## 2 Analysis of Antenna Pattern Errors
### Cross-Polar Antenna Pattern  
Equation (1) represents the visibility function when only the co-polar patterns are considered. If nonzero cross-polar antenna patterns are taken into account, the signal received by the antenna element k in the array is as follows:  
\[ \begin{split}& M_{k}^{X}(t)=\left[\begin{array}{cc}b_{k}^{x}(t)&b_{k}^{y}(t) \end{array}\right]\left[\begin{array}{c}C_{Xk}\\ X_{Xk}\end{array}\right]\\ & M_{k}^{Y}(t)=\left[\begin{array}{cc}b_{k}^{x}(t)&b_{k}^{y}(t) \end{array}\right]\left[\begin{array}{c}X_{Yk}\\ C_{Yk}\end{array}\right]\end{split}  \tag{13}\]  
where \(C_{X,Y}\) is the co-polar pattern and \(X_{X,Y}\) is the cross-polar pattern. \(b^{x}(t)\) and \(b^{y}(t)\) represent two orthogonal signals. For FPASR, the correlations between the output of antenna \(k\) and \(j\) are as follows:  
\[ \begin{split}& V_{kj}^{XX}=<M_{k}^{X}(t)M_{k}^{X}(t)>\\ & V_{kj}^{XY}=<M_{k}^{X}(t)M_{j}^{Y}(t)>\\ & V_{kj}^{YX}=<M_{k}^{X}(t)M_{k}^{X}(t)>\\ & V_{kj}^{YY}=<M_{k}^{Y}(t)M_{j}^{Y}(t)>\end{split}  \tag{14}\]  
\(V_{kj}^{XX}\) and \(V_{kj}^{YY}\) are the correlations of the antennas k and j operating in the same polarization. \(V_{kj}^{XY}\) and \(V_{kj}^{YX}\) are the correlations operating in different polarizations. The horizontal and vertical BT distribution of the scene and the complex polarimetric BT distribution of the scene constitute the fully polarimetric visibility function. The fully polarimetric discretized visibility function equation can be written in terms of the fully polarimetric G-matrix as  
\[ \begin{split}\begin{bmatrix}V^{XX}\\ V^{YY}\\ V^{XY}\end{bmatrix}=\begin{bmatrix}G_{XX}^{CC}&G_{XX}^{XX}&G_{XX}^{CX}&G_{XX}^{XC}\\ G_{YY}^{XX}&G_{YY}^{CC}&G_{YY}^{CC}&G_{YY}^{XX}\\ G_{XY}^{CC}&G_{XY}^{CC}&G_{YY}^{CC}&G_{YY}^{XX}\\ G_{YY}^{CC}&G_{XY}^{CC}&G_{YY}^{XX}&G_{YY}^{CC}\\ \end{bmatrix}\end{bmatrix}\begin{bmatrix}T_{B}^{XX}\\ T_{B}^{YY}\\ T_{B}^{YY}\\ T_{B}^{YY}\\ T_{B}^{YY}\end{bmatrix}\end{split}  \tag{15}\]The elements of the fully polarimetric G-matrix in the above equation are as follows:  
\[ \begin{array}{l}G^{RS}_{pq}(\xi,\eta)=\frac{R^{P}_{k}(\xi,\eta)S^{\eta}_{j}(\xi, \eta)}{\sqrt{\Omega_{k}\Omega_{j}}\sqrt{1-\xi^{2}-\eta^{2}}}\widetilde{\tau}(- \frac{u\xi+v\eta}{f_{c}})e^{-j2\pi(u\xi+v\eta)}\Delta^{\sigma}_{\xi}\Delta\eta\\ \xi^{2}+\eta^{2}\leq 1\end{array}  \tag{16}\]  
where \(R^{P}_{k}(\xi,\eta)\) represents the pattern of the co-polar (C) or cross-polar (X) of the antenna element k at p polarizations. \(S^{d}_{j}(\xi,\eta)\) represents the pattern of the co-polar (C) or cross-polar (X) of the antenna element \(j\) at \(q\) polarizations. \(S^{q\prime}_{j}(\xi,\eta)\) is the conjugate of \(S^{q}_{j}(\xi,\eta)\). It is essential to emphasize that the G-matrix required in Equation (16) must be computed over the complete unit circle. In cases where the cross-polar pattern is zero, the fully polarimetric G-matrix in Equation (15) has zero values except for the diagonal. However, since the cross-polar antenna pattern of a real system cannot be zero, all terms in the fully polarimetric G-matrix are nonzero. Furthermore, given that the co-polar pattern significantly exceeds the cross-polar pattern, it follows that the diagonal values of the fully polarimetric G-matrix are the largest. And \(G^{CC}_{pq}>G^{CX}_{pq}\approx G^{XC}_{pq}>G^{XX}_{pq}\). For \(V^{XY}\) and \(V^{YX}\), \(G^{CX}_{pq},G^{XC}_{pq}\) are multiplied by \(T^{XX}_{B},T^{YY}_{B}\). Therefore, the cross-polarization has a non-negligible contribution to both \(V^{XY}\) and \(V^{YX}\). If only the co-polar pattern is considered in the BT reconstruction, the contribution of the cross-polar pattern is equivalent to adding an additive error to the visibility function. The visibility function error caused by the cross-polar pattern is as follows:  
\[