# HGPT2: An ERA5-Based Global Model to Estimate Relative Humidity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. ERA5 Relative Humidity

^{9}simulations for the entire grid. We used the Wexler formulation to calculate the saturated water vapor pressure (${e}_{s}$) and the water vapor pressure ($e$). The Wexler formulations are based on more recent experimental data than most used formulations due to their numerical simplicity and have gained considerable international acceptance. The Wexler equation to compute the saturation vapor pressure over water is given by [35]:

- g
_{0}= −2.8365744 × 10^{3} - g
_{1}= −6.028076559 × 10^{3} - g
_{2}= 1.954263612 × 10^{1} - g
_{3}= −2.737830188 × 10^{−2} - g
_{4}= 1.6261698 × 10^{−5} - g
_{5}= 7.0229056 × 10^{−10} - g
_{6}= −1.8680009 × 10^{−13} - g
_{7}= 2.7150305

- k
_{0}= −5.8666426 × 10^{3} - k
_{1}= 2.232870244 × 10^{1} - k
_{2}= 1.39387003 × 10^{−2} - k
_{3}= −3.4262402 × 10^{−5} - k
_{4}= 2.7040955 × 10^{−8} - k
_{5}= 6.7063522 × 10^{−1}

- A
_{0}= −1.6302041 × 10^{−1} - A
_{1}= 1.8071570 × 10^{−3} - A
_{2}= −6.7703064 × 10^{−6} - B
_{3}= 8.5813609 × 10^{−9} - B
_{0}= −5.9890467 × 10^{1} - B
_{1}= 3.4378043 × 10^{−1} - B
_{2}= −7.7326396 × 10^{−4} - B
_{3}= 6.3405286 × 10^{−7}

- A
_{0}= −6.0190570 × 10^{−2} - A
_{1}= 7.3984060 × 10^{−4} - A
_{2}= −3.0897838 × 10^{−6} - A
_{3}= 4.3669918 × 10^{−9} - B
_{0}= −9.4868712 × 10^{1} - B
_{1}= 7.2392075 × 10^{−1} - B
_{2}= −2.1963437 × 10^{−3} - B
_{3}= 2.4668279 × 10^{−6}

#### 2.2. HGPT2 Relative Humidity

#### 2.3. HGPT2 Zenith Wet Delay

#### 2.4. HGPT2 Precipitable Water Vapor

#### 2.5. HGPT2 Height-Dependent Values Correction

^{−1}), we introduced the lapse rate latitudinal variation, given by [42]:

## 3. Results and Discussion

^{2}), which was approximately the 1° × 1° horizontal grid. Figure 2 shows the average of the standard deviation calculated for each amplitude. For the mean value, variations up to 28% can be found at the Andes and the west coast of the African continent, and about 20% on the west coast of the United States and Australian coasts, among other places, but with a greater incidence on the land-sea boundaries where complex atmospheric interactions occur (Figure 2a). The annual, semi-annual, and quarterly amplitudes display almost the same spatial pattern (Figure 2b–d). Although the semi-annual and quarterly amplitudes show higher variability in regions far from the coast, all three amplitudes show a higher variability incidence in the south of the Asian continent, mainly between the Himalayas and the north of the Taklamakan Desert, the African continent, and the west coast of the United States (Figure 2b–d). Combining all amplitudes, we found variations up to 48% in cells of 1° × 1° horizontal resolution, which are not considered by the other models in the literature, that can have a significant impact on the ZTD and ZWD estimation. Such variations justify the use of a full ERA5 spatial resolution for applications that require the highest precision in estimating relative humidity.

## 4. Accuracy Assessment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Global distribution of mean coefficients for relative humidity: (

**a**) mean values (%); (

**b**) mean annual amplitude contribution (%); (

**c**) mean semi-annual contribution (%); and (

**d**) mean quarterly contribution (%).

**Figure 2.**Maximum spatial variability for each relative humidity periodicity, calculated using a window of 4 × 4 grid points, approximately a 1° × 1° horizontal grid: (

**a**) mean values (%); (

**b**) annual amplitude (%); (

**c**) semi-annual amplitude (%); (

**d**) quarterly amplitude (%).

**Figure 3.**Relative humidity at two distinct locations with different climatological and topographical characteristics. In blue: over the Sahara Desert (λ = 0°, ϕ = 25°N), and in red: over the Amazon Rainforest (λ = 65°W, ϕ = 0°) during 1 year (2020). The bluish shaded area represents the diurnal cycle, obtained using Equation (7). The dashed line represents the model daily mean, and the continuous line represents the daily mean of ERA5 simulations.

**Figure 4.**Example of ZTD, ZWD, and PWV maps calculated on August 1, 2020, 12:00 UTC: (

**a**) ZTD obtained from the sum of the two components (ZHD and ZWD), in cm; (

**b**) ZWD obtained using the Saastamoinen wet model (see Equation (8)), in cm; and (

**c**) PWV calculated using the ZWD and a constant of proportionality ($\mathsf{\Pi},$ second term of Equation (9)), in mm.

**Figure 5.**Comparison of 1 year (2020) of ERA5 RH (modeled from surface dew point temperature, pressure, and temperature) with HGPT2 RH; (

**a**) mean RMSE values (in %); (

**b**) mean bias values, positive (negative) values indicate an overestimation (underestimation) of the HGPT2 model (in %); and (

**c**) mean Pearson correlation coefficient values.

**Figure 6.**Comparison of 1 year (2020) of ERA5 PWV (calculated by ECMWF) with HGPT2 PWV; (

**a**) mean RMSE values (in mm); (

**b**) mean bias values, positive (negative) values indicate an overestimation (underestimation) of the HGPT2 model (in mm); (

**c**) mean Pearson correlation coefficient values; and (

**d**) mean RMSE values obtained by comparing 5 years of MODIS monthly data and HGPT2 estimates.

**Figure 7.**Weight-root-mean-square (WRMS) variation between the two runs (with and without HGPT2) for the station height. Green circles show stations with a positive impact, red circles stations with a negative impact, and black squares stations with no impact. The circle size indicates the magnitude of the impact: negative for values from −0.8 to −0.1 mm, and positive values from 0.1 to 0.7 mm.

**Table 1.**Mean RMSE, bias, and correlation coefficient (ρ) for GPT2w, GPT3 (two grid resolution), and HGPT2, calculated using 282 met-files with RH observations for 1 year (2020), hourly data.

GPT2W (5° × 5°) | GPT2W (1° × 1°) | GPT3 (5° × 5°) | GPT3 (1° × 1°) | HGPT2 (0.25° × 0.25°) | |
---|---|---|---|---|---|

RMSE (%) | 20.38 | 19.47 | 20.41 | 19.42 | 16.01 |

BIAS (%) | 2.65 | 1.71 | 2.65 | 1.67 | 1.42 |

ρ [−1, 1] | 0.26 | 0.28 | 0.25 | 0.27 | 0.61 |

**Table 2.**Processing main parameters in the GAMIT/GLOBK software. The main difference is the source of the meteorological observation. The first run uses the default options; in the second run, the surface pressure, temperature, and relative humidity at each GNSS station location is provided by the HGPT2 model through met-files. Each run has data from the 70 days selected. The bold option indicates the differences between runs.

FIRST RUN | SECOND RUN | |
---|---|---|

GNSS system | GPS | GPS |

Reference system | ITRF14 | ITRF14 |

Orbits | IGS Final | IGS Final |

Met obs source | GPT3 (1° × 1°) | Met-files (HGPT2) |

Ionospheric model | L3 ^{1} | L3 ^{1} |

Atmospheric tidal loading | Observation level ^{2} | Observation level ^{2} |

Ocean tide loading | FES2004 model ^{3} | FES2004 model ^{3} |

Dry mapping function | GMF ^{4} | GMF ^{4} |

Wet mapping function | GMF | GMF |

Elevation cutoff angle | 10° | 10° |

Interval zenith delays | every 2-hours | every 2-hours |

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**MDPI and ACS Style**

Mateus, P.; Mendes, V.B.; Plecha, S.M.
HGPT2: An ERA5-Based Global Model to Estimate Relative Humidity. *Remote Sens.* **2021**, *13*, 2179.
https://doi.org/10.3390/rs13112179

**AMA Style**

Mateus P, Mendes VB, Plecha SM.
HGPT2: An ERA5-Based Global Model to Estimate Relative Humidity. *Remote Sensing*. 2021; 13(11):2179.
https://doi.org/10.3390/rs13112179

**Chicago/Turabian Style**

Mateus, Pedro, Virgílio B. Mendes, and Sandra M. Plecha.
2021. "HGPT2: An ERA5-Based Global Model to Estimate Relative Humidity" *Remote Sensing* 13, no. 11: 2179.
https://doi.org/10.3390/rs13112179