# Precise Orbit Solution for Swarm Using Space-Borne GPS Data and Optimized Pseudo-Stochastic Pulses
## 3 Swarm Orbit Strategy  
ZD reduced-dynamic method is chosen in this article. Given that high-precise clock products and GPS orbits are essential, we choose 5 s rate clock data [31] and final GPS ephemeris [32], which are offered by the Center for Orbit Determination in Europe (CODE) Analysis Center. EIGEN-684 is a high-quality gravity field model which is used to improve the accuracy of a gravity field model in the Swarm ZD POD. Swarm ZD GPS code and phase observations with 10 s sampling interval are presented. A series of measures are used to select high-quality on-board GPS observation. For example, the elevation cut-off angle is set at 3\({}^{\circ}\), and the signal-to-noise ratio is set as a threshold of 5. Moreover, high ionospheric activity can easily lead to cycle slips. Thus, in order to better detect and repair cycle slips and provide good GPS data for improving Swarm orbits, we choose TurboEdit and COMBINED algorithm, which are widely used in LEO space-borne dual-frequency GPS data [33; 34]. A development orbit strategy of the ZD reduced-dynamic approach is applied to estimate the orbit parameters and relevant parameters (e.g., the values of phase ambiguities and receiver clock error corrections). Pseudo-stochastic parameters are expressed by pseudo-stochastic pulses in radial (R), along-track (T) and cross-track (N) directions with a time interval and a priori STD. Many scholars have conducted research on the reduced-dynamic POD of LEO: CHAMP, GRACE and GOCE. Pseudo-stochastic pulses are derived from experience. For example, under the condition \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Description** \\ \hline Force models & \\ Mean Earth gravity & EIGEN-684 (European Improved Gravity model of the Earth by New techniques) [16] \\ N-body & JPL DE405(Jet Propulsion Laboratory Solar System Ephemeris) [28] \\ Relativity & IERS 2010 (International Earth Rotation Service 2010) [29] \\ Solid-earth tides & IERS 2010 [29] \\ Ocean tides & FES2004(Modelling the global ocean tides: modern insights from FES2004) [30] \\ \hline GPS observation models & \\ Space-borne GPS data & code and phase observation, 10 s sampling interval \\ GPS orbits & Center for Orbit Determination in Europe (CODE) final GPS precise orbit, 15 min \\  & sampling interval \\ GPS clock & CODE final precise clock, 5 s sampling interval \\ GPS phase model & ig98.ack \\ Elevation cut-off & 3\({}^{\circ}\) \\ \hline Estimated parameters & \\ Six initial conditions & \(a,e,i,\Omega,\omega,T_{0}\) \\ Swarm clock bias & Bias epoch-wise \\ Ambiguity parameter & ZD ambiguity estimation \\ Pseudo-stochastic pulses & time interval and a priori standard deviation (STD) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The model standards adopted for Swarm zero-differenced (ZD) reduced-dynamic precise orbit determination (POD). that the time interval is 15 min and a priori STD is \(10^{-3}\) mm/s, the space environment of each LEO is different, applying previous empirical values to Swarm ZD reduced-dynamic POD may not obtain the best orbit solution. Therefore, the optimized pseudo-stochastic pulses of in-depth analysis and research are important in this article. Here, we offer a set of optimized empirical values to obtain better orbit solutions. Based on the following two considerations, Swarm PSOs released by ESA are considered as reference orbits in the paper. On one hand, Swarm PSOs released by ESA used piecewise constant acceleration in a reduced-dynamic orbit determination approach. Independent satellite laser ranging validation indicates that Swarm PSOs have accuracies over 2 cm [3]. Therefore, ESA released reliable Swarm PSOs. In the article, Swarm PSOs released by ESA are compared and analyzed. On the other hand, results of Swarm orbit accuracy are analyzed comprehensively from radial, along-track, cross-track and 3D directions. Therefore, results are compared with Swarm PSOs released by ESA to carry out follow-up related research.