Rapid Deformation Identification and Adaptive Filtering with GNSS TDCP Under Different Scenarios and Its Application in Landslide Monitoring

Abstract

Global navigation satellite system (GNSS) real-time kinematic (RTK) has been widely applied in landslide monitoring and warning, since it can provide real-time and high-precision three-dimensional deformation information in all weather and all the time. The Kalman filter is often adopted for parameter estimation in GNSS RTK positioning since it can effectively suppress the observational noise and improve the positioning accuracy and reliability. However, the discrepancy between the empirical state model in the Kalman filter and the actual state of the monitoring object could lead to large positioning errors or even the divergence of the Kalman filter. In this contribution, we propose a novel rapid deformation identification and adaptive filtering approach with GNSS time-differenced carrier phase (TDCP) under different scenarios for landslide monitoring. We first present the methodology of the proposed TDCP-based rapid deformation identification and adaptive filtering approach for GNSS RTK positioning. The effectiveness of the proposed approach is then validated with a simulated displacement experiment with a customized three-dimensional displacement platform. The experimental results demonstrate that the proposed approach can accurately and promptly identify the rapid between-epoch deformation of more than approximately 1.5 cm and 3.0 cm for the horizontal and vertical components for the monitoring object under a complex observational environment. Meanwhile, it can effectively suppress the observational noise and thus maintain mm-to-cm-level monitoring accuracy. The proposed approach can provide high-precision and reliable three-dimensional deformation information for GNSS landslide monitoring and early warning.

1. Introduction

Landslides are damaging and deadly geological hazards that affect many countries. Their monitoring and early warning is therefore crucial for preventing and controlling the casualties and property losses [1,2,3,4,5]. Global navigation satellite system (GNSS) has been successfully and widely applied in landslide monitoring and warning over the past few decades, since it can provide real-time and high-precision three-dimensional deformation information in all weather and all the time [4,6,7,8,9,10,11].

The two most commonly used approaches for GNSS landslide monitoring are the static relative positioning and the real-time kinematic (RTK) positioning techniques. The application of network RTK in landslide monitoring has been investigated recently but is still in the development stage [4,12,13]. The GNSS static relative positioning technique was first used in landslide monitoring and could achieve mm-level accuracy with hours of continuous observations [14]. However, it is in fact a post-processing positioning mode and thus cannot capture the real-time deformation or surface displacement of the landslides. It is therefore only suitable for the long-term monitoring of the landslide that is in the steady state.

With the development of GNSS technology, the RTK positioning has been gradually applied to realize the real-time, continuous, and long-term monitoring of the landslides, and has currently become the most widely used GNSS landslide-monitoring technology [4,15,16]. The RTK positioning usually adopts the double-differenced (DD) model to eliminate or largely reduce the common errors in the satellite, receiver, and signal propagation path, and retain the integer nature of the ambiguities. The real-time centimeter- or even millimeter-level DD ambiguity-fixed solutions can therefore be easily obtained for short baselines [14,17]. However, due to the impact of observational noise and other unmodeled errors, the achievable accuracy of RTK positioning is inferior to that of the static relative positioning, and spike-like anomalies often contain in the obtained displacement series, especially under complex observational environments that is often the case for GNSS landslide-monitoring applications [4].

In order to suppress the impact of observational noise and enhance the accuracy and reliability of RTK positioning, the Kalman filter is often adopted for parameter estimation since it can effectively integrate the predicted-state information with the observation information [18,19]. Specifically, the empirical-state model of a random walk with a relatively small process noise can be usually adopted to constrain the three-dimensional coordinate states of the monitoring station for the landslide that is in a steady state. However, the landslide is not always in a steady state; it usually experiences three stages of deformation, i.e., initial deformation, stable deformation, and accelerated deformation [12]. The deformation of the landslide exhibits different characteristics at different stages. The landslide will experience rapid and significant deformation if it is in the accelerated deformation stage. In that case, the three-dimensional coordinates of monitoring station will experience rapid and significant variation between adjacent epochs. The inappropriate empirical process noise of the Kalman filter will lead to discrepancy between the empirical-state model in the Kalman filter and the actual state of the monitoring station, which will further result in large positioning errors or even the divergence of the Kalman filter [19]. As a consequence of this, the deformation of the landslide cannot be exactly detected and may lead to missed detection of the landslide. Therefore, it is critical to promptly and accurately identify the rapid and large deformation information of the landslide.

The time-differenced carrier phase (TDCP) algorithm can provide accurate between-epoch displacement based on carrier-phase observations from a single station. When no cycle slip occurs, TDCP can eliminate the integer ambiguities and largely reduce most of the errors (such as the ionospheric and tropospheric errors) that are contained in carrier-phase observations, and therefore provides high-precision displacements (or position variations) between consecutive epochs with millimeter-level accuracy for a single GNSS receiver. The TDCP algorithm has been previously widely applied to determine the velocity of the moving vehicles [20,21,22,23]. As a matter of fact, the displacement that identified by the TDCP algorithm can also be input into the state equation of Kalman filter to adaptively adjust the state transition of the coordinates of monitoring station, and thus maintain estimates that are consistent with the actual state of the landslide.

Therefore, in this contribution, we propose a novel rapid deformation identification and adaptive filtering approach with TDCP under different scenarios for landslide monitoring. In this approach, the environmental condition of the monitoring station is first identified according to the quality indicators of the TDCP solution, such as the maximum and root-mean-square (RMS) errors of the posteriori residuals, etc. Then, whether rapid deformation occurs is identified according to the magnitude of the epoch-by-epoch TDCP displacement under the identified environmental conditions. Finally, the obtained between-epoch displacement is introduced as an additional information to adaptively adjust the process noise of the state equation in the Kalman filter and thus keep it consistent with the between-epoch deformation of the landslide. The proposed approach can accurately and promptly identify the rapid deformation and the monitoring object under a complex observational environment. Meanwhile, it can effectively suppress the observational noise and thus provide high-precision three-dimensional deformation information for GNSS landslide monitoring and early warning. In this contribution, we will first present the fundamental principle and procedure of the proposed TDCP-based rapid deformation identification and adaptive filtering approach applied to different scenarios. Then, the performance of the proposed approach will be evaluated with a simulated displacement experiment with a customized three-dimensional displacement platform.

The remainder of this article is organized as follows. Section 2 introduces the mathematical model of GNSS RTK positioning for short baselines. In Section 3, we first present the principle of a Kalman filter applied to GNSS RTK positioning and the approach of displacement determination with TDCP, and then present the procedure of the TDCP-based rapid deformation identification and adaptive Kalman filtering for RTK positioning under different scenarios. In Section 4, we evaluate the performance of the proposed approach with a simulated displacement experiment. Finally, some conclusions and discussions are summarized in Section 5.

2. Mathematical Model of GNSS RTK Positioning for Short Baseline

Currently, the GNSS RTK positioning has been widely used in the real-time high-precision deformation monitoring of landslides. By using the DD model, the short-baseline RTK positioning technology can eliminate the satellite-specific and receiver-specific errors such as the satellite and receiver clock errors, the satellite and receiver initial-phase biases, and the satellite and receiver code and phase hardware delays, etc. Moreover, the ionospheric and tropospheric delays are negligible for short baselines. It can therefore achieve real-time centimeter- or even millimeter-level baseline vectors once the DD ambiguities are successfully resolved [17]. In this section, we will present the mathematical model of GNSS RTK positioning for short baselines.

For the reference and monitoring stations $b$, $r$ and satellites $s_p$, $s_q$ (from GNSS $s$), the code and carrier-phase DD observation equation at frequency $j$ is given as:

$$\begin{array}{l}\mathbf{\nabla }\mathbf{\Delta }{P}_{br,j}^{s_p{s}_q}=\mathbf{\nabla }\mathbf{\Delta }{\rho }_{br,j}^{s_p{s}_q}+\mathbf{\nabla }\mathbf{\Delta }{e}_{br,j}^{s_p{s}_q}\\ {\mathbf{\nabla }\mathbf{\Delta }}_{br,j}^{s_p{s}_q}=\mathbf{\nabla }\mathbf{\Delta }{\rho }_{br,j}^{s_p{s}_q}+{\lambda }_j\cdot \mathbf{\nabla }\mathbf{\Delta }{N}_{br,j}^{s_p{s}_q}+\mathbf{\nabla }\mathbf{\Delta }{\epsilon }_{br,j}^{s_p{s}_q}\end{array}$$

(1)

where $\mathbf{\nabla }\mathbf{\Delta }$ is the DD operator; $P$ and $\varphi$ are the code and carrier-phase observations in meters, respectively; $\rho$ is the geometric distance between the satellite and receiver antenna; ${\lambda }_j$ is the wavelength corresponding to frequency $j$; $N$ is the integer ambiguity; and $e$ and $\epsilon$ are the sum of observational noise and unmodelled errors for code and phase, respectively.

Assuming that the reference and monitoring stations simultaneously track $n$ satellites ($s_1,s_2,\cdots s_n$) on $f$ frequencies ($j=1,2,\cdots f$), the linearized functional model and stochastic model for short baseline RTK positioning is then given as [24]:

$$\begin{array}{l}\mathrm{E}\left[\begin{array}{c}\mathbf{\nabla }\mathbf{\Delta }\mathit{P}\\ \mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi }\end{array}\right]=\left[\begin{array}{cc}\mathit{G}& \mathbf{0}\\ \mathit{G}& \mathit{\Lambda }\end{array}\right]\left[\begin{array}{c}\mathbf{\Delta }\mathit{X}\\ \mathbf{\nabla }\mathbf{\Delta }\mathit{N}\end{array}\right]\\ \mathrm{D}\left[\begin{array}{c}\mathbf{\nabla }\mathbf{\Delta }\mathit{P}\\ \mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi }\end{array}\right]=\left[\begin{array}{cc}{\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{P}}& \mathbf{0}\\ \mathbf{0}& {\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi }}\end{array}\right]\end{array}$$

(2)

where $\mathrm{E}\left[\cdot \right]$ and $\mathrm{D}\left[\cdot \right]$ are the expectation and dispersion operators, respectively. $\mathbf{\Delta }X$ are the expectation and dispersion operators, respectively., $\mathbf{\nabla }\mathbf{\Delta }\mathit{P}={\left[\mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_1,\cdots ,\mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_f\right]}^{\mathrm{T}}$ is the DD code vector with $\mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_j=\left[\mathbf{\nabla }\mathbf{\Delta }{P}_{br,j}^{s_1{s}_2},\cdots ,\mathbf{\nabla }\mathbf{\Delta }{P}_{br,j}^{s_1{s}_n}\right]$, $\mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi }={\left[\mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_1,\cdots ,\mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_f\right]}^{\mathrm{T}}$ is the DD phase vector with $\mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_j=\left[\mathbf{\nabla }\mathbf{\Delta }{\varphi }_{br,j}^{s_1{s}_2},\cdots \mathbf{\nabla }\mathbf{\Delta }{\varphi }_{br,j}^{s_1{s}_n}\right]$, and $\mathbf{\nabla }\mathbf{\Delta }\mathit{N}={\left[\mathbf{\nabla }\mathbf{\Delta }{\mathit{N}}_1,\cdots ,\mathbf{\nabla }\mathbf{\Delta }{\mathit{N}}_f\right]}^{\mathrm{T}}$ is DD ambiguity vector with $\mathbf{\nabla }\mathbf{\Delta }{\mathit{N}}_j=\left[\mathbf{\nabla }\mathbf{\Delta }{N}_{br,j}^{s_1{s}_2},\cdots ,\mathbf{\nabla }\mathbf{\Delta }{N}_{br,j}^{s_1{s}_n}\right]$. The entries of the design matrix are given as:

$$\begin{array}{l}\mathit{G}={\mathit{e}}_f\mathbf{\otimes }\left(\mathit{D}\cdot \mathit{g}\right) \mathrm{with} \mathit{D}=\left[\begin{array}{cc}-{\mathit{e}}_{n-1}& {\mathit{I}}_{n-1} \end{array}\right]\\ \mathrm{\Lambda }=\mathrm{diag}\left[{\lambda }_1,\cdots ,{\lambda }_f\right]\mathbf{\otimes }{\mathit{I}}_{n-1} \end{array}$$

(3)

where $\mathbf{\otimes }$ denotes the Kronecker product, ${\mathit{I}}_{n-1}$ is the $\left(n-1\right)\times \left(n-1\right)$ identity matrix, ${\mathit{e}}_{n-1}$ is the vector of $\left(n-1\right)$ 1s, $\mathit{g}={\left[{\mathit{g}}_{s_1}^{\mathrm{T}},\cdots ,{\mathit{g}}_{s_n}^{\mathrm{T}}\right]}^{\mathrm{T}}$ is the geometry matrix that contains the receiver–satellite unit direction vectors from the monitoring station to $n$ satellites, and $\mathit{D}$ is the transformation matrix.

Provided that the observations at each frequency are independent and have the same precision, and the phase and code observations are independent, the entries of the positive definite variance–covariance (VC) matrix are then given as:

$$\begin{array}{l}{\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi }}=\mathit{D}\cdot ({\mathit{I}}_f\mathbf{\otimes }\mathit{W})\cdot {\mathit{D}}^{\mathrm{T}},{\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{P}}=\mathit{D}\cdot ({\mathit{I}}_f\mathbf{\otimes }{\mathit{W}}_P)\cdot {\mathit{D}}^{\mathrm{T}}\\ \mathit{W}=\mathrm{diag}\left[2\cdot {\sigma }_{,s_1}^2,\cdots ,2\cdot {\sigma }_{,s_n}^2\right],{\mathit{W}}_P=\mathrm{diag}\left[2\cdot {\sigma }_{P,s_1}^2,\cdots ,2\cdot {\sigma }_{P,s_n}^2\right]\end{array}$$

(4)

where ${\sigma }_{\varphi ,s_q}^2\left(q=1,\cdots ,n\right)$ and ${\sigma }_{P,s_q}^2\left(q=1,\cdots ,n\right)$ are undifferenced phase and code deviations. In this contribution, the widely used elevation-dependent model is adopted, which is given as [25]:

$${\sigma }^2=a^2+b^2/{\sin }^2\left(\theta \right)$$

(5)

where $\sigma$ is the standard deviation of undifferenced observations. $a$ and $b$ are the empirical coefficients, and they are set as 3 mm and 0.3 m for phase and code observations, respectively, $\theta$ is the elevation angle of the satellite.

For multi-GNSS observations, we adopt the classical loosely combined inter-system model, in which DD observations are created within each GNSS separately [17]. As a representative example, the linearized mathematical model for combined Global Positioning System (GPS) and BeiDou navigation satellite system (BDS) RTK positioning can be further given as:

$$\begin{array}{l}\left[\begin{array}{c}\mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_g\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_b\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_g\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_b\end{array}\right]=\left[\begin{array}{ccc}{\mathit{G}}_g& \mathbf{0}& \mathbf{0}\\ {\mathit{G}}_b& \mathbf{0}& \mathbf{0}\\ {\mathit{G}}_g& {\mathit{\Lambda }}_g& \mathbf{0}\\ {\mathit{G}}_b& \mathbf{0}& {\mathit{\Lambda }}_b\end{array}\right]\left[\begin{array}{c}\mathbf{\Delta }\mathit{X}\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{N}}_g\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{N}}_b\end{array}\right]\\ \mathrm{D}\left[\begin{array}{c}\mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_g\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{P}}_b\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_g\\ \mathbf{\nabla }\mathbf{\Delta }{\mathit{\varphi }}_b\end{array}\right]=\left[\begin{array}{cccc}{\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{P},g}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{P},b}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi },g}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& {\mathit{Q}}_{\mathbf{\nabla }\mathbf{\Delta }\mathit{\varphi },b}\end{array}\right]\end{array}$$

(6)

where the subscript “g” and “b” represent the GPS and BDS, respectively.

3. Rapid Deformation Identification and Adaptive Kalman Filtering with TDCP Under Different Scenarios

3.1. Parameter Estimation with Kalman Filter

The Kalman filter is a set of mathematical equations that provides an efficient and computational means to recursively obtain the optimal-state estimates of a process, by integrating the state model (or information) with the observation model (or noisy observations) [26,27]. The Kalman filter has been applied in extensive research and applications such as GNSS positioning [28,29,30,31,32,33,34].

The state equation (without the system control vector for simplicity) and observation equation of the Kalman filter are given as:

$$\begin{array}{l}{\mathit{X}}_k={\mathbf{\Phi }}_{k,k-1}{\mathit{X}}_{k-1}+{\mathit{\omega }}_k\\ {\mathit{Z}}_k={\mathit{H}}_k{\mathit{X}}_k+{\mathbf{\Delta }}_k\end{array}$$

(7)

where ${\mathit{X}}_k$ and ${\mathit{X}}_{k-1}$ represent the state vectors in the current epoch $k$ and previous epoch $k-1$, respectively; ${\mathbf{\Phi }}_{k,k-1}$ represents the state-transition matrix applied to project the previous state ${\mathit{X}}_{k-1}$ to the current state ${\mathit{X}}_k$; ${\mathit{Z}}_k$ represents the observation vector in the current epoch $k$; ${\mathit{H}}_k$ is the corresponding design matrix; and ${\mathit{\omega }}_k$ and ${\mathbf{\Delta }}_k$ are the process noise vector and the measurement noise vector, respectively. The process noise and the measurement noise are assumed to be independent with each other, white, and with normal probability distribution.

In the short baseline RTK model, the estimated state ${\mathit{X}}_k$ includes the three-dimensional baseline coordinates of the monitoring station and the DD ambiguities, i.e., ${\mathit{X}}_k={\left[\begin{array}{c}\mathbf{\Delta }\mathit{X}\\ \mathbf{\nabla }\mathbf{\Delta }\mathit{N}\end{array}\right]}_k$. The DD ambiguities are time constant if no cycle slip occurs. Moreover, the three-dimensional baseline coordinates remain stable or change only slightly when the monitoring object is in a steady state. The parameters states are therefore modeled as rank walk and the transition matrix is an identity matrix, i.e., ${\mathbf{\Phi }}_{k,k-1}=\mathit{I}$.

The Kalman filter is typically implemented in two steps, i.e., the time update step and measurement update step [27]. In the time update step, the state vector and its VC matrix in the current epoch are predicted by using their counterparts in the previous epoch with the state equation as:

$$\begin{array}{l}{\widehat{\mathit{X}}}_{k,k-1}={\widehat{\mathit{X}}}_{k-1}\\ {\widehat{\mathit{P}}}_{{\widehat{X}}_{k,k-1}}={\widehat{\mathit{P}}}_{X_{k-1}}+{\mathit{Q}}_{{\mathit{\omega }}_k}\end{array}$$

(8)

where ${\widehat{\mathit{X}}}_{k,k-1}$ and ${\widehat{\mathit{P}}}_{{\widehat{\mathit{X}}}_{k,k-1}}$ denote the predicted state vector and its VC matrix in the current epoch, respectively; ${\mathit{P}}_{{\widehat{\mathit{X}}}_{k-1}}$ is the VC matrix of ${\widehat{\mathit{X}}}_{k-1}$; and ${\mathit{Q}}_{{\mathit{\omega }}_k}$ is the system process noise matrix.

In the measurement update step, the predicted states are adjusted with the raw measurements in the current epoch to obtain the posteriori state estimates, which is given as the following measurement update equation:

$$\begin{array}{l}{\widehat{\mathit{X}}}_k={\widehat{\mathit{X}}}_{k,k-1}+{\mathit{K}}_k\cdot {\mathit{r}}_k\\ {\mathit{P}}_{{\widehat{\mathit{X}}}_k}=(\mathit{I}-{\mathit{K}}_k{\mathit{H}}_k){\mathit{P}}_{{\widehat{\mathit{X}}}_{k,k-1}}{(\mathit{I}-{\mathit{K}}_k{\mathit{H}}_k)}^{\mathrm{T}}+{\mathit{K}}_k{\mathit{Q}}_{{\mathbf{\Delta }}_k}{\mathit{K}}_k^{\mathrm{T}}\end{array}$$

(9)

with

$$\begin{array}{l}{\mathit{K}}_k={\mathit{P}}_{{\widehat{\mathit{X}}}_{k,k-1}}{\mathit{H}}_k^{\mathrm{T}}{\left[{\mathit{H}}_k{\mathit{P}}_{{\widehat{\mathit{X}}}_{k,k-1}}{\mathit{H}}_k^{\mathrm{T}}+{\mathit{Q}}_{{\mathbf{\Delta }}_k}\right]}^{-1}\\ {\mathit{r}}_k={\mathit{Z}}_k-{\mathit{H}}_k{\widehat{\mathit{X}}}_{k,k-1}\end{array}$$

(10)

where ${\widehat{\mathit{X}}}_k$ and ${\widehat{\mathit{P}}}_k$ are the estimated state vector for the current epoch and its corresponding VC matrix, respectively. ${\mathit{K}}_k$ is the Kalman gain, which minimizes the trace of the VC matrix of the posteriori state estimates; ${\mathit{r}}_k$ is the measurement innovation, which reflects the discrepancy between the predicted measurement (${\mathit{H}}_k{\widehat{\mathit{X}}}_{k,k-1}$) and the actual measurement (${\mathit{Z}}_k$); ${\mathit{Q}}_{{\mathbf{\Delta }}_k}$ is the observation noise VC matrix. It is obvious that the filtered estimate of the current epoch ${\widehat{\mathit{X}}}_k$ depends on both the system process noise ${\mathit{Q}}_{{\mathit{\omega }}_k}$ and the measurement noise ${\mathit{Q}}_{{\mathbf{\Delta }}_k}$. As the system process noise approaches zero, the Kalman gain ${\mathit{K}}_k$ weights the measurement innovation less heavily (or weights the predicted-state estimates more heavily). In that case, the filtered-state estimates depend more on the predicted-state estimates rather than raw observations.

The state equation of Kalman filter (Equations (7) or (8)) describes the variation in the state vector between adjacent epochs, which in practice can often be obtained with useful models by analytical formulations and laboratory-based measurements. For GNSS landslide monitoring with RTK positioning, a relatively small empirical process noise will be given for the DD ambiguities ($\mathbf{\nabla }\mathbf{\Delta }\mathit{N}$) considering that they are time constant while no cycle slip occurs. Moreover, for the three-dimensional baseline coordinates ($\mathbf{\Delta }\mathit{X}$), they are often assumed to remain unchanged or only exhibit insignificant variation when the monitoring object is in a steady state and no displacement occurs. A relatively small empirical process noise is therefore adopted to capture this feature. In that case, the filtered-state estimates depend more on the predicted-state estimates than the raw observations, and the impact of the observational noise on the estimates is suppressed. The improved accuracy and reliability of RTK positioning can therefore be achieved.

However, when rapid deformation occurs in the landslide monitoring objects, the baseline coordinates will change significantly between adjacent epochs. In that case, if the state equation in Equation (8) with a relatively small system process noise is still used, a significant discrepancy will exist between the predicted state of the Kalman filter and the actual state. The filtered state estimates will therefore largely depend on the predicted-state estimates in the current epoch or the filtered-state estimates in the previous epoch, and results in a large positioning error, incorrectly fixed ambiguities, and even the divergence of Kalman filter. As a consequence of this, the estimated states cannot capture the rapid deformation that occurs in the landslide-monitoring objects.

For GNSS landslide monitoring with Kalman filter-based RTK positioning, it is therefore critical to promptly and accurately identify the rapid and large deformation information of the landslide, and set appropriate process noise for the position states in the Kalman filter.

3.2. Between-Epoch Displacement Determination with TDCP

The TDCP algorithm has been widely used in real-time velocity determination for GNSS receivers. Through the differencing of the phase observations between adjacent epochs, the TDCP algorithm can provide between-epoch displacement (or position variations) with millimeter-level accuracy for a single station [20,23]. In this contribution, the displacement between adjacent epochs that obtained by the TDCP algorithm will be used to identify the rapid displacement of the monitoring landslide. Furthermore, the identified displacement will be input into the time update step of the Kalman filter to adaptively adjust the filtering process.

The mathematical model of the TDCP method can be given as the following.

Assuming that no cycle slip occurs in the phase observations, the TDCP observation equation can be obtained by the first-order difference in phase observations between adjacent epochs:

$$\mathbf{\Delta }{\mathit{\varphi }}_{r,j}^s=\mathbf{\Delta }{\rho }_r^s-\mathbf{\Delta }d{t}^s+\mathbf{\Delta }d{t}_r+\mathbf{\Delta }{\epsilon }_{r,j}^s$$

(11)

where $\mathbf{\Delta }$ is the first-order difference between epochs, i.e., $\mathbf{\Delta }\left(·\right)={\left(·\right)}_k-{\left(·\right)}_{k-1}$; $d{t}^s$ and $d{t}_r$ are the satellite and the receiver clock bias in meters, respectively; ${\epsilon }_{r,j}^s$ is the sum of observational noise and other unmodelled errors. The integer ambiguity is canceled out in the absence of cycle slips. Moreover, since the variation in ionospheric and tropospheric delays between adjacent epochs is generally negligible, they are omitted in the formula for simplification.

The geometric term $\mathbf{\Delta }{\rho }_r^s$ in Equation (11) can be rewritten as:

$$\mathbf{\Delta }{\rho }_r^s={\mathit{g}}_{r,k}^s\cdot \left({\mathit{X}}_k^s-{\mathit{X}}_k^r\right)-{\mathit{g}}_{r,k-1}^s\cdot \left({\mathit{X}}_{k-1}^s-{\mathit{X}}_{k-1}^r\right)$$

(12)

where ${\mathit{X}}_k^s$ and ${\mathit{X}}_k^r$ are the three-dimensional geometric vectors for the receiver and satellite in the current epoch k, respectively, ${\mathit{X}}_{k-1}^s$ and ${\mathit{X}}_{k-1}^r$ are their counterparts in the previous epoch $k-1$, respectively; ${\mathit{g}}_{r,k}^s$ and ${\mathit{g}}_{r,k-1}^s$ are the receiver–satellite unit direction vectors in epochs $k$ and $k-1$, respectively, which are given as:

$${\mathit{g}}_{r,k}^s={\displaystyle \frac{{\mathit{X}}_k^s-{\mathit{X}}_k^r}{‖{\mathit{X}}_k^s-{\mathit{X}}_k^r‖}},{\mathit{g}}_{r,k-1}^s={\displaystyle \frac{{\mathit{X}}_{k-1}^s-{\mathit{X}}_{k-1}^r}{‖{\mathit{X}}_{k-1}^s-{\mathit{X}}_{k-1}^r‖}}$$

(13)

The position of the rover receiver in the current epoch $k$ can be expressed as its position in the previous epoch $k-1$ plus its displacement between adjacent epochs, i.e., ${\mathit{X}}_k^r={\mathit{X}}_{k-1}^r+\mathbf{\Delta }{\mathit{X}}^r$. Therefore, the geometric term $\mathbf{\Delta }\rho$ can be further rewritten as:

$$\begin{array}{l}\mathbf{\Delta }{\rho }_r^s={\mathit{g}}_{r,k}^s\cdot \left[{\mathit{X}}_k^s-\left({\mathit{X}}_{k-1}^r+\mathbf{\Delta }{\mathit{X}}^r\right)\right]-{\mathit{g}}_{r,k-1}^s\cdot \left({\mathit{X}}_{k-1}^s-{\mathit{X}}_{k-1}^r\right)\\ =\left({\mathit{g}}_{r,k}^s\cdot {\mathit{X}}_k^s-{\mathit{g}}_{r,k-1}^s\cdot {\mathit{X}}_{k-1}^s\right)-\left({\mathit{g}}_{r,k}^s\cdot {\mathit{X}}_{k-1}^r-{\mathit{g}}_{r,k-1}^s\cdot {\mathit{X}}_{k-1}^r\right)-{\mathit{g}}_{r,k}^s\cdot \mathbf{\Delta }{\mathit{X}}^r\\ =\mathbf{\Delta }{\mu }_r^s-{\mathit{g}}_{r,k}^s\cdot \mathbf{\Delta }{\mathit{X}}^r\end{array}$$

(14)

where $\mathbf{\Delta }{\mu }_r^s=\left({\mathit{g}}_{r,k}^s\cdot {\mathit{X}}_k^s-{\mathit{g}}_{r,k-1}^s\cdot {\mathit{X}}_{k-1}^s\right)-\left({\mathit{g}}_{r,k}^s\cdot {\mathit{X}}_{k-1}^r-{\mathit{g}}_{r,k-1}^s\cdot {\mathit{X}}_{k-1}^r\right)$.

The TDCP equation is therefore given as:

$$\mathbf{\Delta }{\overline{\mathit{\varphi }}}_{r,j}^s=\mathbf{\Delta }{\mathit{\varphi }}_{r,j}^s-\mathbf{\Delta }{\mu }_r^s+\mathbf{\Delta }d{t}^s=-{\mathit{g}}_{r,k}^s\cdot \mathbf{\Delta }{\mathit{X}}^r+\mathbf{\Delta }d{t}_r+\mathbf{\Delta }{\epsilon }_{s,j}^r$$

(15)

The linearized TDCP observation for $n$ satellites ($s_1,s_2,\cdots s_n$) on frequency $j$ is then given as:

$$\begin{array}{l}\mathrm{E}\left[\mathbf{\Delta }{\overline{\mathit{\varphi }}}_{r,j}^s\right]=\left[\begin{array}{cc}-{\mathit{g}}_{r,k}^s& {\mathit{e}}_n\end{array}\right]\left[\begin{array}{c}\mathbf{\Delta }{\mathit{X}}^r\\ \mathbf{\Delta }d{t}_r\end{array}\right]\\ \mathrm{D}\left[\mathbf{\Delta }{\overline{\mathit{\varphi }}}_{r,j}^s\right]=\mathrm{diag}\left[2\cdot {\sigma }_{,s_1}^2,\cdots ,2\cdot {\sigma }_{,s_n}^2\right]\end{array}$$

(16)

where $\mathbf{\Delta }{\overline{\mathit{\varphi }}}_{r,j}^s={\left[\mathbf{\Delta }{\overline{\varphi }}_{r,j}^{s_1},\cdots ,\mathbf{\Delta }{\overline{\varphi }}_{r,j}^{s_n}\right]}^{\mathrm{T}}$, ${\mathit{g}}_{r,k}^s={\left[{\left(\mathbf{\Delta }{\mathit{g}}_r^{s_1}\right)}^{\mathrm{T}},\cdots ,{\left(\mathbf{\Delta }{\mathit{g}}_r^{s_n}\right)}^{\mathrm{T}}\right]}^{\mathrm{T}}$.

The estimated parameters in the TDCP equation include the three-dimensional displacement ($\mathbf{\Delta }{\mathit{X}}^r$) and the variation in receiver clock bias ($\mathbf{\Delta }d{t}_r$) between the current and previous epochs, respectively. They can be estimated together with their VC matrix using the least-squares estimator.

3.3. Procedure of Rapid Deformation Identification and Adaptive Kalman Filtering with TDCP

In the Kalman filtering algorithm, it is critical to match the predicted state to the actual state of the monitoring objects. The accurate between-epoch displacement that is obtained from TDCP algorithm can be used to identify whether large and rapid deformation occurs between the previous and current epochs. Therefore, we introduce the between-epoch displacement that is obtained from the TDCP method as an additional piece of information to adaptively adjust the state transition of the Kalman filter. Since the achievable accuracy of the TDCP displacement largely depends on the observational environment, a scenario identification strategy is also carried out to reduce their impact on the rapid deformation detection, which will be further analyzed in Section 4.1.

The proposed TDCP-based rapid deformation identification and adaptive Kalman filtering approach applied to different scenarios mainly includes three steps, i.e., the identification of observational scenarios with TDCP, the displacement detection with TDCP, and the adaptive adjustment of the Kalman filter. We first identify the environmental condition of the monitoring station according to the quality indicators of the TDCP solution, and set the corresponding threshold of TDCP displacement detection. Then, we identify whether rapid deformation occurs for the monitoring object in the current epoch according to the magnitude of the epoch-by-epoch TDCP displacement under the identified condition. Finally, the process noise in the state equation of Kalman filter is adjusted according to the state of the obtained displacement, so as to keep the predicted state consistent with the actual state.

(1) The identification of observational scenarios with TDCP. The between-epoch three-dimensional displacement ($\mathbf{\Delta }{\widehat{\mathit{X}}}^r$) as well as its VC matrix are first obtained epoch-wise with the TDCP method. To obtain robust and high-precision TDCP displacement results, a combined TDCP quality-control algorithm, including the a priori consistency check of different types of observations, and the IGG-III weight function-based robust estimation with the least-squares estimator are used. Moreover, the dual-frequency TurboEdit method and the joint method of Doppler combined with TDCP for single-frequency observations are used for cycle-slip detection. Then, the quality indicators of the TDCP solutions, such as the number of satellites that are involved in the TDCP estimation and the corresponding PDOP, and the quality indicators of robust estimation, as well as the statistics of posterior residuals, are computed and counted. The observational condition is then identified according to the value (or accumulated value) of the obtained quality indicators of the TDCP solutions.

(2) Displacement detection with TDCP. The obtained TDCP displacement is first converted from the earth-centered earth-fixed coordinate system to the topocentric coordinate system, i.e., $\mathbf{\Delta }{\widehat{\mathit{X}}}_{NEU}^r=\mathit{R}\cdot \mathbf{\Delta }{\widehat{\mathit{X}}}^r$ with $\mathit{R}$ as the transition matrix. Then, we determine whether a significant between-epoch deformation occurs is identified according to the magnitude of the TDCP displacement. $C_{NEU}=\left\{C_N,C_E,C_U\right\}$ is denoted as the threshold of the displacement in the N, E, and U components. An obtained displacement larger than this threshold is regarded as significant deformation; otherwise, it is regarded as noise that is contained in the TDCP solutions. According to the statistics of the TDCP displacement accuracy, different thresholds are adopted for the open-sky and complex observational environments, respectively. In this contribution, the empirical threshold vales of 6, 6, and 8 mm are set for the open-sky environment. For the complex observational environment, they are set to 12, 12, and 24 mm.

(3) The adaptive adjustment of the Kalman filter. $\mathbf{\Delta }{\widehat{\mathit{X}}}_{NEU}^r=\left\{\mathbf{\Delta }{\widehat{\mathit{X}}}_N^r,\mathbf{\Delta }{\widehat{\mathit{X}}}_E^r,\mathbf{\Delta }{\widehat{\mathit{X}}}_U^r\right\}$ is denoted as the obtained displacement in meters in the N, E, and U components in the current epoch. The power-spectrum density of the process noise for the positions in the Kalman filter is denoted as ${\sigma }_{{\mathit{\omega }}_k}$ in the unit of $m\cdot s^{-1/2}$. The scale factors for adjusting the process noise of ${\sigma }_{{\mathit{\omega }}_k}$ is given as:

$${\alpha }_d=\left\{\begin{array}{cc}\mathbf{\Delta }{\widehat{\mathit{X}}}_d^r/({\sigma }_{{\mathit{\omega }}_k,d}\cdot \sqrt{\mathbf{\Delta }t})& , \left|\mathbf{\Delta }{\widehat{\mathit{X}}}_d^r\right|>C_d\\ 1& , \left|\mathbf{\Delta }{\widehat{\mathit{X}}}_d^r\right|\le C_d\end{array}\right. \left(d=N,E,U\right)$$

(17)

where ${\alpha }_d\left(d=N,E,U\right)$ are the scale factors in the N, E, and U components, respectively, and $\Delta t$ is the time interval between consecutive epochs in unit of second.

The VC matrix of the adjusted process noise of the position is then given as:

$$\begin{array}{l}\begin{array}{c}{\mathit{Q}}_{{\mathit{\omega }}_k,NEU}\end{array}=\left[\begin{array}{ccc}{({\alpha }_N\cdot {\sigma }_{{\mathit{\omega }}_k,N})}^2\cdot \Delta t& 0& 0\\ 0& {({\alpha }_E\cdot {\sigma }_{{\mathit{\omega }}_k,E})}^2\cdot \Delta t& 0\\ 0& 0& {({\alpha }_U\cdot {\sigma }_{{\mathit{\omega }}_k,U})}^2\cdot \Delta t\end{array}\right]\end{array}$$

(18)

By substituting Equation (18) into Equation (8), the VC matrix of the predicted position states in the Kalman filter in the current epoch is then given as:

$${\widehat{\mathit{P}}}_{{\widehat{\mathit{X}}}_{k,k-1},NEU}={\widehat{\mathit{P}}}_{{\mathit{X}}_{k-1},NEU}+{\mathit{Q}}_{{\mathit{\omega }}_{k,NEU}}={\widehat{\mathit{P}}}_{{\mathit{X}}_{k-1},NEU}+{\mathit{R}}^{-1}{\mathit{Q}}_{{\mathit{\omega }}_k,NEU}{\left({\mathit{R}}^{-1}\right)}^{\mathrm{T}}$$

(19)

The adjusted process noise of the position state in Kalman filter is consistent with the detected between-epoch deformation of the monitoring object. The critical issue that the filtering results cannot reflect the actual deformation due to the inappropriate constraint of the states can therefore be avoided, and reliable positioning results can be obtained by measurement updating of the Kalman filtering.

The flowchart of the proposed TDCP-based rapid deformation identification and adaptive Kalman filtering approach under different scenarios is shown in Figure 1.

4. Experimental Results and Validation

In this section, we will first evaluate and compare the accuracy and reliability of TDCP displacement solutions under both open-sky and complex environments, so as to determine the thresholds for the classification of different observational environments and rapid deformation detection. Then, the effectiveness of the proposed method is evaluated with a simulated displacement experiment that is carried out using a customized three-dimensional displacement platform, in aspects of the rapid deformation identification and monitoring accuracy.

4.1. Characteristics of the TDCP Displacements Under Different Scenarios

The thresholds for TDCP quality classification and displacement detection are critical parameters for the application of the proposed approach. However, the GNSS navigation signals are susceptible to being affected by the complex observational environment, which will result in poor observational quality such as frequent gross errors and cycle slips. As a consequence of this, the accuracy and reliability of the obtained TDCP displacement will be decreased, and may result in the misjudgment or missing judgment of the displacements.

In this section, we will evaluate the accuracy and reliability of TDCP displacement solutions under both open-sky and complex monitoring environments, so as to provide a reference for the determination of the thresholds for the classification of different observational environments and identification of the rapid deformation. The involved quality indicators include (1) the number of satellites that involved in the TDCP estimation (NSAT) and the corresponding position dilution of precision (PDOP); (2) the quality indicators of robust estimation with TDCP, including the number of rejected observations whose corresponding posterior residual or normalized posterior residual exceeds the threshold (denoted as Resi-v and Resi-nv), and the number of down-weighted observations (denoted as DW); and (3) the statistics of posterior residuals, including the RMS and maximum of posterior residuals (denoted as RMS-V and MaxResi), and the posterior unit weight variance (Sigma0).

Raw GPS and BDS (including the BeiDou regional navigation satellite system (BDS-2) and BeiDou global navigation satellite system (BDS-3)) data with a 1 s sampling interval from an open-sky and a complex experiment are collected and processed. The open-sky and complex experiments were both carried out on the roof of a building at the campus of Wuhan University, and on 21 December and 20 December 2023 GPS time (GPST), respectively. BDStar M66-Lite receivers and HX-GNSS500 antennas were used in both the two experiments. The observational devices and observational environments are shown in Figure 2. Detailed information about the two experiments is listed in

Table 1. Detailed information of the open-sky and complex experiments.

Option	Open-Sky Experiment	Complex Experiment
Location	On the roof of a building at the campus of Wuhan University	On the roof of a building at the campus of Wuhan University
Time	21 December 2023 GPST	20 December 2023 GPST
Duration	About 12 h	About 12 h
Receiver/antenna of the monitoring station	BDStar M66-Lite/HX-GNSS500	BDStar M66-Lite/HX-GNSS500
Sampling interval	1 s	1 s
Observations	GPS and BDS (BDS-2/BDS-3)	GPS and BDS (BDS-2/BDS-3)

.

The obtained three-dimensional displacement series with TDCP for the open-sky and complex experiments are shown in Figure 3. The number of satellites and the corresponding PDOP, as well as the quality indicators of robust estimation with TDCP for the two experiments, are shown in Figure 4. The statistics of the posterior residuals for the two experiments are shown in Figure 5. The corresponding statistics is listed in

Table 2. The statistics of quality indicators for the open-sky and complex experiments.

Statistics	Open-Sky Experiment (mm)	Complex Experiment (mm)
CEP95 of TDCP displacement in the E component	2.1	3.4
CEP95 of TDCP displacement in the N component	2.4	3.4
CEP95 of TDCP displacement in the U component	6.4	8.8
Maximum of Sigma0	309.9	651.7
Maximum of RMS-V	10.6	19.5
Maximum of MaxResi	48.8	77.3

. It is obviously observed that the TDCP displacement series of the open-sky experiment remains stable, while that of the complex experiment tends to fluctuate over a greater range. For the open-sky experiment, the circular error probabilities of 95% (CEP95) are 2.1, 2.4, and 6.4 mm in the east (E), north (N), and up (U) components, respectively, while they are 3.4, 3.4, and 8.8 mm for the complex experiment. Moreover, compared with the open-sky environment, more frequent changes in the number of satellites and PDOP, more abnormal observations identified by the robust estimation (Resi-v, Resi-nv, and DW), and larger statistics of posterior residuals (sigma0, RMS-V, and MaxResi) are observed for the complex experiment. Specifically, the maximums of sigma0, RMS-V, and MaxResi are 309.9, 10.6, and 48.8 mm for the open-sky experiment, respectively, while they are 651.7, 19.5, and 77.3 mm for the complex experiment, respectively. Additionally, for the complex experiment, the periods of fluctuation in the curve of these quality indicators are basically consistent with those in the TDCP displacement series. These results demonstrate that the observations are susceptible to being affected by the complex observational environment, which will result in poor observational quality, and decrease the accuracy and reliability of the obtained TDCP displacements. Therefore, in order to implement rapid displacement detection with TDCP, it is necessary to first identify the observational environment of the monitoring station and set an appropriate displacement detection threshold according to the accuracy of the obtained TDCP displacement under different scenarios.

According to the above-mentioned experimental results, the observational environment of the monitoring station can be identified by using the quality indicators of robust estimation and the statistics of posterior residuals. Meanwhile, the corresponding rapid displacement detection threshold can be set according to the achievable accuracy of the TDCP displacement under different monitoring scenarios. The proposed TDCP-based rapid deformation identification can be successfully implemented with these critical thresholds. In this contribution, the monitoring station is considered to be under a complex environment if the following conditions are satisfied: (1) Sigma0 > 300 mm, (2) RMS-V > 10 mm, (3) MaxResi > 50 mm, and (4) Resi-v, Resi-nv, and DW > 1. Additionally, according to the three CEP95 criterion and the empirical experiences, the corresponding rapid displacement detection thresholds are set as 6/6/8 mm and 12/12/24 mm for the open-sky and complex observational environments, respectively, which have been introduced previously in Section 3.3.

4.2. Performance of the Proposed Method

The performance of the proposed method is evaluated with a raw dataset that was carried out in a simulation experiment with a three-dimensional displacement platform under complex observational environment. Similarly, the experiment was also carried out on the roof of a building at the campus of Wuhan University, but on 15 January 2024 GPST. The location of this experiment is very close to that of the complex experiment in Section 4.1. The same BDStar M66-Lite receiver and HX-GNSS500 antenna as the complex experiment in Section 4.1 were used for the monitoring station. A Trimble Alloy receiver and a HX-GNSS500 antenna was used for the reference station. The reference and monitoring stations were separated by approximately 26.0 m. Detailed information about the experiment is listed in

Table 3. Detailed information of the experiment.

Option	Information Details
Location	On the roof of a building at the campus of Wuhan University
Time	15 January 2024 GPST
Duration	About 2 h
Receiver/antenna of the reference station	Trimble Alloy/HX-GNSS500
Receiver/antenna of the monitoring station	BDStar M66-Lite/HX-GNSS500
Baseline length	About 26.0 m
Sampling interval	1 s

. The observational devices and environments of the monitoring station is shown in Figure 6.

Moreover, a three-dimensional displacement platform as shown in Figure 7 was used in the experiment. The GNSS antenna of the monitoring station was installed on the three-dimensional displacement platform. As shown in Figure 6 and Figure 7, the GNSS antenna mounted on the platform can be moved in three orthogonal directions independently by operating the three micrometer screws that are equipped on it. The measurement ranges of the two micrometer screws (or the displacement of the platform) are 10 cm and 4.5 cm in the horizontal and vertical axes, respectively. The resolutions of the three micrometer screws are all 1 mm. The three-dimensional displacement platform was placed on a tripod with leveling and centering. Moreover, one of the horizontal axes was manually placed facing the north, so that the platform can independently move in the east (E), north (N), and up (U) directions, respectively.

Our attempt to carry out this experiment is to simulate the landslide that is in the accelerated deformation stage or the stage before collapsing, in which stage the landslide will experience rapid and significant deformation. Therefore, during the observation of the experiment, five instances of simulated displacement on each direction were carried out by operating the micrometer screws.

Table 4. Detailed information of simulated displacement in the experiment.

Time (GPST Week and Second)	Direction	Displacement (cm)
2297 95,788	E	−1.35
2297 97,093	N	1.35
2297 98,416	U	3.0
2297 99,783	E	1.35
2297 101,057	N	−1.35

lists the detailed information of the simulated rapid displacements in the experiment. Note that the durations of operations are short enough to ensure that all displacements occur within one epoch (or 1 s) to simulate the occurrence of rapid deformation.

Three different processing schemes (i.e., KF-EMP, KF-NONE, and KF-TDCP) as listed in

Table 5. Strategies of the Kalman filtering for different schemes.

Time (GPST Week and Second)	KF-EMP	KF-NONE	KF-TDCP
Initial process noise of the position	$1\times {10}^{-5} \mathrm{m}\cdot {\mathrm{s}}^{-1/2}$	None	$1\times {10}^{-5} \mathrm{m}\cdot {\mathrm{s}}^{-1/2}$
Constraint of the process noise	Strong constraint	None	Adaptive adjustment with the proposed method

are adopted and compared to show the benefits of the proposed TDCP-based rapid deformation identification and adaptive Kalman filtering method. The three schemes differ in the prediction strategy of the estimated positions in the time update step. In the scheme KF-EMP (Kalman filter with empirical process noise of the position), the three-dimensional position state is updated with their counterparts in the previous epoch. A relatively small empirical system process noise ($1\times {10}^{-5}\mathrm{m}\cdot {\mathrm{s}}^{-1/2}$) is adopted to suppress the impact of the observational noise. This strategy is effective when no deformation occurs throughout the entire observation time span. However, it is inappropriate when large deformation occurs. In Scheme KF-NONE (Kalman filter with no constraint of the position), the three-dimensional position as well as its VC matrix is epoch-wise initialized with the single-point positioning (SPP) solution, which means that no constraint is imposed on the position estimate in the time update step. In Scheme KF-TDCP (adaptive Kalman filtering with a constraint from TDCP), the proposed TDCP-based rapid deformation identification and adaptive Kalman filtering approach is adopted.

In the data processing of the three schemes, GPS L1/L2 and BDS (including BDS-2 and BDS-3) B1I/B3I observations are used. The elevation cutoff angle is set to 10° and the sampling interval is set to 1 s. The signal to noise ratio (SNR) cutoff is set to 25 dBHz, which means that the observations with a corresponding SNR below 25 dBHz are rejected in the preprocessing stage. The ambiguities are resolved with the popular least-squares ambiguity decorrelation adjustment (LAMBDA) method [35], and the ratio test is adopted for ambiguity validation [36]. Details about the fundamental processing strategies are listed in

Table 6. Fundamental data-processing strategies.

Option	Details
Observation	GPS L1/L2, BDS (BDS-2/BDS-3) B1I/B3I
Ephemeris	Broadcast
Atmospheric delays	Neglected
Elevation cutoff angle	10°
SNR cutoff	25 dBHz
Sampling interval	1 s

.

The obtained displacement series of the monitoring station with the TDCP algorithm are shown in Figure 8. As shown, the TDCP-based observational scenario recognition procedure identifies that the monitoring station is located under a complex environment, and the displacement detection thresholds corresponding to the complex environments are therefore adopted, i.e., 12, 12, and 24 mm for the E, N, and U components, respectively (represented by the pink and black dashed lines). The five simulated rapid displacements in the experiment are accurately detected (represented by the gray longitudinal stripes). When the displacement platform moves in a certain direction, the obtained TDCP displacement in the same direction will be larger than the threshold. These results indicate that the TDCP-based rapid displacement detection method can promptly and accurately identify the significant between-epoch deformation of the monitoring station under complex environment, thus providing effective displacement information for the adaptive adjustment of the process noise for the position in the Kalman filtering.

The displacement series in E, N, and U components for the displacement platform using Schemes KF-TDCP, KF-EMP, and KF-NONE are shown in Figure 9 and Figure 10. As shown in Figure 9, the filtering solution series of Scheme KF-EMP exhibit smaller between-epoch fluctuations and smoother curve. However, the rapid displacement is not identified with Scheme KF-EMP. As a consequence of this, the inappropriate process noise of the position is not adjusted when large deformation occurs. The displacement series therefore reconverges after a long time (about 1000 to 2000 s) before it tends to be consistent with the actual situation. As shown in Figure 10, the filtering solution series of Scheme KF-NONE can reflect the actual deformation trend of the monitoring object when rapid displacement occurs. However, the solutions are significantly affected by the observational noise, and significant fluctuations therefore appear in the displacement curve. Outliers larger than 1.0 cm even appear for some epochs, which is not consistent with the actual state of the monitoring object, and may lead to misjudgment and false alarms in the early warning of the landslides.

For Scheme KF-TDCP, which adopts our proposed TDCP-based rapid displacement identification and adaptive filtering approach, since no significant displacement is detected in the period without simulated displacement, the relatively tight between-epoch constraints for the position remain unchanged; the impact of observational noise and other residual errors on the solutions is therefore effectively reduced, and the displacement curve remain smooth. When rapid displacement occurs, Scheme KF-TDCP accurately identifies the significant displacement, and then adaptively adjusts the process noise of the position in the Kalman filter to keep the predicted state consistent with the actual state. The filtering solutions can therefore promptly and accurately reflect the simulated displacement. It should be noted that there exist small trends and discrepancies in the displacement series in the N and U components for the three schemes. This may be caused by the following two reasons: (1) The experiment is carried out with a three-dimensional displacement platform, which might not be very precise. The three axes of the displacement platform may deviate from the true E, N, and U directions, and the manual operation may also be inaccurate; (2) the experiment was carried out in a complex environment, which may lead to severe multipath errors and thus contaminate the achieved RTK positioning results.

The discrepancies between the obtained displacement series from the KF-EMP, KF-NONE, and KF-TDCP schemes with respect to the simulated true displacements are shown in Figure 11, and the corresponding statistics are listed in

Table 7. The RMS errors of obtained displacements from the KF-EMP, KF-NONE, and KF-TDCP schemes with respect to the simulated true displacements.

Scheme	E (mm)	N (mm)	U (mm)
KF-EMP	5.7	5.6	21.5
KF-NONE	5.5	3.2	19.1
KF-TDCP	4.8	2.6	15.2

. It is observed that the displacement errors of the proposed method fluctuate within a smaller range than that of the KF-EMP and KF-NONE methods. Compared with the KF-EMP and KF-NONE methods, the proposed KF-TDCP method can remarkably reduce the RMS errors of the obtained displacement solutions. The RMS errors of obtained displacements for KF-EMP and KF-NONE are 5.7 mm/5.6 mm/21.5 mm and 5.5 mm/3.2 mm/19.1 mm, respectively. When the proposed KF-TDCP method is adopted, they are reduced to 4.8 mm/2.6 mm/15.2 mm. It should be noted that the benefits of the proposed KF-TDCP method over the KF-EMP and KF-NONE methods may be more obvious if the experiment can be carried out more accurately.

In conclusion, during the entire observational period, the proposed approach can accurately and promptly identify the rapid displacement of the monitoring body, and thus reduce the probability of missing alarm. Meanwhile, it can effectively suppress the observational noise, and thus retain the monitoring accuracy and reduce the risk of false alarm. The proposed approach can provide high-precision and reliable three-dimensional deformation information for GNSS landslide monitoring and early warning under complex environments.

5. Conclusions and Discussions

In this contribution, we propose a novel approach of TDCP-based rapid deformation identification and adaptive Kalman filtering under different scenarios for GNSS landslide monitoring. The effectiveness of the proposed approach is validated with a simulated deformation experiment using a customized three-dimensional displacement platform. The raw GPS L1/L2 and BDS-2/BDS-3 B1I/B3I observations are processed and the performance is evaluated in terms of rapid deformation identification and monitoring accuracy. The experimental results demonstrate that the displacement errors of the proposed method fluctuate within a smaller range than those of the KF-EMP and KF-NONE methods. Compared with the classical approaches of Kalman filtering with the relatively small empirical process noise of the position or with no constraint of the position, the proposed approach can accurately and promptly identify and reflect the rapid between-epoch (sampling interval of 1 s) deformation of more than approximately 1.5 cm and 3.0 cm for the horizontal and vertical components for the monitoring object under a complex observational environment. Meanwhile, it can effectively suppress the observational noise and thus maintain mm-to-cm-level monitoring accuracy. The proposed approach can provide high-precision and reliable three-dimensional deformation information for GNSS landslide monitoring and early warning.

The performance of our proposed approach largely depends on the reliability of the displacement that was obtained from TDCP; it is therefore necessary to further improve the reliability and integrity of the TDCP algorithm. Moreover, the observational condition is various, but only two kinds of scenarios (open-sky and complex) are considered in this contribution, more different scenarios as well as the corresponding classification strategies should therefore be considered to further evaluate the performance and improve the applicability of the proposed method. In addition, future work can also be carried out to combine the observations from Inertial Measurement Unit (IMU) sensors to realize a more accurate and sensitive identification of the deformation of the monitoring station in challenging conditions.