## 3 Ranging Error Source Analysis and Modeling
### Analysis and Modeling of LEO Satellite Orbit Disturbance
According to Table 2 and combined with the above selection rules on perturbation factors, we focus on the Earth's non-spherical perturbation and atmospheric resistance perturbation. Mathematical models of the Earth's nonspherical perturbation and atmospheric drag perturbation are very complicated. In addition, in view of the satellite orbit calculation, the accuracy of the perturbation calculation is not high and the speed of the perturbation calculation is required to be faster. In view of the various uncertainties of the drag effect itself, it is necessary to rely on measured data and statistical analysis to reduce it. However, SpaceX has not yet completed global deployment and relevant information is not publicly available; therefore, it is difficult for us to obtain public information about SpaceX to establish the Earth's nonspherical perturbation. Therefore, we refer to the research results of LEO satellite orbit perturbation in reference [36] and analyze the following simplified models of the Earth's nonspherical perturbation and atmospheric resistance perturbation:  
1 Earth nonspherical perturbation model  
The change in the position offset caused by the nonspherical gravitational perturbation of the Earth is periodic and its period is close to the operating period of the LEO satellite. However, the main trend of the offset is an approximately linear increase and the LEO satellites increase by approximately 32 km for each revolution. We give the following mathematical model after parameter coupling:  
\[F_{NS} =\{\alpha[\sin(\frac{2\pi}{T_{\text{LEO}}}t+\theta)(1+kt)]+N \alpha_{0}\}*(1+\kappa*\sigma_{rand}) \tag{21}\]  
where the value of \(\alpha\) is based on the offset of the satellite position after 24 h of the Earth's nonspherical gravity in the reference [36] and the value is 478.1518787694562 km; \(\theta\) represents the initial phase of the perturbation; here, we only consider the case of \(\theta=0\); \(T_{\text{LEO}}=108\) min, is SpaceX's satellite operation period; \(\alpha_{0}=32\) km is the orbit increment of the LEO satellite per revolution; \(N\) is the number of revolutions; here, we take \(N=14\); and the term \((1+kt)\) is mainly the linear increment in the simulated offset. According to reference [36], we can find that \(k\) is approximately equal to 0.3320 km/min. In addition, regarding some of the orbital interference caused by internal or external interference, we might also use random noise \(\sigma_{rand}\) with a mean value of 0 and a variance of 1 to simulate and use the selected perturbation accuracy \(\kappa*\) to describe its amplitude. 2 Atmospheric drag perturbation model  
The increase in position offset caused by atmospheric drag perturbation is nonlinear and the growth rate will gradually increase over time, which gradually strengthens the influence of atmospheric drag perturbation. Therefore, we fit 160 sets of simulation data extracted from reference [36] and the fitting results are as follows:  
\[F_{\text{LEO\_{drag}}}=0.003612+1.048\times 10^{-5}t+2.388\times 10^{-6}t^{2} \tag{22}\]  
Then, the mathematical model of the influence caused by the atmospheric drag perturbation after parameter coupling is:  
\[F_{DG}=F_{\text{LEO\_{drag}}}*(1+\kappa*\sigma_{rand}) \tag{23}\]Finally, we bring Equations (21) and (23) into Equation (16) and the total perturbation due to the aspherical perturbation of the Earth and the perturbation of atmospheric drag is:  
\[ \begin{array}{ll}\ddot{r}_{0}&=F_{NS}+F_{DG}\\ &=\{[a\sin(\frac{2\pi}{T_{\text{LEO}}}t+\theta)(1+kt)+Na_{0}]+F_{\text{LEO}\_ drag}\}*(1+\kappa*\sigma_{rand}).\end{array}  \tag{24}\]