Keywords

1 Introduction

In this contribution we provide an analysis of the capabilities of almost-instantaneous ambiguity resolution enabled precise point positioning (PPP-RTK) using only a few epochs of GNSS observations. With the high precision of the carrier phase observations, centimeter-level positioning results are immediately obtained once the phase ambiguities are correctly resolved. A main obstacle for fast and reliable ambiguity resolution are the ionospheric delays in the user’s GNSS observations. A time-to-first-fix the ambiguities of around \(30\,\mathrm {min}\) is generally reported for GPS-only solutions (Geng et al. 2011; Zhang et al. 2019), which can to some extent be shortened when combining systems (Li and Zhang 2014; Geng and Shi 2017; Li et al. 2018). Faster solutions are also possible when external ionospheric corrections are provided (Teunissen et al. 2010; Banville et al. 2014), but these have to be at the level of at most a few centimeters for a clear gain in terms of the convergence time (Psychas et al. 2018). Such a precision is currently not possible with global ionospheric models but requires corrections from nearby reference stations, which limits the field of applications. We therefore focus on the case without a-priori ionospheric information.

A typical example for kinematic GPS L1/L2 PPP-RTK is shown in Fig. 1 using data recorded at the station PERT in Perth, Australia, during April 1, 2022. While satellite clock and bias corrections are applied, no corrections for the atmospheric delays are used, so that tropospheric and ionospheric delays have to be estimated. The ambiguities are fixed once the failure rate drops to below \(0.1\%\), indicated by the black vertical line after slightly more than \(30\,\mathrm {min}\). The ambiguity-float solution reaches the sub-meter level after several minutes, but the ambiguity-fixed solution is directly at the centimeter level.

Fig. 1
figure 1

GPS L1/L2 positioning errors of the station PERT for a kinematic PPP-RTK example during April 1, 2022. The time of ambiguity fixing is indicated by the black vertical line

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The considered key aspects to obtain a similar performance within only a few epochs are (1) the mean square error (MSE)-optimal best integer-equivariant (BIE) estimator, introduced in Teunissen (2003), which does not ‘fix’ the ambiguities to integers but rather weights different candidates, (2) a multi-GNSS solution using GPS, Galileo, BDS, and QZSS, and (3) a proper weighting of the satellite clock and bias corrections in order to obtain realistic observation models. Simulations and real data analyses are used to demonstrate the impact of these three aspects. We show that centimeter-level horizontal positioning errors are reached within one and two epochs in \(87.6\%\) and \(99.7\%\) during an exemplary day.

2 Multi-GNSS PPP-RTK: Experimental Setup and Formal Analysis

The multi-GNSS PPP-RTK performance is analyzed using one day of simulated and real \(30\,\mathrm {s}\) GPS (G) L1/L2, Galileo (E) E1/E5a, BDS (C) B1/B3, and QZSS (J) L1/L2 data in the area of Perth, Australia, during April 1, 2022.

The single-system undifferenced, uncombined GNSS code and carrier phase observations \(p_{r,f}^s\) and \(\varphi _{r,f}^s\) between the user receiver r and satellite s on frequency f are modeled as

$$\displaystyle \begin{aligned}{} \mathrm{E} [p_{r,f}^s] &= {\mathbf{g}}_r^{s,\mathrm{T}}\Delta{\mathbf{x}}_r + dt_r - dt^s +m_r^s \tau_r + \mu_f i_r^s \\ &\qquad \qquad + d_{r,f} - d^s_{,f} \\ \mathrm{E} [\varphi_{r,f}^s] &= {\mathbf{g}}_r^{s,\mathrm{T}}\Delta{\mathbf{x}}_r + dt_r - dt^s +m_r^s \tau_r - \mu_f i_r^s \\ &\qquad \qquad + \lambda_f ( \delta_{r,f} - \delta^s_{,f} +a_{r,f}^s ), \end{aligned} $$
(1)

with the expectation operator \(\mathrm {E}[\cdot ]\), the satellite-to-receiver unit vector \({\mathbf {g}}_r^s\), the incremental user coordinates \(\Delta {\mathbf {x}}_r\), the receiver and satellite clock offsets \(dt_r\) and \(dt^s\), the residual zenith tropospheric delay \(\tau _r\) with the mapping function \(m_r^s\), the ionospheric slant delay \(i_r^s\) with the coefficients \(\mu _f= \lambda _f^2/\lambda _1^2\) depending on the wavelengths \(\lambda _f\), the frequency specific receiver and satellite code biases \(d_{r,f}\) and \(d_{,f}^s\), the respective phase biases \(\delta _{r,f}\) and \(\delta _{,f}^s\), and the carrier phase integer ambiguities \(a_{r,f}^s\).

Most GNSS parameters as given in (1) cannot be determined in an absolute sense, but only as linear combinations with other parameters. The external satellite clock and phase bias corrections \(d\tilde {t}^s\) and \(\tilde {\delta }_{,f}^s\) are defined as

$$\displaystyle \begin{aligned}{} d\tilde{t}^s &= dt^s + d^s_{\mathrm{IF}} - dt_1 - d_{1,\mathrm{IF}} \\ \tilde{\delta}_{,f}^s &= \delta_{,f}^s - (d_{\mathrm{IF}}^s - \mu_f d_{\mathrm{GF}}^s -d_{1,\mathrm{IF}} + \mu_f d_{1,\mathrm{GF}})/\lambda_f \\ &\qquad \qquad \qquad \qquad - \delta_{1,f} - a_{1,f}^s, \end{aligned} $$
(2)

i.e., the satellite clock corrections also contain the clock offset of the reference receiver and ionosphere-free (IF) combinations of the code biases, and the satellite phase bias corrections contain IF and geometry-free (GF) combinations of the code biases as well as phase biases and ambiguities of the reference receiver. The corrections are either assumed deterministic or are computed by a single reference station on an epoch-by-epoch basis, for which the station NNOR (\(88.5\,\mathrm {km} \) distance to PERT) is used in the real data experiments.

The ‘rover’ station PERT is assumed kinematic with no constraints on the relative movement. After removing the PPP-RTK corrections (2) from (1), the estimable versions of its parameters are given by

$$\displaystyle \begin{aligned} d\tilde{t}_r &= dt_{1r} + d_{1r,\mathrm{IF}} \\ \tilde{i}_r^s &= i_r^s + d_{r,\mathrm{GF}} - d_{\mathrm{GF}}^s \\ \tilde{\delta}_{r,f} &= \delta_{1r,f} - (d_{1r,\mathrm{IF}}-\mu_f d_{1r,\mathrm{GF}} )/\lambda_f +a_{1r,f}^1 \\ \tilde{a}_{r,f}^s &= a_{1r,f}^s - a_{1r,f}^1,\qquad s\neq 1, \end{aligned} $$
(3)

with \((\cdot )_{1r} = (\cdot )_r - (\cdot )_1\). The code biases \(d_{r,f}\) and \(d_{,f}^s\) are absorbed by the clock and ionosphere parameters via their IF and GF combinations, and \(\varDelta {\mathbf {x}}_r\) and \(\tau _r\) are directly estimable. In a multi-GNSS solution, the receiver clock offset \(d\tilde {t}_r\) and phase biases \(\tilde {\delta }_{r,f}\) are estimated per constellation, and a separate pivot satellite is chosen for the ambiguity parameters \(\tilde {a}_{r,f}^s\). We note that the residual tropospheric zenith delay \(\tau _r\), using the global mapping function (Boehm et al. 2006), as well as the ionospheric slant delays \(\tilde {i}_r^s\) are estimated at the user receiver and are assumed unlinked in time, so that the results are valid for any ionospheric activity.

Figure 2 shows the average formal ambiguity-float positioning precision of the east component with the very weak single-epoch, single-station corrections (solid lines) and with the strongest possible, i.e., deterministic, corrections (dashed lines). Although the benefit of combining multiple systems is significant, we cannot expect centimeter-level results within a few epochs even in the four-system case with deterministic corrections. The ambiguity-fixed solutions, on the other hand, would already provide values of below \(1\,\mathrm {cm}\) even in the GPS-only case after one epoch and with single-station corrections. The average times-to-first-fix presented in Table 1, however, show that even in the best case of a multi-GNSS solution with deterministic corrections, more than seven minutes are needed. The fixing criterion is an integer bootstrapping failure rate of \(0.1\%\) or lower (Teunissen 1998). Combining systems generally implies lower failure rates for ambiguity resolution and should lead to shorter fixing times. At the same time, rising satellites – which occur more often with more systems – cause additional parameters and extend the convergence time. In our analysis, the first aspect dominates when switching from GPS-only to the two-system case, whereas when switching from the two-system to the four-system case the second aspect has a larger impact, cf. Table 1.

Fig. 2
figure 2

Average formal ambiguity-float kinematic PPP-RTK positioning precision of the east component with single-epoch, single-station corrections (solid) and with deterministic corrections (dashed)

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Table 1 Average time-to-first-fix in \([\mathrm {min}]\) for kinematic PPP-RTK with single-epoch, single-station corrections and with deterministic corrections. The fixing criterion is an integer bootstrapping failure rate of \(0.1\%\) or lower
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3 PPP-RTK with Best Integer-Equivariant Estimation

An alternative for the ambiguity-float and ambiguity-fixed solutions is given by the BIE estimator (Teunissen 2003). Let \(\hat {\mathbf {a}}\in \mathbb {R}^n\) and \({\mathbf {Q}}_{\hat {\mathbf {a}}} \in \mathbb {R}^{n\times n}\) be the float solution of the ambiguity vector \(\mathbf {a}\in \mathbb {Z}^n\) and its covariance matrix. For normally distributed data, the BIE ambiguity estimates \(\bar {\mathbf {a}}\) are the weighted sum of integers

$$\displaystyle \begin{aligned}{} \bar{\mathbf{a}} = \sum_{\mathbf{z}\in\mathbb{Z}^n} \mathbf{z} \frac{\exp \left( -\frac{1}{2} \left\| \hat{\mathbf{a}} - \mathbf{z} \right\|^2_{{\mathbf{Q}}_{\hat{\mathbf{a}}}} \right) } {\sum_{\mathbf{u}\in\mathbb{Z}^n} \exp \left( -\frac{1}{2} \left\| \hat{\mathbf{a}} - \mathbf{u} \right\|^2_{{\mathbf{Q}}_{\hat{\mathbf{a}}} } \right) }. \end{aligned} $$
(4)

When implementing (4), the infinite sums are replaced by sums over the finite set of integers contained within an ellipsoidal region around \(\hat {\mathbf {a}}\). The BIE positioning solution follows from the conditional least-squares estimator assuming the ambiguities given by \(\bar {\mathbf {a}}\). The BIE results are MSE-optimal, meaning that they are always at least as good as the ambiguity-float or any ambiguity-fixed solution in that sense. The BIE estimator automatically adapts to the strength of the underlying model – without the need to define a fixing criterion. It is identical to the ambiguity-float solution for very poor precision of \(\hat {\mathbf {a}}\) and converges to the ambiguity-fixed solution for very high precision of \(\hat {\mathbf {a}}\) (Teunissen 2003). Further, as the BIE results are MSE optimal, they can serve as a benchmark for analyzing the theoretically best possible performance of any GNSS model.

An extension of the BIE principle for elliptically contoured distributions is provided in Teunissen (2020), and a sequential scalar approximation of the BIE estimator is proposed in Brack et al. (2014). A performance analysis of the BIE estimator for single-baseline RTK positioning is given in Odolinski and Teunissen (2020) for low-cost receivers and in Yong et al. (2022) for smartphone receivers.

In order to gain some insight into the basic properties of the BIE estimator, we consider a simulated kinematic GPS+Galileo PPP-RTK example with single-station corrections. The horizontal positioning errors after six epochs are shown in Fig. 3 for \(10{,}000\) samples together with their root mean square (RMS) errors. The ambiguity-float solution (gray) is normally distributed with an uncertainty at the few-decimeter level. The ambiguity-fixed solution using the integer least-squares estimator is at the sub-centimeter level with correct ambiguity estimates (green) and can otherwise have large errors (red). The BIE solution (blue) is less likely to result in very large errors than the ambiguity-fixed solution, but also has a smaller probability of very small positioning errors. It is generally more concentrated around the true position than the ambiguity-fixed solution, which is also reflected by the smallest RMS errors of \(2.6\,\mathrm {cm}\) and \(2.3\,\mathrm {cm}\) for the east and north components.

Fig. 3
figure 3

Simulated horizontal positioning errors for kinematic GPS+Galileo PPP-RTK after six observation epochs with single-epoch, single-station corrections. The ambiguity-float solution is shown in gray, the ambiguity-fixed solution in green and red for correct and incorrect ambiguity estimates, and the BIE solution in blue

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Figure 4 shows the average simulated RMS east positioning error of the considered kinematic PPP-RTK positioning example for the first ten minutes after initialization using different systems with single-station and deterministic corrections. As already observed in Fig. 2, the ambiguity-float solutions cannot provide centimeter-level results within such a short convergence time. For the GPS+Galileo case, centimeter-level positioning results are obtained with the ambiguity-fixed and BIE estimators after slightly more than five minutes with single-station corrections and after around three minutes with deterministic corrections. In the four-system case, sub-decimeter results are obtained within one minute (two epochs) and sub-centimeter results within one and a half minutes (three epochs) with single-station corrections, which can both be reduced by around half a minute with deterministic corrections. The BIE results are always RMS-optimal. It is noted that although the ambiguity-fixed and BIE RMS errors are often very close, the error characteristics of both estimators can still be quite different, cf. Fig. 3.

Fig. 4
figure 4

Average simulated RMS east positioning error for kinematic PPP-RTK with single-epoch, single-station corrections (solid) and with deterministic corrections (dashed)

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From the above simulation results we can expect centimeter-level horizontal PPP-RTK results with four systems within only a few observation epochs. Real-data PPP-RTK results of the rover station PERT with satellite clock and phase bias corrections from the station NNOR are shown in Fig. 5 for the \(24\,\mathrm {h}\) of April 1, 2022, using one and two observation epochs. The horizontal RMS positioning errors of the BIE solution (shown in blue) are at the one-decimeter level after one epoch and at the centimeter level already after only two epochs. A positioning error of less than \(3\,\mathrm {cm}\) for the horizontal components is obtained in \(87.6\%\) and \(99.7\%\) of the cases, respectively. The corresponding ambiguity-fixed solutions show larger RMS errors caused by incorrect ambiguity estimates, but also have a higher probability of very small positioning errors, as can be seen in the zoom plot for one epoch.

Fig. 5
figure 5

Horizontal positioning errors of the station PERT for kinematic GPS+Galileo+BDS+QZSS PPP-RTK with corrections from the station NNOR. The ambiguity-float solution is shown in gray, the ambiguity-fixed solution in red, and the BIE solution in blue

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4 Neglecting the Uncertainty of the PPP-RTK Corrections

So far, the PPP-RTK corrections have been applied to the user observations together with their full covariance information. In this way, the user obtains a realistic description of his stochastic observation model, and the corrected observations are weighted with their actual inverse covariance matrix in the least-squares adjustment, leading to minimum-variance parameter estimates with a realistic description of their precision. Neglecting the uncertainty of the corrections can, therefore, not only result in an increased failure rate when fixing the ambiguities, but also in unrealistic formal success rates as computed from the precision of the float ambiguity estimates (Psychas et al. 2022). The latter is particularly problematic, as a user might have too much confidence that the ambiguities can be resolved correctly, while in fact the success probability could be quite poor.

In the context of BIE ambiguity estimation, we face a similar problem: As neglecting the uncertainty of the corrections can have an impact on both \(\hat {\mathbf {a}}\) and its covariance matrix, suboptimal weights of the integer candidates might be obtained when computing the BIE ambiguity estimates \(\bar {\mathbf {a}}\) in (4) and the MSE-optimality of the positioning solution might be lost.

Figure 6 shows the magnitude of the three-dimensional PPP-RTK errors of the station PERT using two consecutive observation epochs with the BIE estimator, where the uncertainty of the corrections from the station NNOR is either included by means of their full covariance matrix as before (red), or completely neglected (blue). We can see that neglecting the uncertainty of the corrections generally leads to larger positioning errors, most notably around \(1\,\mathrm {h} \,40\,\mathrm {min}\) with an increase of more than \(3\,\mathrm {m}\). The corresponding empirical RMS positioning errors are given in Table 2 for the east, north, and up components, and show an increase of up to \(67\%\) when neglecting the uncertainty of the corrections.

Fig. 6
figure 6

Three-dimensional BIE positioning errors of the station PERT for kinematic GPS+Galileo+BDS+QZSS PPP-RTK using two consecutive epochs. The precision of the corrections from the station NNOR is fully considered (red) or completely neglected (blue)

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Table 2 Empirical BIE east, north, and up RMS positioning errors of the station PERT for kinematic GPS+Galileo+BDS+QZSS PPP-RTK using two epochs in \([\mathrm {cm}]\). The precision of the corrections from the station NNOR is fully considered or completely neglected
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5 Conclusion

An analysis of the PPP-RTK performance with the current GNSS constellations in the absence of atmospheric corrections was provided. As ionospheric delay parameters have to be estimated in this case, fast and reliable ambiguity resolution is difficult, as was demonstrated in the beginning of this contribution. In order to achieve almost-instantaneous centimeter-level results, the use of the BIE estimator in a multi-GNSS solution was proposed. As the BIE positioning results are MSE-optimal, they can also be interpreted as the limits of the positioning performance of a given model.

PPP-RTK examples with a different selection of systems were analyzed through simulations, where rather weak single-epoch, single-station corrections and ‘perfect’ deterministic corrections were applied. The results showed that centimeter-level RMS positioning errors within a few (one to three) epochs can indeed be achieved when combining all four considered GNSS, even with single-station corrections.

An analysis of real GNSS data from the station PERT with corrections from NNOR confirmed these results. The empirical east and north RMS positioning errors after two epochs are \(2.2\,\mathrm {cm}\) and \(1.4\,\mathrm {cm}\) when combining GPS, Galileo, BDS, and QZSS data.

It was further demonstrated how the user positioning performance is degraded when neglecting the uncertainty of the PPP-RTK corrections, caused by unrealistic assumptions on the user’s stochastic observation model. A significant increase of the RMS positioning errors was observed, reaching \(67\%\) for the up component.

Besides the BIE estimator, another alternative to conventional ambiguity fixing in weak models is partial ambiguity resolution. As demonstrated in Brack et al. (2021) for multi-GNSS single-baseline RTK positioning, it enables similar convergence times to reach centimeter-level results when ionospheric delays are estimated.

A more detailed version of this study is published in Brack et al. (2023).