## 4 Results  
### Fully Polarimetric Simulation  
In order to analyze the antenna pattern error, an end-to-end simulator for FPASR has been developed. The antenna array is a Y-shaped array of 55 elements, with 18 elements in one arm. The position of each antenna element in the array is illustrated in Figure 2. The ideal normalized co-polar and cross-polar antenna patterns are depicted in Figure 3. The antenna patterns for both polarizations are similar. A Fresnel model of the ocean has been employed as the observation scene for simulation. Figure 4 shows the four expected polarimetric BTs: \(\{T^{XX},T^{YY},T^{3},T^{4}\}\). \(T^{3}\) and \(T^{4}\) are the third and fourth Stokes parameters, defined as \(T^{3}=2\text{Re}(T^{XY})\), \(T^{4}=2\text{Im}(T^{XY})\). The BT error \(\Delta T\) is computed within a circle \(\xi^{2}+\eta^{2}<0.3\). The calculation of \(\Delta T\) is as follows:  
\[\Delta T=\sqrt{\frac{1}{Num}\sum_{k=1}^{Num}\left[invT(x_{k})-T(x_{k})\right]^ {2}} \tag{27}\]  
where \(T\) indicates the original BT. \(invT\) is the reconstructed BT obtained by visibility function. \(Num\) is the number of pixels in the circle \(\xi^{2}+\eta^{2}<0.3\). Table 1 presents the antenna pattern errors taken into account in the simulation. Backlobe antenna patterns and cross-polar antenna patterns are also taken into account. The BT of the cold sky observed by the backlobe is set to 2.7 K. The simulation results are shown in Figures 5-7. It is clear that the image quality deteriorates with the introduction of antenna pattern errors, as shown in Figure 5. To mitigate the impact of these errors on BT, the two-step correction method described in the previous section is adopted. The corrected results are shown in Figure 6, where the reconstructed BT images after correction exhibit a closer resemblance to the original BT image. For comparison, the FIT method was also simulated. In the simulation of the FTT method, the simulator simulates the FPASR's observation of cold sky and uses the observation data of cold sky to correct the antenna pattern error. The results are shown in Figure 7. The BT error results are summarized in Table 2. The qualitative results confirm that the two-step correction method is effective in correcting the antenna pattern error and has a better correction effect than the FIT method. Figure 4: Simulated ocean polarimetric scene in the alias-free field of view. (**a**) Ocean scene in X polarization \(T^{XX}\); (**b**) ocean scene in Y polarization \(T^{YY}\); (**c**) ocean scene in T3 polarization \(T^{3}\); (**d**) ocean scene in T4 polarization \(T^{4}\). Figure 3: Normalized antenna pattern. (**a**) Co-polar antenna pattern; (**b**) cross-polar antenna pattern. Figure 2: Array arrangement. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Antenna Pattern Errors** & **Error** \\ \hline Amplitude differences of the antenna & 5\% \\ patterns & 5\({}^{\circ}\) \\ Phase differences of antenna patterns & 5\({}^{\circ}\) \\ Amplitude error of antenna pattern & 1\% \\ measurement & 1\({}^{\circ}\) \\ Phase error of antenna pattern & 1\({}^{\circ}\) \\ measurement & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Antenna pattern errors for simulation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Delta T\) & **X Polarization** & **Y Polarization** & **T3 Polarization** & **T4 Polarization** \\ \hline Before calibration (K) & 12.32 K & 14.54 K & 4.84 K & 4.51 K \\ After calibration with the two-step & & 2.64 K & 2.58 K & 3.18 K & 3.14 K \\ correction method (K) & & & & \\ After calibration with the FTT method (K) & 3.56 K & 3.31 K & 4.84 K & 4.51 K \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The BT error \(\Delta T\) in four polarimetric BTs. Figure 5: Simulation results with antenna pattern error. (**a**) \(T^{XX}\), (**b**) \(T^{YY}\), (**c**) \(T^{3}\), (**d**) \(T^{4}\). Figure 6: Simulation results after correction with the two-step correction method. (**a**) \(T^{XX}\), (**b**) \(T^{YY}\), (**c**) \(T^{3}\), (**d**) \(T^{4}\).