## 2 Analysis of Antenna Pattern Errors  
The relationship between the system input and output can be established for FPASR, where the BT distribution serves as the system input, while the visibility function represents the system output. After canceling the contribution of the Corbella term addressed in [8; 9],and ignoring the antenna backlobe and cross-polar contributions [15], the relationship between the visibility function and the BT can be articulated as follows [16]:  
\[ \begin{split}& V_{kj}^{pq}(u,v)=\\ &\iint\limits_{\xi^{2}+\eta^{2}<1}T_{B}^{pq}(\xi,\eta)\frac{F_{nk}^ {p}(\xi,\eta)F_{nj}^{qq}(\xi,\eta)}{\sqrt{\Omega_{k}\Omega_{j}}\sqrt{1-\xi^{ 2}-\eta^{2}}}\widetilde{r}(-\frac{u\xi+v\eta}{f_{c}})e^{-j2\pi(u\xi+v\eta)}d \xi d\eta\end{split}  \tag{1}\]  
where \(p,q=\{\)X, Y\(\}\) are the polarization configuration of antennas. \(T_{B}^{pq}(\xi,\eta)\) is the BT of the scene. \(u=(x_{j}-x_{k})/\lambda_{0},v=(y_{j}-y_{k})/\lambda_{0}\) are the normalized spatial frequencies corresponding to the pair of antennas \(k\) and \(j\). \((x_{j},y_{j})\) and \((x_{k},y_{k})\) are two antenna positions, respectively. \(V_{kj}^{pq}(u,v)\) is the visibility function. \(F_{nk}^{pq}(\xi,\eta)\) denotes the normalized antenna co-polar voltage patterns at p polarization, while \(F_{nj}^{q}(\xi,\eta)\) represents the normalized antenna co-polar voltage patterns at q polarization. \(F_{nj}^{qq}(\xi,\eta)\) is the conjugate of \(F_{nj}^{q}(\xi,\eta)\). \(\Omega_{k}\) and \(\Omega_{j}\) are the corresponding solid angles of the antenna. \(\xi=\sin\theta\cos\varphi\), \(\eta=\sin\theta\sin\varphi\) are the director cosines with respect to the (X, Y) axes. \(\widetilde{r}(-\frac{u\xi+v\eta}{f_{c}})\) is the fringe-washing function, which is determined by the frequency responses of two receivers and accounts for spatial decorrelation effects [17; 18]. To reconstruct the scene BT from the measured visibility function, the so-called G-matrix method can be used [19; 20]. The G-matrix represents the impulse response matrix of the system, with its elements being functions of the antenna pattern. The elements of the G-matrix can be written as  
\[G_{kj}^{pq}(\xi,\eta)=\frac{F_{nk}^{p}(\xi,\eta)F_{nj}^{qq}(\xi,\eta)}{\sqrt{ \Omega_{k}\Omega_{j}}\sqrt{1-\xi^{2}-\eta^{2}}}\widetilde{r}(-\frac{u\xi+v \eta}{f_{c}})e^{-j2\pi(u\xi+v\eta)}\Delta\xi\Delta\eta \tag{2}\]  
where \(\Delta\xi\Delta\eta\) is the elementary area in \((\xi,\eta)\). The reconstructed BT inv\(T\) can be obtained by solving the G-matrix and the measured visibility function  
\[\mathbf{V}=\mathbf{GT}\Rightarrow\mathbf{invT}=(\mathbf{G_{S}})^{-1}\mathbf{V} \tag{3}\]  
where \(\mathbf{V}\) represents the visibility vector; \(\mathbf{G}\) represents the G-matrix; \(\mathbf{T}\) represents the BT vector. \(\mathbf{invT}\) is the reconstructed BT vector. Note that the left \(\mathbf{G}\) in the above equation has the same expression as the right \(\mathbf{G_{S}}\), but their respective grid ranges are different. The grid points \((\xi,\eta)\) corresponding to \(\mathbf{G}\) fall inside the unit circle, while the grid points \((\xi,\eta)\) corresponding to \(\mathbf{G_{S}}\) are reciprocal to the ones in \((u,v)\), which means that \(\mathbf{G_{S}}\) is a subset of \(\mathbf{G}\)[21]. The expression of the G-matrix indicates that the input data required for its construction include the following: (1) the antenna pattern of all elements in the array; (2) the antenna position of all elements in the array; (3) the fringe-washing function. The antenna position is the reference position of the antenna phase pattern. In general, it is assumed that the antenna positions are equal to the nominal values. For wide-beam antennas of synthetic aperture radiometers, this position is typically close to the phase center of the antenna [21]. The fringe-washing function can be measured by injecting the correlated noise signals with different delay times. However, this paper uses a simpler formulation and does not consider the fringe-washing function.