The ERA5 atmospheric reanalysis data, provided by ECMWF (https://cds.climate.copernicus.eu/datasets, last access: 23 September 2025), is the fifth-generation global climate reanalysis dataset. This dataset provides hourly surface-level parameters and pressure-level data with a horizontal resolution of 0.25°×0.25° (latitude×longitude) and a vertical resolution of 37 levels. ERA5 data can provide high-resolution and relatively complete surface-level and pressure-level data and are thus expected to be widely used in the future. The radiosonde station data can be downloaded for free from the University of Wyoming (http://weather.uwyo.edu/upperair/sounding.shtml, last access: 23 September 2025). This product provides meteorological layered data and surficial parameters such as PWV from the ground to the near-Earth space (an altitude of approximately 30 km) and provides radiosonde data twice a day (00:00 and 12:00 UTC); these data are often used as reference values for model verification tasks. The ERA5 gridded data from 2012 to 2017 and the radiosonde data in 2017 on the global scale were used to analyse and develop the model in this study.

Tm is the key parameter used to convert ZWD into PWV. Using atmospheric reanalysis data, radiosonde data and other data, highly accurate Tm information can be obtained by integral calculation. In addition, the modelling method can also obtain Tm values at a high accuracy and with a high calculation efficiency. The specific Tm integral calculation formula is expressed as follows: 1Tm=∫hbothtopeTdH∫hbothtopeT2dH, where hbot and htop are, respectively, the heights at the bottom and top of the integration calculation, Tm is the atmospheric weighted mean temperature at hbot, e is the water vapour pressure (hPa), T is the temperature (K) and H is the height (m). The height corresponding to Tm is denoted by hbot. Different values of hbot correspond to different values of Tm. Therefore, the layered Tm can be calculated using Eq. (1).

The modelling methods used to calculate Tm can be divided into two categories: (1) Tm models based on measured meteorological parameters, of which the most representative is the Bevis model (Tm=70.2+0.72Ts), namely, the Tm linear regression model, and (2) non-meteorological parameter Tm models, of which the most classical is the GPT series model. The GPT3 model, the latest model in the GPT series, has a high accuracy in calculating global Tm. The GPT3 model formula used to calculate Tm can be expressed as follows: 2TmGPT3=a0+a1cos⁡2πDOY365.25+b1sin⁡2πDOY365.25+a2cos⁡4πDOY365.25+b2sin⁡4πDOY365.25, where TmGPT3 denotes Tm calculated by the GPT3 model, a0 denotes the average annual value of Tm, a1 and b1 denote the annual cycle coefficient of Tm, a2 and b2 denote the semiannual cycle coefficient of Tm and DOY denotes the day of the year.

PWV refers to the total water vapour content of a vertical column per unit area in the atmosphere and can be converted from ZWD using the following formula: 3PWV=Π×ZWD, where Π denotes the PWV conversion factor. This conversion factor can be expressed as follows: 4Π=106ρwRv(k2′+k3/Tm), where Rv denotes the water vapour gas constant, k2′ and k3 are constants (k2′=22.97 KhPa-1 and k3=375463 K2hPa-1) and the other parameters are described above. Therefore, Tm is the key parameter in the GNSS PWV inversion.

To facilitate the subsequent test of the accuracy of the Tm values calculated using the atmospheric reanalysis data and the performance of the new Tm model, this study uses the bias and root-mean-square error (RMSE) as the accuracy evaluation indicators. The formulas are expressed as follows: 5bias=1N∑i=1N(TmMi-TmRi),6RMSE=1N∑i=1N(TmMi-TmRi)2, where N denotes the number of samples, TmMi denotes the calculated value of the atmospheric reanalysis data or model and TmRi denotes the reference value.

Given the discernible variations in topography and the significant range of height at a global scale, there can be considerable disparities between the height of atmospheric reanalysis data grid points and the actual heights of the ground-based GNSS receiver (named target height), so it is necessary to adjust Tm vertically. The previous study has analysed this topic and concluded that there is an approximately linear relationship between the layered Tm data and hbot (Huang et al., 2019). To further analyse the variation in Tm with height, six representative ERA5 reanalysis data grid points ((60° N, 90° W), (60° N, 90° E), (0°, 90° W), (0°, 90° E), (60° S, 90° W) and (60° S, 90° E)) were selected globally to analyse the layered Tm data and corresponding height data on 1 January 2017. The results are shown in Fig. 1.

Figure 1 shows that the layered Tm data of six representative ERA5 reanalysis data grid cells exhibit approximate linear change relationships with height. Moreover, the layered Tm data gradually decrease with increasing height. Therefore, the slope of the fitting line represents the vertical adjustment rate of Tm, and this relation can be expressed as

7Tm=γ×δh+l, where γ denotes the vertical adjustment rate of Tm, δh denotes the height difference and l denotes a constant.

To investigate the variation relationship between the vertical adjustment rate of Tm and time at the global scale, six representative ERA5 reanalysis data grid points were selected to calculate the vertical adjustment rate of Tm from 2012 to 2016. Furthermore, the time series for the vertical adjustment rate of the daily mean Tm from 2012 to 2016 was obtained and used to achieve seasonal fitting by the cosine function of the annual and semiannual periods. The results are shown in Fig. 2.

Figure 2 shows the obvious seasonal variations in the vertical adjustment rate of Tm calculated using the ERA5 reanalysis data at six representative grid points. From Fig. 2c and d, the annual and semiannual variations in the vertical adjustment rate of Tm are relatively slight at the grid points located on the Equator. However, From Fig. 2e and f, the vertical adjustment rate of Tm at the grid points located in the high-latitude areas of the Southern Hemisphere exhibit relatively large variation ranges and show obvious annual and semiannual variations, whereas those in the high-latitude areas of the Northern Hemisphere show slight variation ranges and obvious annual and semiannual cycle variations from Fig. 2a and b. The main reason for these results is that most of the high-latitude areas of the Southern Hemisphere are oceans and are thus not affected by complex climates.

Hence, a clear seasonal pattern is evident in the vertical adjustment rate of Tm, and the variation patterns vary across different regions. The vertical adjustment rate of Tm was then calculated with a temporal resolution of 1 h from 2012 to 2016 at the global scale. The annual mean value, annual cycle amplitude, semiannual cycle amplitude and daily cycle amplitude of the vertical adjustment rate of Tm were calculated by using Eq. (8) at selected grid points at the global scale. The utilised formula is expressed as follows: 8γ=A0+A1cos⁡2πDOY365.25+A2sin⁡2πDOY365.25+A3cos⁡4πDOY365.25+A4sin⁡4πDOY365.25+A5cos⁡2πHOD24+A6sin⁡2πHOD24, where γ is the vertical adjustment rate of Tm; A0 is the annual mean value of the vertical adjustment rate of Tm; (A1, A2) are the annual cycle coefficients for the vertical adjustment rate of Tm; (A3, A4) are the semiannual cycle coefficients for the vertical adjustment rate of Tm; (A5, A6) are the daily cycle coefficients for the vertical adjustment rate of Tm; DOY is the day of the year; and the hour of the day (HOD) is the UTC time. The above coefficients were calculated at each grid point based on a least-square adjustment by using all selected grid points in the world from 2012 to 2016. The results are shown in Fig. 3.

As shown in Fig. 3, a strong correlation was found between the annual mean Tm vertical adjustment rate and latitude. Regarding the annual cycle amplitude of the vertical adjustment rate of Tm, obvious annual cycle amplitude values were observed in most land areas, especially over the Antarctic continent, though these amplitudes were relatively small in the ocean and coastal areas located in the middle and low latitudes of the Northern and Southern Hemispheres. In addition, a sea–land difference was observed in the semiannual cycle amplitude of the Tm vertical adjustment rate. The daily cycle amplitude of the vertical adjustment rate of Tm remained at approximately 0.06 Kkm-1. Since the daily variation in the vertical adjustment rate of Tm can be overshadowed by the annual and semiannual variations, we focused on optimising and simplifying the model coefficients when developing the Tm vertical adjustment rate model.

The above analysis demonstrated that the variation law of the vertical adjustment rate of Tm differs spatially. This makes it difficult to accurately grasp the variation law of the vertical adjustment rate of Tm in developing a global uniform model for the vertical adjustment rate of Tm. Therefore, we presents a solution to the issue of coefficient redundancy that can occur when developing a model from individual grid points. Specifically, a sliding-window algorithm was introduced to develop the Tm vertical adjustment rate model, leading to optimised coefficients and improved accuracy, stability and applicability of the model. Note that the sliding-window algorithm has been used in the previous study, which exhibits a superior performance (Huang et al., 2019).

The ERA5 reanalysis data with a horizontal resolution of 0.25°×0.25° were selected as the data source to develop the model in this study. We divided global segments into regular windows with the same horizontal resolution as the ERA5 reanalysis data. The specific process was as follows: starting from the first window, by using the data of nine gridded points in each window, the model coefficients of the corresponding window were calculated in order from west to east and from north to south and stored at the geometric centre of the corresponding window. Finally, all the coefficients for the global Tm vertical adjustment rate model were obtained. As shown in Fig. 4, blue dots denote ERA5 gridded points, and red dots denote centre points of the window; the red rectangle denotes the size of the sliding window.

To investigate the influence of the window size on the model precision and optimise the model coefficients as much as possible, three different window sizes, with resolutions of 0.5°×0.5°, 1°×1° and 2°×2°, were selected to develop the model. The window with the resolutions of 0.5°×0.5°, 1°×1° and 2°×2° contains 9, 25 and 49 gridded points, respectively. As mentioned above, it was necessary to consider the characteristics of the annual and semiannual cycles when developing the model. Therefore, the formula of the global Tm vertical adjustment rate model in each window can be expressed as follows: 9γi=A0i+A1icos⁡2πDOY365.25+A2isin⁡2πDOY365.25+A3icos⁡4πDOY365.25+A4isin⁡4πDOY365.25, where i is the number of windows; γi is the vertical adjustment rate of Tm in the ith window; A0i is the annual mean value of the vertical adjustment rate of Tm in the ith window; (A1i, A2i) are the annual cycle coefficients of the vertical adjustment rate of Tm in the ith window; (A3i, A4i) are the semiannual cycle coefficients of the vertical adjustment rate of Tm in the ith window; DOY is the day of the year. We also use 10TmT=TmS-γi(HT-HS), where TmT is the Tm value at the target height; TmS is the Tm value at the surface; γi is the vertical adjustment rate of Tm at the window where the target point is located; HT is the height of the target point; HS is the height at surface.

The five coefficients required in the Tm vertical adjustment rate model in all windows of the world were calculated by the least-squares adjustment using ERA5 analysis. Then, the above coefficients were stored at the geometric centres of the windows with resolutions of 0.5°×0.5°, 1°×1° and 2°×2°. Finally, a global real-time and high-precision Tm vertical adjustment rate model was developed and named NGGTm-H (this model contains three models with different resolutions: NGGTm-H1, NGGTm-H2 and NGGTm-H3). The Tm vertical adjustment was calculated by combining Eqs. (9) and (10) and using the position and DOY.

To validate the precision and applicability of the spatial interpolation method using the NGGTm-H model at the global scale, the Tm data collected at 378 radiosonde stations around the world in 2017 were used as reference values. The altitude of radiosonde stations ranges from 0 to 4500 m, mostly within 2000 m. The integrated surface Tm data at four grid points were adjusted to the heights of the radiosonde stations. Then, the adjusted Tm values at these four grid points were interpolated to the positions of the radiosonde stations using the inverse distance-weighted method. The statistical results of the bias and RMSE values of the spatially interpolated Tm values from all radiosonde stations are listed in Table 1.

From Table 1, as the resolution of the model increased, the mean bias of the NGGTm-H model gradually decreased. The mean bias of the NGGTm-H1 model was smallest, at 0.12 K. Compared to those of the NGGTm-H2 model and the NGGTm-H3 model, the mean bias of the NGGTm-H1 model was reduced by only 0.02 and 0.03 K, respectively. Positive mean biases with relatively small absolute values were obtained for the NGGTm-H model at the three resolutions taking Tm data from radiosonde stations as reference values. The main reason for these results was that the majority of radiosonde stations are located in land areas. The vertical adjustment values of Tm obtained using the NGGTm-H model were slightly larger in land areas but smaller in marine areas than the reference values. However, a small number of radiosonde stations distributed in marine areas was susceptible to the influence of marine climate, resulting in the vertical adjustment values of the model being apparently smaller than the reference values. Therefore, the positive biases were smaller than the absolute value of the negative biases. In addition, the precision of the NGGTm-H1 model showed the best result with a mean RMSE of 1.18 K. Thus, NGGTm-H1 model had a certain improvement compared to the NGGTm-H2 and NGGTm-H3 models.

The NGGTm-H model was described in Sect. 3, it can vertically adjust the given Tm at the surface to the target height. In order to directly obtain Tm at any height, it is necessary to develop a Tm model whose hbot is at the surface (named as surface Tm model). Analysing the spatiotemporal characteristics of Tm is crucial for developing Tm models. Relevant studies have shown that Tm undergoes diurnal variations (Sun et al., 2019). To further analyse the temporal characteristics in Tm in depth at the global scale, we calculated the annual mean value, annual cycle amplitude, semiannual cycle amplitude, daily cycle amplitude and semidaily cycle amplitude at all grid points using the least-squares adjustment using surface gridded Tm data calculated from all the ERA5 reanalysis data recorded from 2012 to 2016 worldwide. The results are shown in Fig. 5.

As shown in Fig. 5, strong correlations were found between the annual mean Tm value and latitude and between the annual Tm cycle amplitude and latitude. The semiannual cycle amplitude of Tm also exhibited a certain correlation with latitude, and a certain sea–land difference was observed. In summary, Tm not only undergoes significant annual and semiannual variations but also experiences significant daily and semidiurnal variation.

As mentioned above, it was necessary to consider the time-varying vertical adjustment rate and detailed temporal variations when developing high-precision global Tm models. Therefore, a new global Tm model considering time-varying vertical adjustment rate was developed which used the integrated surface Tm of ERA5 reanalysis recorded from 2012 to 2016 on the basis of NGGTm-H1 model. Because of the significant variations in the horizontal direction of Tm compared to vertical adjustment rate according to Figs. 5a and 3a, it was necessary to develop surface Tm models at each gridded point instead of using sliding windows. This new Tm model is a hybrid-grid model, as the surface Tm model was developed at each gridded point and the NGGTm-H1 model was developed at the geometric centre of the sliding window. The model formulae are expressed as follows: 11TmG=B0+B1cos⁡2πHOD24+B2sin⁡2πHOD24+B3cos⁡4πHOD24+B4sin⁡4πHOD24,12Bi=bi0+bi1cos⁡2πDOY365.25+bi2sin⁡2πDOY365.25+bi3cos⁡4πDOY365.25+bi4sin⁡4πDOY365.25, where TmG is the Tm value at the gridded points; Bi is the daily variation coefficient of Tm; HOD is the UTC time. After Eq. (12) was used to expand Eq. (11), bij (i=0,1,2,3,4 and j=0,1,2,3,4), which represents the 25 coefficient terms of the model, was calculated. DOY is the day of the year.

The 25 model coefficients were calculated by the least-squares adjustment at all global reanalysis data grid points, which used the surface gridded Tm data with a temporal resolution of 1 h. The above coefficients were stored at the grid points with a horizontal resolution of 0.25°×0.25°. Finally, the NGGTm model considering time-varying vertical adjustment rate was developed. The input parameters for this model are location and time only, which makes it convenient for users. Here, we introduce the use of NGGTm. First, users need to find the window where target points are in, extract the five model coefficients of Eq. (9) for the centre point of the window, and input the DOY to calculate γ. Second, users need to find the four surrounding grid points, extract 100 model coefficients of the surface Tm model (Eqs. 11 and 12) for four grid points, and input HOD and DOY into Eqs. (11) and (12) to calculate the Tm at surface. Then values of Tm at the surface are vertically adjusted to the height of the target point using Eq. (10). Finally, Tm at the target point is obtained using inverse distance-weighted interpolation using the four adjusted Tm.

In this section, to validate the accuracy of the new model, The NGGTm model was used to calculate the Tm values at all of the grid points at the global scale, which compared with the Bevis and GPT3 model. The surface gridded Tm data with a temporal resolution of 1 h derived from the ERA5 reanalysis data in 2017 were selected as reference values. We defined the GPT3 model with two horizontal resolutions of 1°×1° and 5°×5° as GPT3-1 and GPT3-5, respectively, which makes it convenient to describe. The Ts data required by the Bevis model to calculate Tm were derived from the GPT3-1 model. The statistical results are listed in Table 2, and shown in Figs. 6 and 7.

From Table 2, it can be seen that the mean bias of the Bevis model was 0.66 K, which indicated that the Tm values calculated by the Bevis model were all larger than the reference values. The mean biases of the GPT3-5 model and the GPT3-1 model were -0.30 and -0.28 K, respectively, which demonstrated that the Tm values calculated by the GPT3 model were slightly smaller than the reference values. The mean bias of the NGGTm model was only -0.09 K, which was the smallest absolute mean bias value among all the analysed models. This result shows that the Tm values calculated by this model were close to the reference values overall, which demonstrated that the NGGTm model performed better than the other models. In terms of the variation ranges of the bias, the bias variation range of the GPT3-1 model shows improvement compared to that of the GPT3-5 model, which had the largest bias variation range. The main reason for the above results may be that the GPT3 model did not consider the influence of height in its calculation of Tm, which resulted in the relatively large bias in the calculated Tm values in high-elevation areas. The variation range of the bias for the GPT3-1 model was smaller than that of the GPT3-5 model, which indicated that improving the model resolution can help improve the stability of the model. Compared with the Bevis model, GPT3-5 model and GPT3-1 model, the variation range of the bias of the NGGTm model was extremely small, ranging from -1.35 to 1.59 K, which indicated that the stability of the NGGTm model was better than those of the other analysed models. In addition, the mean RMSE of the NGGTm model was only 2.84 K, which exhibited improvements of 0.5, 0.18 and 0.06 K over the Bevis model, the GPT3-5 model and the GPT3-1 model, respectively. These results show that the Tm values calculated by the NGGTm model had the highest precision among all analysed models. In terms of the variation ranges of RMSE, the variation ranges of RMSE for the GPT3-5 model and GPT3-1 model were larger than those of the other models. The RMSE variation range of the GPT3-1 model was smaller than that of the GPT3-5 model. Compared with the other models, the RMSE of the NGGTm model had the smallest variation range, ranging from 0.72 to 6.33 K, which demonstrated that the precision and stability of the NGGTm model were better than those of the other analysed models.

Figure 6 shows that the absolute bias values were relatively small for the Bevis model in the mid-latitude areas. The main reason for this result may be the Bevis model was developed based on radiosonde data in North America. Larger absolute bias values were observed for the GPT3-5 model in relatively high-elevation areas, such as the Qinghai-Tibet Plateau, western South America, and parts of Antarctica. The main reason for this result may be that the GPT3 model did not take any Tm vertical adjustment into account. The absolute bias values of the GPT3-1 model were smaller than those of the GPT3-5 model in most parts of the world. Although relatively large absolute bias values were still shown for the GPT3-1 model in relatively high-elevation areas, a significant improvement can be seen. Therefore, the performance and stability of the model can be significantly improved by increasing the resolution of the model. The bias of the NGGTm model remained at approximately 0 K, significantly better than those of the other models. In conclusion, the NGGTm model shows excellent stability and applicability at the global scale.

From Fig. 7, relatively large RMSEs obtained for the Bevis model are shown in some areas, which indicated that the Bevis model performed poorly in the Qinghai-Tibet Plateau, northeastern Asia, the coasts of western and southwestern Africa, the Arctic Ocean and Antarctica. Relatively large RMSEs obtained for the GPT3-5 model are shown in western North America, western South America and the Qinghai-Tibet Plateau. The GPT3-1 model had a certain accuracy improvement compared with the GPT3-5 model in these areas. Furthermore, increasing the resolution of the model can improve the model precision of calculating Tm. The NGGTm model had small RMSEs around the world, which demonstrated higher precision especially at low latitudes. The NGGTm model performed significantly better than the Bevis model, the GPT3-5 model and GPT3-1 model.

To further validate the performance of the NGGTm model, the Tm data from 378 radiosonde stations around the world in 2017 were selected as reference values. The precision of the NGGTm model when calculating Tm at these stations was validated and compared with the other three models. The Ts data required by the Bevis model to calculate Tm were derived from the radiosonde stations. The statistical results are listed in Table 3, and shown in Figs. 8 and 9.

From Table 3, the mean bias of the NGGTm model was only 0.10 K, with the smallest absolute value among the analysed models. The bias range of the NGGTm model was also the smallest, ranging from -4.31 to 3.78 K, which demonstrated that the NGGTm model performed better than the other models. In addition, the mean RMSE of the NGGTm model was only 3.30 K, which exhibited improvements of 0.27 K (8 %), 0.35 K (11 %) and 0.18 K (5 %) over the Bevis model, GPT3-5 model and GPT3-1 model, respectively. The RMSE range of the NGGTm model was the smallest, ranging from 0.99 to 5.17 K, indicating that the NGGTm model had the best precision and stability at the global scale.

From Fig. 8, the Bevis model showed relatively obvious negative biases in low latitudes and obvious positive biases in middle and high latitudes, with a trend of increasing absolute biases from low latitudes to high latitudes. The GPT3-5 model and GPT3-1 model performed similarly, with relatively large absolute bias values on the Qinghai-Tibet Plateau and in western North America because the GPT3 model did not consider the relationship between Tm and height. The absolute bias values of the NGGTm model were relatively small at the global scale, with values of approximately 0 K. These results demonstrated that the stability of the NGGTm model was better than those of the other analysed models at the global scale.

Figure 9 shows all models exhibited relatively small RMSEs at low latitudes and relatively large RMSE values at high latitudes, with a trend of increasing RMSEs from low latitudes to high latitudes. The main reason for this result may be the seasonal variation in Tm is strengthened with increasing latitude. In addition, the GPT3-5 model showed relatively large RMSEs at a few radiosonde stations on the Qinghai-Tibet Plateau, whereas the GPT3-1 model exhibited a certain improvement for the reasons mentioned above. The NGGTm model still had a significant improvement compared with other models at the global scale, which demonstrated that the NGGTm model had the best precision and stability.

Since there are strong correlations between Tm and both elevation and latitude, to further analyse the relationship between the precision of Tm calculated by the four models and the elevation variation, the 378 radiosonde stations around the world were divided into five intervals with an elevation span of 500 m for each interval. The bias and RMSE values at these 378 radiosonde stations were then calculated according to the above intervals. The results are shown in Fig. 10.

From Fig. 10, the Bevis model exhibited a positive correlation between bias and elevation. The main reason for this result may be that the Bevis model was developed by using radiosonde data collected in low-elevation North America, leading to the poor applicability in relatively high-elevation areas. The GPT3-5 model and GPT3-1 model exhibited negative biases in all elevation intervals. Whereas the NGGTm model exhibited relatively small absolute biases in all elevation intervals, especially those below 2000 m. Therefore, the NGGTm model exhibited extremely significant stability in all elevation intervals compared with other models at the global scale. In addition, the Bevis model showed smaller RMSEs than the GPT3-5 model at elevations below 1500 m. The RMSEs of the GPT3-1 model were smaller than those of the GPT3-5 model at all elevation intervals, which further indicated that increasing the resolution of the model can improve the precision and stability of the results. The RMSEs of the NGGTm model were smaller than those of the Bevis model, GPT3-5 model and GPT3-1 model in all elevation intervals. In conclusion, the NGGTm model showed the best precision and stability compared with the other analysed models in all elevation intervals.

To further analyse the relationship between the precision of four models and the latitude variation, the 378 radiosonde stations around the world were divided into several intervals with a latitude interval of 15°. Few radiosonde stations are located at high latitudes. The high-latitude areas in the Northern and Southern Hemispheres were divided into intervals with latitude intervals of 15°. The bias and RMSE values corresponding to the 378 radiosonde stations were calculated according to the above intervals. The results are shown in Fig. 11.

From Fig. 11, the Bevis model obtained relatively large absolute biases in most latitude ranges, which exhibited significantly positive biases in high latitudes and significantly negative biases in low latitudes. The GPT3-5 and GPT3-1 models exhibited negative biases with relatively small absolute values at most latitudes and negative biases with relatively large absolute values in the high-latitude areas of the Southern Hemisphere, which can be observed especially for the GPT3-5 model. The NGGTm model exhibited small absolute biases in all latitude ranges. In addition, all models showed small RMSEs in low-latitude areas but relatively large RMSEs in high-latitude areas. The RMSEs gradually increased with increasing latitude for all models. Compared to the other models, the NGGTm model had the best accuracy in all latitude ranges, especially in the high-latitude areas of the Southern Hemisphere. In summary, the NGGTm model showed a high accuracy and stability for calculating Tm at all latitudes.