A Grid Model of Direct Conversion between Zenith Tropospheric Delay and Precipitable Water Vapor in Tropical Regions

10 To obtain high-accuracy post and real-time precipitable water vapor (PWV) with simple process when 11 the measured meteorological parameters are unavailable, a grid direct conversion model of zenith 12 tropospheric delay (ZTD) and PWV in tropical regions (named CZP) is proposed with the consideration 13 of the characteristics of spatiotemporal changes based on ERA5 reanalysis data from 2016 to 2019. The 14 results show that the CZP model has good performance at each grid points compared with ERA5 PWV. 15 The comparison results with the GNSS PWV of 61 IGS stations in tropical regions show that the mean 16 bias and root mean square (RMS) of CZP GNSS PWW are no more than 1.1 mm and 1.4 mm, respectively.


Introduction
Atmospheric water vapor plays a crucial role in the hydrological cycle and energy transfer process (Trenberth et al. 2005).Water vapor exhibits higher spatial-temporal variability than other greenhouse gases and can cause changes in surface and air temperatures by absorbing and releasing long-wave radiation, having an impact on the global warming, La Nina, and other climate phenomena.Therefore, real-time and accurate PWV information is an important prerequisite for climate change research and numerical weather forecasting (Zhao et al. 2020).
At present, the methods of monitoring PWV include radiosonde (RS), solar photometer, water vapor radiometer (WVR), and infrared remote sensing (Zhang et al. 2018).Radiosonde observations can provide high-accuracy PWV and are generally used to verify the PWV values retrieved by other methods, but its temporal resolution is low and cannot be used for real-time water vapor monitoring (Trent et al. 2019).Although solar photometer and WVR show PWV values with high temporal resolution and high accuracy, they are too expensive and easily damaged to be promoted in practical applications (Sobrino et al. 2008).Infrared remote sensing can obtain continuous and wide-area PWV information, but it is susceptible to cloud cover and rainfall.In addition, the reanalysis data have been produced based on the data described above and can provide decades of stable water vapor data across the globe (Zhang et al. 2018;Zhang et al. 2019).The most commonly used one is ERA5 released by the European Center for Medium-Range Weather Forecasts (ECMWF).The feasibility and high accuracy of ERA5 PWV have been demonstrated by many scholars (Zhang et al. 2013;Wang et al. 2020;Huang et al. 2021;Ren et al. 2022;Wang et al. 2022).Compared with other means of obtaining PWV, GNSS technology shows many advantages such as high accuracy, high temporal resolution, low cost, long-term stability and continuity and it has been widely applied in extreme weather forecasting and real-time water vapor monitoring (Bevis et al. 1992;Hagemann et al. 2003;Zhang et al. 2019;Zhao et al. 2022;Zhang et al. 2019).
In conventional GNSS PWV calculation, measured surface pressure (Ps) and surface temperature (Ts) are needed to calculate the zenith hydrostatic delay (ZHD) and conversion coefficient.Unfortunately, most GNSS stations are not equipped with meteorological sensors.To overcome the lack of meteorological parameters, many scholars have used alternative methods.One approach is to use empirical meteorological models, such as the global pressure and temperature (GPT) series models and the global weighted mean temperature (GWMT) model and so on (Boehm et al. 2007;Yao et al. 2012;Böhm et al. 2015;Landskron and Böhm 2018), but the accuracies of these models are limited by the inaccurate characterization of the spatial-temporal variation of the parameters and using multiple empirical models simultaneously leads to error accumulation.Another approach is to directly interpolate pressure and temperature from atmospheric reanalysis products (Vey et al. 2009;Zhang et al. 2017).
However, the meteorological parameters of the reanalysis data need to be pre-downloaded, as a result, the storage burden is heavy and the process is complicated.Meanwhile, this approach is difficult to be applied in real-time water vapor calculations.
Therefore, scholars have tried to explore a better method-establishing a high-precision direct conversion model of ZTD and PWV.Wang et al. (2007) analyzed the correlation between GNSS ZTD and PWV and established a model for directly estimating PWV by ZTD in Wuhan.The accuracy of the model-derived PWV at the WHDH station is verified by GNSS PWV and RS PWV with the RMS of 2.23 mm and 4.45 mm, respectively.Yi et al. (2017) conducted a correlation analysis on the ZTD and PWV at TWTF station and established a year-round and seasonal conversion model between ZTD and PWV.The RMS of the model tested using both precipitation and non-precipitation data from each season is less than 1.5 mm.Although many efforts have been made to obtain high-accuracy PWV when measured meteorological data is unavailable, the above two alternative solutions still have limitations and existing conversion models can only be applied to specific small areas and lack consideration of time-varying characteristics.The water vapor in tropical regions is characterized by high content and changes rapidly, which plays an important role in global climate change and extreme weather forecasting.Therefore, this study builds a grid model of direct conversion between ZTD and PWV over tropical regions (called CZP model) with the resolution of 0.25° × 0.25° fully considering the temporal variation and differences in spatial distribution using ERA5 hourly products from 2016 to 2019.
The paper is organized as follows.The data and methods are described in Section 2. The construction procedure of CZP model is introduced in Section 3. The evaluations of CZP are presented in Section 4. Followed by the conclusions in Section 5.

Data Description
ERA5 is the fifth generation of ECMWF reanalysis dataset, which can provide global surface and atmospheric temperature, pressure and other grid products with a temporal resolution of 1-hour and a horizontal resolution of 0.25° × 0.25° from 1940 to present (Hersbach et al. 2020;Jiang et al. 2020).In this paper, the temperature, pressure, specific humidity and geopotential height products of the surface and all pressure levels in tropical regions from 2016 to 2019 with a horizontal resolution of 0.25 ° × 0.25 ° and a temporal resolution of 12 h (00:00 12:00 UTC) provided by ERA5 reanalysis data are used for modeling.The single-level PWV product directly provided by ERA5 is also used for the validation of the CZP model.In addition, the final ZTD products supplied by the IGS are utilized to retrieve GNSS PWV and verify the accuracy of real-time ZTD.The radiosonde data are from the University of Wyoming and the temporal resolution is 12-hour.The distribution of 61 GNSS stations and 12 co-located radiosonde stations in the tropics is shown in Fig. 1.The quality control strategies implemented for removing the unqualified radiosonde stations are as follows (He et al. 2017;Zhang et al. 2019): 1) the height should be at least 10 km for the top level； 2) the gaps of pressure profiles should be less than 200 hPa during continuous recording； 3) the height difference between two successive pressure levels should be less than 2 km； 4) the number of available profiles should be greater than 400 throughout the year.

Retrieval of PWV based on ERA5 reanalysis data and radiosonde observations ERA5 ZTD
In this study, the ZTD at each grid point of the surface level and the pressure level is determined by integrating the refractivity at their respective heights.The integral formula of the ZTD is as follows.

6
( 1) 10 ZTD n Nds where n is the refractive index and N is the total refractivity, which can be obtained using the following formula: where 1 k , 2 k and 3 k are constants with values of 1 77.604 , P is the atmospheric pressure, T is the temperature, e is the water vapor pressure and q is the specific humidity.In order to obtain more accurate ZTD, the Saastamoinen (1972) model is used to calculate the ZHD above the top pressure level of the grid point and its result is added to the integrated ZTD.

ERA5 PWV and RS PWV
ERA5 PWV and RS PWV can be obtained from an integral of the meteorological data as follows (Zhai and Eskridge 1997): where q is the specific humidity, p is the pressure and subscript i and 1 i + denote the i th and 1 i + th pressure level, respectively, g is the gravity acceleration of Earth.

GNSS ZTD
In this paper, two types of GNSS ZTD are used.One is the final ZTD products provided by the IGS, and the other is real-time ZTD.The detailed strategies and models used in real-time PPP for ZTD estimation are shown in Table 1.Mapping function GMF

GNSS PWV
The basic procedure of PWV retrieval by GNSS technology is as follows (Askne and Nordius 1987): where  is the conversion coefficient, the ZWD is the zenith wet delay, calculated by separating ZHD from ZTD.For the GNSS stations with measured meteorological parameters, the ZHD is achieved by bringing the pressure directly into Saastamoinen model.For the GNSS stations without measured pressure data, it is necessary to interpolate the surface grid pressure data of ERA5 to GNSS stations based on empirical formulas (Suparta et al. 2013), and then obtain ZHD through Saastamoinen model.
In order to reduce the effect of height differences, the GNSS PWV is extrapolated to the height of the radiosonde stations using the empirical PWV vertical correction model (Zhao et al. 2020): where PWV are the corresponding PWV values at the heights of 1 h and 2 h , respectively.
In this paper, EGM2008 model is used to unify the height system to reduce the impact of height difference on the evaluation results (Pavlis et al. 2013).

Construction of ZTD and PWV grid direct conversion model
In this paper, the correlation between ERA5 surface ZTD and PWV in tropical regions from 2016 to 2019 is first analyzed and the results are shown in Fig. 2. 01 where 0 a is intercept and 1 a is the linear regression coefficient between PWV and ZTD.The fitting results of the linear model show significant seasonal residuals, so the Fast Fourier Transform (FFT) is introduced to perform spectral analysis on the residuals.The results of three random grid points are shown in Fig. 3.
where 0 a is the mean value of the PWV and 1 a is the linear regression coefficient between PWV and ZTD, 12 ( , ) bb and 34 ( , ) bb are the annual and semi-annual periodicity coefficient sets of the PWV residual errors, respectively.The coefficients in the formula (8) are solved by the least squares method.
On the basis of considering the periodicity, to further expand the vertical application space of the model and weak the influence of large height difference on PWV estimation, the coefficient values of each pressure layer are calculated using the ZTD and PWV from the ERA5 pressure-level, and then the relationship between the coefficient and height is investigated.This study only analyzes the data below 5 km since the PWV values above 5 km is very small, the results of three grid points are shown in Fig. 4. corrections are applied to these four coefficients in this paper.The formula of the linear variation of the coefficients a 0 and a 1 with height calculated by the least squares method is given: where h is the height, 0  and 1  are the vertical lapse rate of 0 a and 1 a , respectively.0 a and 1 a are constants.
When this model is used to calculate the GNSS PWV, only the ZTD, day of year (DOY) and target location information are required.

Validation of the accuracy of post and real-time CZP PWV
In this section, the ERA5 PWV, the GNSS PWV at 61 GNSS stations and RS PWV from co-located 12 radiosonde stations are calculated to verify the accuracy and applicability of the post CZP PWV in tropical regions in 2020.In addition, the accuracy of real-time CZP PWV of 5 co-located stations is validated by being compared with PWV based on GPT3 with respect to RS PWV.

Validation of the accuracy of the post CZP PWV based on ERA5 PWV
Taking the ERA5 surface PWV as reference, the bias and RMS of the CZP PWV at grid points are shown in the Table 2 and Fig. 5. PWV.Fig. 5 shows that the bias and RMS values are relatively small in most regions.The bias is negative in the northern Indian Ocean and mid-Atlantic, while it is significantly positive in the other regions.The RMS is larger in the southern North America, the western coast of South America, southeastern Africa and southern Asia at the sea-land interface and the maximum appears near the Andes Mountains, where the RMS is more than 2 mm.The reason may be that the ZTD and PWV of sea-land interface change dramatically affected by El Niño phenomenon and tropical zone maritime climate.Overall, the CZP PWV has a high agreement with the ERA5 PWV and shows good stability and high accuracy in tropical regions.The GNSS PWV values are used to evaluate the CZP GNSS PWV retrieved from the ZTD data of 61 GNSS stations in 2020.The statistical results are presented in Table 3 and Fig. 6.Table 3 shows that the mean values of bias and RMS are 1.06 and 1.34 mm for 11 GNSS stations and those of the 50 GNSS stations are 1.05 and 1.30 mm.It can be seen from Fig. 6 that the overall bias values are small and mostly between 0 and 1.2 mm.Accordingly, the RMS values are mainly range from 0.8 to 1.5 mm.However, CHPG and BAKO stations located at the sea-land interface show relatively large RMS, which is consistent with the comparison results of ERA5 PWV mentioned above.The results of the two sets of GNSS PWV both show that the maximum values of bias and RMS of the CZP GNSS PWV do not exceed 2.5 mm and the mean values are less than 1.4 mm.Fig. 7 shows that the variation trend of CZP GNSS PWV agrees well with the corresponding GNSS PWV at four stations with relatively complete data.From the subfigures on the right, the residual errors of four stations are stable and small, mostly concentrated from 0 to 2 mm.It is indicated that the accuracy of the CZP GNSS PWV is high.The accuracy of the CZP GNSS PWV is further evaluated taking the PWV retrieved from 12 co-located radiosonde stations and the results are given in Table 4.  From Fig. 8, it can be found that there is a significant linear correlation between the CZP GNSS PWV and RS PWV.The correlation coefficients of 75% stations are greater than 0.90.Compared with other stations, the correlation coefficient of 91165 station is smaller.The reason may be that the station is located in the Hawaiian Islands, which is affected by the tropical rainforest climate and the highpressure zone in the North Pacific Ocean, with frequent atmospheric activity and rapidly varying PWV.
Another reason may be that there is a significant height difference between the GNSS station and the radiosonde station.As illustrated in Fig. 9, even if the PWV values of 94120, 91212, and 45004 stations located in the coastal regions display complicated spatiotemporal changes due to geographic and climatic conditions, the CZP GNSS PWV and RS PWV at the four random stations still show a good consistency.
Thus, the CZP model can be used to retrieve the PWV in the tropical regions with high accuracy and good spatiotemporal applications.

Validation of the accuracy of the real-time CZP PWV based on RS PWV
According to the parameter estimation strategy listed in Table 1, the real-time ZTD at 5 GNSS stations in tropical regions from July 1 to 14, 2023 is calculated.The real-time ZTD is evaluated using the IGS final ZTD and the results are shown in Table 5. respectively.The magnitude of the ZTD error can cause an error of approximately 1.9 mm in the GNSS PWV, so it is considered that the real-time ZTD meets the accuracy requirements.
Two real-time GNSS PWV products are calculated based two different methods.One approach is to introduce the real-time ZTD into the CZP model to obtain the real-time CZP PWV.The other approach is to retrieve the PWV using meteorological parameters derived from the GPT3 model, where the ZWD is separated from the real-time ZTD.Two real-time GNSS PWV are assessed using the RS PWV as references.The results are shown in Table 6 and Fig. 10.

Fig. 1
Fig. 1 Distribution of stations in tropical regions.Red pentagram and blue circles represent GNSS

Fig. 3
Fig. 3 Time series of linear model-derived PWV and ERA5 PWV (top) and residual errors series

Fig. 4
Fig. 4 Variation of a 0 (top) and a 1 (bottom) with height at three random grid points

Fig. 5
Fig. 5 (a)Bias and (b)RMS of the CZP PWV with respect to ERA5 PWV in tropical regions in 2020

Fig. 6
Fig. 6 (a) Bias and (b) RMS of the CZP GNSS PWV with respect to GNSS PWV in tropical regions for

Fig. 7
Fig. 7 Time series of PWV (left) and residual errors (right) of the CZP model with respect to GNSS

Fig. 8
Fig. 8 Correlation between CZP GNSS PWV and RS PWV for 12 co-located stations in 2020 252 Fig. 10, the bias values of two real-time GNSS PWV are negative for most stations and the RMS values of the real-time CZP GNSS PWV at all 5 stations are smaller than those of the real-time PWV based on GPT3 model.The RMS values at the ANMG, BRAZ and HKSL stations do not exceed 2.5 mm, which further illustrates the feasibility of the CZP model for real-time PWV inversion.The KOKV and PIMO stations show larger RMS values, and the reason may be that the significant height difference between the GNSS and the radiosonde station, which has a crucial effect on the PWV even though the height difference is corrected.

Fig. 10
Fig. 10 (a) RMS and (b) bias of two real-time GNSS PWV at 5 co-located stations in tropical regions

Table 1
Strategies and models used in the real-time PPP

Table 2
Table 2 lists that the mean values of bias and RMS between CZP PWV and ERA5 PWV are 0.06 and 0.67 mm, respectively.This indicates that the CZP PWV is highly consistent with ERA5

Table 2
Statistical results of the bias and RMS for CZP PWV using ERA5 PWV as reference in 2020

Table 3
Statistical results of the bias and RMS between CZP GNSS PWV and GNSS PWV at 61 GNSS

Table 4
Statistical results of the CZP GNSS PWV with respect to RS PWV at 12 co-located stations in As shown in Table4, the mean values of the bias and RMS of the CZP GNSS PWV are 1.61 and 3.33 mm, and the maximum values are 4.06 and 6.01 mm, respectively.In addition, the mean value of correlation coefficients is 0.93, indicating that the CZP GNSS PWV and RS PWV show a strong correlation.The correlation between CZP GNSS PWV and RS PWV is shown in Fig.8.

Table 5
Statistics results of real-time ZTD at 5 GNSS stations in tropical regions from July 1 to 14, Table5indicates that the mean values of bias and RMS of real-time ZTD are -3.25 and 12.59 mm,

Table 6
Statistics results of two real-time GNSS PWV at 5 co-located stations in tropical regionsIn Table6, for the real-time CZP GNSS PWV, the mean bias and RMS are -0.55 and 3.55 mm, respectively.For the real-time PWV based on GPT3 model, the mean bias and RMS are -2.53 and 4.08 mm, respectively.This means that the CZP GNSS PWV has a higher accuracy than that of conventional models in real-time PWV inversion.From