# Centimeter-Level Precise Orbit Determination for the Luojia-1A Satellite Using BeiDou Observations
## 5 Luojia-1A Satellite POD Results and Discussion
### Precise Orbit Determination with BeiDou Observations
However, the GEO satellite observations of the Luojia-1A satellite contribute to more than 40% observations. Excluding the GEO satellite observations will dramatically reduce the number of effective observations and lead to a negative effect on the reliability of the POD results. To determine the precise orbit of the Luojia-1A satellite, the satellite state vector can be expressed as follows ([PERSON] et al., 2020):  
\[x(t)=(\mathbf{r}(t),v(t),p,q)^{T} \tag{4}\]  
where \(\mathbf{r}(t)\) and \(\mathbf{v}(t)\) are the position and the velocity of the LEO satellite at epoch \(t\). \(p\) represents the force model parameter, e.g., the scaling factors of the perturbing accelerations acting on the LEO satellite. \(q\) is the pseudo-stochastic orbit parameter, which compensates for the inaccuracy force model. The observations equation can be briefly described as  
\[\mathbf{z}=\mathbf{h}(\mathbf{x}_{0})+\varepsilon \tag{5}\]  
where \(\mathbf{z}\) is the n-dimensional observation vector and \(\mathbf{x}_{0}\) is the initial state vector. The recursive least-square estimator is adopted in LEO POD and the weighted least-squares can be written as:  
\[\Delta x_{0}^{\text{I}sq}=\Big{(}H^{T}WH\Big{)}^{-1}\Big{(}H^{T}W\Delta z \Big{)} \tag{6}\]  
where the \(H\) matrix is the partial derivatives of the observations concerning the state vector at the reference epoch \(t_{0}\), \(W\) is the weighting matrix, and \(\Delta z=z-\mathbf{h}\big{(}x_{0}^{ref}\big{)}\). Generally, the weight of different satellites observations is usually set as the same:  
\[P=\frac{\sigma_{0}^{2}}{\sigma_{i}^{2}} \tag{7}\]  
where the \(\sigma_{0}\) is a priori unit weight factor, which is 0.001m in this study. However, as for the BeiDou system, the precision of GEO satellites orbit is not as good as that of IGSO and MEO satellites. To adopt the GEO satellite observations to improve the reliability of the Luojia-1A precise orbit and reduce the effect from the orbit error of GEO satellites, the strategy of estimation that reduces the weight of GEO satellites observations are proposed in this study. To fully discover the effect of the reduced weight strategy, six different weighting strategies are designed and listed in Table 4. In this study, the IGSO  
Figure 8: The average day boundary of BDS orbit products. \begin{table}
\begin{tabular}{c c c c} \hline  & **GEO** & **IGSO** & **MEO** \\ \hline
3D DBD & 82.28 & 26.74 & 17.00 \\ \hline \end{tabular}
\end{table} 
Table 3: The average 3D day boundary of different orbital type of BeiDou satellites (Unit: cm). and MEO satellites are treated as the equal-weighted, and the weight of GEO satellite observations is relative to the IGSO and MEO satellite observations; all the IGSO and MEO satellite observations are set unit weight. The precise orbit determination results of the Luojia-1A satellite with six different weighting strategies are listed in Table 5, and the equal weight of all satellites strategy is used as the reference; the LEO POD precision can be improved by re-weighting the GEO satellite observations. According to the results, the strategy that reduces the weight of GEO satellites can improve the BDS based Luojia-1A satellite POD significantly. As the weight reduces to 1/49, the improvement of precision can reach 44.21%. Moreover, as the weight of GEO satellites is reduced, the precision of Luojia-1A orbit is also improved gradually. Luojia-1A satellite orbit precision can be improved to 2.316 cm in the 3D direction by reducing the GEO satellite weight down to 1/900, in which the RMS of radial direction is better than 1 cm and the improvement is 63.43% refers to the equal-weight strategy. The POD precision will not be continuously improved as the GEO satellite weight reduces to 1/2500. Therefore, 1/900 is considered as the optimal GEO weighting strategy for the Luojia-1A satellite POD. The overlapping orbit comparison result of the Luojia-1A satellite precise orbit is shown in Figure 9, which represents the LEO orbit computed from the different GEO weighting strategies.