# High-Resolution Ionosphere Corrections for Single-Frequency Positioning

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Global Ionosphere Map Generation

#### 2.1. Modeling Coordinate System

#### 2.2. Spherical Harmonics Model

#### 2.3. B-Spline Model

#### 2.4. VTEC Products

- Product type 1: estimated series coefficients ${\widehat{c}}_{k}\left({t}_{s}\right)$ with $k=0,\dots ,N$. It is assumed that a user has an appropriate converter for evaluating the respective series expansion (4). In that way, the corrections can directly be calculated by the user.
- Product type 2: VTEC grid values $\widehat{VTEC}({\phi}_{l},{\lambda}_{r},{t}_{s})$ for $l=1,\dots ,L$ latitudes ${\phi}_{l}$ and $r=1,\dots ,R$ longitudes ${\lambda}_{r}$, where L is the number of points along the meridians and R is the number of grid points along the circles of latitudes with arbitrary sampling $\mathsf{\Delta}\mathsf{\Phi}={\phi}_{l+1}-{\phi}_{l}$ and $\mathsf{\Delta}\mathsf{\Lambda}={\lambda}_{r+1}-{\lambda}_{r}$, respectively. This implies that a user has to interpolate between the provided grid points in order to obtain the required correction.

#### 2.5. Dissemination Formats for Positioning

#### 2.5.1. IONEX Format

#### 2.5.2. Coefficient Based Format

#### 2.5.3. Requirements for GIM Formats

- …
- For precise positioning in post processing mode: to be more specific, as long as the data are not immediately used by a user, the size of the data is less important and the transformation may take longer. Which means, the VTEC information can be prepared with high resolution and quality and be provided to the user. It does not matter which dissemination strategy is chosen and through which platform (FTP, streaming) the information reaches the user.
- …
- For precise navigation and positioning in RT: in this case, the selection of possible dissemination formats is limited. Hence, the user must receive the ionospheric information within seconds and with high precision, in order to correct the GNSS measurements used for positioning. Currently, only the SSR VTEC message is used for this purpose. Consequently, the dissemination strategy with Product type 1 has to be chosen.

## 3. Methodology

- The transformation should be done with high precision, i.e., since B-splines and SHs are characterized by different features, a high degree ${n}_{max}$ of the SH expansion needs to be considered.
- The time which is needed for the transformation is limited. Hence, the processing time including the generation of pseudo-observations—the evaluation of Equation (18)—and the estimation of the SH coefficients (cf. Equation (20)) should be completed within seconds.

#### 3.1. Point Distribution on a Sphere

#### 3.2. Pseudo Observations from B-spline Model Output

#### 3.3. Estimation of SH Coefficients from Reuter Grid

## 4. Results and Discussion

#### 4.1. Validation

#### 4.1.1. dSTEC Analysis

#### 4.1.2. GIM Performance in Single-Frequency PPP

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Klobuchar, J.A. Ionospheric Time-Delay Algorithm for Single-Frequency GPS Users. IEEE Trans. Aerospace Electron. Syst.
**1987**, AES-23, 325–331. [Google Scholar] [CrossRef] - Petit, G.; Luzum, B. IERS Conventions. Technical Report. 2010. Available online: www.iers.org (accessed on 21 December 2020).
- Hernández-Pajares, M.; Juan, J.M.; Sanz, J.; Argón-Àngel, A.; Garcia-Rigo, A.; Salazar, D.; Escudero, M. The ionosphere: Effects, GPS modeling and benefits for space geodetic techniques. J. Geod.
**2011**, 85, 887–907. [Google Scholar] [CrossRef] - Langley, R.B. Propagation of the GPS Signals. In GPS for Geodesy; Teunissen, P.J.G., Kleusberg, A., Eds.; Springer: Berlin/Heidelberg, Germany, 1998; pp. 111–149. [Google Scholar] [CrossRef]
- Ciraolo, L.; Azpilicueta, F.; Brunini, C.; Meza, A.; Radicella, S. Calibration errors on experimental slant total electron content (TEC) determined with GNSS. J. Geod.
**2007**, 81, 111–120. [Google Scholar] [CrossRef] - Schaer, S.; Gurtner, W.; Feltens, J. IONEX: The IONosphere Map EXchange Format Version 1; Technical Report; Astronomical Institute, University of Berne: Bern, Switzerland, 1998. [Google Scholar]
- The International GPS Service (IGS): An interdisciplinary service in support of Earth sciences. Adv. Space Res.
**1999**, 23, 631–653. [CrossRef] - Feltens, J.; Schaer, S. IGS Products for the Ionosphere; Technical report; European Space Operation Centre and Astronomical Institue of the University of Berne: Bern, Switzerland, 1998. [Google Scholar]
- Villiger, A. (Ed.) International GNSS Service Technical Report 2018; Technical report, IGS Central Bureau and University of Bern; Bern Open Publishing: Bern, Switzerland, 2018. [Google Scholar]
- Li, Z.; Yuan, Y.; Wang, N.; Hernández-Pajares, M.; Huo, X. SHPTS: Towards a new method for generating precise global ionospheric TEC map based on spherical harmonic and generalized trigonometric series functions. J. Geod.
**2015**, 89, 331–345. [Google Scholar] [CrossRef] - Hernández-Pajares, M.; Juan, J.M.; Sanz, J. New approaches in global ionospheric determination using ground GPS data. J. Atmos. Sol. Terr. Phys.
**1999**, 61, 1237–1247. [Google Scholar] [CrossRef] - Hernández-Pajares, M.; Juan, J.M.; Sanz, J.; Orus, R.; Garcia-Rigo, A.; Feltens, J.; Komjathy, A.; Schaer, S.C.; Krankowski, A. The IGS VTEC map: A reliable source of ionospheric information since 1998. J. Geod.
**2009**, 83, 263–275. [Google Scholar] [CrossRef] - Orus, R.; Hernández-Pajares, M.; Juan, J.; Sanz, J. Improvement of global ionospheric VTEC maps by using kriging interpolation technique. J. Atmos. Sol.-Terr. Phys.
**2005**, 67, 1598–1609. [Google Scholar] [CrossRef] - Erdogan, E.; Schmidt, M.; Seitz, F.; Durmaz, M. Near real-time estimation of ionosphere vertical total electron content from GNSS satellites using B-spline in a Kalman Filter. Ann. Geophys.
**2017**, 35, 263–277. [Google Scholar] [CrossRef] - Goss, A.; Schmidt, M.; Erdogan, E.; Görres, B.; Seitz, F. High-resolution vertical total electron content maps based on multi-scale B-spline representations. Ann. Geophys.
**2019**, 37, 699–717. [Google Scholar] [CrossRef] [Green Version] - Goss, A.; Schmidt, M.; Erdogan, E.; Seitz, F. Global and Regional High-Resolution VTEC Modelling Using a Two-Step B-Spline Approach. Remote Sens.
**2020**, 12, 1198. [Google Scholar] [CrossRef] [Green Version] - Erdogan, E.; Schmidt, M.; Goss, A.; Görres, B.; Seitz, F. Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content. Remote Sens.
**2020**, 12, 1822. [Google Scholar] [CrossRef] - Hernández-Pajares, M.; Roma-Dollase, D.; Krankowski, A.; García-Rigo, A.; Orús-Pérez, R. Methodology and consistency of slant and vertical assessments for ionospheric electron content models. J. Geod.
**2017**, 19, 1405–1414. [Google Scholar] [CrossRef] - Caissy, M.; Agrotis, L.; Weber, G.; Pajares, M.; Hugentobler, U. The international GNSS real-time service. GPS World
**2012**, 23, 52–58. [Google Scholar] - RTCM. IGS State Space Representation (SSR). Technical Report. 2020. Available online: https://files.igs.org/pub/data/format/igs_ssr_v1.pdf (accessed on 21 December 2020).
- Li, Z.; Wang, N.; Hernández-Pajares, M.; Yuan, Y.; Krankowski, A.; Liu, A.; Zha, J.; García-Rigo, A.; Roma, D.; Yang, H.; et al. IGS real-time service for global ionospheric total electron content modeling. J. Geod.
**2020**, 94, 1–16. [Google Scholar] [CrossRef] - Schmidt, M.; Dettmering, D.; Mößmer, M.; Wang, Y. Comparison of spherical harmonic and B-spline models for the vertical total electron content. Radio Sci.
**2011**, 46, RS0D11. [Google Scholar] [CrossRef] - Schmidt, M.; Dettmering, D. Using B-spline expansions for ionosphere modeling. In Handbook of Geomathematics, 2nd ed.; Freeden, W., Nashed, M.Z., Sonar, T., Eds.; Springer: Berlin/Heidelberg, Germany, 2015; pp. 939–983. [Google Scholar] [CrossRef]
- Roma-Dollase, D.; Hernández-Pajares, M.; Krankowski, A.; Kotulak, K.; Ghoddousi-Fard, R.; Yuan, Y.; Li, Z.; Zhang, H.; Shi, C.; Wang, C. Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle. J. Geod.
**2017**, 92, 691–706. [Google Scholar] [CrossRef] [Green Version] - Takasu, T.; Yasuda, A. Development of the low-cost RTK-GPS receiver with an open source program package RTKLIB. In International Symposium on GPS/GNSS; International Convention Center: Jeju, Korea, 2009. [Google Scholar]
- Reuter, R. Über Integralformeln der Einheitsphäre und Harmonische SPlinefunctionen. Ph.D. Thesis, RWTH Aachen, Aachen, Germany, 1982. [Google Scholar]
- Freeden, W.; Grevens, T.; Schreiner, M. Constructive Approximation on the Sphere; Oxford University Press: Oxford, UK, 1998. [Google Scholar]
- Laundal, K.; Richmond, A. Magnetic Coordinate Systems. Space Sci. Rev.
**2017**, 206, 27–59. [Google Scholar] [CrossRef] [Green Version] - Lyche, T.; Schumaker, L. A multiresolution tensor spline method for fitting functions on the sphere. SIAM J. Sci. Comput.
**2001**, 22, 724–746. [Google Scholar] [CrossRef] [Green Version] - Eicker, A. Gravity Field Refinement by Radial Basis Functions from In-situ Satellite Data. Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany, 2008. [Google Scholar]
- Schaer, S. Mapping and Predicting the Earth’s Ionosphere Using the Global Positioning System. Ph.D. Thesis, University of Berne, Bonn, Germany, 1999. [Google Scholar]
- Prol, F.; Camargo, P.; Monico, J.; Muella, M. Assessment of a TEC calibration procedure by single-frequency PPP. GPS Solut.
**2018**, 22, 35. [Google Scholar] [CrossRef] [Green Version] - Dow, J.; Neilan, R.E.; Rizos, C. The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J. Geod.
**2009**, 83. [Google Scholar] [CrossRef]

**Figure 1.**Polynomial B-spline of level ${J}_{1}=3$ (

**left**). The blue-colored splines indicate the endpoint interpolating feature at the boundaries of the interval. In the right panel, the trigonometric B-spline of level ${J}_{2}=2$ (

**right**) are represented. Here, the blue-colored spline as well as the green-colored spline indicate the wrapping around at the boundaries of the 0° and 360° meridian. The red-colored B-spline functions are of shift parameters ${k}_{1}=3$ and ${k}_{2}=3$ for the polynomial and trigonometric B-splines, respectively.

**Figure 2.**Reuter grids with values $\gamma =16$ in panel (

**a**), $\gamma =25$ in panel (

**b**) and $\gamma =35$ in panel (

**c**).

**Figure 3.**Schematic representation of the transformation and the following generation of the products of type 1 and 2 and the typically used data formats SSR VTEC message and IONEX, respectively.

**Figure 4.**Original global ionosphere map (GIM) generated by the B-spline model with levels ${J}_{1}=5$ and ${J}_{2}=3$ in the

**left**column. In the

**right**column are the pseudo observations on a Reuter grid with $\gamma =21$.

**Figure 5.**SH GIM (

**top panels**) and deviation maps (

**bottom panels**) of the 1. (

**left column**), 3. (

**middle column**) and 5. (

**right column**) test case from Table 4 for September 8, at 12:00 UT.

**Figure 6.**Change in relative RMS values (

**left**) and RMS values (

**right**), for the period from 2 September 2017 (DOY 245) to 12 September 2017 (DOY 255); temporal sampling intervals of 10 min.

**Figure 7.**Global distribution of 9 receiver stations that are used for the dSTEC validation and for comparison of the generated GIMs.

**Figure 8.**Differences $S\left({t}_{s}\right)$ in the 3D position between the position determined by Precise Point Positioning (PPP) and ionospheric corrections calculated by different GIMs and the actual position. Left side shows the differences for the stations BOGT, WTZR and ASPA during the 2 September 2017 (DOY 245) and the right side the corresponding differences for 8 September 2017 (DOY 251) with different scaling of the y-axis. On the x-axis, the time in UT is depicted in hours.

Latency Type | Final | Rapid | Ultra-Rapid | Near Real-Time | Real-Time |
---|---|---|---|---|---|

Data | post-processed | daily | hourly | data-streams | data-streams |

Data Latency | 2–3 weeks | >1 day | >1 h | real-time | real-time |

Product Latency | 2–3 weeks | >1 day | >1 h | <15 min | <1 min |

**Table 2.**Sampling intervals $\mathsf{\Delta}{\phi}_{l}$ and $\mathsf{\Delta}{\lambda}_{l}$ of the Reuter grid points for different values of $\gamma $ and the corresponding number V of Reuter grid points. Required max. sampling intervals $\mathsf{\Delta}{\phi}_{SH}$ and $\mathsf{\Delta}{\lambda}_{SH}$ and the number of unknown coefficients of spherical harmonics (SHs) with different values for ${n}_{max}$.

Reuter Grid | Spherical Harmonics | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\gamma $ | 16 | 21 | 25 | 31 | 35 | ${n}_{max}$ | 15 | 20 | 24 | 30 | 34 |

$\mathsf{\Delta}{\phi}_{l}$, $\mathsf{\Delta}{\lambda}_{l}$ | 11.25° | 8.57° | 7.2° | 5.8° | 5.14° | $\mathsf{\Delta}\phi $, $\mathsf{\Delta}\lambda $ | 12 | 9 | 7.5 | 6 | 5.3 |

V | 317 | 563 | 797 | 1225 | 1561 | N | 256 | 441 | 625 | 961 | 1224 |

Polynomial B-Splines | Trigonometric B-Splines | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${J}_{1}$ | 1 | 2 | 3 | 4 | 5 | 6 | ${J}_{2}$ | 1 | 2 | 3 | 4 | 5 | 6 |

${n}_{max}$ | 3 | 5 | 9 | 17 | 33 | 63 | ${n}_{max}$ | 3 | 6 | 12 | 24 | 48 | 96 |

${L}_{\phi}$ | 120° | 72° | 40° | 21° | 10.9° | 5.7° | ${L}_{\lambda}$ | 120° | 60° | 30° | 15° | 7.5° | 3.75° |

**Table 4.**Numerical and statistical results to estimate the quality and feasibility for the different cases.

1. Case | 2. Case | 3. Case | 4. Case | 5. Case | |
---|---|---|---|---|---|

$\gamma $ | 16 | 21 | 25 | 31 | 35 |

V | 317 | 563 | 797 | 1225 | 1561 |

${n}_{max}$ | 15 | 20 | 24 | 30 | 34 |

N | 256 | 441 | 625 | 961 | 1224 |

$\tilde{\mathsf{\Delta}t}$ | 1.43 s | 1.85 s | 3.12 s | 4.99 s | 8.23 s |

rel. RMS [%] | 9.23 | 5.83 | 4.19 | 2.54 | 1.83 |

RMS [TECU] | 1.31 | 0.83 | 0.60 | 0.36 | 0.26 |

${\delta}_{max}$ [TECU] | 8.22 | 5.91 | 5.23 | 2.22 | 1.79 |

${\delta}_{min}$ [TECU] | −8.06 | −6.29 | −4.21 | −2.23 | −1.9 |

${\delta}_{mean}$ [TECU] | 0.016 | 0.0012 | 0.014 | 0.0033 | 0.003 |

**Table 5.**Comparison of GIMs with different characteristics, i.e., model type, degree of series expansion (${J}_{1}$, ${J}_{2}$, ${n}_{max}$), temporal output sampling intervals ($\mathsf{\Delta}T$) and latency until provision. For each of the investigated GIMs, the RMS of the dSTEC analysis is given.

GIM | othg | o15g | o20g | o24g | o30g | o34g | codg | uqrg |
---|---|---|---|---|---|---|---|---|

Model | B-Splines | SHs | SHs | SHs | SHs | SHs | SHs | Voxel & Kriging |

${n}_{max}$ | 15 | 20 | 24 | 30 | 34 | 15 | n.a. | |

${J}_{1}$/${J}_{1}$ | 5/3 | |||||||

$\mathsf{\Delta}T$ | 10 min | 10 min | 10 min | 10 min | 10 min | 10 min | 60 min | 15 min |

Latency | <3 h | <3 h | <3 h | <3 h | <3 h | <3 h | >1 week | >1 day |

Reference | [15] | Section 3 | Section 3 | Section 3 | Section 3 | Section 3 | [31] | [13] |

RMS [TECU] | 0.91 | 1.18 | 1.05 | 1.00 | 0.93 | 0.92 | 1.01 | 0.85 |

**Table 6.**RMS and $\overline{S}$ values of deviations $S\left({t}_{s}\right)$ for September 2 (DOY 245) and 8 (DOY 251), 2017 at the stations BOGT, WTZR and ASPA. The maximum and minimum values of the RMS and $\overline{S}$ are marked in red and green, respectively. The RMS and $\overline{S}$ of the transformed versions are bold if they are lower than the values of the original GIM, ‘othg’.

Value | DOY | othg | o15g | o20g | o24g | o30g | o34g | codg | uqrg | |
---|---|---|---|---|---|---|---|---|---|---|

BOGT | RMS [TECU] | 245 | 1.07 | 1.35 | 1.17 | 1.09 | 1.05 | 1.05 | 1.29 | 0.90 |

251 | 2.91 | 3.19 | 3.12 | 3.03 | 2.91 | 2.86 | 3.14 | 2.98 | ||

$\overline{S}$ [TECU] | 245 | 0.90 | 1.13 | 1.00 | 0.92 | 0.89 | 0.89 | 1.05 | 0.77 | |

251 | 2.22 | 2.44 | 2.34 | 2.31 | 2.23 | 2.20 | 2.35 | 2.18 | ||

WTZR | RMS [TECU] | 245 | 0.49 | 0.55 | 0.53 | 0.52 | 0.50 | 0.48 | 0.51 | 0.55 |

251 | 0.49 | 0.90 | 0.57 | 0.54 | 0.50 | 0.49 | 0.51 | 0.62 | ||

$\overline{S}$ [TECU] | 245 | 0.45 | 0.50 | 0.49 | 0.48 | 0.46 | 0.44 | 0.46 | 0.50 | |

251 | 0.44 | 0.53 | 0.52 | 0.47 | 0.44 | 0.44 | 0.47 | 0.57 | ||

ASPA | RMS [TECU] | 245 | 0.81 | 2.53 | 0.98 | 0.92 | 0.85 | 0.92 | 1.13 | 1.19 |

251 | 2.18 | 3.12 | 2.59 | 2.67 | 2.17 | 2.11 | 2.65 | 1.80 | ||

$\overline{S}$ [TECU] | 245 | 0.69 | 1.37 | 0.86 | 0.79 | 0.73 | 0.77 | 0.95 | 0.94 | |

251 | 1.53 | 2.10 | 1.76 | 1.84 | 1.54 | 1.48 | 1.64 | 1.40 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Goss, A.; Hernández-Pajares, M.; Schmidt, M.; Roma-Dollase, D.; Erdogan, E.; Seitz, F.
High-Resolution Ionosphere Corrections for Single-Frequency Positioning. *Remote Sens.* **2021**, *13*, 12.
https://doi.org/10.3390/rs13010012

**AMA Style**

Goss A, Hernández-Pajares M, Schmidt M, Roma-Dollase D, Erdogan E, Seitz F.
High-Resolution Ionosphere Corrections for Single-Frequency Positioning. *Remote Sensing*. 2021; 13(1):12.
https://doi.org/10.3390/rs13010012

**Chicago/Turabian Style**

Goss, Andreas, Manuel Hernández-Pajares, Michael Schmidt, David Roma-Dollase, Eren Erdogan, and Florian Seitz.
2021. "High-Resolution Ionosphere Corrections for Single-Frequency Positioning" *Remote Sensing* 13, no. 1: 12.
https://doi.org/10.3390/rs13010012