## 2 Methods
### Regional Atmospheric Enhancement Rapid Precise Positioning System  
The architecture of the regional atmospheric enhancement rapid precise positioning system is illustrated in Figure 1. It is divided into three parts: server side, atmosphere product generation, and user side. Among these, accurate atmospheric correction information is crucial for achieving fast and accurate PPP-RTK ambiguity resolution. After fixing the ambiguity at the reference station, the atmospheric information at the reference station can be accurately determined. This allows the atmospheric information at the user station to be derived through atmospheric modeling of the reference station, thereby improving the speed of ambiguity resolution at the user station. The atmospheric enhancement at the rover is performed using inverse distance interpolation. Assuming there are three reference stations around the rover, the process for inverse distance interpolation is as follows [32]:  
\[\left[I_{U}^{\xi,\mathcal{S}}\right]=\left[\frac{1}{d_{R_{1}}}\quad\cdots\quad \frac{1}{d_{R_{1}}}\right]\left[ \begin{matrix}\frac{1}{D}&0&0\\ 0&\cdots&0\\ 0&0&\frac{1}{D}\end{matrix} \right]\left[ \begin{matrix}I_{1}^{\xi,\mathcal{S}} \\ \cdots\\ I_{i}^{\xi,\mathcal{S}}\end{matrix} \right] \tag{3}\]  
where \(I_{U}^{\xi,\mathcal{S}}\) is the ionospheric error of satellite \(p\) on rover station \(U\); \(d_{R_{i}}\) represents the distances from the three reference stations \(i\) to the rover; and \(D\) is the sum of the reciprocal distances from the rover to the three reference stations. In this way, it can effectively ensure that the interpolation coefficient is 1, ensuring that the code hardware biases of the satellite and receiver ends included in the ionosphere at the rover remain unchanged. Therefore, the code hardware biases in the ionosphere of the rover can still be absorbed by clock bias or eliminated by a single difference from the satellite; thus, the fixed ambiguity of the rover is not affected. The inverse distance weighting method is also used for interpolation of tropospheric delay; however, due to the strong correlation between tropospheric errors and height, the influence of height on the troposphere needs to be considered when performing tropospheric interpolation. When there is a significant difference in height between the reference stations and the rover, it is necessary to convert the tropospheric height before interpolation. The GPT3 (Global Pressure and Temperature 3) model can be used to solve the tropospheric error between different heights and perform height correction; alternatively, the vertical reduction coefficient can be used for height conversion [33]. In this article, the vertical reduction coefficient is used. [PERSON] et al. (2013) used years of reprocessed atmospheric data to fit and obtain a prior exponential function model as follows [34]:  
\[ZTD_{h}=ZTD_{h_{0}}\cdot\exp(k(h-h_{0})) \tag{4}\]  
In the formula, \(k=-1.3137\times 10^{-4}\). By inputting the height of the troposphere with the target height, the conversion of tropospheric height can be completed. The interpolated tropospheric and ionospheric products have high accuracy, but there are still centimeter-level errors, and even after correction, parameter estimation is still necessary. If correction is directly made in the observation equation, it leads to inaccurate stochastic models and causes problems in parameter estimation. To avoid this issue, the influence of ionospheric and tropospheric correction is added numerically to the original observation equation in the form of constraint equations, as follows:  
\[E(\Delta p_{r,i}^{\xi,\mathcal{S}})=H\cdot x_{r}+c\cdot(\widetilde{d}t_{r}- \widetilde{d}^{\xi,\mathcal{S}})+m_{r}^{\xi,\mathcal{S}}\cdot T_{r}^{\xi, \mathcal{S}}+u_{i}\cdot\widetilde{T}_{r,i}^{\xi,\mathcal{S}}+M_{i,r}^{\xi, \mathcal{S}} \tag{5}\]  
Figure 1: The architecture of the regional atmospheric enhancement rapid precise positioning system.