## 1 Introduction  
Tropospheric delay is the main error source in Global Navigation Satellite System (GNSS) positioning and it is closely correlated with the positioning accuracy along up direction. [PERSON] et al. [1] illustrated that the zenith tropospheric delay (ZTD) product provided by International GNSS Service (IGS) can improve the positioning accuracy of both float and fixed solutions. In addition, tropospheric delay based on ground-based GNSS is an important data source in meteorology, which can be used to determine the distribution of atmospheric water vapor, and used for short-term weather and rainstorm disaster forecast [2]. Different from the relative positioning, we can obtain high-accuracy un-differenced tropospheric delay using the precise point positioning (PPP) technique with high efficiency. It has been demonstrated to be a promising technology in real-time positioning and meteorological applications [3]. In traditional PPP data processing, the initial tropospheric delay is first calculated by an empirical model, then the residual tropospheric delay is used as an estimated parameter. There are mainly three methods for acquiring initial tropospheric value: (1) the traditional empirical tropospheric correction model; (2) the numerical weather model (NWM)-based empirical tropospheric model; (3) the real-time NWM tropospheric model. The empirical tropospheric correction model and NWM tropospheric model can reflect the seasonal variation of the troposphere. However, it cannot reflect the spatial and temporal characteristics of the troposphere in the short term, especially under extreme meteorological conditions. Therefore, scholars proposed applying the real-time NWM tropospheric model for PPP research. [PERSON] et al. [4] proposed a four-system PPP method based on the NWM model; the experimental results showed that the tropospheric parameters obtained by NWM can improve the convergence times in the N/E/U direction by 20%/32%/25%, respectively, and the corresponding positioning accuracy can be improved by 2.5%/12.1%/18.7%. [PERSON] et al. [5] obtained zenith tropospheric delay (ZTD) and horizontal gradients from the NWM model to enhance BeiDou PPP; the results showed that the convergence times in the east and up direction were increased by 60% and 66.7%, respectively. [PERSON] et al. [6] proposed BeiDou and BeiDou/GPS-combined PPP based on the NWMmodel, and the results show that the convergence time of a single BeiDou PPP can be decreased by 80.6%, and the positioning accuracy can be improved by 34%. Apart from the tropospheric products from the NWM model, researchers have conducted in-depth studies on the estimation and application of the troposphere based on PPP technology. [PERSON] et al. [7] analyzed the tropospheric accuracy obtained by using real-time multi-GNSS orbit and clock products; the results showed that the accuracy of tropospheric products can be better than 1 cm. [PERSON] et al. [8] further adopted real-time ambiguity-fixed PPP for the tropospheric estimation, the root-mean-square (RMS) of the troposphere was 4.5 mm and 7.1 mm, respectively. It was demonstrated that the multi-mode GNSS ambiguity-fixed technique can improve the accuracy of troposphere compared to the GPS signal positioning. The obtained tropospheric products from PPP can be used for regional tropospheric modeling based on the GNSS reference station network. [PERSON] et al. [9] analyzed the accuracy of tropospheric delay from the regional GNSS network. The results showed that the standard deviation of the real-time tropospheric error was 8.8 mm and the RMS was 8.8 mm; PPP verification showed that the convergence time was significantly shortened. [PERSON] and [PERSON] et al. [10; 11] evaluated the use of the regional tropospheric model to enhance real-time PPP, and illustrated that the convergence time of PPP could be greatly shortened. Apart from the PPP application, high-precision tropospheric delay can be used for estimating precipitable water vapor (PWV). [PERSON] et al.