pit bottom (southern open-pit) is −175 m and the slope angle of the stope is 46–56.5°. Highway 3 runs between the eastern slope of the North Open Pit Mine and the ancient copper Mine Site Museum [35,45].

Located in the south bank of the middle and lower reaches of the Yangtze River, the study area is a typical subtropical continental monsoon climate, characterized by cold winters and hot summers, four distinct seasons, abundant rainfall and annual average rainfall of 1382.6 mm. The annual rainfall distribution is uneven. Due to the influence of the monsoon trough, most rainfall events occur between April and August, with that rainfall accounting for about 67~85.2% of the annual rainfall [45].

Geologically, the structure of the study area is interleaved and superimposed by the NW folds and compressional faults formed in the Indosinian period and the NE folds and compressional faults formed in the Yanshanian period. These two groups of structures with different formation periods and properties control the spatial distribution of the deposit and ore body. A large area of the study area is covered by a Quaternary clay layer. The exposed floor layer of the mining area is mainly the marble and dolomite marble of the Lower Triassic Daye Formation and the tubreccia and tuff of the Lower Cretaceous Dasi Formation. The area is rich in mineral resources, complex geological structure, a large number of copper, iron, gold and other metallic mineral resources and limestone, lapis lazuli, marble and other non-metallic mineral resources. It is an important part of the middle and lower reaches of the Yangtze River metallogenic belt [35,45].

3.2. Landslide Inventory Map

Creating a landslide cataloguing map is the key step and a prerequisite for landslide susceptibility evaluation [18]. They provide information on the spatial distribution of the current landslide location, as well as the known dates of landslides and the types of landslides that have left visible traces in an area. This is helpful to deeply understand the relationship between the occurrence of landslides and their causes in the past [9,46]. Therefore, an accurate landslide cataloguing map is very important for landslide susceptibility evaluation. In this study, landslide catalogues of the study area were produced through extensive field investigation, review of historical and documentary landslide records and visual interpretation of Google Earth images. A total of 12 landslides have been identified, as shown in Figure 3c. The total area of the landslide is 21,787.83 m², accounting for about 0.39% of the total area of the study area, in which the minimum and maximum area of the landslide are 439.17 m² and 5388.67 m² respectively.

As can be seen from Figure 3, the landslide in the study area is mainly distributed on the high and steep slope of the east side of the north open-pit mine, showing the characteristics of flake distribution. Due to the joint action of open-pit mining in the early stage and underground mining in the late stage, the slope of the east side of the North open-pit mine and Site Museum tend to further aggravate the deformation, which seriously threatens the protection of the Tonglvshan ancient copper mine site.

3.3. Landslide Conditioning Factors (LCFs)

Landslide susceptibility evaluation is to determine the most favorable combination of influencing factors for slope instability based on the analysis of geological environmental conditions, so as to predict the possibility of landslide occurrence in a certain area [47,48]. Selecting a reasonable LCF is the key step of the landslide susceptibility evaluation model, which will directly affect the accuracy and precision of the susceptibility evaluation results. However, due to the complexity of different regional geological environments, the existing landslide susceptibility evaluation lacks unified criteria for the selection of LCF [9,49]. In order to obtain a relatively reasonable LCF, the LCF can be determined according to the engineering geological characteristics and scale of the study area, expert knowledge reserve or experience, scientific and technological literature review [14,35,50] and data availability. Based on the above criteria, on the premise of ensuring the objectivity and accuracy of the selected evaluation factors, this study preliminarily selected 10 evaluation index factors in the four categories of topography, geological conditions, hydrological environment and human engineering activities by combining the field geological survey and literature review. Among them, topography includes elevation, slope, aspect and relief amplitude; geological conditions include lithology and distance of fault; the quality of hydrological environment includes the topographic wetness index (TWI); human engineering activities include distance of road, land-cover/land-use (LCLU) and distance of draft (

Table 2. Characteristic details of landslide conditioning factors.

S. No.	Conditioning Factor	Data Source	Resolution/Scale	Characteristic Description
1	Lithology	Geological Map	1:5000	Stratum lithology is an important material basis for the formation and development of landslides, which directly affects the physical and mechanical properties of slopes and plays a decisive role in the stability of slopes [46] (Figure 4a).
2	Aspect	ALOS-PALSAR DEM	12.5 m	Different slope directions have different solar radiation intensities, resulting in different evaporation of surface water, weathering of rocks and vegetation coverage, which indirectly affect the physical and mechanical properties of rock and soil [48] (Figure 4b).
3	Slope	ALOS-PALSAR DEM	12.5 m	The slope provides an empty surface for the formation of the landslide, which has different effects on surface runoff, groundwater recharge/discharge and stress distribution characteristics of the landslide. The bigger the slope, the easier the landslide [40] (Figure 4c).
4	Relief Amplitude	ALOS-PALSAR DEM	12.5 m	Relief amplitude is the difference between the highest and lowest elevation in a particular topographic unit, which can reflect the features of topographic relief and is a quantitative index to describe the types of geomorphology [51] (Figure 4d).
5	Distance of draft	Geological Map	1:5000	The historical landslide events in the study area are mainly distributed on the high and steep slope of the east side of the north open pit of Tonglvshan Mine. In order to evaluate the impact of underground mining on the high and steep slope, this paper uses the literature [52,53] method to calculate the tunnel buffer zone using the European distance (Figure 4f).
6	Elevation	ALOS-PALSAR DEM	12.5 m	Elevation not only reflects the topographic conditions that directly control the weathering rate and vegetation coverage, but also affects the rainfall intensity that controls the occurrence of landslides [29] (Figure 4g).
7	Distance of fault	Geological Map	1:5000	Geological structure controls the development of joints and fissures in the slope, resulting in the slope being cut into pieces, affecting the development of weak structural planes in the rock mass [17] (Figure 4h).
8	TWI	ALOS-PALSAR DEM	12.5 m	TWI comprehensively reflects the impact of terrain and soil characteristics on the water distribution of slope. The higher TWI value may be related to the higher probability of landslide [17] (Figure 4i).
9	Distance of road	Google Earth Image	30 m	The road distribution data is vectorized by Google Earth Image, and the European distance is used to calculate the buffer zone of the road [22]. (Figure 4i)
10	LULC	Literature [54]	-	Obtain and vectorize the current land use map of the site protection area from the literature [14,54]. (Figure 4j).

). However, this study does not consider the influence of inducing factors such as rainfall and earthquake. On one hand, these environmental variables are highly sensitive to time changes, and there is no record of the occurrence of such landslides in the current landslide list. On the other hand, due to the limitation of the size of the study area, it cannot reflect the difference of the influence of inducing factors such as rainfall and earthquake [37]. Therefore, this study does not consider the influence of these two factors.

For the acquisition of environmental factors under different spatial resolutions, this paper resamples the 12.5 m resolution DEM and the thematic map of related environmental factors and correlation spatial analysis. Details and thematic charts of the selected influencing factors are shown in Table 2 and Figure 4.

4. Landslide Susceptibility Zonation Mapping

4.1. Establishment of Comprehensive Evaluation Index System

Landslide susceptibility evaluation is to determine the most favorable combination of influencing factors for slope instability on the basis of the analysis of landslide breeding conditions, so as to predict the possibility of landslide occurrence in a certain area [48]. First of all, according to the existing research on landslide sensitivity factors and the characteristics of the study area, a total of 10 LCF in 4 categories, including topography, geological conditions, hydrological environment and human engineering activities, are preliminarily determined (as shown in Table 2 and Figure 4). Second, it is necessary to extract the factors which have greater correlation with landslide disaster from the above influencing factors to construct the comprehensive evaluation index system of landslide susceptibility. The establishment of comprehensive evaluation index system mainly includes two aspects: factor state correlation analysis and factor state classification.

4.1.1. Multicollinearity Analysis

At present, there is a lack of unified standards for the selection of landslide susceptibility evaluation indexes [55]. The engineering geology analogy method is usually used to determine the landslide susceptibility evaluation index system in the study area. Although the factors that are closely related to landslide disasters are selected, they may not be completely independent of each other, but rather have a certain correlation. If we blindly pursue more evaluation factors without correlation analysis, the evaluation results may not be more accurate [56]. Therefore, in order to ensure the mutual independence of the selected evaluation factors and reduce the interference of redundant information, it is necessary to conduct correlation test for the selected factors.

Pearson’s correlation coefficient (PCC) [55] was used for multicollinearity analysis of the selected factors. Firstly, the attributes of selected landslide influencing factors are derived from ArcGIS software and sorted out. Secondly, the sorted data are imported into SPSS software, and the PCC test results of each influencing factor were obtained by using the correlation analysis tool.

Table 3. Pearson’s correlation coefficient analysis of the selected evaluation factors.

Factors	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$
$X_1$	1
$X_2$	0.071	1
$X_3$	0.129	0.180	1
$X_4$	0.128	0.164	0.693	1
$X_5$	−0.423	−0.073	−0.280	−0.267	1
$X_6$	−0.166	−0.011	0.400	0.382	−0.062	1
$X_7$	−0.330	−0.035	−0.215	−0.209	0.654	0.017	1
$X_8$	0.056	−0.056	0.023	0.024	−0.119	0.063	−0.055	1
$X_9$	−0.102	−0.037	−0.312	−0.308	0.349	−0.178	0.334	−0.055	1
$X_{10}$	0.398	0.034	0.097	0.105	−0.326	−0.357	−0.232	0.132	−0.099	1

Note: $X_1$—Lithology, $X_2$—Aspect, $X_3$—Slope, $X_4$—Relief amplitude, $X_5$—Distance of draft, $X_6$—Elevation, $X_7$—Distance of fault, $X_8$—TWI, $X_9$—Distance of road, $X_{10}$—LULC, $Sig.\le 0.05$.

shows the Pearson correlation test results of each evaluation factor obtained based on the statistical analysis software SPSS. PCC is widely used to measure the degree of correlation between two variables $X$ and $Y$, and its value $R$ is between −1 and 1, where $R=0$ R = indicates that there is no correlation between the two variables. When $0.3\le \left|R\right|<0.5$ and $Sig.\le 0.05$, the two factors can be considered to be weakly correlated. When $0.5\le \left|R\right|<0.7$ and $Sig.\le 0.05$, a moderate correlation between the two factors can be considered. When $\left|R\right|>0.7$ and $Sig.\le 0.05$, there is a strong correlation between the two factors. The calculation results of SPSS software show that $\left|R\right|<0.7$ and $Sig.\le 0.05$ of the selected factors (Table 3). Therefore, it can be inferred that the selected evaluation factors are mutually exclusive and independently contribute to the occurrence of landslides in the study area.

4.1.2. Factor Status Grading

The most important step in landslide susceptibility evaluation is to establish the classification standard of the landslide susceptibility evaluation index for the quantitative comprehensive evaluation of regional landslide susceptibility. At present, the commonly used classification methods of susceptibility evaluation index include: the empirical judgment method [35,57], the weight distribution method [1,18,55] and the statistical analysis method [51,58]. Among them, statistical analysis is the most commonly used index state classification method in landslide susceptibility evaluation at present, but this method needs enough historical landslide statistical information to ensure the effectiveness and accuracy of evaluation results. Although the empirical judgment method and the weight distribution method are greatly affected by the subjective factors, they are suitable for the situation of insufficient information of historical landslide statistics. Obviously, the number of historical landslide events collected in this study area is too small to meet the calculation accuracy requirements of statistical analysis. Therefore, this paper combined this with the characteristics of the study area itself to use empirical judgment method to grade the risk status of evaluation indicators.

The factor status classification of evaluation index means that the factor of a single index is divided into multiple second-level states according to certain grading standards [59]. At present, there is no unified standard for the classification of landslide hazard evaluation grade. Based on the previous landslide hazard evaluation grade classification standard [35,37], this study uses a single factor method to divide landslide hazard into 5 evaluation grades; namely, the evaluation grade space is $U=\left\{C_1,C_2,C_3,C_4,C_5\right\}$, in which $C_1~C_5$ corresponding risk grades are very low risk (Level I), low risk (Level II), medium risk (Level III), high risk (Level IV) and very high risk (Level V). The evaluation index system includes qualitative indexes and quantitative indexes. In order to facilitate the realization of regional landslide risk zoning, the quantitative values of each evaluation index are averaged. For quantitative indicators such as elevation, slope and distance to fault, measured values extracted by the Arcgis10.2 software are directly selected for expression. For qualitative indicators such as formation lithology, slope direction and land use, the quantitative evaluation values of qualitative indicators are divided into 5 levels, and successively assigned 0.1, 0.3, 0.5, 0.7 and 0.9 [35,37]. The landslide risk assessment index system and classification standards in the study area are shown in

Table 4. Grading standard of landslide vulnerability evaluation index state.

Primary Evaluation Index	Secondary Evaluation Index	Evaluation Criteria
		Very Low (Level I)	Low (Level II)	Moderate (Level III)	High (Level IV)	Very High (Level V)
terrain	Elevation/m	0–30	30–45	45–60	60–75	>75
	Slope/(°)	0–15	15–25	25–35	35–45	>45
	Aspect	0.1	0.3	0.5	0.7	0.9
	Relief amplitude	0–1	1–2	2–3	3–4	>4
Engineering geological characteristics	Lithology	0.1	0.3	0.5	0.7	0.9
	Distance of fault	>550	250–550	150–250	50–150	0–50
Hydrological environment indicators	TWI	0–4	4–8	8–12	12–16	>16
Human engineering activity	Distance of draft	>350	250–350	150–250	50–150	0–50
	Distance of road	>200	150–200	100–150	50–100	0–50
	LULC	0.1	0.3	0.5	0.7	0.9

.

4.2. Determination of the Single-Index Measure Evaluation Matrix

Based on the definition of unascertained measure function of single index and the grading standard of each evaluation index in Table 4, the unascertained measure function of each evaluation index in regional landslide risk assessment is constructed respectively; thus, the unascertained measure value of the 10 evaluation indexes and the single-index measure function graph of each evaluation index could be obtained. Figure 5a–g shows the single-index measure function of 7 quantitative indexes, such as slope, relief amplitude, distance of draft, elevation, distance of fault, topographic wetness index and distance of road. Figure 5h–i shows the single-index measure function of lithology, aspect, land-cover/land-use (LCLU) and three other qualitative indicators.

The study area is divided into 56,051 units in a 10 m × 10 m grid. The attributes of each pixel evaluation index are extracted by ArcGIS (version 10.2) software. Then, the sorted measured values of each pixel evaluation index are substituted into the corresponding single-index measure function (see Figure 5) to obtain the single-index measure evaluation matrix of each evaluation unit.

Table 5. Measured value of evaluation index of prediction unit.

S. NO.	Evaluation Index Value
	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$
1	0.300	0.418	1.889	0.092	1017.085	20.472	459.061	10.214	404.607	0.300
2	0.300	0.686	4.184	0.188	1010.142	21.162	451.926	3.990	393.039	0.300
3	0.300	0.608	5.482	0.253	1002.215	22.769	444.167	6.627	380.138	0.300
4	0.300	0.593	6.531	0.299	992.839	24.550	435.396	4.509	366.239	0.300
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
56048	0.300	0.244	1.702	0.082	1349.518	17.899	1204.751	7.007	385.811	0.300
56049	0.300	0.344	3.324	0.157	1404.219	18.720	1260.274	6.884	398.824	0.300
56050	0.300	0.504	4.034	0.179	1392.165	18.495	1245.841	6.018	402.689	0.300
56051	0.300	0.286	1.844	0.087	1379.279	18.388	1232.126	2.785	401.474	0.300

shows the measured values of evaluation indicators of some prediction units. Taking prediction unit 1 as an example, its single-index measure evaluation matrix $z_1$ is shown in Equation (30).

$$z_1=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0.411& 0.589& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0.394& 0.606& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0.947& 0.054& 0\\ 0& 1& 0& 0& 0\end{array}\right]$$

(30)

4.3. Determination of Dynamic Comprehensive Weights

Due to the different contribution of each evaluation index to the occurrence of landslide disaster, the weight assigned to each evaluation index is also different. In this paper, the subjective weight and objective weight of each evaluation index are determined based on the fuzzy AHP method and entropy weight method, and the two are optimized through the minimum information entropy principle, so as to obtain the final dynamic comprehensive weight of each evaluation index. First, geotechnical engineering experts are invited to make a pairwise comparison of each evaluation index by using the three-value judgment method, and the fuzzy judgment matrix is constructed by combining the triangular fuzzy number algorithm. The fuzzy judgment matrix of ten evaluation indexes is shown in

Table 6. Pairwise comparison matrix of the LCFs.

Factors	Lithology	Aspect	Slope	Relief Amplitude	Distance of Draft	Elevation	Distance of Fault	TWI	Distance of Road	LULC
Lithology	(1,1,1)	(1,2,3)	(1,2,3)	(3,4,5)	(1,2,3)	(1,2,3)	(3,4,5)	(3,4,5)	(2,3,4)	(2,3,4)
Aspect	(1/3,1/2,1)	(1,1,1)	(1/2,1,1)	(2,3,4)	(1/2,1,1)	(1,1,2)	(2,3,4)	(2,3,4)	(1,2,3)	(1,2,3)
Slope	(1/3,1/2,1)	(1,1,2)	(1,1,1)	(2,3,4)	(1,1,2)	(1/2,1,1)	(2,3,4)	(2,3,4)	(1,2,3)	(1,2,3)
Relief amplitude	(1/5,1/4,1/3)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1,1,1)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1/2,1,1)	(1,2,3)	(1/3,1/2,1)	(1/3,1/2,1)
Distance of draft	(1/3,1/2,1)	(1,1,2)	(1/2,1,1)	(2,3,4)	(1,1,1)	(1,1,2)	(2,3,4)	(2,3,4)	(1,2,3)	(1,2,3)
Elevation	(1/3,1/2,1)	(1/2,1,1)	(1,1,2)	(2,3,4)	(1/2,1,1)	(1,1,1)	(2,3,4)	(2,3,4)	(1,2,3)	(1,2,3)
Distance of fault	(1/5,1/4,1/3)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1,1,2)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1,1,1)	(1/2,1,1)	(1/3,1/2,1)	(1/3,1/2,1)
TWI	(1/5,1/4,1/3)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1/2,1/2,1)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1,1,2)	(1,1,1)	(1/3,1/2,1)	(1/3,1/2,1)
Distance of road	(1/4,1/3,1/2)	(1/3,1/2,1)	(1/3,1/2,1)	(1,2,3)	(1/3,1/2,1)	(1/3,1/2,1)	(1,2,3)	(1,2,3)	(1,1,1)	(1/2,1,1)
LULC	(1/4,1/3,1/2)	(1/3,1/2,1)	(1/3,1/2,1)	(1,2,3)	(1/3,1/2,1)	(1/3,1/2,1)	(1,2,3)	(1,2,3)	(1,1,2)	(1,1,1)

. Second, based on the fuzzy judgment matrix, Equations (16)–(20) are used to calculate the initial weights of each evaluation index, and the results are shown in

Table 7. The calculation of degree possibility for and subjective weight of LCFs.

Factors		$\mathit{V}\{{\tilde{\mathit{S}}}_{\mathit{i}}(\mathit{i}={\mathit{X}}_1,{\mathit{X}}_2\cdots {\mathit{X}}_{10})\}$
		${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$
$V\{{\tilde{S}}_j(j=X_1,X_2\cdots X_{10})\}$	$X_1$	-	0.710	0.723	0.061	0.723	0.710	0.000	0.000	0.388	0.419
	$X_2$	1.000	-	0.000	0.355	0.000	0.000	0.275	0.267	0.680	0.700
	$X_3$	1.000	0.000	-	0.342	1.000	1.000	0.259	0.251	0.675	0.696
	$X_4$	1.000	1.000	1.000	-	0.000	0.000	0.904	0.863	1.000	1.000
	$X_5$	1.000	0.000	0.000	0.342	-	0.000	0.259	0.251	0.675	0.696
	$X_6$	1.000	0.000	0.000	0.355	0.000	-	0.275	0.266	0.680	0.700
	$X_7$	1.000	1.000	1.000	1.000	1.000	1.000	-	0.949	1.000	1.000
	$X_8$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	-	1.000	1.000
	$X_9$	1.000	1.000	1.000	0.721	1.000	1.000	0.633	0.609	-	0.000
	$X_{10}$	1.000	1.000	1.000	0.713	1.000	1.000	0.623	0.599	0.000	-
$\min \{V({\tilde{S}}_i\ge {\tilde{S}}_j)\}$		1.000	0.710	0.723	0.061	0.723	0.710	0.259	0.251	0.388	0.419

. After standardization, the final subjective weight of each evaluation index is obtained. The subjective weights of each evaluation index are shown in

Table 8. Subjective weight after normalization.

Factors	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$
Weight	0.191	0.135	0.138	0.012	0.138	0.135	0.049	0.048	0.074	0.080

.

Then, taking the evaluation unit as the research object, the objective weights ($V_i=(V_{i1},V_{i2}\cdots ,V_{i10}), i=1,2,\cdots ,56051$) are calculated from the single index measure evaluation matrix by using Equations (22) and (23). Matlab software is used to calculate the objective weight value of each evaluation unit in the whole research area. Taking evaluation unit 1 as an example, the objective weight of each evaluation index in this unit is:

$$V_1=(0.111, 0.064, 0.111, 0.111, 0.111, 0.111, 0.065, 0.096, 0.111, 0.111)$$

(31)

Finally, Equation (26) is used to optimize the combination of subjective and objective weights calculated by the fuzzy AHP and entropy weight method, so as to obtain dynamic comprehensive weights that change with the specific situation of the evaluation unit. The whole fusion process of subjective weight and objective weight of each evaluation unit is carried out in the Matlab environment. By combining Table 8 and Equation (31), the dynamic comprehensive weight of prediction unit 1 can be obtained as follows:

$$w_1=(0.153, 0.098, 0.130, 0.038, 0.130, 0.129, 0.059, 0.071, 0.095, 0.099)$$

(32)

4.4. Landslide Susceptibility Zoning Based on Improved UM-SPA Model

According to the improved UM-SPA model, the single-index measure evaluation matrix of each prediction unit and its corresponding dynamic comprehensive weight are substituted into Equation (27), and the multi-index comprehensive measure evaluation vector of each prediction unit in the study area can be obtained. Taking prediction unit 1 as an example, the single index measure matrix $z_1$ and dynamic comprehensive weight w₁ of prediction unit 1 are substituted into Equation (27), then the multi-index comprehensive measure vector of prediction unit 1 is $V_1=(V_{i1},V_{i2}\cdots ,V_{i10}), i=1,2,\cdots ,56051$. Similarly, the multi-index comprehensive measure evaluation vector of other prediction units can be obtained in Excel environment.

As mentioned above, this study divides the regional landslide susceptibility zoning into 5 levels, namely $t=5$. According to the equipartition principle, the 5-element connection number u is divided into 5 equal parts on the interval $[-1,1]$, namely $[-1.0,-0.6), [-0.6,-0.2), [-0.2,0.2), [0.2,0.6)$ and $[0.6,1.0]$. The corresponding relationship between the landslide susceptibility level and the judgment interval is shown in

Table 9. Classification of landslide hazard.

Susceptibility Level	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ
Grade description	Very low	Low	Moderate	High	Very high
Judgment interval	(0.6, 1.0]	(0.2, 0.6]	(−0.2, 0.2]	(−0.6, −0.2]	[−1.0, −0.6]

. Additionally, the component matrix of connection degree $E={[1,p_{i1},p_{i2},\cdots ,p_{i(t-2)},q]}^T$ can be expressed as $E={[1.0,0.5,0,-0.5,-1.0]}^T$ by using Equation (29). Finally, the multi-index comprehensive measure evaluation vector is substituted into Equation (28) to construct the five-element connection degree of landslide susceptibility assessment in the prediction unit. Taking prediction unit 1 as an example, the five-element correlation number of the overall index evaluation of landslide risk regionalization in this unit is:

$${\mu }_5=0.469+0.359p_1+0.166p_2+0.005p_3=0.646$$

(33)

As can be seen from Table 9, the landslide risk zoning level in prediction unit 1 is level I, belonging to the medium risk area. Similarly, landslide susceptibility levels of other prediction units can be evaluated in the Excel environment. The final evaluation results are shown in Figure 6.

5. Results and Discussion

By constructing the UM-SPA model based on dynamic comprehensive weight, the landslide susceptibility zoning level values of 56,051 grid units in the Tonglvshan mining area of Daye can be determined. Then, according to the landslide susceptibility zoning value of each prediction unit, the attribute connection function is used to assign values to each prediction unit in Arcgis (Version 10.2) software, and the landslide hazard zoning map of the whole mining area is generated. Figure 6 shows the zoning diagram of landslide susceptibility in the Tonglvshan mining area of Daye.

5.1. Results Analysis

As shown in Figure 6, the landslide susceptibility map of the entire study area is divided into five levels: Very low, Low, Moderate, High and Very high. From the level distribution of susceptibility zoning, the very high susceptibility area is mainly distributed in the high slope area on the east side of the north open pit of the Tonglvshan Mine, the scattered distribution area on the west side slope shoulder of the north open pit, the south side and the north side of the north open pit and the north area of the south open pit. The slope of this area is high and steep, which is greatly affected by human engineering activities, and the area is about 0.3252 km². The high susceptibility areas are mainly concentrated in the west side slope of the north open-pit, the east side slope of the south open-pit, the east side of the Tonglvshan Ancient Copper Mine Site Museum and part of the No. I and No. II dump, which covers an area of about 0.5693 km². The medium susceptibility area is mainly distributed in the south open-pit side slope and pit bottom area, the low hills in the study area (such as No. I and No. II dump), and has a good overlap with the slope of 21–40° area, the area is about 1.0669 km². The very low susceptibility area and low susceptibility area are mainly distributed in areas with gentle terrain, small slope and far away from open pit (such as tailings ponds in the north and east of the study area and paddy fields in the north-east corner), with a total area of 3.6437 km². The landslide susceptibility assessment results show (as shown in Figure 9) that 5.8% and 10.16% of the Tonglvshan mining area are divided into the extremely high susceptibility area and high susceptibility area, respectively. The low susceptibility area and very low susceptibility area account for 30.87% and 34.14% of the total area of the study area, respectively. The remaining 19.03% of the area is classified as a medium susceptibility zone. In addition, it has been also noted that the density of landslides in different susceptibility areas does not increase with the increase of landslide susceptibility zoning level.

5.2. Model Validation

The receiver operating characteristic (ROC) curve has been widely used in the performance evaluation of LSZ models [9,19,60] as a common method to evaluate the goodness of fit of classification. It is a comprehensive index reflecting the sensitivity and specificity of continuous variables and the composition method to reveal the relationship between the two. The ROC curve is plotted by taking the true positive rate (sensitivity) as the y-axis and the false positive rate (1-specificity) of different cut-off thresholds as the x-axis. The area under the curve (AUC) is a common index to test the accuracy of the model, and is used as an evaluation standard to measure the quality of the LSZ model, and its value is between 0.5 and 1.0. The closer the curve AUC value is to 1, the better the classification effect of the model is.

To prepare an ROC curve, the validation dataset that contained landslide events and non-landslide events should be prepared in advance. Accordingly, for this study, 12 known landslides and the same number of randomly generated non-landslides are selected to produce ROC curve. Then, the LSI is spatially associated with the landslide validation datasets and the ROC curve is drawn. The ROC curve of the developed model is shown in Figure 7. The AUC value for the LSZ map produced using the improved UM-SPA method is 0.777, which means that the overall success rate of LSZ map is 77.7%. The results show that the model can be used to draw the landslide prone zoning map.

5.3. Discussion

Combining UM theory, the dynamic comprehensive weighting method based on the fuzzy AHP-entropy weight method and SPA theory, an improved UM-SPA coupling model for landslide susceptibility assessment is proposed in this study. The results show that the UM-SPA model based on dynamic comprehensive weight is an effective tool to evaluate the landslide susceptibility zonation of the Tonglvshan mining area in Daye City. Figure 6 is the landslide susceptibility zonation map of the Tonglvshan mining area in Daye, which indirectly shows the potential landslide occurrence in the whole mining area. The ten thematic layers, lithology, aspect, slope, relief amplitude, distance of draft, elevation, distance of fault, TWI, distance of road and LULC are used to classify the study area into different landslide susceptibility zones. From Figure 9, it is observed that 15.96% of the Tonglvshan mining area is classified as high and very high susceptibility zones, whereas 65.01% of the Tonglvshan mining area is divided into low and very low susceptibility zones. The remaining 19.03% of the study area is classified as moderate susceptibility zones. High and extremely high landslide prone areas are mainly distributed in the north and south open pit areas of the Tonglvshan Mine and its surrounding areas (Figure 6). This is mainly attributed to the complex geological structure conditions near the open pit, steep terrains and underground goaf.

The accuracy of predictive models is considered a major concern for most environmental modeling applications, including the LSZ model. The essence of the LSZ model is a multi-criteria decision analysis method. The multi-criteria decision-making process will inevitably be affected by decision-makers’ views, especially in the process of data quantization and standard weighting, thus affecting the accuracy of evaluation results. However, the conventional LSZ method usually assigns a fixed weight to the whole study area, which cannot reflect the difference of the status of control factors in different locations in the study area, which obviously reduces the accuracy and rationality of the evaluation results. Given this, this study aims to develop an UM-SPA model based on dynamic comprehensive weights to explore the landslide susceptibility zonation of the Tonglvshan mining area. The dynamic comprehensive weighting method considers the influence of subjective and objective weights, which not only weakens the influence of subjective factors, but also makes full use of the original information of spatial data. In order to minimize the subjective arbitrariness and uncertainty in the conventional AHP process, this study introduced fuzzy AHP to calculate the subjective weight of the evaluation index. Mallick [19] has proved that fuzzy AHP is more suitable to obtain expert preferences than AHP, so as to obtain subjective weights with higher accuracy. Taking the evaluation unit as the research object, we use the entropy weight method to get the objective weight of each evaluation unit in the whole research area. Through the principle of minimum information entropy, the dynamic comprehensive weight, which changes with the evaluation unit, can be obtained. The results show that the proposed UM-SPA model based on dynamic comprehensive weights has good accuracy, and the AUC value is 0.777 (Figure 7).

To verify the superiority of the proposed model, it is compared with conventional AHP [9] and Entropy-FAHP [29]. The landslide susceptibility map produced using the conventional AHP and Entropy-FAH is shown in Figure 8. Figure 9 shows the area coverage of five landslide susceptibility levels of the three calculation models, and the paired comparison matrix of conventional AHP analysis is shown in

Table 10. Pairwise comparison matrix and normalized weight of ten LCFs using the conventional AHP.

LCF	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$
$X_1$	1
$X_2$	1/2	1
$X_3$	1/2	1	1
$X_4$	1/3	1/3	1/5	1
$X_5$	1/2	4	1	5	1
$X_6$	1/2	1/2	1/3	4	1	1
$X_7$	1/3	1/2	1/3	2	1/3	1/4	1
$X_8$	1/3	1/3	1/3	1	1/5	1/4	1/2	1
$X_9$	1/3	1/2	1/3	3	1/5	1/5	1	3	1
$X_{10}$	1/3	1/2	1/2	3	1/3	1/3	2	1/2	2	1
$w_i$	0.188	0.111	0.151	0.031	0.180	0.133	0.049	0.042	0.533	0.062

Note: Consistency Ratio: CR = 0.069 < 1.

. Past studies [29] propose that the Entropy-FAHP model also adopts dynamic comprehensive weight to give weight to evaluation indicators, so the subjective and objective weights of the Entropy-FAHP model quoted in this study are provided by conventional AHP and improved UM-SPA method, respectively. The comparative analysis of Figure 6 and Figure 8 shows that the distribution of landslide susceptibility of the three models is similar, meaning that the regions with large susceptibility levels are distributed near the open-pit slope, while the regions with small susceptibility levels are distributed at the edge of the study area, and there are only some differences in local areas. Compared with the conventional AHP model, the susceptibility areas in the landslide susceptibility diagram drawn by the improved UM-SPA model are more concentrated, as shown in Figure 9a,b. The difference lies in that the extremely high susceptibility areas of the conventional AHP model are mainly distributed in the east and west side slopes of the north open pit. The highly prone area of the improved UM-SPA model is mainly concentrated in the east side slope of north open pit, which is also consistent with the geological disaster investigation results. In addition, the results of high susceptibility partitioning between the two models are also significantly different. The reason for this difference is that the conventional AHP model assigns a fixed weight to the whole research area, but cannot reflect the differences of influencing factors in different locations in the research area, thus reducing the accuracy and rationality of the final evaluation results. At the same time, in contrast to the Entropy-FAHP model with dynamic comprehensive weighting, both high and low landslide susceptibility regions are highly similar, as shown in Figure 9a,c. This also proves the superiority of the dynamic comprehensive weighting method in the study of landslide prone regionalization. As can be seen from Figure 7, the AUC value for the LSZ maps produced using the conventional AHP and Entropy-FAHP model are 0.698 and 0.764, The AUC value for the LSZ maps produced using the conventional AHP and Entropy-FAHP Model are 0.698 and 0.764, respectively. Obviously, the improved UM-SPA model has higher prediction accuracy in the study of landslide susceptibility zonation.

Although the UM-SPA model based on dynamic comprehensive weights shows considerable potential in predicting landslide prone regionalization, there are some limitations in this study. First, it should be noted that a tricky problem faced by this study and other landslide susceptibility assessment methods is how to scientifically and reasonably select LCFs. The lack of unified criteria for LCFs selection will further limit the application of LSM in engineering practice. Second, the selection of unascertained measure function will directly affect the construction of a single-index measure evaluation matrix of each evaluation unit, and then affect the calculation of objective weight. Despite these limitations, it can be concluded that the research results of this paper will be helpful to minimize the occurrence and management of landslide susceptibility in the mining area, and will be conducive to the long-term protection of the Tonglvshan ancient copper mine site. Therefore, future studies will further improve the shortcomings of the computational model in order to obtain a more accurate LSZ map.

6. Conclusions

Landslides are the most common geological hazard in the Tonglvshan mining area, which seriously threatens the safety of surrounding residents and the Tonglvshan ancient copper mine site. To effectively reduce the landslide risk of the mining area and protect the safety of the Tonglvshan ancient copper mine site, this study aims to combine the dynamic comprehensive weighting method based on the fuzzy AHP-entropy weight method with UM theory and SPA theory, and propose an improved UM-SPA coupled landslide susceptibility evaluation model. The improved UM-SPA model shows significant advantages. First, the dynamic comprehensive weight method can dynamically adjust the comprehensive weight according to the measured values of each factor in each evaluation unit, which overcomes the shortcomings of the traditional LSZ method in assigning fixed weights to the whole study area. Second, the dynamic comprehensive weighting method can not only reflect the opinions of experts and decision makers, but also make full use of the original information of spatial data. Finally, the improved UM-SPA model can more effectively determine the landslide susceptibility grade.

The landslide susceptibility zonation of the whole study area is divided into five grades: very low vulnerability, low vulnerability, model vulnerability, high vulnerability and very high vulnerability. Higher landslide susceptibility areas are mainly distributed in the north and south open pit areas of the Tonglvshan Mine and its surrounding areas. The results show that 5.8% and 10.16% of the Tonglvshan mining area are classified into extremely highly vulnerable areas and highly vulnerable areas, respectively. The low and extremely low susceptibility areas account for 30.87% and 34.14% of the total area of the study area, respectively. The remaining 19.03% of the area is divided into medium-prone areas. The verification results using ROC curves show that the improved UM-SPA model performs well in developing the LSZ map of the study area. To verify the superiority of the proposed model, it is compared with the conventional AHP and the Entropy-FAHP model using the same datasets. The AUC values of three predication models are 0.777, 0.764 and 0.698, respectively. The verification results show that the improved UM-SPA model has higher prediction accuracy in LSZ research.

The research results show that the developed model can produce landslide susceptibility zoning maps with high reliability. This study can be used as the basis for decision-makers and engineers to initially understand the landslide vulnerability risk in the Tonglvshan mining area, and reduce the hazards and losses caused by existing and future landslides by taking appropriate preventive and mitigation measures. Therefore, these results can provide a reference for emergency planning, disaster reduction and prevention decision-making in the Tonglvshan mining area.