## 3 Correction Method for Antenna Pattern Errors
### Correction of Antenna Pattern Additive Errors  
The first step is to correct the visibility function additive errors caused by the backlobe antenna pattern and the cross-polar antenna pattern. This additive error can be alleviated by subtracting a model visibility function from the measured visibility function. The model visibility function is composed of the contribution of the backlobe antenna pattern and the cross-polar antenna pattern. It is founded on an artificial BT distribution and the antenna pattern measured in the grounded anechoic chamber. Notably, the closer the artificial BT distribution to the observation scene, the better the correction effect is. The contribution of the backlobe can be calculated from the artificial BT distribution associated with the backlobe and its corresponding backlobe antenna pattern:  
\[ \begin{bmatrix}\Delta V_{back}^{XXX}\\ \Delta V_{back}^{YYX}\\ \Delta V_{back}^{XXX}\\ \Delta V_{back}^{XXX}\\ \Delta V_{back}^{XXX}\end{bmatrix} = \begin{bmatrix}H_{XX}^{CC}&H_{XX}^{XX}&H_{ XX}^{CX}&H_{XX}^{XX}\\ H_{YY}^{XX}&H_{YY}^{XX}&H_{YY}^{XX}\\ H_{XY}^{XX}&H_{XY}^{XX}&H_{XY}^{XX}\\ H_{XY}^{XX}&H_{XX}^{XX}&H_{XX}^{XX}\\ H_{XX}^{XX}&H_{XX}^{XX}&H_{XX}^{XX}&H_{XX}^{XX}\\ \end{bmatrix}\begin{bmatrix}T_{M,back}^{XX}\\ T_{M,back}^{YYY}\\ H_{M,back}^{XXX}\\ T_{M,back}^{XXX}\\ T_{M,back}^{XXX}\\ \end{bmatrix}  \tag{20}\]  
where \(T_{M,back}^{pq}\) represents the artificial BT distribution that corresponds to the scene observed through the backlobe antenna pattern. The computation of the H-matrix elements closely  
Figure 1: The flowchart of the correction. resembles that of the G-matrix, with the distinction that the H-matrix is derived using the backlobe pattern. Its expression is as follows:  
\[ \begin{split} H^{RS}_{\text{PQ}}(\xi,\eta)=\frac{R^{p}_{back,k}( \xi,\eta)S^{pq^{+}}_{back,j}(\xi,\eta)}{\sqrt{\Omega_{k}\Omega_{j}}\sqrt{1-\xi^{ 2}-\eta^{2}}}\tilde{\tau}(-\frac{u_{0}^{2}+v\eta}{f_{c}})e^{-\beta\pi(u_{0}^{2 }+v\eta)}\Delta\xi\Delta\eta\\ \xi^{2}+\eta^{2}\leq 1\end{split}  \tag{21}\]  
where \(R^{p}_{back,k}(\xi,\eta)\) represents the backlobe pattern of the co-polar (C) or cross-polar (X) of the antenna element \(k\) at \(p\) polarizations. The contribution of the cross-polar antenna pattern can be calculated as follows:  
\[ \begin{split}\begin{bmatrix}\Delta V^{XX}_{cross}\\ \Delta V^{YY}_{cross}\\ \Delta V^{XY}_{cross}\\ \Delta V^{XY}_{cross}\\ \end{bmatrix}=\begin{bmatrix}0&G^{XX}_{XX}&G^{XX}_{XX}&G^{XX}_{XX}\\ G^{XX}_{YY}&0&G^{XX}_{YY}&G^{XX}_{YY}\\ G^{XX}_{YY}&G^{XX}_{XY}&0&G^{XX}_{YY}&T^{XY}_{M}\\ G^{XX}_{YY}&G^{XX}_{YY}&G^{XX}_{YY}&0&G^{XX}_{YY}&T^{XY}_{M}\\ G^{XX}_{YY}&G^{XX}_{YY}&G^{XX}_{YY}&0&0\\ \end{bmatrix}\end{bmatrix}\end{split}  \tag{22}\]  
The G-matrix required in Equation (22) is calculated in the same way as in Equation (16). The grid points \((\xi,\eta)\) fall inside the complete unit circle, unlike the one used in the image reconstruction process, which is only computed in the fundamental period. \(T^{pq}_{M}\) represents the artificial BT distribution. The observation scenes of the backlobe and the cross-polarization are divided into two regions: sky and Earth. The BT of the sky adopts a default value of 2.7 K. It is convenient to divide the artificial BT of the Earth into two components: (1) land and (2) ocean. The BT of the land pixel is set to a constant value, while that for ocean pixels employs a simplified Fresnel model to represent polarimetric temperatures. In nominal mode, the artificial BT distribution corresponding to the backlobe \(T^{pq}_{M,back}\) is the sky, while that of the cross-polarization \(T^{pq}_{M}\) is mainly the Earth. In cold sky mode, \(T^{pq}_{M,back}\) is mainly the BT of the Earth, and \(T^{pq}_{M}\) is the BT of the sky. The measured visibility function subtracted the contribution of the backlobe and cross-polarization:  
\[