Intelligent Safety Monitoring of Reservoir Slopes: A Multi-Point Deformation Prediction Approach Considering Spatiotemporal Lag Effects

Abstract

Reservoir water level fluctuations and rainfall drive bank slope deformation, typically exhibiting spatiotemporal lags. Existing prediction models often fail to characterize these complex coupled relationships and rely on manual variable selection, causing information loss and reduced performance. To address these issues, a novel multi-point prediction model for reservoir bank slope deformation based on lag-aware clustering (LAC), referred to as LAC-MOGP, is proposed in this study. First, the Maximal Information Coefficient (MIC) quantifies the lag between influencing factors and deformation for objective factors screening. Next, an improved dynamic time warping (DTW) algorithm with a lag-difference-constrained matching window is combined with affinity propagation (AP) to cluster monitoring points based on asynchronous temporal correlations. Finally, a DTW-based similarity weighting scheme is embedded into a multi-output Gaussian Process (MOGP) kernel to refine covariance modeling and predict deformation within each cluster. Validated using observations from the Jinlongshan slope of the Ertan arch dam, the proposed model outperformed traditional methods in prediction accuracy and long-term stability. Achieving the lowest average root mean square error (2.677 mm) and average mean absolute error (2.325 mm), the LAC-MOGP model demonstrates significant effectiveness and practical applicability for reservoir slope deformation forecasting.

1. Introduction

Landslides and slope instabilities constitute major global geohazards. In hydraulic engineering contexts, dynamic loading induced by reservoir impoundment and rainfall serves as the primary trigger for bank slope destabilization. Historical catastrophic failures—such as those at the Vajont Dam in Italy [1], the Grand Coulee Dam in the United States [2], and various bank slopes within the Three Gorges Reservoir area in China [3,4,5]—highlight the strong correlation between water level fluctuations, intense rainfall, and subsequent slope failures [6,7]. Because surface deformation serves as the most direct macroscopic indicator of internal mechanical degradation, continuous displacement monitoring remains essential for evaluating reservoir bank stability [8,9].

Current displacement analysis frameworks broadly fall into model-driven and data-driven categories. Model-driven approaches, grounded in engineering geology and solid mechanics, offer high interpretability regarding complex physical processes such as seepage–stress coupling and nonlinear boundary responses [10,11,12,13,14,15,16]. Nevertheless, their heavy reliance on precise geological boundary conditions and initial geomechanical parameters restricts their adaptability in field scenarios characterized by high spatial heterogeneity and uncertainty.

Conversely, data-driven models bypass explicit physical assumptions to extract kinematic patterns directly from monitoring series, offering strong site adaptability. Although traditional statistical tools (e.g., ARIMA, Kalman filters) [17,18,19] struggle to capture nonlinear deformation behaviors, machine learning algorithms—including Support Vector Machines (SVMs) [20,21], Random Forests (RFs) [22], and Long Short-Term Memory (LSTM) networks [23,24]—provide robust nonlinear mapping. To further enhance accuracy, recent studies have increasingly employed hybrid frameworks, such as utilizing Variational Mode Decomposition (VMD) to separate displacement series into trend and periodic components before applying neural networks [25,26].

However, a fundamental challenge persists: the kinematic response of slopes to reservoir water levels and rainfall inherently exhibits a spatially heterogeneous temporal lag [27,28,29,30,31]. Although recent studies have quantified these lag times using grey relational or wavelet analyses [32,33,34], or captured inter-sensor spatial dependencies via graph neural networks and spatial clustering [35,36], current methodologies rarely integrate both the asynchronous temporal lags and the inter-point spatial covariance within a unified predictive architecture [37].

To address this limitation, this study develops a predictive framework that incorporates the spatiotemporal lag effects of hydrological triggers to enhance the accuracy and long-term stability of multi-point deformation modeling. The principal contributions are as follows:

(a)

A constrained lag-aware clustering method is proposed, which explicitly incorporates deformation response lags into similarity computation. The MIC is utilized to estimate the lag time of each monitoring point relative to water level fluctuations and rainfall. The maximum lag is then employed as the global constraint for DTW alignment paths, and the distance matrix is constructed for improved clustering.

(b)

A DTW-similarity-weighted MOGP modeling approach is developed. A weight matrix is constructed using DTW distances and embedded into the MOGP kernel function to account for heterogeneous deformation responses among monitoring points. This modification enhances the joint covariance structure of multiple outputs, enabling more accurate spatial modelling and inter-point information transfer.

(c)

The proposed LAC-MOGP model is applied to multi-point deformation prediction of reservoir bank slopes. Comparative experiments with traditional methods demonstrate that the proposed method yields higher predictive precision long-term stability, demonstrating its practical viability for field monitoring scenarios.

The remainder of this paper is organized as follows: Section 2 introduces the methodological framework, covering MIC-based correlation analysis, the enhanced DTW-AP clustering approach, and the modified MOGP prediction model. Section 3 outlines the study area and the monitoring dataset. Section 4 reports the prediction experiments and compares the proposed method with baseline models. Section 5 summarizes the main findings and highlights directions for future research.

2. Model Development

This section introduces the analytical architecture of the proposed LAC-MOGP framework. As outlined in the introduction, the core methodology sequentially encompasses: (1) extracting the temporal lag between slope deformation and hydrological triggers via the MIC; (2) partitioning the monitoring points using a constrained DTW-AP clustering algorithm; and (3) predicting multi-point displacements via a MOGP model, structurally enhanced by embedding inter-point similarities into its coregionalization covariance matrix.

2.1. MIC-Based Estimation of Temporal Lag

To quantify the temporal delays between slope deformation and hydrological triggers, the Maximal Information Coefficient (MIC) [38] is employed to extract complex nonlinear dependencies. By evaluating temporally shifted sequences of environmental factors against the displacement series, this approach identifies significant triggers and their optimal lag times for subsequent regression modeling.

For the temporal analysis, the lag interval step was set to 30 days. This monthly resolution aligns with regional monsoon rhythms and reservoir operational cycles. Furthermore, it effectively filters out the high-frequency stochastic noise inherent in daily GNSS measurements. Informed by historical geomechanical stability assessments of the study area, the maximum temporal search boundary was conservatively defined as 180 days for both cumulative rainfall and reservoir level fluctuations. These expanded search limits represent a two-to-threefold increase relative to historically observed empirical lags. This conservative configuration guarantees the extraction of prolonged delayed responses during the operational phase of the reservoir [28,32,33,34].

During feature screening, candidate factors yielding maximum MIC scores above 0.3 are retained. Statistically, the finite sample sizes of field monitoring data establish an inherent background noise floor. The empirical null distribution of the MIC typically ranges from 0.15 to 0.25 under these conditions [38,39]. To securely bypass this baseline, a rigid threshold of 0.3 is enforced. This specific value is widely adopted as an optimal boundary in recent landslide prediction studies [40,41,42]. Such a criterion rigorously filters out spurious background correlations while preserving genuine hydrological dependencies. Ultimately, this screening reduces the dimensionality of the input space and identifies the optimal temporal lag for each monitoring point.

2.2. Improved DTW-AP-Based Clustering for Monitoring Points

This section presents a spatial clustering strategy that accounts for temporal asynchrony among deformation monitoring points, enabling a structured understanding of the spatiotemporal response of slopes to variations in reservoir water levels and rainfall. The resulting clustering structure reflects the similarity of deformation patterns among monitoring points and provides a solid foundation for subsequent multi-point joint modelling.

2.2.1. Inter-Point Similarity Measurement Based on Improved DTW

DTW is a widely used technique for measuring the similarity between time series with differing lengths or misaligned temporal axes. Figure 1 illustrates a typical alignment path obtained through DTW in the context of slope deformation monitoring. In monitoring of slope deformation, the displacement time series often exhibit periodicity and autocorrelation while also being susceptible to noise and external disturbances. However, due to its unconstrained search space, the standard DTW algorithm may yield overly flexible alignments that pair segments lacking physical relevance. This can significantly reduce the interpretability and reliability of similarity assessments between monitoring points. A widely adopted improvement strategy is the Sakoe–Chiba band [43], which restricts the alignment path to a symmetric region around the main diagonal of the distance matrix. This constraint ensures that each point is matched only within a predefined temporal neighborhood, reducing extreme deviations in the alignment path. However, the bandwidth is often empirically set as 10% of the sequence length, lacking a clear connection to the causal characteristics of the slope deformation.

To address this issue, a physically informed enhancement to the DTW constraint mechanism is proposed based on prior analysis of lagged deformation responses. As described in Section 2.1, MIC is employed to estimate the maximum lag DT for each monitoring point concerning external environmental factors. It is assumed that the lag difference between any two points does not exceed DT, and DT is adopted as the temporal window width to constrain the DTW alignment path. In this way, sequence matching is restricted to the maximum potential physical delay range while excluding geologically unrealistic matches. Compared to traditional empirically defined constraints, the proposed method enables improved interpretability in inter-point similarity measurement, thereby providing a more robust foundation for clustering and multi-point modelling.

2.2.2. AP-Based Clustering of Slope Monitoring Points

Slope deformation is typically characterized by spatial and temporal correlations, with the deformation trends of monitoring points reflecting latent similarities across regions. The AP clustering algorithm [44] is introduced in this study to enable the effective identification of representative deformation patterns and facilitate data-driven zoning. Representative points are automatically identified by iteratively updating two sets of messages, namely responsibility and availability, and clustering of slope monitoring points is subsequently achieved (Figure 2).

A key modification in this study is the integration of the DTW distance matrix (D) into the similarity matrix (S). Specifically, the similarity $s(i,k)$ between monitoring points i and k is defined as the negative DTW distance ($s(i,k)=-d_{i,k}$), which directly influences the responsibility update rule:

$$r(i,k)=s(i,k)-{\max }_{k^{\prime }\ne k}\left\{a\left(i,k^{\prime }\right)+s\left(i,k^{\prime }\right)\right\}$$

(1)

where $r(i,k)$ indicates the relative preference of point i for choosing k as its cluster center; $s(i,k)$ represents the similarity between point i and point k, which is defined as the negative DTW distance for monitoring sequences. This formulation ensures that clusters are formed based on lag-aware temporal alignments rather than rigid Euclidean proximity. $a\left(i,k^{\prime }\right)$ represents the current availability of candidate k′; $s(i,i)$ denotes the self-similarity or “preference” (p) of point i. The preference parameter p governs the granularity of the resulting partitions; higher p values generally yield a greater number of clusters. To ensure model objectivity and avoid arbitrary manual bias, p is assigned as the median of the similarity matrix $S=[s_{ij}]$—a robust initialization strategy that allows the algorithm to automatically determine the optimal cluster count. The stability of this configuration is further validated via a sensitivity analysis in Section 4.1.2.

2.3. Overview and Optimization of MOGP

In reservoir bank monitoring, observation sequences typically exhibit spatial coherence and shared physical drivers. MOGP [45] effectively exploits these cross-correlations to enhance predictive generalization. However, despite the preliminary partitioning provided by DTW-AP clustering, intra-cluster heterogeneity persists, as individual monitoring points contribute unequally to the underlying regional deformation signal.

Traditional MOGP based on the Linear Model of Coregionalization (LMC) often assumes a uniform coupling strength across outputs, which can lead to interference from weakly correlated sensors and degrade precision at more relevant locations. To mitigate this, we propose a covariance-refinement mechanism that embeds DTW-based similarity weights into the MOGP kernel. This approach enables a more refined description of inter-point relationships by adjusting the covariance strength based on the temporal similarity of deformation sequences.

For brevity, the standard mathematical derivations of MOGP, including the definition of the coregionalization matrix $B$ and the input kernel $k$, are detailed in Appendix A. The primary contribution of this study lies in the structural optimization of B via a spatiotemporal weight matrix W. Following the Squared Exponential (RBF) paradigm, the elements of $W=[w_{ij}]$ are defined as:

$$w_{ij}^{(\mathrm{DTW})}=\exp \left(-{\displaystyle \frac{d_{ij}^2}{\gamma }}\right)$$

(2)

where $d_{ij}^2$ is the DTW distance between the deformation sequences of points i and j, and $\gamma$ is a scaling parameter set as the mean of all pairwise DTW distances. This RBF-based weight formulation is theoretically motivated by two requirements: first, it provides a smooth, monotonic mapping from DTW distances to the similarity space; second, it ensures that the resulting Hadamard product within the covariance structure maintains strict positive definiteness—a prerequisite for robust Gaussian Process inference. Compared to alternative linear or step-function mappings, the RBF paradigm better characterizes the non-linear decay of spatial correlations while preserving the mathematical stability of the multi-output framework.

To modulate the covariance structure, the similarity weights are explicitly integrated into the standard coregionalization matrix B via the Hadamard product (element-wise multiplication, $\odot$). The revised coregionalization matrix $B^{\ast }$ is formulated as:

$$B^{\ast }=B\odot W$$

(3)

According to the Schur Product Theorem, the Hadamard product of two positive semi-definite matrices remains positive semi-definite. The full covariance matrix $K$ of the revised MOGP model is constructed using the Kronecker product (denoted by $\otimes$). This operation integrates the revised coregionalization matrix $B^{\ast }$ with the input kernel matrix $K_x$:

$$K=B^{\ast }\otimes K_x$$

(4)

This mathematical integration explicitly modulates the information-sharing strength based on actual kinematic similarities while strictly preserving the mathematical validity required for MOGP convergence.

2.4. Construction Framework of the LAC-MOGP Model

LAC-MOGP model is proposed for predicting multi-point slope deformation based on monitoring data. The overall framework is illustrated in Figure 3, and the implementation steps are described as follows:

(1)

Determination of Lag Time. MIC is calculated between each candidate environmental factor and the deformation response at each monitoring point. Factors with MIC values exceeding 0.3 are selected as explanatory factors. The maximum lag among the selected factors is employed to define the unified lag time window.

(2)

Computation of Inter-Point Similarity. The improved DTW algorithm is employed to quantify the temporal similarity between all pairs of monitoring points.

(3)

Clustering of Monitoring Points. AP algorithm is applied to perform unsupervised clustering on deformation points, which enables the classification of monitoring points into sub-regions with similar spatiotemporal responses.

(4)

Training of the Modified MOGP Prediction Model. The data set is divided into training and testing subsets. Based on the X and Y, the MOGP model with DTW-based weighted covariance is trained within each cluster.

(5)

Model Validation and Evaluation. The proposed model is validated through predictive experiments on multi-point slope displacement data and compared with representative methods. Evaluation metrics include the coefficient of determination (R²), root mean squared error (RMSE), and additional metrics such as the averaged MAE (aMAE), averaged MSE (aMSE), and averaged RMSE (aRMSE), to assess both fitting and forecasting performance.

3. Case Study

This section briefly introduces the selected real-world reservoir bank slope project, the dataset, and an overview of the associated environmental variables.

3.1. Study Area

The Jinglongshan reservoir bank slope, situated along the middle reaches of the Yalong River in Sichuan Province, China, was selected as the case study area. The mountain’s macro-topography is relatively gentle, with significantly weathered surface rock layers forming a stepped platform and mild slope landforms. The terrain is relatively gentle below an elevation of 1400 m, with an average slope of approximately 25°. Above 1400 m, however, the slope becomes markedly steeper, ranging from 35° to 45°. The Yalong River flows past the slope toe in the S60°E direction, and the slope exhibits a typical dip-slope structure. Notable deformation has been observed between 980 m and 1400 m, where the slope body shows poor structural integrity. As shown in Figure 4, the slope lies near the Ertan Dam, with a horizontal distance ranging from 600 m to 1300 m. The slope is massive, with a longitudinal extent of approximately 700 m along the river, a relative height between 500 m and 700 m, and a surface area of roughly 7.0 × 10⁵ m².

3.2. Dataset

The original GNSS monitoring network consisted of 18 observation stations, designated as Jiao 1 through Jiao 18. Points Jiao 1 to Jiao 4 were located at the lowest elevations and served primarily to monitor the slope stability during the construction and initial impoundment phases. Following the reservoir impoundment, these four points were submerged beneath the normal operational water level. Consequently, the remaining 14 points (Jiao5–Jiao18), which offer continuous time-series data and comprehensive spatial coverage of the unsubmerged slope, were selected to analyze the long-term operational deformation characteristics. The spatial distribution of these monitoring sites is shown in Figure 4, where red markers denote GNSS points, and the black contour lines indicate the topographic elevations. The elevation of each measuring point is listed in

Table 1. Elevation data of selected GNSS monitoring points.

Monitoring Point	Elevation (m)	Monitoring Point	Elevation (m)	Monitoring Point	Elevation (m)
Jiao5	1261.86	Jiao10	1436.68	Jiao15	1274.41
Jiao6	1232.14	Jiao11	1698.21	Jiao16	1372.4
Jiao7	1293.06	Jiao12	1458.8	Jiao17	1537.81
Jiao8	1342.2	Jiao13	1231.22	Jiao18	1595.08
Jiao9	1326.91	Jiao14	1330.09

.

To better understand the environmental influences on slope movement, reservoir water level and daily rainfall data also collected, from March 1998 to December 2019. These environmental variables, along with the displacement time series data of each monitoring point, are presented in Figure 5 and Figure 6, respectively. The displacement history indicates that significant slope deformation began shortly after reservoir impoundment commenced on 1 May 1998. This was especially pronounced during the initial impoundment stages in 1998 and 1999, during which several monitoring points recorded abrupt increases in both displacement magnitude and rate. Although the rate of deformation has gradually decreased since 1999, it has not ceased entirely. The slope is currently undergoing steady-state creep, suggesting that potential movement persists.

4. Results and Discussion

4.1. Spatiotemporal Analysis of Monitoring Points Considering Lag Effects

4.1.1. Screening of Explanatory Factors and Estimation of Lag Time

To account for the delayed response of slope deformation to reservoir water level fluctuations and rainfall, the MIC between each candidate historical environmental factor and the displacement data are calculated. The parameters are set as follows: grid resolution α is 0.6, and the minimum bin width c is 15. These values are adopted directly from the optimal heuristic baseline established in the original MIC framework [39] to ensure robust dependency detection without arbitrary manual tuning. The candidate factors of model are expressed in Equation (5). In the equation, H₀ represents the reservoir water level on the observation day, while H_p–q denotes the average reservoir level from p to q days before the observation date. Similarly, R₀ is the rainfall on the observation day, and R_p–q denotes the average daily rainfall from p to q days prior to the observation. $\theta =t/100$, where t denotes the number of days from the starting date to the monitoring date. The linear term $\theta$ represents a constant trend, while the logarithmic term $\ln \theta$ is physically motivated to characterize the non-linear rheological creep of the rock mass, which progressively decelerates over time according to classical empirical creep laws [46].

$$\delta =\left[\begin{array}{l}\begin{array}{c}\left.{\delta }_H\right| H_0,H_{0–30},H_{30–60},H_{60–90},H_{90–120},H_{120–150},H_{150–180}\end{array}\hfill \\ \begin{array}{l}\left.{\delta }_R\right| R_0,R_{0–30},R_{30–60},R_{60–90},R_{90–120},R_{120–150},R_{150–180}\hfill \end{array}\hfill \\ \begin{array}{c}\left.{\delta }_{\theta }\right| \theta ,\ln \theta \end{array}\hfill \end{array}\right]$$

(5)

Candidate factors with C_MIC values greater than 0.3 were selected as explanatory factors and incorporated into the predictive model. Taking Jiao5 as an example, the MIC analysis results are presented in

Table 2. MIC correlation analysis between candidate factors and displacement at Jiao5.

Number	Candidate Factor	CMIC	Is the Correlation Significant	Number	Candidate Factor	CMIC	Is the Correlation Significant
1	H0	0.305	Yes	9	R0–30	0.302	Yes
2	H0–30	0.419	Yes	10	R30–60	0.265	No
3	H30–60	0.536	Yes	11	R60–90	0.286	No
4	H60–90	0.444	Yes	12	R90–120	0.216	No
5	H90–120	0.313	Yes	13	R120–150	0.223	No
6	H120–150	0.246	No	14	R150–180	0.196	No
7	H150–180	0.252	No	15	θ	0.912	Yes
8	R0	0.307	Yes	16	lnθ	0.978	Yes

.

According to Table 2, the earliest explanatory factors identified are H_90–120 and R_0–30, indicating that for Jiao5, a lag time of 120 days for reservoir water level and 30 days for rainfall should be considered. The same MIC-based correlation analysis was applied to the remaining monitoring points. The lag times corresponding to the majority of monitoring points within each cluster were integrated. The final lag time for different clusters are summarized in

Table 3. The lag time estimation results for different points and clusters.

Cluster	Monitoring Points	Lag Time of Water Level (d)	Lag Time of Rainfall (d)	Integrated Lag Time of Water Level (d)	Integrated Lag Time of Rainfall (d)
Cluster 1	Jiao 5	120	30	120	30
	Jiao 9	90	30
	Jiao 13	120	60
	Jiao 16	90	30
Cluster 2	Jiao 8	120	30	90	30
	Jiao 10	90	30
	Jiao 14	90	30
	Jiao 15	90	30
Cluster 3	Jiao 7	120	30	120	30
	Jiao 11	90	30
	Jiao 12	90	30
	Jiao 17	120	30
	Jiao 18	120	60
Cluster 4	Jiao 6	120	30	120	30

. It can be seen that the maximum lag time did not exceed 120 days. Consequently, the constraint on the DTW matching window was set to 120 days.

The correlation between reservoir fluctuations and surface displacement was systematically evaluated across the entire monitoring network. While Table 3 provides a quantitative summary of the lag times for all 14 points, Jiao 9 is selected for detailed graphical analysis to provide an intuitive visualization of these temporal dependencies. This point is located at a mid-elevation on the left flank of the slope, and its cumulative deformation magnitude aligns with the mean macroscopic trend observed across the site. Due to the long span of the full monitoring sequences, the resulting data density and overlapping matching paths made visualization difficult. Data from March 1998 to March 2006 were selected for analysis, as shown in Figure 7.

Both the initial impoundment period and the corresponding deformation response period are highlighted. Figure 7 reveals a significant consistency between the displacement at Jiao9 and reservoir water level fluctuations, though with a temporal lag of 60–90 days. A distinct pattern of delayed deformation response is evident during the initial impoundment in May 1998. The slope began to respond approximately 90 days after the start of impoundment. This response relationship is well captured by the DTW matching path, demonstrating the effectiveness of the proposed DTW improvement method.

To further validate the effectiveness of the proposed temporal constraints, Jiao 11 and Jiao 18 were selected as a representative comparative pair. Both points are situated in the same spatial zone (Cluster 3) and share highly similar macroscopic deformation trends. This spatial proximity provides an optimal controlled baseline to evaluate the performance of different matching algorithms. Their respective constrained and unconstrained alignment paths are presented in Figure 8 and Figure 9. According to Figure 8, it is evident that although the variation patterns of the two points are quite similar, the displacement response at Jiao11 consistently leads that of Jiao18 by approximately 30 days. In contrast, Figure 9 illustrates the DTW matching path generated by the traditional DTW method. The maximum matching offset exceeds 420 days, particularly before March 2004. This exceeds the maximum lag time of 120 days derived from water level analysis and is inconsistent with empirical knowledge from previous studies [3,34,35,36]. Such discrepancies suggest that unrestricted DTW, which relies solely on the mathematical similarity of displacement sequences, fails to account for the physical interpretation of lag effects.

Ultimately, the temporal asynchrony observed between Jiao11 and Jiao18 is not an isolated phenomenon. Corroborated by the comprehensive data in Table 3, relative spatiotemporal lags are widely prevalent across the entire monitoring network due to local geological and topographical heterogeneities. These universal asynchronous behaviors fundamentally underscore the necessity of employing the proposed DTW method for global lag feature extraction prior to multi-point predictive modeling.

4.1.2. Monitoring Point Zoning

The pairwise DTW distances between all monitoring points were calculated by the improved DTW algorithm, and got the distance matrix D and the corresponding similarity matrix S. A heatmap of the matrix S is presented in Figure 10. As shown, Jiao6 exhibits the lowest similarity with all other points. When combined with the displacement data presented in Figure 6, it becomes evident that Jiao6 recorded the largest overall displacement and the most pronounced increasing trend among all monitoring points.

Next, the improved DTW-AP algorithm was applied to perform clustering. To further visualize the inter-point relationships, the cluster results among monitoring points were dimensionally reduced to two dimensions. The final results are visualized using Multidimensional Scaling (MDS) in Figure 11, which intuitively illustrates the spatial distribution of similarity among monitoring points and the resulting clustering structure. As illustrated in Figure 11, the 14 monitoring points were classified into four clusters, each occupying a distinct spatial region in the MDS plot. Cluster 1 comprises Jiao5, Jiao9, Jiao13, and Jiao16; Cluster 2 includes Jiao8, Jiao10, Jiao14, and Jiao15; Cluster 3 consists of Jiao7, Jiao11, Jiao12, Jiao17, and Jiao18; while Cluster 4 contains only Jiao6. It is clear from the plot that Jiao6 is positioned farthest from all other points, which supports and visually reinforces the earlier conclusion based on Figure 10. It should be noted that points within the same cluster are not necessarily grouped in the visually closest manner. Instead, they are often distributed in diagonally banded area. This phenomenon may be attributed to the dimensionality reduction inherent in the MDS plot, which approximates the original high-dimensional similarity structure. As a result, the two-dimensional distances can only partially reflect the true DTW-based similarities.

Figure 12 presents the spatial distribution of the four monitoring point clusters, overlaid on the slope topography. Cluster 1, Cluster 2, and Cluster 3 are spatially separated and located in the lower-left, central, and upper regions of the slope, respectively, while Cluster 4 is situated closest to the slope toe. The spatial patterns reflect the combined influences of elevation, geological conditions, and anti-slide engineering measures implemented during the construction period on the slope deformation behavior: (a) Cluster 3 points are primarily located at higher elevations, where the influence of reservoir water level fluctuations is minimal, resulting in relatively small surface displacements. (b) Clusters 1 and 2 are distributed across mid-elevation zones but exhibit notable geological differences: Cluster 1 is situated in ancient landslide zones characterized by soil slopes, while Cluster 2 falls within rock creep zones dominated by mixed soil-rock slopes. (c) Cluster 4 comprises only Jiao 6, which is closest to the slope toe and experiences the largest cumulative displacement due to pronounced reservoir-induced effects. Although this point is currently in a stable deformation phase, it remains a priority for future monitoring. In contrast, Jiao 5, which is also near the slope toe, displays significantly smaller displacement as a result of toe-weighting stabilization measures implemented before reservoir impoundment. Consequently, Jiao 5 was grouped into Cluster 1 instead of Cluster 4. These findings further validate the physical plausibility and engineering relevance of the clustering results.

In order to validate the effectiveness of the clustering results, MIC was adopted to evaluate the nonlinear and asynchronous similarity between each pair of monitoring point sequences. The pairwise MIC similarity values were visualized by heatmap, as shown in Figure 13. The monitoring points along axes are ordered according to their assigned cluster labels. It allows points within the same cluster to be displayed in contiguous blocks. From Figure 13, it is apparent that the intra-cluster regions generally exhibit warmer colors, indicating higher similarity. Conversely, inter-cluster regions located outside the dashed boxes are mostly represented in cooler colors, indicating lower correlation. The computed average MIC values for each cluster are 0.925 for Cluster 1, 0.918 for Cluster 2, 0.930 for Cluster 3, and 0.978 for Cluster 4. These intra-cluster averages are all higher than the overall average MIC value computed across all monitoring point pairs. This result reveals that the proposed clustering method captures both spatial and temporal correlation patterns, thus verifying its validity and robustness.

To validate the choice of the AP clustering preference parameter, a sensitivity analysis was conducted. This parameter strictly governs the resultant cluster count. The default median initialization (50th percentile of the similarity matrix) was benchmarked against the 25th and 75th percentiles. Assigning the preference to the 25th percentile aggregated the network into only three clusters. Conversely, the 75th percentile partitioned the monitoring points into six isolated groups.

As illustrated in Figure 14, when evaluated using the proposed LAC-MOGP model, the median preference yields the lowest average predictive RMSE across all monitoring points. This quantitative result confirms that the 50th percentile optimally balances the grouping of kinematically similar points. It effectively preserves the spatial information-sharing capacity of the MOGP framework while explicitly preventing both under-clustering and over-partitioning.

4.2. Prediction Using LAC-MOGP Model

The proposed LAC-MOGP model was employed for the displacement prediction of monitoring points within each cluster. The monitoring dataset is divided with the first 90% of the sequence for training, corresponding to data prior to 1 November 2018. And the remaining 10%, after that date, is used for testing.

The prediction model is first trained with the training set and then validated with the testing set. Figure 15 presents a comparative result of the monitored and predicted displacement sequences for each monitoring point. The segment of the sequence corresponding to the test set is highlighted in blue shading. It is evident that the LAC-MOGP model effectively captures the temporal evolution trends of displacement at each monitoring point, demonstrating commendable fitting accuracy.

The modeling results for each monitoring point are summarized in

Table 4. Modeling accuracy of the proposed LAC-MOGP model at each monitoring point.

Monitoring Point	R2	RMSE (mm)	Monitoring Point	R2	RMSE (mm)
Jiao 5	0.996	4.526	Jiao 12	0.994	3.133
Jiao 6	0.993	10.562	Jiao 13	0.996	3.885
Jiao 7	0.993	3.522	Jiao 14	0.998	2.793
Jiao 8	0.996	3.162	Jiao 15	0.997	3.022
Jiao 9	0.993	5.237	Jiao 16	0.993	4.585
Jiao 10	0.998	1.600	Jiao 17	0.994	2.807
Jiao 11	0.992	3.728	Jiao 18	0.990	4.969

. It is evident that the R² for all points exceed 0.99, indicating outstanding overall model performance. Although the RMSE for Jiao 6 reached 10.562 mm, this is attributed to the fact that Jiao 6 recorded the largest measured displacement of 642.694 mm, which is more than twice the maximum displacement recorded at other points. For all other monitoring points, the RMSE values remained below 10 mm. Notably, the RMSE-to-maximum-displacement ratios for all points are below 2%, further confirming the high prediction accuracy of the proposed model.

While the reported performance metrics indicate strong predictive capability, these values should also be interpreted within the specific kinematic context of the studied slope. Certain landslides exhibit step-like or fluctuation-dominated behaviors. In these cases, the displacement variance is primarily driven by complex, episodic responses to rainfall or reservoir variations, rather than being anchored by a dominant, continuous monotonic baseline [40]. In contrast, the deformation in this study area is continuously shaped by a logarithmically decelerating rheological adjustment following the initial reservoir impoundment. From a statistical perspective, this massive multi-year cumulative trend a substantial portion of the total variance of the displacement series, which elevates the baseline of the coefficient of determination (R²). Consequently, the exceptionally high R² values also reflect this physically smooth, reservoir-induced kinematic adjustment.

4.3. Performance Comparison of LAC-MOGP with Baseline Models

To evaluate the performance of the proposed model, a comparative analysis was conducted with several classical methods, including Single-Output Gaussian Process (SOGP), BPNN, Least Squares Support Vector Machine (LSSVM), and RF. To ensure a fair comparison, all models are constructed using the same explanatory factors and monitoring sequences.

(1)

Performance comparison within the same conditions

Taking monitoring points from Cluster 2 as example, which is located in the central region of the slope. Their displacement series are fitted and predicted using LAC-MOGP and other baseline models under the same conditions. The hyperparameters for these models were determined through standardized optimization procedures tailored to each model’s architecture. Specifically, for the BPNN, RF, and LSSVM models, a systematic grid search combined with 5-fold cross-validation was employed on the training set to identify the optimal hyperparameter combinations. For the SOGP and MOGP based models, the hyperparameters were optimized using Maximum Marginal Likelihood Estimation (MLE). The optimal model hyperparameters are determined as follows: The SOGP model kernel is RBF function, the kernel width is 100 and the sparse point limit is 29. For the BPNN, the layers are 2, with 32 and 24 neurons in the first and second layer, respectively. The epochs are 500, and the learning rate is 0.01. For the LSSVM, the penalty coefficient C is 10, and the kernel function parameter σ is 0.05. For RF model, the number of decision trees is 100, with the maximum depth of 25, and the leaf node sample size of 19. Notably, these baseline models are constructed individually for each point, whereas the proposed model adopts a multi-output approach. All models are trained with the first 90% of the monitoring data as the training set, and the remaining 10% used for validation.

Figure 16 presents the comparative fitting and prediction results for Jiao 8, Jiao 10, Jiao 14, and Jiao 15 within Cluster 2. The left column in each subfigure shows the overall modeling performance, while the right column highlights the detailed prediction performance in the testing set. From Figure 17, several observations can be made as follows: (a) All models achieved satisfactory fitting of the overall displacement trend and fluctuation pattern in the training set. (b) The accuracy of long-term predictions generally degraded compared to short-term predictions, with varying degrees of performance across monitoring points. (c) For the four monitored points during the prediction phase, only the proposed LAC-MOGP model can capture the observed fluctuations to a certain extent. Other models failed to correctly capture the local dynamic features of the monitoring sequences. Overall, the proposed LAC-MOGP model demonstrated superior predictive capability across all evaluation criteria.

The evaluation metrics are calculated for each point in Cluster 2. The detailed results are summarized in

Table 5. Prediction metrics of five models for monitoring points in Cluster 2.

Monitoring Point	Metrics	LAC-MOGP	SOGP	BPNN	LSSVM	RF	Rel. Reduction 1
Jiao 8	MAE	2.476	4.651	3.982	3.784	4.495	34.57%
	MSE	7.751	24.269	36.393	39.829	22.029	64.81%
	RMSE	2.784	4.974	6.033	6.311	4.694	40.69%
Jiao 10	MAE	1.769	4.518	4.864	9.810	7.007	60.85%
	MSE	4.433	29.791	35.310	120.654	55.158	85.12%
	RMSE	2.106	5.472	5.942	10.984	7.427	61.51%
Jiao 14	MAE	3.465	6.490	4.620	10.497	7.736	25.00%
	MSE	14.610	50.643	31.549	134.941	71.626	53.69%
	RMSE	3.822	6.775	5.617	11.616	8.463	31.96%
Jiao 15	MAE	1.592	5.658	6.562	7.175	6.161	71.86%
	MSE	3.989	37.948	65.958	68.184	43.606	89.49%
	RMSE	1.997	5.609	8.121	8.257	6.603	64.40%

Notes: ¹ “Rel. Reduction” represents the percentage decrease in the respective error metric achieved by LAC-MOGP compared to the best-performing baseline model for that specific monitoring point.

, and the comparative performance is visualized by radar charts, as shown in Figure 17. From the results, the following analysis can be made:

(a)

The prediction metrics obtained by SOGP, BPNN, and RF are generally close in most cases, indicating comparable predictive capabilities among these classic single-point models. Although slight fluctuations exist across different monitoring points, they effectively capture the general deformation trend.

(b)

Among the baseline models, LSSVM exhibited the weakest predictive performance. Its overall error metrics were noticeably higher than those of the alternative methods. While LSSVM often serves as an effective standard regression tool, the empirical results indicate a limited capacity to map the complex, non-stationary local dynamics of this specific GNSS time series. Consequently, its generalization capability proved inferior to the ensemble mechanism of RF and the probabilistic architectures of SOGP and the proposed model.

(c)

The modified LAC-MOGP model consistently outperformed all baseline models across all monitoring points. As quantified in Table 5, LAC-MOGP achieved substantial error reductions compared to the best-performing baselines, with the RMSE decreasing by 31.96–64.40%. This enhanced accuracy is attributed to two primary mechanisms: First, the introduction of DTW-based lag estimation and AP-based clustering enables the model to effectively characterize the asynchronous temporal response and spatial similarity of the slope. Second, unlike the isolated learning of baseline models, the MOGP framework leverages the covariance structure among monitoring sequences. This facilitates the identification of inter-point dependencies and effectively filters out local noise by borrowing information from neighboring points, thereby significantly enhancing both prediction accuracy and long-term stability.

(2)

Performance comparison with different prediction lengths

To further assess the stability and robustness of the five models, their performance was compared across test sets of different lengths. Using the monitoring points in Cluster 2 as an example, test set lengths are set as 5% and 15% of the full dataset, respectively. The detailed results are presented in

Table 6. Prediction accuracy metrics for five models with different prediction lengths.

Indicators		LAC-MOGP	SOGP	BPNN	LSSVM	RF
5%	aMAE	1.390	2.293	3.170	4.691	5.417
	aMSE	3.675	9.573	17.960	43.869	34.927
	aRMSE	1.668	3.268	3.973	5.538	5.800
10%	aMAE	2.325	5.329	5.007	7.817	6.350
	aMSE	7.696	35.663	42.302	90.902	48.105
	aRMSE	2.677	5.708	6.428	9.292	6.797
15%	aMAE	3.194	5.844	6.713	10.791	7.216
	aMSE	11.430	58.192	64.906	134.575	60.341
	aRMSE	3.314	6.067	7.849	11.451	7.601

, and the trends corresponding to different prediction lengths are visualized in Figure 18.

As shown in Table 6 and Figure 18, the performance metrics, including aMAE, aMSE, and aRMSE, for the five models varied within the ranges of 1.390 to 10.791, 3.675 to134.575, and 1.668 to 11.451, respectively. It is evident that all accuracy metrics increased as the prediction lengths extended, indicating a general degradation in predictive performance with longer forecast intervals. Among the models, the proposed approach consistently achieved the lowest values across all metrics and prediction lengths, demonstrating superior robustness. Notably, the degradation in its performance was relatively limited when the forecast interval increased from 5% to 15%, confirming its strong generalization ability and adaptability to long-term predictions.

4.4. Computational Cost and Scalability

To evaluate the practical applicability of the proposed method, the computational cost was analyzed using a personal laptop (Intel Core i7-13650HX CPU, 16 GB RAM) running MATLAB 2016a. To ensure a fair comparison, all baseline models utilized the same optimally lagged input variables identified by the MIC analysis. Therefore, the global MIC preprocessing time is excluded from the model execution metrics.

The execution times for training and predicting a single monitoring point were systematically recorded and averaged. Specifically, the average execution times per single monitoring point are 5.035 s for BPNN, 0.627 s for RF, 0.048 s for LSSVM, and 4.868 s for the standard SOGP. In comparison, the proposed LAC-MOGP framework requires 26.783 s per point. This duration encompasses both the DTW-AP clustering overhead and the MOGP training process. Although the matrix inversion operations within the spatial covariance structure introduce an additional computational burden relative to isolated point models, the total execution time remains strictly on the scale of seconds. Considering that the sampling frequency of real-world reservoir slope monitoring typically does not exceed once per day, this computational cost is highly efficient and well within the acceptable limits for real-time early warning applications.

Regarding scalability, a standard global MOGP typically suffers from cubic computational complexity, denoted as $\mathcal{O}(M^3{N}^3)$, where M is the number of sensors and N is the sequence length. The proposed framework structurally mitigates this limitation through the DTW-AP clustering mechanism, which acts as a dimensionality reduction strategy. By partitioning the M monitoring points into K independent sub-regions (with $m_k$ points in each cluster), the overall complexity is reduced to ${\displaystyle \sum _{k=1}^K\mathcal{O}}(m_k^3{N}^3)$. This divide-and-conquer architecture significantly lowers the memory requirements and ensures the model’s scalability for large-scale monitoring networks.

5. Conclusions

A novel LAC-MOGP model is proposed to address the challenges in predicting spatially heterogeneous and temporally asynchronous deformation patterns in reservoir slopes. The accuracy, robustness, and reasonability of LAC-MOGP are demonstrated by a reservoir bank slope. The main conclusions can be summarized as follows:

(1)

A systematic approach was established for identifying the delayed response between reservoir water level variations and slope deformation. Effective explanatory factors and lag times are estimated for each monitoring point by MIC. It enabled the model to capture essential time-dependent influence patterns and define a meaningful DTW alignment window.

(2)

An improved DTW-based similarity measure is introduced and incorporated into AP clustering framework to partition the monitoring points. The clustering results revealed clear regional deformation patterns. The effectiveness and physical interpretability of the clustering scheme are confirmed by MIC similarity.

(3)

A covariance-enhanced MOGP regression model is developed by embedding DTW-based similarity weights into the LMC kernel structure. The proposed LAC-MOGP model demonstrated superior prediction performance compared to classic models, the lowest aRMSE of 2.677 mm and the lowest aMAE of 2.325 mm. Meanwhile, it maintains the favorable accuracy across different prediction lengths.

While the proposed prediction model demonstrates excellent prediction performance, several areas warrant further improvement. Future work will focus on the following directions: (1) The model framework will be extended to incorporate more influencing factors such as temperature, wet/dry cycle injury or parameters derived from remote sensing, improving model adaptability to complex slope environments. (2) Considering the evolving nature of slope deformation mechanisms, dynamic model updating strategies will be investigated to integrate newly acquired monitoring data for real-time prediction and risk assessment.