# Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content

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## Abstract

**:**

## 1. Introduction

## 2. Global VTEC Representation Based On B-Splines

#### Coordinate System

## 3. GNSS Ionospheric Observables

- Observations with an elevation angle of less than 10${}^{\circ}$ were rejected from the data to avoid contributions from likely very noisy measurements.
- To mitigate the errors due to multi-path effects on pseudo-range measurements, which are generally inversely proportional to the satellite elevation angle, only the observations with elevation angles larger than 20${}^{\circ}$ are included in the determination of the leveling bias (3).
- Moreover, an elevation dependent weighting function, e.g., ${w}_{i}=\mathrm{sin}({e}_{i})$, is introduced to leverage the influence of more precise observations. ${e}_{i}$ means the elevation angle of the $i\mathrm{th}$ observation along the arc.
- A pre-processing algorithm was developed for the recursive processing of GNSS data acquired as hourly data blocks broadcasted, e.g., by the IGS global data centers. To reach a maximum number of observations and a high pre-processing quality, data blocks with a moving window of 3 h length are considered. For instance, to apply the pre-processing procedures to a new acquired data set between ${t}_{k}$ and ${t}_{k}+1h$, a data set with a window length of 3 h extending from ${t}_{k}-2h$ to ${t}_{k}+1h$ is considered for a more accurate computation of the leveling bias.

## 4. Adaptive Estimation of Global Ionospheric VTEC

#### 4.1. Model Definition

#### 4.2. Measurement Model

#### 4.3. Prediction Model

#### 4.4. Adaptive Filtering Using Method of Variance-Components (VC) Estimation

#### 4.5. Constrained Filtering

## 5. Results and Discussions

#### 5.1. Filter Settings

#### 5.2. Validation Methods and Data Sets

#### 5.3. Comparisons to IGS and IAACs

#### 5.3.1. dSTEC Analysis

#### 5.3.2. Altimetry Comparisons

#### 5.4. Evaluation of the Proposed Approach

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The panels in the top row show the VTEC maps from IGS final GIMs drawn in an Earth-fixed coordinate system at 02:00, 08:00, 14:00 and 20:00 UTC for 17 March 2015. The corresponding maps of estimated B-spline coefficients for each of the VTEC maps are illustrated in the bottom row; the B-spline coefficients ${d}_{{k}_{1},{k}_{2}}^{{J}_{1},{J}_{2}}({t}_{k})$ refer to resolution levels; ${J}_{1}=5$ and ${J}_{2}=3$ for latitude and longitude, respectively.

**Figure 2.**Correlation analysis of global mean values of VTEC maps and B-spline coefficients; the mean values of the IGS VTEC maps computed with a 2 h temporal resolution during the year 2015 (red dots) as well as the mean values of estimated B-spline coefficients maps (blue dots).

**Figure 3.**Exemplary process noise parameters; (

**a**) Distribution of the B-spline coefficients on 17 March 2015, 12:00 UTC, and the maps of the corresponding process noise parameters for the coefficients ${C}_{1,{d}_{i},k}$ (

**b**) and ${C}_{2,{d}_{i},k}$ (

**c**).

**Figure 4.**RMS values from the dSTEC analysis at the GNSS stations depicted in panel (

**a**). The results refer to the data sets covering the days between (

**b**) DOY 41 and DOY 110 of year 2015 and (

**c**) DOY 222 and DOY 293 of year 2017. The label “othg” stands for the presented approach.

**Figure 5.**Global VTEC maps with six hours sampling interval generated by the approach presented in this study: (

**a**) 16 March 2015, the day before the main phase of the St. Patrick storm (

**b**) 17 March 2015, the St. Patrick storm day and (

**c**) 18 March 2015, the day after the main phase of the St. Patrick storm.

**Figure 6.**Comparison of DGFI-TUM’s VTEC values with the IAAC solutions in terms of RMS deviations. The daily RMS deviations are shown in (

**a**) for the data sets of 2015 and in (

**b**) for the year of 2017. (

**a**) includes comparisons with respect to the altimeter VTEC data from the Jason-2 mission whereas (

**b**) refers to the data from the Jason-3 mission. The label “othg” stands for the presented approach.

**Figure 7.**Comparison of “othg” and “TC4” solutions in terms of RMS deviations; (

**a**) RMS values from the dSTEC analysis at the GNSS stations depicted in panel (

**a**) of Figure 4; (

**b**) daily RMS deviations with respect to the altimeter VTEC data from the Jason-2 mission. The results refer to the data set covering the days between DOY 42 and DOY 110 of year 2015.

Product Label | VCE | Process Noise Model | Test Definition | Product Quality (in TECU) |
---|---|---|---|---|

othg | Enabled | Enabled | presented approach | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{OTHG}}=1.62$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{OTHG}}=5.3$ |

TC1 | Disabled | Disabled | ${\sigma}_{{y}_{\mathrm{GPS}},k}^{2}={\sigma}_{{y}_{\mathrm{GPS},\mathrm{NOM}}}^{2}$, ${\sigma}_{{y}_{\mathrm{GLO}}}^{2}={\sigma}_{{y}_{\mathrm{GLO},\mathrm{NOM}}}^{2}$, ${\sigma}_{{d}_{i},k}^{2}={\sigma}_{{d}_{\mathrm{NOM}}}^{2}$ | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{TC}1}=1.66$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{TC}1}=5.4$ |

TC2 | Disabled | Enabled | ${\sigma}_{{y}_{\mathrm{GPS}},k}^{2}={\sigma}_{{y}_{\mathrm{GPS},\mathrm{NOM}}}^{2}$, ${\sigma}_{{y}_{\mathrm{GLO}},k}^{2}={\sigma}_{{y}_{\mathrm{GLO},\mathrm{NOM}}}^{2}$, ${\mathbf{P}}_{{y}_{\mathrm{GPS}},k}={\mathbf{P}}_{{y}_{\mathrm{GLO}},k}=\mathbf{I}$ | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{TC}2}=1.94$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{TC}2}=5.9$ |

TC3 | Disabled | Enabled | ${\sigma}_{{y}_{\mathrm{GPS}},k}^{2}={\sigma}_{{y}_{\mathrm{GLO},\mathrm{NOM}}}^{2}$, ${\sigma}_{{y}_{\mathrm{GLO}},k}^{2}={\sigma}_{{y}_{\mathrm{GLO},\mathrm{NOM}}}^{2}$ | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{TC}3}=1.75$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{TC}3}=5.5$ |

TC4 | Enabled | Disabled | ${\sigma}_{{d}_{i},k}^{2}={\sigma}_{{d}_{\mathrm{NOM}}}^{2}$ | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{TC}4}=1.69$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{TC}4}=5.4$ |

TC5 | Enabled | Enabled | ${C}_{1,k}={C}_{2,k}=1$ in Equation (20) | ${\mathrm{RMS}}_{\mathrm{dSTEC},\mathrm{TC}5}=1.76$, ${\mathrm{RMS}}_{\mathrm{ALT},\mathrm{TC}5}=5.6$ |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/rs12111822

**AMA Style**

Erdogan E, Schmidt M, Goss A, Görres B, Seitz F. Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content. *Remote Sensing*. 2020; 12(11):1822.
https://doi.org/10.3390/rs12111822

**Chicago/Turabian Style**

Erdogan, Eren, Michael Schmidt, Andreas Goss, Barbara Görres, and Florian Seitz. 2020. "Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content" *Remote Sensing* 12, no. 11: 1822.
https://doi.org/10.3390/rs12111822