# Ionosphere Model for European Region Based on Multi-GNSS Data and TPS Interpolation

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## Abstract

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## 1. Introduction

^{16}el/m

^{2}, and it is equivalent to 0.162 m of L1 signal delay). This is one of the reasons why spherical harmonics expansion (SHE) is used for the global and regional TEC parameterization [15,16,17]. The smoothing effect of SHE undoubtedly results in the low accuracy of the ionospheric models. Also, the ionosphere models often use GPS-only data. Another important aspect is using a single layer model (SLM) ionosphere approximation and its associated relatively simple mapping function [18,19]. This results in rather low relative accuracy of publically available models that amounts to 20–30%, as was shown in Hernández-Pajares et al. [20]. In the aforementioned study, the authors compared several existing and publically available ionosphere models to satellite altimeter data, and the results showed that the accuracy of absolute vertical TEC (vTEC) was on the level of 4–5 TECU (15–25% relative).

## 2. Methodology

#### 2.1. Carrier Phase Bias Estimation

#### 2.2. TEC Calculation Procedure

#### 2.3. TEC Modelling by TPS

_{TPS}values are determined for the data points $\left\{{\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right\}$. Deviations of these TPS-modelled values from the given TEC data allow for root mean error (RMS) calculation. The received value is compared with an a priori accuracy (${\mathsf{\sigma}}_{\mathrm{a}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}$) of TEC data. If these values are significantly different from each other, the parameter $\mathsf{\alpha}$ is modified and a new TPS function is determined. In another case, the ellipsoidal grid $\left\{{\mathsf{\varphi}}_{\mathrm{G}},{\mathsf{\lambda}}_{\mathrm{G}}\right\}$ is mapped onto the plane by means of UTM projection and TEC

_{TPS}values (${\mathsf{\varphi}}_{\mathrm{G}},{\mathsf{\lambda}}_{\mathrm{G}}$) are calculated. Finally, TEC

_{TPS}values are merged with the ellipsoidal grid. The approximation parameter $\mathsf{\alpha}$ depends strongly on the data to be modelled. We use values of the order of 10

^{−10}for TEC data smoothing and interpolation.

## 3. Numerical Results and Discussion

#### 3.1. Self-Consistency Analysis

- Firstly, a geometry-free linear combination (L4) of carrier phase observations is formed for each continuous data arc (red line, Figure 3 left panel);
- next, sTEC for the same satellite arc extracted from a tested model (GIM_sTEC) is calculated (blue line, Figure 3 left panel);
- then, carrier phase bias is estimated by fitting carrier phase data (L4) into GIM_sTEC, resulting in calibrated sTEC (L4_sTEC) (red line, Figure 3 right panel).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic diagram of vertical total electron content (vTEC) modelling by thin plate splines (TPS).

**Figure 2.**Variations of ΣKp and Dst indices during March 2015. The vertical red lines highlight the seven test days (14–20 March 2015).

**Figure 3.**Scheme of the L4 data fitting into Global Ionosphere Maps’ (GIM) slant TEC (sTEC) (sTEC calibration).

**Figure 5.**Example TEC maps derived from the UWM-rt1 model on a quiet day before the storm (

**A**), the stormy day (

**B**) and one day after the storm (

**C**) all at 11:00 GPST. Dark lines represent tracks of the GPS PRN30 satellite observed by POTS station.

**Figure 6.**sTEC for selected PRN arcs from calibrated geometry-free L4 (L4_sTEC—red) and selected GIMs (GIM_sTEC—blue) on the quiet day before the storm (

**A**), the stormy day (

**B**) and one day after the storm (

**C**).

**Table 1.**Root mean error (RMS) of post fit residuals for the analyzed TEC maps (TECU). The stormy day is marked with bold font. UWM-rt1—our regional maps, IGS—International GNSS Service global maps, UQRG—high-rate maps provided by Technical University of Catalonia, JPL—global maps from Jet Propulsion Laboratory, CODG—global maps from Center for Orbit determination in Europe, ESA—global maps from European Space Agency.

DOY | UWM-rt1 | IGS | UQRG | JPL | CODG | ESA |
---|---|---|---|---|---|---|

73 | 0.44 | 0.81 | 0.94 | 0.91 | 0.61 | 0.95 |

74 | 0.36 | 0.85 | 0.95 | 0.94 | 0.63 | 1.00 |

75 | 0.38 | 0.83 | 1.00 | 0.95 | 0.62 | 0.84 |

76 | 0.61 | 1.67 | 1.32 | 1.81 | 1.14 | 2.36 |

77 | 0.22 | 0.71 | 0.54 | 0.78 | 0.52 | 0.82 |

78 | 0.24 | 0.84 | 0.61 | 0.91 | 0.56 | 1.20 |

79 | 0.15 | 0.76 | 0.55 | 0.86 | 0.42 | 0.79 |

UWM-rt1 | IGS | UQRG | JPL | CODG | ESA |
---|---|---|---|---|---|

0.34 | 0.92 | 0.84 | 1.02 | 0.64 | 1.14 |

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**MDPI and ACS Style**

Krypiak-Gregorczyk, A.; Wielgosz, P.; Borkowski, A.
Ionosphere Model for European Region Based on Multi-GNSS Data and TPS Interpolation. *Remote Sens.* **2017**, *9*, 1221.
https://doi.org/10.3390/rs9121221

**AMA Style**

Krypiak-Gregorczyk A, Wielgosz P, Borkowski A.
Ionosphere Model for European Region Based on Multi-GNSS Data and TPS Interpolation. *Remote Sensing*. 2017; 9(12):1221.
https://doi.org/10.3390/rs9121221

**Chicago/Turabian Style**

Krypiak-Gregorczyk, Anna, Pawel Wielgosz, and Andrzej Borkowski.
2017. "Ionosphere Model for European Region Based on Multi-GNSS Data and TPS Interpolation" *Remote Sensing* 9, no. 12: 1221.
https://doi.org/10.3390/rs9121221