# A BDGIM-Based Phase-Smoothed Pseudorange Algorithm for BDS-3 High-Precision Time Transfer
## 2 BDGIM-Based Phase-Smoothed Pseudorange Algorithm  
### Bdgim  
With a revised BDGIM that has 9 broadcast ionospheric parameters and 17 non-broadcast ionospheric parameters, BDS-3 was finally finished in 2020 after the project started in 2017 [19; 20; 21]. The ionospheric non-broadcast parameters are solidified to the receiver terminals, while the broadcast parameters are communicated via navigation messages. Twenty-four sets of coefficients corresponding to 24 h are provided at once by the BDGIM, which is updated once every day. The ionospheric delay can be obtained by using the following formula:  
\[I=M_{F}\frac{40.28\times 10^{16}}{f^{2}}\cdot\left[A_{0}+\sum\limits_{i=1}^{9} \alpha_{i}A_{i}\right] \tag{1}\]  
where \(M_{F}\) is the mapping function, which can project the ionospheric delay from the station's zenith direction to that of the satellites and stations in their line-of-sight direction; f denotes the frequency of the signal; \(A_{0}\) is the predicted ionospheric delay determined from the ionospheric pierce point (IPP) and observation time; \(\alpha_{i}(i=1\sim 9)\) denotes the ionospheric broadcast parameter; and \(A_{i}(i=1\sim 9)\) denotes the Legendre function value associated with the geomagnetic latitude and longitude. The specific calculation of the symbols in Equation (1) is shown below.  
The mapping function \(M_{F}\) can be expressed as follows:  
\[M_{F}=\frac{1}{\sqrt{1-\left(\frac{R}{R+H}\cdot\cos(Ele)\right)^{2}}} \tag{2}\]  
where \(H\) is the height of the ionosphere, equal to 400 Km; \(R\) is the radius of the Earth, equal to 6,378,137 m; and \(Ele\) is the satellite elevation angle.  
\[A_{0}=\sum\limits_{j=1}^{17}\beta_{j}B_{j} \tag{3}\]  
\[\left\{\begin{array}{l}\beta_{j}=a_{0,j}+\sum\limits_{k=1}^{12}\left(a_{k,j} \cdot\cos\omega_{k}t_{p}+b_{k,j}\cdot\sin\omega_{k}t_{k}\right)\\ \omega_{k}=\frac{2\pi}{T_{k}}\end{array}\right. \tag{4}\]  
where \(a_{0,j},~{}a_{k,j}\), and \(b_{k,j}\) represent the non-broadcast parameters of the BDGIM; \(t_{p}\) stands for the odd hour of the day; and \(T_{k}\) denotes the forecast period.  
\[B_{j}=\left\{\begin{array}{l}N_{nj,mj}\cdot P_{nj,mj}(\sin\sigma^{\prime}) \cdot\cos(m_{j}\cdot\lambda^{\prime})~{}m_{j}\geq 0\\ N_{nj,mj}\cdot P_{nj,mj}(\sin\sigma^{\prime})\cdot\sin(-m_{j}\cdot\lambda^{ \prime})~{}m_{j}<0\end{array}\right. \tag{5}\]  
\[\left\{\begin{array}{cccccccccccc}n_{i}=&0&1&1&1&2&2&2&2&2&2\\ m_{i}=&0&0&1&-1&0&1&-1&2&-2\end{array}\right. \tag{6}\]  
where \(N_{nj,mj}\) represents the normalized function; \(P_{nj,mj}\) represents the standard Legendre function; and \(\varphi^{\prime}\) and \(\lambda^{\prime}\) represent the geomagnetic latitude and longitude of the IPP in the sun-fixed coordinate system, respectively. \(A_{i}\) is calculated in the same way as \(B_{j}\).  
Then, the vertical total electron content (VTEC) in the station's zenith direction can be expressed as  
\[VTEC=A_{0}+\sum\limits_{i=1}^{9}\alpha_{i}A_{i} \tag{7}\]In addition, the following restrictions are imposed in order to avoid negative VTECs calculated by the BDGIM:  
\[VTEC_{\max}=\begin{cases}\max(\alpha_{0}/10,VTEC)&\alpha_{0}\geq 35\\ \max(\alpha_{0}/8,VTEC)&20\leq\alpha_{0}<35\\ \max(\alpha_{0}/6,VTEC)&12\leq\alpha_{0}<20\\ \max(\alpha_{0}/4,VTEC)&else\end{cases} \tag{8}\]