# Analysis and Correction of Antenna Pattern Effects in AMAO Spaceborne SAR Images
## 2 Antenna Pattern Characteristics
### Antenna Pattern Model  
Based on the antenna coordinates established above, this section presents a theoretical model of the antenna pattern, which explicitly represents the 2D angular dependence of the elevation pattern of a squint beam and the effect on azimuth pattern due to beam rotation.  
Accurate antenna modeling requires detailed on-ground measurements, onboard characteristic monitoring, and external verification. For convenience, the theoretical model of the planar antenna pattern is used for analytical analysis, usually written as [12]  
\[G(u,v)=G_{0}\cdot\text{sinc}^{2}\bigg{[}\pi\frac{D_{e}}{\lambda}(v-v_{{}_{B}}) \bigg{]}\cdot\text{sinc}^{2}\bigg{[}\pi\frac{D_{a}}{\lambda}(u-u_{{}_{B}}) \bigg{]}\, \tag{6}\]  
where \(G_{0}\) is the antenna gain, \(\lambda\) is the wavelength of the center frequency, \((u_{{}_{B}},v_{{}_{B}})\) is the boresight direction (subscript \(B\) denotes boresight), and \(D_{a}\) and \(D_{e}\) are the antenna sizes in azimuth and elevation directions, respectively. There are three distinctive factors in Equation (6). From left to right, the first accounts for the antenna gain, \(G_{0}\); the second accounts for the EAP, denoted as \(G_{el}\); and the last accounts for the azimuth antenna pattern (AAP), denoted as \(G_{az}\).  
After variable substitution with Equation (2), the antenna pattern is written approximately as  
\[G(\theta,\phi)\simeq G_{0}\cdot\text{sinc}^{2}\bigg{[}\pi\frac{D_{e}}{\lambda} (\sin\theta-\sin\theta_{{}_{B}})\cos\phi\bigg{]}\cdot\text{sinc}^{2}\bigg{[} \pi\frac{D_{a}}{\lambda}(\phi-\phi_{{}_{B}})\cos\phi\bigg{]}. \tag{7}\]  
The approximate relation of Equation (7) is based on the fact that for spaceborne SAR antennas, the azimuth width of the physical beam is very narrow (i.e., \(\phi\) is very close to \(\phi_{{}_{B}}\)), so two approximate relations are found: \(\sin[(\phi-\phi_{{}_{B}})/2]\simeq(\phi-\phi_{{}_{B}})/2\) and \(\cos[(\phi+\phi_{{}_{B}})/2]\simeq\cos\phi\). In Equation (7), the azimuth angle, \(\phi\), is a free variable. The elevation angle, \(\theta\), is a constant for a given point target and a variable for different targets across the swath. The antenna beam should always point to the swath centerline, so \(\theta_{{}_{B}}\) is a constant, while \(\phi_{{}_{B}}\) can be a variable in the case of beam rotation.  
AMAO SAR antenna beam is often steered to a rotation center to achieve high resolution and/or wide swath. A factor is defined to describe the beam azimuth rotation:  
\[k_{\phi}=\frac{\phi-\phi_{c}}{\phi-\phi_{{}_{B}}}\,. \tag{8}\]It can be deduced that the synthetic angle is \(k_{\phi}\) times the width \(\phi_{bw}\) of the physical beam. With Equation (1) and the expression of physical beamwidth, \(\phi_{bw}=0.886\lambda/(D_{a}\cos\phi_{c})\), the factor \(k_{\phi}\) can also be related to azimuth resolution by  
\[k_{\phi}=\frac{\phi_{\Delta}}{\phi_{bw}}\simeq\frac{D_{a}}{1.77\rho_{a}}\,. \tag{9}\]  
Moreover, the beam rotation factor is also related to the imaging mode. For instance, one stands for the Stripmap, greater than one for the sliding Spotlight, and between one and zero stands for the TopSAR. When \(k_{\phi}\) is large enough, the AAP of a rotating beam is approximately expressed by  
\[G_{az}(\phi)=\text{sinc}^{2}\bigg{[}\pi\frac{D_{a}}{\lambda k_{\phi}}(\phi- \phi_{c})\cos\phi\bigg{]}\,. \tag{10}\]  
Compared with the AAP factor in Equation (7), which has two variables for a given point target in the case of beam rotation, Equation (10) has the advantage of only one free variable \(\phi\) in it. In this way, one does not need to know how the antenna beam is steered, but only needs to know the azimuth resolution to derive the antenna pattern.