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Review and principles of PPP-RTK methods

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Abstract

PPP-RTK is integer ambiguity resolution-enabled precise point positioning. In this contribution, we present the principles of PPP-RTK, together with a review of different mechanizations that have been proposed in the literature. By application of \(\mathcal {S}\)-system theory, the estimable parameters of the different methods are identified and compared. Their interpretation is essential for gaining a proper insight into PPP-RTK in general, and into the role of the PPP-RTK corrections in particular. We show that PPP-RTK is a relative technique for which the ‘single-receiver user’ integer ambiguities are in fact double-differenced ambiguities. We determine the transformational links between the different methods and their PPP-RTK corrections, thereby showing how different PPP-RTK methods can be mixed between network and users. We also present and discuss four different estimators of the PPP-RTK corrections. It is shown how they apply to the different PPP-RTK models, as well as why some of the proposed estimation methods cannot be accepted as PPP-RTK proper. We determine analytical expressions for the variance matrices of the ambiguity-fixed and ambiguity-float PPP-RTK corrections. This gives important insight into their precision, as well as allows us to discuss which parts of the PPP-RTK correction variance matrix are essential for the user and which are not.

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References

  • Allison T (1991) Multi-observable processing techniques for precise relative positioning. In: Proceedings ION GPS-91. Albuquerque, New Mexico, 11–13 September, pp 715–725

  • Baarda W (1973) S-transformations and criterion matrices. Tech. rep., Netherlands Geodetic Commission, Publ. on Geodesy, New Series, vol 5(1), Delft

  • Bertiger W, Desai SD, Haines B, Harvey N, Moore AW, Owen S, Weiss JP (2010) Single receiver phase ambiguity resolution with GPS data. J Geod 84(5):327–337

    Article  Google Scholar 

  • Beutler G, Bock H, Dach R, Fridez P, Gade A, Hugentobler U, Jaggi A, Meindl M, Mervart L, Prange L, Schaer S, Springer T, Urschl P, Walser P (2007) Bernese GPS software version 5.0. Astron. Inst., Univ. of Bern, Bern, Switzerland

  • Bisnath S, Collins P (2012) Recent developments in precise point positioning. Geomatica 66(2):103–111

    Article  Google Scholar 

  • Bisnath S, Gao Y (2008) Current state of precise point positioning and future prospects and limitations. In: Observing our changing earth, IAG Symp 133:615–623

  • Collins P (2008) Isolating and estimating undifferenced GPS integer ambiguities. In: Proceedings ION NTM, pp 720–732

  • Collins P, Lahaye F, Heroux P, Bisnath S (2008) Precise point positioning with ambiguity resolution using the decoupled clock model. In: Proceedings of the 21st international technical meeting of the satellite division of the Institute of Navigation (ION GNSS 2008), pp 1315–1322

  • de Jonge PJ (1998) A processing strategy for the application of the GPS in networks. PhD thesis, Delft University of Technology, Publication on Geodesy, 46, Netherlands Geodetic Commission, Delft

  • Ge M, Gendt G, Rothacher M, Shi C, Liu J (2008) Resolution of GPS carrier-phase ambiguities in precise point positioning (PPP) with daily observations. J Geod 82(7):389–399

    Article  Google Scholar 

  • Geng J (2011) Rapid integer ambiguity resolution in GPS precise point positioning. PhD thesis, University of Nottingham, UK

  • Geng J, Bock Y (2013) Triple-frequency GPS precise point positioning with rapid ambiguity resolution. J Geod 87(5):449–460

    Article  Google Scholar 

  • Geng J, Meng X, Dodson A, Teferle F (2010) Integer ambiguity resolution in precise point positioning: method comparison. J Geod 84(9):569–581

    Article  Google Scholar 

  • Geng J, Teferle FN, Meng X, Dodson AH (2011) Towards PPP-RTK: ambiguity resolution in real-time precise point positioning. Adv Space Res 47(10):1664–1673

    Article  Google Scholar 

  • Geng J, Shi C, Ge M, Dodson AH, Lou Y, Zhao Q, Liu J (2012) Improving the estimation of fractional-cycle biases for ambiguity resolution in precise point positioning. J Geod 86(8):579–589

    Article  Google Scholar 

  • Hernandez-Pajares M, Juan JM, Sanz J, Colombo OL (2000) Application of ionospheric tomography to real-time GPS carrier-phase ambiguities resolution, at scales of 400–1,000 km and with high geomagnetic activity. Geophys Res Lett 27(13):2009–2012

    Article  Google Scholar 

  • Hofmann-Wellenhof B, Lichtenegger H, Wasle E (2008) GNSS: global navigation satellite systems: GPS, Glonass, Galileo, and more. Springer, New York

    Google Scholar 

  • Jonkman N, Teunissen P, Joosten P, Odijk D (2000) GNSS long baseline ambiguity resolution: impact of a third navigation frequency. In: Geodesy Beyond 2000, IAG Symp 121, pp 349–354

  • Khodabandeh A, Teunissen PJG (2014) Single-epoch GNSS array integrity: an analytical study. IAG Symp 142, accepted for publication

  • Kouba J, Heroux P (2001) Precise point positioning using IGS orbit and clock products. GPS solut 5(2):12–28

    Article  Google Scholar 

  • Lannes A, Prieur JL (2013) Calibration of the clock-phase biases of GNSS networks: the closure-ambiguity approach. J Geod 87(8):709–731

    Article  Google Scholar 

  • Lannes A, Teunissen PJG (2011) GNSS algebraic structures. J Geod 85(5):273–290

    Article  Google Scholar 

  • Laurichesse D (2011) The CNES real-time PPP with undifferenced integer ambiguity resolution demonstrator. In: Proceedings of the ION GNSS, pp 654–662

  • Laurichesse D, Mercier F (2007) Integer ambiguity resolution on undifferenced GPS phase measurements and its application to PPP. In: Proceedings of the 20th international technical meeting of the satellite division of the Institute of Navigation (ION GNSS 2007), pp 839–848

  • Laurichesse D, Mercier F, Berthias J, Broca P, Cerri L, CNES F (2009) Integer ambiguity resolution on undifferenced GPS phase measurements and its application to PPP and satellite precise orbit determination. Navigation 56(2):135–149

    Article  Google Scholar 

  • Li T, Wang J, Laurichesse D (2013a) Modeling and quality control for reliable precise point positioning integer ambiguity resolution with GNSS modernization. GPS Solut, pp 1–14

  • Li X, Ge M, Zhang H, Wickert J (2013b) A method for improving uncalibrated phase delay estimation and ambiguity-fixing in real-time precise point positioning. J Geod 87(5):405–416

    Article  Google Scholar 

  • Li X, Ge M, Lu C, Zhang Y, Wang R, Wickert J, Schuh H (2014) High-rate GPS seismology using real-time precise point positioning with ambiguity resolution. Geoscience and remote sensing, IEEE transactions on, pp 1–15. doi:10.1109/TGRS.2013.2295373

  • Loyer S, Perosanz F, Mercier F, Capdeville H, Marty JC (2012) Zero-difference GPS ambiguity resolution at CNES-CLS IGS analysis center. J Geod 86(11):991–1003

    Article  Google Scholar 

  • Mervart L, Lukes Z, Rocken C, Iwabuchi T (2008) Precise point positioning with ambiguity resolution in real-time. In: Proceedings of ION GNSS, pp 397–405

  • Mervart L, Rocken C, Iwabuchi T, Lukes Z, Kanzaki M (2013) Precise point positioning with fast ambiguity resolution-prerequisites, algorithms and performance. In: Proceedings of ION GNSS, pp 1176–1185

  • Odijk D (2002) Fast precise GPS positioning in the presence of ionospheric delays. Ph.D. thesis, Delft University of Technology, Publication on Geodesy, 52, Netherlands, Geodetic Commission, Delft

  • Odijk D, Teunissen PJG (2008) ADOP in closed form for a hierarchy of multi-frequency single-baseline GNSS models. J Geod 82(8):473–492

    Article  Google Scholar 

  • Odijk D, Teunissen PJG, Zhang B (2012) Single-frequency integer ambiguity resolution enabled GPS precise point positioning. J Survey Eng 138(4):193–202

    Article  Google Scholar 

  • Odijk D, Arora BS, Teunissen PJG (2014a) Predicting the success rate of long-baseline GPS + Galileo (partial) ambiguity resolution. J Navigat 1–17. doi:10.1017/S037346331400006X

  • Odijk D, Teunissen PJG, Khodabandeh A (2014b) Single-frequency PPP-RTK: theory and experimental results. IAG Symp Earth Edge: Sci Sustain Planet 139:571–578

  • Rao CR (1973) Linear statistical inference and its applications, vol 2. Wiley, New Jersey

    Book  Google Scholar 

  • Shi J (2012) Precise point positioning integer ambiguity resolution with decoupled clocks. PhD thesis, University of Calgary, Canada

  • Shi J, Gao Y (2013) A comparison of three PPP integer ambiguity resolution methods. GPS Solut (published online)

  • Teunissen PJG (1985) Generalized inverses, adjustment, the datum problem and S-transformations. In: Grafarend EW, Sanso F (eds) Optimization and design of geodetic networks. Springer, Berlin

    Google Scholar 

  • Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70(1–2):65–82

    Article  Google Scholar 

  • Teunissen PJG (1997a) A canonical theory for short GPS baselines. Part I: the baseline precision. J Geod 71(6):320–336

    Article  Google Scholar 

  • Teunissen PJG (1997b) A canonical theory for short GPS baselines. Part IV: precision versus reliability. J Geod 71(9):513–525

    Article  Google Scholar 

  • Teunissen PJG (1997c) The geometry-free GPS ambiguity search space with a weighted ionosphere. J Geod 71(6):370–383

    Article  Google Scholar 

  • Teunissen PJG (1997d) On the GPS widelane and its decorrelating property. J Geod 71(9):577–587

    Article  Google Scholar 

  • Teunissen PJG (1998) Success probability of integer GPS ambiguity rounding and bootstrapping. J Geod 72(10):606–612

    Article  Google Scholar 

  • Teunissen PJG (1999) An optimality property of the integer least-squares estimator. J Geod 73(11):587–593

    Article  Google Scholar 

  • Teunissen PJG (2000) Adjustment theory: an introduction. Delft University Press, series on Mathematical Geodesy and Positioning

  • Teunissen PJG (2002) The parameter distributions of the integer GPS model. J Geod 76(1):41–48

    Article  Google Scholar 

  • Teunissen PJG (2012) A-PPP: array-aided precise point positioning with global navigation satellite systems. Signal Process IEEE Trans 60(6):2870–2881

    Article  Google Scholar 

  • Teunissen PJG, de Bakker PF (2012) Single-receiver single-channel multi-frequency GNSS integrity: outliers, slips, and ionospheric disturbances. J Geod 1–17

  • Teunissen PJG, Khodabandeh A (2013) BLUE, BLUP and the Kalman filter: some new results. J Geod 87(5):461–473

    Article  Google Scholar 

  • Teunissen PJG, Odijk D, Zhang B (2010) PPP-RTK: results of CORS network-based PPP with integer ambiguity resolution. J Aeronaut, Astronaut Aviat 42(4):223–229

    Google Scholar 

  • Wubbena G (1989) The GPS adjustment software package-GEONAP-concepts and models. In: International geodetic symposium on satellite positioning l, Las Cruces, New Mexico, 13–17 March, pp 452–461

  • Wubbena G, Schmitz M, Bagg A (2005) PPP-RTK: precise point positioning using state-space representation in RTK networks. In: Proceedings of ION GNSS, pp 13–16

  • Zhang B, Teunissen PJG, Odijk D (2011) A novel un-differenced PPP-RTK concept. J Navig 64(S1):S180–S191

    Article  Google Scholar 

  • Zhang X, Li P (2013) Assessment of correct fixing rate for precise point positioning ambiguity resolution on a global scale. J Geod 87(6):579–589

    Article  Google Scholar 

  • Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, Webb FH (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–5017

Download references

Acknowledgments

This work has benefitted from the many fruitful PPP-RTK discussions we had with our colleagues from the Curtin GNSS Research Centre. The first author is the recipient of an Australian Research Council (ARC) Federation Fellowship (project number FF0883188). This work has been done in the context of the Positioning Program Project 1.01 “New carrier phase processing strategies for achieving precise and reliable multi-satellite, multi-frequency GNSS/RNSS positioning in Australia” of the Cooperative Research Centre for Spatial Information (CRC-SI). All this support is gratefully acknowledged.

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Correspondence to A. Khodabandeh.

Appendix

Appendix

Proof of Lemma 2

As the phase observations in (74) are reserved for the ambiguities \(\tilde{a}_r\), the float solutions of \(d\tilde{t}_r\) and \(\tilde{\iota }_r\) are, respectively, determined as the IF and GF combinations of the code observations only, that is

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}} = -[\mu _\mathrm{\scriptscriptstyle IF}^T\otimes D_m^T]p_r,\\ &{}\!\!\!\displaystyle \hat{\tilde{\iota }}_{r,\scriptscriptstyle \mathrm{GF}} = +[\mu _\mathrm{\scriptscriptstyle GF}^T\otimes D_m^T] p_r \end{array} \end{aligned}$$
(114)

which gives the first expression of (76). The second expression (76) follows by substituting the preceding equations into

$$\begin{aligned}&\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}} = [\Lambda ^{-1}\otimes D_m^T]\nonumber \\&\quad \times \left[ \phi _r+[e\otimes I_{m-1}]d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}} +[\mu \otimes I_{m-1}]\hat{\tilde{\iota }}_{r,\scriptscriptstyle \mathrm{GF}}\right] , \end{aligned}$$
(115)

together with the identity \(e\mu _\mathrm{\scriptscriptstyle IF}^T=I_2-\mu \mu _\mathrm{\scriptscriptstyle GF}^T\).

An application of the (co)variance propagation law to (76) gives (77). \(\square \)

Proof of Lemma 3

Upon resolving the DD ambiguities \(\check{\tilde{z}}_r\) (\(r=1,\ldots ,n\)), with \(\check{\tilde{z}}_1=0\), \(2(m-1)(n-1)\) redundant model’s misclosures contribute to the estimation procedure, namely

$$\begin{aligned} t_{\tilde{z}_r} = \hat{\tilde{z}}_r - \check{\tilde{z}}_r,\quad r=2,\ldots ,n \end{aligned}$$
(116)

According to the least-squares conditional adjustment, the unbiased estimators \(\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}\) and \(d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}}\) are corrected by the above misclosures to provide the best linear unbiased estimators (BLUEs) \(\check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}\) and \(d\check{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}}\) (Teunissen 2000). Adding the corrections, the BLUEs must remain unbiased and get uncorrelated with the underlying misclosures, see Teunissen and Khodabandeh (2013, p. 463). The unique corrections must, therefore, fulfill two conditions: (1) they must be zero-mean and (2) their covariances with the misclosures must be identical to those between the float estimators and the misclosures with a negative sign. Proposing the following corrections

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \epsilon _{\tilde{a}_r} = \check{\tilde{z}}_r-\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}+\frac{1}{n}\sum _{j=1}^n (\hat{\tilde{a}}_{j,\scriptscriptstyle \mathrm{GF}}-\check{\tilde{z}}_j)\\ &{}\!\!\!\epsilon _{d\tilde{t}_r} = -Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} (\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}-\check{\tilde{a}}_{r,\scriptscriptstyle GF}) \end{array} \end{aligned}$$
(117)

their zero-mean property follows, respectively, from

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \mathsf {E}(\hat{\tilde{a}}_{j,\scriptscriptstyle \mathrm{GF}}) = \check{\tilde{z}}_j-\tilde{\delta },\quad j=1,\ldots ,n\\ &{}\!\!\!\displaystyle \mathsf {E}(\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}) = \mathsf {E}(\check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}), \end{array} \end{aligned}$$
(118)

while the second-property follows, respectively, from

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle Q_{\epsilon _{\tilde{a}_r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} = -Q_{\hat{\tilde{a}}_{r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} \\ &{}\!\!\!\displaystyle Q_{\epsilon _{d\tilde{t}_r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} = -Q_{d\hat{\tilde{t}}_{r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} \end{array} \end{aligned}$$
(119)

Accordingly, the fixed solutions are obtained as

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}} = \hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}+\epsilon _{\tilde{a}_r}\\ &{}\!\!\!\displaystyle d\check{\tilde{t}}_{r} = d\hat{\tilde{t}}_{r}+\epsilon _{d\tilde{t}_r} \end{array} \end{aligned}$$
(120)

which proves (78). Applying the (co)variance propagation law to (78) gives (79). \(\square \)

Proof of Table 3

We first prove the geometry-free results where again use is made of the GNSS misclosure concept (Khodabandeh and Teunissen 2014). In the \(k\)-epoch case, as the ambiguities are assumed constant in time, any (co)variance matrix \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle GF}\) is corrected according to the least-squares conditional adjustment as

$$\begin{aligned} Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] = Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}-[\frac{k-1}{k}]Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle GF} Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} Q_{\hat{\tilde{a}}_r \hat{y}}^{\scriptscriptstyle \mathrm{GF}} \end{aligned}$$
(121)

Setting \(\hat{y}=\hat{\tilde{a}}_r\), the above equation is specialized to

$$\begin{aligned} Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k] = \frac{1}{k}Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}} \end{aligned}$$
(122)

This gives the expressions of \(Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) and \(Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) by setting \(\hat{x}=\hat{\tilde{a}}_{r,{\scriptscriptstyle \mathrm{GF}}}\) and \(\hat{x}=d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}\), respectively. The expression of \(Q_{d\hat{\tilde{t}}_rd\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) follows by setting \(\hat{x}=\hat{y}=d\hat{\tilde{t}}_r\) in (121), together with the identity

$$\begin{aligned} Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}} Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} Q_{\hat{\tilde{a}}_r d\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}= Q_{d\hat{\tilde{t}}_rd\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}-c_{\check{\rho }}^2 C_s \end{aligned}$$
(123)

We now prove the geometry-based results. To do so, we first formulate the \([k(m-1)-\nu ](n-1)\) redundant misclosures brought by the geometry-based model (cf. Table 2). The geometry parametrization \(\Delta \tilde{x}_r\) gives \((m-1-\nu )(n-1)\) misclosures as

$$\begin{aligned} t_{\hat{g}_r}\!=\! (D_m^TG) ^{\bot T} [\frac{1}{k}\sum \limits _{i=1}^k (d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(i)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(i))],\; r\!=\!2,\ldots ,n\nonumber \\ \end{aligned}$$
(124)

while the time-invariance assumption on \(\Delta \tilde{x}_r\) gives \((k-1)(m-1)(n-1)\) misclosures as

$$\begin{aligned} t_{i,r}= [d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(i)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(i)]-[d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(1)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(1)]\nonumber \\ \end{aligned}$$
(125)

for \(r=2,\ldots ,n\) and \(i=2,\ldots ,k\).

Since the two misclosure vectors \(t_{\hat{g}}=[t_{\hat{g}_2}^T,\ldots ,t_{\hat{g}_n}^T]^T\) and \(t=[t_{2,2}^T,\ldots ,t_{2,n}^T,\ldots ,t_{k,n}^T]^T\) are uncorrelated, any (co)variance matrix \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k]\) is corrected to \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GB}}[k]\) as

$$\begin{aligned} Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GB}}[k]&= Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] -Q_{\hat{x}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k]\nonumber \\&-Q_{\hat{x}t}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{tt}^{-1} Q_{t\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] \end{aligned}$$
(126)

Accordingly, the expressions of \(Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GB}}[k]\), \(Q_{d\hat{\tilde{t}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GB}}[k]\) and \(Q_{d\hat{\tilde{t}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GB}}[k]\) follow through the identities

$$\begin{aligned} Q_{\hat{\tilde{a}}_rt_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 \Lambda ^{-1}e_\mu e_\mu ^T \Lambda ^{-1} \otimes \tilde{C}_s\nonumber \\ Q_{d\hat{\tilde{t}}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 e_\mu ^T \Lambda ^{-1} \otimes \tilde{C}_s\nonumber \\ Q_{d\hat{\tilde{t}}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}d\hat{\tilde{t}}}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 \otimes \tilde{C}_s \end{aligned}$$
(127)

as well as

$$\begin{aligned} Q_{d\hat{\tilde{t}}t}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{tt}^{-1} Q_{td\hat{\tilde{t}}}^{\scriptscriptstyle \mathrm{GF}}[k] = \frac{k-1}{k}\frac{n-1}{n}c_{\check{\rho }}^2 \otimes C_s \end{aligned}$$
(128)

with \(Q_{\hat{\tilde{a}}_rt}^{\scriptscriptstyle \mathrm{GF}}[k]=0\). \(\square \)

Proof of Tables 4 and 5

The proof goes along the same lines as the proof of Table 3, so it will not be presented here. \(\square \)

The exact of value of \(\gamma \) The exact value of \(\gamma \), introduced in Lemma 5, can be stated as

$$\begin{aligned} \gamma =\frac{(f_1^3-f_2^3)^2}{f_1^2f_2^2(f_1+f_2)^2+\epsilon \,\eta } \end{aligned}$$
(129)

with

$$\begin{aligned} \eta = \left[ \frac{f_1+f_2}{f_1-f_2}\right] ^2 (f_1^2+f_2^2)(f_1^4+f_2^4) \end{aligned}$$
(130)

The approximation, given in Lemma 5, follows by neglecting \(\epsilon \,\eta \), compared to the first term in the denominator of (129). \(\square \)

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Teunissen, P.J.G., Khodabandeh, A. Review and principles of PPP-RTK methods. J Geod 89, 217–240 (2015). https://doi.org/10.1007/s00190-014-0771-3

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