## 5 Conclusions  
In this study, the rapid ABP determination and thermally induced BPE analysis method in support of the development of the onboard INR system needed by the GEO mechanical scanning microwave radiometer were studied using the example offset Cassegrain antenna with a planar folding mirror proposed for China's future GEO microwave radiometer. The main contributions of this study are as follows. Figure 10: The impacts of different thermal translation errors at different levels on the BPE. The impact was measured by the RMSE of the ABPs over a scan period of the ECF. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Thermal** & \(x_{\text{SSR},ma}\) & \(y_{\text{SSR},ma}\) & \(z_{\text{SSR},ma}\) & \(x_{\text{PSR},ma}\) & \(y_{\text{PSR},ma}\) & \(z_{\text{PSR},ma}\) & \(x_{\text{MR},ma}\) & \(y_{\text{MR},ma}\) & \(z^{\text{MR},ma}\) \\ \hline
**RMSE of ABPs** (\(\upmu\)rad) & 0.0 & 0.0 & 50.8 & 91.2 & 170.6 & 164.9 & 124.9 & 202.3 & 159.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The maximum impact of each thermal translation error on the BPE at the level of \(\pm\)1.0 mm. First, we adapted the traditional MO method for modeling the equivalent optical path of the microwave antennas with complicated configurations and deduced the ideal and actual ABP determination models for the examined GEO antenna. Compared with the classical electromechanical coupling principle, the MO-based ABP determination model avoids the complex calculations such as finite element thermoanalysis and PO approximation, so it is advantageous in terms of speed performance and technical implementability. Thus, it is more suitable for the in-orbit INR technique (for future development) needed by the GEO microwave radiometer. When INR is turned on, the deduced ideal ABP determination model can be used in the 2-D beam scan planning and the direct geolocation of the acquired radiometric data. To verify the overall performance of the MO-based ABP determination model, the comparative experiment for ABP determination was conducted by setting the GRASP 10.3 software as the evaluation benchmark. The experimental results show that the MO-based ABP determination model yielded a significant speed improvement (\(\sim\)4,730,000 times at 54 GHz) relative to the GRASP software, and the maximum ABP calculation error of the MO-based model was only \(5.16\times 10^{-2}\)\(\upmu\)rad within the 360\({}^{\circ}\) scan cycle of the ECF, which can be completely ignored with respect to the NWP operational requirements. Another contribution of this study was to propose an MO-based method for fast calculation of the thermally induced BPE, which established a direct connection between the reflector TDEs and the resulting BPE. With the aid of this MO-based BPE computing method, the impacts of the reflector TDEs on the BPE were quantified on a case-by-case basis, and those TDEs that had a significant impact on the BPE were thus identified. The experimental results provided general conclusions for the thermal distortion compensation scheme design of the classical offset Cassegrain reflector antenna with a planar folding mirror as follows:  
1. The misalignment angles of the yaw axis (focal axis) of the MR and FSR had no impact on the thermally induced BPE, and the misalignment angle of the yaw axis (surface normal) and the roll and pitch axial translation of the SSR (i.e., the planar folding mirror) also had no impact on the BPE. 2. For thermal misalignment errors, the roll and pitch misalignment angles of the MR had the greatest impacts on the BPE, followed by the thermal misalignment errors of the FSR, and the thermal misalignment errors of the SSR had the lowest impact. The impacts of MR's thermal misalignment errors were by one order of magnitude substantially higher than those of the thermal misalignment errors of the FSR and SSR. 3. For the thermal translation errors, their impacts on the BPE had the same order of magnitude.