Modeling Dependence Structures in Hydrodynamic Landslide Deformation via Hierarchical Archimedean Copula Framework: Case Study of the Donglinxin Landslide

Abstract

This study proposes a hierarchical Archimedean copula (HAC) framework to model the complex dependence structures in hydrodynamic landslide deformations, with a focus on the Donglinxin (DLX) landslide. Hierarchical Archimedean copulas, compared to elliptical copulas, offer greater flexibility by requiring fewer parameters while maintaining broader applicability. The HAC model, combined with pseudo-maximum likelihood estimation (PMLE), is applied to analyze the interdependencies among the landslide-related variables, such as monthly displacement increments, reservoir water level fluctuations, groundwater variations, and precipitation. A case study of the DLX landslide demonstrates the model’s ability to quantify the critical aspects of landslide deformation, including variable correlations, risk thresholds, conditional probabilities, and return periods. The analysis reveals a strong hierarchical dependence between monthly displacement increments and reservoir water level drops. The model also provides valuable insights into the potential risk factors, helping to optimize landslide monitoring and early-warning systems for more effective disaster mitigation.

1. Introduction

Hydrodynamic landslides induced by extreme precipitation events, abrupt reservoir water-level fluctuations, and the associated groundwater variations have emerged as critical geological hazards jeopardizing hydropower infrastructures. These cascading hydrological processes profoundly disrupt the geo-environmental equilibrium surrounding hydroelectric facilities, posing imminent threats to both human safety and critical civil structures. Particularly noteworthy are the nonlinear interdependencies between the reservoir operation parameters (water level fluctuation rate), meteorological drivers (precipitation intensity), and hydrogeological responses (groundwater table dynamics). Establishing the quantitative relationships among these triggering factors through multivariate analysis could substantially enhance the real-time monitoring systems and enable proactive early-warning mechanisms for landslide risk mitigation [1,2,3,4,5,6].

The copula framework, introduced by Sklar in 1959, offers an effective method for modeling multivariate dependencies, especially when the relationships between the variables are nonlinear or asymmetric [7,8,9]. Copula models have been widely used in hydrology and geotechnical engineering to model the interactions between variables such as precipitation, reservoir water levels, and landslide displacement. For instance, t-copulas have been used to assess the landslide displacement in response to fluctuating precipitation and reservoir water levels, while Gumbel copulas have been applied to predict landslide hazard probabilities based on rainfall thresholds [10,11,12,13,14]. However, most of these studies have relied on elliptical copulas, which, despite their effectiveness, have limitations in modeling complex, high-dimensional dependencies. Specifically, elliptical copulas require the estimation of many parameters in high-dimensional cases and constrain the marginal distributions to elliptical forms, which reduces their flexibility.

The existing studies primarily utilize elliptical copulas and Archimedean copulas (ACs), which possess advantageous mathematical properties but inherent limitations. Multidimensional elliptical copulas require excessive parameter estimation and restrict marginal distributions to elliptical forms. Conversely, multidimensional ACs enforce exchangeability constraints and identical dependence structures across components. To address these limitations, this study introduces hierarchical Archimedean copulas (HACs) [15,16,17], which offer a reduced parametric complexity and enhanced flexibility in modeling heterogeneous dependence structures compared to conventional ACs.

The current methodologies face additional challenges in accurately characterizing the marginal distributions of hydrodynamic landslide variables [18,19]. Traditional maximum likelihood estimation (MLE) necessitates simultaneous parameter estimation for both marginal distributions and copula functions. In contrast, pseudo maximum likelihood estimation (PMLE) eliminates the parametric assumptions for marginal distributions, thereby reducing the estimation errors [20,21,22,23].

This study proposes an HAC model integrated with PMLE to investigate the correlations among the hydrodynamic landslide variables. The framework is applied to analyze the correlation structures, conditional probabilities, risk thresholds, and deformation return periods. These advancements provide a robust technical foundation for improving hydrodynamic landslide monitoring, early warning systems, and disaster prevention strategies.

2. Proposed Method

Theory

The basic concept of an HAC is to construct a hierarchical structure of Archimedean copulas, coupling the marginal distribution functions from the bottom to the top in a stepwise manner to form a hierarchical structure. Before introducing the HAC, the definition of a copula and Archimedean copula will be introduced.

For the k-dimensional continuous joint distribution function F, its corresponding copula function $C:{\left[\mathrm{0,1}\right]}^k\to \left[\mathrm{0,1}\right]$ can be expressed as:

$$C\left(F_1\left(x_1\right),F_2\left(x_2\right),\dots ,F_k\left(x_k\right)\right)=f\left(x_1,x_2,\dots x_k\right)$$

(1)

where $x_1,x_2,\dots ,x_k\in R$, $F_1\left(x_1\right),F_2\left(x_2\right),\dots ,F_k\left(x_k\right)$ is a marginal distribution function and continuous.

Define the following function class:

$$\mathsf{\Omega }=\{\phi :\left[0,\mathrm{\infty }\right)\to [\mathrm{0,1}\left]\left|\phi \left(0\right)=1,\phi \left(\mathrm{\infty }\right)=0,{(-1)}^j{\phi }^j\ge 0,j=\mathrm{1,2},\dots \right.\right\}$$

The k-dimensional Archimedean copula $C:{\left[\mathrm{0,1}\right]}^k\to \left[\mathrm{0,1}\right]$ is:

$$C\left(u_1,u_2,\dots ,u_k\right)=\phi ({\phi }^{-1}\left(u_1\right)+{\phi }^{-1}\left(u_2\right)+\dots +{\phi }^{-1}\left(u_k\right))$$

(2)

where $\phi \in \mathsf{\Omega }$ is called generating functions, and different generating functions constitute different Archimedean copula functions. There are many common Archimedean copula functions, such as Clayton copula, Frank copula, and Gumbel copula, and the generating functions can be written as follows [15]:

$${\phi }_{\mathrm{G}\mathrm{u}\mathrm{m}}(t;\theta )=(-\ln t{)}^{\theta }$$

(3)

$${\phi }_{\mathrm{C}\mathrm{l}\mathrm{a}}\left(t;\theta \right)={\displaystyle \frac{1}{\theta }}\left(t^{-\theta }-1\right)$$

(4)

$${\phi }_{\mathrm{F}\mathrm{r}\mathrm{a}}\left(t;\theta \right)=-\ln {\displaystyle \frac{{\mathrm{e}}^{-\theta t}-1}{{\mathrm{e}}^{-\theta }-1}}$$

(5)

where θ denotes the parameter to be estimated.

From Equation (2), the Archimedean copula satisfies the commutative law, i.e., for arbitrary $\left(u_1,u_2,\dots ,u_k\right),C\left(u_1,u_2,\dots ,u_k\right)=C\left(u_{j1},u_{j2},\dots ,u_{jk}\right),j_s\ne j_v,s,v=\mathrm{1,2},\dots ,k$. As mentioned above, this condition is too strong in practical applications, so many scholars have proposed a more general HAC. Take a fully nested HAC as an example, defined as:

$$\begin{gathered} C(u_1,u_2,\dots ,u_k)= \\ {\phi }_{k-1}\{{\phi }_{k-1}^{-1}°\left\{{\phi }_{k-2}\left[\dots \left({\phi }_2^{-1}°{\phi }_1\left[{\phi }_1^{-1}\left(u_1\right)+{\phi }_1^{-1}\left(u_2\right)\right]+{\phi }_2^{-1}\left(u_3\right)\right)+\dots +{\phi }_{k-2}^{-1}\left(u_{k-1}\right)\right]\right\}+{\phi }_{k-1}^{-1}(u_k\left)\right\} \end{gathered}$$

(6)

where ${\phi }_1,{\phi }_2,\dots ,{\phi }_k\in \mathsf{\Omega },{\phi }_{k-i}^{-1}\circ {\phi }_{k-j}\in {\mathsf{\Omega }}^{\mathrm{*}},i<j,i,j=\mathrm{1,2},\dots ,k$. As per the definition of $\mathsf{\Omega }$, ${\mathsf{\Omega }}^{\mathrm{*}}$ can be defined as below:

$${\mathsf{\Omega }}^{\mathrm{*}}=\left\{\omega :[0,\mathrm{\infty })\to \left[\mathrm{0,1}\right]|\omega \left(0\right)=0,\omega \left(\mathrm{\infty }\right)=\mathrm{\infty },{\left(-1\right)}^j{\omega }^{\left(j\right)}\ge 0,j=\mathrm{1,2},\dots \right\}$$

The HAC establishes dependencies between random variables in a progressive manner. As can be seen from Equation (6), the lowest-level copula function is generated by ${\phi }_1$ with the parameter ${\theta }_1$, which represents the correlation between $u_1$ and $u_2$ such that $z_1=C\left(u_1,u_2\right)={\phi }_1[{\phi }_1^{-1}\left(u_1\right)+{\phi }_1^{-1}\left(u_2\right)]$. The copula function of the second layer is generated by ${\phi }_2$ with the parameter ${\theta }_2$ such that $z_2=C\left(z_1,u_3\right)={\phi }_2[{\phi }_2^{-1}\left(z_1\right)+{\phi }_2^{-1}\left(u_3\right)]$, indicating the correlation between $z_1$ and $u_3$. By analogy to the $k-1$ layer, the generating function is ${\theta }_{k-1},$ and the corresponding parameter is ${\theta }_{k-1}$ [24]. Here, ${\phi }_i$ can be a generating function of different function types, or it can be a generating function of the same function type, but the parameters must be satisfied such that ${\theta }_1>{\theta }_2>\dots {>\theta }_{k-1}$. In Equation (6), if ${\phi }_i$ and ${\phi }_{j }\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}$, $i\ne j$, i.e., the function type of the generating function is the same, and the corresponding parameters, ${\theta }_i$ and ${\theta }_j,$ are also the same, then the copula function represents a partially nested HAC. In Equation (6), if the type of generating function and the corresponding parameters are the same for both the generating functions used, then Equation (6) degenerates into Equation (2), i.e., ACs, which can be regarded as a special case of the HAC.

Figure 1 shows the hierarchical structure of the Archimedean copula (AC) models. The process begins by constructing marginal distributions for each variable, which are then connected through copulas. In a fully nested HAC structure, the first level captures the dependencies between two variables using a copula, such as the Gumbel copula. Each subsequent level introduces additional copulas to model the dependencies between the variables at higher levels, progressively expanding the correlation structure. This stepwise nesting allows for greater flexibility in modeling complex, nonlinear interdependencies. The copula parameters are estimated using pseudo-maximum likelihood estimation to minimize the estimation errors.

Compared with ACs, an HAC can characterize a more general dependency structure, and at each node a marginal distribution function can be obtained, which establishes the dependencies between the variables on its branches. For a k-dimensional HAC, take any l variables, $2\le l<k$, and it still constitutes an HAC [8]. For example, in Equation (6), let $u_k=1$, and since, ${\phi }_{k-1}^{-1}\left(1\right)=0$, then, ${C\left(u_1,u_2,\dots ,u_{k-1},1\right)=\phi }_{k-2}\{\dots \left({\phi }_2^{-1}°{\phi }_1\left[{\phi }_1^{-1}\left(u_1\right)+{\phi }_1^{-1}\left(u_2\right)\right]+{\phi }_2^{-1}\left(u_3\right)\right)+\dots +{\phi }_{k-2}^{-1}\left(u_{k-1}\right)\}$, i.e., the joint distribution of $u_1,u_2,\dots ,u_{k-1}$ constitutes a fully nested HAC in the $k-1$ dimension.

Theoretically, the optimal structure of an HAC with a dimension of k needs to be determined, but as k increases, there are $2^k-k-1$ possible structures, and the screening process will be complicated [15]. In practical applications, the fully nested HAC is usually used as an alternative structure, and the function type of its generating function is assumed to be the same, for the following reasons: (1) the fully nested HAC structure is more general, and any partially nested HAC can be regarded as its special case; see Figure 1; (2) in the time series modeling of landslide deformation, if the aim is to improve the estimation accuracy, the correlation parameters between the influencing factors can be considered to be completely different when the generating function is the same. In view of this, a fully nested HAC is used to study the correlation of landslide deformation.

3. Model Construction

3.1. Data Preprocessing

To obtain the marginal distributions, the empirical distribution function is used to represent the marginal distributions of random variables. Because all of these variables are discrete, the marginal distribution cannot be used to build the copula model [25]. Hierarchical transform and linear interpolation methods have been used to solve this problem. In [26], an effective transform method that defines the following function given a known discrete random variable X was proposed:

$$D\left(x,k\right)=P\left(X<x\right)+k P\left(X=x\right)$$

(7)

where x ∈ R and k follows a uniform distribution and is independent of x.

3.2. Parameter Estimation

Pseudo-maximum likelihood estimation was selected to estimate the HAC parameters [20]. If a probability density function of $c(u_1,u_2,\dots ,u_k;{\theta }_1,{\theta }_2,\dots ,{\theta }_k)$, joint distribution function of $F(x_1,x_2,\dots ,x_k)$, and joint density function of $f\left(x_1,x_2,\cdots ,x_k\right)$ are assumed, then the following is obtained:

$$f\left(x_1,x_2,\cdots ,x_k\right)=c\left(F_1\left(x_1;{\alpha }_1\right),F_2\left(x_2;{\alpha }_2\right)\right.\left.\cdots ,F_k\left(x_k;{\alpha }_k\right);{\theta }_1,{\theta }_2,\cdots ,{\theta }_k\right){\prod }_{i=1}^k{ f}_i\left(x_i;{\alpha }_i\right)$$

(8)

where ${\alpha }_i,i=\mathrm{1,2}\dots , k,$ denote the parameters to be estimated for the marginal distributions, and ${\theta }_i,i=\mathrm{1,2}\dots , k,$ denote the parameters to be estimated for the copula function.

If an empirical distribution function is used to replace the marginal distribution, the expression becomes:

$${\widehat{F}}_n\left(t\right)={\displaystyle \frac{1}{n+1}}{\sum }_{i=1}^n{1}_{x_i\le t}$$

(9)

where ${\widehat{F}}_n\left(t\right)$ denotes the unbiased estimate of the empirical distribution function $F_n\left(t\right)$.

The copula log-likelihood function is given by:

$$L_2\left({\theta }_1,{\theta }_2,\cdots ,{\theta }_k\right)={\sum }_{j=1}^n\lg c\left(F_1\left(x_{1j}\right)\right.\left.F_2\left(x_{2j}\right),\cdots ,F_k\left(x_{kj}\right);{\theta }_1,{\theta }_2,\cdots ,{\theta }_k\right)$$

(10)

Solving the above expression yields the pseudo-maximum likelihood estimation of ${\theta }_i,i=\mathrm{1,2}\dots , k,$

$$\left({\widehat{\theta }}_1,{\widehat{\theta }}_2,\dots ,{\widehat{\theta }}_k\right)=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}\ln L\left({\theta }_1,{\theta }_2,\dots ,{\theta }_k\right)$$

(11)

To find the optimal HAC model, three methods, i.e., the root mean square error (RMSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC), are employed to quantitatively evaluate the goodness of fit. These are expressed below:

$$d\left(C^{\mathrm{e}\mathrm{s}\mathrm{t}},C^{\mathrm{e}\mathrm{m}\mathrm{p}}\right)=\sqrt{{\sum }_{i=1}^n{\left(C^{\mathrm{e}\mathrm{s}\mathrm{t}}\left(U_i\right)-C^{\mathrm{e}\mathrm{m}\mathrm{p}}\left(U_i\right)\right)}^2}$$

(12)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{d{\left(C^{\mathrm{e}\mathrm{s}\mathrm{t}},C^{\mathrm{e}\mathrm{m}\mathrm{p}}\right)}^2}{n}}}$$

(13)

$$\mathrm{A}\mathrm{I}\mathrm{C}=-n\ln \left({\displaystyle \frac{d{\left(C^{\mathrm{e}\mathrm{s}\mathrm{t}},C^{\mathrm{e}\mathrm{m}\mathrm{p}}\right)}^2}{n}}\right)+2k$$

(14)

$$\mathrm{B}\mathrm{I}\mathrm{C}=-n\ln \left({\displaystyle \frac{d{\left(C^{\mathrm{e}\mathrm{s}\mathrm{t}},C^{\mathrm{e}\mathrm{m}\mathrm{p}}\right)}^2}{n}}\right)+k\ln n$$

(15)

where $C^{\mathrm{e}\mathrm{s}\mathrm{t}}$ denotes the estimated copula model, $C^{\mathrm{e}\mathrm{m}\mathrm{p}}$ denotes the empirical copula model, n denotes the sample size, and k is the number of parameters in the model.

3.3. Conditional Probability, Risk Threshold, and Return Period

The conditional probability of the influencing factors of the monthly displacement increment is derived from the HAC model to quantify the risk of hydrodynamic landslide, and the thresholds of variables are extracted based on the value-at-risk (VaR) theory. These can be expressed as follows:

$$F\left(y∣X=x\right)=C\left(V\le v∣U=u\right)={\displaystyle \frac{\partial C\left(u,v\right)}{\partial u}}$$

(16)

$${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(x\right)={\widehat{F}}^{-1}\left(x\right)$$

(17)

The conditional return period, which describes the average number of time intervals required for a variable to have a value greater than or equal to a specific value, is given by:

$$T(y\mid X=x)={\displaystyle \frac{1}{1-F(y\mid X=x)}}={\displaystyle \frac{1}{1-\frac{\partial C(u,v)}{\partial u}}}$$

(18)

where C is the correlation models for the monthly displacement increment based on copula functions, x is the value of each variable, α is the confidence level (0.95 in this study), and ${\widehat{F}}^{-1}\left(x\right)$, $\widehat{F}\left(x\right)$ is the cumulative probability density function.

4. A Case Study: DLX Landslide

4.1. Data Source and Description

The ancient DLX landslide is located in Jianhe Country, Guizhou Province, and it is situated on the upper reaches of the Sanbanxi Hydropower Station reservoir. This landslide is approximately 0.8 km from Liuchuan Town, as shown in Figure 2a, and it mainly comprises landslide deposits. The topographic map is shown in Figure 2b. The total volume of the DLX landslide deposit is approximately 2.07 × 10⁷ m³, with a total area of around 3.83 × 10⁵ m². If it leads to a wave surge damming disaster chain, it poses a significant threat to Jianhe County. After Sanbanxi Hydropower Station started storing water, landslide deformation began to appear with a rising tendency. Meanwhile, the deformation showed an obvious upward trend from April to October, which is when precipitation is concentrated. An increasing reservoir water level raises the groundwater level inside the landslide, which increases the pore pressure and driving forces. Likewise, a decrease in the reservoir water level removes fine particles from slip soils, and the high seepage pressure acting outward inside the landslide destabilizes the slope toe. Rainwater infiltrates the landslide through cracks, which increases the pore water pressure and reduces the mechanical properties.

Based on the above analysis, the following variables were selected to construct the HAC model: monthly displacement increment, monthly reservoir water level drop and rise, monthly groundwater level drop and rise, and monthly precipitation. For the monitoring points, we selected DLX05 for monitoring the displacement because it had the longest monitoring time and largest deformation, and we selected ZK14 for monitoring the groundwater level because it was the closest monitoring point available to DLX05 as the original data. Figure 3 shows the monitoring curves of the displacement, reservoir water level, groundwater level, and precipitation, and Figure 4 shows the work flowchart.

4.2. Correlation Structure Among the Variables of the Hydrodynamic Landslide

The Gumbel, Clayton, and Frank copulas were selected as the generating functions to construct the HAC models, and

Table 1. Estimated parameters of models.

Parameters	Gumbel-HAC	Clayton-HAC	Frank-HAC
θ1	1.142	0.283	1.130
θ2	1.209	0.428	1.594
θ3	1.218	0.436	1.654
θ4	1.252	0.445	1.884
θ5	1.267	0.535	1.971

and

Table 2. Goodness-of-fit test of the models.

Type	Gumbel-HAC	Clayton-HAC	Frank-HAC
AIC	−647.701	−469.100	−476.101
BIC	−637.884	−459.283	−466.284
RMSE	0.022	0.062	0.060

present the results of the parameter estimation and goodness-of-fit tests. The correlation structure and interpretation of the variables are as shown in Figure 5 and

Table 3. Interpretation of variables.

Variables	Items
u1	Monthly displacement increment
u2	Monthly reservoir water level drop
u3	Monthly reservoir water level rise
u4	Monthly groundwater level drop
u5	Monthly groundwater level rise
u6	Monthly precipitation

, respectively.

The fitting results of the three generating functions of Gumbel, Clayton, and Frank are as shown in Figure 5, and the correlation structure has six levels. The monthly displacement increment and monthly reservoir water level drop were in the fifth layer, and the monthly reservoir water level rise, monthly groundwater level drop, monthly groundwater level rise, and monthly precipitation were in the fourth, third, second, and first layers, respectively.

Table 2 shows that the Gumbel-HAC model had the smallest goodness-of-fit value. Therefore, this paper asserts that the correlation of variables characterized by the Gumbel generating function is more reliable.

To clarify the correlations among the hydrodynamic landslide variables, the generalized joint probability densities were extracted, as shown in Figure 6. It shows the generalized joint probability densities and contour plots for the hydrodynamic landslide variables at different hierarchical levels. Each contour plot visualizes the dependency between the variables at a specific layer of the HAC model. The plots for higher layers (such as the fifth and fourth layers) exhibit strong upper-tail dependence, indicating a significant correlation during extreme events (e.g., large displacement increments corresponding to significant drops in the reservoir water levels). In contrast, the lower-tail dependencies are weaker, which means that for lower values of displacement or water level fluctuations, the correlation is much less pronounced. The contours reveal a high degree of correlation between the variables along the upper tail of the distribution, suggesting that the extreme values of one variable (e.g., reservoir water level drops) are strongly associated with large displacement increments. This hierarchical structure of dependence reflects the progressive nature of the landslide deformation process, where larger deformations tend to occur alongside the substantial changes in the reservoir’s water level.

Based on Table 1 and Figure 5 and Figure 6 for further analysis, the correlation between the monthly displacement increment and monthly reservoir water level drop is in the fifth layer, indicating the strongest correlation between these two variables. The DLX landslide is mainly composed of silty clay and rubble, and its shear strength decreases after being immersed in water. The drop in the reservoir water level will form an outward seepage force inside the slope, taking away the fine-grained soil in the slope. After many cycles of rise and drop in the reservoir water level, the soil structure becomes loose, and the slope deforms significantly.

4.3. Conditional Probability and Risk Analysis

To quantify the risk of hydrodynamic landslide, the cumulative conditional probability curves of the monthly displacement increment with other variables are extracted from the HAC model, as shown in Figure 7 and

Table 4. Probability distributions of the monthly displacement increment with different variables.

Variable Levels	Monthly Displacement Increment (mm)
	u1 ≤ 2	2 < u1  ≤ 8	u1 > 8
Monthly reservoir water level drops by 10 m	0.1964	0.6454	0.1582
Monthly reservoir water level drops by 15 m	0.1528	0.6294	0.2178
Monthly reservoir water level drops by 20 m	0.1180	0.5739	0.3081
Monthly reservoir water level rises by 10 m	0.2126	0.6355	0.1519
Monthly reservoir water level rises by 15 m	0.1756	0.6324	0.1920
Monthly reservoir water level rises by 20 m	0.1518	0.6139	0.2343
Monthly groundwater level drops by 5 m	0.2175	0.6338	0.1487
Monthly groundwater level drops by 7.5 m	0.1746	0.6308	0.1946
Monthly groundwater level drops by 10 m	0.1547	0.6158	0.2295
Monthly groundwater level rises by 5 m	0.2346	0.6286	0.1368
Monthly groundwater level rises by 7.5 m	0.2028	0.6359	0.1613
Monthly groundwater level rises by 10 m	0.1711	0.6290	0.1999
Monthly precipitation of 100 mm	0.2264	0.6189	0.1547
Monthly precipitation of 150 mm	0.2053	0.6223	0.1724
Monthly precipitation of 200 mm	0.1835	0.6173	0.1992

. The DLX landslide monthly displacement increment data are arranged in order of magnitude to create a new sequence. The lower quartile was 2 mm and the upper quartile was 8 mm. A monthly displacement increment of less than 2 mm is defined as a small deformation, between 2 and 8 mm is a medium deformation, and greater than 8 mm is a large deformation. The following levels of the variables were also, respectively, defined as small, medium, and large: monthly reservoir water level drops and rises of 10, 15, and 20 m; monthly groundwater level drops and rises of 5, 7.5, and 10 m; and monthly precipitation of 100, 150, and 200 mm.

As the level of a variable increased, the cumulative conditional probability curve became closer to the exponential function curve, and the gradient in the second half of the curve increased. The occurrence probability of medium deformation reached a maximum value of 0.6454 when the monthly reservoir water level drop was 10 m. Among the deformations, the medium deformation had the highest occurrence probability for a given combination of different variable levels. When the monthly reservoir water level dropped from 10 to 20 m, the occurrence probability of small deformation changed from 0.1964 to 0.1180, and that of large deformation changed from 0.1582 to 0.3081. For a given variable, increasing the level decreased the occurrence probability of the small deformation and increased that of the large deformation. These results coincide with the situation at the DLX landslide.

Table 5. Risk thresholds of variables.

Variables	VaR and Corresponding Displacement Increments
Monthly reservoir water level drop	15.52 m (3.26 mm)
Monthly reservoir water level rise	19.64 m (2.84 mm)
Monthly groundwater level drop	6.33 m (2.59 mm)
Monthly groundwater level rise	12.94 m (2.28 mm)
Monthly precipitation	227.71 mm (2.07 mm)

presents the risk thresholds of the variables. The displacement increments corresponding to the VaR of the groundwater level and precipitation were generally smaller than those of the reservoir water level, which indicates that the landslide deformation had a stronger response to changes in the reservoir water than in the groundwater and precipitation. The displacement increment corresponding to the VaR also indicated that the landslide deformation had a stronger response to a reservoir water level drop than to a reservoir water rise. When the five variables exceeded their respective VaR, the risk of a large deformation increased. In such situations, it is necessary to strengthen the monitoring of various indicators of landslide deformation and take corresponding emergency measures to prevent the occurrence of landslide disasters.

4.4. Conditional Return Period

Table 6. Return period distribution of monthly displacement increment with other variables.

Variable Levels	Monthly Displacement Increment (mm)
	2	4	6	8
Monthly reservoir water level drops by 10 m	1.25	1.71	2.74	6.47
Monthly reservoir water level drops by 15 m	1.19	1.54	2.29	4.92
Monthly reservoir water level drops by 20 m	1.13	1.36	1.82	3.25
Monthly reservoir water level rises by 10 m	1.27	1.75	2.82	6.59
Monthly reservoir water level rises by 15 m	1.22	1.60	2.42	5.27
Monthly reservoir water level rises by 20 m	1.17	1.46	2.07	4.03
Monthly groundwater level drops by 5 m	1.28	1.77	2.87	6.73
Monthly groundwater level drops by 7.5 m	1.22	1.62	2.48	5.46
Monthly groundwater level drops by 10 m	1.18	1.49	2.14	4.28
Monthly groundwater level rises by 5 m	1.28	1.82	2.95	6.82
Monthly groundwater level rises by 7.5 m	1.25	1.68	2.62	5.80
Monthly groundwater level rises by 10 m	1.20	1.56	2.29	4.71
Monthly precipitation of 100 mm	1.29	1.80	2.87	6.46
Monthly precipitation of 150 mm	1.26	1.70	2.65	5.80
Monthly precipitation of 200 mm	1.22	1.60	2.38	4.92

presents the return period distribution of monthly displacement increments with other variables. Overall, the return period was longer for large deformations than for small and medium deformations. For a given variable, increasing the level decreased the return period. The return period for a large deformation was shorter with a reservoir water level drop than with an equivalent reservoir water level rise. When the monthly displacement increment was increased from 2 to 8 mm, the return periods of the reservoir water level dropping and rising 20 m increased to 2.12 and 2.86 months, respectively. This indicates that the DLX landslide can easily transition to large deformations when the reservoir water level drops.

Increasing the monthly groundwater drop shortened the return period and increased the risk of a large landslide deformation. A groundwater drop of 10 m had a shorter return period than a groundwater rise of 10 m. For a displacement increment of 8 mm, increasing the monthly precipitation from 100 mm to 200 mm decreased the return period from 6.46 months to 4.92 months. This indicates that a large deformation can easily occur at the DLX landslide with heavy precipitation.

5. Conclusions

In this study, we introduced a HAC framework integrated with PMLE to model the complex interdependencies in hydrodynamic landslide deformations. Based on the analytical results, the following conclusions are drawn:

(1)

Hydrodynamic landslide variables exhibit significant tail-dependent correlations, with a distinct asymmetry in their joint probability distributions. Specifically, the variables show pronounced upper-tail dependence and weaker lower-tail dependence. This reflects the complex nature of hydrodynamic landslide deformation, where extreme events (e.g., large displacement increments) are strongly associated with extreme changes in reservoir water levels, but lower-level fluctuations show less pronounced correlation.

(2)

The strongest pairwise correlation was identified between monthly displacement increments and monthly reservoir water level drops. This finding underscores the critical role of reservoir water level fluctuations in influencing landslide deformation, with large displacement increments closely linked to significant water level drawdowns. The results suggest that monitoring and managing the reservoir water levels is crucial for assessing and mitigating the landslide risks in the affected regions.

(3)

The HAC model demonstrated its ability to effectively quantify the deformation correlations at the DLX landslide, aligning closely with the field observations. The VaR metrics and conditional return period analyses offer valuable insights for optimizing the landslide monitoring schemes, formulating targeted prevention measures, and enhancing the risk management strategies. These findings provide a solid foundation for improving the early-warning systems and disaster mitigation efforts in reservoir-affected regions.