# Analysis of Wide-Lane Ambiguities Derived from Geometry-Free and Geometry-Based Precise Point Positioning Models and Their Implication for Orbit and Clock Quality

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. WL Ambiguities Derived from the GF Model

**H**is the coefficient matrix derived from Equation (5) only for ambiguity parameters as follows:

^{s}is the elevation of the satellite. According to the error propagation law, the variance-covariance of ambiguity terms can be derived as:

## 3. WL Ambiguities Derived from the GB Model

## 4. Impact of Orbit and Clock Residual Errors on Ambiguity Estimation

#### 4.1. Orbit and Clock Residual Error Being Time-Constant Bias

#### 4.2. Orbit and Clock Residual Error Being Time-Variant Error

## 5. Experiment and Results Analysis

#### 5.1. Analysis of GB and GF Ambiguities

#### 5.2. Orbit and Clock Residual Error Effects

## 6. Simulation Study of Orbit/Clock Error Effects

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zumberge, J.F.; Heflin, M.B.; Jefferson, D.C.; Watkins, M.M.; Webb, F.H. Precise point positioning for the efficient and robust analysis of GPS data from large networks. J. Geophys. Res.
**1997**, 102, 5005–5017. [Google Scholar] [CrossRef] - Kouba, J.; Heroux, P. Precise point positioning using IGS orbit products. GPS Solut.
**2001**, 5, 12–28. [Google Scholar] [CrossRef] - Geng, J.; Teferle, F.N.; Shi, C.; Meng, X.; Dodson, F.N.; Liu, J. Ambiguity resolution in precise point positioning with hourly data. GPS Solut.
**2009**, 13, 263–270. [Google Scholar] [CrossRef] - Muellerschoen, R.J.; Iijima, B.; Meyer, R.; Bar-Server, Y.; Accad, E. Real-time point-positioning performance evaluation of single-frequency receivers using NASA’s global differential GPS system. In Proceedings of the ION GNSS 17th international Technical Meeting of the Satellite Division, Long Beach, CA, USA, 21–24 September 2004; pp. 1874–1880. [Google Scholar]
- Leandro, R.; Landau, H.; Nitschke, M.; Glocker, M.; Seeger, S.; Chen, X.; Deking, A.; Tahar, M.B.; Zhang, F.; Ferguson, K.; et al. RTX positioning: The next generation of cm-accurate real-time GNSS positioning. In Proceedings of the 24th International Technical Meeting of the Satellite Division of the ION, Portland, OR, USA, 20–23 September 2011; pp. 1460–1475. [Google Scholar]
- Li, X.; Ge, M.; Dai, X.; Ren, X. Accuracy and reliability of multi-GNSS real-time precise positioning: GPS, GLONASS, BeiDou, and Galileo. J. Geod.
**2015**, 89, 607–635. [Google Scholar] [CrossRef] - Ge, M.; Gendt, G.; Rothacher, M.; Shi, C.; Liu, J. Resolution of GPS carrier phase ambiguities in precise point positioning (PPP) with daily observations. J. Geod.
**2008**, 82, 389–399. [Google Scholar] [CrossRef] - Laurichesse, D.; Mercier, F.; Berthias, J.P. Zero-difference integer ambiguity fixing on single frequency receivers. In Proceedings of the ION ITM-2009, Anaheim, CA, USA, 26–28 January 2009; pp. 2460–2469. [Google Scholar]
- Collins, P.; Bisnath, S.; Lahaye, F.; Heroux, P. Undifferenced GPS ambiguity resolution using the decoupled clock model and ambiguity datum fixing. Navigation
**2010**, 57, 123–135. [Google Scholar] [CrossRef] - Geng, J.; Shi, C.; Ge, M.; Dodson, A.H.; Lou, Y.; Zhao, Q.; Liu, J. Improving the estimation of fractional-cycle biases for ambiguity resolution in precise point positioning. J. Geod.
**2012**, 86, 579–589. [Google Scholar] [CrossRef] - Li, X.; Li, X.; Yuan, Y.; Zhang, K.; Zhang, X.; Wickert, J. Multi-GNSS phase delay estimation and PPP ambiguity resolution: GPS, BDS, GLONASS, Galileo. J. Geod.
**2017**. [Google Scholar] [CrossRef] - Wübbena, G.; Schmitz, M.; Bagge, A. PPP-RTK: Precise point positioning using state-space representation in RTK networks. In Proceedings of the ION GNSS 18th International Technical Meeting of the Satellite Division, Long Beach, CA, USA, 13–16 September 2005; pp. 2584–2594. [Google Scholar]
- Teunissen, P.J.G.; Odijk, D.; Zhang, B. PPP-RTK: Results of CORS network-based PPP with integer ambiguity resolution. J. Aeronaut. Astronaut. Aviat.
**2010**, 42, 223–230. [Google Scholar] - Li, X.; Ge, M.; Zhang, H.; Wickert, J. A method for improving uncalibrated phase delay estimation and ambiguity-fixing in real-time precise point positioning. J. Geod.
**2013**, 87, 405–416. [Google Scholar] [CrossRef] - Chen, G.; Zhao, Q. Near-field surface displacement and permanent deformation induced by the Alaska Mw 7.5 earthquake determined by high-rate real-time ambiguity-fixed PPP solutions. Chin. Sci. Bull.
**2014**, 59, 4781–4789. [Google Scholar] [CrossRef] - Hatch, R. The Synergism of GPS code and carrier measurements. In Proceedings of the Third International Symposium on Satellite Doppler Positioning, Las Cruces, NM, USA, 8–12 February 1982; Volume 2, pp. 1213–1231. [Google Scholar]
- Melbourne, W. The case for ranging in GPS-based geodetic systems. In Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Rockville, MD, USA, 15–19 April 1985; pp. 373–386. [Google Scholar]
- Wübbena, G. Software developments for geodetic positioning with GPS using TI-4100 code and carrier measurements. In Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Rockville, MD, USA, 15–19 April 1985; pp. 403–412. [Google Scholar]
- Teunissen, P.J.G.; Khodabandeh, A. Review and principals of PPP-RTK methods. J. Geod.
**2015**, 89, 217–240. [Google Scholar] [CrossRef] - Montenbruck, O.; Steigenberger, P.; Prange, L.; Deng, Z.; Zhao, Q.; Perosanz, F.; Romero, I.; Noll, C.; Stürze, A.; Weber, G.; et al. The multi-GNSS experiment (MGEX) of the international GNSS service (IGS)—Achievements, prospects and challenges. Adv. Space Res.
**2017**, 59, 1671–1697. [Google Scholar] [CrossRef] - Kazmierski, K.; Sosnics, K.; Hadas, T. Quality assessment of multi-GNSS orbits and clocks for real-time precise point positioning. GPS Solut.
**2018**, 22, 11. [Google Scholar] [CrossRef] - Dilssner, F.; Springer, T.; Schoenemann, E.; Zandbergen, R.; Enderle, W. Side-effects of a bad attitude: How GNSS spacecraft orientation error affect solar radiation pressure modelling. In Proceedings of the EGU 2015, Vienna, Austria, 12–17 April 2015. [Google Scholar]
- Kouba, J. A simplified yaw-attitude model for eclipsing GPS satellites. GPS Solut.
**2009**, 13, 1–12. [Google Scholar] [CrossRef] - Lou, Y.; Zhang, W.; Wang, C.; Liu, J. The impact of orbital errors on the estimation of satellite clock errors and PPP. Adv. Space Res.
**2014**, 54, 1571–1580. [Google Scholar] [CrossRef] - Pan, S.; Chen, W.; Jin, X.; Shi, X.; He, F. Real-time PPP based on the coupling estimation of clock bias and orbit error with broadcast ephemeris. Sensors
**2015**, 15, 17808–17826. [Google Scholar] [CrossRef] [PubMed] - Teunissen, P. Basic Observation Equations. In Springer Handbook of Global Navigation Satellite Systems, 2nd ed.; Teunissen, P., Montenbruck, O., Eds.; Springer Nature: Cham, Switzerland, 2017; pp. 561–573. ISBN 978-3-319-42926-7. [Google Scholar]
- Khodabandeh, A.; Teunissen, P.J.G. An analytical study of PPP-RTK corrections: Precision, correlation and user-impact. J. Geod.
**2015**, 89, 1109–1132. [Google Scholar] [CrossRef]

**Figure 2.**GF (blue) and GB (red) L1, L2 and WL ambiguities (

**left**) and their corresponding formal errors (

**right**) of PRN 09 computed from a single receiver (CEBR) observations and ESA final orbit and clock products.

**Figure 3.**GF (blue) and GB (red) L1, L2 and WL ambiguities (

**left**) and their corresponding formal errors (

**right**) of PRN 09 computed using a single receiver (REDU) observations and ESA final orbit clock products.

**Figure 4.**Differences of GB and GF WL ambiguities (red) and their formal errors (blue) of PRN 09 from the receivers CEBR (

**top**) and REDU (

**bottom**).

**Figure 5.**Differences of GB and GF WL ambiguities computed with ESA final orbit and clock products at site KOKV.

**Figure 6.**Various differences of PRN 09 WL ambiguities and orbit clock products computed at the receiver CEBR (

**left**) and REDU (

**right**): (1) differences of GB ambiguities computed with final and real-time orbit clock products are indicated in red in top panels; (2) differences of GF ambiguities computed with final and real-time orbit clock products are indicated in blue in top panels; (3) differences of GB and GF ambiguities computed with final orbit clock products (OC1) are indicated in blue in bottom panels; (4) differences of GB and GF ambiguities computed with real-time orbit clock products (OC2) are indicated in red in bottom panels; (5) differences of orbit clock impacts in LOS directions between final and real-time orbit clock products in green.

**Figure 7.**Differences of GB-GF WL ambiguities computed with real-time orbit and clock products at Site KOKV.

**Figure 8.**Variations of final orbit/clock and real-time orbit/clock differences projected into the LOS at Site KOKV for all visible GPS satellites.

**Figure 9.**GB-GF WL ambiguities computed with real-time orbit clock products for GPS PRN 09 at all selected sites.

**Figure 10.**Recomputed GB-GF WL ambiguities minus the previous ones in Figure 5 with the simulated constant bias (

**top**) and the time-variant errors (

**bottom**) added into orbit clock of PRN 30 at Site KOKV.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, G.; Liu, S.; Zhao, Q. Analysis of Wide-Lane Ambiguities Derived from Geometry-Free and Geometry-Based Precise Point Positioning Models and Their Implication for Orbit and Clock Quality. *Sensors* **2018**, *18*, 1760.
https://doi.org/10.3390/s18061760

**AMA Style**

Chen G, Liu S, Zhao Q. Analysis of Wide-Lane Ambiguities Derived from Geometry-Free and Geometry-Based Precise Point Positioning Models and Their Implication for Orbit and Clock Quality. *Sensors*. 2018; 18(6):1760.
https://doi.org/10.3390/s18061760

**Chicago/Turabian Style**

Chen, Gang, Sijing Liu, and Qile Zhao. 2018. "Analysis of Wide-Lane Ambiguities Derived from Geometry-Free and Geometry-Based Precise Point Positioning Models and Their Implication for Orbit and Clock Quality" *Sensors* 18, no. 6: 1760.
https://doi.org/10.3390/s18061760