# An ERA5-Based Hourly Global Pressure and Temperature (HGPT) Model

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## Abstract

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## 1. Introduction

## 2. HGPT Model Formulation

#### 2.1. Global Temperature and Pressure

#### 2.2. Zenith Hydrostatic Delay (ZHD)

#### 2.3. Weighted Mean Temperature (${T}_{m}$)

## 3. Data and Model Computation

^{9}simulations. Following the time-segmentation concept, we extracted from the 1-h temporal resolution time series the simulations at the same hour for each time series, obtaining 24 time series with a 24-h temporal resolution at each grid point. The linear coefficient values were obtained using a linear regression model. The amplitude and initial phase coefficients were obtained using a Fourier analysis. After detrending and removing the mean of the time series, a fast Fourier transform (FFT) was applied. The amplitudes were obtained after identifying the frequency that corresponded to each periodicity and the initial phase was given by the inverse tangent of the FFT. Both methods were applied at each hour and grid point (about 25 million) for temperature and pressure. All temperature coefficients were saved in an external binary file, but only the grid for a requested hour was loaded at every run of the HGPT subroutine to save time and improve efficiency. Likewise, the pressure coefficients were also saved in a binary file, and as for temperature, only the grid for the requested hour was loaded. The temperature and pressure coefficients obtained were referred to as the ERA5 surface elevation grid. The ERA5 vertical datum was the mean sea level and was based on the WGS84 Earth gravitational model (EGM 96) geoid [25,28]. The HGPT model uses the pressure and temperature lapse rate (see Xu [2], page 56) to convert pressure and temperature to the desired height. For GNSS data processing, we needed to consider that the GNSS heights are referenced to the WGS84 ellipsoid. To transform the GNSS ellipsoidal heights to the mean sea level system such that they become geoid-based, the following relation is used:

## 4. Results and Discussion

## 5. Accuracy Assessment

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Global distribution of mean coefficients for surface air temperature. The globe was divided into a 0.25° × 0.25° grid. (

**a**) $a$ mean values (K), (

**b**) $b$ mean values (trend) (K/day), (

**c**) mean annual amplitude contribution (K), (

**d**) mean semi-annual contribution (K), (

**e**) mean quarterly contribution (K); and (

**f**) mean impact of introducing a linear trend versus using only the mean temperature, in percentage.

**Figure 2.**Maximum spatial variability for each surface air temperature periodicity, which was calculated using a window of 4 × 4 grid points, approximately a 1° × 1° horizontal grid: (

**a**) maximum spatial variability for the annual amplitude (K), (

**b**) maximum spatial variability for the semi-annual amplitude (K), and (

**c**) maximum spatial variability for the quarterly amplitude (K).

**Figure 3.**Surface air temperature at two distinct locations (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°) for a one-year timespan (2018). The bluish shaded area represents the diurnal cycle. The dashed line represents the model daily mean and the continuous line represents the daily mean of all ERA5 simulations (24 time series, each one represents one hour).

**Figure 4.**Global distribution of mean coefficients for surface pressure. The globe was divided into a grid of 0.25° × 0.25°. (

**a**) $a$ mean values (hPa), (

**b**) $b$ mean values (trend) (hPa/day), (

**c**) contribution from the annual mean amplitude (hPa), (

**d**) contribution from the semi-annual mean amplitude (hPa), and (

**e**) mean impact of introducing a linear trend versus using only the mean pressure (%).

**Figure 5.**Maximum spatial variability for each surface pressure periodicity, which were calculated using a window of 4 × 4 grid points, approximately a 1° × 1° horizontal grid: (

**a**) maximum spatial variability for the annual amplitude (hPa) and (

**b**) maximum spatial variability for semi-annual amplitude (hPa).

**Figure 6.**(

**a**,

**b**) $\alpha $ and $\beta $ regression coefficients, obtained using 20 years of monthly ERA5 data, which were used to estimate the weighted mean temperature (${T}_{m}$) from the surface air temperature (${T}_{s}$). (

**c**) Pearson correlation coefficient between the ${T}_{m}$ obtained using ERA5 numerical integration at each grid point and ${T}_{s}$. (

**d**) RMSE between ${T}_{m}$ obtained using $\alpha $, $\beta $, and ${T}_{s}$, and the ${T}_{m}$ obtained using ERA5’s numerical integration (see Equation (5)). The estimation of the coefficients in the tropic regions could generate artefacts, especially over the sea, when ${T}_{s}$ did not vary over a range large enough to get a satisfactory estimation of the parameters.

**Figure 7.**Comparison of 20 years of ERA5 surface air temperature and pressure with the modeled values using Equations (1) and (2): (

**a**) mean RMSE for surface air temperature (K), (

**b**) mean RMSE for surface pressure (hPa), (

**c**) mean bias for surface air temperature (K), and (

**d**) mean bias for surface pressure (hPa). Note that a positive bias value indicates an hourly global pressure and temperature (HGPT) model overestimation and a negative bias value indicates an underestimation.

**Figure 8.**Global distribution of the mean RMSE values: (

**a**) mean RMSEs for surface air temperature at 00:00 UTC (K) and (

**b**) mean RMSEs for surface pressure at 00:00 UTC (hPa).

**Figure 9.**Accuracy spatial distribution for ${T}_{m}$ considering a global standard deviation of 1.1 K for surface air temperature and ignoring possible errors in $\alpha $ and $\beta $.

**Table 1.**Statistical analysis. RMSE and bias values (with standard deviation) between the HGPT model and radiosondes and the original signal (ERA5 simulations) and radiosondes, for 00:00 and 12:00 UTC. A negative bias indicates overestimation by the HGPT or ERA5 models and a negative bias indicates an underestimation.

00:00 UTC | 12:00 UTC | |||||||
---|---|---|---|---|---|---|---|---|

RMSE | Bias | RMSE | Bias | |||||

HGPT | ERA5 | HGPT | ERA5 | HGPT | ERA5 | HGPT | ERA5 | |

T (K) | 2.9 ± 1.6 | 1.7 ± 0.7 | 0.5 ± 2.1 | −0.1 ± 1.2 | 2.8 ± 1.5 | 1.6 ± 0.7 | 0.7 ± 2.0 | −0.2 ± 1.9 |

P (hPa) | 6.5 ± 2.5 | 3.8 ± 3.2 | −1.1 ± 3.8 | −0.7 ± 4.7 | 6.4 ± 3.1 | 3.7 ± 3.4 | −1.1 ± 4.1 | −0.9 ± 4.7 |

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**MDPI and ACS Style**

Mateus, P.; Catalão, J.; Mendes, V.B.; Nico, G.
An ERA5-Based Hourly Global Pressure and Temperature (HGPT) Model. *Remote Sens.* **2020**, *12*, 1098.
https://doi.org/10.3390/rs12071098

**AMA Style**

Mateus P, Catalão J, Mendes VB, Nico G.
An ERA5-Based Hourly Global Pressure and Temperature (HGPT) Model. *Remote Sensing*. 2020; 12(7):1098.
https://doi.org/10.3390/rs12071098

**Chicago/Turabian Style**

Mateus, Pedro, João Catalão, Virgílio B. Mendes, and Giovanni Nico.
2020. "An ERA5-Based Hourly Global Pressure and Temperature (HGPT) Model" *Remote Sensing* 12, no. 7: 1098.
https://doi.org/10.3390/rs12071098