# Analysis and Correction of Antenna Pattern Effects in AMAO Spaceborne SAR Images
## 3 Antenna Pattern Effects in Images  
This section establishes a practical analytical model of the antenna pattern effect on SAR image intensities. The model is succinctly represented by two factors derived from the 2D antenna pattern, a linear coefficient of EAP variation with squint angle and the centroid shift of the azimuth signal. The radiometric SAR measurement is based on the SAR radar equation [13], which relates the image intensity to target backscattering characteristics and the SAR system parameters. Conventionally, the elevation pattern is included in the SAR radar equation, and the azimuth pattern is not. This is problematic for AMAO SAR because it does not account for the squint angle dependence of the elevation pattern and the AAP nonideality. Moreover, future high-resolution AMAO spaceborne SARs have long synthetic apertures and strong beam-steering capabilities with squint angle up to tens of degrees [7; 8], thus involving higher variations. For a given point target, the received signal magnitude is given by  
\[|s(t)|=A\sqrt{\sigma}\,\frac{G(\theta,t)}{R^{2}(\theta,t)}\,, \tag{13}\]  
where \(A\) is a magnitude coefficient that contains parameters considered to be constant in the radar equation, and \(\sigma\) is the target RCS. The azimuth variation of target RCS within  
Figure 4: The squint angle dependence of the EAP at the swath edge. the synthetic aperture is neglected. The integrated energy of the point target signal is then given by  
\[\mathcal{E}=\int_{t_{1}}^{t_{2}}\lvert s(t)\rvert^{2}\mathrm{d}t=A^{2}\sigma \int_{t_{1}}^{t_{2}}\frac{G^{2}(\theta,t)}{R^{4}(\theta,t)}\mathrm{d}t\,, \tag{14}\]  
where \(t_{1}\) and \(t_{2}\) are the start and end times of the SAR acquisition. Conventionally, the effects of EAP and the slant range are normalized using a correction function derived at the reference squint angle (usually the squint angle of the beam center when crossing the scene center). After that, the signal energy becomes  
\[\mathcal{E}^{\prime}=A^{2}\sigma\int_{t_{1}}^{t_{2}}\frac{G^{2}(\theta,t)}{G_{ ref}^{2}(\theta)}\frac{R_{ref}^{4}(\theta)}{R^{4}(\theta,t)}\mathrm{d}t\,, \tag{15}\]  
where the subscript \(ref\) signifies the reference. Two facts are apparent from Equations (5) and (12): the azimuth variations in EAP and slant range are gradual and almost linear over the narrow synthetic angle. On these bases, the first-order approximation is used  
\[\frac{G_{\theta}^{2}(\theta,t)}{G_{ref}^{2}(\theta)}\frac{R_{ref}^{4}(\theta) }{R^{4}(\theta,t)}\approx 1+b(\theta)t\,, \tag{16}\]  
where \(b(\theta)\) is a linear coefficient that depends on the elevation angle, which summarizes the linear variations in EAP and slant range with the squint angle. Using Equation (16), a convenient expression of signal energy can be further deduced from Equation (15)  
\[\mathcal{E}^{\prime} =A^{2}\sigma\int_{t_{1}}^{t_{2}}\frac{G_{0}^{2}G_{el}^{2}(\theta,t )G_{az}^{2}(t)}{G_{ref}^{2}(\theta)}\frac{R_{ref}^{4}(\theta)}{R^{4}(\theta,t) }\mathrm{d}t \tag{17}\] \[=A^{2}G_{0}^{2}\sigma\int_{t_{1}}^{t_{2}}\frac{G_{0}^{2}(\theta,t )}{G_{ref}^{2}(\theta)}\frac{R_{ref}^{4}(\theta)}{R^{4}(\theta,t)}G_{az}^{2}(t )\mathrm{d}t\] \[\approx A^{2}G_{0}^{2}\sigma\int_{t_{1}}^{t_{2}}\left[1+b(\theta)t \right]G_{az}^{2}(t)\mathrm{d}t\] \[=A^{2}G_{0}^{2}\sigma\left[1+b(\theta)\frac{\int_{t_{1}}^{t_{2}}t \,G_{az}^{2}(t)\mathrm{d}t}{\int_{t_{1}}^{t_{2}}G_{az}^{2}(t)\mathrm{d}t} \right]\int_{t_{1}}^{t_{2}}G_{az}^{2}(t)\mathrm{d}t\,.\]  
For an accurately calibrated AMAO SAR image sequence, any variation in the overall system gain should be compensated, i.e., the system gain should equal one. From Equation (17), despite the normalization using the correction function derived at the reference squint angle (see Equation (15)), there are still radiometric variations in AMAO images, given by  
\[\Delta=b(\theta)\mu\,, \tag{18}\]  
where, for convenience,  
Figure 4:0  
represents the shift of the signal centroid from the reference point.