# Establishment of a Real-Time Local Tropospheric Fusion Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multisource Tropospheric Data

#### 2.1.1. Zenith Tropospheric Delay Obtained by GNSS Processing

#### 2.1.2. Zenith Tropospheric Delay Obtained by the Saastamoinen Model

#### 2.1.3. Zenith Tropospheric Delay Obtained by the GPT2w Model

_{0}is the mean value, A

_{1}and B

_{1}are the annual cycle amplitudes, A

_{2}and B

_{2}are the semi-annual cycle amplitudes, doy is the day of the year. In Equation (5), ${k}_{2}^{\prime}$ and k

_{3}are the atmospheric refractive index constants provided by Thayer [29], where ${k}_{2}^{\prime}$= 16.529 k·mb

^{−1}and k

_{3}= 3.776 × 10

^{5}k·mb

^{−1}. Furthermore, T

_{m}is the atmospheric weighted average temperature, g

_{m}is the gravity acceleration, R

_{d}is the gas ratio of dry air constant values 287.058 J·kg

^{−1}·K

^{−1}, e

_{s}is the vapor pressure, β is calculated by Equation (6). The β parameter is one of the weather parameters provided by GPT2w, whose solution requires the fitting of meteorological profile data at the station.

#### 2.2. Methods for Establishing the Local Tropospheric Fusion Model

#### 2.2.1. Tropospheric Fusion Modeling

_{i,j,k}are the ZTD observations at the station. The parameters to be solved are the ten fitting model coefficients, the Saastamoinen systematic bias, and the GPT2w systematic bias. The usual practice is to consider the systematic biases as model unknowns, and to estimate them together with the tropospheric fusion model coefficients using a least squares approach.

_{0}, a

_{1},……a

_{9}; δ

_{S}is the systematic bias between the Saastamoinen model and the fusion model, and δ

_{G}is the systematic bias between the GPT2w model and the fusion model. We assume that there is no systematic bias between the GNSS-based model and the fusion model.

#### 2.2.2. Precise Weights Determination

- The prior weights of the different observed values, i.e., the initial values of the weight of each type of observation (P
_{1}, P_{2}, …), are assigned. - A first adjustment is made, to obtain the values of ${V}_{i}^{T}{P}_{i}{V}_{i}$.
- In accordance with Equations (10) or (11) for the first-time variance component estimation, the first value of the unit weight variance ${\hat{\sigma}}_{{0}_{i}}^{2}$ of various observations is obtained, and then the weights are determined according to the following formula:$${\hat{P}}_{i}=\frac{c}{{\hat{\sigma}}_{{0}_{i}}^{2}{P}_{i}^{-1}}$$

#### 2.3. Data Description and Processing Strategy

## 3. Results and Discussion

#### 3.1. Verification of the Systematic Bias Estimation

#### 3.2. Verification of the Zenith Tropospheric Delay

#### 3.2.1. Active Troposphere Condition

#### 3.2.2. Quiet Troposphere Condition

#### 3.3. Impact of the dDstribution of Modeling Station Elevations on Model Precision

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Time series of daily average ZTD in Hong Kong and Systematic bias of Saastamoinen and GPT2w model in two cases.

**Figure 8.**(

**a**–

**c**) Differences (cm) between ZTDs provided by the fusion model and GNSS processing on the day 202/203/204 UTC 12:00.

Data source | Period | CORS Station | Weather Station | GPT2w |
---|---|---|---|---|

Grid Points | ||||

Hong Kong | 20 July 2015~26 July 2015 1 August 2015~7 August 2015 | 15 | 14 | 4 (1°×1°) 21.5°~22.5° 113.5°~114.5° |

DOY | Helmert | Comprehensive | ||
---|---|---|---|---|

Bias (cm) | RMS (cm) | Bias (cm) | RMS (cm) | |

201.00 | 0.00 | 1.33 | 0.04 | 1.28 |

202.00 | 0.35 | 1.84 | 0.22 | 1.73 |

203.00 | −0.40 | 1.10 | −0.28 | 1.13 |

204.00 | −0.23 | 1.05 | −0.06 | 1.05 |

205.00 | −0.06 | 1.58 | 0.22 | 1.59 |

206.00 | 0.58 | 1.98 | 0.53 | 1.93 |

207.00 | 0.82 | 2.02 | 0.70 | 1.55 |

213.00 | 0.97 | 2.86 | 0.60 | 2.19 |

214.00 | 0.55 | 1.82 | 0.50 | 1.53 |

215.00 | −0.11 | 1.31 | 0.13 | 1.21 |

216.00 | −0.28 | 1.30 | 0.15 | 1.30 |

217.00 | -0.62 | 1.97 | -0.16 | 1.41 |

218.00 | -0.23 | 1.40 | -0.06 | 1.10 |

219.00 | -0.12 | 2.71 | -0.18 | 1.38 |

Mean | 0.0871 | 1.7336 | 0.1679 | 1.4557 |

DOY | Fusion-Model | PPP-Model | GPT2w | SAAS | ||||
---|---|---|---|---|---|---|---|---|

bias | RMS | bias | RMS | bias | RMS | bias | RMS | |

201 | 0.04 | 1.28 | −0.19 | 1.43 | −8.59 | 8.66 | −13.02 | 13.10 |

202 | 0.27 | 1.82 | −0.44 | 1.43 | −6.55 | 6.59 | −11.08 | 11.22 |

203 | −0.29 | 1.15 | 0.14 | 0.81 | −7.12 | 7.17 | −8.90 | 9.04 |

204 | −0.06 | 1.04 | 0.20 | 0.89 | −6.52 | 6.57 | −8.67 | 8.78 |

205 | 0.22 | 1.60 | −0.93 | 2.03 | −3.99 | 4.11 | −6.10 | 6.41 |

206 | 0.52 | 1.91 | −0.55 | 2.01 | −1.83 | 2.27 | −4.75 | 5.11 |

207 | 0.70 | 1.56 | −0.05 | 1.51 | −0.01 | 1.03 | −2.37 | 2.90 |

mean | 0.20 | 1.48 | −0.26 | 1.44 | −4.94 | 5.20 | −7.84 | 8.08 |

DOY | Fusion-Model | PPP-Model | GPT2w | SAAS | ||||
---|---|---|---|---|---|---|---|---|

bias | RMS | bias | RMS | bias | RMS | bias | RMS | |

213 | 0.61 | 2.20 | −0.26 | 2.01 | 7.37 | 7.47 | 2.79 | 4.44 |

214 | 0.49 | 1.49 | −0.56 | 1.49 | 9.72 | 9.75 | 5.68 | 5.88 |

215 | 0.14 | 1.20 | −0.15 | 1.03 | 10.15 | 10.18 | 6.34 | 6.57 |

216 | 0.16 | 1.32 | 0.15 | 1.15 | 8.07 | 8.11 | 2.64 | 3.27 |

217 | −0.16 | 1.41 | 0.24 | 1.12 | 7.75 | 7.87 | 2.98 | 3.87 |

218 | −0.04 | 1.08 | −0.09 | 0.90 | 7.09 | 7.16 | 2.04 | 3.49 |

219 | −0.24 | 1.42 | −0.16 | 0.86 | 6.34 | 6.40 | 0.08 | 4.87 |

mean | 0.14 | 1.45 | −0.12 | 1.22 | 8.07 | 8.13 | 3.22 | 4.63 |

DOY | Site Involved in Modeling | HKNP | HKST | ||||||
---|---|---|---|---|---|---|---|---|---|

bias | RMS | bias | RMS | ||||||

201 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.90 m) | HKMW(194.94 m) | 2.10 | 2.75 | 1.29 | 1.64 |

202 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 3.91 | 4.68 | 2.34 | 2.75 |

203 | HKKS(44.69 m) | HKKT(34.54 m) | HKLT(125.89 m) | HKMW(194.94 m) | HKNP(350.67 m) | 0.27 | 0.37 | 0.85 | 1.43 |

204 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.41 | 0.44 | 1.44 | 2.11 |

205 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 2.83 | 3.04 | 2.17 | 2.40 |

206 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 3.91 | 4.71 | 2.60 | 2.90 |

207 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 1.89 | 3.44 | 2.60 | 2.82 |

213 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 3.74 | 5.61 | 3.26 | 3.49 |

214 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKMW(194.94 m) | 3.36 | 3.94 | 2.44 | 2.73 |

215 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.03 | 0.04 | 2.20 | 2.73 |

216 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.04 | 0.06 | 2.09 | 2.69 |

217 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.01 | 0.04 | 1.23 | 2.47 |

218 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.00 | 0.03 | 1.21 | 1.96 |

219 | HKKS(44.69 m) | HKKT(34.54 m) | HKLM(8.53 m) | HKLT(125.89 m) | HKNP(350.67 m) | 0.01 | 0.02 | -1.43 | 2.95 |

Station | Height (m) | DOY203 | |
---|---|---|---|

bias | RMS | ||

HKKS | 44.692 | 0.13 | 0.40 |

HKKT | 34.538 | −0.59 | 0.73 |

HKLT | 125.89 | 0.02 | 0.42 |

HKMW | 194.936 | 0.17 | 0.29 |

HKNP | 350.666 | 0.27 | 0.37 |

HKOH | 166.375 | 0.69 | 1.48 |

HKPC | 18.083 | −1.03 | 1.50 |

HKSC | 20.204 | −0.56 | 1.17 |

HKSL | 95.266 | 0.42 | 0.56 |

HKSS | 38.684 | −0.66 | 0.89 |

HKST | 258.687 | 0.85 | 1.43 |

HKTK | 22.497 | −1.09 | 1.35 |

HKWS | 63.762 | −0.08 | 0.52 |

T430 | 41.29 | -1.10 | 1.20 |

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**MDPI and ACS Style**

Yao, Y.; Xu, X.; Xu, C.; Peng, W.; Wan, Y.
Establishment of a Real-Time Local Tropospheric Fusion Model. *Remote Sens.* **2019**, *11*, 1321.
https://doi.org/10.3390/rs11111321

**AMA Style**

Yao Y, Xu X, Xu C, Peng W, Wan Y.
Establishment of a Real-Time Local Tropospheric Fusion Model. *Remote Sensing*. 2019; 11(11):1321.
https://doi.org/10.3390/rs11111321

**Chicago/Turabian Style**

Yao, Yibin, Xingyu Xu, Chaoqian Xu, Wenjie Peng, and Yangyang Wan.
2019. "Establishment of a Real-Time Local Tropospheric Fusion Model" *Remote Sensing* 11, no. 11: 1321.
https://doi.org/10.3390/rs11111321