1. Introduction
The troposphere, which ranges from the ground to the beginning of the stratosphere, is a layer of atmosphere near the Earth. Air in the troposphere accounts for 75% of the total atmospheric mass. The troposphere’s height over China is approximately 10 km. The global navigation satellite system (GNSS) provides high-precision three-dimensional coordinates and speed. Tropospheric delay limits the accuracy of space geodesy [
1]. GNSS technology inevitably generates errors in its applications owing to the tropospheric delay, which seriously affects the accuracy of navigation, positioning, and interferometric synthetic aperture radar (InSAR) [
2]. The zenith tropospheric delay (ZTD) consists of the zenith hydrostatic delay (ZHD) and the zenith wet delay (ZWD). Owing to the particularities of the troposphere, the ZHD with a stable variation law can be accurately modeled. Unlike ZHD, ZWD, which is caused by precipitable water vapor (PWV), is difficult to model accurately [
3]. In addition, ZTD can be a useful signal for PWV inversion [
4]. Therefore, research on ZTD modeling is of considerable significance for navigation, positioning, InSAR monitoring, and PWV inversion [
5].
Existing ZTD models include meteorological and non-meteorological parameter models. The meteorological parameter models rely on measured meteorological parameters, whereas the non-meteorological parameter models only require the input of spatiotemporal information. The Hopfield [
6], Saastamoinen [
7], and Black models [
8] are the main meteorological parameter models. Hopfield established a ZTD model called the Hopfield model using data from 18 radiosonde stations worldwide; this model requires meteorological parameters. Saastamoinen established the Saastamoinen model based on the U.S. Standard Atmospheric Model (SAM), which also requires input values such as temperature, pressure, and station location information. The Black model was developed as a new-generation model that follows the Hopfield model, which also requires the input of temperature and pressure. The accuracy of the abovementioned models can reach a centimeter level, provided that the measured meteorological parameters are available [
9]. However, the measured meteorological parameters can only be obtained from stations equipped with meteorological sensors. The distribution of these stations with low spatial resolution is uneven, and there is a time delay, which considerably limits the ability of meteorological parameter models to realize real-time applications. Therefore, real-time calculation of ZTD using non-meteorological parameter models has become a considerable challenge that must be solved, and it has also become a hot research topic for scholars [
10,
11]. The TropGrid and GPT series models were established to realize a higher temporal resolution [
12,
13,
14]. Owing to the GPT2 model only calculating certain meteorological parameters, Böhm et al. [
15] established the GPT2w model using monthly ERA-Interim data, which adds two output values: water vapor decline rate and atmospheric weighted mean temperature. The global pressure and temperature 3 (GPT3) model is a new-generation model that follows the GPT2w model and improves the empirical mapping function coefficients [
16,
17,
18]. The GPT3 model, which has the ability to output comprehensive meteorological parameters, can calculate ZTD by combining the Saastamoinen and Askne models. The GPT3 model provides two horizontal resolutions of 1° × 1° and 5° × 5°, which need to be further improved. To improve the applicability of the ZTD model globally, studies have been conducted on high-precision global ZTD modeling, which considers multiple factors. Li et al. [
19] established an IGGtrop model that considers latitude, longitude, elevation, and annual period, which attained a further improvement in the global ZTD model’s accuracy. In addition, Huang et al. [
20] proposed a new global ZTD grid model based on a sliding window algorithm with different spatial resolutions, a novel model which shows excellent performance compared to the GPT3 and UNB3m models and significantly optimizes model parameters. However, the above models cannot capture the diurnal variation of ZTD, and the temporal resolution of these models needs to be further improved.
Recently, the calculation of the ZTD using the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalysis (ERA5) and the Second Modern-Era Retrospective Analysis for Research and Applications (MERRA-2) has received much attention [
21,
22,
23]. These atmospheric reanalysis data exhibited high-precision results when they were validated by other reference data [
24]. Therefore, the ERA5 and MERRA-2 datasets are expected to be used widely in the future. Because these atmospheric reanalysis grid data exhibit a high spatiotemporal resolution, the grid point data around the target point can be interpolated to calculate the data at the user’s position with high precision. However, ZTD varies much more vertically than horizontally. Direct interpolation produces large errors in regions with an undulating terrain. Therefore, to solve the aforementioned problems, it is necessary to develop a high-precision model to adjust the ZTD vertically. The selection of a fitting function for the variation in layered ZTD is an important research topic. Therefore, extensive studies have been conducted on ZTD vertical profile functions [
25]. The vertical variations of the ZTD are typically modeled using polynomials [
26,
27] and negative exponential functions [
28,
29,
30]. Zhu et al. [
31] developed a segmented global ZTD vertical profile model (GZTD-P) with different spatial resolutions that considers the time-varying characteristics of the ZTD vertical variation factor to address the limitations of a single function in expressing the ZTD vertical profile, which shows better performance compared to the GPT3 model. However, the GZTD-P model can only vertically adjust ZTD from the starting height to the target height and cannot directly calculate ZTD at the target position, which limits its application. Sun et al. [
32] proposed a global ZTD model, the GZTDS model, which considers delicate periodic variations by adopting a nonlinear function. The GSTDS model was developed as a new-generation model that follows the GZTDS model, and the performance of the new model deteriorated as the zenith angle increased. Hu and Yao [
33] adopted the Gaussian function to fit the vertical ZTD and then established a ZTD vertical profile model that considers seasonal variation, which shows good performance in both time and space. The model achieved good results on a global scale. The model was developed using the monthly mean ZTD with horizontal resolutions of 5° × 5° provided by ERA-Interim. Its adaptability needs to be further improved in regions with greatly undulating terrain and complex climates. Zhao et al. [
34] proposed a high-precision ZTD model that considers the height effect on ZTD after analyzing the relationship between the ZTD periodic residual term and the height of the GNSS station at different seasons. Although the aforementioned models have demonstrated their respective advantages, it is necessary to conduct further research on the more delicate vertical and temporal variations in the ZTD over China.
China is characterized by greatly undulating terrain, large latitudinal spans, various climate types, and large diurnal atmospheric differences [
35,
36,
37]. Under such conditions, the ZTD exhibits complex variations in both time and space. Existing ZTD models have difficulty meeting the requirements of applications in China with the aforementioned characteristics. Therefore, a detailed investigation of the spatiotemporal variations of the ZTD and the selection of better functions are of considerable significance for applications in GNSS and InSAR. Our aim was to develop a NGZTD model that considers the time-varying vertical adjustment and delicate diurnal variations of ZTD. To attain this objective, this paper is organized as follows.
Section 2 introduces the data and methods for calculating ZTD and analyzes the spatiotemporal characteristics of Gaussian coefficients and surface ZTD.
Section 3 develops a NGZTD-H model to vertically adjust ZTD, and the vertical interpolation accuracy of the model is validated using ERA5 and radiosonde data. In addition, the NGZTD model is developed to directly calculate ZTD, and its accuracy is validated using GNSS and radiosonde data.
Section 4 discusses the main error sources of the NGZTD model.
Section 5 contains the Conclusions. The research framework is shown in
Figure 1. The proposed model compensates for the limitations of existing ZTD models in China. The new model is expected to be applied to precise point positioning (PPP) and InSAR atmospheric correction to improve their monitoring accuracy.
4. Discussion
As shown in
Table 2,
Table 4 and
Table 5, the mean RMSE of the vertical interpolation for the NGZTD-H model was 1.70 cm when the ERA5 data were used as the reference value, whereas the mean RMSE of the NGZTD model for the direct calculation of the ZTD exceeded 3 cm when both the GNSS data and radiosonde data were used as the reference value. Unlike the vertical interpolation validation for the NGZTD-H model in
Section 3.1.1, the starting ZTD for the NGZTD model in
Section 3.2.1 and
Section 3.2.2. was obtained by a model considering the seasonal and diurnal variations of ZTD, whereas the starting ZTD in
Section 3.1.1. was obtained by the integral method. In addition, the NGZTD model required horizontal interpolation to calculate the ZTD at GNSS and radiosonde stations in
Section 3.2.1 and
Section 3.2.2. Based on the above analysis, the errors of the NGZTD model may be attributed to the starting ZTD and horizontal interpolation.
To validate the above error sources, the GNSS-derived ZTD in 2018 over China was used to validate the spatial interpolation accuracy for the NGZTD-H model. First, the Gaussian coefficients
b and
c were determined. Second, the ZTD on the ERA5 grid points was vertically adjusted to the GNSS station height using the NGZTD-H model. It should be noted that the starting ZTD on the ERA5 grid points was obtained by the integral method. Finally, the adjusted ZTD was interpolated horizontally to the GNSS station employing an inverse distance interpolation method. A comparison of the interpolated results with the GNSS-derived ZTD validated the effectiveness of the NGZTD-H model. Because the elevations of GNSS stations are geodesic heights and the elevations of ERA5 grid points are geopotential heights, it is necessary to unify the elevations of these two products before vertically adjusting the ZTD. In this study, we employed the EGM2008 model [
45] to convert elevations of the ERA5 data to geodesic elevations.
The spatial interpolation accuracy of the NGZTD-H model was compared with that of the GPT3 model. The mean bias of the GPT3 model was −1.14 cm (
Table 6). This indicated the GPT3 model had certain systematic errors. The bias intervals of the two models were similar; both ranged from −3 to 1 cm. The mean RMSE of the NGZTD-H and GPT3 models were 1.48 cm and 1.79 cm. Compared with the GPT3 model, the accuracy of the NGZTD-H model improved by 0.31 cm (17%). Therefore, the NGZTD-H model has higher accuracy for the spatial interpolation of the ZTD.
Figure 15 shows the stable results for NGZTD-H model at each GNSS station in China. Its bias was approximately 0 cm in most regions and reached −3 cm at individual stations in Southwest China, which indicated that the model struggles to express the vertical variation of ZTD in these regions. However, the result of the GPT3 model shows a large negative bias in most regions. The RMSE of the NGZTD-H model was smaller at high latitudes. However, the RMSE of the GPT3 model was greater than 2 cm at most GNSS stations. Therefore, compared with the GPT3 model, the NGZTD-H model has better stability in spatial interpolation.
The mean RMSE of spatial interpolation for the NGZTD-H model was 1.48 cm, which was apparently smaller than that of the NGZTD model in
Section 3.2.1 and
Section 3.2.2. The validation of spatial interpolation accuracy for the NGZTD-H model used ZTD obtained by the integral method as the starting ZTD and required horizontal interpolation. This indicated that the errors of the NGZTD model mainly derive from the starting ZTD rather than from horizontal interpolation.
As shown in
Table 2 and
Table 6, the spatial interpolation accuracy (the mean RMSE was 1.48 cm) of the NGZTD-H model is higher than the vertical interpolation accuracy (the mean RMSE was 1.70 cm). The validation of spatial interpolation accuracy for the NGZTD-H model required horizontal interpolation, whereas this is not required for vertical interpolation in
Section 3.1.1. This further indicated that the errors caused by horizontal interpolation were relatively small. According to the above analysis, the accuracy of ZTD vertical adjustment was higher in flat-terrain regions and lower in western China with undulating terrain. The GNSS stations are fewer in western China, whereas the distribution of ERA5 grid points is uniform, which resulted in a higher number of GNSS stations with smaller errors. Therefore, the accuracy of the spatial interpolation for the NGZTD-H model was higher than that of the vertical interpolation in
Section 3.1.1.
5. Conclusions
To overcome the limitations of existing ZTD models that lack an appropriate vertical adjustment function and are unsuitable for use in China in light of its complex climate and greatly undulating terrain, this study proposed an NGZTD model considering time-varying vertical adjustment based on the Gaussian function and diurnal variation. First, a ZTD vertical adjustment model, NGZTD-H, was developed that can adjust the ZTD from the starting height to the target height. Then, an NGZTD model that considers the seasonal and delicate diurnal variation in the ZTD was developed to estimate the ZTD directly. The vertical interpolation accuracy of the NGZTD-H model was validated, and the model exhibited stability for different geographic regions, pressure layers, latitudes, seasons, and elevations. The NGZTD model was validated using GNSS-derived ZTD and layered ZTD at radiosonde stations, and the results show that the NGZTD model performed better than the GPT3 model in time and space dimensions. In particular, in terms of capturing diurnal variation, the NGZTD model can effectively detect diurnal variation in the ZTD. Because the NGZTD model adopted the Gaussian function to adjust the ZTD vertically, the new model exhibited a more stable performance than the GPT3 model in western and southwestern China, where the terrain is undulating. Seasonal variations in Gaussian coefficients need to be considered. Using constant Gaussian coefficients will generate large errors. Finally, the error sources of the NGZTD model were discussed, which indicated that the starting ZTD was the main error source. In the future, we will further overcome the error sources of the NGZTD model to improve its accuracy and apply the proposed NGZTD model to precise point positioning and InSAR technology. The proposed model is expected to provide a reference for studying tropospheric parameters.