# Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. GNSS Observation Equations

#### 2.2. PPP-RTK Network

#### 2.3. Prediction of Ionospheric Corrections

#### 2.4. PPP-RTK User

#### 2.4.1. Ionosphere-Float Model

#### 2.4.2. Ionosphere-Weighted Model

## 3. Results and Analysis

#### 3.1. Data and Processing Strategy

#### 3.2. PPP-RTK Network Corrections

#### 3.3. Real-Time PPP-RTK Performance

#### 3.3.1. Convergence Time

#### 3.3.2. Positioning Accuracy

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic principle of the satellite-by-satellite approach used in predicting user-specific slant ionospheric corrections per satellite and per epoch.

**Figure 2.**Geographic distribution of the selected CORS receivers in North Carolina used for the PPP-RTK network and user processing. The network receivers are classified in groups of four to form networks of varying inter-station distance and are denoted by red, green, magenta and yellow triangles in ascending order by distance. The remaining three receivers, denoted by blue dots, represent the user stations.

**Figure 4.**(

**a**) Satellite phase bias estimates on L1 (in cycles) and (

**b**) their formal standard deviations for all GPS satellites during the selected day. Each color represents a different GPS satellite.

**Figure 5.**(

**a**) Differences (in meters) of the undifferenced regionally network-derived user-predicted slant ionospheric delays and their counterparts estimated by the user NCWL, and (

**b**) their between-satellite single-differenced results. Each color represents a different GPS satellite. This ionospheric correction prediction is referred to network $\#2$ (mean station spacing of 115 km). The empirical mean and STDs were calculated for the complete 24-h time-series.

**Figure 6.**Time-series of the GPS dual-frequency ionosphere-float (

**a**) PPP and (

**b**) Full integer ambiguity resolution (FAR)-based PPP-RTK kinematic user position for station NCWL with respect to its ground-truth. The empirical means and STDs are calculated for the estimated positions after 2 h. A zoom-in window during the first 2 h is provided.

**Figure 7.**Time-series of (

**a**) the GPS dual-frequency ionosphere-weighted FAR-based PPP-RTK kinematic user position for station NCWL with respect to its ground-truth, and (

**b**) its associated network-user scaled DCB estimate. The ionospheric corrections were determined from network #2. The empirical means and STDs are calculated for the estimated positions after 2 h. A zoom-in window during the first 2 h is provided.

**Figure 8.**Convergence behavior of the horizontal radial positioning errors for (

**a**) 50% of the FAR-based solutions, (

**b**) 90% of the FAR-based solutions, (

**c**) 50% of the PAR-based solutions, and (

**d**) 90% of the PAR-based solutions of all user stations as a function of time since the processing start. The processing window has been re-initialized every 1 min within the selected day for all available solutions and networks.

**Figure 9.**Convergence time of the horizontal radial position errors to 10 cm as a function of the network density for both FAR and PAR, based on 50% and 90% of the sample solutions.

**Figure 10.**Horizontal positioning accuracy (90th percentile) at the first epoch and 1, 5, 10, 20, 30, 40, 50, 60 min since start for the PAR-based PPP-RTK user solutions.

**Table 1.**Estimable dual-frequency PPP-RTK network parameters and their interpretation using the Common Clocks $\mathcal{S}$-system (the symbol p denotes the pivot satellite/receiver if it is used as superscript/subscript).

Estimable Parameter | Interpretation |
---|---|

Receiver clock | $d{\tilde{t}}_{r\ne p}=d{t}_{pr}+{d}_{pr,\mathrm{IF}}$ |

Satellite clock | $d{\tilde{t}}^{s}=(d{t}^{s}+{d}_{,\mathrm{IF}}^{s})-(d{t}_{p}+{d}_{p,\mathrm{IF}})$ |

Ionospheric slant delay | ${\tilde{\iota}}_{r}^{s}={\iota}_{r}^{s}+{d}_{r,\mathrm{GF}}-{d}_{,\mathrm{GF}}^{s}$ |

Receiver phase bias | ${\tilde{\delta}}_{r\ne p,j}={\delta}_{pr,j}-\frac{1}{{\lambda}_{j}}({d}_{pr,\mathrm{IF}}-{\mu}_{j}{d}_{pr,\mathrm{GF}})+{a}_{pr,j}^{p}$ |

Satellite phase bias | ${\tilde{\delta}}_{,j}^{s}={\delta}_{,j}^{s}-\frac{1}{{\lambda}_{j}}\left([{d}_{,\mathrm{IF}}^{s}-{d}_{p,\mathrm{IF}}]-{\mu}_{j}[{d}_{,\mathrm{GF}}^{s}-{d}_{p,\mathrm{GF}}]\right)-{\delta}_{p,j}-{a}_{p,j}^{s}$ |

Phase ambiguity | ${\tilde{a}}_{r\ne p,j}^{s\ne p}={a}_{pr,j}^{s}-{a}_{pr,j}^{p}$ |

**Table 2.**Changes in parameter estimability and interpretation in the PPP-RTK user model due to the introduction of external ionospheric corrections.

Estimable Parameter | Interpretation |
---|---|

Receiver phase bias | ${\tilde{\delta}}_{u,j}={\delta}_{pu,j}-\frac{1}{{\lambda}_{j}}({d}_{pu,\mathrm{IF}}-{\mu}_{j}{d}_{p{p}^{\prime},\mathrm{GF}})+{a}_{pu,j}^{p}$ |

Receiver code bias | ${\tilde{d}}_{u,\mathrm{GF}}={d}_{u,\mathrm{GF}}-{d}_{{p}^{\prime},\mathrm{GF}}$ |

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**MDPI and ACS Style**

Psychas, D.; Verhagen, S. Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations. *Sensors* **2020**, *20*, 3012.
https://doi.org/10.3390/s20113012

**AMA Style**

Psychas D, Verhagen S. Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations. *Sensors*. 2020; 20(11):3012.
https://doi.org/10.3390/s20113012

**Chicago/Turabian Style**

Psychas, Dimitrios, and Sandra Verhagen. 2020. "Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations" *Sensors* 20, no. 11: 3012.
https://doi.org/10.3390/s20113012