## 5 Simulation  
### Validating the GDOP Using Monte Carlo Simulation Experiments  
A scenario in which a low-orbiting optical sensor observes a high-orbiting target was designed. The classical orbit elements of the optical sensor orbit were (7000 km, 0.01, 170\({}^{\circ}\), 50\({}^{\circ}\), 0\({}^{\circ}\), 60\({}^{\circ}\)), and those of the target orbit were (40,000 km, 0, 60\({}^{\circ}\), 0\({}^{\circ}\), 60\({}^{\circ}\), 40\({}^{\circ}\)). The orbital data were simulated over the coming 48 h, with the line-of-sight not being blocked by the Earth as the visible condition. The specific judgment conditions are shown in Figure 1. The target is blocked by the Earth when \(r\sin\varphi<R_{\rm Earth}\) and \(\varphi<\frac{\pi}{2}\). Here, \(r\) is the sensor orbit radius; \(R_{\rm Earth}\) is the radius of the Earth; and \(\varphi\) is the angle subtended by the Earth, the sensor, and the target. Then, 400 s continuous observation data were selected in different visible arc segments. The observation interval was 10 s, and the observation error was five arc seconds. The GDOPs of different observable arc segments obtained through simulations using Equation (10) are shown in Figure 2. Monte Carlo simulations were performed 100 times under each observation condition, following which, the covariance matrix and the GDOP were calculated. The results of the simulation experiment are shown in Figure 3. It is worth noting that the situation of the UVM's iterative process not converging occurred in the simulation experiments. If the iterative process of the UVM does not converge, the data are discarded. The iterative process is very complex. [PERSON] [20] characterized the convergence behaviours of the orbit determination algorithm applied to the three angles-only data. There is no perfect method to determine whether UVM applied to multiple angles-only data converges so far. As shown in Figure 2, it is reasonable to calculate the GDOP using the analytic form of the target state obtained by assuming that the values of \(f_{t}\), \(g_{t}\), and \(\rho_{t}\) under short-arc conditions are constant. It can also be seen that the orbit determination accuracy varies periodically at different visible arc segments when observing the same target continuously. The period is approximately the orbital period of the optical sensor located in a lower orbit. Figure 1: Judgment conditions for visibility. was 400 s, and the observation error was 5 arc seconds. The GDOPs corresponding to \(0<\alpha_{\text{sensor}}<2\pi\) and \(0<\alpha_{\text{Target}}<2\pi\) were simulated as the starting observation times at different values of \(i\). This corresponds to the accuracy of the initial orbit determination for all possible observation geometries under the specified radius of the target and optical sensor. It can be used to express the effect of the observation geometry on initial orbit determination. The simulation results are shown in Figure 4.  
Figure 4: _Cont_. differences in the GDOP for the same angle between orbital planes. Further, the GDOP is significantly larger when the right ascensions of the target and the optical sensor are close to \([0^{\circ},0^{\circ}]\), \([180^{\circ},180^{\circ}]\). The declination of the optical sensor is 0 when the right ascension of the optical sensor is 0\({}^{\circ}\), 180\({}^{\circ}\) and 360\({}^{\circ}\). At these times, the right ascension and declination of the target and the optical sensor are the same; the right ascension and declination of the target and the optical sensor in the natural coordinate system are also the same. It is important to note that, if we do not consider Earth blocking, it can be summarized that the GDOP is significantly larger when the target and the optical sensor are both close to the orbital plane intersection line than the other situations. The values of GDOP differ by more than 10 times. The actual calculation of the initial orbit with a large GDOP often results in iterative divergence.