# Analysis and Correction of Antenna Pattern Effects in AMAO Spaceborne SAR Images
## 2 Antenna Pattern Characteristics  
### Geometric Properties  
The antenna pattern represents a 2D sensitivity in space. In this section, the coordinates of the antenna pattern are established and linked to the SAR observation coordinates. Then, the geometric properties in the three-dimensional antenna coordinates are analyzed, including azimuth, elevation, and range distance. The imaging geometry of AMAO spaceborne SAR is shown in Figure 1. Vector \(V\) is the relative velocity of the radar to the scatterer, including the effect of the Earth's rotation, so that it is perpendicular to the zero Doppler plane. The slant-range vector, \(R\), forms the slant-range plane with vector \(V\). The observation direction from radar to scatterer is measured in the two planes--the off-nadir angle \(\theta_{N}\) in the zero-Doppler plane, and the squint angle \(\phi\) in the slant-range plane (i.e., the angle between the slant-range vector and the zero Doppler plane). Figure 1: AMAO spaceborne SAR imaging geometry. Figure 2 shows the antenna pattern coordinates. The azimuth and elevation axes are in the antenna plane, perpendicular to each other. The broadside axis is normal to the antenna plane. For convenience and simplicity, the azimuth axis is assumed to be parallel to the relative velocity vector so that convenient relations link the antenna pattern coordinates and the SAR observation coordinates. Then, the elevation plane formed by the elevation axis and the broadside axis is the zero-Doppler plane in observation geometry, and the azimuth plane formed by the azimuth axis and the slant-range vector is the slant-range plane. Consequently, the line-of-sight direction \((\theta,\phi)\) in antenna coordinates is related to \((\theta_{N},\phi)\) in observation geometry, by the relation \(\theta=\theta_{N}-\theta_{F}\), where \(\theta_{F}\) is the off-nadir angle of antenna broadside in observation geometry. The antenna-scatterer relative motion is in the slant-range plane, i.e., the azimuth plane. Therefore, the instantaneous squint angle is a free variable for a given on-ground point target, while the elevation angle is a constant. The synthetic angle, \(\phi_{\Delta}\), is the change in the squint angle of a scatterer from entering to exiting the antenna footprint, related to the azimuth resolution \(\rho_{a}\) by  
\[\phi_{\Delta}=\frac{\lambda}{2\rho_{a}\cos\phi_{c}}\,, \tag{1}\]  
where the subscript \(c\) signifies a target's beam center crossing. In the high-resolution case, the synthetic angle can be much larger than the width of the physical beam due to the beam rotation. For example, 0.25 m X-band resolution (the same as TerraSAR-X NG [11]) needs a synthetic angle of 3.44 degrees. The larger synthetic angle implies the higher variations involved. The antenna pattern is often defined in the \((u,v)\) coordinates  
\[ \begin{split} u&=\sin\phi\,,\\ v&=\cos\alpha=\sin\theta\cos\phi\,,\end{split}  \tag{2}\]  
where \(\alpha\) is the elevation off-axis angle between the slant-range vector and the elevation axis. From Equation (2), both \(u\) and \(v\) vary with the free variable \(\phi\), indicating a 2D variation  
Figure 2: Antenna pattern coordinates. of the line of sight in antenna pattern coordinates. The variations in (\(u\), \(v\)) coordinates are given by  
\[\Delta u\simeq 0.5\lambda/\rho_{a}\, \tag{3}\] \[\Delta v\simeq(0.5\lambda/\rho_{a})|\sin\theta\tan\phi_{c}|\.\]  
From Equation (3), the variation in the \(v\) coordinate is related to three parameters--the azimuth resolution, the elevation angle, and the squint angle. It is low in the low-squint case and emerges in the high-resolution, high-squint case. The slant-range distance changes with the azimuth time, given by  
\[R(\theta,t)=\sqrt{R_{c}^{2}(\theta)+(Vt)^{2}-2 VtR_{c}(\theta)\sin\phi_{c}}\, \tag{4}\]  
where the free variable \(t\) represents the azimuth time. At the beam center crossing time \(t=0\), the slant-range distance is \(R_{c}(\theta)\).