# Three Dual-Frequency Precise Point Positioning Models for the Ionospheric Modeling and Satellite Pseudorange Observable-Specific Signal Bias Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. General Observations

#### 2.2. DFPPP1: Dual-Frequency Ionosphere-Float PPP Model

- ${\mathit{\xi}}_{DF1}={\left[\mathrm{Z}\mathrm{W}{\mathrm{D}}_{r}(t),d{\overline{t}}_{r}(t),\mathit{\tau},{\mathit{a}}_{2}\right]}^{T}$, $\mathrm{Z}\mathrm{W}{\mathrm{D}}_{r}(t)$ denotes the tropospheric zenith wet delay (ZWD), $d{\overline{t}}_{r}(t)$ denotes the receiver clock offset, $\mathit{\tau}={\left[{\overline{I}}_{r,1}^{1}(t),\cdots ,{\overline{I}}_{r,1}^{m}(t)\right]}^{T}$, ${\mathit{a}}_{2}={\left[{\overline{N}}_{r,i}^{1}(t),{\overline{N}}_{r,i}^{2}(t),\cdots ,{\overline{N}}_{r,j}^{m}(t)\right]}^{T}$;
- ${\mathit{e}}_{m}$ denotes m-dimension row vector, in which all values are 1;
- ${\mathit{I}}_{m}$ denotes m-dimension identity matrix;
- ${\mathit{M}}_{r}={\left[m{f}_{r,j}^{1}(t),\cdots ,m{f}_{r,j}^{m}(t)\right]}^{T}$ denotes the design matrix of the tropospheric wet mapping function;
- ${\mathit{n}}_{2}={\left[1,-1\right]}^{T}$; ${\mathit{\mu}}_{2}={\left[{\mu}_{i},{\mu}_{j}\right]}^{T}$; ${\mathit{z}}_{2}={\left[0,1\right]}^{T}$;
- ${\mathit{q}}_{2}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({q}_{i}^{2},{q}_{j}^{2})$, in which ${q}_{i}$ denotes the ratio of the observation noise on ith frequency.
- ${\mathit{Q}}_{r}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\delta}_{p}^{2},{\delta}_{\varphi}^{2})$ denotes the corresponding observation precision matrix in the vertical direction, and ${\mathit{Q}}_{m}$ denotes the elevation diversity cofactor matrix;
- $\otimes $ denotes the Kronecker product.

#### 2.3. DFPPP2: Dual-Frequency Ionosphere-Free PPP Model

#### 2.4. DFPPP3: Dual-Frequency UofC PPP Model

#### 2.5. Ionospheric Modeling and OSB Estimation

#### 2.6. Analysis of PPP Approaches

## 3. Results and Analysis

#### 3.1. Data Processing Strategy

^{−4}m

^{2}/s in DFPPP1 solution. The ionospheric single- and multi-layer MFs are both applied to evaluate the experimental performance. Other error items in the data processing strategies can refer to Su et al. [39].

#### 3.2. Analysis of the Ionospheric Observables from PPP

#### 3.3. Analysis of the Estimated VTEC

#### 3.4. Analysis of the Estimated BDS Satellite Pseudorange OSB

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviation

BDGIM | BeiDou Global Ionospheric delay correction Model |

BDS | Beidou Navigation Satellite System |

CCL | Carrier-to-Code Leveling |

DCB | Differential Code Bias |

DFPPP1 | Dual-frequency ionosphere-float PPP |

DFPPP2 | Dual-frequency ionosphere-free PPP |

DFPPP3 | Dual-frequency UofC PPP |

DOY | Day Of Year |

GEO | Geostationary Earth Orbit |

GFZ | Deutsches GeoForschungsZentrum |

GIM | Global Ionospheric Map |

GNSS | Global Navigation Satellite System |

GTSF | Generalized Trigonometric Series Function |

IGS | International GNSS Service |

IGSO | Inclined GeoSynchronous Orbit |

IPP | Ionospheric Pierce Point |

MCCL | Modified Carrier-to-Code Leveling |

MEO | Medium Earth Orbit |

MF | Mapping Function |

MGEX | Multi-GNSS EXperiment |

MSLM | Modified Single-Layer Model |

NTCM | Neustrelitz TEC Model |

OSB | Observable-specific Signal Bias |

PNT | Positioning, Navigation and Timing |

PPP | Precise Point Positioning |

RMS | Root Mean Square |

SLM | Single-Layer Model |

SPR | Satellite Plus Receiver |

STEC | Slant Total Electron Content |

STD | STandard Deviation |

TEC | Total Electron Content |

TECU | Total Electron Content Unit |

VTEC | Vertical Total Electron Content |

## References

- Hoque, M.; Jakowski, N. A new global model for the ionospheric F2 peak height for radio wave propagation. Ann. Geophys.
**2012**, 30, 797–809. [Google Scholar] [CrossRef] [Green Version] - Jin, S.; Su, K. PPP models and performances from single-to quad-frequency BDS observations. Satell. Navig.
**2020**, 1, 16. [Google Scholar] [CrossRef] - Yang, Y.; Mao, Y.; Sun, B. Basic performance and future developments of BeiDou global navigation satellite system. Satell. Navig.
**2020**, 1, 1. [Google Scholar] [CrossRef] [Green Version] - Jin, S.; Jin, R.; Kutoglu, H. Positive and negative ionospheric responses to the March 2015 geomagnetic storm from BDS observations. J. Geod.
**2017**, 91, 613–626. [Google Scholar] [CrossRef] - Anđić, D. Impact of sampling interval on variance components of epoch-wise residual error in relative GPS positioning: A case study of a 40-km-long baseline. Geod. Geodyn.
**2021**, 12, 368–380. [Google Scholar] [CrossRef] - Hoque, M.; Jakowski, N. An alternative ionospheric correction model for global navigation satellite systems. J. Geod.
**2015**, 89, 391–406. [Google Scholar] [CrossRef] - Klobuchar, J.A. Ionospheric time-delay algorithm for single-frequency GPS users. IEEE Trans. Aerosp. Electron. Syst.
**1987**, AES-23, 325–331. [Google Scholar] [CrossRef] - Nava, B.; Coisson, P.; Radicella, S. A new version of the NeQuick ionosphere electron density model. J. Atmos. Sol. Terr. Phys.
**2008**, 70, 1856–1862. [Google Scholar] [CrossRef] - Yuan, Y.; Wang, N.; Li, Z.; Huo, X. The BeiDou global broadcast ionospheric delay correction model (BDGIM) and its preliminary performance evaluation results. Navigation
**2019**, 66, 55–69. [Google Scholar] [CrossRef] [Green Version] - Jin, S.; Han, L.; Cho, J. Lower atmospheric anomalies following the 2008 Wenchuan Earthquake observed by GPS measurements. J. Atmos. Sol. Terr. Phys.
**2011**, 73, 810–814. [Google Scholar] [CrossRef] - Liu, T.; Zhang, B.; Yuan, Y.; Zhang, X. On the application of the raw-observation-based PPP to global ionosphere VTEC modeling: An advantage demonstration in the multi-frequency and multi-GNSS context. J. Geod.
**2020**, 94, 1. [Google Scholar] [CrossRef] - Psychas, D.; Verhagen, S.; Liu, X.; Memarzadeh, Y.; Visser, H. Assessment of ionospheric corrections for PPP-RTK using regional ionosphere modelling. Meas. Sci. Technol.
**2018**, 30, 014001. [Google Scholar] [CrossRef] [Green Version] - Chen, L.; Yi, W.; Song, W.; Shi, C.; Lou, Y.; Cao, C. Evaluation of three ionospheric delay computation methods for ground-based GNSS receivers. GPS Solut.
**2018**, 22, 125. [Google Scholar] [CrossRef] - Ciraolo, L.; Azpilicueta, F.; Brunini, C.; Meza, A.; Radicella, S. Calibration errors on experimental slant total electron content (TEC) determined with GPS. J. Geod.
**2007**, 81, 111–120. [Google Scholar] [CrossRef] - Zha, J.; Zhang, B.; Yuan, Y.; Zhang, X.; Li, M. Use of modified carrier-to-code leveling to analyze temperature dependence of multi-GNSS receiver DCB and to retrieve ionospheric TEC. GPS Solut.
**2019**, 23, 103. [Google Scholar] [CrossRef] - Zhang, B.; Teunissen, P.J.; Yuan, Y.; Zhang, X.; Li, M. A modified carrier-to-code leveling method for retrieving ionospheric observables and detecting short-term temporal variability of receiver differential code biases. J. Geod.
**2019**, 93, 19–28. [Google Scholar] [CrossRef] [Green Version] - Tu, R.; Zhang, H.; Ge, M.; Huang, G. A real-time ionospheric model based on GNSS Precise Point Positioning. Adv. Space Res.
**2013**, 52, 1125–1134. [Google Scholar] [CrossRef] - Zhang, B.; Zhao, C.; Odolinski, R.; Liu, T. Functional model modification of precise point positioning considering the time-varying code biases of a receiver. Satell. Navig.
**2021**, 2, 11. [Google Scholar] [CrossRef] - Li, Z.; Wang, N.; Hernández-Pajares, M.; Yuan, Y.; Krankowski, A.; Liu, A.; Zha, J.; García-Rigo, A.; Roma-Dollase, D.; Yang, H. IGS real-time service for global ionospheric total electron content modeling. J. Geod.
**2020**, 94, 32. [Google Scholar] [CrossRef] - Ren, X.; Zhang, X.; Xie, W.; Zhang, K.; Yuan, Y.; Li, X. Global ionospheric modelling using multi-GNSS: BeiDou, Galileo, GLONASS and GPS. Sci. Rep.
**2016**, 6, 33499. [Google Scholar] [CrossRef] [Green Version] - Liu, T.; Zhang, B.; Yuan, Y.; Li, M. Real-Time Precise Point Positioning (RTPPP) with raw observations and its application in real-time regional ionospheric VTEC modeling. J. Geod.
**2018**, 92, 1267–1283. [Google Scholar] [CrossRef] - Komjathy, A.; Sparks, L.; Mannucci, A.J.; Coster, A. The ionospheric impact of the October 2003 storm event on Wide Area Augmentation System. GPS Solut.
**2005**, 9, 41–50. [Google Scholar] [CrossRef] - Li, M.; Yuan, Y.; Zhang, B.; Wang, N.; Li, Z.; Liu, X.; Zhang, X. Determination of the optimized single-layer ionospheric height for electron content measurements over China. J. Geod.
**2018**, 92, 169–183. [Google Scholar] [CrossRef] - Xiang, Y.; Gao, Y. An enhanced mapping function with ionospheric varying height. Remote Sens.
**2019**, 11, 1497. [Google Scholar] [CrossRef] [Green Version] - Hoque, M.M.; Jakowski, N. Mitigation of ionospheric mapping function error. In Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, USA, 16–20 September 2013. [Google Scholar]
- Su, K.; Jin, S. A novel GNSS single-frequency PPP approach to estimate the ionospheric TEC and satellite pseudorange observable-specific signal bias. IEEE Trans. Geosci. Remote Sens.
**2021**. [Google Scholar] [CrossRef] - Li, Z.; Wang, N.; Liu, A.; Yuan, Y.; Wang, L.; Hernández-Pajares, M.; Krankowski, A.; Yuan, H. Status of CAS global ionospheric maps after the maximum of solar cycle 24. Satell. Navig.
**2021**, 2, 19. [Google Scholar] [CrossRef] - Wang, N.; Li, Z.; Duan, B.; Hugentobler, U.; Wang, L. GPS and GLONASS observable-specific code bias estimation: Comparison of solutions from the IGS and MGEX networks. J. Geod.
**2020**, 94, 74. [Google Scholar] [CrossRef] - Leick, A.; Rapoport, L.; Tatarnikov, D. GPS Satellite Surveying; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Erol, S.; Alkan, R.M.; Ozulu, İ.M.; İlçi, V. Performance analysis of real-time and post-mission kinematic precise point positioning in marine environments. Geod. Geodyn.
**2020**, 11, 401–410. [Google Scholar] [CrossRef] - Lee, H.; Rizos, C. Position-domain hatch filter for kinematic differential GPS/GNSS. IEEE Trans. Aerosp. Electron. Syst.
**2008**, 44, 30–40. [Google Scholar] [CrossRef] [Green Version] - Dyrud, L.; Jovancevic, A.; Ganguly, S. Ionospheric measurement with GPS: Receiver techniques and methods. In Proceedings of Proceedings of the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2007), Fort Worth, TX, USA, 25–28 September 2007; pp. 2313–2323. [Google Scholar]
- Yasyukevich, Y.; Mylnikova, A.; Vesnin, A. GNSS-based non-negative absolute ionosphere total electron content, its spatial gradients, time derivatives and differential code biases: Bounded-variable least-squares and taylor series. Sensors
**2020**, 20, 5702. [Google Scholar] [CrossRef] - Li, B.; Ge, H.; Shen, Y. Comparison of ionosphere-free, UofC and uncombined PPP observation models. Acta Geod. Cartogr. Sin.
**2015**, 44, 734. [Google Scholar] - Moses, M.; Dodo, J.D.; Ojigi, L.M.; Lawal, K. Regional TEC modelling over Africa using deep structured supervised neural network. Geod. Geodyn.
**2020**, 11, 367–375. [Google Scholar] [CrossRef] - Schaer, S.; Socit Helvtique des Sciences Naturelles; Commission Godsique. Mapping and Predicting the Earth’s Ionosphere Using the Global Positioning System; Institut für Geodäsie und Photogrammetrie, Eidg. Technische Hochschule, University of Berne: Berne, Switzerland, 1999; Volume 59. [Google Scholar]
- Yuan, Y.; Ou, J. A generalized trigonometric series function model for determining ionospheric delay. Prog. Nat. Sci.
**2004**, 14, 1010–1014. [Google Scholar] [CrossRef] - Xiang, Y.; Gao, Y.; Shi, J.; Xu, C. Consistency and analysis of ionospheric observables obtained from three precise point positioning models. J. Geod.
**2019**, 93, 1161–1170. [Google Scholar] [CrossRef] - Su, K.; Jin, S.; Jiang, J.; Hoque, M.; Yuan, L. Ionospheric VTEC and satellite DCB estimated from single-frequency BDS observations with multi-layer mapping function. GPS Solut.
**2021**, 25, 68. [Google Scholar] [CrossRef] - Hernández-Pajares, M.; Juan, J.; Sanz, J.; Orus, R.; Garcia-Rigo, A.; Feltens, J.; Komjathy, A.; Schaer, S.; Krankowski, A. The IGS VTEC maps: A reliable source of ionospheric information since 1998. J. Geod.
**2009**, 83, 263–275. [Google Scholar] [CrossRef] - Schaer, S.; Beutler, G.; Rothacher, M.; Springer, T.A. Daily global ionosphere maps based on GPS carrier phase data routinely produced by the CODE Analysis Center. In Proceedings of the 1996 IGS Analysis Center Workshop, Silver Spring, MD, USA, 19–21 March 1996. [Google Scholar]

**Figure 2.**Slant ionospheric delay at station ULAB with the DFPPP1, DFPPP2 and DFPPP3 models on DOY 288, 2020.

**Figure 3.**Slant ionospheric delay with the elevation variation with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 4.**STD distribution of ionospheric observables differences for the DFPPP2 and DFPPP3 models. The corresponding medium and mean values are also shown.

**Figure 5.**Slant ionospheric delay single difference for the short-baseline stations with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 6.**Average values of the ionospheric delay single difference STDs with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 7.**Estimated ionospheric VTEC for the six stations with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 8.**Estimated ionospheric VTEC for the two short-baseline stations with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 9.**Estimated ionospheric VTEC single-difference for the two short-baseline stations with the DFPPP1, DFPPP2 and DFPPP3 models.

**Figure 10.**Distribution of the RMS, mean bias and STD of the ionospheric VTEC difference for the three PPP models and IGS GIM values.

**Figure 11.**Distribution of the RMS, mean bias and STD of the ionospheric VTEC difference for the DFPPP2 and DFPPP3 models compared with the DFPPP1 model.

**Figure 12.**RMS distribution of the ionospheric VTEC difference for the short-baseline stations for the three PPP models.

**Figure 14.**Average value of the estimated BDS pseudorange OSB by the DFPPP1, DFPPP2 and DFPPP3 models on October 2020.

**Figure 15.**Monthly STD value of the estimated BDS pseudorange OSB by the DFPPP1, DFPPP2 and DFPPP3 models in October 2020. The average STD values are also shown in the figure.

**Figure 16.**Monthly RMS error value of the estimated BDS pseudorange OSB by the DFPPP1, DFPPP2 and DFPPP3 models in October 2020 compared to the CAS product. The average RMS errors of the GEO, IGSO, MEO and all BDS satellites are also shown.

DFPPP1 | DFPPP2 | DFPPP3 | |
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Number of the observation | 4m | 2m + m | 3m + m |

Unknown parameters number | sysNum + 3m + 1 | sysNum + m + 1 + m | sysNum + 2m + 1 + m |

Freedom degrees | m-sysNum-1 | m-sysNum-1 | m-sysNum-1 |

Ionospheric observable biases | Ionospheric observables and SPR DCB | Ionospheric observables, SPR DCB, leveling errors and pseudorange noises | Ionospheric observables, SPR DCB, and carrier phase noises |

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**MDPI and ACS Style**

Su, K.; Jin, S.
Three Dual-Frequency Precise Point Positioning Models for the Ionospheric Modeling and Satellite Pseudorange Observable-Specific Signal Bias Estimation. *Remote Sens.* **2021**, *13*, 5093.
https://doi.org/10.3390/rs13245093

**AMA Style**

Su K, Jin S.
Three Dual-Frequency Precise Point Positioning Models for the Ionospheric Modeling and Satellite Pseudorange Observable-Specific Signal Bias Estimation. *Remote Sensing*. 2021; 13(24):5093.
https://doi.org/10.3390/rs13245093

**Chicago/Turabian Style**

Su, Ke, and Shuanggen Jin.
2021. "Three Dual-Frequency Precise Point Positioning Models for the Ionospheric Modeling and Satellite Pseudorange Observable-Specific Signal Bias Estimation" *Remote Sensing* 13, no. 24: 5093.
https://doi.org/10.3390/rs13245093