## 1 Introduction
For the GEO three-axis stabilized satellite platform, its high orbital height (about 36,000 km) and periodic entry and exit into the Earth's shadow cause antenna reflectors exposed outside the spacecraft to undergo intense alternating heating and cooling by the sun and the cold universe. Therefore, among the sources of disturbances that cause the beam-pointing error (BPE), structural thermal distortions of the antenna reflectors due to the thermal environment around the GEO satellite have the most significant impact on the BPE [21]. To achieve the ABP accuracy required by the GEO microwave radiometer, it is necessary to introduce the Image Navigation and Registration (INR) system [22] that was first applied to the GEO optical meteorological satellites to compensate for the BPE caused by the perturbing effects described previously (especially for the effect corresponding to antenna thermal distortions) while the radiometer is scanning the desired Earth region. From the viewpoint of the INR system, a rapid and accurate method to calculate the ABP under ideal and thermally disturbing conditions and assess the relationship between numerous antenna thermal distortion errors (TDEs) and the associated thermally induced BPE is highly desired. Currently, the most used methodology to study the antenna thermal distortion effects on BPE is the electromechanical coupling principle [23; 24; 25; 26]. First, the thermo-structural finite element model is used to simulate the thermally deformed antenna in a given temperature field. Then, a \"geometric-electromagnetic\" model is established in electromagnetic field simulation software (such as FEKO, HFSS, and GRASP), and the three-dimensional (3-D) far-field antenna pattern and other antenna performance indicators (including the ABP) at the given temperature are calculated using a physical optics (PO) approximation. Theoretically, the electromechanical coupling principle is a rigorous methodology to study the thermal distortion's effects on antenna performance. This approach allows not only an accurate calculation of ABP but also gives the relationship between thermal distortions and other antenna performance indicators (e.g., antenna gain, main lobe width, sidelobe level, beam efficiency, and co- and cross-polarization levels) [26]. However, in the process of in-orbit operation, the electromechanical coupling methodology cannot be performed rapidly enough to maintain pace with the routine INR operations due to its high computational complexity. Therefore, how to accurately describe the beam propagation geometry inside the antenna with or without the presence of antenna reflector thermal distortion effects and analyze the contribution of each thermal distortion on the direction of propagation of the antenna beam has become an urgent fundamental issue that needs to be solved before the GEO microwave radiometer is put into practical application. At the time of this paper, the research on this issue has not been addressed yet; thus, a focused discussion of this issue remains necessary. In this study, by taking China's future GEO microwave radiometer as an example, the rapid ABP determination and BPE analysis method suitable for the mechanically scanned microwave antenna are studied, considering the thermal distortion effects of antenna reflectors. The contributions of this study mainly lie in three aspects:  
1. The method of matrix optics (MO), which is widely used in characterizing the optical path of the visible/infrared sensor, is adapted for modeling the equivalent optical path of the microwave antenna with a much more complicated configuration. 2. The ideal ABP determination model and the model for determining the actual ABP affected by thermal distortions of the antenna reflectors are deduced for the examined GEO microwave antenna.