## 2 Methodology
### Measuring Principle  
For LEO spaceborne SAR with phased array antennas, when the satellite is in orbit, ignoring yawing guidance, the direction of the antenna \(X_{a}\) is generally consistent with the direction of the satellite speed. As shown in Figure 1, the antenna beam squint angle and the azimuth angle are equal. In fact, when the satellite is in orbit, what the ground can directly feel is the squint angle of the antenna beam, and this angle directly affects the azimuth imaging performance. Therefore, the so-called antenna beam azimuth pointing measurement on-orbit is the measurement of this squint angle.  
This paper focuses on the method for measuring the squint angle of the antenna beam in the orbital coordinate system. Furthermore, the beam pointing test is performed while the SAR is in the side looking mode, with the beam squint angle being close to zero.  
Assuming that the spaceborne SAR is orbiting, a ground receiver is placed near the center of the ground imaging area to receive the pulse signal transmitted by the spaceborne SAR. The signal is sampled and recorded at a certain frequency which meets the sampling theorem. We define the closest relative position of the spaceborne SAR and ground receiver as the position where the satellite is right  
Figure 2: Azimuth and elevation angle of the antenna beam.  
over-the-top of the receiver (this moment is denoted as the satellite's ceiling time). At this position, the connection line between the satellite and the ground receiver is approximately perpendicular to the satellite's flight trajectory.  
When the center line of the spaceborne SAR's antenna azimuth is aligned with the ground receiver, this moment is defined as the beam alignment time. As shown in the Figure 3, at the beam alignment time (the corresponding position is point B in Figure 3), if the satellite is right over-the-top of the ground receiver (the corresponding position is point A in Figure 3), the azimuth pointing of the antenna is perpendicular to the satellite flight trajectory, which indicates that the squint angle is 0. After the corrections of range power attenuation and the ground receiver antenna pattern's weighting, the amplitude envelope of the SAR's pulse signal received by the ground receiver is symmetrical about point A (and point B).  
If the azimuth of the SAR's antenna beam deviates from the plane of the vertical satellite flight trajectory with an angle \(\Delta\alpha\), that is, the squint angle is \(\Delta\alpha\), as shown Figure 4, then when the satellite is over-the-top of the ground receiver, corresponding to point A, after the corrections of range power attenuation and the ground receiver antenna pattern's weighting, the amplitude envelope of complex sampling data received by the ground receiver is asymmetrical about this point. When the beam is aligned with the ground receiver at point B, the amplitude envelope is symmetrical about this point. Ignoring the orbital curvature of the satellite, according to the geometric relationship, \(\angle BPA\) is equal to \(\Delta\alpha\), which is the azimuth pointing of the antenna beam.  
Figure 3: The situation where the squint angle of the antenna beam is 0.  
In order to obtain \(\angle BPA\). which is \(\Delta\alpha\) accurately, it is necessary to know the satellite orbit, the position of the ground receiver, the length of the line segment \(AB\). \(\Delta\alpha\) is as (8):  
\[\Delta\alpha=tg^{-1}[\frac{(t_{A}-t_{B})V_{S}}{R_{0}}] \tag{8}\]  
where \(R_{0}\) is the distance between the satellite and the ground receiver at the satellite's ceiling time, \(V_{S}\) is the velocity of the satellite, \(t_{A}\) is the satellite's ceiling time, \(t_{B}\) is the beam alignment time.  
The satellite orbital ephemeris data, the position of the ground receiver, and the velocity of the satellite are all relatively easy to obtain. The key to this method lies in determining \(t_{A}\) and \(t_{B}\).