# Improved Empirical Coefficients for Estimating Water Vapor Weighted Mean Temperature over Europe for GNSS Applications

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methodology

^{5}± 0.0076 K

^{2}/hPa, respectively. ${M}_{w}=\text{}18.0151$ and ${M}_{d}=\text{}28.9644$ are the molar masses of wet and dry air expressed in g/mol.

## 3. Accuracy of the Existing ${\mathit{T}}_{\mathit{m}}\mathbf{-}{\mathit{T}}_{\mathit{s}}$ Coefficients

## 4. Estimation of the New ${\mathit{T}}_{\mathit{m}}\mathbf{-}{\mathit{T}}_{\mathit{s}}$ Coefficients

## 5. Results and Discussion

#### 5.1. Accuracy of the New ${T}_{m}-{T}_{s}$ Coefficients

#### 5.2. Impact of the New Coefficients on the PWV Estimated at RS Stations

#### 5.3. Impact of the New Coefficients on the PWV Estimated at GNSS Stations

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution of the radio-sounding (RS) stations used in the study. The red triangles denote stations from the estimation group, whereas the light violet circles denote stations from the external group. The station number is assigned to the station from the estimation group.

**Figure 2.**RMSE of differences between direct ${T}_{m}$ measurements and ${T}_{m}$ calculated using ${T}_{s}$ and the following coefficients of linear relationship: Bevis (

**a**), BevisRev (

**b**), Mendes (

**c**), Solbrig (

**d**). The dots indicate the location of 109 RS station, while their colors denote RMSE values.

**Figure 3.**Correlation and ${T}_{m}-{T}_{s}$ coefficients for selected RS stations: 07761 (Ajaccio, France) (

**a**), 08430 (Murcia, Spain) (

**b**), 14240 (Zagreb Maksimir, Croatia) (

**c**), and 01415 (Stavanger Sola, Norway) (

**d**). The light blue and light pink dots represent observations at 00:00 and 12:00 UTC, respectively. Fitted lines are shown using blue (night) and red (day) lines. Linear regression coefficients ($a$ and $b$), correlation coefficient (corr.) and RMSE for each case are also shown.

**Figure 4.**RMS values of ${T}_{m}$ estimated using ETm (

**a**), ETm2 (

**b**), ETm4 (

**c**), and ETmPoly (

**d**) coefficients. RMSE calculated in reference to the in situ measurements performed at RS stations in a period since 1996 to the end of 2018. The circles and the triangles represent RS stations belonging to estimation and external groups, respectively.

**Figure 5.**Results of fitting different degree polynomials into ${T}_{m}-{T}_{s}$ linear regression coefficients: $a$ (left) and $b$ (right). Color of the line denotes degree of the polynomial: 3rd—blue, 4th—orange, 5th—green, 6th—red.

**Figure 6.**RMS values of the precipitable water vapor (PWV) estimated using Bevis (left) and ETmPoly (right) coefficients. The PWV derived from RS profiles were adopted as a reference.

**Figure 7.**Differences between PWV estimated using Bevis (blue) and ETmPoly (pink) coefficients and PWV obtained directly from the RS measurements, for 15614 (left) and 16144 (right) RS stations.

**Figure 8.**Differences between PWV values estimated using Bevis and ETmPoly coefficients for EBRE (

**a**), SFER (

**b**), MEDI (

**c**), and KIRU (

**d**) GNSS stations.

**Figure 9.**Standard deviations of differences between global navigation satellite system (GNSS) PWV values estimated using Bevis and ETmPoly coefficients for the seasons: (

**a**) December/January/February (DJA) and (

**b**) June/July/August (JJA).

**Table 1.**Average root mean square error (RMSE) ($\overline{RMSE}$) of the analyzed coefficients calculated based on 109 RS stations.

${\mathit{T}}_{\mathit{m}}=\mathit{a}\cdot {\mathit{T}}_{\mathit{s}}+\mathit{b}$ | |||
---|---|---|---|

Coefficients | a | b | $\overline{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}$ |

Bevis | 0.72 | 70.2 | 3.1 ± 0.4 |

BevisRev | 0.668 | 85.63 | 3.3 ± 0.4 |

Mendes | 0.789 | 50.4 | 3.1 ± 0.5 |

Solbrig | 0.77 | 54.7 | 3.2 ± 0.6 |

${\mathit{T}}_{\mathit{m}}-{\mathit{T}}_{\mathit{s}}\mathbf{Coefficients}$ | |||||
---|---|---|---|---|---|

Name | Hours (UTC) | $\mathit{a}$ | $\mathit{b}$ | $\overline{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}$ | Number of Observations |

ETm | all available | 0.7440 ± 0.0004 | 62.84 ± 0.10 | 3.02 | 967 277 |

ETm2 | 00:00 | 0.8436 ± 0.0006 | 35.88 ± 0.17 | 2.91 | 391 401 |

12:00 | 0.7430 ± 0.0004 | 61.84 ± 0.13 | 2.46 | 381 106 | |

ETm4 | 00:00 | 0.8436 ± 0.0006 | 35.88 ± 0.17 | 2.91 | 391 401 |

06:00 | 0.7997 ± 0.0014 | 48.07 ± 0.40 | 2.95 | 26 160 | |

12:00 | 0.7430 ± 0.0004 | 61.84 ± 0.13 | 2.46 | 381 106 | |

18:00 | 0.7478 ± 0.0011 | 61.00 ± 0.31 | 2.43 | 24 862 |

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**MDPI and ACS Style**

Baldysz, Z.; Nykiel, G. Improved Empirical Coefficients for Estimating Water Vapor Weighted Mean Temperature over Europe for GNSS Applications. *Remote Sens.* **2019**, *11*, 1995.
https://doi.org/10.3390/rs11171995

**AMA Style**

Baldysz Z, Nykiel G. Improved Empirical Coefficients for Estimating Water Vapor Weighted Mean Temperature over Europe for GNSS Applications. *Remote Sensing*. 2019; 11(17):1995.
https://doi.org/10.3390/rs11171995

**Chicago/Turabian Style**

Baldysz, Zofia, and Grzegorz Nykiel. 2019. "Improved Empirical Coefficients for Estimating Water Vapor Weighted Mean Temperature over Europe for GNSS Applications" *Remote Sensing* 11, no. 17: 1995.
https://doi.org/10.3390/rs11171995