## 2 Model for Attitude, Antenna PCO/PCV and Its POD Strategies
### Jason-3 Satellite Attitude
#### 2.1.1 Jason-3 Attitude Model  
In general, the satellite attitude modes include nominal attitude and yaw-steering mode. For Jason satellites, in order to optimize the energy collected on the solar panel, ideal yaw steering should always be sinusoidal, but when beta is low that mean that the satellite would rotate back and forth very fast, exceeding the capacity of the attitude control; the sub-optimal fixed yaw attitude is used. Therefore, the Jason satellite attitude is a mixed mode of fixed yaw and yaw steering. Users who do not want to use quaternions should use an attitude model. The targeted attitude of the Jason-1/2 satellite [16] can be summarized as follows: When beta is higher, the platform is steered around the yaw direction to keep the \(y\) axis of SBS perpendicular to the Sun vector, which adopts yaw-steering mode. When beta is low, the yaw angle is fixed and the \(x\) axis is kept parallel to the along-track direction, which is nominal mode (in the form of fixed yaw used in more LEOs), but this targeted attitude is invariant due to its fixed threshold.  
In the absence of quaternions during the POD, even the near-real-time OD, to obtain attitude information easily, we designed a combined attitude based on experimental analysis as an alternative or revision scheme. The relative formula for the yaw angle is expressed as:  
\[\varphi=\left\{\begin{array}{rl}&\frac{\pi}{2}+(\begin{array}{c}\frac{\pi}{ 2}-\beta\end{array})\cdot\sin\mu,\textit{where }\beta>\beta_{0}\\ &0\,\textit{where}-\beta_{0}\leq\beta\leq\beta_{0}\\ &-\frac{\pi}{2}-(\begin{array}{c}\frac{\pi}{2}+\beta\end{array})\cdot\sin \mu,\textit{where }\beta<-\beta_{0}\end{array}\right. \tag{1}\]  
where \(\varphi\) denotes the yaw angle, that is, the angle between the \(x\) axis of the SBS and the velocity direction; \(\beta\) is the angle between the Sun vector and the satellite orbit plane; \(\beta_{0}\) is the threshold of the attitude transition between yaw steering and fixed yaw; and \(\mu\) is the orbit angle, that is, the geocentric angle between the spacecraft position vector and the orbit midnight, growing with the satellite's motion [30]. The calculations of \(\mu\) and \(\beta\) are provided in [10; 30] and will not be described here.  
In this method, we dealt with some details correspondingly. The acquisition of transition threshold is based on the difference between quaternions and attitude model. When the difference between fixed model and quaternions exceeds a certain value (such as 1 cm), the beta corresponding to this value is considered as the transition threshold of combined model. By introducing a transition threshold into Equation (1) one may get attitude information. The differences between the combined attitude and attitude event file essentially are the difference of attitude transition nodes. Even if the quaternions have update delays, the combined model can maintain the attitude information for a period of time. Similar to an event file, the threshold of combined model varies for different periods and represents the short-term results. For example, the value \(14.6^{\circ}\) is adopted based on the experimental analysis of PCO correction for a certain period of time data in Section 3.1.2. In addition, when \(\beta\) is close to a certain value (such as \(1.06^{\circ}\)), the Jason-3 satellite is flipped around the yaw axis (the \(z\) axis of SBS); the attitudes would be correspondingly dealt with using a rotation matrix. In turn, the transition threshold of combined attitude can provide the basis of data selection for PCO estimation or other researches related to satellite attitude. Based on the combined attitude, attitude information is available easily without event file, which can be applied as a revision of event file to get stable attitude information and orbits.