GNSS/Accelerometer Adaptive Coupled Landslide Deformation Monitoring Technology

Abstract

Global Navigation Satellite System (GNSS) positioning technology has become the most effective method for real-time three-dimensional landslide monitoring. However, the GNSS observation signal is easily affected by the presence of a complex landslide environment with high occlusion and strong interference, in which case its accuracy and reliability cannot meet the requirements of landslide deformation monitoring. Although the accelerometers have strong autonomous working capacities and can complement the GNSS landslide monitoring technology, regular GNSS/accelerometer coupled deformation monitoring technology relies on high-quality GNSS measurement information in order to obtain high-precision accelerometer-reckoned results, derived by accurately estimating the baseline shift error (BSE). When the GNSS signal suffers severe interference, the GNSS monitoring error will be partially absorbed by the BSE component after Kalman filtering, resulting in the divergence of the deformation solution. In this study, an abnormal observation variance inflation model was used to process the simulated landslide monitoring data (GNSS and accelerometer raw observation) for three typical scenes—GNSS signal normally locked, signal partially lost, and short-term interruption. The results were as follows: (1) When the GNSS signal was normally locked, the accuracy was comparable to that of the coupled solution employing an accelerometer (the Root Mean Square (RMS) values in the East (E), North (N) and Upward (U) directions were 0.11 cm, 0.33 cm, and 0.30 cm, respectively). (2) When the GNSS signal was partially lost, the accelerometer could effectively suppress the low-precision float solution of the GNSS, but here, the accuracy of the coupled solution would also decrease with the duration of the floats (the RMS values were E—1.21 cm, N—0.31 cm, and U: 0.58—cm, respectively, when the floats lasted for 10 s, and increased to E—3.09 cm, N—0.39 cm, and U—1.14 cm when they lasted for 20 s, wherein E was the main simulated sliding direction). (3) When the GNSS signal was interrupted for a short time, the accuracy of the coupled solution gradually decreased during continuous interruption, and decreased more quickly during the sliding period of the landslide (when the interruption persisted for 10 s, the RMS values in the simulated landslide stability period were E—0.61 cm, N—0.24 cm, and U—0.25 cm, respectively, while in the simulated landslide sliding period they reached E—4.10 cm, N—6.84 cm, and U—2.30 cm). However, raw observations of the accelerometer could still effectively be used to assist in identifying the real state of the landslide, thereby providing auxiliary information pertinent to early landslide disaster warning.

1. Introduction

GNSS navigation satellite systems (GNSSs) can provide continuous three-dimensional (3D) navigation and positioning results, with characteristics of high precision, all-weather and all-time period suitability. In the static GNSS mode, the high precision positioning solution is available by baseline creation and network adjustment [1,2,3], but it is post-processed and unable to get the real-time (1 Hz) solution. Particularly in real-time kinematic (RTK) mode, double difference carrier phase observations are used to eliminate common errors such as ionosphere, troposphere, and receivers’ clock errors. The positioning accuracy can reach centimeter-level easily after fixing the ambiguity [4,5]. GNSS-RTK has been widely used in landslide deformation monitoring [6,7], and has yielded successful warnings many times [8]. However, the GNSS can be seriously disturbed by factors such as mountain occlusion and vegetation diffraction [9], resulting in significant multipath error and positioning accuracy degradations, making it difficult to obtain accurate and reliable landslide deformation monitoring results [10,11]. The short-term signal interruption caused by poor communication conditions can even lead to false alarms and the absence of early landslide disaster warnings.

Accelerometers, as a kind of inertial sensor, can function without being affected by the external environment and have the characteristics of short-term high accuracy and strong stability. However, the errors of this inertial sensor accumulate over time, leading to a gradual divergence in the reckoned results. Therefore, the coupling of the accelerometer and GNSS could establish effective redundancy and complementarity and could help obtain more continuous and reliable monitoring results. When a landslide enters the critical sliding stage, the accelerometer’s observation could be used to assist in the identification and verification of the landslide’s instability state [12], and thus, ensure the accuracy of the landslide early warning.

At present, many scholars have applied the GNSS/accelerometer coupled method in many fields, such as structural health monitoring and seismic inversion. In the health monitoring of bridges [13,14] and buildings [15,16], accelerometers have been used to obtain information on the high-frequency vibration and displacement of structures [17], which is usually studied in combination with modal decomposition, and has many excellent engineering applications [18,19]. In seismic inversion, a strong seismometer with an accelerometer as the core component plays an important role. While the Baseline Shift Error (BSE) is the main error source in the accelerometer caused by instrument tilt, the co-seismic deformation of the surface can be obtained with a high-precision by coupled GNSS and accelerometer observation after estimating this error accurately [20]. There are established empirical BSE estimation methods that are used to restore co-seismic displacement [21], which have been proven to agree well with GPS monitoring results [22]. Some scholars have also analyzed the contact between the BSE and the installation tilt angle of the accelerometer and proposed a random walk modeling method by treating it as an unknown parameter [23,24]. The above studies paid more attention to the vibration characteristics of the structure, or the instantaneous co-seismic deformation. In structural health monitoring studies, the displacement of the deformed body was very small (1–5 cm), and showed significant periodic vibration characteristics [16]. In earthquake monitoring, no significant displacement is usually seen beforehand, and large destructive displacement (300–500 cm) may occur suddenly between earthquakes [25]. However, the growth process of landslides is relatively slow, and can generally be divided into the four stages of constant velocity, initial acceleration, intermediate acceleration, and impending slip [26]. For example, the Heifangtai loess landslide in the Gansu province of China lasted for nearly four months, from the stages of initial acceleration to instability [8], and was, thus, a typical slow deformation. Obviously, the effects were essentially different from the high-frequency vibration and instantaneous damage to bridges seen in earthquakes. In addition, the complex observation environment in landslides is significantly different from the open environments typically encountered in bridge and earthquake monitoring, which means the regular GNSS/acceleration coupled technology cannot be directly applied in landslide deformation monitoring.

In order to solve the above problem, this study developed a variance inflation model based on GNSS/accelerometer coupled technology and used a simulated deformation platform to simulate the slowly changing deformation characteristics of a landslide. Three typical GNSS landslide monitoring scenes were designed to analyze and demonstrate the effectiveness of this method in landslide deformation monitoring. Our research results can provide reference materials for further GNSS/accelerometer-coupled landslide deformation monitoring and research.

2. Method

2.1. GNSS-RTK Double Difference Observation Model

Real-time GNSS landslide disaster deformation monitoring technology usually adopts the RTK process to obtain 3D deformation displacement data for the landslide body. GNSS-RTK can obtain high-accuracy positioning results when the carrier phase double difference observation model is used to eliminate the common error and the integer ambiguity is resolved by the lambda algorithm. The GNSS carrier and pseudorange observation equations are as follows:

$$\left\{\begin{array}{l}{\varphi }_{rb}^{jk}={\rho }_{rb}^{jk}+d{t}_{rb}-I_{rb}^{jk}+T_{rb}^{jk}+\lambda am{b}_{rb}^{jk}+{\epsilon }_{\varphi }\\ P_{rb}^{jk}={\rho }_{rb}^{jk}+d{t}_{rb}+I_{rb}^{jk}+T_{rb}^{jk}+{\epsilon }_P\end{array}\right.$$

(1)

where ${\varphi }_{rb}^{jk}$ and $P_{rb}^{jk}$ are carrier and pseudorange double difference observations, $\rho$ is the geometric distance between the satellite and the GNSS antenna, $dt$ is the single difference (SD) between two offset GNSS receiver clocks, $I_{rb}^{jk}$ is the double difference (DD) of ionospheric delay, $T_{rb}^{jk}$ is the DD of tropospheric delay, $am{b}_{rb}^{jk}$ is the DD of carrier phase ambiguity, and λ is the carrier wavelength. The subscripts $r$ and $b$ represent the rover and the base station, respectively, while the superscripts $j$ and $k$ represent the jth and kth satellites, respectively, and ε is the measurement residual. It should be noted that when the length of the baseline (distance between rover and base station) is less than 15 km and at the same altitude level, $I_{rb}^{jk}$ and $T_{rb}^{jk}$ are approximately equal to zero [27,28]; when the receivers and antennas of the rover and base stations are the same, $d{t}_{rb}$ is also approximately equal to 0.

In actual RTK processing, the DD integer’s ambiguity is often separated into two single-difference ambiguities between two stations for parameter estimation, which avoids the extra burden of program writing caused by the frequent switching of the reference satellite [29]:

$$am{b}_{rb}^{jk}=am{b}_{rb}^j-am{b}_{rb}^k$$

(2)

where $k$ is the reference satellite, whose altitude angle is the highest of all the visible satellites at a certain moment.

In combination with the Extended Kalman Filter (EKF), the DD observation equation is linearized by calculating the Jacobi matrix, after which the receiver coordinates and the DD ambiguities can be solved.

2.2. Accelerometer Integration

The accelerometer is the core component of many industrial instruments, such as Inertial Measurement Units (IMUs), Strong Motion (SM), intensity meters, etc. It is capable of outputting raw observations at a high frequency (generally greater than 100 Hz). This kind of instrument is often used to capture high-frequency data on the deformation of structures, which can be modeled as variable accelerated motion. Continuous 3D velocity and displacement information can be obtained by primary and double integration of the sampling interval $\Delta t$:

$$\left\{\begin{array}{l}{v}_k=\int a_{k}dt=a_k\Delta t\\ d_k=\int v_{k}dt=\iint a_{k}dt=\frac{1}{2}{a}_k\Delta t^2\end{array}\right.$$

(3)

where $a_k{v}_k{d}_k$ denote the 3D acceleration, the velocity, and the displacement at the $k$ moment, respectively. This is organized into the form of a matrix expression, consistent with the state update equation for Kalman filtering, which can be expressed as follows:

$$X_k=A_k{X}_{k-1}+B_k{U}_k+{\epsilon }_k$$

(4)

where $X_k$ and $X_{k-1}$ are the state parameters at moments $k$ and $k-1$, respectively, $A_k$ is the state transform matrix, $\Delta t$ is the accelerometer sampling interval, $B_k$ is the control input matrix, $U_k$ is the drive input vector, and ${\epsilon }_k$ is the noise term. These can be written in the following form:

$$X_k=\left[\begin{array}{c}{d}_k\\ v_k\end{array}\right]A_k=\left[\begin{array}{cc}1& \Delta t\\ 0& 1\end{array}\right]X_{k-1}=\left[\begin{array}{c}{d}_{k-1}\\ v_{k-1}\end{array}\right]B_k=\left[\begin{array}{c}\frac{1}{2}\Delta t^2\\ \Delta t\end{array}\right]U_k=a_k$$

2.3. GNSS/Accelerometer Adaptively Coupled Monitoring Algorithm

The GNSS/accelerometer adaptively coupled monitoring algorithm is mainly implemented in the Kalman filtering (KF) model, with displacement and velocity as the state estimated parameters and accelerometer reckoning being used for the state time update. This can then be made to output high-precision deformation results by implementing measurement correction with the GNSS displacement solution. It should be noted that accelerometers generally have significant BSE, which often leads to the rapid accumulation of errors and the divergence of the filter solution. In this case, this error term required correction in order to improve the robustness of the filter. Scholars have proposed some methods based on the context of practical application, such as empirical value correction [21] and parameter estimation [23]. Since this error varied continuously during the landslide sliding period due to ground tilt and installation angle, we selected the latter method to model the BSE as a random walk process (Equation (5)). The BSE is treated as a state parameter for Kalman filter and can then be used to close-loop correct the accelerometer’s raw observation during the state prediction process. It should be noted that the accurate estimation of this error is key to obtaining a high-precision coupled solution:

$$u_k=u_{k-1}+w_k$$

(5)

where $u_k$ is the BSE at the $k$ moment and $w_k$ is Gaussian white noise.

The algorithm can be summarized in the following eight steps.

Step 1: Space–time synchronization of GNSS/accelerometers. Synchronizing the accelerometer’s timestamp system to GPST. Establishing the east–north–up (ENU) local coordinate system by taking the first GNSS fixed solution as the origin. Since GNSS positioning results are generally in Earth-centered, Earth-fixed (ECEF) coordinates, they need to be converted to the ENU coordinate system with the help of geodetic methods such as the Bursa model. Here, the accelerometer was mounted on the surface, and the three axes were made to be roughly coincident with the east–north–up direction (see Figure 1), so that the coordinate systems of the two kinds of instruments could be unified. The specific procedure can be found in the literature [20].

Step 2: Calculate the Allan variance of the accelerometer and obtain the random wandering coefficient q. Here, we turn the accelerometer on and leave it to stand for at least 30 min, after which the Allan variance can be calculated with Equation (7). The slope of the log–log curve of the correlation time of τ-Allan variance for the acceleration random walk is 1/2, and the vertical coordinate corresponding to the horizontal coordinate $\sqrt{3}$ is the specific value of this noise term. A detailed explanation of Allan variance can be found in the literature [30].

$${\sigma }^2\left(T\right)=\frac{1}{2\left(N-1\right)}{\displaystyle \sum }_{k=1}^{N-1}{\left[{\overline{\mathsf{\Omega }}}_{k+1}\left(T\right)-{\overline{\mathsf{\Omega }}}_k\left(T\right)\right]}^2$$

(6)

where ${\overline{\mathsf{\Omega }}}_k\left(T\right)$ denotes the average value of the kth group accelerometer observation, the length of which is T.

Step 3: Determine the state vector parameters $X_k$, the state transform matrix, the control input matrix and the input vector of KF. This involves extending the BSE $u_k$ to the estimated states and taking it as a close-loop calibration parameter to correct the raw observations of the accelerometer. The above four vectors and matrix can be written in the following form by combining Equations (4) and (5):

$$X_k=\left[\begin{array}{c}{d}_k\\ v_k\\ u_k\end{array}\right]A_k=\left[\begin{array}{ccc}1& \Delta t& -\frac{1}{2}\Delta t^2\\ 0& 1& -\Delta t\\ 0& 0& 1\end{array}\right]B_k=\left[\begin{array}{c}\frac{1}{2}\Delta t^2\\ \Delta t\\ 0\end{array}\right]U_k=a_k$$

(7)

Step 4: Determine the process noise matrix $Q_k$. Since the accelerometer is the only noise source in the state time update process, its error is mapped to the two dimensions of the velocity and displacement domains through the matrix $b_k$ which can be deduced according to the law of error propagation (Equation (8)):

$$b_k=\left[\begin{array}{c}\frac{1}{2}\Delta t^2\\ \Delta t\\ 1\end{array}\right]Q_k=b_k{\left(q\Delta t\right)}^2{b}_k^T=\left[\begin{array}{ccc}\frac{1}{4}\Delta t^4& \frac{1}{2}\Delta t^3& \frac{1}{2}\Delta t^2\\ \frac{1}{2}\Delta t^3& \Delta t^2& \Delta t\\ \frac{1}{2}\Delta t^2& \Delta t& 1\end{array}\right]{\left(q\Delta t\right)}^2$$

(8)

Step 5: State time update of KF. Since the accelerometer sampling interval $\Delta t$ in $A_k$ and $B_k$ remains unchanged, the filter can be considered to be a linear time-invariant system, and no linearization is required. The time update equations of the state parameters and their covariances can be expressed as Equation (9):

$$\left\{\begin{array}{c}{X}_{k,k-1}=A_k{X}_{k-1}+B_k{U}_k\\ P_{k,k-1}=A_k{P}_{k-1}{A}_k^T+Q_k\end{array}\right.$$

(9)

where $P$ denotes the covariance matrix of the state parameters, $Q$ is the process noise matrix, and $k,k-1$ represents the time update from epoch $k-1$ to epoch $k$.

Step 6: Determine the measurement vector $Z_k$ and the coefficient matrix $H_k$. The measurement equation can be represented by Equation (10):

$$Z_k=H_k{X}_k+{\epsilon }_k$$

(10)

where the measurement vector $Z_k$ only contains the 3D displacement results of GNSS, $Z_k{H}_k$ can be written in the form of Equation (11), and ${\epsilon }_k$ is the measurement residual, which obeys the standard normal distribution ${\epsilon }_k~N\left(0,R^2\right)$.

$$\left\{\begin{array}{l}{Z}_k=\left[d_{k_{GNSS}}\right]\\ H_k={\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T\end{array}\right.$$

(11)

Step 7: Determine the measurement noise matrix R with the variance inflation model. Here, we amplify the variance of the abnormal float solution of GNSS, and adjust the measurement noise adaptively to reduce its influence on the parameter estimation [31]. The measurement noise matrix can be determined by Equations (12) and (13):

$$\left\{\begin{array}{c}{\overline{r_i}}^2={\lambda }_{ii}{r}_i^2\\ {\overline{r_j}}^2={\lambda }_{jj}{r}_j^2\\ \overline{r_{ij}}={\lambda }_{ij}{r}_{ij}\end{array}\right.$$

(12)

where ${\lambda }_{ij}=\sqrt{{\lambda }_{ii}{\lambda }_{jj}}$ is the variance inflation factor, which can be calculated by a weight selection method, such as Huber and IGG3 [32,33]; ${\overline{r_i}}^2{\overline{r_j}}^2\overline{r_{ij}}$ are the equivalent variance and covariance, respectively, and the equivalent measurement noise can be described as a matrix:

$$R=\left[\begin{array}{ccc}{\overline{r_E}}^2& \overline{r_{EN}}& \overline{r_{EU}}\\ \overline{r_{NE}}& {\overline{r_N}}^2& \overline{r_{NU}}\\ \overline{r_{UE}}& \overline{r_{UN}}& {\overline{r_U}}^2\end{array}\right]=\left[\begin{array}{ccc}{\lambda }_{11}{r}_E^2& {\lambda }_{12}{r}_{EN}& {\lambda }_{13}{r}_{EU}\\ {\lambda }_{21}{r}_{NE}& {\lambda }_{22}{r}_N^2& {\lambda }_{23}{r}_{NU}\\ {\lambda }_{31}{r}_{UE}& {\lambda }_{32}{r}_{UN}& {\lambda }_{33}{r}_U^2\end{array}\right]$$

(13)

Step 8: Kalman filter measurement gain update. We implement the Kalman filter measurement update if the current epoch contains GNSS input information. Then, we repeat steps 4–7 on this basis to perform the adaptively coupled GNSS/accelerometer process for the $k+1$ epoch. The measurement update equation can be expressed by Equation (14) [34]:

$$\left\{\begin{array}{l}{K}_k=P_{k,k-1}{H}_k^T{\left(H_k{P}_{k,k-1}{H}_k^T+R_k\right)}^{-1}\\ X_k=X_{k,k-1}+K_k\left(Z_k-H_k{X}_{k,k-1}\right)\\ P_k=\left(I-K_k{H}_k\right)P_{k,k-1}\end{array}\right.$$

(14)

where $K_k$ denotes the gain matrix and $X_k{P}_k$ are the state estimation and its covariance matrix after the Kalman filter measurement gain, respectively.

The process involved in employing the GNSS/accelerometer adaptive coupled deformation monitoring algorithm is summarized in Figure 2.

3. Experiment and Analysis

The experiment was carried out at the GNSS landslide monitoring instrument test site of Chang’an University in China, surrounded by a small number of buildings and some vegetation. The GNSS antenna and the accelerometer were fixed in a rigid body structure, using the simulation deformation platform to ensure that the two instruments would move in the same direction at the same speed during landslide deformation simulation. The installation diagram is shown in Figure 3. We installed another GNSS receiver 10 m from the simulated deformation platform as the base station. The data collection time was from 11:00 to 12:30 (UTC +08:00) on 13 December 2021 (GPS Week: 2188; Second of Week (SOW): 97,200–102,600 s).

The GNSS receivers all used a Unicorecom UB4B0M low-cost board (for detailed technical parameters, please visit https://www.unicorecomm.com/products/detail/1, accessed on 17 June 2022) and a HighGain HG-GOYH7151 Geodetic Antenna (for detailed technical parameters, please visit http://www.highgain.com.cn/productcenter/info.aspx?itemid=137&lcid=21, accessed on 17 June 2022). Since the GNSS receivers and antennas of the rover station and the base station were the same, the offset between the receiver clock error and the antenna phase center could be directly canceled to reduce the estimated parameters when performing the DD observation equation. Considering the universal requirements of landslide monitoring equipment, we selected a low-cost and low-power MEMS accelerometer developed by the Institute of Precision Measurement Science and Technology Innovation of the Chinese Academy of Sciences. This instrument can output three-axis acceleration observations at a frequency of 100 Hz, and its timestamp was synchronized to the GPST by the 1PPS pulse of the GPS timing signal. Its technical performance indicators are shown in

Table 1. Accelerometer performance specifications.

Indicator Entry	Technical Indicators
Channels	≥3(E-W, N-S, U-D)
Range	−19.6 m/s2~19.6 m/s2 (E-W and N-S); −29.6 m/s2~9.8 m/s2 (U-D);
Dynamic range	80 dB (0.1 Hz~20 Hz)
Measurement error	<5% (0.1 Hz~20 Hz)
Frequency band	Low-frequency cutoff frequency: ≤0.01 Hz (−3 dB) High-frequency cutoff frequency: ≥40 Hz (−3 dB, sampling rate 100 sps)
Linearity error	<1%

. The cost of the GNSS and accelerometer were around one thousand RMB, and the power consumption was below 4.5 W.

Before the simulation experiment, the accelerometer was held static for 30 min, and the Allan variance was used to standardize the accelerometer. The log–log curve of the correlation time–Allan variance is shown in Figure 4, from which we can read the random walk coefficient of the three-axis accelerometer, as shown in

Table 2. The calibration results of the three-axis acceleration random walk coefficient (unit: m/s²·sqrt (Hz)).

	E Axis	N Axis	U Axis
Acceleration random walk coefficient	4.54 × 10−5	2.94 × 10−5	2.05 × 10−5

. This parameter can provide a reference for the assignment of the process noise of the Kalman filter.

In order to analyze and verify the effect of using GNSS/accelerometer coupled technology in landslide monitoring, this study simulated three typical GNSS landslide monitoring scenes: GNSS signal normally locked, GNSS signal partially unlocked, and GNSS signal short-term interruption. For each scene, experiments with two states of landslide stability and sliding were designed for the sake of analysis and verification. Due to the duration during the stage of constant velocity, initial acceleration and intermediate acceleration lasts for a long time (usually serval months and even decades) and the ground acceleration values during these stages are very tiny, which cannot be detected by the accelerometer. In this case, we treat these stages as stability for analysis.

3.1. Scene 1: GNSS Signal Normally Fixed

This study designed static (lasting 400 s) and dynamic (lasting 100 s) experiments to simulate the landslide stabilization and sliding phases. We performed short-baseline RTK processing with GPS + BDS dual-frequency observation data and the lambda algorithm for the quick repair of ambiguity and to obtain high-precision GNSS deformation results. When the integer ambiguity was successfully fixed, the deformation results in both directions of the plane were within 1 cm, and the elevation results were within 2 cm. On this basis, the accelerometer was coupled to verify the effect of the adaptively coupled method in this landslide monitoring scene. The information pertaining to the simulation experiment for each period is shown in

Table 3. The record of the simulated deformation in Scene 1.

Static/Dynamic	Time Period	Deformation
Static period	11:18:20–11:25:00 UTC+ 08:00 (SOW 98,300–98,700 s)	NULL
Dynamic period	11:41:40–11:43:20 UTC+ 08:00 (SOW 99,700–99,800 s)	Swipe east about 19.5 cm

.

Simulation of the Landslide Stability Period

When the landslide was stable, the results of the GNSS/accelerometer coupled solution were in good agreement with those of the GNSS (Figure 5), with the former results appearing smoother and closer to the real state of landslide deformation. Since the deformation of the landslide was approximately equal to 0 when it was stable, this study used 0 as the true value to determine the Root Mean Square (RMS) values of the 3D displacements of the GNSS and coupled solutions, respectively, as shown in

Table 4. RMS statistical results when simulating the landslide stability period in Scene 1 (unit: cm).

	RMS_E	RMS_N	RMS_U
GNSS/accelerometer Coupled	0.11	0.33	0.30
GNSS	0.12	0.34	0.33

. The RMS value of the coupled solution was essentially the same as that of GNSS, but the former was closer to the true value derived in the three directions of ENU. Compared with GNSS, the plane direction of the coupled solution was increased by about 5%, and the elevation direction was increased by about 8%, which also proves that the GNSS/accelerometer coupled method could be used in monitoring operations with stable landslides, and it was able to obtain a more stable monitoring sequence than GNSS alone.

$$X_{RMS}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(X_i-\widehat{X}\right)}^2}{n}}$$

(15)

where $X_{RMS}$ is the RMS of the sample $X$, $n$ is the sample number, $X_i$ denotes the i-th sample, and $\widehat{X}$ is the reference value.

Simulating the Landslide Sliding Period

In the data of the landslide sliding period, the process noise of the Kalman filter was appropriately amplified to prevent abnormal situations, such as overfitting or divergence. It should be noted that when the instrument moved in a certain direction, it would cause the accelerometer to tilt and significantly change its BSE, which in turn led to the rapid divergence of the GNSS/accelerometer coupled results. To restrain this divergence phenomenon, this study treated the error term as an estimated parameter modeled with a random walk process, and in this way, obtained accurate coupled results. Figure 6 shows that the accelerometer BSE would have a short convergence time (3–5 s) at the beginning of the solution, and this would also change in the sliding period of the landslide. Through statistical analysis, we found that the 3D Pearson correlation coefficients of the coupled results and GNSS were all above 95% (E—99.9%, N—99.8%, U—96.7%), which denotes a strong correlation. Moreover, when the landslide was sliding, the noise of the coupled solution was more negligible, which is more in line with the real sliding situation.

3.2. Scene 2: GNSS Signal Blocking Unlocked

After the ambiguity was fixed, the monitoring and positioning accuracy of the GNSS could easily reach the centimeter level [35]. However, in the complex environment of a landslide, due to the signal reflection and occlusion caused by mountain vegetation, the GNSS was liable to being affected by external disturbances such as base station outage and multi-path diffraction. For GNSS float solutions, errors of tens of centimeters or even meters often arise. In this study, we designed landslide stabilization and sliding simulation experiments to analyze the contribution of the GNSS/accelerometer coupled method in Scene 2 for two common GNSS anomalies that occur in landslide monitoring, namely, base station outage and multi-path diffraction, respectively.

3.2.1. Scene 2.1 Base Station Outage

Simulate the Landslide Stability Period

In order to analyze the contribution of the GNSS/accelerometer coupled method in the landslide stabilization period in Scene 2.1, this study selected period 1, shown in

Table 5. Records of simulated landslide experiments in four groups of GNSS anomalies.

GNSS Anomalies	Period Index	Time Period	Deformation
Base station outage	Period 1 (stabilization)	11:05:00–11:05:50 UTC + 08:00 (SOW 97,500–97,550 s)	NULL
	Period 2 (Sliding)	11:31:40–11:33:20 UTC + 08:00 (SOW 98,300–98,700 s)	Swipe south about 20.0 cm Keep for about 25 s
Multi-path diffraction	Period 3 (stabilization)	11:10:00–11:12:30 UTC + 08:00 (SOW 97,800–97,950 s)	NULL
	Period 4 (Sliding)	11:41:40–11:43:20 UTC + 08:00 (SOW 99,700–99,800 s)	Swipe east about 19.5 cm Keep for about 30 s

, and simulated the outage of the base station for 25 s. When the GNSS base station broke off, the RTK algorithm often used the observation of the last epoch of the base station for asynchronous calculation [36,37], but here, the accuracy gradually degraded with the continuance of the outage. Figure 7 shows that after a 20 s outage, the ambiguity fixes failed. This accuracy degradation could be directly reflected in the state covariance value corresponding to the displacement. The plane error of the GNSS float solution reached 0.15 m, and the elevation error reached 2 m. The GNSS/accelerometer coupled solution also behaved abnormally when the regular fixed-weight model was used. When this phenomenon occurred in landslide monitoring operations, it was easy to misjudge the disaster warning. For this reason, we adjusted the GNSS measurement noise based on the abnormal observation variance inflation model [31] to improve the accuracy of the coupled solution. In this case, the result was more in line with the real deformation state, as shown in Figure 8. The essence of this method was to adjust the weighting factor of the accelerometer’s and GNSS solution’s results. The state variance of the time update and the process noise determined the weight ratio of the accelerometer, and the measurement noise of the measurement gain determined the weight ratio of the GNSS. It was found through calculation that when the GNSS float solution arose, the variance inflation model could be used to reduce the weight ratio of the GNSS by about 1000 times, so as to give full play to the accelerometer’s characteristics of high accuracy in the short term and reduce the false alarm rate of landslide disaster warning when the GNSS signal lock was partially lost.

We take zero as the reference for counting the RMS of the coupled solution displacement. As shown in

Table 6. RMS statistical results when simulating the landslide stability period in Scene 2.1 (unit: cm).

GNSS Base Station Outage Duration	RMS_E	RMS_N	RMS_U
5 s	0.08	0.02	0.11
10 s	0.08	0.01	0.43
15 s	0.04	0.10	0.65
20 s	0.06	0.12	1.14
25 s *	0.48	0.16	5.97

* indicates that the GNSS results show continuous float solutions from this moment.

, there was no float solution in the first 20 s of base station outage, and the 3D RMS were values basically within 1 cm. When the outage duration reached 25 s, due to the emergence of GNSS float solutions, the RMS increased a little, reaching values of E—0.48 cm, N—0.16 cm, and U—5.97 cm. However, it remained more noticeable than the decimeter-level GNSS float solution.

Simulate the Landslide Sliding Period

In order to analyze the contribution of the GNSS/accelerometer coupled algorithm during the sliding period of the landslide in Scene 2.1, this study selected period 2, shown in Table 5, and artificially simulated a 25 s base station outage, wherein the RTK fixed solution without base station outage was treated as the reference. As shown in Figure 9, the GNSS float solution deviated from the reference by several tens of centimeters, while the GNSS/accelerometer coupled solution was more consistent with the reference. However, the RMS also increased gradually during the outage of the base station. When the outage duration reached 20 s, the 3D RMS values were E—0.9 cm, N—1.3 cm, and U—1.5 cm, and after 25 s, they were E—6.9 cm, N—7.2 cm, and U—4.8 cm. (Figure 10). It could be found that the divergence trend of the coupled solution was more obvious when the landslide was sliding than when it was stable. The reason is that during the sliding period, the BSE of the accelerometer would continuously change, and when the GNSS float solution appeared, the measurement noise was adaptively adjusted to become larger. Therefore, the estimated state of the Kalman filter showed a more time-updated result. Particularly for the state estimation strategy of BSE, random walk modeling was adopted in this study, which meant that the BSE result tended to be more consistent with the results of the previous epoch, which also meant that the accuracy of the BSE was gradually degraded, resulting in the indirect accumulation of errors.

3.2.2. Scene 2.2 Multipath Reflection and Satellite Occlusion

Simulate the Landslide Stability Period

When multipath reflection occurred, due to the reduction in the signal-to-noise ratio (SNR) of the GNSS satellites, the quality of the observation data was degraded, which would eventually lead to the failure of ambiguity fixing and the GNSS float solution. This phenomenon is very common in GNSS landslide monitoring operations [38], and reduces the accuracy of deformation monitoring to a certain extent. It can also give rise to misjudgments of early landslide disaster warnings. In order to analyze the contribution of the GNSS/accelerometer coupled algorithm during the landslide stabilization period in Scene 2.2, period 3 (Table 5) was selected for analysis. The average multipath error in this period was relatively large, mostly above 0.5 m, leading to many discontinuous float solutions (the float rate was about 12.6%), as shown in Figure 11.

The coupled solution better agreed with the reference after using the variance inflation model to adaptively adjust the measurement noise (Figure 12). We selected two periods (SOW: 97,812–97,830 s; SOW: 9789–97,903 s) with relatively more float results and measured their RMS with reference to zero. The statistical results show that the maximum 3D RMS values of the two periods were E—1.00 cm, N—0.57 cm, and U—1.18 cm, and E—3.14 cm, N—3.28 cm, and U—2.82 cm, respectively (Figure 13). Both appeared after continuous (>5 s) float solutions. It is worth noting that when the coupled solution showed a divergent trend, the measurement update of the fixed solution with only one epoch of GNSS helped the result to re-converge, because the GNSS high-precision measurement information re-corrected the BSE of the accelerometer.

Simulate the Landslide Sliding Period

In order to analyze the contribution of the GNSS/accelerometer coupled algorithm during the sliding period in Scene 2.2, this study selected period 4 (Table 5) and simulated a satellite occlusion lasting 15 s (Table 5, period 4). During the occlusion period, only 4–5 GPS satellites with low SNR (below 40 dB) were retained, and the RTK fixed solution without satellite occlusion was also used as the reference, as shown in Figure 14. The results show that extreme satellite occlusion was more likely to lead to the failure of ambiguity fixing. The 3D RMS values of the GNSS float solutions reached up to 1.9 m, while the coupled solution was more consistent with the reference, but still diverged with the continuation of the GNSS float solution. When the GNSS float solution appeared continuously for 20 s, the 3D RMS values of the coupled solution were E—3.09 cm, N—0.39 cm, and U—1.14 cm, respectively (

Table 7. RMS statistical results when simulating landslide sliding period in Scene 2.2 (unit: cm).

Duration of Satellite Occlusion	RMS_E	RMS_N	RMS_U
5 s *	0.82	0.15	0.34
10 s	1.21	0.31	0.58
15 s	1.83	0.31	0.85
20 s	3.09	0.39	1.14

* indicates that the GNSS results contain continuous float solutions from this moment.

). Among these, the E direction was the primary sliding direction of the landslide. Since the accuracy of the accelerometer’s BSE in this direction was degraded, significant errors accumulated.

3.3. Scene 3: GNSS Signal Short-Term Interruption

In the GNSS landslide monitoring operation, the monitoring results were often missing due to equipment failures, imperfect data preprocessing algorithms, etc., which would lead directly to early landslide disaster warning misjudgments. In this study, we designed simulated landslide stability and sliding experiments for Scene 3 (

Table 8. Simulation record of landslide deformation in Scene 3.

Period Index	Time	Simulated Landslide Deformation
Period 1 (Stability)	11:05:00–11:05:50 GMT + 08:00 (SOW 97,500–97,950 s)	NULL
Period 2 (Sliding)	11:35:00–11:36:40 GMT + 08:00 (SOW 99,300–99,400 s)	Swipe north about 32.4 cm

) to analyze the contribution of the GNSS/accelerometer coupled algorithm in this scenario.

Simulate the Landslide Stability Period

In order to analyze the contribution of the GNSS/accelerometer coupled algorithm when the landslide was stable in Scene 3, this study selected period 1 (Table 8) and simulated the interruption period of 97,800–97,820 s, during which the BSE remained unchanged. It should be noted that, before the GNSS interruption, the coupled process had proceeded for 200 s, and the accelerometer’s BSE had been correctly estimated, so the deformation result of the initial epoch was highly accurate. However, this error showed obvious random walk characteristics. If the BSE remained unchanged, the original data of the accelerometer could not be corrected and compensated for during GNSS interruption, which could very likely give rise to the accumulation of errors, as shown in Figure 15. With zero as the reference, we assessed the RMS every 5 s during the interruption period, which diverged over time. As shown in

Table 9. RMS statistical results when simulating the landslide stability period in Scene 3 (unit: cm).

GNSS Interruption Duration	RMS_E	RMS_N	RMS_U
5 s	0.24	0.02	0.05
10 s	0.61	0.24	0.25
15 s	0.92	0.92	0.60
20 s	1.11	1.82	1.04

, the 3D RMS values were E—0.61 cm, N—0.24 cm, and U—0.25 cm when the GNSS was interrupted for 10 s, and reached E—1.11 cm, N—1.82 cm, and U—1.04 cm when the interrupted duration was 20 s. Therefore, when the GNSS is interrupted, the deformation results assessed by the accelerometer cannot be used as the basis for early warning judgments, owing to its strong tendencies. However, in the simulated landslide stability period, the raw observations of the accelerometer did not change significantly when the GNSS was interrupted, and the accuracy was not affected by the interference of external signals. In this case, this system could be used as the basis in another area to avoid early disaster warning misjudgments during short-term GNSS signal interruption.

Simulate the Landslide Sliding Period

In order to analyze the contribution of the GNSS/accelerometer coupled algorithm during the landslide sliding period in Scene 3, this study selected period 2 (Table 8), and simulated a GNSS signal interruption lasting 10 s. As shown in Figure 16, the coupled solution diverged faster than that in the landslide stabilization period. Taking the RTK fixed solution without GNSS interruption as the reference, we measured the RMS of the coupled solution, as shown in

Table 10. RMS statistical results when simulating landslide sliding period in Scene 3 (unit: cm).

GNSS Interruption Duration	RMS_E	RMS_N	RMS_U
5 s	1.08	1.27	0.57
10 s	4.10	6.84	2.30

. The 3D RMS values when the GNSS was interrupted for 10 s were E—4.10 cm, N—6.84 cm, and U—2.30 cm, respectively. The direction with the largest RMS was N, which remained the main sliding direction of the landslide. Therefore, in the period of landslide sliding, the deformation results given by the accelerometer are not suitable for use as the basis for early disaster warning judgments; however, the raw observations of the accelerometer fluctuate significantly, which can give some auxiliary judgment information.

4. Conclusions

Landslides instability is categorized as slow deformation, which is fundamentally different from the high-frequency vibration encountered in bridge monitoring and the co-seismic high-frequency displacement resulting from earthquakes. There are also problems such as GNSS signal occlusion and diffraction, which renders the regular GNSS/accelerometer coupled technology unable to be directly applied to landslide disaster monitoring. To solve this problem, the variance inflation model was used to improve the robustness of the GNSS/accelerometer coupled algorithm. At the same time, this study designed a series of simulated landslide monitoring experiments by means of a simulated deformation platform. The results were as follows:

(1)

When the GNSS signal was normally locked, the accuracy was comparable to that of the coupled solution with an accelerometer, and the 3D RMS values were all within 1 cm.

(2)

When the GNSS signal lock was partially lost, after closed-loop correcting the raw accelerometer observation by estimating the BSE and introducing the variance inflation model, the proposed algorithm could effectively suppress the GNSS low-precision solution and more effectively restore the deformation sequence of the landslide.

(3)

When the GNSS was interrupted for a short time and unable to output any monitoring information, since the BSE cannot be updated by filtering, the accuracy of the results given by the accelerometer would degrade faster when the landslide was sliding than when it was stable. However, the raw observation of the accelerometer could yield the landslide motion state at a high resolution, which could perhaps be used as a real-time identification indicator for early disaster warning.