## 2 Methodology  
### Knowledge and Definition  
Spaceborne SAR antenna beam pointing can be expressed by the line of sight (LOS) of the antenna in the satellite orbital coordinate system \(OX_{o}Y_{o}Z_{o}\), as shown in the Figure 1. The satellite speed direction points to the inside of the paper, which is denoted as the \(X_{o}\) direction. \(Z_{o}\) direction is determined by the center of gravity of the satellite and the center of the earth and the \(Y_{o}\) direction is determined by the right-hand rule. \(V_{LOS}\) is the unit vector of LOS whose direction is that of the centerline of the antenna pattern, along which the antenna pattern has the strongest energy. In general, the antenna pattern is symmetrical about the centerline, for the convenience of description, this paper discusses this situation. The coordinates of the vector \(V_{LOS}\) in the orbital coordinate system are:  
\[S_{o-LOS}=[x_{o},y_{o},z_{o}]^{T} \tag{1}\]  
The beam pointing squint angle is:  
\[\alpha=\frac{\pi}{2}-\cos^{-1}(V_{LOS}\cdot X_{o}) \tag{2}\]  
The beam pointing look angle is:  
\[\beta=\frac{\pi}{2}-\cos^{-1}(V_{LOS}\cdot Y_{o}) \tag{3}\]The squint angle \(\alpha\) and look angle \(\beta\) represent the direction of the beam itself in space, but sometimes it needs to know the direction of the beam in the antenna coordinate system \(O_{a}X_{a}Y_{a}Z_{a}\), i.e., the beam direction represented by the antenna beam azimuth and elevation angle, as shown in Figure 2. In the antenna coordinate system, \(X_{a}\) points to the direction of the length of the antenna, \(Y_{a}\) points to the direction of the height of the antenna, and \(Z_{a}\) points to the direction of the normal of the antenna. The coordinates of the vector \(V_{LOS}\) in the antenna coordinate system are:  
\[S_{a-LOS}=[x_{a},y_{a},z_{a}]^{T} \tag{4}\]  
The beam pointing azimuth angle is:  
\[\phi=\frac{\pi}{2}-\cos^{-1}(V_{LOS}\cdot X_{a}) \tag{5}\]  
The beam pointing elevation angle is:  
\[\varphi=\frac{\pi}{2}-\cos^{-1}(V_{LOS}\cdot Y_{a}) \tag{6}\]  
The following formula gives the conversion relation of the description of the vector \(V_{LOS}\) from the orbital coordinate system to the antenna coordinate system:  
\[S_{a-LOS}=\mathbf{C}_{star2 antenna}\mathbf{C}_{orbit2 star}S_{o-LOS} \tag{7}\]  
where, \(\mathbf{C}_{orbit2 star}\) is the transition matrix from the orbital coordinate system to the satellite body coordinate system measured by the satellite attitude sensors, \(\mathbf{C}_{star2 antenna}\) is the transition matrix from the satellite body coordinate system to the antenna coordinate system measured from the ground. In Appendix A, we give the specific expressions of these two matrices. Figure 1: Antenna beam pointing diagram in the orbital coordinate system. The beam pointing error can be divided into three parts: First, the orbit error, whose contribution is relatively small. Second, the attitude control error. The attitude control system is a closed-loop control system with the satellite attitude sensors as the measurement component because the satellite attitude sensor's accuracy is very high and the contribution of this part of the error is also relatively small. Third, the antenna error, which is the greatest error, including the relative installation error between the satellite attitude sensors and the antenna, and the beam pointing error caused by the antenna's own electromechanical and thermal factors. The antenna is associated with the satellite attitude sensors through the installation matrix \(\mathbf{C}_{antenna2 star}\). Although the installation matrix is accurately measured on the ground, as a result of the launching vibration environment and the stress release in orbit, the actual installation relationship will change and cannot be directly measured. Antenna error is the most important error, which can only be resolved by on-orbit calibration and attitude closed-loop control. However, it is difficult to measure the antenna's own beam pointing error directly on orbit.