Fine Identification of Landslide Acceleration Phase Using Time Logarithm Prediction Method Based on Arc Synthetic Aperture Radar Monitoring Data

Abstract

In the field of slope landslide prevention and monitoring in open-pit mines, addressing the lag issues associated with the traditional GNSS inverse-velocity method, this study introduces a novel strategy that integrates high-spatiotemporal-resolution monitoring data from ArcSAR with a time log model for prediction. The key findings include the following: (1) This strategy utilizes the normal distribution characteristics of deformation velocities to set confidence intervals, accurately identifying the starting point of accelerated deformation. (2) Coupled with coordinate transformation, the time logarithm prediction method was constructed, unifying the units of measurement and resolving convergence issues in data fitting. (3) Empirical research conducted at the Kambove open-pit mine in the Democratic Republic of the Congo demonstrates that this method successfully predicts landslide times four hours in advance, with an error margin of only 0.18 h. This innovation offers robust technical support for slope landslide prevention and control in open-pit mines, enhancing safety standards and mitigating disaster losses.

1. Introduction

Landslide prediction is an indispensable part of geological disaster research, as it directly impacts public safety, ecological environments, and the stability of engineering facilities [1]. It can also cause settlements to be abandoned and lead to the ‘ghost towns’ phenomenon [2,3]. In recent years, frequent landslides occurring globally have resulted in severe casualties and immense economic losses. According to a 2016 study by Haque et al. [4], the annual economic losses due to landslides worldwide amount to approximately USD 19.8 billion, accounting for roughly 17% of the total annual losses from disasters globally. Drought-induced soil desiccation and structural weakening [5], coupled with slope destabilization from flood-induced saturation [6], collectively exacerbate the likelihood of landslides. This underscores the significance of research into landslide prediction. In particular, landslides occurring on the slopes of opencast mines can cause significant damage to mining operation areas and various engineering construction zones. Besides causing casualties, they also bring heavy economic losses to mining enterprises and impact their social image. For instance, on 22 February 2023, a particularly severe landslide accident occurred at an opencast coal mine in Alashan, Inner Mongolia, China, resulting in 53 fatalities, 6 injuries, and direct economic losses of USD 28 million. Landslide prediction faces numerous challenges, including the nonlinearity and uncertainty of slope deformation, the complexity of external factors, and the selection and deployment of monitoring equipment, as well as the choice of prediction models [7,8,9].

With the advancement of technology, more and more advanced techniques are being applied in the mining sector [10,11,12,13,14,15]. Ground-Based Synthetic Aperture Radar (GB-SAR) has surpassed traditional GNSS (Global Navigation Satellite System) monitoring, satellite remote sensing monitoring, and total station monitoring technologies as the preferred monitoring equipment for mine managers due to its advantages of long-term real-time online monitoring, wide coverage, and immunity to weather conditions such as rain, snow, fog, and hail [16,17]. GB-SAR can be classified into sliding-rail GB-SAR, arc GB-SAR, or MIMO GB-SAR based on its technological evolution [18]. Sliding-rail GB-SAR offers high spatial resolution but has limited monitoring range due to restrictions from the mobile platform and relatively low data acquisition efficiency. Arc GB-SAR boasts a wide observation range, short revisit cycles, and rapid response to surface changes. MIMO GB-SAR, however, has higher application costs due to its complex technology and process characteristics [19].

Landslide prediction theory has evolved from deterministic prediction models, such as the earliest empirical formula for landslide prediction proposed by Japanese scholar M. Satio [20] based on creep theory of landslides, to statistical prediction models, like the GM (gray model), Kalman filtering, and curve regression models, and finally to nonlinear prediction models such as BP (Back Propagation) neural networks and catastrophe theory prediction models [21,22]. To date, most research has focused on short-term prediction analysis using displacement monitoring data from the pre-critical accelerated deformation stage of landslides. By deeply mining and analyzing displacement monitoring data, corresponding physical–mechanical or mathematical models are established to understand the deformation characteristics of slopes in real-time, accurately assess the current safety status of slopes, and predict future deformation trends, thereby precisely estimating the specific time of instability and failure of rock and soil masses. Many scholars have dedicated themselves to research on landslide time prediction, leading to a series of observations and repeated experiments based on the strain or displacement variation over time. These empirically derived methods and equations often relate to the intrinsic properties of the rock mass itself [23].

The inverse velocity method (INV) [24,25], a classic approach to landslide prediction, originated in the early 20th century, and has been gradually developed and refined in subsequent research. This method, based on the three-stage model of slope creep theory, particularly focuses on the accelerated deformation stage before landslides, and uses the variation in deformation velocity over time to estimate the timing of landslides. Fukuzono [26] et al. conducted experimental research and derived various relationship patterns between the reciprocal of velocity and time, initially establishing the foundation for the application of the INV in landslide prediction. They believed that the reciprocal of the velocity–time relationship curve can be approximately fitted into a straight line, and the intersection of this line with the time axis represents the time of landslide instability. Barry Voight [27] conducted research on plastic-accelerated creep in soils, rocks, and other materials, concluding that under constant effective stress conditions, the reciprocal of velocity–time fitting curve is actually linear. However, the traditional INV has limitations, such as failing to consider that the velocity during actual landslides is not infinite, resulting in predicted landslide times that may lag behind actual landslide times. Due to the complex heterogeneous nature of rocks, the INV has limitations in certain aspects. It has good adaptability for landslides in accelerated creep states but poor adaptability for progressive and instantaneous landslides. The reason for this is that in practical applications, factors such as rainfall and external force disturbances can cause the reciprocal of the velocity–time fitting curve to exhibit concave or convex properties. Dick, G.J. [16], and Bozzano, F. [24], raised the issue of reselecting the Time of Start Point ($T_{SP}$) after the onset of acceleration (OOA) during the accelerated deformation stage.

Combining the characteristics of arc GB-SAR monitoring data, this paper proposes an improved INV method for landslide time prediction to enhance the accuracy of landslide time predictions. In response to the lag issues present in traditional inverse velocity methods, this paper proposes a time logarithm model and utilizes the normal distribution characteristics of deformation velocities to set confidence intervals, accurately identifying the onset of accelerated deformation. This approach enhances the accuracy of landslide disaster predictions.

2. Arc Synthetic Aperture Radar

ArcSAR (Arc Synthetic Aperture Radar) is a radar system that utilizes arcuate motion to achieve high-resolution imaging. Its working principle is based on the fundamental concept of Synthetic Aperture Radar (SAR), which simulates the effect of a large physical antenna aperture to obtain high-resolution ground images using a smaller actual antenna size. The uniqueness of ArcSAR lies in its arcuate path of motion, where the radar employs a long rotating boom to swing, enabling the low-gain antenna at the end of the boom to move along an arcuate trajectory. By synthesizing high-resolution imaging beams along this arcuate trajectory, ArcSAR is able to cover a large monitoring area in a short period of time, thereby enhancing monitoring efficiency and flexibility.

A picture of the field operation of the ArcSAR arcuate Synthetic Aperture Radar equipment is shown in Figure 1. The hardware performance parameters of the ArcSAR radar are shown in

Table 1. Parameters of Arc Synthetic Aperture Radar equipment.

Parameter Name	Parameter
spatial resolution reaching	ΔR 0.3 m × ΔAz 5 mrad
monitoring distance exceeding	5 km
deformation accuracy	0.1 mm
coverage area	360° × 30°
length, width, and height	1.33 m × 0.42 m × 0.63 m
scanning period	0.5 to 4 min (configurable)

.

The ArcSAR radar monitoring data display and interaction platform adopt a B/S (browser/Server) architecture, allowing users to access the application by simply opening a web browser and entering the website address. It is cross-platform compatible, meaning that users can access real-time monitoring data anytime, anywhere, using desktop computers, laptops, tablets, or mobile phones. This greatly enhances the convenience of using the radar product. The platform provides radar monitoring data scatter plots (intensity maps), which can be processed through interferometry to generate deformation maps. Users can quickly locate dangerous slope areas through the deformation maps.

The software platform supports map import functionality, allowing the import of three-dimensional topographic maps of opencast mine slopes captured by drone aerial photography into the radar monitoring platform for three-dimensional registration, as shown in Figure 2. Additionally, observation points can be manually set on the topographic map, and the system will automatically calculate displacement change data, velocity change data, acceleration change data, and reciprocal velocity change data for each observation point. Compared to GNSS monitoring technology, ArcSAR radar monitoring technology offers advantages such as non-contact, high sampling rate, high precision, and overall regional monitoring. While ArcSAR radar monitoring technology offers numerous advantages, it is important to note its limitations, particularly in terms of the high cost, data processing complexity, and high maintenance difficulty involved in using the equipment.

3. Method

3.1. The $\mathrm{\Delta }\mathrm{t}\text{-}\log (T)$ Logarithmic Model

Current research primarily focuses on utilizing displacement monitoring data from the accelerated deformation phase for short-term landslide prediction. The reciprocal velocity method is applied after the initiation of the accelerated deformation phase, assuming that the deformation velocity tends to infinity at the instant of slope failure. By performing a linear fit of the reciprocal of velocity versus time, a trend line can be obtained, and the intersection of this trend line with the time axis represents the predicted time of failure ($T_f$). Therefore, starting from the Time of Start Point ($T_{SP}$), the linear relationship between the reciprocal of velocity and time can be expressed using a formula. Fukuzono [26] provided a method for calculating the reciprocal of velocity, and Rose, N.D., and Hungr, O., [28] successfully applied this method to an open-pit mine in Nevada. The calculation formula is as follows (1).

$${\displaystyle \frac{1}{V}}=-A\left(T-\right.\left.T_{SP}\right)+B$$

(1)

where $V$ represents the deformation velocity, $T$ represents the time corresponding to different velocities during the landslide process, $T_{SP}$ represents the starting time of the acceleration phase of the slope, and $A$ and $B$ are constants of the fitted curve.

When ${\displaystyle \frac{1}{V}}$ = 0, the time $T$ at this point is the moment of slope failure.

Formula (1) can be further transformed into the velocity-solving Formula (2).

$$V={\displaystyle \frac{1}{-A\left(T-T_{SP}\right)+B}}$$

(2)

By integrating Equation (2), we obtain the solution Formula (3) for displacement S, where C represents the integration constant.

$$S={\displaystyle \frac{-\log \left[-A\left(T-T_{SP}\right)+B\right]}{A}}+C$$

(3)

Based on the three-stage theory of slope creep and sliding, the displacement–time curve exhibits a stage of uniform deformation prior to the accelerated deformation stage. During this stage, the deformation rate of the slope remains basically constant, and the average velocity of the landslide can be assumed as $\overline{\mathrm{v}}$. By applying the physical relationship of displacement, which is obtained by multiplying velocity by the relative time of motion, Equation $\mathrm{\Delta }\mathrm{t}={\displaystyle \frac{S}{\overline{\mathrm{v}}}}$ transforms into a logarithmic function relationship of time ($\mathrm{\Delta }\mathrm{t}\text{-}\log (T)$), as shown in Equation (4); we can derive a logarithmic function relationship between $\mathrm{\Delta }\mathrm{t}$ (relative time) and $T$ (absolute time). This transformation serves to unify the dimensional units of the horizontal and vertical axes of the analysis curve. The purpose of unifying the dimensional units of the coordinates on the analysis curve is to address the issue of convergence in the curve fitting of monitoring data, ensuring that the constants $A$ and $B$, and the $C$ value can be accurately calculated through curve fitting, as shown in Figure 3.

$$\mathrm{\Delta }\mathrm{t}={\displaystyle \frac{-\log \left[-A\left(T-T_{SP}\right)+B\right]}{A\overline{\mathrm{v}}}}+{\displaystyle \frac{C}{\overline{\mathrm{v}}}}$$

(4)

3.2. Selection of $T_{SP}$ for the Slope Deformation Acceleration

Carlà, T [23] and others proposed a method using the intersection of the short-term moving average (SMA) and long-term moving average (LMA) of velocity to calculate the onset of acceleration (OOA). The calculation methods for SMA and LMA values are shown in Equation (5). Based on experience [29], when n = 3, this represents the SMA value; when n = 7, this represents the LMA value. When the short-term average velocity curve forms a positive cross (i.e., the short-term average velocity curve rises above and crosses the long-term average velocity curve), it indicates that the slope has entered the accelerated deformation stage, and this intersection point is defined as the OOA. If there is a negative intersection between SMA and LMA, it indicates that the slope at that location begins to undergo decelerated deformation, accompanied by a Termination of Acceleration (TOA) point. This method enables rapid and automated identification of hazardous areas in slopes.

$$\overline{v}={\displaystyle \frac{v_t+v_{t-1}+\begin{array}{cc}\dots & \dots \end{array}+v_{t-n+1}}{n}}$$

(5)

Francesca Bozzano [24] and others believe that directly using the OOA point as the time of occurrence $T$ in the reciprocal velocity method can lead to low accuracy in landslide time prediction. They suggest that a new starting point should be selected after the OOA point, but they do not provide a specific calculation method.

During the second stage of uniform deformation in slope landslides, the measurement errors of the monitoring data follow a normal distribution. However, the measurement errors of the monitoring data during the acceleration stage do not follow a normal distribution [30,31]. Therefore, the normal distribution confidence interval can be considered for dynamically identifying the Time of Start Point ($T_{SP}$) for the slope deformation acceleration. When the deformation velocity falls outside the confidence interval of the normal distribution, it indicates that the deformation velocity begins to increase significantly. The monitoring data point determined at this time is taken as the $T_{SP}$.

The specific methods are as follows:

• Data Preparation and Processing: Collect continuous deformation velocity data from slope radar monitoring and perform statistical analysis at a certain time step (for example, recording the deformation velocity at regular intervals).
• Calculation of Normal Distribution Parameters: Calculate the expected value ($\mu$) and standard deviation ($\sigma$) of the monitoring data samples. These two parameters constitute the characteristics of the normal distribution. According to the normal distribution, approximately 68.27% of the data fall within the interval ($\mu -\sigma , \mu +\sigma$), and 95.45% of the data fall within the interval ($\mu -2\sigma , \mu +2\sigma$), and so on.
• Confidence Interval Testing: Set a confidence level (such as the commonly used 95%) and determine the width of the corresponding confidence interval (such as 1$\sigma$, 2$\sigma$, or 3$\sigma$). During the monitoring process, check whether the deformation velocity at each monitoring time point falls within this confidence interval.
• Identification of $T_{SP}$: When the deformation velocity points monitored consecutively for several times do not fall within a certain confidence interval of the normal distribution and the deformation velocity shows a clear upward trend, it is thought that the deformation velocity has increased significantly. At this time, the last monitoring data point that falls outside the confidence interval is identified as the $T_{SP}$ for the slope deformation acceleration.
• Dynamic Updating and Confirmation: As time passes, continuously update the monitoring data and repeat the above steps until a series of deformation velocity values are found that continuously exceed the start point of the normal distribution confidence interval, confirming the existence of $T_{SP}$.

3.3. Landslide Time Prediction

After determining the $T_{SP}$, the relationship between the cumulative displacement of the slope and time conforms to the logarithmic function in Equation (4). By applying the logarithmic function model to fit the data points, the parameters $A$, $B$, and $C$ can be obtained. With these parameters, the predicted landslide time point $T_f$ can be calculated, as shown in Figure 4.

The flowchart of the methodology and approach presented in this paper is shown in Figure 5.

4. Case Study

The Kambove open-pit mine [32] is situated on the Katanga Plateau, in the town of Kambove, Katanga Province, in the southeastern part of the Democratic Republic of the Congo (DRC), as shown in Figure 6. The mine extends approximately 2.5 km east to west and 1 km north to south. With an annual production capacity of 1 million tons of ore, the final designed slope boundary height reaches 220 m. The surface rock strata of the Kambove open-pit mine exhibit severe weathering and are characterized by fault distributions. During mining operations, slope failures occur frequently. Based on field investigations, these failures are primarily classified as bedding-plane landslides, which are significantly influenced by mining-induced disturbances and meteorological conditions [33].

The rock formations in the Kambove open-pit mine area consist of a sedimentary sequence composed of clastic rocks, argillaceous rocks, and carbonate rocks. This mining area has undergone multiple nappe tectonic movements, which have superimposed upon each other, resulting in the occurrence of exotic ore deposits within the region. During the period of overthrust tectonic activity, the edges of various rock fragments underwent displacement and compression, forming unconformable sliding surfaces. The lithology of the ore deposit stratigraphy is quite complex, with a significant depth of weathering ranging from completely weathered to highly weathered. Fracture karst (cavities) is relatively developed. The ore bodies are directly water-filled, and some slopes have poor stability, making them highly susceptible to collapse. The landform characteristics of the Kambove open-pit mine are shown in Figure 7, with some slope landslide areas marked by red diagonal lines [34].

Considering the geological conditions and mining processes at the site, there are varying levels of landslide risk around the perimeter of the open-pit mine area. Therefore, a comprehensive slope monitoring scheme using two slope radars has been designed to cover the entire mine area, as shown in Figure 8. The slope radars selected are ArcSAR (Software version number: SSR200K V2.1. The manufacturer of this device is Changsha Smart Mining Inc., Ltd., which is from Changsha, China. This device has been purchased by the mine.) radars, which are deployed at high points in the north and south directions of the mine area. Initially, the radar equipment can be installed in a fixed position. However, as the slope height increases, there may be blind spots in the radar’s field of view. To address this issue, the radars can be deployed at the bottom of the open pit to provide real-time monitoring of high and steep slope areas that pose a risk.

The slope radars are set to collect data every 10 min, resulting in 144 data sets per day. The short-range monitoring distance is set to 0 m, while the long-range monitoring distance is set to 1200 m. The azimuth monitoring angle is approximately 120°, covering a slope length of about 2.5 km. The slope is monitored using a downward-looking angle of −8°. For the landslide risk prediction in the hazardous areas of the Kambove open-pit mine, observation points need to be manually set up based on the displacement deformation map from the previous cycle and the geological structural characteristics of the site, such as the locations of exposed faults, the positions where cracks appear, and the areas with severe weathering, as shown in Figure 9. In the specialized software platform for slope radar monitoring, users can conveniently utilize the zoom function to clearly view deformation hazard zones highlighted by a red point cloud model. Subsequently, users can manually and precisely set deformation observation points within these prominent red areas, assigning each point a rational and easily identifiable name. The software then automatically tracks and records deformation data for these critical observation points, effectively isolating and managing data from different zones. Through the precise monitoring of slope deformation and issuing timely warnings as necessary, the software ensures the accuracy and timeliness of safety monitoring.

Each observation point has a corresponding displacement deformation curve. Based on the displacement deformation curve and the monitoring data from the observation points, landslide time prediction is carried out using the improved reciprocal velocity method.

Displacement data and deformation velocity data of the monitoring points are extracted from the radar software (software version number: SSR200K V2.1.). For convenience in data processing, data are extracted at an interval of one hour. The velocity values of SMA and LMA are then calculated according to Formula (5). The specific numerical values are shown in

Table 2. Relative time, displacement, velocity, and long–short average velocity data.

Relative Time t/h	Displacement/mm	Velocity/(mm/h)	v-SMA/(mm/h)	v-LMA/(mm/h)
1.59	1.75	1.83	-	-
2.58	3.49	1.72	-	-
3.53	4.9	1.73	1.76	-
4.58	5.82	1.14	1.53	-
5.54	7.15	1.64	1.50	-
6.53	8.12	1.24	1.34	-
7.57	8.95	1.05	1.31	1.48
8.53	9.66	1	1.10	1.36
9.58	10.69	1.25	1.10	1.29
10.54	11.42	1.03	1.09	1.19
11.58	12.34	1.14	1.14	1.19
12.54	13.31	1.27	1.15	1.14
13.58	14.17	1.08	1.16	1.12
14.53	15.07	1.21	1.19	1.14

, and complete data are provided in Appendix A,

Table A1. Relative time, displacement, velocity, and long–short average velocity data.

Relative Time t/h	Displacement/mm	Velocity/(mm/h)	v-SMA/(mm/h)	v-LMA/(mm/h)
1.59	1.75	1.83	-	-
2.58	3.49	1.72	-	-
3.53	4.9	1.73	1.76	-
4.58	5.82	1.14	1.53	-
5.54	7.15	1.64	1.50	-
6.53	8.12	1.24	1.34	-
7.57	8.95	1.05	1.31	1.48
8.53	9.66	1	1.10	1.36
9.58	10.69	1.25	1.10	1.29
10.54	11.42	1.03	1.09	1.19
11.58	12.34	1.14	1.14	1.19
12.54	13.31	1.27	1.15	1.14
13.58	14.17	1.08	1.16	1.12
14.53	15.07	1.21	1.19	1.14
15.57	16.36	1.5	1.26	1.21
16.53	17.4	1.33	1.35	1.22
17.58	18.36	1.18	1.34	1.24
18.54	19.05	0.98	1.16	1.22
19.58	20.5	1.65	1.27	1.28
20.54	21.77	1.58	1.40	1.35
21.59	22.69	1.14	1.46	1.34
22.58	23.37	0.94	1.22	1.26
23.54	24.76	1.72	1.27	1.31
24.58	26.39	1.82	1.49	1.40
25.57	29.67	3.57	2.37	1.77
26.57	33.23	3.84	3.08	2.09
27.56	35.53	2.58	3.33	2.23
28.51	37.42	2.24	2.89	2.39
29.59	39.4	2.11	2.31	2.55
30.57	40.96	1.85	2.07	2.57
31.53	42.58	1.95	1.97	2.59
32.57	44.52	2.12	1.97	2.38
33.52	46.35	2.17	2.08	2.15
34.57	48.17	2	2.10	2.06
35.53	49.65	1.81	1.99	2.00
36.58	51.33	1.86	1.89	1.97
37.53	53.04	2.04	1.90	1.99
38.56	54.76	1.94	1.95	1.99
39.52	56.58	2.16	2.05	2.00
40.57	58.66	2.24	2.11	2.01
41.53	60.52	2.2	2.20	2.04
42.57	61.9	1.58	2.01	2.00
43.53	64.16	2.62	2.13	2.11
44.57	65.9	1.93	2.04	2.10
45.53	67.76	2.2	2.25	2.13
46.53	69.29	1.79	1.97	2.08
47.57	71.06	1.95	1.98	2.04
48.53	72.84	2.13	1.96	2.03
49.58	74.93	2.26	2.11	2.13
50.57	77.09	2.44	2.28	2.10
51.52	78.36	1.59	2.10	2.05
52.57	80.34	2.16	2.06	2.05
53.52	82.68	2.71	2.15	2.18
54.59	84.76	2.19	2.35	2.21
55.55	86.5	2.09	2.33	2.21
56.59	88.72	2.37	2.22	2.22
57.55	90.58	2.21	2.22	2.19
58.59	92.81	2.39	2.32	2.30
59.55	94.8	2.34	2.31	2.33
60.6	97.11	2.47	2.40	2.29
61.55	99.14	2.36	2.39	2.32
62.58	102.25	2.59	2.47	2.39
63.54	104.5	2.61	2.52	2.42
64.59	107.27	2.63	2.61	2.48
65.54	109.58	2.64	2.63	2.52
66.59	112.1	2.67	2.65	2.57
67.55	114.49	2.66	2.66	2.59
68.51	116.67	2.67	2.67	2.64
69.56	119.21	2.68	2.67	2.65
70.55	121.48	2.77	2.71	2.67
71.59	124.24	2.9	2.78	2.71
72.55	126.37	3.03	2.90	2.77
73.59	129.33	3.1	3.01	2.83
74.58	132.42	3.38	3.17	2.93
75.54	135.52	3.5	3.33	3.05
76.58	139.06	3.64	3.51	3.19
77.54	142.42	3.77	3.64	3.33
78.53	146.03	3.93	3.78	3.48
79.57	150.14	4.2	3.97	3.65
80.52	153.93	4.23	4.12	3.81
81.57	158.2	4.35	4.26	3.95
82.52	163.75	6.06	4.88	4.31
83.57	170.21	6.44	5.62	4.71
84.53	175.51	5.79	6.10	5.00
85.57	181.3	5.8	6.01	5.27
86.52	187.65	6.96	6.18	5.66
87.56	195.75	8.02	6.93	6.20
88.52	204.86	9.76	8.25	6.98
89.57	216.88	11.77	9.85	7.79
90.52	230.53	14.51	12.01	8.94
91.57	250.81	19.65	15.31	10.92
92.53	280.09	30.82	21.66	14.50
93.57	330.26	48.26	32.91	20.40

.

The displacement, v-SMA, and v-LMA data are plotted into a curve graph, as shown in Figure 10. According to the method proposed by Carlà, T. et al. [23], the starting point of the acceleration phase, OOA, corresponds to a relative time T = 60.6 h on the horizontal axis. By comparing the displacement curves, it was found that at the OOA point, the displacement curve had not yet undergone significant nonlinear or accelerated changes.

Considering that using OOA as the starting point for calculations may result in lower prediction time accuracy, this study proposes using a normally distributed confidence interval to dynamically identify the $T_{SP}$ for the slope deformation acceleration. The $T_{SP}$ point is selected based on whether the velocity variable no longer falls within the confidence interval ($\mu -2\sigma , \mu +2\sigma$), where μ represents the mean and $\sigma$ is calculated using Formula (6), as shown below.

$$\sigma ={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{v}}_{\mathrm{i}}-\overline{v}\right)}^2}}{\mathrm{n}-1}}$$

(6)

Using Table 1 and Formula (6), the curve shown in Figure 11 can be calculated. After the OAA point, the velocity curve monitored by the radar is still within the confidence interval. However, at $T$ = 72.55 h, the velocity curve no longer falls within the velocity confidence interval ($\mu -2\sigma , \mu +2\sigma$), and this point is identified as the $T_{SP}$ point. The $T_{SP}$ point marks the critical transition from the second stage of creep deformation (constant velocity deformation stage) to the third stage (acceleration deformation stage).

Based on Formula (4), a logarithmic curve of $\mathrm{\Delta }\mathrm{t}\text{-}\log (T)$ is plotted, and the values of parameters $A$, $B$, and $C$ are obtained from the simulation results, as shown in Figure 12 and

Table 3. Model parameters, landslide prediction time, and actual landslide time.

Sequence Number	Time/T	A	B	C	Prediction Time/h	Actual Failure Time/h
1	72.55	–	–	–	–
2	73.59	–	–	–	–
3	74.58	–	–	–	–
4	75.54	–	–	–	–
5	76.58	0.059	0.721	64.78	84.75	94.74
6	77.54	0.041	0.695	61.92	87.9	94.74
7	78.53	0.041	0.676	59.12	91.34	94.74
8	79.57	0.031	0.665	57.37	94.56	94.74
9	80.52	0.031	0.654	55.16	95.21	94.74
10	81.57	0.031	0.645	52.4	97.64	94.74
11	82.52	0.031	0.661	56.48	94.27	94.74
12	83.57	0.031	0.681	59.29	92.19	94.74
13	84.53	0.031	0.672	58.53	92.7	94.74
14	85.57	0.031	0.651	56.04	94.2	94.74
15	86.52	0.031	0.641	54.45	95.1	94.74
16	87.56	0.031	0.631	53.71	95.48	94.74
17	88.52	0.031	0.641	54.12	95.28	94.74
18	89.57	0.031	0.649	55.15	94.88	94.74
19	90.52	0.031	0.649	56.08	94.56	94.74
20	91.57	0.031	0.649	56.06	94.56	94.74
21	92.53	0.031	0.649	56.07	94.56	94.74
22	93.57	0.031	0.649	56.08	94.56	94.74

. The monitoring data after the $T_{SP}$ point are fitted using a logarithmic function model. As subsequent monitoring data are input, the landslide prediction time becomes increasingly close to the actual landslide time. Figure 13 shows the actual landslide scene at the site. The landslide mass gradually expands from the front edge towards the back edge, exhibiting the characteristic sliding pattern of ‘progressive retrogression’. The main reason for this is the disturbance caused by underlying mining activities, which leads to tensile stress concentration at the top of the slope in the front edge. This tensile stress concentration results in the tensile fracture and dislocation deformation of the rock and soil mass towards the free face, thereby forming transverse tensile cracks.

The statistical curve of the predicted landslide time from Table 2 and the actual landslide time is shown in Figure 14. Initially, due to the limited number of data points, there was a large error between the predicted time and the actual landslide time. As the monitoring data continued to be updated, at T = 90.52 h, the predicted time curve remained basically unchanged and was almost level with the actual landslide time curve. This means that the actual landslide time can be accurately predicted 4 h in advance, and there is no lag in the prediction time.

We adopted the method proposed by Carlà, T., which involves the intersection of the short-term moving average line and the long-term moving average line, to calculate the starting point OOA of the acceleration stage. Meanwhile, the landslide time was predicted by applying the reciprocal velocity method. The analysis results regarding the predicted landslide time and the actual landslide time are presented in

Table 4. Comparison of landslide time predicted by traditional methods and actual landslide time.

OOA (t = 60.6)	Prediction Time/h	Actual Failure Time/h
Raw data	96.89	94.74
SMA	97.91	94.74
LMA	100.6	94.74

.

As can be observed from the table, when utilizing the OOA point, the short-term moving average (SMA), and the long-term moving average (LMA) to predict the landslide time, a time lag in the predicted time occurred. The reason for this is that the OOA point obtained through the intersection method of the short-term moving average line and the long-term moving average line is actually earlier than the starting moment of the real acceleration deformation stage.

Consequently, the method of identifying the starting point of acceleration by employing the normal distribution confidence interval proposed in this paper proves to be more precise.

The reciprocal velocity method utilizes the velocity monitoring data in the accelerated deformation stage. Its basic assumption is that the deformation velocity of the slope is infinite at the moment of sliding; that is, the intersection point of the linear trend line of the reciprocal value of velocity and time on the time axis is the landslide time. However, in reality, the velocity at the time of a landslide is not infinite, but a value between 0 and infinity. In the calculation case in this paper, the maximum deformation velocity collected using the slope radar before the landslide was 48.26 mm/h. When using the logarithmic function model to predict the landslide time, the calculation error was within 0.54 h to 10 h before the actual moment the landslide began, and the prediction error was within 0.18 h to 4 h before this moment. Overall, the error is relatively small, and the closer it is to the landslide moment, the smaller the error is. The specific data are shown in

Table 5. Error analysis of landslide prediction time and actual landslide time.

Time/T	Prediction Time/h	Actual Failure Time/h	△h
85.57	94.2	94.74	−0.54
86.52	95.1	94.74	0.36
87.56	95.48	94.74	0.74
88.52	95.28	94.74	0.54
89.57	94.88	94.74	0.14
90.52	94.56	94.74	−0.18
91.57	94.56	94.74	−0.18
92.53	94.56	94.74	−0.18
93.57	94.56	94.74	−0.18

.

The landslide early warning method proposed in this paper has a broad application foundation. Included in this, the reciprocal velocity method for predicting landslide time has been applied in both mine slopes and geological landslides of mountains in China and abroad. Moreover, the Tsp point picking method and the time logarithmic model can be popularized and applied in classic creep deformation type slope landslides.

5. Discussion

This study innovatively combines the improved reciprocal velocity method with ArcSAR technology to achieve high-precision prediction of landslide occurrence times. Compared to traditional methods, this model precisely captures the subtle dynamic changes during the accelerated deformation stage of landslides by utilizing the normal distribution confidence interval of velocity parameters, significantly enhancing prediction accuracy. Furthermore, we creatively constructed a $\mathrm{\Delta }\mathrm{t}\text{-}\log (T)$ prediction model and successfully achieved the unitization of monitoring data dimensions through coordinate transformation, effectively solving the previously existing problem of non-convergence in data fitting and further consolidating the stability and predictive performance of the model. When compared to the research findings of Carlà et al. [23], our model demonstrates superior prediction accuracy in practical case applications.

Although the data for this study originate from a specific mining environment, meaning the generalization of its findings may be limited, the slope landslide type studied conforms to the general law of creep deformation stage. This suggests that, with further research and applications in diverse settings, this study could possess potential universal application value. However, it is important to note that future research should deepen this perspective through additional case studies and practical applications to fully assess the method’s universality and effectiveness. It undoubtedly provides a novel perspective and powerful tool for landslide prediction. It is worth noting that the precise fitting of model parameters still relies on a large amount of high-quality monitoring data support. For regions where data acquisition is difficult, targeted research may be required to adapt to local conditions. Looking ahead, the methods proposed in this study have broad application prospects and await deep integration with other monitoring technologies (such as GNSS monitoring, satellite remote sensing, water level monitoring, etc.) and artificial intelligence algorithms (such as deep learning, machine learning, etc.), in order to improve the accuracy and automation level of landslide predictions on a larger scale. At the same time, researchers can actively explore adaptive adjustment mechanisms for model parameters, enabling them to flexibly adapt to different geographical environments and types of landslides, further enhancing their universality and practicality.

6. Conclusions

This study innovatively introduces a method for dynamically identifying the $T_{SP}$ for the slope deformation acceleration. (1) By conducting an in-depth analysis of the normal distribution characteristics exhibited by the deformation velocity random variable in the uniform deformation stage of the monitoring data, we set an appropriate confidence interval to accurately identify the initial moment of accelerated deformation of the landslide, i.e., the $T_{SP}$ location. This improvement helps to capture the signal of the landslide entering the acceleration stage earlier, thereby achieving a timely warning in terms of time prediction. (2) The $\mathrm{\Delta }\mathrm{t}-\log (T)$ logarithmic model is constructed. By transforming the $S\text{-}\log (T)$ coordinate system, the monitoring data are unified into a coordinate system with the same time dimension, solving the problem of fitting convergence of the data in the model, improving the robustness of the model, and ensuring the reliability of the prediction results. (3) By fully leveraging the high-precision and short-cycle monitoring advantages of ArcSAR radar technology, the latest displacement data of the landslide area are obtained in real time, and the parameters of the $\mathrm{\Delta }\mathrm{t}\text{-}\log (T)$ prediction model are continuously updated and corrected. This dynamic adjustment mechanism ensures that the model remains synchronized with the actual evolution state of the landslide, enabling the predicted landslide occurrence time to more accurately approximate the actual landslide time, greatly improving the accuracy of the prediction. At the same time, the all-weather, large-area, and high-precision monitoring capabilities of ArcSAR radar ensure that accurate warning information for various parts of the mine slope can be continuously provided within hours to tens of hours before a potential landslide, providing strong technical support for effectively preventing landslide disasters.

It should be noted that the method presented in this paper is applicable to landslide types characterized by creep deformation, such as translational landslides and progressive retrogressive landslides. However, there are certain limitations in predicting sudden-collapse and flow-type landslides caused by heavy rainfall. Furthermore, we also need to further enhance the robustness of the monitoring data to reduce the impact of noise signals on it. This requires improvements in radar equipment and noise data processing techniques.