## 2 Model for Attitude, Antenna PCO/PCV and Its POD Strategies
### Jason-3 Satellite Attitude
#### 2.1.2 Models of PCO Estimation and PCO Correction  
In general, the differences between the locations of antenna phase center and antenna reference points (ARP) are referred to as the phase center correction [22; 24]; it consists of the PCO vector and PCVs. The antenna PCO can be defined as the vector from the ARP to the mean antenna phase center (MAPC) in the antenna-fixed system, and can also be described as the vector from center of mass (COM) to mean antenna phase center in the SBS. The difference between the two definitions is a vector of the ARP with respect to the COM. The second definition is applied in the researches. The coordinates of a priori PCO are generally provided by ground calibrations [21; 31]. The actual PCO differs from the priori PCO due to ground calibration error, fuel consumption, and deformation caused by in-flight temperature variations [31]. The fuel reduction of the satellite's tank would cause offsets of different magnitudes of the satellite COM. The regular updates of the receiver PCO coordinates are indispensable to ensure high-accuracy orbits. In addition, the schematic diagram of antenna PCO and PCV model is shown as Figure 1.  
The receiver PCO as the unknown parameter is usually introduced into observation equations of the POD. The observation equation involving LEO satellite attitude can be represented as:  
\[L=A\cdot\overrightarrow{x}+\mathrm{B}\cdot\overrightarrow{e}^{T}\cdot ATT_{S }^{l}\cdot\overrightarrow{PCO}+\mathrm{C}\cdot dtr+D\cdot\overrightarrow{n}+ \varepsilon_{L} \tag{2}\]  
where \(L\) denotes the phase observation of the ionosphere-free combination; \(\overrightarrow{x}\) denotes the parameter vectors of the initial orbits and empirical acceleration; \(\overrightarrow{PCO}\) denotes the receiver PCO vector in the SBS or antenna-fixed system; \(\overrightarrow{e}\) denotes the unit vector of the incident direction of navigation satellite signal; \(ATT_{S}^{l}\) denotes the LEO attitude; \(dtr\) denotes the receiver clock error; \(\overrightarrow{n}\) denotes the ambiguity parameter; A, B, C, D is the corresponding coefficient matrix; \(\varepsilon_{L}\) denotes the phase observation noise.  
In general, for fixed-yaw satellites, the x- and y-components of PCO estimation are unreliable and unstable [31; 32]; only the z-direction results of PCO estimation are feasible. Considering the mixed attitude and slightly canted antenna for Jason-3 satellite, the yaw-steering data were selected to estimate the z-direction results of PCO vector. The thresholds of attitude transition would be used to select suitable data.  
The formula of the PCO correction in the inertial system is given by:  
\[\overrightarrow{PCO}=ATT_{S}^{l}\cdot\overrightarrow{PCO} \tag{3}\]  
where \(ATT_{S}^{l}\) denotes the satellite attitude; \(\overrightarrow{PCO}\) denotes the PCO vector in the SBS; and \(\overrightarrow{PCO}\) denotes the PCO correction in the inertial system. The difference in PCO correction based on different attitude modes can provide a defined method for the transition threshold of the combined attitude; it is also a validation way for attitude modes.