## 2 Materials and Methods
### New Dynamic Pseudo-Observation Equation  
[PERSON]'s equation of motion for LEOs in the inertial frame can be expressed as [6]  
\[\overset{\to}{r}~{}=~{}-GM\frac{\overset{\to}{r}}{r^{3}}+\overset{\to}{f} (t,\overset{\to}{r},\overset{\to}{r},p_{1},\cdots p_{d}) \tag{2}\]  
where \(GM\) is the gravitational constant times the Earth's mass; \(r\) is the distance from the satellite to the Earth's center; and \(\overset{\to}{r}\), \(\overset{\to}{r}\), and \(\overset{\to}{r}\) are the position, velocity, and acceleration of the satellite, respectively. \(\overset{\to}{r}\) includes Earth's gravitation, Earth's solid and ocean tidal forces, third body perturbations (e.g., luni-solar gravitation), perturbations due to the relativistic effect, atmospheric drag, solar radiation pressure, and Earth radiation pressure. \(p_{1},\cdots p_{d}\) denote dynamical parameters considered as unknowns, e.g., scaling factors of analytically predicted accelerations.  
In this paper, we propose to use the STP of LEOs as an additional dynamic \"pseudo-observation\". Based on Equation (2), the STP equation for LEOs is derived (details shown in Appendix A) as follows [35]:  
\[\overset{\rightarrow}{r}(t+\Delta t)-2\overset{\rightarrow}{r}(t)+\overset{ \rightarrow}{r}(t-\Delta t)\ =\ \Delta t^{2}\overset{\rightarrow}{\int}\kappa \overset{\rightarrow}{r}(t+\tau^{\prime}\Delta t)d\tau^{\prime} \tag{3}\]  
where  
\[\kappa\ =\ \left\{\begin{array}{ll}1-\tau^{\prime}&0\leq\tau^{\prime}\leq 1\\ 1+\tau^{\prime}&-1\leq\tau^{\prime}<0\end{array}\right.,0\leq\kappa\leq 1\]  
\[\tau^{\prime}\ =\ \frac{\tau-t}{\Delta t},\ \ t\ -\Delta t\leq\tau\leq t+\Delta t\]  
\(\Delta\Delta(*)(t)\ =\ (*)(t+\Delta t)-2(*)(t)+(*)(t-\Delta t)\) is the second-order time-difference operator, in which \((t+\Delta t)\), \((t)\) and \((t-\Delta t)\) refer to three adjacent epochs. \(\Delta\Delta\overset{\rightarrow}{r}(t)\) stands for \(\overset{\rightarrow}{r}(t+\Delta t)-2\overset{\rightarrow}{r}(t)+\overset{ \rightarrow}{r}(t-\Delta t)\), which is the STP.  
Equation (3) directly establishes the relationship between the positions at three adjacent epochs \((\overset{\rightarrow}{r}(t+\Delta t),2\overset{\rightarrow}{r}(t),\overset{ \rightarrow}{r}(t-\Delta t))\) and accelerations \(\overset{\rightarrow}{r}(t)\). According to Equation (3), \(\Delta\Delta\overset{\rightarrow}{r}(t)\) is only related to the in-arc acceleration \(\overset{\rightarrow}{r}(t)\) and the square of the sampling interval \(\Delta t^{2}\). The acceleration of LEOs \(\overset{\rightarrow}{r}(t)\) in the inertial frame, which is the function of position, velocity, and time, can be computed precisely from the known dynamic models with the a priori orbit of LEOs [11; 14; 15; 16]. Moreover, the errors caused by the unmodeled dynamic signals can be treated as the error sources of the STP \"pseudo-observation\", and taken into account by the stochastic model of the STP \"pseudo-observation\". The \(\Delta\Delta\overset{\rightarrow}{r}(t)\) is computed by the numerical integration techniques used in the dynamic/RD method [36], and the practical procedure of the intergral STPs can refer to Section 3.1.2 in [35]. Therefore, combined with the kinematic observation equation, the intergrated STPs of LEOs can be used to estimate the precise positions of LEOs.