## 2 BDS-3 Satellites POD Method
### Satellite Equations of Motion and Observation Models
\(\phi(t,t_{0})\) is the state transfer matrix, which can represent the partial derivatives of the position and velocity of the satellite at any moment concerning the initial partial derivative. \(S(t)\) is a sensitivity matrix, that describes the partial derivatives of the state vector and dynamical parameters. In the actual solution process, the state transfer matrix \(\phi(t,t_{0})\) is usually obtained by the numerical integration method due to the complexity of the relevant perturbing force function on the satellite. The GNSS raw observation equations are expressed in terms of pseudorange and carrier phase, depending on the amount of observation:  
\[P_{i,r}^{s}=\rho_{r}^{s}+c\delta t_{r}-c\delta t^{s}+\mu_{i}I_{r}^{s}+Rm_{P_{i },r}^{s}+\varepsilon_{P_{i}} \tag{13}\]  
\[L_{i,r}^{s}=\rho_{r}^{s}++c\delta t_{r}-c\delta t^{s}-\mu_{i}I_{r}^{s}+\lambda _{i}N_{i,r}^{s}+Rm_{P_{i},r}^{s}+\varepsilon_{L_{i}} \tag{14}\]  
where: \(s\) indicates satellite, \(r\) indicates Earth station, \(\rho_{r}^{s}\) is the geometric distance of the station from the satellite, \(i\) is a pseudorange or a frequency point of the carrier phase, and \(\delta t\) is the clock offset; \(\mu_{i}\) is the ionospheric delay factor at the \(i\) frequency point and \(I_{r}^{s}\) is the ionospheric delay corresponding to the observed quantity; \(\lambda_{i}\) indicates the wavelength of the frequency \(i\), \(N_{i,r}^{s}\) is the ambiguity parameter, \(Rm\) are tropospheric delays, relativistic effects, multipath effects, antenna phasecenter errors, and other errors; and \(\varepsilon_{P_{i}}\) and \(\varepsilon_{L_{i}}\) are the pseudorange and phase observation noise, respectively. Usually, instead of using the original observations directly, various combinations of pseudorange and carrier phase are used to eliminate or attenuate the associated errors. Combinations of measurements that eliminate the effect of the first-order term in the ionosphere are the most widely used. In this case, the combined pseudorange and carrier phase quantities can be expressed as:  
\[ \begin{split} P_{LC,r}^{s}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}} P_{1,r}^{s}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}P_{2,r}^{s}\\ L_{LC,r}^{s}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}L_{1,r}^{s}- \frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}L_{2,r}^{s}\end{split}  \tag{15}\]  
where \(P_{i}\), \(L_{i}\) (\(i\) = 1, 2) indicates pseudorange and carrier phase observations and \(f_{i}\) represents different frequencies. Assume that the coordinates of the station under the Earth-fixed coordinate system are \(X_{r,s}=(x_{r,s},y_{r,s},z_{r,s})\). The first-order Taylor series expansion of the satellite on \(X^{s,s}=(x^{s,s},y^{s,s},z^{s,s})\) reference orbit. \(X^{s,s}\) is defined as the reference orbit of the satellite [17; 18; 29]. According to Equations (1) and (14), the linearized equation can be obtained as:  
\[ \begin{bmatrix}v_{PC}\\ v_{LC}\end{bmatrix} = \begin{bmatrix}-u_{r}^{s}&u_{r}^{s}&1&1&-1&0\\ -u_{r}^{s}&u_{r}^{s}&1&1&-1&1\end{bmatrix} X_{\text{all}}- \begin{bmatrix}l_{PC }\\ l_{LC}\end{bmatrix}  \tag{16}\]  
where \(v_{PC}\) is the pseudorange observation error, \(v_{LC}\) is the carrier phase observation error, and \(l_{PC}\) and \(l_{LC}\) are the difference between the estimate of the pseudorange and the carrier phase observation. The coordinates of the ground-monitoring station in the inertial system can be expressed as \(X_{r,s}^{r}=R\cdot X_{r,s}\). \(R\) denotes the corresponding transformation matrix of the coordinate system, \(u_{r}^{s}=\left[\dfrac{x_{r,s}^{\prime}-x^{s,s}}{\rho_{r,s}^{s}}\quad\dfrac{y_{r, s}^{\prime}-y^{s,s}}{\rho_{r,s}^{s}}\quad\dfrac{z_{r,s}^{\prime}-z^{s,s}}{\rho_{r,s}^{ s}}\right]\), \(X_{all}=\left[dX^{s}\quad dX_{r}\quad dRm\quad t_{r}\quad t^{s}\quad N\right]^{T}\), and \(X_{all}\) is the transpose matrix of all the correction values. \(dX^{s}\) is the correction of the satellite position, velocity, and perturbing force model, etc., concerning the reference orbit.