2.1. Multi-GNSS PPP Functional and Stochastic Model
Precise point positioning may provide displacements of the monitoring point with respect to a global reference frame using a single GNSS receiver. In contrast to the relative approach, in this case, the reference frame is defined by the satellite orbits and clocks [
27,
28]. Recent advances in algorithms, which led to PPP with resolved integer ambiguities, predestine this method to compete with the relative approach in terms of accuracy of the solution [
23,
29]. In this field, we should acknowledge methods such as the decoupled clock model or integer recovery clocks [
30,
31], as well as the method that takes advantage of uncalibrated hardware delays [
32].
The starting point of the PPP methodology applied in this work is the equations of the ionosphere-free linear combination (IF-LC) of phase and pseudorange observations. Their basis lies in undifferenced phase and code observations, which, for frequency
n, can be written as follows:
where
is the wavelength on selected frequency signal;
is the phase observable in cycles;
is the pseudorange in meters;
l and i represent station and satellite, respectively;
is the geometric range between satellite and station;
and are the receiver and satellite clock corrections in seconds, respectively;
is the speed of light in meters per second;
and denote the frequency-dependent receiver and satellite phase delays in cycles, respectively, which include initial and hardware phase biases;
and are the frequency-dependent receiver and satellite code biases in meters, respectively;
α refers to the troposphere mapping function coefficient;
ZTD denotes the zenith tropospheric delay;
denotes ionospheric delay;
is the phase ambiguity term;
denotes the observational noise.
Taking advantage of dual-frequency observations, we can form the ionosphere-free linear combination of phase and pseudorange signals [
28] and eliminate the first order of the ionospheric delay as follows:
where
is the phase observable in the units of meters corresponding to a particular frequency or combination;
,
denote selected frequencies for each constellation (e.g., L1 and L2, E1 and E5a, and B1 and B2 in the case of GPS, Galileo, and BDS constellations).
Functional model
When we integrate multi-constellation observables, we are obliged to account for the time scale inter-system bias. This can be done by selecting one of the GNSS system time scales as the pivot one and estimating the inter-constellation time difference. The other possibility is to parameterize the receiver clock correction combined with receiver hardware delays individually for each GNSS system. In general, the researchers reported comparable performance of both approaches [
33]. Because the latter strategy was employed in this case, the applied multi-constellation PPP functional model is given as follows (Equations (5)–(10)). The equations are derived for multi-constellation signals using superscripts G, E, and C for GPS, Galileo, and BDS, respectively.
where;
i,
j,
k denote selected satellites of GPS, Galileo, and BDS constellations, respectively;
are clock corrections of receiver (
l) combined with the receiver hardware delays corresponding to selected constellation;
denote ionosphere-free non-integer parameters including carrier-phase ambiguity terms coupled with the satellite and receiver phase biases.
As the approximate position of the antenna is known, this can be constrained with an appropriate a priori weight in the processing, providing faster convergence. The a priori variances that constrain the station coordinates correspond to the expected maximal displacements of the GNSS antenna. Other details of the employed PPP processing strategy, including correction models, are given in
Table 1.
Stochastic Model of Observables
For the parameter estimation, we used a sequential least squares adjustment; hence, appropriate stochastic modelling is of high interest. Because the linear combination of original GNSS signals is used, their variances are derived using the law of random error propagation. Taking advantage of GPS L1/L2 frequencies and a priori phase signal variances, the variance of the ionosphere-free linear combination reads as follows:
where
and
are the a priori variances of GPS L1 and L2 phase signals, respectively. Assuming that variances for L1 and L2 are equal for simplification, the variance of the
IF combination reads as follows:
Correspondingly, the variances for other systems and for code pseudoranges are obtained. According to Equation (12), we may expect the noise of combined observables to be a factor-three higher with respect to original observations. The weighting scheme applied for particular slant observables takes into account the initial variance of the IF signal and an elevation dependent function. Such derived values are placed at the diagonal of the variance matrix of observables. The matrix assumes no correlation between the frequencies and the types of measurement (code vs. phase). The a priori coordinates constraining is performed by the introduction of pseudo-observables to the observational model with corresponding weights computed as the inversion of variances of the constrained parameters.
At this point, we should acknowledge the studies on the application of advanced stochastic models for high-rate positioning. In the works of [
7,
34,
35], the authors justified taking into account a cross correlation between dual-frequency signals and a time correlation. Shu et al. [
35] reported a much higher time correlation of high-rate code observables than phase observables. In the latter case, which is of higher significance in PPP, the autocorrelation was weak and reached to 0.17 for the GPS L1 signal at the time lag of 0.2 s. On the other hand, Moschas and Stiros [
34] demonstrated that coordinates derived from 100 Hz GNSS data are correlated for a lag up to 0.05 s. Furthermore, they also indicated the potential mitigation of the temporal correlation by the modification of the phase-locked loop (PLL) bandwidth, for example, from common 25 Hz up to 50–100 Hz. Taking into account these results and the fact that the frequency of the periodic oscillations in our experiment did not exceed 4 Hz, we assume that the temporal correlation distorts the displacements results to a lesser extent. As a result, we employed a simplified and well-known stochastic model without accounting for temporal correlation. Nevertheless, there is no doubt that this issue needs further investigations to improve stochastic modelling and hence reach the highest precision.
2.2. Filtration of PPP Coordinate Time Series for the Extraction of the High-Rate Dynamic Displacements
It is known that the high accuracy of the PPP solution (up 1 cm) may be reached after long (e.g., several hours) static solution [
36]. However, in structural monitoring, the observational session may be much shorter. Moreover, the PPP-derived coordinate time series are still the subject of several un-modelled effects and residual biases, which may affect the estimated position. In this group, the most challenging to handle is the phase multipath. This effect may reach up to several centimeters, which importantly deteriorates the coordinate estimates. The impact of multipath can be mitigated with sidereal filtering, but such an approach requires a session of at least a few days, which is not always available. On the other hand, it should be remarked that these deteriorations are characterized with relatively long periods (several minutes and more) with respect to the analyzed dynamic displacements. Other long-term effects in the PPP time series may be related to the imprecision of satellite orbits and phase center variations, as well as residual impact of the tropospheric delay. Considering the latter, there is no doubt that estimated ZTD parameters represent only an average state of the troposphere, and thus the residuals at particular observables are basically expected. The last important factor, which propagates into the estimated coordinates, is the residual impact of satellites and receiver clocks. In this case, the short-term variations are rather expected. As a result of all of the unwanted errors, we cannot provide the mm-level position with regular PPP, even when employing multi-constellation and multi-frequency observations. However, according to the above, the most disturbing factors are expected to be long-term effects and can be effectively mitigated with high-pass filtering. Hence, such a process was applied to the coordinate time series to handle residual adverse effects and, consequently, to reach high-accuracy displacements. Owing to the known frequency of the simulated motion (3.8 Hz), it was decided to apply a high-pass Butterworth filter with cut-off period set to 2 s. This filter is one of the most commonly used tools in time series processing. Its response as a function of angular frequency
can be written as follows:
where
corresponds to cut-off frequency and
is the order of filter. According to the equation, the filter has no ripple in pass band and is often labelled as maximally flat. The flat shape is realized at the expense of the wide transition region. This undesired effect can, however, be improved through the application of a higher order, which, in our case, was 6.
The high efficiency for seismic waveforms in GNSS estimates of this filter has been recently proven, for example, in the work of [
37]. The selected threshold in the frequency domain allows the elimination of all the above indicated factors excluding short-term variations of coordinates related to the clocks residuals (mostly caused by less stable internal oscillator of receiver). The high-frequency component of these residuals cannot be simply separated from the noise of phase data, which consequently reduces the precision of PPP estimates. Nevertheless, we can assess the theoretical ratio of precisions for filtered time series derived from PPP and relative positioning. Neglecting the impact of residual clocks, the standard deviation of estimates is mainly driven by the phase data noise. If we assume the same precision at both frequencies used for IF combination, the ratio should be close to 1.5. As a result, we expect the precision of the filtered time series for PPP solution to be at the level of a few millimeters.