The risk assessment of landslide hazards in the Badong section of Three Gorges using the variable fuzzy set theory

The risk assessment of landslide hazards has a tremendous influence on people’s lives and property safety; therefore, its investigation is significant. The stratigraphic lithology, degree of weathering, relationship between the structural plane and slope direction, cohesive force, angle of internal friction, severity, average slope degree, height of slope, and type of landslide are adopted as the evaluation factors first. Second, an assessment model is developed based on the variable fuzzy set theory. In addition, the proposed model is utilized to assess the landslide hazards in the Badong section of Three Georges in China. Finally, the results demonstrate that the results derived from the proposed model are consistent with the current specifications; the accuracy rate reaches 83%. The method can determine the risk level of landslide hazards and provide an alternative scheme. Hence, this study can accurately present a new approach for assessing landslide hazards in the future.

1 Introduction

A landslide is a disaster that frequently occurs [1]. Its area of impact can be extensive and very destructive [2]. Landslides can cause massive damage to local infrastructure and even threaten the safety of people [3, 4].

Landslide hazards often occur in the Three Gorges Reservoir area. Until November 2003, landslide hazards occurred 4,638 times, and several thousand landslide hazards could not be controlled according to the relevant statistics [5, 6]. Therefore, how to lessen the economic losses and casualties arising from landslide hazards has become an important issue [2, 7]. The accurate risk level of landslide hazards in the Three Gorges Reservoir area is significant.

Research on the risk assessment and prediction of landslide hazards has become a hot issue [8]. For example, slope, precipitation, and human activities are adopted as the assessment index. Gao et al. [9] assessed the risk level of landslide hazards in the Wanzhou zone, Chongqing, China. Wang et al. [10] established three nonlinear prediction models based on the weight of the joint prediction model to assess the risk level of the Qinglong landslide in Guizhou. Liu et al. [11] performed an investigation on the risk and vulnerability evaluation of landslide hazards in the bank section of the reservoir. Men et al. [12] investigated the reasons for and mechanisms of the Baijia Bao landslide based on geological surveys and rainfall data using numerical simulations in combination with actual monitoring data. Shi et al. [13] analyzed the risk assessment of landslide hazards using GIS technology with existing applications to improve the precision of the evaluation method. Spatial prediction of landslide hazards was investigated using the information model and quantitative analysis, multifactor regression analysis model, and fuzzy discriminant analysis model by Huang et al. [14]. Gu et al. [15] analyzed the landslide hazards in Shiwangmiao, Chongqing, using the intuitionistic fuzzy set-TOPSIS model.

With the development of science and technology, many methods have been used to evaluate landslide hazards, such as the catastrophe theory model [16, 17], neural network model [18], fuzzy comprehensive evaluation, and gray correlation analysis model [19]. The aforementioned methods have fostered the development of an evaluation model for landslide hazards. However, deficiencies still exist [20, 21], such as the complex calculation process and the quantitative or qualitative analytical results [22–24].

To conquer the insufficiency of the aforementioned theories, the variable fuzzy set theory is introduced to assess the risk level of landslide hazards. The method has many merits, such as its algorithm is precise, its operability is very strong in reality, and the grading standards, which are interval forms, can be solved well. For the proposed model, the construction of the relative difference function is depicted as quantitative tools of variable fuzzy sets. It describes the essence for contradictions and unity of opposites of movement and the change criterion about the objective things. The theory confirms the principle of dialectics of nature; therefore, the model improves the traditional fuzzy set model enormously.

This paper is organized as follows: in Section 2, the study area is correlated first; Section 3 presents a new assessment model based on the proposed method ; Section 4 explores the construction of the evaluation model of landslide hazards; Section 5 presents the analysis of the evaluation results derived from the proposed method; and Section 6 summarizes the conclusions.

2 Study area

The Three Gorges Reservoir area is located at 28°32′∼31°44′ North and 105°44′∼31°44′ East. The total length of backwater in the mainstream is 662.9 km. It includes Badong County on the Yangtze River. Badong County is located in southwestern Hubei Province in China and contains the middle and upper streams of the Yangtze River watershed. It is located in Enshi Tujia and Miao Autonomous Prefecture. Its width from east to west is 10.3 km; its length from north to south is 135 km; its total area reaches 3,354 km²; and the survey area is shown in Figure 1.

FIGURE 1

FIGURE 1. Survey area.

The soil material in the landslide is mainly silty clay with crushed stone and crushed stone soil distributed in the whole slide body. The slippage soil is mainly silty clay with gravel breccia. The thickness of the slip band is 0.2 cm. The extraordinary phenomenon of deep slip zone soil is not apparent, but it has less breccia.

The material of the slide bed consists mainly of feldspar quartz sandstone and mudstone of the Jurassic Badong Formation. The occurrence of the bedrock is 260° < 30°. The bedrock surface undulates little and is close to the ground. According to the strata, the slope is reversed.

3 Methodology

Entropy theory and the variable fuzzy set assessment method are combined to develop a comprehensive approach for land hazard assessment. It can depict the inconsistent conception and implication of membership and non-membership functions dynamically; therefore, it is the significant optimization of traditional fuzzy sets.

3.1 The basic definition

Assuming that $\mathrm{F}$ belongs to the domain U, at any $u\in U$, the number ${\mu }_F^0\left(\mathrm{u}\right)$ can be determined in the closed interval. The absolute membership relationship is defined as the relation between $\mathrm{U}$ and $\mathrm{F}$, which can be expressed as follows [25]:

$\begin{array}{l}{\mu }_F^0:\mathbf{U}\to \left[\mathrm{0,1}\right].\\ u|\to {\mu }_F^0\end{array}(1)$

In the domain $\mathbf{U}$, $u\in \mathbf{U}$, there are two opposite fuzzy numbers: $\mathrm{F}$ and ${\mathrm{F}}^c$. For any variable $u$, there are two determined numbers, ${\mu }_F\left(\mathrm{u}\right)$ and ${\mu }_{F^{\mathrm{c}}}\left(\mathrm{u}\right)$, and the relative membership degree of $u$ to $\mathrm{F}$ and ${\mathrm{F}}^{\mathrm{c}}$ is defined as

$\begin{array}{l}{\mu }_F,{\mu }_{F^c}:\mathbf{U}\to \left[\mathrm{0,1}\right]\\ u|\to {\mu }_F\left(u\right),{\mu }_{F^C}\left(u\right)\in \left[\mathrm{0,1}\right]\end{array}.(2)$

Figure 2 depicts the dynamic variable of any number in any closed interval as follows:

FIGURE 2

FIGURE 2. Dynamic variables.

The relative membership degree of $\mathrm{F}$ and ${\mathrm{F}}^{\mathrm{c}}$ meet with ${\mu }_F\left(\mathrm{u}\right)+{\mu }_{F^c}=1$, $0\le {\mu }_F\left(\mathrm{u}\right)\le 1$, and $0\le {\mu }_{F^{\mathrm{c}}}\left(\mathrm{u}\right)\le 1$, and they can be expressed as follows:

$\underset{\sim }{\mathbf{F}}=\left\{u,{\mu }_F\left(u\right),{\mu }_{F^C}\left(u\right)|u\in U\right\},(3)$

where $\underset{\sim }{\mathrm{F}}$ is the opposite fuzzy set. Figure 3 shows its definition.

FIGURE 3

FIGURE 3. Diagram of opposite fuzzy sets.

The attractive and repelled sets ${\mu }_F\left(\mathrm{u}\right)$ and ${\mu }_{F^{\mathrm{c}}}\left(\mathrm{u}\right)$ can likewise be defined as follows:

$D_F\left(u\right)={\mu }_F\left(u\right)-{\mu }_{F^c}\left(u\right).(4)$

When ${\mu }_F\left(\mathrm{u}\right)>{\mu }_{F^{\mathrm{c}}}\left(u\right)$, $0\le D_F\left(\mathrm{u}\right)\le 1$, and when ${\mu }_F\left(\mathrm{u}\right)={\mu }_{F^{\mathrm{c}}}$, $D_F\left(\mathrm{u}\right)=0$, but when ${\mu }_F\left(\mathrm{u}\right)<{\mu }_{F^{\mathrm{c}}}\left(u\right)$, $-1\le D_F\left(\mathrm{u}\right)\le 0$. The mapping of the relative difference function $D_F\left(\mathrm{u}\right)$ can be expressed as follows:

$\begin{array}{l}D:\mathbf{U}\to \left[\mathrm{0,1}\right]\\ u|\to D_F\left(u\right)\in \left[-\mathrm{1,1}\right].\end{array}(5)$

Figure 4 shows the relative difference function of $u$ to $\mathrm{F}$.

FIGURE 4

FIGURE 4. Relative difference function.

3.2 Determining the relative membership degree

$\mathrm{X}$ is a sample set, which is expressed as follows :

$X=\left(x_{ij}\right),(6)$

where $x_{\mathrm{i}\mathrm{j}}$ is the eigenvalue of the index $\mathit{i}$ of sample $j$, $i=1,\mathit{2},...,m;j=\mathrm{1,2},...c$. $\mathrm{c}$ represents the grade of the index; the attractive domain ${\mathrm{I}}_{\mathrm{a}\mathrm{b}}$ can be obtained in Eq. 7.

${\mathbf{I}}_{\mathbf{a}\mathbf{b}}=\left(\left|a_{ij},b_{ij}\right|\right).(7)$

When we enlarge the set ${\mathrm{I}}_{\mathrm{a}\mathrm{b}}$ according to the upper and lower bounds of its adjacent intervals, set ${\mathrm{I}}_{\mathrm{d}\mathrm{e}}$ is expressed as follows:

${\mathbf{I}}_{de}=\left(\left|d_{ij},e_{ij}\right|\right).(8)$

Based on the relevant Ref. [26], the level standard $\mathrm{F}$ of the index is depicted as follows:

$\mathbf{F}=\left[\begin{array}{ccc}{\mathbf{F}}_{\mathbf{11}}& ...& {\mathbf{F}}_{\mathbf{1}\mathbf{j}}\\ ...& ...& ...\\ {\mathbf{F}}_{\mathbf{i}\mathbf{1}}& ...& {\mathbf{F}}_{\mathbf{i}\mathbf{j}}\end{array}\right],(9)$

where the element $F_{\mathrm{i}\mathrm{j}}$ is depicted as follows:

$F_{ij}=\frac{c-j}{c-1}{a}_{ij}+\frac{j-1}{c-1}{b}_{ij},(10)$

when $j=1$, $F_{\mathrm{i}1}=a_{i1}$; when $j=\mathrm{c}$, then, $F_{\mathrm{i}\mathrm{j}}=b_{ic}$; and when $j=\frac{\mathrm{c}+1}{2}$, then, $F_{\mathrm{i}\mathrm{j}}=\frac{a_{ij}+b_{ij}}{2}$.

$X_0\left(a,b\right)$ is defined as the attractive domain, namely, when $0\le D_F\left(\mathrm{u}\right)\le 1$, $\mathrm{X}=\left[d,e\right]$, and it belongs to the upper and lower domain intervals of $X_0\left(X_0\subset \mathrm{X}\right)$. Figure 5 shows their position relationship.

FIGURE 5

FIGURE 5. Drawing of the position relationship.

Therefore, their relative membership degree is depicted in Eqs. 11, 12.

$\{\begin{array}{c}{\mu }_F\left(\mathrm{u}\right)=0.5\left[1+{\left(\frac{x-a}{F-a}\right)}^{\beta }\right];x\in \left[a,F\right]\\ {\mu }_F\left(\mathrm{u}\right)=0.5\left[1-{\left(\frac{x-a}{\mathrm{d}-a}\right)}^{\beta }\right];x\in \left[\mathrm{d},a\right]\end{array}.(11)$

$\{\begin{array}{c}{\mu }_F\left(\mathrm{u}\right)=0.5\left[1+{\left(\frac{x-\mathrm{b}}{F-b}\right)}^{\beta }\right];x\in \left[F,b\right]\\ {\mu }_F\left(\mathrm{u}\right)=0.5\left[1-{\left(\frac{x-b}{e-b}\right)}^{\beta }\right];x\in \left[b,\mathrm{e}\right].\end{array}(12)$

3.3 Determining index weights

(1) It is assumed that sample set $X$ can be depicted as follows:

$X=\left[\begin{array}{cccc}{x}_{11}& x_{12}& ...& x_{1m}\\ x_{21}& x_{22}& ...& x_{2m}\\ ...& ...& ...& ...\\ x_{n1}& x_{n2}& ...& x_{nm}\end{array}\right].(13)$

(2) Sample set $X_{ij}$ is normalized.

3.3.1 The positive index:

$x_{ij}^{\prime }=\frac{x_{ij}-\min \left\{x_{ij},...,x_{nj}\right\}}{\max \left\{x_{1j},...,x_{nj}\right\}-\min \left\{x_{ij},...,x_{nj}\right\}}(14)$

3.3.2 The negative index:

$x_{ij}^{\prime }=\frac{\min \left\{x_{ij},...,x_{nj}\right\}-x_{ij}}{\max \left\{x_{1j},...,x_{nj}\right\}-\min \left\{x_{ij},...,x_{nj}\right\}}(15)$

where $i$ is the number of evaluation schemes, $j$ is the number of evaluation indices, and $x_{ij}$ is the corresponding magnitude.

(3) Determining the proportion of the assessment index.

$b_{ij}=\frac{x_{ij}}{{\displaystyle \sum }_{i=1}^n{x}_{ij}}.(16)$

(4) The entropy is calculated in Eq. 17:

$s_j=-k{\displaystyle \sum }_{i=1}^n{b}_{ij}\ln \left(b_{ij}\right).(17)$

(5) The final weight can be depicted in Eq. 18:

${\omega }_j=\frac{1-s_j}{n-{\displaystyle \sum }_{j=1}^n{s}_j.}(18)$

3.4 Determining the evaluation grade

According to Eqs. 11, 12, and 18 and in combination with the relevant Ref. [26], a synthetic membership degree is shown in Eq. 19:

${\mathrm{v}}_F{\left(\mathrm{u}\right)}_j=\frac{1}{1+{\left(\frac{{\displaystyle \sum }_{i=1}^m{\left[{\omega }_i\left(1-{\mu }_F{\left(u\right)}_{ij}\right)\right]}^l}{{\displaystyle \sum }_{i=1}^m{\left[{\omega }_i{\mu }_F{\left(u\right)}_{ij}\right]}^l}\right)}^{\frac{\mathrm{k}}{l}}}.(19)$

Based on Eq. 19, the synthetic membership degree is calculated as follows:

$V=\left({\mathrm{v}}^{\prime }\right),(20)$

where

$v^{\prime }=\frac{v_F{\left(u\right)}_j}{{\displaystyle \sum }_{j=1}^m{v}_F{\left(u\right)}_j}.(21)$

The evaluation grade $R$ is expressed in Eq. 22.

$R=\left(\mathrm{1,2},...,\mathrm{c}\right)•V.(22)$

3.5 The calculative step and the flow chart

Its calculative process is correlated as follows:

(1) According to the specific data and evaluation standard, the eigenvalue matrix $\mathrm{X}$ and classification matrix $\mathrm{Y}$ are constructed.

(2) The attractive domain ${\mathrm{I}}_{\mathrm{a}\mathrm{b}}$, range matrix ${\mathrm{I}}_{\mathrm{d}\mathrm{e}}$, and point value matrix $\mathrm{F}$ are constructed.

(3) Based on Eqs. 11, 12, the relative membership degree is calculated.

(4) The weights of the landslide hazards using the proposed model are calculated.

(5) The grade eigenvalues $R$ based on the relevant equations are calculated. If $n-0.5\le H\le n+0.5$, then the risk grade is $n$ ($n$ is a nonnegative integer).

Its flow chart is plotted in Figure 6. First, a complete evaluation index system should be constructed before the risk level of landslide hazards is evaluated. Second, entropy–weight theory is adopted to calculate the weight of each evaluation index. Third, the relative membership degree is defined based on the proposed model. Then, the proposed model can determine the risk level of landslide hazards.

FIGURE 6

FIGURE 6. Risk assessment process of landslide hazards.

4 Construction of the evaluation model

4.1 Determining evaluation indices

The Caofang River landslide, Leijia Ping landslide, Daping landslide, Lijia Wan landslide, Zhujia Dian landslide, and Jiaojia Wan landslide in the Badong sections of the Three Gorges Reservoir area are selected as the assessment objects. According to the characteristics of landslides in the Three Gorges Reservoir area, the stratigraphic lithology (X₁), degree of weathering (X₂), relationship between the structural plane and slope direction (X₃), cohesive force (X₄), angle of internal friction (X₅), severity (X₆), average slope degree (X₇), height of slope (X₈), and type of landslide (X₉) are selected as assessment indices. Their original values are shown in Table 1 [27].

TABLE 1

TABLE 1. Data of different assessment indices.

It can be found in Table 1 that X₁, X₂, X₃, and X₉ are qualitative and the other indices are quantitative. To assess the risk level of landslides, the quantitative indices should be transformed into qualitative indices. According to the hierarchy method, the quantitative indices are divided into five groups: excellent, good, moderate, bad, and very bad. The five grades are 5, 4, 3, 2, and 1, respectively. Table 2 is as follows:

TABLE 2

TABLE 2. Comparison sequence of the evaluation index.

According to extensive statistics and analysis and in combination with the geological conditions of assessment objects [16, 28], the risk level of landslides is classified into five classes: safe (I), mildly dangerous (II), dangerous (III), more dangerous (IV), and hazardous (V). Their classification standards are shown in Table 3.

TABLE 3

TABLE 3. Classification of the assessment index.

4.2 Determination of the risk level of the landslide hazard

(1) The construction of the attractive domain, range matrix, and point value matrix.

According to Table 3 and in combination with Eq. 7, the attractive domain ${\mathrm{I}}_{ab}$ is depicted as follows:

${\mathbf{I}}_{\mathbf{a}\mathbf{b}}=\left[\begin{array}{ccccc}\left[\begin{array}{cc}5& 4\end{array}\right]& \left[\begin{array}{cc}4& 3\end{array}\right]& \left[\begin{array}{cc}3& 2\end{array}\right]& \left[\begin{array}{cc}2& 1\end{array}\right]& \left[\begin{array}{cc}1& 0\end{array}\right]\\ \left[\begin{array}{cc}5& 4\end{array}\right]& \left[\begin{array}{cc}4& 3\end{array}\right]& \left[\begin{array}{cc}3& 2\end{array}\right]& \left[\begin{array}{cc}2& 1\end{array}\right]& \left[\begin{array}{cc}1& 0\end{array}\right]\\ \left[\begin{array}{cc}5& 4\end{array}\right]& \left[\begin{array}{cc}4& 3\end{array}\right]& \left[\begin{array}{cc}3& 2\end{array}\right]& \left[\begin{array}{cc}2& 1\end{array}\right]& \left[\begin{array}{cc}1& 0\end{array}\right]\\ \left[\begin{array}{cc}35& 30\end{array}\right]& \left[\begin{array}{cc}30& 25\end{array}\right]& \left[\begin{array}{cc}25& 20\end{array}\right]& \left[\begin{array}{cc}20& 15\end{array}\right]& \left[\begin{array}{cc}15& 0\end{array}\right]\\ \left[\begin{array}{cc}38& 33\end{array}\right]& \left[\begin{array}{cc}33& 28\end{array}\right]& \left[\begin{array}{cc}28& 23\end{array}\right]& \left[\begin{array}{cc}23& 18\end{array}\right]& \left[\begin{array}{cc}18& 0\end{array}\right]\\ \left[\begin{array}{cc}24& 23\end{array}\right]& \left[\begin{array}{cc}23& 22\end{array}\right]& \left[\begin{array}{cc}22& 21\end{array}\right]& \left[\begin{array}{cc}21& 20\end{array}\right]& \left[\begin{array}{cc}20& 0\end{array}\right]\\ \left[\begin{array}{cc}0& 15\end{array}\right]& \left[\begin{array}{cc}15& 25\end{array}\right]& \left[\begin{array}{cc}25& 35\end{array}\right]& \left[\begin{array}{cc}35& 45\end{array}\right]& \left[\begin{array}{cc}45& 90\end{array}\right]\\ \left[\begin{array}{cc}0& 170\end{array}\right]& \left[\begin{array}{cc}170& 200\end{array}\right]& \left[\begin{array}{cc}200& 230\end{array}\right]& \left[\begin{array}{cc}230& 260\end{array}\right]& \left[\begin{array}{cc}260& 500\end{array}\right]\\ \left[\begin{array}{cc}5& 4\end{array}\right]& \left[\begin{array}{cc}4& 3\end{array}\right]& \left[\begin{array}{cc}3& 2\end{array}\right]& \left[\begin{array}{cc}2& 1\end{array}\right]& \left[\begin{array}{cc}1& 0\end{array}\right]\end{array}\right]$

Based on Eq. 8, the matrix $I_{\mathrm{d}\mathrm{e}}$ can be expressed as follows:

${\mathbf{I}}_{\mathbf{d}\mathbf{e}}=\left[\begin{array}{ccccc}\left[\begin{array}{cc}5& 3\end{array}\right]& \left[\begin{array}{cc}5& 2\end{array}\right]& \left[\begin{array}{cc}4& 1\end{array}\right]& \left[\begin{array}{cc}3& 0\end{array}\right]& \left[\begin{array}{cc}2& 0\end{array}\right]\\ \left[\begin{array}{cc}5& 3\end{array}\right]& \left[\begin{array}{cc}5& 2\end{array}\right]& \left[\begin{array}{cc}4& 1\end{array}\right]& \left[\begin{array}{cc}3& 0\end{array}\right]& \left[\begin{array}{cc}2& 0\end{array}\right]\\ \left[\begin{array}{cc}5& 3\end{array}\right]& \left[\begin{array}{cc}5& 2\end{array}\right]& \left[\begin{array}{cc}4& 1\end{array}\right]& \left[\begin{array}{cc}3& 0\end{array}\right]& \left[\begin{array}{cc}2& 0\end{array}\right]\\ \left[\begin{array}{cc}35& 25\end{array}\right]& \left[\begin{array}{cc}35& 20\end{array}\right]& \left[\begin{array}{cc}30& 15\end{array}\right]& \left[\begin{array}{cc}25& 0\end{array}\right]& \left[2\begin{array}{cc}0& 0\end{array}\right]\\ \left[\begin{array}{cc}38& 28\end{array}\right]& \left[\begin{array}{cc}38& 23\end{array}\right]& \left[\begin{array}{cc}33& 18\end{array}\right]& \left[\begin{array}{cc}28& 0\end{array}\right]& \left[\begin{array}{cc}23& 0\end{array}\right]\\ \left[\begin{array}{cc}24& 22\end{array}\right]& \left[\begin{array}{cc}24& 21\end{array}\right]& \left[\begin{array}{cc}23& 20\end{array}\right]& \left[\begin{array}{cc}22& 0\end{array}\right]& \left[\begin{array}{cc}21& 0\end{array}\right]\\ \left[\begin{array}{cc}0& 25\end{array}\right]& \left[\begin{array}{cc}0& 35\end{array}\right]& \left[\begin{array}{cc}15& 45\end{array}\right]& \left[\begin{array}{cc}25& 90\end{array}\right]& \left[\begin{array}{cc}35& 90\end{array}\right]\\ \left[\begin{array}{cc}0& 200\end{array}\right]& \left[\begin{array}{cc}0& 230\end{array}\right]& \left[\begin{array}{cc}170& 260\end{array}\right]& \left[\begin{array}{cc}200& 500\end{array}\right]& \left[\begin{array}{cc}230& 500\end{array}\right]\\ \left[\begin{array}{cc}5& 3\end{array}\right]& \left[\begin{array}{cc}5& 2\end{array}\right]& \left[\begin{array}{cc}4& 1\end{array}\right]& \left[\begin{array}{cc}3& 0\end{array}\right]& \left[\begin{array}{cc}2& 0\end{array}\right]\end{array}\right]$

Based on Eq. 10, the point value matrix $\mathrm{F}$ can be depicted as follows:

$\mathbf{F}=\left[\begin{array}{ccccc}5& 3.75& 2.5& 1.25& 0\\ 5& 3.75& 2.5& 1.25& 0\\ 5& 3.75& 2.5& 1.25& 0\\ 35& 28.75& 27.5& 16.25& 0\\ 38& 31.75& 25.5& 19.25& 0\\ 24& 22.75& 21.5& 20.25& 0\\ 0& 19.5& 30& 42.5& 90\\ 0& 177.5& 215& 252.5& 500\\ 5& 3.75& 2.5& 1.25& 0\end{array}\right]$

(2) Determining the relative membership degree

Based on Table 2 and in combination with Eqs. 11, 12, we should decide whether the evaluation magnitudes are to the left or the right of point $\mathrm{F}$. The data of the Caofang River landslide are adopted for an examination. If $i=1$, then, ${\left[\begin{array}{cc}a& b\end{array}\right]}_{1j}$, ${\left[\begin{array}{cc}d& e\end{array}\right]}_{1j}$, and $\mathrm{F}$ can be depicted as follows:

${\left[\begin{array}{cc}a& b\end{array}\right]}_{1j}=\left(\begin{array}{ccccc}\left[\begin{array}{cc}5& 4\end{array}\right]& \left[\begin{array}{cc}4& 3\end{array}\right]& \left[\begin{array}{cc}3& 2\end{array}\right]& \left[\begin{array}{cc}2& 1\end{array}\right]& \left[\begin{array}{cc}1& 0\end{array}\right]\end{array}\right),$

${\left[\begin{array}{cc}d& e\end{array}\right]}_{1j}=\left(\begin{array}{ccccc}\left[\begin{array}{cc}5& 3\end{array}\right]& \left[\begin{array}{cc}5& 2\end{array}\right]& \left[\begin{array}{cc}4& 1\end{array}\right]& \left[\begin{array}{cc}3& 0\end{array}\right]& \left[\begin{array}{cc}2& 0\end{array}\right]\end{array}\right),$

$F_{1\mathrm{j}}=\left[\begin{array}{ccccc}5& 3.75& 2.5& 1.25& 0\end{array}\right].$

When $x_1=4.5$, $a_{11}=5$, $b_{11}=4$, $d_{11}=5$, $e_{11}=3$, and $F_{11}=5$, then $x_1$ is located in the interval $\left[\begin{array}{cc}{F}_{11}& b_{11}\end{array}\right]$; thus, ${\mu }_F\left({\mathrm{u}}_{11}\right)=0.75$; when $a_{12}=4$, $b_{12}=3$, $d_{12}=5$, $e_{12}=2$, and $F_{12}=3.75$, then $x_1$ is located in the interval $\left[\begin{array}{cc}{d}_{11}& a_{11}\end{array}\right]$; thus, ${\mu }_F\left({\mathrm{u}}_{12}\right)=0.25$; when $a_{13}=3$, $b_{13}=2$, $d_{13}=4$, $e_{13}=1$, and $F_{13}=2.5$, then $x_1$ is located in the out of intervals; thus, ${\mu }_F\left({\mathrm{u}}_{13}\right)=0$.

In the same way, the relative membership degree matrix of the Caofang River landslide can be obtained as follows:

${\mu }_F\left(u_{1j}\right)=\left[\begin{array}{ccccc}0.75& 0.25& 0& 0& 0\\ 0.5& 0.5& 0& 0& 0\\ 0& 0.5& 0.5& 0& 0\\ 0& 0.5& 0.5& 0& 0\\ 0& 0& 0& 0.5& 0.5\\ 0.25& 0.833& 0.25& 0& 0\\ 0.583& 0.417& 0& 0& 0\\ 0.529& 0.471& 0& 0& 0\\ 0& 0.5& 0.5& 0& 0\end{array}\right]$

(3) Determining weight coefficients

Based on Table 2 and in combination with Eq. 16, Table 4 shows the parameter matrix.

TABLE 4

TABLE 4. Synthetic parameters of landslide hazards.

According to Table 4 and Eq. 17, the entropy matrix can be shown in Table 5.

TABLE 5

TABLE 5. Entropy weight matrix.

According to Eq. 18, the weight coefficients are shown in Table 6.

TABLE 6

TABLE 6. Weight coefficient matrix.

(4) Determination of the comprehensive relative membership degree

Based on Eq. 19 and ${\mu }_F\left({\mu }_{1\mathrm{j}}\right)$, the results are calculated in Table 7.

TABLE 7

TABLE 7. Comprehensive relative membership.

Based on Eqs. 20, 21, the comprehensive relative membership degree matrix is normalized in Table 8.

TABLE 8

TABLE 8. Normalization of the comprehensive relative membership degree vector.

(5) Determining the risk level of the landslide hazards.

According to Eq. 22 and Table 8, the ranking value of the Caofang River landslide is shown in Table 9.

TABLE 9

TABLE 9. Feature values.

Similarly, the feature values of the other five landslides are shown in Table 10.

TABLE 10

TABLE 10. Values of the assessment model for the other five landslides.

The results obtained from different methods are contrasted in Table 11.

TABLE 11

TABLE 11. Comparison of results from the different models.

The variable fuzzy set assessment method is used to evaluate the risk level of landslide hazards. Their complete results are shown in Table 10. Table 10 shows that the risk levels of landslide hazards of six different landslides are different. The risk level of the Daping landslide and Lijia Wan landslide is III and that of the rest of the landslides is II. This indicates that the risk level of the Daping landslide and Lijia Wan landslide is dangerous and that of the rest of the landslides is mildly dangerous. Therefore, the necessary consolidation measurements should be performed at the Daping landslide and Lijia Wan landslide. The qualified rate of landslide hazards in all the Badong section of Three Gorges is 67%.

Based on the analytical results of the evaluation method in Table 11, the conclusions obtained from the proposed model are consistent with the site investigations of the five landslides, except for the Zhujia Dian landslide. Its accuracy is 83% in the proposed model, which is higher than that (50%) while using the gray fuzzy comprehensive evaluation theory [27] and that while using level-based weight assessment (67%). In comparison with gray fuzzy comprehensive evaluation theory, the variable fuzzy set assessment method can accurately transmit the risk degree of landslide hazards. Therefore, the conclusions indicate that estimating the risk level of landslide hazards is feasible using the proposed method. Accurate results and details of landslide hazards were obtained. For example, the cohesive force of the Lijia Ping landslide is 33, which should be Grade I based on Table 3. In addition, the membership degree of the other indices obtained by the proposed method is Grade II; therefore, the grade probability of the Lijia Ping landslide at Grade II is more extensive than that at Grades I, IV, III, and V. The risk level probability of the Lijia Ping landslide must be Level II and almost impossible is Level I, IV, III, and V. The risk grade of the Zhujia Dian landslide possibly belongs to Grade III, which is more than that of the Lijia Wan landslide, because the mean ranking feature value (3.1773) of the Zhujia Dian landslide, Level III, is higher than that of the Lijia Wan landslide (2.8162). The conclusions obtained using the proposed method demonstrate the accuracy of the risk level and further determine the ranking of landslide hazards for different landslides at the same grade.

5 Discussion

5.1 Comparison with existing studies

The variable fuzzy set method is provided to assess the risk level of landslide hazards, and the results are good. However, due to a lack of information, the uncertain human mind, and time complexity, decision experts (DEs) cannot provide accurate results for subjective methods such as the gray fuzzy comprehensive evaluation method and level-based weight assessment (LBWA). While the proposed model addresses this concern, it not only considers the unreliability or reliability of the problem but also solves some degrees of uncertainty and ambiguity of the data. Therefore, it has significant advantages over these subjective methods.

5.2 The advantages and limitations of the proposed model

In comparison with the traditional models, the advantages of the variable fuzzy set theory are analyzed as follows:

(1) The variable fuzzy set method can accurately demonstrate the risk degree of landslide hazards using the eigenvalue of level H.

(2) Interval-oriented evaluation, not point assessment, is applied in the proposed model; therefore, the reliability of evaluation outcomes is enhanced, and the quality state of landslide hazards can be discovered with effect.

6 Conclusion

Considering the stratigraphic lithology (X₁), degree of weathering (X₂), relationship between the structural plane and slope direction (X₃), cohesive force (X₄), angle of internal friction (X₅), severity (X₆), average slope degree (X₇), height of slope (X₈), and type of landslide (X₉), the variable fuzzy set theory is applied to evaluate the risk level of landslide hazards in the study.

The conclusions demonstrate that the outcomes obtained using the proposed model are consistent with the site investigations; its accuracy rate reaches 83%. The acceptance rate of landslide hazards for all landslides in the Badong section of Three Gorges is 67%. The proposed method further determines the risk ranking of landslide hazards of different landslides at the same grade. It can accurately demonstrate the risk degree of landslide hazards. Relative to the conventional model, its evaluation process is more reliable and efficient. However, it still has some drawbacks, such as complicated calculation and necessary multiple variable parameters; therefore, it still has significant room for improvement in the future.

In summary, the variable fuzzy set model could offer an alternate route to precisely evaluate the risk grade of landslide hazards.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

YL provided the method and wrote the manuscript; Y-HW provided Figures 1, 2; Q-HW provided Figure 3. X-BG provided the rest work.

Funding

This work is supported by the mineral resources project of Sichuan province (SCKCZY2022-ZC003), the cross project in the Nanyang Institute of Technology (520009), the start-up Foundation (510126), the Sichuan Science and Technology Program (No: 2020YJ0424), Regional Innovation Cooperation Programs of Sichuan Province (2021YFQ0050), Guizhou Provincial Science and Technology Projects [No. (2020)2004], the Scientific and Technological Projects of Henan Province (No: 222102320296), and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2022QYJ02).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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