## 4 Conclusions  
Differing from the traditional dynamic/RD method and the kinematic method, we propose a new reduced-dynamic method for the POD of LEOs using GPS code and carrier phase observations. Given that the STP directly establishes the relationship between the position and acceleration of the LEOs, and can be easily and precisely computed from known dynamic models, we used it as an additional \"pseudo-observation\" equation for the dynamic constraint on the kinematic POD observation equation. Moreover, all the unmodeled dynamic signals are taken into account by the stochastic model of the STP \"pseudo-observations\" in the RD_STP method, rather than being absorbed by the unknown dynamical parameters or pseudo-stochastic orbit parameter. The theoretical and numerical analyses show that, when the accuracy of the a priori force models is \(10^{-5}\) ms\({}^{-2}\) over a short time span (e.g., \(2\Delta t~{}=~{}60\)s), the accuracy of the integrated STPs is better than 0.01 m, which is good enough for the POD of LEOs. Therefore, compared with the traditional dynamic/RD method, the minor perturbation forces with a magnitude less than \(10^{-5}\) ms\({}^{-2}\) (such as luni-solar gravitation, tidal forces, and atmospheric drag) acting on the satellite can be ignored and treated as the error sources of the STP pseudo-observation. Further, only the Earth's gravity field model (such as EGM2008, EIGEN5S) up to degree and order 90 is sufficient for the POD because its omission errors are less than \(10^{-5}\) ms\({}^{-2}\). This outcome means that the dependence of the RD_STP method on Earth's gravity field model is reduced compared with the traditional dynamic/RD method. The POD experiment of GOCE based on the RD_STP method shows that the RD_STP solution is consistent with the RD solution in terms of accuracy and is more accurate and smoother than the KIN solution in the regions near the equator and poles. The 3D RMS of the differences between the RD_STP and RD solutions is 1.93 cm, 2.53 cm, and 2.77 cm for 1 s, 10 s, and 30 s sampling, respectively. These experimental results indicate that the RD_STP method is effective and can be an alternative for the POD of LEOs. In addition, if \(\sigma_{\Delta\Delta t_{L}}~{}=~{}\infty\), the RD_STP method would be exactly equivalent to the kinematic method, which means that the RD_STP solutions have nothing to do with the a priori dynamic model. Thus, the RD_STP method can adjust the influence of the dynamic information on the POD solutions by modifying the stochastic model of the STP pseudo-observations, and might also be an alternative to estimating the orbits in the context of gravity field determination such as the highly-reduced-dynamic orbits [7]. Conceptualization, H.W. and X.X. ; Data curation, H.W. and K.K. ; Formal analysis, H.W. ; Funding acquisition, H.W., J.L. and X.X. ; Investigation, H.W. ; Methodology, H.W. and J.L. ; Resources, H.W. and J.L. ; Software, H.W., S.Z. and K.K. ; Validation, X.X. ; Writing--original draft, H.W. and X.X. ; Writing--review & editing, H.W. and X.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the Foundation for Innovative Research Groups at the National Natural Science Foundation of China (grant no. 41774020,42074019), the German Academic Exchange Service (DAAD) Thematic Network Project (grant no. 57421148), and the fellowship of China National Postdoctoral Program for Innovative Talents (grant no. BX20200251). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mean-RMS (cm)** & **R** & **T** & **N** & **3D** \\ \hline KIN-10 s & 2.63 & 2.28 & 2.42 & 4.25 \\ KIN-30 s & 2.74 & 2.42 & 2.55 & 4.47 \\ STPRD-10 s & 1.52 & 1.33 & 1.50 & 2.53 \\ STPRD-30 s & 1.69 & 1.46 & 1.63 & 2.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean RMS of the differences between the solutions (KIN-10 s, KIN-30 s, RD_STP-10 s, and RD_STP-30 s) and the RD solutions for DOY 318–332, 2009.