Introduction and motivation

The Global Navigation Satellite System (GNSS) is a powerful tool that may reliably characterize dynamic displacements in Structural Health Monitoring (SHM) and seismology (Avallone et al. 2011). Over recent years, we have witnessed efforts focused on the development of dedicated processing algorithms, e.g., those based on a time-differencing concept (Colosimo et al. 2011; Paziewski et al. 2018; Zheng et al. 2022) and adaptation of the existing positioning methods, such as Real Time Kinematics (RTK) or Precise Point Positioning (PPP) (Li et al. 2019; Paziewski et al. 2020) aiming at retrieving dynamic displacements. As a result, with high-rate GNSS, it is now achievable to reach displacement accuracy of millimeter-level over a short period (Psimoulis et al. 2018).

Still, one of the limitations of GNSS is its lower sampling rate compared to conventional seismic sensors such as accelerometers. Current GNSS receivers usually acquire signals with a 20 Hz sampling rate; some record observations up to 100 Hz (Larocca et al. 2014; Paziewski et al. 2019). Such high-rate data may be subject to non-negligible time correlation (Moschas and Stiros 2015; Shu et al. 2017). Strong motion seismographs operate with a higher sampling rate, typically 200–300 Hz; however, numerical integration of accelerations performed to recover displacements may introduce drifts to the displacement time series due to hysteresis, sensor tilt, and rotation (Boore 2001; Allen and Kanamori 2003). A response to such constraints may be a fusion of GNSS with accelerometer data to provide a combined solution of high reliability, accuracy, and sampling rate (Crowell et al. 2013; Dahmen et al. 2020). The groundwork for further advances in this field was laid by Smyth and Wu (2007) and Bock et al. (2011), who employed a Kalman filter to integrate acceleration records with GNSS-derived displacements from collocated instruments. Such an approach is termed a loosely coupled one. These studies were followed, e.g., by Geng et al. (2013), who presented a so-called tightly coupled model incorporating accelerations into the state transition matrix in Kalman filtering, and Xin et al. (2021), who introduced a new real-time seismogeodetic instrument that combines GNSS and accelerometers records.

Despite the progress in PPP algorithms, relative positioning (RP) is still recognized as the most precise GNSS positioning method. It benefits from the double-difference (DD) model, which eliminates or mitigates most biases and propagation errors. RP also outperforms PPP in terms of time-to-fix, allowing instantaneous solutions in real-time; therefore, it is commonly used in seismogeodesy and SHM (Shen et al. 2019).

Conventionally, the RP model is fed with GNSS observations of the same sampling rate from the rover and reference receivers. On many occasions, e.g., due to expense reduction, high-rate GNSS observations from the monitoring stations are not accompanied by such data from the reference sites. Then, Continuous Operation Reference Stations (CORS) data of low rates are only available. Collecting high-rate GNSS observations by a dense multi-constellation CORS network also significantly increases the amount of data to be transmitted and processed, constraining the possibility of real-time operation. As a result, we face a problem of processing high-rate rover and low-rate reference station GNSS observations.

A traditional approach to handling asynchronous data is downsampling the high-rate GNSS observations. Such a trivial solution could be acceptable for many surveying and navigation applications. Still, for seismogeodesy and SHM, this would lead to an unacceptable loss of information on high-frequency movements. More advanced approaches have been proposed to deal with asynchronous GNSS data in demanding applications. Wang et al. (2011) developed a new strategy to handle such data in cm-level kinematic positioning. The validation with an airplane flight experiment based on 1 s rover and 60 s reference station data exhibited several centimeter degradations of the positioning accuracy. We also acknowledge the contributions of Zhang et al. (2015) and Du et al. (2019), who modified RTK models to handle asynchronous GNSS observations being an effect of outages at reference stations. More recently, Dong et al. (2020) presented an asynchronous RTK and time-differenced carrier phase integration method for high-rate positioning with a moving baseline.

Unlike previous studies, this one is devoted to processing asynchronous GNSS observations of a high sampling rate to detect dynamic displacements of low-scale, specifically up to cm-level, and the first to analyze the impact of asynchronous high-rate GNSS data on coupled GNSS and accelerometer model performance. We examine if such a multi-sensor model fed with low sampling rate observations from a reference station may provide competitive accuracy to that based on synchronous high-rate GNSS data. As the satellite clock and ephemeris are the possible main contributors to the error budget, we also analyze the susceptibility of the method to this error source. We highlight that our goal is to retrieve short-term and low-scale dynamic displacements, which are valid for structural and seismic movements. Consequently, medium and long-term movements are not accounted for since these are commonly not considered in dynamic displacement monitoring (Bezcioglu et al. 2022).

In the next section, we present the method of coupled accelerometer and asynchronous high-rate GNSS data processing, aiming to retrieve the dynamic displacements. Next, we describe the experiment designed for method validation in a controllable environment with a shake table and discuss the results. Finally, in the last section, we conclude the study.

Methods

Data processing consists of three main steps:

  • High-rate kinematic relative GNSS positioning based on asynchronous observations.

  • Integration of GNSS-derived displacements with accelerometer records using Kalman filter in a loosely coupled model (Bock et al. 2011).

  • Backward filtration of the displacements using the Rauch Tung Striebel (RTS) smoother (Rauch et al. 1965).

Relative GNSS positioning with asynchronous observations

Let us recall observation equations of carrier phase and code signals emitted by the satellite n at epoch \(T_{0}\) and acquired at arriving time \(t_{0}\) by the receiver r (Teunissen and Kleusberg 1998):

$$\begin{aligned} \Phi_{r}^{n} \left( {t_{0} } \right) & = \rho_{r}^{n} \left( {T_{0} } \right) + \lambda N_{r}^{n} \left( {t_{0} } \right) + c\left[ {\delta t_{r} \left( {t_{0} } \right) - \delta t_{r}^{n} \left( {T_{0} } \right)} \right] - \delta_{ion,r}^{n} \left( {t_{0} } \right) + \delta_{trop,r}^{n} \left( {t_{0} } \right) \\ & \quad + \delta \varphi_{r} \left( {t_{0} } \right) - \delta \varphi^{n} \left( {T_{0} } \right) + E^{n} \left( {t_{0} } \right) + \epsilon_{r,\Phi }^{n} \left( {t_{0} } \right) \\ P_{r}^{n} \left( {t_{0} } \right) & = \rho_{r}^{n} \left( {T_{0} } \right) + c\left[ {\delta t_{r} \left( {t_{0} } \right) - \delta t_{r}^{n} \left( {T_{0} } \right)} \right] + \delta_{ion,r}^{n} \left( {t_{0} } \right) + \delta_{trop,r}^{n} \left( {t_{0} } \right) + d_{r} \left( {t_{0} } \right) - d^{n} \left( {T_{0} } \right) \\ & \quad + E^{n} \left( {t_{0} } \right) + \epsilon_{r,P}^{n} \left( {t_{0} } \right) \\ \end{aligned}$$
(1)

where \(\Phi\) and \(P\) are phase and code observations in meters, respectively; \(\rho\) stands for the geometric distance between the satellite and receiver; \(\lambda\) refers to the signal wavelength of the GNSS; \(N\) is the integer ambiguity of the phase observable in cycles; \(c\) is the speed of light in meters per seconds; \(\delta t_{r}\) denotes the receiver clock correction, while \(\delta t_{r}^{n}\) is the satellite clock correction in seconds; \(\delta_{ion}\) and \(\delta_{trop}\) are the slant ionospheric and tropospheric delays in meters, respectively; \(\delta \varphi_{r}\) and \(\delta \varphi^{n}\) are the receiver and satellite initial phase biases in meters; \(d_{r}\) and \(d^{n}\) correspond to the receiver and satellite biases of code observations in meters, respectively; \(E\) refers to ephemeris error; finally, \(\epsilon\) is the observation noise with lumped multipath effect.

If the rover and reference receivers acquire the same satellite signal at different (asynchronous) arriving epochs, namely \(t_{0}\) and \(t_{1}\), then the single-differenced (SD) observation equations between two stations and a satellite read:

$$\begin{aligned} \Phi_{rs}^{n} \left( {t_{0} ,t_{1} } \right) & = \rho_{rs}^{n} \left( {T_{0} ,T_{1} } \right) + \lambda N_{rs}^{n} \left( {t_{0} ,t_{1} } \right) + \delta \varphi_{rs} \left( {t_{0} ,t_{1} } \right) + c\left[ {\delta t_{rs} \left( {t_{0} ,t_{1} } \right) - \delta t_{rs}^{n} \left( {T_{0} ,T_{1} } \right)} \right] \\ & \quad - \delta_{ion,rs}^{n} \left( {t_{0} ,t_{1} } \right) + \delta_{trop,rs}^{n} \left( {t_{0} ,t_{1} } \right) + E^{n} \left( {t_{0} ,t_{1} } \right) + \epsilon_{rs,\Phi }^{n} \left( {t_{0} ,t_{1} } \right) \\ P_{rs}^{n} \left( {t_{0} ,t_{1} } \right) & = \rho_{rs}^{n} \left( {T_{0} ,T_{1} } \right) + d_{rs} \left( {t_{0} ,t_{1} } \right) + c\left[ {\delta t_{rs} \left( {t_{0} ,t_{1} } \right) - \delta t_{rs}^{n} \left( {T_{0} ,T_{1} } \right)} \right] + \delta_{ion,rs}^{n} \left( {t_{0} ,t_{1} } \right) \\ & \quad + \delta_{trop,rs}^{n} \left( {t_{0} ,t_{1} } \right) + E^{n} \left( {t_{0} ,t_{1} } \right) + \epsilon_{rs,P}^{n} \left( {t_{0} ,t_{1} } \right) \\ \end{aligned}$$
(2)

In SD observations, satellite-specific phase and code biases are canceled as they are relatively stable. The receiver-specific code and initial phase biases are, in turn, eliminated when forming DD observations. Moreover, if the baseline is short, approximately < 10 km, and the dynamics of the ionosphere and troposphere are sufficiently small, atmospheric propagation errors may also be neglected (Zhang et al. 2015). Then, DD between the observations of pairs of receivers (r, s) and satellites (n, m) collected at asynchronous arrival epochs \(\left( {t_{0} ,t_{1} } \right)\) may be given in a simplified form:

$$\begin{aligned} & \Phi_{rs}^{nm} \left( {t_{0} ,t_{1} } \right) = \rho_{rs}^{nm} \left( {T_{0} ,T_{1} } \right) + \lambda N_{rs}^{nm} \left( {t_{0} ,t_{1} } \right) - c\delta t_{rs}^{nm} \left( {T_{0} ,T_{1} } \right) + E^{nm} \left( {t_{0} ,t_{1} } \right) + \epsilon_{rs,\Phi }^{nm} \left( {t_{0} ,t_{1} } \right) \\ & P_{rs}^{nm} \left( {t_{0} ,t_{1} } \right) = \rho_{rs}^{nm} \left( {T_{0} ,T_{1} } \right) - c\delta t_{rs}^{nm} \left( {T_{0} ,T_{1} } \right) + E^{nm} \left( {t_{0} ,t_{1} } \right) + \epsilon_{rs,P}^{nm} \left( {t_{0} ,t_{1} } \right) \\ \end{aligned}$$
(3)

The DD observational model of asynchronous observations still includes satellite clock and ephemeris errors. These factors are not accounted for in the respective model of synchronous observations, which reads as follows:

$$\begin{aligned} & \Phi_{rs}^{nm} \left( {t_{0} } \right) = \rho_{rs}^{nm} \left( {T_{0} } \right) + \lambda N_{rs}^{nm} \left( {t_{0} } \right) + \epsilon_{rs,\Phi }^{nm} \left( {t_{0} } \right) \\ & P_{rs}^{nm} \left( {t_{0} } \right) = \rho_{rs}^{nm} \left( {T_{0} } \right) + \epsilon_{rs,P}^{nm} \left( {t_{0} } \right) \\ \end{aligned}$$
(4)

The model given in (3) is fed with the instantaneous high-rate GNSS observations of the rover receiver and the historical low-rate measurements from the reference one. The observations from the reference station are repeatedly used until up-to-date observations are synchronous with those from the rover receiver. For synchronous epochs, model (4) is used. As a result, the solution is provided with a rate equal to that of rover observations. Such three-dimensional high-rate displacements are then transformed into the local topocentric frame and filtered. We use a high-pass Digital Butterworth filter with a cut-off period set to 10 s to deprive of low-frequency components, as we focus on dynamic displacements up to centimeter-level only.

Comparing (3) and (4), we discover that the displacement time series derived from the asynchronous observations processing are subject to residual satellite clock and ephemeris errors. As satellite clock error differs significantly between consecutive epochs, it also impacts solution accuracy when considering observations from the rover and reference stations with different sampling rates. This error is strictly dependent on the satellite clock stability. Zhang et al. (2015) reported millimeter interstation distance errors induced when the difference between arrival time at the receivers is a dozen seconds. Similarly, an ephemeris error also introduces non-negligible distortions. The authors also determined that the interstation distance error is 2 mm when the receiver’s time arrival difference is 1 s, assuming a 1 m error of broadcast ephemeris and a baseline length of 10 km. If time arrival asynchrony reaches 10 s, this error will be twofold. Fortunately, due to invariable geometry between receivers and satellites, the impact of these errors on coordinate estimates is of low dynamics.

Loosely coupled integration of GNSS and accelerometer data for displacement monitoring

Typically, accelerometer data are offered with a much higher sampling rate than GNSS. A method to integrate both solutions is a dynamic model through estimation and prediction with a Kalman filter. Then, a discrete dynamic system reads as follows (Bock et al. 2011):

$$x_{k + 1} = A_{k} x_{k} + B_{k} u_{k} + \alpha_{k}$$
(5)

where \(x\) refers to the state vector; \(A\) denotes a matrix that defines the system dynamics; \(B\) is an input matrix, while \(u\) is a vector describing the system's inputs, which in this case means acceleration records. \(\alpha\) refers to the vector of system noise with zero mean and the covariance matrix Q, and finally, subscript \(k\) is the epoch number.

We employ a 2-element state vector that comprises displacements (\(d\)) and velocities (\(v\)).

$$x_{k} = \left[ {\begin{array}{*{20}c} {d_{k} } & {v_{k} } \\ \end{array} } \right]^{T}$$
(6)

At discrete epochs, GNSS displacements are fed the measurement process:

$$z_{k} = H_{k} x_{k} + \beta_{k}$$
(7)

where \(z\) is a vector of GNSS-derived displacements; \(H\) refers to a design matrix, and \(\beta\) is the measurement noise vector with zero mean and covariance matrix R.

The system is provided with accelerometer records assumed to be constant between epochs. Then, discrete system model elements are defined as follows:

$$A = \left[ {\begin{array}{*{20}c} 1 & {\tau_{a} } \\ 0 & 1 \\ \end{array} } \right],\quad B = \left[ {\begin{array}{*{20}c} {\tau_{a}^{2} /2} \\ {\tau_{a} } \\ \end{array} } \right],\quad H = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right]$$
(8)

and the covariance matrices of process and measurement noise read:

$$Q = \left[ {\begin{array}{*{20}c} {q\tau_{a}^{3} /3} & {q\tau_{a}^{2} /2} \\ {q\tau_{a}^{2} /2} & {q\tau_{a} } \\ \end{array} } \right],\quad R = \frac{r}{{\tau_{d} }}$$
(9)

where q and \(r\) are the noise variances of the acceleration records and GNSS-derived displacements, correspondingly. Sampling intervals of accelerations and GNSS displacements are denoted with \(\tau_{a}\) and \(\tau_{d}\), respectively.

The integration is performed independently for each coordinate component with the initialization, prediction, and corrections steps so as to handle different sampling rates of GNSS displacements and acceleration records (Smyth and Wu 2007; Bock et al. 2011):

  • Initialization of the state vector and state covariance matrix (\(P\)):

    $$x_{0} = 0,P_{0} = I$$
    (10)

  • Time update (prediction):

    $$P_{k + 1}^{ - } = A_{k} P_{k} A_{k}^{T} + Q_{k}$$
    (11)
    $$\hat{x}_{k + 1}^{ - } = A_{k} \hat{x}_{k} + B_{k} u_{k}$$
    (12)

  • Measurement update (correction):

    $$\hat{x}_{k + 1} = \hat{x}_{k + 1}^{ - } + K_{k + 1} \left( {z_{k + 1} - H_{k + 1} \hat{x}_{k + 1}^{ - } } \right)$$
    (13)
    $$P_{k + 1} = \left( {I - K_{k + 1} H_{k + 1} } \right)P_{k + 1}^{ - }$$
    (14)

    in which Kalman gain (\(K_{k + 1}\)) is obtained according to the formula:

    $$K_{k + 1} = P_{k + 1}^{ - } H_{k + 1}^{T} \left( {H_{k + 1} P_{k + 1}^{ - } H_{k + 1}^{T} + R} \right)^{ - 1}$$
    (15)

    The advantage of the presented integration model is providing a coupled solution with a sampling rate equal to that of accelerometer records, which is typically at least ten times higher than that of GNSS.

Backward filtration

Operation in post-processing or near real-time modes allows for backward filtration, which enhances the accuracy through noise reduction. Such smoothing may be efficiently accomplished with the Rauch–Tung–Striebel algorithm (Rauch et al. 1965). Therefore, the whole integration process combines forward filtration performed with the Kalman filter and backward smoother realized in the following steps. The smoothed covariances and estimates are obtained with the equations, namely:

$$P_{k} = P_{k}^{f} - F_{k} \left( {P_{k + 1}^{f - } - P_{k + 1} } \right)F_{k}^{T}$$
(16)
$$\hat{x}_{k} = \hat{x}_{k}^{f} + F_{k} \left( {\hat{x}_{k + 1} - \hat{x}_{k + 1}^{f - } } \right)$$
(17)

in which \(P_{k + 1}^{f - }\), \(P_{k}^{f}\) denote a priori and a posteriori covariance matrices obtained from the Kalman filter, and the smoother gain is defined as:

$$F_{k} = P_{k}^{f} A\left( {P_{k + 1}^{f - } } \right)^{ - 1}$$
(18)

As a result of the procedure, the accuracy of the estimates from past epochs is improved.

Experiment design

We aim to assess the implications of asynchronous GNSS observation for retrieving dynamic displacements by integrating high-rate GNSS and accelerometers. For this purpose, a shake table equipped with a GNSS receiver and accelerometer simulated a set of dynamic movements (Fig. 1). We have generated four uniaxial East–West harmonic motions of frequencies up to 3.5 Hz and low-scale amplitudes in the range of 5–10 mm (Table 1). GPS, Galileo, and BDS data were collected at 20 Hz by the CHC I80 GNSS rover receiver, while the accelerations were recorded at 200 Hz by the Guralp CMG-5 T accelerometer. Another 20 Hz CHC I80 GNSS receiver located at a distance of 60 m served as a reference station for a relative kinematic solution. Linear Variable Differential Transformer (LVDT), in turn, provided the ground truth displacements, operating at sub-mm level accuracy and 100 Hz sampling rate.

Fig. 1
figure 1

GNSS receiver and accelerometer mounted to the shake table during data collection

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Table 1 Frequency and amplitude of the harmonic motion for each test case simulated by the shake table
Full size table

The details of the employed correction and stochastic models used in GNSS data processing are summarized in Table 2. In the Kalman integration, we adopt the weight ratio between GNSS displacements and accelerometer records of 100:1 (Xin et al. 2021).

Table 2 Summary of the processing strategy and correction models
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The displacement time series are obtained with three processing strategies based on GNSS and accelerometer data:

  1. 1.

    GNSS-only,

  2. 2.

    GNSS combined with accelerometer records (GNSS&acc.),

  3. 3.

    GNSS combined with collocated accelerometer records and backward filtered with the Rauch–Tung–Striebel algorithm (GNSS&acc. + RTS).

Strategies #1 and #2 may be considered intermediate steps of the proposed full-step processing method (#3). The recovered displacements are compared to the benchmark ones provided by the LVDT sensor.

We intentionally downsample GNSS data at the reference station to assess the implications of asynchronous GNSS data processing. As a result, the observations of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 15, 20, and 30 s are used. We note that always 0.05 s (20 Hz) GNSS data from the rover receiver and 200 Hz accelerometer records are used at the monitoring station. The impact of satellite orbits and clocks on asynchronous observation processing is verified by performing the computations twofold in each case, namely with broadcast or precise satellite products. The outcomes are assessed in coordinate displacement and time–frequency domains.

Assessment in the displacement domain

Our primary goal in this section is to assess the implications of downsampling GNSS data at a reference station in the displacement domain. We also analyze the benefits of using sensor integration and the susceptibility of the method to satellite orbit and clock error.

Benefit from GNSS and accelerometer signals integration

Figures 2 and 3 depict the responses in a coordinate domain on the four harmonic motions triggered by the shake table. At this time, the results were derived using 200 Hz accelerometer and synchronous 20 Hz GNSS observations at rover and reference receivers supported with broadcast ephemerides. Figure 2 presents the topocentric displacement time series for the period during which the frequencies of the harmonic motions were in the range of 0.25–0.60 Hz that correspond to motions #1, #2, and #3. Figure 3, in turn, shows the corresponding results for the #4 motion, characterized by the highest frequency of 3.5 Hz. Moreover, in the middle panels of Figs. 2 and 3, which correspond to the East coordinate component, we show the benchmark LVDT displacements.

Fig. 2
figure 2

Displacement time series derived from 20 Hz GNSS and 200 Hz accelerometer data processing during simulated harmonic motions #1, #2, #3. Note that GNSS&acc. + RTS displacements coincide with the LVDT ones

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Fig. 3
figure 3

Displacement time series derived from 20 Hz GNSS and 200 Hz accelerometer data processing during simulated harmonic motion #4

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The plots confirm that all processing strategies allow the detection of the simulated waves well; however, with a different noise. Such an effect is mainly pronounced for the North and Height components deprived of displacements. What transpires from the displacement time series is that the combined solutions, namely GNSS&acc. and GNSS&acc. + RTS, are always less noisy compared to the GNSS-only one. As expected, the height component is characterized by a higher noise than horizontal ones. However, as the harmonic motion was simulated in a uniaxial East–West direction, the North and Height components are not further analyzed.

The displacement time series provided in Figs. 2 and 3 and the focus on the example of 10 s periods shown in Fig. 4 give us the first impression of the performance divergence between the GNSS-only and multi-sensor solutions. As can be read from these plots, the displacement time series obtained with the GNSS&acc. + RTS strategy perfectly coincides with the benchmark ones determined with LVDT. One may also notice that the displacements yielded by the GNSS-only solution are the noisiest among all analyzed ones. The effect is especially pronounced for the harmonic motion of the highest frequency, as shown in the bottom panel of Fig. 4.

Fig. 4
figure 4

Examples of 10 s-long displacement time series for the East component derived from 20 Hz GNSS and 200 Hz accelerometer data processing. The displacement responses on simulated harmonic motions #1, #2, #3, and #4 are presented from top to bottom, respectively

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We examine the standard deviations (STD) of the displacement errors for a more rigorous analysis. The error is here defined as the difference between the displacements provided by the assessed models and LVDT. The statistics in Table 3 support our first findings from the displacement time series analysis as, in most cases, the integrated GNSS&acc + RTS solution exhibits the lowest and GNSS-only the highest STDs. A fusion of GNSS with an accelerometer brings about a 50% reduction in the displacement error STD compared to a GNSS-only solution. Filtration with the RTS algorithm allows further drop in the STDs, even to the value below 1 mm, as in the case of harmonic motion # 1.

Table 3 STDs of displacements errors computed as the difference between LVDT and GNSS, GNSS&acc., or GNSS&acc. + RTS solutions. Broadcast ephemerides and synchronous 20 Hz GNSS data at rover and reference receivers were used in the processing
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Figure 4 also gives us the impression that GNSS-derived displacements during motion #4 are overestimated. Indeed, the GNSS-only solution during this movement provides the displacements of the highest error reflected in approximately two times greater STD than during the three previous waveforms (Table 3). An explanation of such an effect may be deduced from Fig. 5, in which we demonstrate residuals of phase measurements obtained as detrended DD observations built between GPS PRN 25 and PRN 29 satellites. We used the 22,200–22,206 s-of-day period when the rover antenna was stationary, and the satellites were tracked at high elevations, over 60°. Such a situation should provide low-noise observations with no displacement response. Unexpectedly, DD phase residuals are subject to a wave-like effect with a frequency that seems to be close to that of the harmonic motion #4. For a convenient frequency comparison, a sample GNSS-only displacement response to the harmonic motion #4 during the period of 23,130–23,136 s-of-day is recalled. Indeed, Fourier transformation revealed that the dominating frequencies of the DD phase residuals fit the range of 3.6–5.6 Hz. It thus becomes possible to attribute the overestimation of GNSS-derived displacements during the #4 movement as resulting from the superposition of two oscillations, namely the shake table-generated and the ones present in phase noise.

Fig. 5
figure 5

Time series of DD phase noise for the sample static antenna period and example of GNSS-only derived displacement time series during the harmonic motion#4

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Impact of downsampling of GNSS data at reference station on the accuracy of displacement retrieving

We turn a spotlight on the performance of displacement retrieving when different sampling rates of GNSS observations at the reference station are used. In this regard, Fig. 6 depicts the time series of displacement errors for the harmonic motion of 0.25 Hz frequency and 5 mm amplitude. The consecutive panels correspond to the results when different sampling rates are used at the reference station, starting in the top panel with the synchronous 0.05 s data and ending with 30 s in the bottom.

Fig. 6
figure 6

Displacement error time series for the harmonic motion #1 of 0.25 Hz and 5 mm amplitude using different intervals of GNSS observations at the reference station

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A first glance at the displacement error time series confirms the apparent benefit of GNSS and accelerometer fusion, as integrated solutions always exhibit lower noise than the GNSS-only one. Looking further at the plots, we may discover that downsampling GNSS observations at the reference station for the GNSS-only solution is justified merely to 0.2 s without a noticeable error gain. One can also find that integrated solutions, namely GNSS&acc. and GNSS&acc. + RTS, are less prone to the adverse effect of the low sampling rate of GNSS data at the reference station. Nonetheless, even integrated solutions are subject to high displacement divergences, pronounced in evident low-frequency effects, if we downsample GNSS observations at the reference station to the 10 s interval and more.

A more comprehensive view of the implications of GNSS data downsampling is provided in Fig. 7. The plot depicts the STDs of displacement errors as a function of the observation interval at the reference station. We observe that the GNSS&acc. + RTS solution always provides the lowest and GNSS-only the highest STDs of displacement errors irrespectively on the sampling rate of observations at the reference station. More importantly, the statistics clearly show how using the GNSS data at the reference station with an interval sparser than 10 s decreases displacement accuracy.

Fig. 7
figure 7

STDs of displacements errors computed as the difference between LVDT and GNSS, GNSS&acc., or GNSS&acc. + RTS solutions using broadcast ephemerides. The #1–4 harmonic motion results are presented in consecutive panels

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In contrast, the plot also proves that with an integrated GNSS&acc. + RTS model, it is feasible to provide dynamic displacements of 2 mm error STD even when GNSS data are downsampled to 15 s. Going further, if one aims to achieve 1 mm displacement accuracy, the GNSS data at the reference station should not be downsampled below the level of 2 s.

Since GNSS&acc. scheme is an intermediate step of the full processing, we do not examine its performance further. Therefore, we now focus on the fusion of GNSS & accelerometer backward filtered with the RTS algorithm and its advantages over the GNSS-only solution. We adopt a 2 mm threshold and inspect a ratio of displacement errors falling into the defined bin. This ratio, as a function of the observation sampling rate at the reference station, is depicted in Fig. 8.

Fig. 8
figure 8

Ratio of displacement errors fitting the accuracy threshold of 2 mm

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The results demonstrate that GNSS&acc. + RTS is characterized by at least an 80% ratio of displacement errors fitting the adopted threshold when using GNSS data to the reference station of 10 s interval or denser. This ratio mostly ranges from 90 to 100% when the interval of GNSS data at the reference stations is denser than 2 s. In contrast, using sparse GNSS observations at the reference station, specifically below 10 s, results in a considerable drop to 60–70% in the ratio of displacement errors meeting the adopted criterion. Such an effect is in line with the displacement error STDs given in Fig. 7. According to such results, we may conclude that regional CORS networks, which usually collect 1 s GNSS observations, can be efficiently employed as reference sites in SHM.

Impact of satellite orbits and clocks

Satellite orbits and clocks are important sources of errors in relative positioning with asynchronous GNSS observations (Zhang et al. 2015). Therefore, we examine the susceptibility of the presented method to the quality of satellite products. In Fig. 9, we present STDs of displacement errors for GNSS-only and GNSS&acc. + RTS solutions. The results are distinguished by taking the source of orbits and clocks as a criterion. In most cases, the solution based on precise products outperforms that utilizing broadcast ephemerides since applying precise orbits and clocks reduces displacement errors. Such a benefit is, however, pronounced practically purely for the GNSS-only solution. In this case, the application of precise orbits brings even a 30% reduction in displacement error STD, depending on the interval and simulated motion. We recall that the GNSS-derived displacement time series are subject to a high-pass Butterworth filter eliminating a low-frequency impact of satellite orbit and clock uncertainty. Consequently, the improvement from precise products is seen for the solutions with reference station GNSS data interval below the cut-off period applied in filtering, that is, 10 s. In the opposite case, the outcomes are partly ambiguous. We attribute such results to an artificial effect of the filtration process and a natural shift in time series between the last epoch with asynchronous data and the new synchronized one. The impact of this factor cannot be fully controlled with a classical filtering scenario; thus, in such a case, the results may not reveal the advantage of precise orbits and clocks.

Fig. 9
figure 9

STDs of displacements errors computed as the difference between LVDT and GNSS, or GNSS&acc. + RTS solutions. The results for broadcast (brd) and precise (sp3) orbits are distinguished. In the top and bottom panels, we show the results for the harmonic motions #1 and #3, respectively

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When it comes to the GNSS&acc. + RTS model, the profits from precise products are less explicit than for the GNSS-only one. It means that the combined solution is less susceptible to the quality of satellite orbits and clocks than GNSS-only, thanks to the multi-sensor integration and RTS smoothing.

Suppose we confront the STDs for the GNSS&acc. + RTS and GNSS-only solutions. On this basis, we will finally confirm the finding on the outperformance of the combined solution over a single sensor one. The former model results in lower displacement error STDs compared to the latter. Specifically, the combined model offers a drop in displacement error STD by 17–72% and 14–69% compared to the GNSS-only solution when the broadcast or precise satellite products are utilized, respectively.

Analysis in the time–frequency-amplitude domain

We employ Fourier transformation (FT) and Least-Squares Wavelet Analysis (LSWA) (Ghaderpour and Pagiatakis 2017) to study the displacement errors in the time–frequency-amplitude domain. The analysis in the coordinate domain has justified the feasibility of employing asynchronous GNSS observations. Therefore, we begin with the challenging case when the observations at the reference station are downsampled to a 5 s interval, but the rover data retain the original 0.05 s interval. Figure 10 visualizes the respective FT spectra, and Table 4 summarizes the peak frequency responses for such a case.

Fig. 10
figure 10

Fourier transformation spectra of the LVDT, GNSS-only, and GNSS&acc. + RTS displacement time series in the left, middle, and right panels, respectively. The results for the harmonic motions #1, #2, #3, and #4 are presented in consecutive rows from top to bottom. Asynchronous GNSS observations of 0.05 s and 5 s intervals at the rover and reference stations are used, respectively

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Table 4 Peak frequency detected with Fourier transformation from the displacement time series obtained with LVDT, GNSS, and integrated GNSS&acc. + RTS solutions. The GNSS observations at the reference station are downsampled to a 5 s interval
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For all simulated harmonic motions, the detected dominant frequencies agree well with the LVDT-derived values (Fig. 10). The maximal divergence reaches only 0.005 Hz for the GNSS-only solution when considering the #4 motion of 3.502 Hz frequency and 5 mm amplitude. What is also pronounced, integrating GNSS and accelerometer data allows better signal recovery than GNSS only solution; as for the combined solution, the detected peak frequencies, in most cases, perfectly agree with those of LVDT (Table 4).

Figure 11 shows the spectrograms of the displacement time series during the two motions #3 and #4 of the highest frequency, computed with LSWA (Ghaderpour 2021). The results reveal a meaningful noise reduction for the GNSS&acc. + RTS solution compared to the GNSS-only one, manifested by a lack of weak artificial signatures covering the entire spectrum of the latter solution. The spectrograms related to the harmonic motion #4 of 3.5 Hz frequency given in the bottom panel of Fig. 11 also accentuate that GNSS&acc. + RTS better recovers the actual frequency and amplitude than GNSS-only. The combined solution outcomes are stable and consistent with the benchmark parameters provided by LVDT, whereas GNSS-only results exhibit noticeable variations over time. Moreover, the fusion solution eliminates the artificial overestimation of the amplitude seen for the GNSS-only solution during the #4 motion. We find such benefits of GNSS&acc. + RTS as crucial for extracting the dynamic displacements in SHM.

Fig. 11
figure 11

LSWA spectrograms of the LVDT, GNSS-only, and GNSS&acc. + RTS displacement time series (see time series in the bottom two panels of Fig. 4). The results for harmonic motions #3 and #4 are presented in the top and bottom rows, respectively. In the processing are used asynchronous GNSS observations of 0.05 s and 5 s intervals at the rover and reference stations, respectively

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The impact of GNSS reference station data downsampling is further assessed by investigating the retrieved amplitudes of the simulated waveforms. Figure 12 depicts FT-derived amplitudes as the function of the GNSS interval at the reference station. In most cases, the amplitude error does not exceed 0.5 mm. Practically always, it is less than 1 mm. However, we discover higher amplitude divergences for the GNSS-only solution when the #4 harmonic motion is considered. In such a case, the amplitude error reaches 2–3 mm. Such an effect could be expected as we attribute it to the superposition of the oscillations in phase observation residuals exhibited in Fig. 5 with the simulated ones. Fortunately, that case also clearly illustrates how integrating GNSS with accelerometers allows better recovery of the sin-wave displacement amplitude than the GNSS-only solution. Specifically, after the fusion of GNSS with the accelerometer and smoothing with the RTS algorithm, the amplitude error drops to 0.5 mm (Fig. 12, bottom panel).

Fig. 12
figure 12

FT-derived amplitudes of the peak frequencies as a function of the GNSS data interval at the reference station

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The impact of GNSS data downsampling in the amplitude domain is less pronounced but still present. This statement holds true especially for the GNSS-only solution in the cases of harmonic motions #2 and #3, as the detected peak amplitudes exhibit increasing errors along with more resampled GNSS observations at the reference stations.

Conclusions

We investigated the impact of downsampling the high-rate GNSS data at the reference station on dynamic displacement retrieval when combining relative GNSS positioning and accelerometer records. First, we presented the processing methods and characterized the asynchronous GNSS data processing. Then, we evaluated the method performance based on the shake-table experiment. The results led to the following findings.

We demonstrated that downsampling high-rate GNSS observations at the reference station was justified merely to 0.2 s without a noticeable gain in solution noise for the model based only on GNSS data. Incorporating accelerometer records reduced displacement errors by half compared to the GNSS-only solution. Further accuracy improvement was obtained by applying RTS backward filtration. Consequently, it was justifiable to downsample high-rate GNSS data at the reference station even to the level of 2 s and preserve the displacement error below 1 mm with an integrated GNSS and accelerometer solution filtered with the RTS algorithm. Such results lead us to the general conclusion that regional CORS networks collecting 1 s observations can be efficiently used as references in SHM. Thus, it will halve the cost of SHM as there will be no need to install a reference GNSS station at the same data collection sampling rate as the rover station.

The impact of satellite orbits and clocks was mainly pronounced for the GNSS-only solution as a fusion of GNSS and accelerometer filtered with RTS was little susceptible to the adverse effect of low-quality satellite ephemerides.

Time–frequency domain analysis revealed that for the integrated solution, it was feasible to precisely recover the dominant frequencies of the simulated harmonic motions and the peak amplitudes with errors even below 0.5 mm. We also demonstrated that the implications of GNSS data downsampling at the reference station were less pronounced for the time–frequency domain than for the coordinate one.

We showed that a fusion of asynchronous GNSS data and accelerometer filtered with RTS could better reconcile dynamic low-scale motions than a GNSS-only solution or simply integrate an accelerometer. Moreover, the method retrieved low-scale dynamic displacements with millimeter precision and provided the solution with a much higher sampling rate than the GNSS-based one.