## 2 Materials and Methods
### Function Model of the RD_STP Method  
Combining Equations (1) and (3), the observation model of the RD_STP method for the POD of LEOs is expressed as follows:  
\[ \begin{array}{ll}P_{c,L}^{G_{k}}(t)&=\overset{\rightarrow}{e}_{L}^{G_{k}}( t)\cdot d\overset{\rightarrow}{r}_{L}(t)-cdt_{L}(t)+\varepsilon_{P_{c,L}}^{G_{k}}(t)\\ L_{c,L}^{G_{k}}(t)&=\overset{\rightarrow}{e}_{L}^{G_{k}}(t)\cdot d \overset{\rightarrow}{r}_{L}(t)-cdt_{L}(t)+\lambda_{c}N_{c,L}^{G_{k}}+ \varepsilon_{L_{c,L}}^{G_{k}}(t)\\ \Delta\Delta d\overset{\rightarrow}{r}_{L}(t)&=\Delta\Delta\overset{ \rightarrow}{r}_{L}(t)-\Delta\Delta\overset{\rightarrow}{r}_{L,0}(t)\\ &=\Delta t^{2}\overset{\rightarrow}{\int}\ (1-|\tau^{\prime}|)\overset{ \rightarrow}{r}_{M}(t+\tau^{\prime}\Delta t)d\tau^{\prime}-\Delta\Delta \overset{\rightarrow}{r}_{L,0}(t)+\varepsilon_{\Delta\Delta\overset{ \rightarrow}{r}_{L}}(t)\end{array}  \tag{4}\]  
where \(P_{c,L}^{G_{k}}(t)\) and \(L_{c,L}^{G_{k}}(t)\) are ionosphere-free linear combinations; \(\Delta\Delta\overset{\rightarrow}{r}_{L}\) is integrated via the right part of Equation (3); \(\overset{\rightarrow}{r}_{M}\) is the acceleration of the LEOs; and \(\overset{\rightarrow}{r}_{L,0}(t)\) is an a priori LEO orbit, e.g., from a code solution. Most of the force (e.g., Earth gravitation, luni-solar gravitation, tide forces) acting on LEOs is only function of time and position. Thus, the STPs \(\Delta\Delta\overset{\rightarrow}{r}_{L}\) can be computed precisely with the known dynamic models and an a priori LEO orbit \(\overset{\rightarrow}{r}_{L,0}(t)\). Linearizing Equation (4) yields the following:  
\[y_{L}^{G_{k}}\ =\ {B_{L}^{G_{k}}\cdot dX_{L}^{G_{k}}+\varepsilon_{L}^{G_{k}}} \tag{5}\]  
where  
are the partial derivatives of the a priori orbit \(\overset{\rightarrow}{r}_{\text{L,0}}(t)\) to the satellite's initial conditions and the dynamical parameters, respectively. The errors caused by the unmodeled dynamic signals are absorbed by the unknown dynamical parameters or pseudo-stochastic orbit parameters. It is obvious that both the STP equation Equation (3) in the RD_STP method and Equation (7) in the traditional dynamic/RD method are based on [PERSON]'s equation of motion. However, compared with the dynamic/RD method, the essence of the RD_STP method is that the STPs are taken as an additional dynamic \"pseudo-observation\" with errors to constrain the GPS observations. Moreover, all the unmodeled dynamic signals are treated as the error sources of the STP pseudo-observation, rather than absorbed by the unknown dynamical parameters or pseudo-stochastic orbit parameters. The main characteristics of the RD_STP method for the POD of LEOs are summarized in Table 1. For comparison, the characteristics of the kinematic, dynamic, and RD POD methods are also given in Table 1 and shown in Figure 1. As shown, the RD_STP method has four main differences compared with the dynamic and RD methods. First, the RD_STP method integrates the acceleration over a very short arc (e.g., \(2\Delta t\ =\ 60\) s) to compute the STPs, while the dynamic and RD methods usually use the long-arc integration (e.g., 24 h or 30 h) [11], Thus, the short-arc integration reduces the accuracy requirements for the dynamic models. Second, the known minor perturbation forces (e.g., Earth's solid and ocean tidal forces, third body perturbation, and the non-conservative forces) can be ignored if the sampling interval is less than 30 s. Third, an Earth's gravity field model with low degree and order (e.g., up to degree and order 90) is sufficient, even for LEOs. Certainly, similar to the dynamic/reduced-dynamic method, the perturbation forces (<10\({}^{-5}\) ms\({}^{-2}\)) can be taken into account for integrating the STPs into the proposed RD_STP method as well, if a stronger dynamic constraint on the POD solutions is desired. Finally, there are no dynamic parameters or pseudo-stochastic parameters to be estimated in the proposed method.