## 1 Introduction  
Low Earth orbit (LEO) satellites are considered as key technologies for space missions due to their advantages of flexibility, redundancy, efficiency, and low cost. To fulfill scientific mission requirements, precise absolute or relative positions are required. The precise orbit determination (POD) capability, based on spaceborne Global Positioning System (GPS) observations, has been successfully verified on the TOPEX/Poseidon satellite [1; 2]. The reduced dynamic POD technique [3; 4] combined with the strength of GPS observations has shown its advantages in sampling rate and accuracy. Since then, more GPS receivers have been deployed on LEO satellites to meet the needs of various scientific missions for high precision orbit. The quality of global navigation satellite system (GNSS) derived LEO orbits has steadily improved thanks to numerous improvements in the GNSS orbit and clock products provided by the International GNSS Service (IGS) [5], in the dynamic background models, and in modeling the carrier phase observations. LEO orbit accuracy of 1-3 cm can be fulfilled with float ambiguity (FA) [6; 7]. With GPS precise orbit and clock products provided by IGS, CHAMP orbit calculated with ionosphere free (IF) observations achieves the accuracy of centimeter level [8]. GRACE orbit determined using GPS has been proven by independent satellite laser ranging (SLR) and K-band ranging (KBR) data, and 1 cm radial orbit accuracy has been achieved [9]. JASON's GPS derived orbit is evaluated based on SLR residuals, dynamic orbit produced by SLR/DORIS data, and altimeter crossover tests [10; 11], and the sub-centimeter radial accuracy is verified. Similar results are also obtained in other LEO satellite missions, such as GOCE [12; 13], Sentinel [14; 15], TerraSAR-X [7], HY [16], and FY-3 [17; 18] satellites. Ambiguity resolution (AR) and in-flight antenna phase center variation (PCV) calibration are essential to fully exploit the precision of GPS observations for POD. In order to fix the carrier phase ambiguities, a Kalman filter modeling the relative spacecraft dynamics has been developed for the GRACE mission and the double difference (DD) ambiguities are resolved to fully exploit the inherent measurement accuracy. Finally, the resulting GRACE orbit matches the KBR measurements with an accuracy of 1 mm [19]. A GRACE POD based on undifferenced and doubly differenced GPS data is studied [6]. Different baselines, including the space baseline between two GRACE satellites, the space-ground baselines consisting of LEO satellites and ground stations, and both types of baselines together, are processed. Results show that fixing of the GPS DD ambiguities has a significant impact on the space baseline [6]. With the orbit solution constrained by resolved DD ambiguities, a GRACE baseline accuracy of 2 mm is also achieved [20]. The double difference ambiguity resolution (DD AR) approach requires two satellites to construct the baseline. Several methods have been developed to resolve the integer ambiguity for the single receiver user. The idea of a single receiver integer ambiguity resolution forms the basis of PPPRTK. Taking the ionospheric delay as the unknown parameter, the common clock model [21; 22] and distinct clocks model [23; 24] are proposed to perform the single receiver ambiguity resolution. For IF formulation, the integer recovery clock (IRC) model [25; 26], decoupled satellite clock (DSC) model [27], and uncalibrated phase delay (UPD) or fractional cycle bias (FCB) model [28; 29] are presented. The S-system theory is applied by [PERSON] to establish the linkage among different PPP-RTK methods [30; 31]. Their differences are shown to lie (a) in the choice of S-basis; (b) in the choice of parameterization; (c) in the choice of whether or not to eliminate the ionospheric delay. With wide-lane (WL) and phase biases provided by the Jet Propulsion Laboratory (JPL), the single receiver AR of GRACE and JASON-2 satellites is realized [32].