Examining the Controls on the Spatial Distribution of Landslides Triggered by the 2008 Wenchuan Ms 8.0 Earthquake, China, Using Methods of Spatial Point Pattern Analysis

Abstract

Landslide risk management contributes to the sustainable development of the region. Understanding the spatial controls on the distribution of landslides triggered by earthquakes (EqTLs) is difficult in terms of the prediction and risk assessment of EqTLs. In this study, landslides are regarded as a spatial point pattern to test the controls on the spatial distribution of landslides and model the landslide density prediction. Taking more than 190,000 landslides triggered by the 2008 Wenchuan Ms 8.0 earthquake (WcEqTLs) as the research object, the relative density estimation, Kolmogorov–Smirnov testing based on cumulative distribution, receiver operating characteristic curve (ROC) analysis, and Poisson density modeling are comprehensively applied to quantitatively determine and discuss the different control effects of seven factors representing earthquakes, geology, and topography. The distance to the surface ruptures (dSR) and the distance to the epicenter (dEp) show significant and strong control effects, which are far stronger than the other five factors. Using only the dSR, dEp, engineering geological rock group (Eg), and the range, a particularly effective Poisson model of landslide density is constructed, whose area under the ROC (AUC) reaches 0.9244 and whose very high-density (VHD) zones can contain 50% of landslides and only comprise 3.9% of the study areas. This research not only deepens our understanding of the spatial distribution of WcEqTLs but also provides new technical methods for such investigation and analysis.

1. Introduction

Landslides are ubiquitous in the hills and mountains [1,2], and in many areas, they cause significant human, societal, economic, and environmental damage and costs [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and hinder the sustainable development of society. Earthquake-triggered landslides (EqTLs) are the most common and destructive geological disaster caused by earthquakes [22]. According to rough statistics, in some major earthquakes, the loss of life and property caused by earthquake landslides can account for more than 50% of the total losses of the earthquakes [23,24,25]. Obtaining information about earthquake landslides is the key to quickly organizing rescue operations to save lives and property [25]. The reliable anticipation of landslides and their consequences is, thus, of primary importance [1]. EqTLs are one of the most devastating secondary disasters [26,27,28]. The evaluation of earthquake hazards and risks must include the hazards related to EqTLs [29]. The related problems of EqTLs have become an important research field today [12]. Understanding the spatial pattern of EqTLs is helpful to better understand the complex interaction mechanism between them and parameters such as earthquakes, topography, and lithology [27]. It is of great significance to understand the genetic mechanism of EqTLs to improve the spatial prediction method of earthquake landslides [30]. It is very important to understand the spatial distribution of EqTLs [12,24,31,32], that is, which areas are most likely to have landslides in future, so as to reduce the harm in future earthquakes.

On 12 May 2008, the Wenchuan Ms 8.0 earthquake in China (WcEq) triggered more than 190,000 landslides (Figure 1) [33]. About 20,000 people were killed in the earthquake [28]. It also created an extraordinary natural laboratory, which provides unprecedented landslide data [28,34]. Since the WcEq, a lot of research has been carried out on the spatial distribution characteristics of landslides [30,31,35,36], the correlation between landslides and earthquake parameters, geological and geomorphological conditions [30,33,34,36,37,38,39], comparisons of different landslide susceptibility models [40], and the formation mechanism of typical large-scale landslides [41,42].

Testing whether the landslide density depends on the specified spatial covariate is an important basis for investigating the driving factors of EqTLs and screening the predictors of EqTL susceptibility in potential earthquake areas, which is of great significance for earthquake prevention and disaster reduction and needs formal testing. Although it is not a new proposition to analyze the dependence of landslides triggered by the 2008 Wenchuan Ms8.0 earthquake in China (WcEqTLs) on earthquake factors, most of the previous methods are discrete statistical methods [30,31,33,34,36,38,45]. Firstly, they disperse the factors into finite categories and, then, count the frequency of landslide points in each category, the density of landslide points (the ratio of landslide points in a factor classification area), or the density of landslide areas (the ratio of landslide area in a factor classification area). However, there are some shortcomings in these discrete methods: (1) dividing the factor into discrete segments makes it impossible to count the distribution of landslides within the segments, and the information of the dependency within the segments will inevitably be lost; (2) the significance test of the dependency is lacking; (3) it fails to provide a standard for measuring the strength of dependence; (4) there is still great uncertainty in choosing the cut-off point of continuous numerical factor classification. In order to overcome the above shortcomings, this study seeks a statistical method based on the spatial point pattern to analyze.

The spatial distribution and the prediction modeling of landslides are very important. The landslide susceptibility model is a basic part of the landslide risk management strategy [46]. The above method of preprocessing factors into discrete classes is also commonly used in landslide susceptibility modeling and evaluation, such as the evidence weight method [8,15,47,48], the frequency ratio method [47], and the information method [49]. In previous studies, in order to estimate the possible location of landslides in the future, researchers used the existence/non-existence regression model [50]. This model assumes that landslides occur in a fixed area and are distributed according to the Bernoulli distribution. This distribution describes the probability of binary results, that is, whether there is a landslide in the specified drawing unit. Sensitivity roughly describes the probability of observing at least one landslide in a given mapping unit [50]. One of the limitations of traditional and most current models of statistical methods is that they only predict whether the analysis unit predicts landslides (or not), without considering the number of landslides predicted by each analysis unit. However, the number of landslides is often closely related to the magnitude of landslide damage. If the number of landslides per unit area is considered, richer and more accurate information can be obtained [51].

In this paper, a statistical method is proposed to investigate the dependence of the landslide density on covariates. The relative distribution, the significance evidence, and the strength of the correlation between the landslide density and covariates are analyzed by using the exact values of the covariate at each landslide point instead of discretizing the covariate first. By estimating the relative distribution of the landslide density to covariates, the spatial density distribution characteristics of WcEqTLs depending on the covariates are described. By counting the cumulative distribution function of the precise value of the covariates at each landslide point, the statistical test of relying on significant evidence is carried out by using the Kolmogorov–Smirnov statistics. The receiver operating characteristic curve (ROC) and the area under the ROC (AUC) are used to quantify the strength of the dependence effect [52,53,54,55,56]. This is the first main research content of this study. In order to overcome the inherent problem that the traditional landslide sensitivity model cannot predict the number of landslides, this study seeks to construct a Poisson model for predicting the spatial density of landslides. This is the second main research content in this study.

2. Data

The WcEq occurred along the Longmenshan thrust belt at the eastern margin of the Tibetan Plateau (Figure 1), in the northwest of Chengdu, China, with a focal depth of 14 km and the epicenter located at 31.0° N and 103.4° E. Seismological data indicate that the earthquake initiated in the southern Longmenshan (Figure 1) and propagated unilaterally toward the northeast on a ~33° dipping fault for ~300 km [43,57,58]. The earthquake triggered 197,481 landslides in a range of about 46,000 km² [33]. According to satellite images, the landslide is drawn into polygons, and some verification and field investigations were carried out [33]. We extracted the geometric center point of the landslide polygon, which constitutes the WcEqTLs data used in this study.

This study draws the earthquake surface ruptures (SRs) and the earthquake epicenter (Ep). These data come from the field investigation results of research institutions of the China Earthquake Administration after the earthquake [43,44], and their quality is reliable. The distance from the SR (dSR) and the distance from the Ep (dEp) are compiled by the Euclidean distance tool in QGIS software (the version is 3.16.3-Hannover). They are raster files with a pixel size of 30 m × 30 m. This study analyzes these two factors because some researchers suggest that earthquake vibration is the main cause of landslides [24], and the distance from the source is a convenient covariant for studying the spatial distribution of earthquake landslides [29,59,60]. The distance from the seismic source can, thus, be regarded as the key parameter to assess the influence of ground shaking on the spatial pattern of landslides, with the additional advantage that it may suitably deconvolve amplitude and frequency effects [29]. Massey et al. [61] found that the distance from the SR was a better predictor of the landslide probability than the PGA derived from the ShakeMap.

The exposed lithology map is vectorized from the 1:250,000 regional geological map, which comes from China Geological Survey and is the highest-quality regional lithologic data in China [62]. It is combined into thirteen types according to the engineering geological rock group (Eg) (Figure 2). They are loose gravel soil (10); sandstone (21); conglomerate (22); sandstone, mudstone, and shale (23); mudstone, shale, and siltstone (24); limestone and dolomite (31); marl, argillaceous limestone, and argillaceous dolomite (32); slate (41); phyllite, schist, and gneiss (42); marble (44); basalt (51); granite (61); and other intrusive rocks (63).

This study collects the digital elevation model data (Elv) from the NASADEM [63]. The Elv is a raster file with a pixel size of 30 m × 30 m. This study converts the slope (Slp), the aspect (Asp), and the topographic range (range) from the Elv (Figure 2) via the topography tools in SAGAGIS software (the version is 2.3.2). The range is obtained by calculating the difference between the maximum value and the minimum value of 9 × 9 pixels (270 m × 270 m) around each pixel. The Slp, the Asp, and the range are raster files with a pixel size of 30 m × 30 m.

In order to compare the method based on the spatial point pattern evaluation with the statistical method of discrete covariates, this study divides the dSR, the dEp, the Elv, the Slp, the Asp, and the range into 10 categories in QGIS software (the version is 3.16.3-Hannover). In order to not lose generality, the natural breaks method, commonly used in geomorphology, is used to discretize the data.

3. Methods

3.1. Relative Distribution Estimate of Landslide Density on a Covariate

The uniformity of the landslide density can reflect the dependence of landslide spatial change on external factors. When the density of landslide changes in space, it is actually a function of spatial position. This study estimates this parameter from the covariate by the spatial point pattern analysis method. Different from the previous research methods, this study does not use the discrete segments of the continuous numerical covariate to calculate the landslide density of each segment but assumes that the landslide strength is a function of the continuous numerical covariate. Taking the dSR as an example, in any spatial position $u$, let $\lambda \left(u\right)$ be the landslide density, and $dSR\left(u\right)$ is the value of dSR at the spatial position $u,$ respectively. It is assumed that the following formula holds $\lambda \left(u\right)=\rho \left(dSR\left(u\right)\right)$, where $\rho$ is the estimation of the average density $\lambda \left(u\right)$ at all positions $u$. $\rho$ is the parameter we want to study, which quantifies how the landslide density depends on the $dSR\left(u\right)$ and reflects the preference of landslide for a particular value in the dSR. That is to say, $\rho$ is a landslide density prediction index with $dSR\left(u\right)$ as the covariate. $\rho$ is the ratio of two probability densities, where the numerator is the covariant density at the landslide point and the denominator is the covariant density at the random position. Taking the $dSR\left(u\right)$ as an example, the $\widehat{\rho }$ (estimated of the $\rho$) is calculated using the following formula [64]:

$$\widehat{\rho }={\displaystyle \frac{1}{\left|W\right|G^{\prime }\left(d\right)}}\sum _i\kappa \left(dSR\left(u_i\right)-d\right)$$

(1)

where $u_i$ is the location of the landslide point $i$; the $dSR\left(u_i\right)$ is the value of the $dSR$ at the position $u_i$ of landslide point $i$; the $\left|W\right|$ is the area of the study area; the $\kappa$ is a one-dimensional smoothing kernel function of the $dSR\left(u_i\right)$; approximate the $G^{\prime }\left(d\right)$ by smooth estimation of the differential $G\left(d\right)$; the $G\left(d\right)$ is the ratio of the cumulative distribution of the $dSR\left(u\right)$ in the study area space $W$ to the whole study area: $G\left(d\right)={\displaystyle \frac{1}{\left|W\right|}}{\int }_{u\in W}^{ }\mathbf{1}\left\{dSR\left(u\right)\le d\right\}\mathrm{d}u$, where $\mathbf{1}\left\{dSR\left(u\right)\le d\right\}$ means that if $dSR\left(u\right)\le d$ is true, it is equal to 1; otherwise, it is equal to 0.

The result of the relative distribution estimation is a graph of landslide density estimated value $\rho$ versus the covariant value, and the 95% confidence band of statistics under the assumption of the non-uniform Poisson point process quantifies the size of the estimation error.

3.2. Spatial Cumulative Distribution Function Test (CDF-Test)

In this study, the spatial cumulative distribution function test (CDF test) method of the complete spatial randomness (CSR) is used to evaluate the strength of evidence that WcEqTLs’ density depends on the covariate. This is also a spatial point pattern analysis method. At the same time, aiming at the deficiency of the factor discrete statistical method, a method based on the covariate exact value test is realized; that is, the significance test is carried out by comparing the distribution of covariate values observed at landslides with that at all spatial positions in the observation window. Taking the dSR as an example, the principle is that, if the spatial distribution of landslide points is random, then the landslide points are actually a random sample of spatial positions in the observation window, and then the $dSR$’s value, the $dSR\left(u_i\right)$, at the landslide point $i$ should be random samples of covariant values at all spatial positions in the observation window. The specific method is to compare the accurate value distribution of covariates at landslide points with an accurate value distribution of covariates when landslide points are assumed to be completely random in each analysis range. Taking the $dSR$ as an example, the cumulative distribution function is used to test whether the exact value data of covariate, ${dSR}_1$, …, ${dSR}_n$, follow the specified probability distribution $F_0\left(dSR\right)$. It is assumed that the exact value of covariates at the landslide point is independent and comes from the common distribution of cumulative distribution function $F$. The $F_0$ is the cumulative distribution function of the exact value of covariate when the landslide points are assumed to be completely randomly distributed. Put forward the original hypothesis, $H_0:F_0\equiv F$, and the alternative hypothesis, $H_1:F_0\not\equiv F$. Firstly, the $\widehat{F}\left(dSR\right)$, the cumulative distribution function of landslide points, is calculated, that is, the fraction of the number of landslide points whose covariate is less than or equal to $d$ to the total number of landslide points, $\widehat{F}\left(dSR\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n\mathbf{1}\left\{dSR\left(u_i\right)\le d\right\}$, where the formula $\mathbf{1}\left\{dSR\left(u_i\right)\le d\right\}$ indicates that if $dSR\left(u_i\right)\le d$ is true, the formula is equal to 1; otherwise, it is equal to 0. Then, based on the difference between the $\widehat{F}$ and the $F_0$, it is tested as a statistic. The $\widehat{F}$ is greater than the $F_0$; that is, the positive rather than the negative spatial correlation between a group of landslide positions and a group of covariates is very important in landslide prediction and risk assessment, because it shows that covariates represent the reasonable control of a group of landslides [65]. The covariant value of the maximum positive difference between them is also very important in landslide prediction and risk assessment, because it means that the proportion of landslides within the distance of the source is obviously higher than that under the assumption of the completely random distribution [65].

We used Kolmogorov–Smirnov (K–S) statistics to determine the significance of its spatial correlation [66,67], that is, the significance of the spatial evidence. Kolmogorov–Smirnov test statistic is the maximum vertical distance between curves of the $\widehat{F}\left(dSR\right)$ and the $F_0\left(dSR\right)$, which was used to evaluate the spatial correlation between the deposits distribution and the distance to faults [65,67,68]:

$$D=\begin{array}{c}\mathrm{m}\mathrm{a}\mathrm{x}\\ dSR\end{array}\left|\widehat{F}\left(dSR\right)-F_0\left(dSR\right)\right|$$

(2)

The result graph shows the cumulative distribution function curves actually observed and assumed, as expected, outputs the maximum vertical difference between the two functions, and reports the p-value of the hypothesis test.

3.3. Poisson Model of Landslide Density

The Poisson point process is a basic point process to describe the natural event of landslides. The spatial distribution of landslides accords with the Poisson probability distribution. The modeling method of landslide density in this study assumes that landslide events originate from the spatial intensity function $\mathsf{\lambda }\left(u\right)$. The explanation of this intensity function is that an average number of landslide events $\mathsf{\lambda }\left(u\right)$ is observed around each spatial position $u$. For any domain $W$ in space, the average value of observed landslide events can be obtained by integrating ${\int }_W\mathsf{\lambda }\left(u\right)\mathrm{d}u$. The natural distribution of the landslide event count is the Poisson distribution given by the integral [64]. In this study, covariates are used to fit the density model of the Poisson point process; that is, density is a logarithmic linear function of covariates [64].

$$\mathsf{\lambda }\left(u\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\alpha +{\beta }_1{Z}_1\left(u\right)+\dots +{\beta }_i{Z}_i\left(u\right)\right)$$

where $\alpha$ and ${\beta }_i$ are the parameters to be estimated, and $Z_i\left(u\right)$ is a covariant.

This model provides more information for the general planners than the general sensitivity results, and it not only predicts the possible location of potential landslides in similar earthquakes but also predicts the actual number of landslides. A similar method has been successfully applied to the prediction of landslides [1,50,69]. In addition, the Poisson model in this study has two new features. Firstly, the susceptibility result of the binary existence/non-existence can be easily deduced. Secondly, it can be derived from one analysis unit to another in a simple and consistent way without rebuilding the model.

This model can easily derive the true probability of any region of interest. According to the basic probability property of the Poisson point process, the number of landslide events $N\left(B\right)$ in the limited area $B$ obeys a Poisson distribution, that is, $\mathrm{P}\left(N\left(B\right)=k\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-E\left(B\right)\right){E\left(B\right)}^k/k!$, $k=0,1.$ Then, the probability that landslides will not occur can be expressed as $\mathrm{P}\left(N\left(B\right)=0\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-E\left(B\right)\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\int }_B\mathsf{\lambda }\left(u\right)\mathrm{d}u\right)$. Therefore, the probability of at least one landslide event in the area $B$ can be conveniently written as $\mathrm{P}\left(N\left(B\right)\ge 1\right)=1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\int }_B\mathsf{\lambda }\left(u\right)\mathrm{d}u\right)$, and the traditional classical binary existence/non-existence probability result can be conveniently obtained. Therefore, the probabilistic framework based on the point process in this study is much richer and more complete than the dichotomy existence/non-existence hypothesis based on the fine-scale grid (pixel resolution) or specific analysis unit.

This model can also directly derive the comprehensive strength of any interest region without readjusting the model. According to the basic assumption of the Poisson model, if $B_1$ and $B_2$ represent two disjoint spatial regions, then $N\left(B_1\right)$ and $N\left(B_2\right)$ are independent. As mentioned above, the process intensity $\mathsf{\lambda }\left(u\right)$ of landslide points is defined as the density of landslides around the specific spatial position $u$. The comprehensive intensity $N\left(B\right)={\int }_B\mathsf{\lambda }\left(u\right)\mathrm{d}u$ is the number of landslides expected to occur in the unit $B$. Different from the classical landslide sensitivity method based on the Bernoulli regression model, its fitting strongly depends on the predefined and fixed mapping unit selection [50], and the intensity function $\mathsf{\lambda }\left(u\right)$ is continuously defined and additive in the space. Therefore, we can describe the cumulative number of landslides consistently at different spatial resolutions. The pixel intensity $\widehat{\mathsf{\lambda }}\left(u\right)$ can be estimated, that is, the estimated number of landslides within each pixel. Then, the $N\left(B\right)$ can be obtained by summing the pixels in any one drawing unit $B$.

3.4. Receiver Operating Characteristic Curve (ROC) and the Area Under ROC (AUC)

This study uses the receiver operating characteristic curve (ROC) and the area under ROC (AUC) [52,53,54,55,56] to quantify the strength of dependence effects and the prediction accuracy of models [53,54]. The AUC is a number between 0 and 1, with a value close to 1 indicating very strong action, a value of 0.7 to 0.9 indicating strong action, and a value of 0.5 indicating no action. The horizontal axis of the ROC graph represents the proportion of the range in which the exact value of covariate is less than the eigenvalue in the whole analysis range, and the vertical axis represents the proportion of landslide points in the range in which the exact value of covariate is less than the eigenvalue. If the result curve is close to or lower than the diagonal, it shows that the accurate value of covariate has no influence on the density of landslides. When there is a strong effect density, the resulting curve will be located at the upper-left corner of the diagonal.

3.5. Analysis Process

The method framework includes the following steps:

(1)

Collecting and preprocessing data. WcEqTLs are compiled as a ppp format dataset suitable for the R package spatstat [64,70]. The Slp, the Asp, and the range are compiled on the basis of the Elv by the topography tools in SAGAGIS software (the version is 2.3.2).

(2)

Examining the controls. It is implemented by using the following functions of the R package spatstat (the version is 2.3) [64,70]. Read landslides data and factor raster data into the R package spatstat; then, call the rohat( ) function for the relative density estimation, call the quadrantcount( ) and the intensity( ) function for the relative density estimation of discrete data, call the cdf.test.ks( ) function for the significance of controls, and call the roc( ) and the auc( ) functions to quantify the strength of dependence effects.

(3)

Modeling. The ppm function in the R package spatstat (the version is 2.3) is applied.

4. Results

4.1. The Relative Distribution Estimation of Landslide Density on a Covariant

4.1.1. dSR

The WcEqTLs’ spatial distribution relates to the dSR, and the $\mathsf{\rho }$ varies greatly with the dSR. The landslide density is high near the SR and decreases rapidly with an increase in the dSR, indicating that the density distribution of WcEqTLs depends on the spatial distribution of the dSR. The landslide density can be divided into higher or lower than the average density of the study area (2.6 landslides/km²) at about 32.8 km. The curve in Figure 3 describes the correlation distribution between the landslide density and the dSR in detail. Figure 3 shows that the density of WcEqTLs rapidly decreases from the peak and tends to the average density distribution in the whole study area at $\mathrm{d}\mathrm{S}\mathrm{R}\approx 32.8 \mathrm{k}\mathrm{m}$, decreasing with an increase in the dSR. The curve shows obvious negative exponential correlation characteristics, which is consistent with the trend reported by other studies [30,37,71]. In the dSR range, the estimated minimum and maximum landslide density is 0~18.55 landslides/km². The narrow envelope of the 95% confidence interval (a gray shade of the black curve in Figure 3) indicates that the estimation error of the landslide density depends on the dSR being small at each covariant value. The statistical curve in Figure 3A clearly displays the relative distribution of WcEqTLs’ density on the dSR, but it loses information of two dimensions. Figure 3B provides the spatial density distribution. As can be seen from Figure 3B, the area with a high landslide density is close to the SR, with a narrow width and a clear boundary between high- and low-density distribution areas. The landslide density in the section $\mathrm{d}\mathrm{S}\mathrm{R}\approx 32.8 \mathrm{k}\mathrm{m}$ tends to the average strength of the whole study area. Figure 3 displays more information about the estimated relative distribution of WcEqTL density on the dSR. The results remind geotechnical engineers to pay special attention to the dSR when analyzing and predicting the distribution of WcEqTLs and also remind planners to pay attention to the near-SR areas, especially to avoid these areas in the risk assessment and the prevention zoning.

The results of the discrete method overestimate the landslide density, and there are only two categories above the average density. The spatial distribution information of WcEqTLs relative to the dSR obtained by the method in this study is more abundant and complete.

4.1.2. dEp

The density of WcEqTLs also depends on the dEp, and the $\mathsf{\rho }$ varies significantly with the dEp. It is high near the Ep and decreases sharply with an increase in the dEp. Landslide density can be divided into about 101 km before and after being higher or lower than the average density. The WcEqTLs density decreases rapidly from the peak and tends to the average density. The narrow envelope of the 95% confidence interval shows that the error of estimated density on the dEp is small at each covariant value. In the dEp range, the estimated minimum and maximum landslide density is 0~82 landslides/km². The information of two-dimensional space is lost in Figure 4A. Figure 4B shows more information about the spatial distribution of landslide density on the dEp. Geotechnical engineers should pay attention to the application of the above information and provide an influence index from large to small with the dEp from near to far.

The analysis of the discrete method underestimates the landslide density in the 0~50 km section and overestimates the landslide density in the 50~100 km section. There are only two categories above the average density. The spatial distribution information of WcEqTLs relative to the dEp obtained by the method in this study is more abundant and complete.

4.1.3. Topographical and Geomorphological Factors

The relative distribution of the WcEqTLs density on the elevation (Elv) shows that the landslide density is higher than the average in a range of 950~3100 m (Figure 5A), and when the elevation is less than 650 m, the landslide density decreases sharply with the elevation decreasing. In the area above 3100 m, the landslide density decreases with elevation. The statistical results of the natural breaks method are almost completely underestimated, especially in the area where the landslide density is higher than the average. It is quite different from the estimated value of this study and is outside the 95% CI statistical error interval.

The distribution of the WcEqTLs density relative to the range shows a nearly monotonous positive change; that is, the landslide density increases with an increase in the range (Figure 5B). It changes significantly in a range of 0~180 m and is stable in a range of 180 m, and the landslide density in the 90 m interval is higher than the average. The estimated results of the discrete method are mostly lower than that of this study’s method, and they are outside the 95% CI statistical error interval.

The distribution of the WcEqTLs’ density is generally positive relative to the Slp, but in the steep slope above 56°, the landslide density decreases with an increase in the slope (Figure 5C). The statistical method of discrete classification fails to capture the downward trend of landslide density in steep slope areas.

Relative to the Asp, the characteristics of landslide density distribution are not obvious, and the estimated landslide density has little difference in all values, which is less than 3.5 landslides/km² (Figure 5D). The landslide density in the aspect of 50~220° is higher than the average.

The above results suggest that geotechnical engineers should give positive effects to the following factors in the risk assessment of WcEqTLs: 950~3100 m of the Elv, >100 m of the range, 40~60 of the Slp, and 50~220 of the Asp; they should also remind planners to avoid these areas when planning.

The Eg is a classification factor. It is analyzed using a traditional discrete statistical method. According to the landslide density, rocks can be divided into four categories (Figure 6A,B): (1) The high-density category (14.0~15.1 landslides/km²), including granite (61) and other intrusive rocks (63). (2) The medium-density category (3.7~4.3 landslides/km²), including sandstone, mudstone, and shale (23); basalt (51); limestone and dolomite (31). (3) The low-density category (1.8~2.4 landslides/km²), including phyllite, schist, an gneiss (42); mudstone, shale, and siltstone (24). (4) The very-low-density category (0.1~0.8 landslides/km²), including other rock formations except those mentioned above.

4.2. The Results of Dependence Significance Test

There is strong evidence that the spatial distribution of WcEqTLs’ density depends on the dSR and the dEp (Figure 7,

Table 1. A statistical table of the difference between the observed distribution and the assumed CSR distribution in Figure 7.

dSR (km)	Pro (Obs) (%)	Pro (CSR) (%)	Pro (Obs)/Pro (CSR)
65	99	55	1.8
41	95	34.5	2.8
23.2	80	19.3	4.1
10	50	8.6	5.8
dEp (km)	Pro (Obs) (%)	Pro (CSR) (%)	Pro (Obs)/Pro (CSR)
256	99	77	1.3
212	95	65	1.5
115	80	38.5	2.1
56.5	50	12	4.2

and

Table 2. The statistical results of dependence significance test.

Covariate	K–S	p	Covariate	K–S	p
dSR	D = 0.6294	p = 0 ***	dEp	D = 0.4329	p = 0 ***
Range	D = 0.2541	p = 0 ***	Slp	D = 0.2107	p = 0 ***
Elv	D = 0.1872	p = 0 ***	Asp	D = 0.0625	p = 0 ***

*** means that statistics are significant.

). p = 0 *** means very significant. Figure 7 graphically illustrates the test results. It shows the significant difference of the landslide density depending on the dSR between the observation statistical curve (the solid black line) and the CSR hypothetical distribution statistical curve (the red dashed line). Thus, 50% of WcEqTLs are distributed in $dSR\le 10 \mathrm{k}\mathrm{m}$, which is about 5.88-times higher than 8.5% of the CSR hypothesis distribution; 80% of WcEqTLs are distributed in $dSR\le 23.2 \mathrm{k}\mathrm{m}$, which is about 4.2-times higher than 19% of the CSR hypothesis distribution; 95% of WcEqTLs are distributed in $dSR\le 41 \mathrm{k}\mathrm{m}$, which is about 2.81-times higher than 34.5% of the CSR hypothesis distribution (Table 1). Table 2 gives K–S statistics values and p = 0 ***, revealing the significant difference between observation statistics and CSR hypothetical distribution statistics.

Figure 7 shows the significant difference of the landslide density depending on the dEp. Further, 50% of WcEqTLs are distributed in $dEp\le 56.5 \mathrm{k}\mathrm{m}$, which is about 4.2-times higher than the 12% of the CSR hypothesis distribution; 80% of WcEqTLs are distributed in $dEp\le 115 \mathrm{k}\mathrm{m}$, which is about 2.1-times higher than 38.5% of the CSR hypothesis distribution; 95% of WcEqTLs are distributed in $dEp\le 212 \mathrm{k}\mathrm{m}$, which is about 1.5-times higher than that of 65% of the CSR hypothesis (Table 1). Table 2 gives K–S statistics values and p = 0 ***, revealing the significant difference between observation statistics and CSR hypothetical distribution statistics.

The CDF curves of terrain factors (the Elv, the Slp, the range, and the Asp) are quite different from those of the dSR and the dEp (Table 2, Figure 8). In the CDF curve of the range and the Slp, the actual landslide observation curve is always lower than the CSR statistical curve. The observed curves of the Elv and the Asp are lower than the CSR statistical curve in the low-value domain and turn over in the medium-high value domain. The K–S statistic D of terrain factors and geological morphological factors is much less than that of the dSR and the dEp, indicating that the difference between the actual cumulative distribution of landslides and the assumed CSR distribution is less than that of the dSR and the dEp.

The K–S statistic D represents the difference between the actual observed distribution and the assumed CSR distribution, thus representing the significance of the spatial correlation. According to the K–S statistic D, the covariate of each factor is in descending order: the dSR (0.6294), the dEp (0.4329), the range (0.2541), the Slp (0.2107), the Elv (0.1872), and the Asp (0.0625).

4.3. Strength Assessment of Dependence Effect (ROC and AUC)

The AUC value of WcEqTLs density depends on the dSR is 0.88, which indicates that the strength of the dependence effect is very strong (Figure 9). The ROC diagram (Figure 9) shows that when the distance from the nearest SR is less than the specified distance, 50% of the landslides can be found in 8.6% of the study area, 80% in 19.3% of the study area, and 95% in 34.5% of the study area.

The AUC value of WcEqTLs’ density depends on a dEp of 0.795 (Figure 9), indicating a strong dependency effect. The ROC diagram (Figure 9) shows that when the distance from the nearest epicenter is less than the specified distance, 50% of the landslides can be found in 12% of the areas, 80% in 38.5% of the areas, and 95% in 65% of the areas.

AUC values of the Eg, the range, the Slp, the Elv, and the Asp are 0.6721, 0.6739, 0.6434, 0.5406, and 0.5156, which are far less than those of the dSR and the dEp.

4.4. Model Structure

According to the AUC from high to low, this study selects covariates to join the model in turn and uses the AUC of the model as the index to compare and optimize the model (

Table 3. Model structure. Bold content indicates the best model of each round.

Model	Structure	Covnames	AUC
M1	M1	$\mathbf{d}\mathbf{S}\mathbf{R}$, $\mathbf{d}\mathbf{E}\mathbf{p}$	0.901
M2	M1 + $\mathbf{E}\mathbf{g}$	$\mathbf{d}\mathbf{S}\mathbf{R}$, $\mathbf{d}\mathbf{E}\mathbf{p}$, $\mathbf{E}\mathbf{g}$	0.9222
M3	M1 + $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$	0.9162
M4	M1 + $\mathrm{S}\mathrm{l}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{S}\mathrm{l}\mathrm{p}$	0.9142
M5	M1 + $\mathrm{E}\mathrm{l}\mathrm{v}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{l}\mathrm{v}$	0.9108
M6	M1 + $\mathrm{A}\mathrm{s}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{A}\mathrm{s}\mathrm{p}$	0.9014
M7	M2 + $\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{g}\mathbf{e}$	$\mathbf{d}\mathbf{S}\mathbf{R}$, $\mathbf{d}\mathbf{E}\mathbf{p}$, $\mathbf{E}\mathbf{g}$, $\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{g}\mathbf{e}$	0.9244
M8	M2 + $\mathrm{S}\mathrm{l}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{S}\mathrm{l}\mathrm{p}$	0.9235
M9	M2 + $\mathrm{E}\mathrm{l}\mathrm{v}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{E}\mathrm{l}\mathrm{v}$	0.9223
M10	M2 + $\mathrm{A}\mathrm{s}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{A}\mathrm{s}\mathrm{p}$	0.9222
M11	M7 + $\mathrm{S}\mathrm{l}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$, $\mathrm{S}\mathrm{l}\mathrm{p}$	0.9244
M12	M7 + $\mathrm{E}\mathrm{l}\mathrm{v}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$, $\mathrm{E}\mathrm{l}\mathrm{v}$	0.9244
M13	M7 + $\mathrm{A}\mathrm{s}\mathrm{p}$	$\mathrm{d}\mathrm{S}\mathrm{R}$, $\mathrm{d}\mathrm{E}\mathrm{p}$, $\mathrm{E}\mathrm{g}$, $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$, $\mathrm{A}\mathrm{s}\mathrm{p}$	0.9244

, Figure 10). (1) The first model M1 selects the dSR and the dEp, and the prediction accuracy AUC is 0.901, which is 0.02~0.1 higher than the AUC of the dSR or the dEp. (2) M2 composes three covariates, the dSR, the dEp, and the Eg, whose AUC is 0.9222, which is 0.02 higher than the AUC of M1. (3) M7 composes four covariates, the dSR, the dEp, the Eg, and the range, whose AUC is 0.9244, 0.002 higher than the AUC of M2. (4) This study continues to test and add three other covariates on the basis of M7, but the AUC of new models could not be improved.

The parameters of M7 are shown in

Table 4. Coefficients of the model M7.

Covnames	Estimate	S.E.	CI95.lo	CI95.hi	Ztest	Zval
(Intercept)	−1.1824	0.0653	−1.3105	−1.0544	***	−18.0958
dSR	−0.0591	0.0002	−0.0594	−0.0588	***	−339.4553
dEp	−0.0078	0.0000	−0.0078	−0.0077	***	−193.9179
Eg10	1.1323	0.0768	3.4817	3.7829	***	47.2668
Eg21	2.7108	0.0795	2.5549	2.8667	***	34.0840
Eg22	3.3142	0.0769	3.1635	3.4649	***	43.1003
Eg23	3.6108	0.0660	3.4815	3.7401	***	54.7383
Eg24	3.9010	0.0659	3.7718	4.0303	***	59.1534
Eg31	4.2633	0.0657	4.1345	4.3921	***	64.8504
Eg32	6.1521	1.0022	4.1878	8.1164	***	6.1385
Eg41	3.7900	0.0671	3.6585	3.9215	***	56.4888
Eg42	4.3620	0.0656	4.2335	4.4906	***	66.4983
Eg44	4.2957	0.0755	4.1478	4.4436	***	56.9293
Eg51	4.4285	0.0677	4.2957	4.5612	***	65.3798
Eg61	4.9024	0.0657	4.7737	5.0311	***	74.6572
Eg63	4.8789	0.0659	4.7497	5.0082	***	73.9804
Range	0.0070	0.0001	0.0069	0.0071	***	133.0385

*** means that statistics are significant.

. The 95% confidence interval (95% CI) of the coefficient does not contain 0 (Table 4). These coefficients are effective. The coefficient test also shows that the coefficient is statistically significant (Table 4).

4.5. Model Prediction Performance and Comparison

Based on the ROCs of M1, M2, and M7, this study compiled a landslide distribution zoning map (

Table 5. The statistical table of landslide density zoning areas by M1.

Sub-Regions	Area of Sub-Regions (%)	Total Area of Sub-Regions (%)	Landslides (%)	Total Landslides (%)
VHD	6.2	6.19	50	50
HD	9.2	15.41	30	80
MD	14.8	30.25	15	95
LD	24.4	54.69	4	99
VLD	45.3	100.00	1	100

,

Table 6. The statistical table of landslide density zoning areas by M2.

Sub-Regions	Area of Sub-Regions and the Change from M1 (%)	Total Area of Sub-Regions (%)	Landslides (%)	Total Landslides (%)
VHD	4.1% (↓ 2.1)	4.1%	50	50
HD	7.4% (↓ 1.8)	11.5%	30	80
MD	14.1% (↓ 0.7)	25.6%	15	95
LD	25.0% (↑ 0.6)	50.6%	4	99
VLD	49.4% (↑ 4.1)	100.0%	1	100

↓ means decrease. ↑ means increase.

and

Table 7. The statistical table of landslide density zoning areas by M7.

Sub-Regions	Area of Sub-Regions and the Change from M2 (%)	Total Area of Sub-Regions (%)	Landslides (%)	Total Landslides (%)
VHD	3.9% (↓ 0.2)	3.9%	50	50
HD	6.7% (↓ 0.7)	10.6%	30	80
MD	13.7% (↓ 0.4)	24.3%	15	95
LD	25.0% (—)	49.3%	4	99
VLD	50.7% (↑ 1.3)	100.0%	1	100

↓ means decrease. ↑ means increase. — means no change.

and Figure 11).

This method uses the success rate to determine that the cumulative landslide area exceeds the cumulative area that is considered vulnerable [72], which can improve the readability of the map. Take M7 as an example to show the results. Its AUC is 0.9244. The very-high-density (VHD) areas comprise only 3.9% of the study area and contain 50% of the landslides. The high-density (HD) areas comprise 6.7% of the study area and contain 30% of the landslides (Table 5, Table 6 and Table 7 and Figure 11). The medium-density (MD) area, the low-density (LD) area, and the very-low-density (VLD) area comprise 13.7%, 25.0%, and 50.7% of the study area, respectively, and contain 15%, 4%, and 1% of landslides. Sub-region areas (%) of the VHD, the HD, and the MD are 2.3, 2.5, and 1.1 lower than M1 and they are 0.2, 0.7, and 0.4 lower than M2, respectively. Therefore, the HD and the VHD contain 80% of landslides and only comprise 10.6% of the study area. It is not difficult to find that the prediction result diagram of M7 is more detailed than that of M1 or M2, which improves the accuracy of the model.

5. Discussion

5.1. Summary of Method Novelties

This study quantitatively evaluates the spatial correlation between WcEqTLs and different factors by using the analysis method based on the spatial point pattern, the relative spatial distribution estimation, the cumulative distribution test, and the ROC analysis. The spatial distribution characteristics of WcEqTLs are described in detail, and the statistical significance evidence and the spatial correlation strength are quantized. The continuous numerical analysis method adopted in this study can quantitatively evaluate the spatial correlation, the significance, and the strength on the basis of accurate data analysis. Compared with the discrete statistical method, this method can reveal the relative distribution and the correlation between landslides and factors in more detail, reducing subjectivity. Although there are many other studies [33,34,36,37,71], it is a point of progress and innovation to complete the formal statistics, and it provides quantitative evaluation indicators. The Poisson model of landslide density fundamentally deviates from the classical susceptibility mapping method based on the relative probability, and it provides more details and complete information on the spatial distribution of landslides. The new method is effective and can be extended to similar research.

5.2. Control Effects of Covariates

This study shows that the density of WcEqTLs significantly depends on the dSR and the dEp, and the dependence effect is very strong. The relative relationship of the WcEqTL density with the dSR and the dEp can preliminarily explain their spatial correlation. Figure 3 and Figure 4 clearly describe the mean value and the 95% confidence interval of landslide density estimation with the dSR and the dEp, revealing that the distribution of WcEqTLs has significant preference compared with the dSR and the dEp. These statistical results are more continuous than dividing the dSR and the dEp into discrete segments, and it is difficult to miss the detailed features of the relationship between them. The $\rho$ changes greatly with the change in the dSR and the dEp, indicating that the landslide density changes greatly with the change in the dSR and the dEp, and the landslide density is obviously uneven. The density of WcEqTLs has a strong preference on the dSR. The spatial correlation between landslide density and the dEp is complex. In the 0~68 km section, there is a strong spatial correlation between them. When the distance exceeds 68 km, the curve fluctuates strongly, indicating that the correlation between them may be greatly affected by other factors. This may be one reason leading some studies to conclude that the correlation between them is very poor [34,71], and some other studies came to the conclusion that there is a complex correlation between them [33,36].

Although there seems to be evidence in Figure 3 and Figure 4 that the density of landslides depends on the dSR, a formal test can evaluate the significance of this evidence. This significance test was not found in previous research based on the discrete method. The test result of the K–S has a very significant deviation from the null hypothesis, and p-values are 0, which shows high statistical significance. The invalid assumption is that landslides occur independently of each other, and their spatial distribution has not changed relative to the dSR or the dEp. The ROC and the AUC quantified the spatial correlation strength between different covariates and WcEqTLs. The results show that the spatial correlations of WcEqTLs with the dSR and the dEp are strong, and these two covariates carry rich and key information to control the spatial distribution of WcEqTLs; they are strong in discriminating the spatial density of landslides. It can also be said that both covariates are key control factors of WcEqTLs.

The spatial correlation strengths of WcEqTLs with the dSR and the dEp are high, and their AUCs reach 0.88 and 0.795. They carry important information reflecting the spatial distribution of WcEqTLs, and they are key covariates of WcEqTLs. Their forms are simple and they are easy to obtain. This helps to improve the modeling work. The dSR and the dEp describe the macro-pattern that the spatial density of WcEqTLs decreases exponentially from the epicenter to the northeast and from near SR to far SR. The AUC of model M1 reaches 0.901, which shows that the model captures the overall trend of the spatial distribution of WcEqTLs. Xu et al. [33] also held that WcEqTLs are correlated with the dSR, the dEp, and the Eg, but this study gives more systematic, quantitative, and objective evidence through the relative distribution estimation, the significant evidence test, and the correlation effect strength. This is progress in the field.

The difference in the relative density of landslide reflects the strong spatial correlation between WcEqTLs and the Eg. A landslide density map of the engineering geological rock groups was drawn (Figure 6, the right), which further shows the two-dimensional situation of the relative distribution and provides more information than one-dimensional statistics. Relative to the Eg, the landslide density of WcEqTLs has obvious spatial division. Rocks with high landslide density concentrate in the northwest of the SR and 90 km northeast of the epicenter. Rocks with medium landslide density are distributed in the southeast of the SR in a strip shape. Rocks with low landslide density distribute on both sides of the SR, with a large area of sheet distribution on the northwest side, and a long thin strip interwoven with rock formations with medium landslide density on the southeast side. Rocks with very low landslide density generally divide into two parts; one locates in the southeast of the SR, and the other locates in the northwest of the SR, about 60 km away from the SR. The difference in landslide density in different rock groups is 15.1 landslides/km². The AUC of the spatial correlation between the Eg and WcEqTLs is 0.6721, which is relatively high.

Geotechnical engineers often use topographical covariates in landslide risk assessment, and these covariates often affect the spatial distribution of landslides. For different covariates, the biggest difference in the landslide density reflects the influence of covariates on the spatial distribution of landslides. The landslide density difference in the elevation is only about 5 landslides/km², which is relatively small. The landslide density difference in the Slp is 5.6 landslides/km². The landslide density difference of the range is 8 landslides/km². The landslide density difference of the Asp is only 3.5 landslides/km². In addition, the strength index AUC of the spatial correlation between WcEqTLs and covariates further provides evidence. According to AUCs from high to low, the sequence of these covariates is the range, the Eg, the Slp, the Elv, and the Asp, and their AUCs are 0.6739, 0.6721, 0.6434, 0.5406, and 0.5156, respectively. Although the Asp has little influence on the spatial distribution of landslides, its results still provide some useful information. The control effect of the Asp is not obvious, but the relatively developed landslide at 50~220° also shows that landslides may be affected by the direction of principal stress in the crust and the direction of the hanging wall thrust or the direction of seismic wave propagation. Both the range and the Slp can represent the ups and downs of the terrain and often affect the distribution of landslides. Their change trend of the landslide density is similar, and both of them are positively correlated. However, there is a big difference in the AUC between them. The range is relatively high. It reflects that the range has a greater influence on the spatial distribution of landslides in the study area. It reminds geotechnical engineers that in the landslide risk assessment in the study area, priority should be given to the range, rather than the Slp, as a covariant representing terrain fluctuation.

Therefore, based on the above results, this study holds that the dSR and the dEp constitute a reasonable control on the spatial distribution of WcEqTLs. The spatial correlations of WcEqTLs with the Eg, the range, and the Slp are moderate. These three covariates are important controlling factors of WcEqTLs. The spatial correlations of WcEqTLs with the Elv and the Asp are very low, and their AUCs are close to the AUC of the CSR (the AUC is 0.5). These two covariates are not important controlling factors of WcEqTLs. This result has practical guiding significance for geotechnical engineers to evaluate the risk of EqTLs and planners to carry out land use planning for natural risk management.

5.3. Discussion on the Prediction Results of the Model and Explanation of the Covariates

The predicted result of M1 shows that the predicted landslide density distribution presents an annular pattern of the exponential decrease from the epicenter to the northeast and from near SR to far SR. The AUC of M1 reaches 0.901, which is excellent, and 15.4% of the areas (the VHD and the HD) capture 80% of the landslides, indicating that M1 or the dSR and the dEp captures the key pattern of the spatial distribution of WcEqTLs. This is consistent with the investigation results of the Wenchuan earthquake. According to Xu et al. [43,44], the Wenchuan earthquake spread from the epicenter along the SR to the northeast. M1 captures this effect. It also shows that the dSR and the dEp well represent WcEqTLs, and they are simple and effective substitutes for WcEqTLs. This understanding is consistent with the above results that spatial correlation strengths of WcEqTLs with the dSR and the dEp are as high as 0.88 and 0.7952. From the perspective of the prediction, it is once again proved that the dSR and the dEp are key factors to control the spatial distribution of WcEqTLs.

The spatial correlation analysis between WcEqTLs and dSR (Section 4.1, Section 4.2 and Section 4.3) shows that with an increase in the dSR, the landslide density decreases rapidly, which is significant in the whole study area. However, Figure 3B also shows that it is obviously not enough to estimate the landslide distribution only through the dSR. The dSR can capture the feature that the earthquake-induced vibration function decreases rapidly from near SR to far SR [29], but it cannot capture the feature that the earthquake-induced vibration function propagates from the epicenter to the northeast and gradually decreases [43,44]. This also leads to the fact that only using the dSR to predict will significantly underestimate the southwest section of the SR and significantly overestimate the northeast section. What the dEp can capture is precisely the characteristics that the earthquake vibration spreads from the epicenter to the northeast and gradually decreases. The model M1 constructed by the dSR and the dEp well reflects the overall pattern of WcEqTLs’ spatial distribution and the overall pattern of the WcEq vibration. Therefore, the prediction accuracy of M1 reaches AUC = 0.901.

Compared with M1, the improvement in M2’s prediction result is reflected in the prediction results diagram: (1) It inherits the macro model that the landslide density captured by M1 decreases exponentially from the epicenter to the northeast and from near SR to far SR. (2) The macro structure improves partially, mainly by reducing the actual low-density areas of the VHD and the HD in M1, reducing the misjudgment of M1 and improving the prediction accuracy of the model. ① On the southeast side of the southwest section of the SR, sub-regions of the VHD and the HD in the user-defined coordinate (−10~120 km, −30~20 km) predicted by M1 are greatly reduced and are re-predicted as the VLD by M2. ② On the southeast of the northeast section of the SR, HD sub-regions in the user-defined coordinate (120~200 km, −20~0 km) predicted by M1 were re-predicted as MD sub-regions, and some original MD sub-regions were re-predicted as LD sub-regions, which is more in line with the actual landslide distribution. Therefore, the prediction effect of M2 is obviously improved compared with M1, whose AUC increases by 0.02, and the area ratio of VHD and HD sub-regions for predicting 80% of landslides reduces to 11.5, which is 3.9% lower than M1. The improvement in M2 compared with M1 is due to the addition of the covariate Eg, so the Eg is an important covariate to predict the spatial distribution of WcEqTLs. From the epicenter to the northeast 40 km, the earthquake has the strongest effect and continues to the northeast, gradually weakening. The SR displacement also shows that the thrust with large vertical displacement is dominant in the 40 km section, the strike–slip–thrust with the large vertical displacement is dominant in the 40~120 km section, and the slip–strike–thrust is dominant in the 120~240 km section [43]. There is a great spatial coupling between the distribution of the Eg and the above distribution (the right of Figure 6). For example, the granite and other intrusive rocks with high landslide density distribute on the northwest side of the SR, and from the epicenter to the northeast 90 km section. Rocks with medium-low landslide density distribute on the northwest side of the northeast section of the SR and the southeast side of the SR, while alluvial rocks with low landslide density distribute on the southeast side of the SR. It is believed that the Eg may carry the control information of WcEqTLs and effectively adjust the distribution of landslides in these areas. Of course, the mechanical properties of different rocks and soils also affect the occurrence of landslides.

The improvement in M7 compared with M2 is mainly reflected in the “fine” particle size. On the basis of inheriting the patch of M2 prediction results, the main feature of the prediction result map is to analyze the landslide density changes within and between patches in the pixel-resolution accuracy, which is smoother and more precise. After adding the range to the model, the accuracy of the model improves to a certain extent. The AUC of the model further improves by 0.002, and the area ratio of VHD and HD sub-regions for predicting 80% of landslides reduces to 10.6, which is 0.9% lower than M2 and 4.8% lower than M1. The above improvement in M7 compared with M2 is due to the increase in the range covariate, so the range is an important covariate for predicting the spatial distribution of WcEqTLs.

Generally speaking, the dSR and the dEp make the model M1 effectively capture the macro-pattern of the WcEqTL spatial distribution, and the ring-shaped structure is remarkable. After adding the Eg, M2 subdivides sub-regions and still presents a big patch. The model M7, which continues to add the range, effectively realizes the fine prediction in patch partition.

These results are of practical value. The high-accuracy prediction can support geotechnical engineers and planners to take a step towards better risk management. The selection of landslide control factors and the prediction and the evaluation of landslide density distribution completed in this study are not only important tasks for landslide disaster prevention but are also important contents for supporting regional sustainable development. The method flow based on the landslide spatial point pattern and the statistical method is put forward, which is beneficial to improving scientific and accurate results and can be popularized and used in similar earthquake areas and works.

6. Conclusions

(1)

This study proposes a new method framework based on the point space pattern analysis to test the controls on the spatial distribution of landslides and model the landslide density prediction.

(2)

The dSR and the dEp show significant and strong control effects, which are far stronger than the other five factors. They carry the key information of the spatial pattern of WcEqTLs; that is, the landslide density decreases exponentially from the epicenter to the northeast and from near SR to far SR. The Elv and the Asp are weak in controlling the spatial distribution of WcEqTLs.

(3)

Using only four covariates, the dSR, the dEp, the Eg, and the range, this study establishes an excellent Poisson model of landslide density, whose AUC is 0.9244.

(4)

The research in this study not only deepens the understanding of the WcEqTL spatial distribution but also provide new technical methods for such investigations and analyses. At the same time, this study can provide a useful reference for the sustainable development of the study area, especially for the prevention and control of regional landslides.