# Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning

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## Abstract

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## 1. Introduction

- 1.
- Increasing the observation time span and/or applying recursive estimation, particularly when the atmospheric residuals become part of the state estimate.
- 2.
- Applying Partial Ambiguity Resolution (PAR), which relaxes the integer estimation over the complete set of ambiguities to a subset instead.

## 2. RTK and the Mixed Model

#### Mixed Model Estimation

**Float****solution:**- during the first step of Equation (6a), the integer nature of the carrier ambiguities is neglected and, instead, a conventional WLS for real-valued parameters is employed. The result of this estimation is denoted as float solution, whose distribution is described by$$\left[\begin{array}{c}\widehat{\mathbf{a}}\\ \widehat{\mathbf{b}}\end{array}\right]\sim \mathcal{N}\left(\left[\begin{array}{c}\widehat{\mathbf{a}}\\ \widehat{\mathbf{b}}\end{array}\right],\left[\begin{array}{cc}{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}& {\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{b}}}\\ {\mathbf{Q}}_{\widehat{\mathbf{b}}\widehat{\mathbf{a}}}& {\mathbf{Q}}_{\widehat{\mathbf{b}}\widehat{\mathbf{b}}}\end{array}\right]\right),$$
**Integer****Ambiguity****Resolution**:- the second minimization problem Equation (6b) constitutes the Integer Least Squares (ILS) adjustment, a real-to-integer mapping $\mathcal{S}:{\mathbb{R}}^{n}\to {\mathbb{Z}}^{n}$ such that$$\stackrel{\u02c7}{\mathbf{a}}=\mathcal{S}\left(\widehat{\mathbf{a}}\right),\phantom{\rule{4pt}{0ex}}\stackrel{\u02c7}{\mathbf{a}}\in {\mathbb{Z}}^{n},$$$$\underset{{P}_{s,IB}}{\underbrace{P\left({\stackrel{\u02c7}{\mathbf{a}}}_{IB}=\mathbf{a}\right)}}=\prod _{i=1}^{n}\left(2\mathsf{\varphi}\left(\frac{1}{2{\sigma}_{{\widehat{a}}_{i|I}}}\right)-1\right)\le \underset{{P}_{s,ILS}}{\underbrace{P\left({\stackrel{\u02c7}{\mathbf{a}}}_{ILS}=\mathbf{a}\right)}}$$After the float estimation, the ambiguities’ covariance matrix ${\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}$ presents a high correlation among ambiguities which hinders the IAR process (i.e., integer rounding (IR) and IB performance is jeopardized, while ILS increases its computational load). To overcome this limitation, integer reparametrizations, also known as Z-transformations, are typically applied. The general class $\mathcal{Z}$ of Z-transformations is$$\mathcal{Z}=\left\{\left.\mathbf{Z}\in {\mathbb{Z}}^{n,n}\phantom{\rule{4pt}{0ex}}\right|\phantom{\rule{4pt}{0ex}}\mathbf{Z}=\pm 1\right\},$$$$\stackrel{\u02c7}{\mathbf{z}}=arg\underset{\mathbf{z}\in {\mathbb{Z}}^{n}}{min}{\u2225\widehat{\mathbf{z}}-\mathbf{z}\u2225}_{{\mathbf{Q}}_{\widehat{\mathbf{z}}\widehat{\mathbf{z}}}}^{2},\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}\widehat{\mathbf{z}}=\mathbf{Z}\widehat{\mathbf{a}},\phantom{\rule{4pt}{0ex}}{\mathbf{Q}}_{\widehat{\mathbf{z}}\widehat{\mathbf{z}}}=\mathbf{Z}{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}{\mathbf{Z}}^{\top},$$IAR also includes a validation step to determine the reliability of the integer estimate. Thus, an integer solution is accepted only if the success rate is sufficiently high or the validity test is passed. Thus, the integer mapping Equation (8) can be described in a more flexible way as$$\mathcal{S}\left(\widehat{\mathbf{a}}\right)=\left\{\begin{array}{cc}\stackrel{\u02c7}{\mathbf{a}}\in {\mathbb{Z}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}\mathcal{T}(\xb7)\le {\mu}_{0},\hfill \\ \widehat{\mathbf{a}}\in {\mathbb{R}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{otherwise},\hfill \end{array}\right.$$$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{MD}}\left(\widehat{\mathbf{a}}\right)=\left\{\begin{array}{cc}\stackrel{\u02c7}{\mathbf{a}}\in {\mathbb{Z}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}\mathcal{T}\left({\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}\right)\le {P}_{0},\hfill \\ \widehat{\mathbf{a}}\in {\mathbb{R}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$$${\mathcal{S}}_{\mathrm{DD}}\left(\widehat{\mathbf{a}}\right)=\left\{\begin{array}{cc}\stackrel{\u02c7}{\mathbf{a}}\in {\mathbb{Z}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}\mathcal{T}\left(\widehat{\mathbf{a}},{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}\right)\le {\mu}_{0},\hfill \\ \widehat{\mathbf{a}}\in {\mathbb{R}}^{n}\hfill & \phantom{\rule{4pt}{0ex}}\mathrm{otherwise},\hfill \end{array}\right.$$$${\mathcal{T}}_{\mathrm{RT}}=\frac{{\u2225\widehat{\mathbf{a}}-\stackrel{\u02c7}{\mathbf{a}}\u2225}_{{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}}^{2}}{{\u2225\widehat{\mathbf{a}}-\overline{\mathbf{a}}\u2225}_{{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}}^{2}}\le {\mu}_{\mathrm{RT}},\phantom{\rule{1.em}{0ex}}{\mathcal{T}}_{\mathrm{DT}}={\u2225\widehat{\mathbf{a}}-\overline{\mathbf{a}}\u2225}_{{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}}^{2}-{\u2225\widehat{\mathbf{a}}-\stackrel{\u02c7}{\mathbf{a}}\u2225}_{{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}}^{2}\le {\mu}_{\mathrm{DT}},$$
**Fixed solution**:- the last minimization (6c) improves the vector of real-valued parameters $\widehat{\mathbf{b}}$ upon the knowledge of the integer ambiguities $\stackrel{\u02c7}{\mathbf{a}}$, driving to high accuracy positioning, denoted as fixed solution. The mean and covariance for the fixed solution, $\stackrel{\u02c7}{\mathbf{b}}$, ${\mathbf{Q}}_{\stackrel{\u02c7}{\mathbf{b}}\stackrel{\u02c7}{\mathbf{b}}}$ are based on the projection of the integer ambiguities into the position domain, as$$\begin{array}{c}\hfill \stackrel{\u02c7}{\mathbf{b}}=\widehat{\mathbf{b}}-{\mathbf{Q}}_{\widehat{\mathbf{b}}\widehat{\mathbf{a}}}{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}^{-1}\left(\widehat{\mathbf{a}}-\stackrel{\u02c7}{\mathbf{a}}\right),\end{array}$$$$\begin{array}{c}\hfill {\mathbf{Q}}_{\stackrel{\u02c7}{\mathbf{b}}\stackrel{\u02c7}{\mathbf{b}}}={\mathbf{Q}}_{\widehat{\mathbf{b}}\widehat{\mathbf{b}}}-{\mathbf{Q}}_{\widehat{\mathbf{b}}\widehat{\mathbf{a}}}{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{a}}}^{-1}{\mathbf{Q}}_{\widehat{\mathbf{a}}\widehat{\mathbf{b}}},\end{array}$$

## 3. Partial Ambiguity Resolution Strategies

Algorithm 1: Data-Driven PAR |

#### Precision-Driven PAR Scheme

Algorithm 2: Precision-Driven PAR |

## 4. CRB for the PAR Mixed Model

#### New Insights from the CRB for the PAR Mixed Model

- Bounds: (i) CRB
_{Real}corresponds to the standard CRB associated to the float solution; (ii) CRB${}_{Real/Integer}$ (30) refers to the FAR solution, that is, fixing the complete set of ambiguities; and (iii) CRB${}_{Real/Intege{r}_{PD-PAR}}$ (33) is the bound corresponding to the PD-PAR scheme where only a subset of ambiguities is resolved. - Methods: (i) LS refers to the float solution estimate, (ii) ILS${}_{FAR}$ is the estimator that tries to fix all ambiguities, and (iii) ILS${}_{PD-PAR}$ is the new PAR scheme proposed in this article. Notice that the horizontal line $\alpha =$ 5 cm is the specific precision constraint considered in this experiment.

- (1)
- For low code-noise levels (${\sigma}_{c}<0.2\left[m\right]$ for the FAR and $<0.5\left[m\right]$ for the PD-PAR), the so-called asymptotic regime, the ILS performance for both FAR and PD-PAR schemes coincides with CRB${}_{Real/Integer}$. This confirms that a correct ILS which considers only a successful IAR is asymptotically efficient. Obviously, there is a slight performance degradation when not fixing all the ambiguities, that is, the asymptotic performance of PD-PAR is slightly larger than with a correct FAR.
- (2)
- At high code-noise levels (${\sigma}_{c}>1\left[m\right]$), the RMSE performance of both ILS coincides with the float solution LS. In other words, after a certain level of noise, trying to fix the ambiguities is useless, and a correct IAR is never achieved.
- (3)
- The region between the asymptotic convergence to the mixed real/integer bound and the unconstrained float region is the so-called threshold region. Such a threshold provides information on the optimal receiver operation conditions. It is remarkable to see that the PD-PAR threshold appears for larger noise levels when compared to the FAR scheme, which implies that the PD-PAR provides a more reliable single-epoch IAR solution.

## 5. Evaluation Results

#### 5.1. Simulation Results

#### 5.2. Validation with Real Data

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Morton, Y.J.; van Diggelen, F.; Spilker, J.J., Jr.; Parkinson, B.W.; Lo, S.; Gao, G. Position, Navigation, and Timing Technologies in the 21st Century: Integrated Satellite Navigation, Sensor Systems, and Civil Applications; Wiley-IEEE Press: Hoboken, NJ, USA, 2021. [Google Scholar]
- Teunissen, P.J.G. A new class of GNSS ambiguity estimators Artif. Satell.
**2002**, 37, 111–119. [Google Scholar] - Langley, R.B.; RTK GPS. GPS World 1998, 70–75. Available online: http://www2.unb.ca/gge/Resources/gpsworld.september98.pdf (accessed on 20 January 2020).
- Odolinski, R.; Teunissen, P.J.G.; Odijk, D. Combined BDS, Galileo, QZSS and GPS single-frequency RTK. GPS Solut.
**2014**. [Google Scholar] [CrossRef] - Teunissen, P.J.G.; de Jonge, P.J.; Tiberius, C.C.C.J.M. The LAMBDA-Method for Fast GPS Surveying. In Proceedings of the International Symposium “GPS Technology Applications”, Bucharest, Romania, 26–29 September 1995; pp. 203–210. [Google Scholar]
- Verhagen, S. Integer ambiguity validation: An open problem? GPS Solut.
**2004**, 8, 36–43. [Google Scholar] [CrossRef] - Williams, N.; Wu, G.; Closas, P. Impact of positioning uncertainty on eco-approach and departure of connected and automated vehicles. In Proceedings of the 2018 IEEE/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, USA, 23–26 April 2018; pp. 1081–1087. [Google Scholar]
- Kassas, Z.M.; Closas, P.; Gross, J. Navigation systems panel report navigation systems for autonomous and semi-autonomous vehicles: Current trends and future challenges. IEEE Aerosp. Electron. Syst. Mag.
**2019**, 34, 82–84. [Google Scholar] [CrossRef] - Liu, X.; Ribot, M.Á.; Gusi-Amigó, A.; Closas, P.; Garcia, A.R.; Subirana, J.S. RTK Feasibility Analysis for GNSS Snapshot Positioning. In Proceedings of the 33rd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2020), Online, 22–25 September 2020; pp. 2911–2921. [Google Scholar]
- Liu, X.; Ribot, M.Á.; Gusi-Amigó, A.; Rovira-Garcia, A.; Subirana, J.S.; Closas, P. Cloud-Based Single-Frequency Snapshot RTK Positioning. Sensors
**2021**, 21, 3688. [Google Scholar] [CrossRef] - Parkins, A. Increasing GNSS RTK availability with a new single-epoch batch partial ambiguity resolution algorithm. GPS Solut.
**2011**, 15, 391–402. [Google Scholar] [CrossRef] - Takasu, T.; Yasuda, A. Kalman-filter-based integer ambiguity resolution strategy for long-baseline RTK with ionosphere and troposphere estimation. In Proceedings of the 23rd International Technical Meeting of the Satellite Division of the Institute of Navigation 2010, Portland, OR, USA, 21–24 September 2010. [Google Scholar]
- Teunissen, P.J.G.; Odijk, D. Ambiguity dilution of precision: Definition, properties and application. Proc. ION GPS
**1997**, 1, 891–899. [Google Scholar] - Henkel, P.; Günther, C. Partial integer decorrelation: Optimum trade-off between variance reduction and bias amplification. J. Geod.
**2010**, 84, 51–63. [Google Scholar] [CrossRef] - Brack, A.; Günther, C. Generalized integer aperture estimation for partial GNSS ambiguity fixing. J. Geod.
**2014**, 88, 479–490. [Google Scholar] [CrossRef] - Brack, A. On reliable data-driven partial GNSS ambiguity resolution. GPS Solut.
**2015**, 19, 411–422. [Google Scholar] [CrossRef] - Brack, A. Partial ambiguity resolution for reliable GNSS positioning—A useful tool? IEEE Aerosp. Conf. Proc.
**2016**, 1–7. [Google Scholar] [CrossRef] - Brack, A. Partial Carrier-Phase Integer Ambiguity Resolution for High Accuracy GNSS Positioning. Ph.D. Thesis, Technische Universität München (TUM), München, Germany, 2019. [Google Scholar]
- Closas, P.; Fernández-Prades, C.; Fernández-Rubio, J.A. Cramér-Rao bound analysis of positioning approaches in GNSS receivers. IEEE Trans. Signal Process.
**2009**, 57, 3775–3786. [Google Scholar] [CrossRef] - Das, P.; Ortega, L.; Vilà-Valls, J.; Vincent, F.; Chaumette, E.; Davain, L. Performance limits of GNSS code-based precise positioning: GPS, galileo & meta-signals. Sensors
**2020**, 20, 2196. [Google Scholar] [CrossRef] - Bar-Shalom, Y.; Li, X. Estimation and Tracking-Principles, techniques, and Software; Artech House, Inc.: Norwood, MA, USA, 1993. [Google Scholar]
- Bar-Shalom, Y.; Li, X.R.; Kirubarajan, T. Estimation with Applications to Tracking and Navigation; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2001. [Google Scholar] [CrossRef]
- Teunissen, P.J.G. Success probability of integer GPS ambiguity rounding and bootstrapping. J. Geod.
**1998**, 72, 606–612. [Google Scholar] [CrossRef] [Green Version] - Hassibi, A.; Boyd, S. Integer parameter estimation in linear models with applications to GPS. In Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 13 December 1996. [Google Scholar] [CrossRef] [Green Version]
- Teunissen, P.J.G. An optimality property of the integer least-squares estimator. J. Geod.
**1999**. [Google Scholar] [CrossRef] - Medina, D.; Ortega, L.; Vilà-Valls, J.; Closas, P.; Vincent, F.; Chaumette, E. Compact crb for delay, doppler, and phase estimation—A pplication to gnss spp and rtk performance characterization. IET Radar Sonar Navig.
**2020**, 14, 1537–1549. [Google Scholar] [CrossRef] - Medina, D.; Vilà-Valls, J.; Chaumette, E.; Vincent, F.; Closas, P. Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model. Signal Process.
**2021**, 179. [Google Scholar] [CrossRef] - Teunissen, P.J.G.; Montenbruck, O.E. Springer Handbook of Global Navigation Satellite Systems; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef]
- Eueler, H.J.; Goad, C.C. On optimal filtering of GPS dual frequency observations without using orbit information. Bull. Géodésique
**1991**, 65, 130–143. [Google Scholar] [CrossRef] - Medina, D.; Gibson, K.; Ziebold, R.; Closas, P. Determination of Pseudorange Error Models and Multipath Characterization under Signal-Degraded Scenarios. In Proceedings of the 31st International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2018), Miami, FL, USA, 24–28 September 2018. [Google Scholar] [CrossRef] [Green Version]
- Wielgosz, P. Quality assessment of GPS rapid static positioning with weighted ionospheric parameters in generalized least squares. GPS Solut.
**2011**, 15, 89–99. [Google Scholar] [CrossRef] - Paziewski, J.; Wielgosz, P. Investigation of some selected strategies for multi-GNSS instantaneous RTK positioning. Adv. Space Res.
**2017**, 59, 12–23. [Google Scholar] [CrossRef] - Odijk, D. Weighting Ionospheric Corrections to Improve Fast GPS Positioning Over Medium Distances. In Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000), Salt Lake City, UT, USA, 19–22 September 2000. [Google Scholar]
- de Jonge, P.; Tiberius, C. Integer Ambiguity Estimation with the Lambda Method. In GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications; Springer: Berlin/Heidelberg, Germany, 1996; pp. 280–284. [Google Scholar] [CrossRef]
- Verhagen, S.; Teunissen, P.J.G. The ratio test for future GNSS ambiguity resolution. GPS Solut.
**2013**, 17, 535–548. [Google Scholar] [CrossRef] - Wang, L.; Verhagen, S. A new ambiguity acceptance test threshold determination method with controllable failure rate. J. Geod.
**2015**, 89, 361–375. [Google Scholar] [CrossRef] [Green Version] - Verhagen, A.A.; Teunissen, P.J.G.; van der Marel, H.; Li, B. GNSS ambiguity resolution: Which subset to fix? In Proceedings of the International Global Navigation Satellite Systems Society, Sydney, Australia, 15–17 November 2011.
- Odijk, D.; Arora, B.S.; Teunissen, P.J. Predicting the success rate of long-baseline GPS+Galileo (Partial) ambiguity resolution. J. Navig.
**2014**, 67, 385–401. [Google Scholar] [CrossRef] [Green Version] - Brack, A. Optimal Estimation of a Subset of Integers with Application to GNSS. Artif. Satell.
**2016**, 51, 123–134. [Google Scholar] [CrossRef] [Green Version] - Fernandez-Hernandez, I.; Senni, T.; Calle, D.; Cancela, S.; Vecchione, G.A.; Seco-Granados, G. Analysis of High-Accuracy Satellite Messages for Road Applications. IEEE Intell. Transp. Syst. Mag.
**2020**, 12, 92–108. [Google Scholar] [CrossRef] - Heßelbarth, A.; Medina, D.; Ziebold, R.; Sandler, M.; Hoppe, M.; Uhlemann, M. Enabling Assistance Functions for the Safe Navigation of Inland Waterways. IEEE Intell. Transp. Syst. Mag.
**2020**, 12, 123–135. [Google Scholar] [CrossRef] - Blanco-Delgado, N.; Nunes, F.D. Satellite selection method for multi-constellation GNSS using convex geometry. IEEE Trans. Veh. Technol.
**2010**, 59, 4289–4297. [Google Scholar] [CrossRef] - Pany, T.; Dampf, J.; Bär, W.; Winkel, J.; Stöber, C.; Fürlinger, K.; Closas, P.; Garcia-Molina, J. Benchmarking CPUs and GPUs on embedded platforms for software receiver usage. In Proceedings of the 28th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2015), Tampa, FL, USA, 14–18 September 2015; pp. 3188–3197. [Google Scholar]
- Dampf, J.; Pany, T.; Bär, W.; Winkel, J.; Stöber, C.; Fürlinger, K.; Closas, P.; Garcia-Molina, J. More than we ever dreamed possible: Processor technology for GNSS software receivers in the year 2015. Inside GNSS
**2015**, 10, 62–72. [Google Scholar] - Brack, A. Reliable GPS + BDS RTK positioning with partial ambiguity resolution. GPS Solut.
**2017**, 21, 1083–1092. [Google Scholar] [CrossRef]

**Figure 1.**Right side, IGS MGEX station POTS0 in Potsdam, Germany, as reference station (red mark). Right side, skyplot of the synthetic GNSS scenario.

**Figure 2.**RMSE and CRB as a function of the code noise standard deviation, ${\sigma}_{c}$, for FAR and PD-PAR schemes illustrating the difference between both asymptotic regions.

**Figure 3.**Comparison of the single-epoch IAR experimental success rate for FAR and PD-PAR schemes. At high code-noise levels ${\sigma}_{c}>1\left[m\right]$, an IAR is still achieved with a precision-aided PAR scheme approach.

**Figure 5.**Model-driven (MD), data-driven (DD), and precision-driven (PD) performance analysis. (

**a**) Experimental success rate. (

**b**) Number of fixed ambiguities.

**Figure 6.**Comparison with different ambiguity resolution schemes for one GNSS constellation only. (

**a**) GPS (L1 + L2) experimental success rate. (

**b**) Galileo (E1 + E5a) experimental success rate.

**Figure 9.**Comparative of success rate ${P}_{s}=1-{P}_{F}$. Number of fixed ambiguities and precision for FAR, Data-Driven PAR, and Data-Driven+precision-aided PAR schemes. (

**a**) GPS (L1 + L2) + GAL (E1 + E5a) performance comparative for a DD-PAR scheme when FAR fails (no fixed solution). (

**b**) GPS (L1 + L2) + GAL (E1 + E5a) performance comparative for the proposed DD+PD PAR scheme when FAR or DD-PAR fails (no fixed solution). Furthermore, PD-PAR guarantees a fixed solution whose fixed positioning errors (and precision) respect the criteria for minimum required positioning precision $\alpha $.

**Figure 11.**Horizontal (NE) position scatter plots along with the success rate $1-{P}_{f}$ for 12 h of real data (PERT-CUT0) only for fixed solutions. (

**a**) DD-PAR performance analysis for 12 h of real data. (

**b**) PD-PAR performance analysis for 12 h of real data.

**Table 1.**Wavelengths and zenith-referenced code and carrier standard deviations for GPS and Galileo observations.

GPS | Galileo | |||
---|---|---|---|---|

L1 | L2 | E1 | E5a | |

$\lambda $ (cm) | 19.03 | 24.42 | 19.03 | 25.48 |

${\sigma}_{c}$ (cm) | 37 | 28 | 35 | 28 |

${\sigma}_{\varphi}$ (mm) | 2 | 2 | 2 | 2 |

Ambiguity Resolution Method | Fix Ratio (%) |
---|---|

FAR | 88.33 |

$D{D}_{PAR}$ | 98.60 |

$P{D}_{PAR}$ | 100 |

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**MDPI and ACS Style**

Castro-Arvizu, J.M.; Medina, D.; Ziebold, R.; Vilà-Valls, J.; Chaumette, E.; Closas, P.
Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning. *Remote Sens.* **2021**, *13*, 2904.
https://doi.org/10.3390/rs13152904

**AMA Style**

Castro-Arvizu JM, Medina D, Ziebold R, Vilà-Valls J, Chaumette E, Closas P.
Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning. *Remote Sensing*. 2021; 13(15):2904.
https://doi.org/10.3390/rs13152904

**Chicago/Turabian Style**

Castro-Arvizu, Juan Manuel, Daniel Medina, Ralf Ziebold, Jordi Vilà-Valls, Eric Chaumette, and Pau Closas.
2021. "Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning" *Remote Sensing* 13, no. 15: 2904.
https://doi.org/10.3390/rs13152904