The New Improved ZHD and Weighted Mean Temperature Models Based on GNSS and Radiosonde Data Using GPT3 and Fourier Function

Abstract

Compared to the zenith hydrostatic delay (ZHD) obtained from the Saastamonien model based on in-situ measured meteorological (IMM) data and radiosonde-derived weighted mean temperature ($T_m$), the ZHD and $T_m$ deviations of the GPT3 model have shown obvious periodic trends. This article analyzed the seasonal variations of GPT3-ZHD and GPT3-$T_m$ during the 2016–2020 period in the Yangtze River Delta region, and the new improved ZHD and $T_m$ models were established by the multi-order Fourier function. The precision of the improved-ZHD model was verified using IMM-ZHD products from 7 GNSS stations during the 2016–2020 period. Furthermore, the precisions of improved $T_m$ and precipitable water vapor (PWV) were verified by radiosonde-derived $T_m$ and PWV in the 2016–2019 period. Compared with the IMM-ZHD and GNSS-PWV products, the mean Bias and RMS of GPT3-ZHD are −0.5 mm and 2.1 mm, while those of GPT3-PWV are 2.7 mm and 11.1 mm. Compared to the radiosonde-derived $T_m$, the mean Bias and RMS of GPT3-$T_m$ are −0.8 K and 3.2 K. The mean Bias and RMS of the improved-ZHD model from 2019 to 2020 are −0.1 mm and 0.5 mm, respectively, decreasing by 0.4 mm and 1.6 mm compared to the GPT3-ZHD, while those of the improved-$T_m$ are −0.6 K and 2.7 K, respectively, decreasing by 0.2 K and 0.5 K compared to GPT3-$T_m$. The mean Bias and RMS of PWV calculated by GNSS-ZTD, improved-ZHD, and improved-$T_m$ are 0.5 mm and 0.6 mm, respectively, compared to the GNSS-PWV, decreasing by 2.2 mm and 10.5 mm compared to the GPT3-PWV. It indicates that the improved ZHD and $T_m$ models can be used to obtain the high-precision PWV. It can be applied effectively in the retrieval of high-precision PWV in real-time in the Yangtze River Delta region.

1. Introduction

Sufficient atmospheric water vapor is one of the necessary conditions for precipitation formation, and water vapor detection plays an important role in weather forecasting, disaster monitoring and global climate change monitoring [1]. GNSS precipitable water vapor (GNSS-PWV) can be used to reflect the atmospheric water vapor variations in GNSS meteorology. It has the potential to forecast severe weather phenomena [2,3,4] and examine the effects of climate change [5,6]. Previous studies [3,7,8,9,10] have shown that there will be severe rainstorms in the downward trend process after GNSS-PWV reaches its peak value. Benevides et al. [11] proposed that the accuracy of weather forecasting could be improved after analyzing 3D distribution variations of PWV [12,13,14,15,16].

The inversion process of GNSS-PWV requires the zenith wet delay (ZWD) and the water vapor conversion coefficient ($K$). ZWD can be obtained by subtracting the zenith hydrostatic delay (ZHD) from the zenith total delay (ZTD) [17]. ZHD is the main delay for the GNSS signals transmitted in the neutral atmosphere, accounting for more than 90% of ZTD. It means that the precision of ZHD will indirectly affect the precision of ZWD. The weighted mean temperature ($T_m$) is one of the important parameters to calculate $K$ for the conversion of ZWD to PWV [18]. The ZHD and $T_m$ play a crucial role in obtaining high-precision real-time GNSS-PWV [19,20].

Currently, the commonly used ZHD models can be divided into two categories. One is the empirical models based on the measured meteorological parameters, including the Saastamonien [21], Hopfield [22], and Black [23] models. For example, the Saastamonien model uses meteorological sensors to measure surface pressure and can calculate the ZHD at a millimeter level. However, it is difficult to obtain the measured meteorological parameters in real time at any place in the world, which limits the application of tropospheric delay models that need measured meteorological parameters with respect to GNSS meteorology. In this case, the use of atmospheric data to establish regional or global real-time tropospheric delay models has been widely concerned, such as UNB3m [24], EGNOS [1] and GPT [25,26,27,28]. ERA-Interim [29] and ERA5 [30] are also used to interpolate surface meteorological parameters at the GNSS stations. At the same time, a series of studies have achieved fruitful results in this research area. Ghaffari Razin and Voosoghi [31] use Bernese GNSS software and Saastamoinen model to calculate the ZTD and ZHD, then the ZWD obtained by subtracting the ZHD from the ZTD is modeled by two different machine learning methods, which can obtain PWV with high accuracy. A site-specific ZHD model was established by collecting an atmospheric vertical profile from radiosondes stations. It provides an error about 0.19 mm, which can be used to accurately estimate PWV [32]. Yang, et al. [33] analyzed the global performance of the three most commonly used ZHD models, the best temperature and pressure models were established by evaluating the influence of different modeling factors and the meteorological parameters estimated by the above-mentioned models. Based on the ERA5 reanalysis data of the European Centre for Medium-Range Weather Forecasts (ECMWF), Mateus, et al. [34] developed a one-hour global air pressure and temperature model (HGPT) to provide pressure, temperature, ZHD, and $T_m$. Climate studies, GNSS meteorology and other atmospheric research can significantly benefit from it.

The $T_m$ is a necessary parameter to calculate the K value and plays a key role in atmospheric water vapor conversion. Among various $T_m$ calculation methods, the accuracy of radiosonde-derived $T_m$ is the highest, but it is difficult to popularize due to the spatiotemporal limitation of radiosondes [35,36,37]. Therefore, Bevis used the ground surface temperature ($T_s$) from the profiles of vapor partial pressure and dewpoint temperature of North American radiosondes over a two-year period to establish a global mean temperature model in 1992 ($T_m$ = 0.72 $T_s+7$0.2) [38]. However, due to the influence of location, time, and other factors, the regional accuracy of Bevis model is inconsistent. In general, the systematic deviation is greater than 4.0 K, and even greater than 8.0 K in some areas [39,40]. When encountering bad weather, it may even lead to a significant deviation of GNSS-PWV [41,42]. Considering the linear relationship between $T_m$ and $T_s$, many scholars have established different regional $T_m$ models (RTM) based on local radiosondes [40,42,43,44]. The RTM established in Hong Kong can control the deviation within 4.0 K, which is superior to the Bevis model [45]. Singh, et al. [46] have found that the site specific $T_m$ model is better than the developed regional $T_m$ model and global model at New Delhi and Patiala. Elhaty, et al. [47] use radiosonde profiles from four stations situated in Egypt during 2015–2016 and Bevis linear regression method to develop a new $T_m$ model. Several RTMs using one factor ($T_s$) have been established in China [45,48,49,50]. Li and Mao [48] deeply studied the monthly coefficient of RTM in eastern China. Guo, et al. [51] established a good annual single factor RTM model based on sounding data in the Yangtze River Delta region. Based on the above research, the researchers established many multi-factor RTM models [52,53,54]. Considering the influence of pressure ($P_s$) and water vapor pressure ($e_s$) on $T_m$, Gong [52] analyzed the relationship between meteorological elements using the data of 123 radiosonde stations in China, and established multi-factor models in different climatic regions, effectively improving the accuracy of single factor models. However, the models established by Wang, Song, Dai, and Cao [53] show that there is little difference in accuracy between the single factor model and the multi-factor model in Hong Kong. According to the above linear regression models, the precision of non-linear RTM between $T_m$ and $T_s$ proposed and established by Yao, et al. [55], which is slightly better than that of linear RTM. Zhu, et al. [56] established a non-linear $T_m$ model for China with elevation corrections, which provides a significant correction effect for $T_m$ in the vertical direction. Lan, Zhang, and Geng [39] adopt the sliding average method to calculate the correlation coefficient between $T_s$ of the ECMWF and $T_m$ from the “GGOS Atmosphere”. Compared to the $T_m$-$T_s$ relation of Bevis model, the $T_m$ Grid model shows higher precision. Most researchers will analyze its precision comparing the PWV calculated by different ZHD and $T_m$ models [57,58].

As mentioned above, when GNSS stations lack meteorological instruments, GPT series models are usually used to obtain real-time meteorological parameters. Boehm, Heinkelmann and Schuh [25] first proposed GPT in 2007, which can provide the pressure and temperature at any geographical location on the earth surface. Lagler, Schindelegger, Böhm, Krásná, and Nilsson [28] developed the GPT2 by combining the GPT with global mapping function (GMF), which can provide more meteorological parameters. Böhm, Möller, Schindelegger, Pain and Weber [26] introduced the vertical gradient of water vapor pressure and $T_m$ to establish GPT2w on the basis of GPT2 in 2015. The GPT3 model is the latest version of the GPT series models, it can provide not only parameters from GPT2w, but also empirical gradient grids and is one of the most accurate and widely used tropospheric delay models [27]. Many studies have shown that the GPT3 model can provide high-precision ZTD and horizontal gradient information on a global scale, however, due to the limitation of terrain and other conditions, the GPT3 model based on European Centre for Medium-Range Weather Forecasts (ECMWF) data cannot be perfectly applied to any area [59,60,61]. Therefore, in a specific time and area, the precision of GPT3 model may not meet the requirements of some high-precision GNSS-PWV applications

Based on the GNSS products and radiosondes data in the Yangtze River Delta region during the 2016–2020 period, this paper analyzed the seasonal variations of the $T_m$ and ZHD of GPT3 model in the Section 2, and then the Fourier function was used to establish the improved ZHD and $T_m$ models in the Section 3. Meanwhile, the precisions of these improved ZHD and $T_m$ models were verified by comparing them with the GNSS and radiosondes products in the Section 3. The Section 4 is the conclusion.

2. Data Sources and Methodology

2.1. Data Sources

Data from seven GNSS stations in the Yangtze River Delta region from 2016 to 2020 [62], mainly including parameters such as altitude, pressure, temperature, ZTD, ZHD and PWV (time resolution is one hour) were used to analyze the precision of corresponding parameters from the GPT3 model. Among them, the deviation between GPT3-ZHD and IMM-ZHD during the 2016–2018 period was used to establish the improved-ZHD model based on the Fourier function, and IMM-ZHD from 2016 to 2020 was used to verify the precision of the improved-ZHD model.

Data from seven radiosonde stations in the Yangtze River Delta region from 2016 to 2019 were obtained from Wyoming State University (http://weather.uwyo.edu/upperair/sounding.html, accessed on 15 April 2022), and it include meteorological parameters such as pressure, temperature and elevation (time resolution is 12 h). The PWV and $T_m$ can be obtained by the integration method, and used as a reference to analyze the precision of GPT3-$T_m$ and its inversion precision of PWV. The deviation of the GPT3-$T_m$ and the radiosonde-derived $T_m$ obtained by the integration method during the 2016–2018 period was used to establish an improved $T_m$ model based on the Fourier function, and the data from 2016 to 2019 were used to verify the precision of the improved $T_m$ model. Figure 1 is the location information of 7 GNSS stations and seven radiosonde stations.

2.2. Methodology

2.2.1. T_m Calculation

The $T_m$ is calculated by the numerical integration method using the radiosondes selected in Figure 1. The radiosonde-derived $T_m$ has high calculation precision and is easy to realize [63]. It will be used as the reference or true value to evaluate the precision of GPT3-$T_m$.

$$\left\{\begin{array}{c}{T}_m=\frac{\int \left(e/T\right)dz}{\int \left(e/T^2\right)dz} \\ e=6.112\times e^{\frac{17.62t_d}{243.12+t_d}}\end{array}\right.$$

(1)

where, z is the elevation of the layer (km), e is the surface water vapor pressure (hPa), and $t_d$ is the surface dew point temperature (°C).

2.2.2. GNSS-PWV and GPT3-PWV Calculations

GNSS has the advantages of all-weather, continuous and global coverage, it can provide high-precision ZTD except for navigation and positioning. According to the multi-frequency GNSS observations, the GNSS-ZTD time series can be obtained using precise point positioning (PPP) [64]. Firstly, the real-time clock error of satellites can be obtained by the data of GNSS reference stations and IGU orbit from IGS, then, then the time series of real-time ZTD will be calculated by the PPP routine of RTKLib on the basis of the real-time clock error of satellites and IGU products [3].

GNSS-ZTD consists of IMM-ZHD and GNSS-ZWD ($\mathrm{ZTD}=\mathrm{ZWD}+\mathrm{ZHD}$), and the IMM-ZHD can be calculated by the $\mathrm{Saastamonien}$ model based on the latitude, air pressure and elevation of the sites [21], the formula is as follows.

$$\left\{\begin{array}{c}\mathrm{IMM}-\mathrm{ZHD}=0.0022768\times \frac{P_c}{f\left({\phi }_c,H_c\right)} \\ f\left({\phi }_c,H_c\right)=1-0.002 66\cos 2{\phi }_c-0.000 28H_c\end{array}\right.$$

(2)

where, $P_c$ is the pressure of the station (hPa), ${\phi }_c$ is the latitude of the station ($°),H_c$ is the elevation of the station (km). ZWD obtained by ZTD minus ZHD and the GNSS-PWV is determined by the following expression.

$$\left\{\begin{array}{l} \mathrm{PWV}=K\times \mathrm{ZWD}\\ K=\frac{{10}^5}{R_v\left(k_3/T_m+k_2^{\prime }\right)} \end{array}\right.$$

(3)

where, $K$ is the atmospheric water vapor conversion coefficient, $R_v$ is the gas constant of the water vapor ratio, $R_v=461\left(\mathrm{J}·\mathrm{K}·{\mathrm{kg}}^{-1}\right)$; $k_2^{\prime }$ and $k_3$ are the atmospheric refractive index constants; $K_3=3.776\times {10}^5\left({\mathrm{K}}^2·{\mathrm{hpa}}^{-1}\right)$; $k_2^{\prime }=k_2-k_1·\left(m_v/m_d\right)\left(\mathrm{K}·{\mathrm{hpa}}^{-1}\right)$; $k_1=77.604 \left(\mathrm{K}·\mathrm{hPa}\right); k_2=64.79 \left(K·\mathrm{hPa}\right); m_v=18.0152 \left(\mathrm{g}·{\mathrm{mol}}^{-1}\right); m_d=28.9644 \left(g·{\mathrm{mol}}^{-1}\right)$; The $T_m$ is obtained from the measured meteorological data according to Equation (1).

GPT3-PWV and GNSS-PWV adopt the same calculation method as Equation (3), but the difference is that $T_m$ and ZWD for GPT3-PWV are obtained from the GPT3 model, which is more convenient.

2.2.3. Fourier Function

The Fourier function is a special triangular series proposed by Fourier, a French mathematician. Its characteristics and advantages are that complex functions can be transformed into linear combinations of simple trigonometric functions, which can accurately approximate any complex function, achieving the purpose of studying complex functions when given the appropriate orders, coefficients, and frequency [65]. The basic formula is as follows.

$$y=f\left(x\right)=a_0+{\displaystyle \sum }_{n=0}^m\left(a_n\cos \left(n\times x\times w\right)+b_n\sin \left(n\times x\times w\right)\right)$$

(4)

where, $m$ is the orders, $w$ is the frequency, and $a_n$ and $b_n$ are the coefficients.

2.2.4. Statistical Method

The mean Bias and the root mean square (RMS) are used to evaluate the precision of the meteorological parameters and improved models as follows.

$$\left\{\begin{array}{c}\mathrm{Bias}=\frac{{{\displaystyle \sum }}_{i=1}^n\left(x_{model,i}-x_{true,i}\right)}{n} \\ \mathrm{RMS}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(x_{model,i}-x_{true,i}\right)}^2}{n}}\end{array}\right.$$

(5)

where, $X_{model,i}$ and $X_{true,i}$ represent parameter value of the models and truth, respectively, and the $n$ is the number of samples.

3. Results and Discussions

The GPT series model (http://ggosatm.hg.tuwien.ac.at/DELAY/, accessed on 10 January 2022) is one of the popular open-source tropospheric delay models by adding empirical meteorological parameters. It only needs to provide the approximate coordinates of the station (Latitude, Longitude, Height) and day of year (DOY), and the values of meteorological parameter at any location all over the world can be obtained. The GPT3 model, as the latest model of the GPT series, has added gradients in two directions (N for north and E for east), and is completely consistent with the Vienna mapping function (VMF3), which can meet various purposes for meteorological and climate research. Its formula is as follows.

$$r\left(t\right)=A_0+A_1\cos \left(2\pi \frac{doy}{365.25}\right)+B_1\sin \left(2\pi \frac{doy}{365.25}\right)+A_2\cos \left(4\pi \frac{doy}{365.25}\right)+B_2\sin \left(4\pi \frac{doy}{365.25}\right)$$

(6)

where, r(t) are the values of the meteorological parameters at the grid point; $A_0$ is the mean value; $\left(A_1,B_1\right)$, the amplitudes of the annual cycle; and $\left(A_2,B_2\right)$, the amplitudes of the semi-annual cycle. Coefficients can be obtained by combining external grid files $\left(\mathrm{gpt}{3}_1.\mathrm{grd}\right)$ with bilinear interpolation.

The GPT3 model uses the Saastamonien model to calculate ZHD as shown in Equation (1). The $\mathrm{Asken}\&\mathrm{Nordius}$s model [66] is used to calculate the ZWD. The GPT3 model also provides some empirical meteorological parameters that are inconvenient to measure directly. These meteorological parameters can be used to calculate GPT3-ZWD; the calculation formula of GPT3-PWV is the same as Equation (2).

3.1. The Improved ZHD Model

The comparison of GPT3-ZHD and IMM-ZHD of four GNSS stations at Anqing, Bengbu, Lishui and Lianyungang from 2016 to 2020 is shown in Figure 2. It can be seen that ZHD shows obvious seasonal periodical changes, and the overall trend of the two types of ZHD is similar. The ZHD value reaches the valley value in June, while it reaches its peak value in December. Affected by location and altitude, the peak value (or valley value) of ZHD at each station generally has a millimeter to centimeter deviation.

3.1.1. The Establishment of an Improved ZHD Model

Based on the analysis of GPT3-ZHD, experiments and analysis show that the Fourier function can be effectively used for model fitting. It will perform different fitting effects for the deviation of GPT3-ZHD by using different orders of the Fourier function. By comparison, the accuracy of the third order Fourier is much better than that of the first order, which is close to the fourth order, and the coefficient of the third order is simpler than that of the fourth order, so we chose the third order. Therefore, the third-order Fourier function is adopted to establish an improved ZHD model in this paper. Its calculation formula is as follows.

$$f\left(x\right)=a_0+a_1\cos xw+b_1\sin xw+a_2\cos 2xw+b_2\sin 2xw+a_3\cos 3xw+b_3\sin 3xw$$

(7)

where, $a_0$, $a_1$, $b_1$, $a_2$, $b_2$, $a_3$, $b_3$, $w$, represent the coefficients, $x$ is the day of year, and $f\left(x\right)$ is the deviation fitting curves based on the third-order Fourier function.

The GPT3-ZHD deviations of the 7 GNSS stations in the Yangtze River Delta region from 2016 to 2018 were used in Equation (7). It can obtain the model coefficients of the third-order Fourier function as shown in

Table 1. The Fourier coefficients of the GPT3-ZHD bias in the Yangtze River Delta region during 2016–2018.

Parameter	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{b}}_1$	${\mathit{a}}_2$	${\mathit{b}}_2$	${\mathit{a}}_3$	${\mathit{b}}_3$	$\mathit{w}$
ZHD	−0.3909	−0.2525	−2.676	−0.2953	0.3294	0.00059	−0.0097	0.01718

, which is a numerical description of the distribution characteristics of the GPT3-ZHD deviation. Figure 3 is the fitting curve of the ZHD deviation, it can be seen that the deviation distribution of GPT3-ZHD at seven stations is relatively small and the difference is mainly within 2 mm. The fitting curve agrees well with the deviation of each GNSS station, and the fitting deviation is mostly within 0.5 mm.

3.1.2. Precision Analysis

The improved ZHD model established in the previous section can be used to predict the ZHD value (improved-ZHD) at GNSS stations from 2016 to 2020. Moreover, the article analyzed the precision of the improved ZHD model using the IMM-ZHD as the true value, and verified the weakening effect of the seasonal periodic deviation in Figure 4. It is clear that the periodic deviation of improved-ZHD at most stations has been greatly decreased, but the periodic fluctuation of improved-ZHD at Bengbu Station still has a small amplitude, which is related to the large gap between its GPT3-ZHD deviation and the overall trend of the deviation distribution. In general, most of the deviations of the improved ZHD model are maintained within 0.5 mm, and the precision of the improved-ZHD is greatly improved compared to the GPT3 model.

Table 2. The precision of improved ZHD in the Yangtze River Delta region during the 2016–2020 period (mm).

Sites	GPT3-ZHD				Improved-ZHD
	2016–2018		2019–2020		2016–2018		2019–2020
	Bias	RMS	Bias	RMS	Bias	RMS	Bias	RMS
Anqing	−0.1	2.0	−0.2	2.1	0.3	0.4	0.4	0.4
Bengbu	0.1	2.2	−0.0	2.2	0.6	0.7	0.6	0.7
Jiande	−0.8	2.0	−1.0	1.8	−0.4	0.5	−0.4	0.7
Lishui	−0.2	2.1	−0.4	2.1	0.2	0.4	0.2	0.4
Lianyungang	−0.7	2.2	−1.0	2.3	−0.4	0.4	−0.4	0.4
Shanghai	−0.4	1.9	−2.2	2.3	0.0	0.4	−0.4	0.4
Wenzhou	−0.9	1.9	−1.0	2.0	−0.5	0.8	−0.4	0.5
Average	−0.4	2.0	−0.8	2.1	−0.0	0.5	−0.1	0.5

shows the precision statistics of the improved ZHD model from 2016 to 2020. It can be seen from the table that the Bias and RMS of the improved ZHD model based on the Fourier function from 2019 to 2020 are −0.1 mm and 0.5 mm, respectively, which are 0.7 mm and 1.6 mm better than that of GPT3-ZHD, respectively. The overall precision of the improved-ZHD is significantly improved and relatively stable at each station. Therefore, the improved ZHD model has a higher precision and applicability than the GPT3 model in the Yangtze River Delta region.

3.2. The Improved $T_m$ Model

Comparisons between the GPT3-$T_m$ and the radiosonde-derived $T_m$ (as a true value) at the four radiosondes in Anqing, Sheyang, Fuyang and Nanjing from 2016 to 2019 are shown in Figure 5. It can be seen that the $T_m$ show obvious seasonal periodic changes and the overall change trends of the two types are similar, which are lower in spring and winter and higher in summer and autumn.

3.2.1. The Establishment of an Improved $T_m$ Model

Based on the analysis of GPT3-$T_m$, the deviations of GPT3-$T_m$ from seven radiosondes in the Yangtze River Delta region from 2016 to 2018 are substituted into Equation (7), and the coefficients of third order Fourier function model of GPT3-$T_m$ deviations can be obtained as shown in

Table 3. Fourier coefficients of the GPT3-$T_m$ bias in the Yangtze River Delta region during the 2016–2018 period.

Parameter	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{b}}_1$	${\mathit{a}}_2$	${\mathit{b}}_2$	${\mathit{a}}_3$	${\mathit{b}}_3$	$\mathit{w}$
$T_m$	−0.8249	−1.247	1.301	−0.2511	−0.2311	0.2893	−0.9035	0.01729

.

The bias distribution of GPT3-$T_m$ of the seven radiosondes is shown in Figure 6. It can be seen that the $T_m$ deviations of nearly all radiosondes are maintained within $\pm$10 K, but the deviation distribution is relatively discrete and exhibits obvious periodicity. A fitted curve has also been shown in the figure. It can describe the deviations distribution of GPT3-$T_m$ well.

3.2.2. Precision Analysis

The improved $T_m$ model established in the previous section can be used to predict the $T_m$ at seven radiosonde stations from 2016 to 2019. Furthermore, the article analyzed the precision of the improved $T_m$ model using the radiosonde-derived $T_m$ as true value, and verified the weakening effect of the seasonal periodic deviation of GPT3-$T_m$ in Figure 7. It is clearly shown that the improved-$T_m$ and GPT3-$T_m$ are generally similar, and their deviations are both within $\pm$10 K compared to the radiosonde-derived $T_m$. However, the deviations of improved-$T_m$ are more concentrated, indicating that the improvement effect is more obvious.

Table 4. Precision of the GPT3-$T_m$ and the improved-$T_m$ in the Yangtze River Delta during the 2016–2019 period (K).

Sites	$\mathbf{GPT}3-{\mathit{T}}_{\mathit{m}}$				$\mathbf{Improved}-{\mathit{T}}_{\mathit{m}}$
	2016–2018		2019		2016–2018		2019
	Bias	RMS	Bias	RMS	Bias	RMS	Bias	RMS
Anqing	−0.2	3.0	−1.1	2.9	0.4	2.8	−0.3	2.7
Sheyang	−1.1	3.4	−1.7	3.3	−0.5	2.8	−0.9	2.7
Fuyang	−0.5	3.2	−1.0	2.9	0.2	2.9	−0.3	2.7
Nanjing	−0.8	3.2	−1.5	3.1	−0.2	2.8	−0.7	2.6
Hangzhou	−1.0	3.4	−1.8	3.4	−0.3	2.8	−1.0	2.8
Quzhou	0.1	3.0	−0.9	2.7	0.7	2.9	−0.2	2.5
Shanghai	−1.9	3.4	−1.9	3.4	−0.4	2.8	−1.1	2.6
Average	−0.8	3.2	−1.4	3.1	−0.0	2.8	−0.6	2.7

is the precision statistics of the improved-$T_m$ and the GPT3-$T_m$ from 2016 to 2019. It can be seen that the mean Bias and RMS of improved-$T_m$ are −0.6 K and 2.7 K in 2019, respectively, which are improved by 0.8 K and 0.4 K, respectively, compared to that of GPT3-$T_m$. Although the improvement of RMS is small, the bias of each station decreases steadily. Therefore, the improved $T_m$ model in the paper has a higher and more stable precision than the GPT3 model in the Yangtze River Delta region.

3.3. The PWV Based on Improved ZHD and $T_m$ Models

The time series variation of GPT3-PWV and GNSS-PWV in Anqing (a), Bengbu (b), Lishui (c), and Lianyungang (d) during the 2016–2020 period is shown in Figure 8. Both GPT3-PWV and GNSS-PWV have obvious seasonal periodic changes, due to the influence of the subtropical monsoon in the Yangtze River Delta region, where rain and heat are in the same period, so the PWV is lower in spring and winter and higher in summer and autumn.

GNSS-ZTD, improved-ZHD, and improved-$T_m$ can be used integrated to calculate the improved-PWV at seven GNSS stations in the Yangtze River Delta region from 2016 to 2020 in the paper, the precision improvement of improved-PWV calculated as follows.

$$\left\{\begin{array}{c}\mathrm{PWV}=K\times \left(GNS{S}_{ZTD}-Improve{d}_{ZHD}\right) \\ K=\frac{{10}^5}{R_v\left(k_3/Improve{d}_T{}_m+k_2^{\prime }\right)} \end{array}\right.$$

(8)

where, $Improve{d}_T{}_m$ and $Improve{d}_{ZHD}$ come from the improved $T_m$ and ZHD models based on Fourier functions in Section 3.1 and Section 3.2. $GNS{S}_{ZTD}$ are obtained from GNSS, other parameters are the same as Equation (2).

The bias of GPT3-PWV and improved-PWV in Anqing (a), Bengbu (b), Lishui (c) and Lianyungang (d) during the 2016–2020 period is shown in Figure 9. The improved-PWV has greatly eliminated the seasonal periodic deviation of GPT3-PWV, indicating that the differences between the improved-PWV and GNSS-PWV are small and are highly consistent with each other.

The precision statistics of GPT3-PWV and improved-PWV are shown in

Table 5. Precision of GPT3-PWV and improved-PWV in the Yangtze River Delta during 2016–2020 (mm).

Sites	GPT3-PWV		Improved-PWV
	Bias	RMS	Bias	RMS
Anqing	2.9	11.3	0.6	0.6
Bengbu	1.3	10.2	0.5	0.6
Jiande	3.3	11.4	0.4	0.4
Lishui	1.8	11.2	0.5	0.5
Lianyungang	1.6	9.2	0.4	0.4
Shanghai	5.2	13.3	0.7	0.8
Wenzhou	3.0	11.0	0.7	0.7
Average	2.7	11.1	0.5	0.6

. It can be seen that the mean Bias and RMS of improved-PWV are 0.5 mm and 0.6 mm, respectively, which are 2.2 mm and 10.5 mm higher than that of GPT3-PWV, and the precision of GPT3-PWV is greatly improved. The precision of improved-PWV is very close to that of GNSS-PWV, and it can be used for real-time high-precision inversion of PWV in the Yangtze River Delta region.

In order to further verify the inversion precision of improved-PWV, Anqing and Shanghai that have co-located radiosondes and GNSS stations are selected to compare improved-PWV and GNSS-PWV with radiosonde-derived PWV from 2016 to 2019. The bias of GNSS-PWV and improved-PWV is shown in Figure 10. The vacancy in the Figure 10b is caused by the lack of radiosonde and GNSS data after March 2019 in Shanghai station.

The deviation distribution of the improved-PWV and GNSS-PWV is in good agreement, and most deviations of the PWV are kept within $\pm$10 mm, indicating that the precision of the improved-PWV and GNSS-PWV are equivalent.

Table 6. Precision of the GNSS-PWV and the improved-PWV at the co-located GNSS and radiosonde stations in Anqing and Shanghai during the 2016–2019 period (mm).

Sites	Improved-PWV		GNSS-PWV
	Bias	RMS	Bias	RMS
Anqing	−1.2	3.9	−1.8	4.2
Shanghai	−1.0	3.4	−1.5	3.7
Average	−1.1	3.7	−1.6	4.0

shows the precision statistics of improved-PWV and GNSS-PWV. Compared with the radiosonde-derived PWV, the mean Bias and RMS of the improved-PWV are −1.1 mm and 3.7 mm, respectively, which are 0.5 mm and 0.3 mm better than that of the GNSS-PWV, and the precision is slightly improved. It can be seen that the precision of PWV based on improved ZHD and $T_m$ models is slightly better than that of GNSS-PWV, and they are roughly equivalent on the whole, which is further verified and demonstrated that the improved ZHD and $T_m$ models can be used to obtain real-time high-precision PWV comparable to the GNSS-PWV precision.

4. Conclusions

The precision of the GPT3 model (ZHD, $T_m$ and PWV) in the Yangtze River Delta region was first analyzed with reference to GNSS and radiosondes products. Aiming at the problem that the ZHD and $T_m$ from GPT3 model have obvious seasonal periodic deviations, the third-order Fourier function was used to establish improved ZHD and $T_m$ models, and their precision were analyzed and verified. The main research conclusions are as follows.

The mean biases of the GPT3-ZHD, $T_m$ and PWV are −0.5 mm, −0.8 K, and 2.7 mm, respectively, and the mean RMS of those are 2.1 mm, 3.2 K, and 11.1 mm, respectively. Compared to the reference values of the GNSS products and radiosondes, the ZHD and PWV deviations have obvious seasonal variations. Specifically, the deviation of ZHD is negative in spring and winter, but positive in summer and autumn, and the deviation of PWV is smaller in spring and winter, but larger in summer and autumn.

Compare with the GPT3 model, the mean Bias and RMS of the improved-ZHD based on Fourier function from 2019 to 2020 are −0.1 mm and 0.5 mm, respectively, improved by 0.7 mm and 1.6 mm, while the mean Bias and RMS of the improved-$T_m$ in 2019 are −0.6 K and 2.7 K, respectively, which are 0.8 K and 0.5 K better than GPT3-$T_m$. The precision of two models is improved slightly.

The mean Bias and RMS of the improved-PWV based on GNSS-ZTD and the improved ZHD and $T_m$ models are 0.5 mm and 0.6 mm, respectively, which are 2.2 mm and 10.5 mm better than that of GPT3-PWV, and the overall precision is improved greatly. Compared to the radiosonde-derived PWV, the mean Bias and RMS are −1.1 mm and 3.7 mm, respectively, which are 0.5 mm and 0.3 mm higher than that of the GNSS-PWV, and the precision of the two methods performs similarly. Therefore, the improved ZHD and $T_m$ models based on GPT3 and Fourier function established in the Yangtze River Delta region can be used for real-time high-precision PWV inversion.