# 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
 \tag{3}\]  
where \(\mathbf{e}\rho(t_{r},t^{s})\) is the unit vector from the receiver antenna to the GEO transmitter antenna, \(\mathbf{e}A(t^{s})\), \(\mathbf{e}\mathbf{C}(t^{s})\), and \(\mathbf{e}\mathbf{R}(t^{s})\) are unit vectors parallel to the GEO orbit along-track, cross-track, and radial directions, respectively, which are used to convert the coordinates of constant offsets from the orbital reference frame to the conventional inertial frame, \(dA\), \(dC\), and \(dR\) have the same physical unit as \(\rho(t_{r},t^{s})\), \(\mathbf{R}(t^{s})\) and \(\mathbf{V}(t^{s})\) denote the position and velocity vectors of the GEO in the inertial frame at time \(t^{s}\), \(\mathbf{r}(t_{r})\) is the position vector of receiver antenna in the inertial frame at time \(t_{r}\). Equation (2) indicates that the partial derivatives of range observations with respect to the GEO orbit biases can be expressed using \(\mathbf{e}\rho^{\mathrm{T}}(t_{r},t^{s})\mathbf{e}A(t^{s})\), \(\mathbf{e}\rho^{\mathrm{T}}(t_{r},t^{s})\mathbf{e}\mathbf{C}(t^{s})\), and \(\mathbf{e}\rho^{\mathrm{T}}(t_{r},t^{s})\mathbf{e}\mathbf{R}(t^{s})\), respectively. These variables represent the projective decomposition of the unit vector from receiver to satellite in theorbital reference frame, which are also known as the corresponding direction cosines. Consequently, the partial derivatives can be expressed as Equation (4):  
\[ \begin{split}\epsilon\rho^{\mathrm{T}}(t_{r},t^{s})eA(t^{s})& =\sin\theta(t_{r},t^{s})\cos\alpha(t_{r},t^{s})\\ \epsilon\rho^{\mathrm{T}}(t_{r},t^{s})eC(t^{s})&= \sin\theta(t_{r},t^{s})\sin\alpha(t_{r},t^{s})\\ e\rho^{\mathrm{T}}(t_{r},t^{s})eR(t^{s})&= \cos\theta(t_{r},t^{s})\end{split}  \tag{4}\]  
where \(\theta(t_{r},t^{s})\) is the angle between the vector from receiver to satellite and the GEO orbit radial direction, it is also known as the zenith of line-of-sight from receiver to satellite, \(\alpha(t_{r},t^{s})\) is the azimuth of vector from receiver to satellite in a planar coordinate system with base vector \(eA(t^{s})\) and \(eC(t^{s})\). Equation (4) shows that the impact factors of orbit errors in the satellite orbital reference frame on range errors depend on the values of zenith \(\theta(t_{r},t^{s})\) and azimuth \(\alpha(t_{r},t^{s})\) of the line-of-sight. A negative impact factor means that the positive orbit error will cause the negative range error. An impact factor of one unit means that both the orbit error and the caused range error have the same size and sign. Figure 2 shows the series variation during 14 July 2017 of zenith and azimuth angles of the line-of-sight from positioning antenna of FY-3C GNOS receiver to transmitter antenna of BeiDou GEO. The zenith angle varies in the range of 0\({}^{\circ}\)-10\({}^{\circ}\) while the azimuth angle can vary in the range of 0\({}^{\circ}\)-360\({}^{\circ}\). The corresponding partial derivatives of range observations with respect to the GEO orbit biases in the GEO orbital frame illustrate the sensitivity indicators of the range errors along the line-of-sight from FY-3C to the satellite on the specific orbit errors. By using the situation that the BeiDou C04 tracked by FY3C GNOS receiver during July 2017 as an example, Figure 3 shows the distribution for impact factors of orbit biases in along-track, cross-track, and radial on line-of-sight range errors from the satellite view of C04. As indicated in Figure 3, the coverage area on Earth where C04 can be observed by FY-3C is the service region of BeiDou C04 on FY-3C. It ranges in longitude from 80\({}^{\circ}\)E to 120\({}^{\circ}\)W and in latitude from 80\({}^{\circ}\)S to 80\({}^{\circ}\)N. The central projection of the service region for BeiDou C04 is 160\({}^{\circ}\)E on the equator. This is very close to the theoretical values according to the maximum zenith 10\({}^{\circ}\) for the average altitude 850 km of FY-3C. The impact factors of C04 orbit biases on range errors show south-north, east-west, and central symmetries in orbit along-track, cross-track, and radial, respectively.