# Regiomontan: A Regional High Precision Ionosphere Delay Model and Its Application in Precise Point Positioning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Estimating the Ionospheric Delay from GNSS Observations

^{2}and a height of the slant signal path [31]. The $STEC$ is measured in total electron content units (TECU), where 1 TECU accounts for 10

^{16}electrons/m

^{2}.

## 3. Development of a Regional High Precision Ionosphere Delay Model

#### 3.1. Taylor Approximation of the Single-Layer Model

#### 3.2. Parameter Estimation Using Code-Leveled Carrier-Phase Observations

#### 3.3. Specifications of the Output Files in IONEX Format

## 4. Applying the Regiomontan Model to Precise Point Positioning

#### 4.1. Classical PPP Model

#### 4.2. Uncombined PPP Model with Ionospheric Constraint

## 5. Results

#### 5.1. Validating Regiomontan via Ionosphere-Free Linear Combination

#### 5.2. PPP Results with Regiomontan

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**GNSS station network used for processing the Regiomontan model. Red triangles depict the permanent reference stations selected for this purpose. The blue dot indicates the center of the Taylor expansion ${P}_{0}$.

**Figure 2.**Station DCB values $DC{B}_{R}$ of the Regiomontan station network over four consecutive GNSS weeks in December 2019. The $DC{B}_{R}$ values mostly vary only a few centimeters during this period. The error bars indicate the standard deviation of the least squares adjustment.

**Figure 3.**Behavior of the Regiomontan model parameters on $doy$ 335, 2019: (

**a**) The top panel depicts the behavior of $VTE{C}_{0}$ in TECU. The error bars illustrate the formal errors of the least squares adjustment magnified by the factor of 10${}^{2}$. (

**b**) The middle panel depicts the behavior of the first derivatives of $VTEC$ with respect to latitude (green) and longitude (blue) in TECU/rad. Formal errors are magnified by the factor of 10${}^{3}$. (

**c**) The bottom panel depicts the behavior of the second derivatives of $VTEC$ with respect to latitude (green), longitude (blue), and the mixed term (red) in TECU${}^{2}$/rad${}^{2}$. Formal errors are magnified by the factor of 10${}^{3}$.

**Figure 4.**Range residuals for station WOFU on $doy$ 153, 2019. Green indicates residuals smaller than 0.5 m, yellow residuals smaller than 1 m, and red residuals larger than 1 m: (

**a**) Regiomontan; (

**b**) IGS final TEC grid (IGS); (

**c**) IGS rapid TEC grid (IGR); and (

**d**) Klobuchar. The Klobuchar model overestimates the ionospheric delay during the daytime. The other models yield very similar results. All observations down to an elevation angle of 5° are included.

**Figure 5.**Range residuals for station WOFU on $doy$ 335, 2019. Green indicates residuals smaller than 0.5 m, yellow residuals smaller than 1 m, and red residuals larger than 1 m: (

**a**) Regiomontan; (

**b**) IGS final TEC grid (IGS); (

**c**) IGS rapid TEC grid (IGR); and (

**d**) Klobuchar. All four models yield the same level of accuracy in winter as they do in summer. All observations down to an elevation angle of 5° are included.

**Figure 6.**Percentage of converged solutions after [1, 2.5, 5, 7.5, 10, 15, 20] min for: the height component (

**top**); and the horizontal position (

**bottom**). The height of a bar corresponds to the percent of convergence periods which have reached convergence at this point in time.

**Figure 7.**68% (

**left**) and 95% (

**right**) quantiles of the coordinate error in the height component (

**top**) and the horizontal position error (

**bottom**) for the different PPP solutions.

**Figure 8.**Histogram of the difference between the estimated and modeled ionospheric delay of GPS and Galileo for: (

**a**) Regiomontan; (

**b**) IGS; and (

**c**) CODE for June and December 2019.

**Figure 9.**(

**Top**) Estimated receiver DCBs for GPS (red) and Glonass (cyan) of the PPP solution using the uncombined model with Regiomontan as ionospheric constraint for December 2019. The CODE estimation of the receiver DCBs is shown in green. (

**Bottom**) Histogram of the difference to the CODE estimation with the corresponding standard deviation (std) and bias.

**Figure 10.**Horizontal position error for a PPP solution calculated on the next day for the station PFA3 (29–31 December 2019) using single-frequency GPS observations, the IGS ultra-rapid product, and Regiomontan. The 95% quantile is shown in black and the 68% quantile in grey.

Specification | Value |
---|---|

# of maps | 25 (00:00 UTC–00:00 UTC) |

Interval | 3600 s |

Latitude: min, max | 30°, 70° |

Longitude: min, max | −20°, 45° |

Spatial resolution | 1° × 1° |

**Table 2.**Statistics of the range residuals $(P1+dIon)-P3$ for all 22 stations of the Regiomontan station network. All observations down to an elevation angle of 5° are included. The results are listed for GPS Week 2056 ($doy$ 153–159) and GPS Week 2082 ($doy$ 335–341).

GPS Week 2056 | GPS Week 2082 | |||||||
---|---|---|---|---|---|---|---|---|

Residuals | Regiomontan | IGS | IGR | Klobuchar | Regiomontan | IGS | IGR | Klobuchar |

<0.5 m | 52.0% | 49.3% | 49.2% | 24.7% | 53.9% | 52.5% | 51.0% | 13.0% |

<1.0 m | 76.4% | 74.0% | 73.9% | 45.3% | 79.7% | 78.7% | 77.6% | 30.1% |

<1.5 m | 87.3% | 85.1% | 85.1% | 61.0% | 90.0% | 89.4% | 88.6% | 49.0% |

Setting | Value |
---|---|

Stations | GRAZ, PFA3, LINZ, SBG2, TRF2 |

Period | June 2019, December 2019 |

GNSS | GPS, Glonass (weighted 1:1) |

GPS Observations | L1, L2 |

Glonass Observations | G1, G2 |

Processing mode | undifferenced observations, static receiver |

Observation interval | 30 s, reset solution: every full hour |

Raw observation noise | code = 30 cm, phase = 2 mm |

Observation weighting | Elevation weighted ($sin{\left(elev\right)}^{2}$) |

Cutoff angle | elevation: 5° |

Satellite Orbits, Clocks and DCBs | CODE final products [43] |

Satellite and receiver antenna | IGS Antex igs14.atx [44] |

Troposphere model | VMF3 [45], residual ZWD is estimated |

Reference coordinates | EUREF [46] |

Adjustment | Kalman-Filter |

Receiver clock, time offset | white noise |

Phase Ambiguities | float, constant |

Cycle-Slip Detection | dL1-dL2 |

Correction models | Phase wind-up [47], solid earth tides [48], relativistic effects |

**Table 4.**Mean of 3D coordinate accuracy in centimeter for specific points in time after the last reset.

5 min | 10 min | 15 min | 30 min | 45 min | |
---|---|---|---|---|---|

IF LC | 45.2 | 18.1 | 8.7 | 4.9 | 3.8 |

REGIO | 39.6 | 16.6 | 8.0 | 4.8 | 3.8 |

IGS | 39.1 | 16.5 | 8.0 | 4.7 | 3.8 |

CODE | 39.0 | 16.4 | 7.9 | 4.8 | 3.8 |

**Table 5.**Statistics of the difference between the estimated and modeled ionospheric delay for the PPP solution using different ionosphere models as background model for June and December 2019.

GNSS | GPS | Glonass | ||||
---|---|---|---|---|---|---|

Model | REGIO | IGS | CODE | REGIO | IGS | CODE |

Month | June | |||||

<12.5 cm | 42.6% | 43.4% | 49.7% | 27.5% | 30.7% | 31.8% |

<25 cm | 69.1% | 71.4% | 76.6% | 50.6% | 56.8% | 57.9% |

<50 cm | 89.7% | 92.0% | 93.4% | 77.4% | 85.7% | 86.2% |

Month | December | |||||

<12.5 cm | 41.2% | 43.9% | 47.5% | 30.2% | 34.6% | 41.7% |

<25 cm | 68.7% | 73.2% | 74.9% | 53.8% | 60.1% | 68.3% |

<50 cm | 90.1% | 93.0% | 92.2% | 80.1% | 87.6% | 89.5% |

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**MDPI and ACS Style**

Boisits, J.; Glaner, M.; Weber, R.
Regiomontan: A Regional High Precision Ionosphere Delay Model and Its Application in Precise Point Positioning. *Sensors* **2020**, *20*, 2845.
https://doi.org/10.3390/s20102845

**AMA Style**

Boisits J, Glaner M, Weber R.
Regiomontan: A Regional High Precision Ionosphere Delay Model and Its Application in Precise Point Positioning. *Sensors*. 2020; 20(10):2845.
https://doi.org/10.3390/s20102845

**Chicago/Turabian Style**

Boisits, Janina, Marcus Glaner, and Robert Weber.
2020. "Regiomontan: A Regional High Precision Ionosphere Delay Model and Its Application in Precise Point Positioning" *Sensors* 20, no. 10: 2845.
https://doi.org/10.3390/s20102845