## 2 Open-Loop Tracking and Inversion Algorithm
### Mountain-Based Atmospheric Retrieval Algorithm  
The GNSS signal can generally be expressed as follows:  
\[L_{i}^{s,r}(t)=\rho^{s,r}(t)+c(s^{s}(t)+\sigma^{r}(t))+exL_{i}(t)+m^{s,r}(t)+n ^{s,r}(t)+N^{s,r}(t) \tag{1}\]  
\[exL_{i}^{s,r}(t)=Ion_{i}^{s,r}(t)+Att_{i}^{s,r}(t) \tag{2}\]  
where \(c\) is the speed of light, and \(L_{i}^{s,r}(t)\) is the observed phase of the transmitter (\(s\)) and receiver (\(r\)) of the i-th frequency at time (\(t\)). \(\rho^{s,r}(t)\) is the geometric distance between transmitter (\(s\)) and receiver (\(r\)) at time \(t\). \(\sigma^{s}(t)\), \(\sigma^{r}(t)\) are the clock difference of the transmitter (\(s\)) and receiver (\(r\)) at time (\(t\)), respectively. \(m_{i}^{s,r}(t)\) is the local multipath error [35; 36]; \(n_{i}^{s,r}(t)\) is the residual phase. \(N_{i}^{s,r}(t)\) is the ambiguity of the whole cycle. The excess phase \(exL_{i}^{s,r}(t)\) of transmitter \(s\) and receiver (\(r\)) of the \(i\)-th frequency at time (\(t\)) is generally defined as the sum of neutral atmospheric delay \(Att_{i}^{s,r}(t)\) and ionospheric delay \(Ion_{i}^{s,r}(t)\)[36; 37; 38; 39; 40]. Based on Internet GNSS Servers (IGS) products and low-orbit satellite precision orbit determination products, it is possible to obtain a sufficiently accurate distance \(\rho^{s,r}(t)\) from the navigation satellite transmitter to the low-orbit satellite receiver and its clock difference \(\sigma^{s}(t)\). The local multipath error can be ignored in mountain-based occultation. The receiver clock error \(\sigma^{r}(t)\) can generally be eliminatedby selecting a higher altitude navigation star as a reference and applying the satellites' phase difference. As a result, high-precision excess phase observations of the occultation can be obtained, which only include the influence of the ionosphere and atmosphere through the path. In the process of spaceborne neutral atmosphere inversion, the combination of different frequency point bending angle sequences can better eliminate the ionospheric influence. In extracting the excess phase, only the ionospheric influence of the reference star can be eliminated. In the mountain-based atmospheric occultation inversion, the atmospheric state information from the ground to the top of the mountain can be obtained through the relationship between the impact height observed by the positive and negative altitude angle. As shown in Figure 1, observation of negative elevation angle (light and dark green thick line, bending angle is \(\alpha_{-}\)) can be regarded as the Abel integral (light green thick line) and positive elevation angle (light blue thick line, bending angle is \(\alpha_{+}\)) from the top to the impact height, namely [39; 40; 41; 42]:  
\[\alpha_{-}(a)=-\alpha_{+}-2a\int_{a}^{n_{1}r_{1}}\frac{\frac{dmn}{dx}}{\sqrt{ x^{2}-a^{2}}}dx \tag{3}\]  
Let  
\[\widetilde{\alpha}(a)=\alpha_{-}(a)-\alpha_{+}(a)=-2a\int_{a}^{n_{1}r_{1}} \frac{\frac{dmn}{dx}}{\sqrt{x^{2}-a^{2}}}dx, \tag{4}\]  
Figure 1: Schematic diagram of mountain base inversion. M stands for the top of the mountain, where the antenna and receiver are installed, and Global Navigation Satellite System (GNSS) stands for the navigation satellite we want to receive, that is, the navigation satellite that generates the occultation event. then the refractive index \(n\) of the impact height \(\alpha\) (\(n_{1}\) is the peak refractive index, \(\xi\) is the maximum impact height of the occultation) is:  
\[n(a)=n_{1}exp\Bigg{(}\frac{1}{\pi}\int_{a}^{n_{1}r_{1}}\frac{\overline{\alpha}( x)}{\sqrt{\xi^{2}-a^{2}}}dx\Bigg{)} \tag{5}\]  
Thus, the refractive index profile from the ground to the top of the mountain can be obtained from the mountain-based occultation data [42; 43]. In the above process, the influence of the atmosphere or ionosphere on the observation data above the top of the mountain can be eliminated in the process of searching for the symmetrical impact height and its combination.