# Precise Orbit Determination of FY-3C with Calibration of Orbit Biases in BeiDou GEO Satellites
## 2 BeiDou Orbit Error Analysis and Mitigation
### Mitigation of BeiDou GEO Orbit Errors  
Reducing GEO orbit errors is similar to the phenomenon that errors in the orbit radial and along-track of the GPS can be mostly compensated by estimating the satellite clock offsets with a small size tracking station network. The small size is sufficient because of the small changes in the line-of-sight between whole stations and the satellite [25; 26]. Furthermore, the systematic constant biases in GEO orbit products can still be absorbed mainly in the PPP of static or low kinematic stations by estimating phase ambiguity parameters owing to the almost unchanged line-of-sight direction. Errors in the GEO clock products are not considered separately due to the strong correlation between the satellite clock offsets and radial errors in orbits. Besides, there is a visible improvement in the positioning performance of PPP for ground stations in the Asia-Pacific region by including BeiDou GEO. The batch least square processing strategy is adopted to adjust the estimated parameters in observation equations with the constraints of the LEO orbit dynamic models. The unknown parameters associated with the BeiDou phase observation equation for one GEO are expressed as Equation (1):  
\[\rho(t_{r},t^{s})=\rho(\mathbf{r_{0}},\mathbf{v_{0}},\mathbf{p},\ \mathbf{PCO},\ \delta t(t_{r}),AMB,\ dA,\ dc,\ dR) \tag{1}\]  
where \(t_{r}\) and \(t^{s}\) are the time of signal reception by receiver and the time of signal transmission by GEO transmitter, the symbol followed by \(t_{r}\) or \(t^{s}\) means that the physical quantity it represents changes over time \(t_{r}\) or \(t^{s}\), \(\rho(t_{r},t^{s})\) is the phase observation of GEO measured by the onboard receiver, \(\mathbf{r_{0}}\) and \(\mathbf{v_{0}}\) are the initial position and velocity vectors of the LEO center of mass at the reference time in a chosen conventional inertial frame, \(\mathbf{p}\) is the parameter collection associated with the carefully selected dynamic models such as atmospheric drag, solar radiation pressure, and additional empirical forces, etc. \(\mathbf{PCO}\) is the phase center offset vector of the LEO receiver antenna, \(\delta t(t_{r})\) is the LEO receiver clock offset at time \(t_{r}\), \(AMB\) is the constant phase ambiguity bias of the GEO in a tracking arc without cycle slips, \(dA\), \(dC\), and \(dR\) denote the calibrated constant biases in the GEO orbit along-track, cross-track, and radial components, respectively, which represent the average differences between the GEO true reference orbit and the published orbit. More detailed considerations of the BeiDou observation models are listed in Table 2 of Section 3.1. The partial derivatives of the carrier-phase and code pseudo-range observations with respect to the orbit biases of the GEO in the conventional inertial frame can be expressed as Equations (2) and (3):  
\[\partial\rho(t_{r},t^{s})=\mathbf{e}\rho^{\mathrm{T}}(t_{r},t^{s})(\mathbf{e}A(t^{s} )\partial dA+\mathbf{e}\mathbf{C}(t^{s})\partial d\mathbf{C}+\mathbf{e}\mathbf{R}(t^{s})\partial d \mathbf{R}) \tag{2}\]  
\[ \begin{array}{c}\mathbf{e}\rho(t_{r},t^{s})=\frac{\mathbf{R}(t^{s})-\mathbf{r}(t_{r})}{ \left\|\mathbf{R}(t^{s})-\mathbf{r}(t_{r})\right\|}\\ \mathbf{e}\mathbf{C}(t^{s})=\frac{\mathbf{R}(t^{s})\times\mathbf{V}(t^{s})}{\left\|\mathbf{R}(t^{s })\times\mathbf{V}(t^{s})\right\|}\\ \mathbf{e}\mathbf{R}(t^{s})=\frac{\mathbf{R}(t^{s})}{\left\|\mathbf{R}(t^{s})\right\|}\\ \mathbf{e}\mathbf{A}(t^{s})=\mathbf{e}\mathbf{C}(t^{s})\times\mathbf{e}\mathbf{R}(t^{s})\end{array}