## 3 Orbit Determination Method for GEO Satellite Based on RDSS Data
### Principles of Calibrating Time Delay for RDSS Equipments Based on SLR Data  
For calibrating RDSS equipment time delay based on SLR data, first the SLR data and RDSS data are preprocessed, then a reasonable dynamical model is selected, and, finally, SLR and RDSS data are jointly used for orbit determination. The estimated parameters are the satellite orbit, dynamic parameters, and the RDSS equipment time delay. All GEO satellites are processed together. The normal equations of the SLR data and the RDSS data are formed, respectively. The former only contains the partial derivatives of the satellite orbital parameters and dynamic parameters, while the latter contains the partial derivatives of the satellite orbital parameters, the dynamic parameters, and the partial derivatives of the RDSS time delay parameters. The final normal equations are formed by superpositioning both normal equations, and the joint orbit determination is completed through the least square method. The process flow chart of calibrating RDSS equipment time delay based on SLR data is shown in Figure 2.  
For SLR observation \(r_{m}^{i}\) and RDSS four-way ranging observation \(\rho_{CU}^{ij}\), with the error term removed, the observation equations are as follows:  
\[\begin{array}{l}r_{m}^{i}=2*\left|\mathbf{X}_{m}(t)-\mathbf{X}^{i}(t)\right| \\ \rho_{CU}^{ij}=\left|\mathbf{X}_{C}(t)-\mathbf{X}^{i}(t)\right|+\left|\mathbf{ X}_{U}(t)-\mathbf{X}^{i}(t)\right|+\left|\mathbf{X}_{U}(t)-\mathbf{X}^{i}(t) \right|+\left|\mathbf{X}_{U}(t)-\mathbf{X}^{i}(t)\right|+c\cdot\tau_{all}\end{array} \tag{3}\]  
where \(\mathbf{X}_{m}(t)\) is the coordinates of the SLR station \(m\), \(\mathbf{X}_{C}(t)\) and \(\mathbf{X}_{U}(t)\) are the coordinates of RDSS MCC station \(k\) and user \(l\), respectively, which are known quantities; \(\mathbf{X}^{i}(t)\) and \(\mathbf{X}^{i}(t)\) are the orbital positions of satellites \(i\) and \(j\), respectively; \(\tau_{all}\) is sum of the time delays of the out-station beam and the in-station beam.  
The error equation can be concluded by linearising Formula (4).  
\[V(t)=\begin{bmatrix}\frac{\partial r_{m}^{i}}{\partial\mathbf{X}(t_{0})}&0&0 \\ \frac{\partial r_{CU}^{i}}{\partial\mathbf{X}(t_{0})}&\frac{\partial\rho_{CU}^ {ij}}{\partial\mathbf{X}^{i}(t_{0})}&1\end{bmatrix}\begin{bmatrix}\mathbf{X}^ {i}(t_{0})\\ \mathbf{X}^{i}(t_{0})\\ c\cdot\tau_{all}\end{bmatrix}-L(t) \tag{4}\]  
where \(\mathbf{X}^{i}(t_{0})\) and \(\mathbf{X}^{i}(t_{0})\) are the initial orbital positions, velocities, and dynamic model parameters of satellites \(i\) and \(j\), respectively. Based on Equation (3), the RDSS equipment time delay is estimated using the least-squares batch method.  
Because SLR observation is subject to weather conditions, it is impossible to achieve all-weather observations. Therefore, the POD method by SLR and RDSS data cannot be a routine method for GEOs. While RDSS data can be obtained in real time, this paper adopts the SLR data to calibrate the time delay of the RDSS. RDSS data are used only for GEO satellite orbit determination, for which only the parts corresponding to RDSS data in Formulas (1) and (2) are used.