## 1 Introduction  
The aperture synthesis radiometer uses an antenna array to observe the target scene [1]. The complex cross-correlations of signals, referred to as visibilities, are collected by every pair of antennas in the array. The reconstructed BT of the scene can be obtained from the visibility function. The fully polarimetric aperture synthesis radiometer (FPASR) combines aperture synthesis technology with fully polarimetric information acquisition. FPASR is capable of measuring four Stokes parameters, thereby enhancing the polarization detection capabilities beyond those offered by dual polarimetric radiometers [2, 3, 4]. FPASR plays an important role in Earth remote sensing. It improves the retrieval ability of important geophysical parameters such as sea surface wind speed, wind direction, sea surface salinity, and soil moisture [5, 6, 7]. The integration of full polarization and aperture synthesis technology not only enhances the capabilities of current remote-sensing systems but also opens up new fields for multidimensional data fusion and interdisciplinary research. The antenna plays a vital role in FPASR, exerting a substantial influence on the system's performance. The maximum aperture of the antenna array dictates the spatial resolution, while the minimum spacing establishes an alias-free field of view. The sensitivity is closely related to the minimum antenna spacing and array arrangement. The antenna pattern serves as a critical auxiliary data set for the BT reconstruction of FPASR. In an ideal case (equal antenna and neglecting the fringe-washing function), the BT can be obtained by solving the visibility function through inverse Fourier transform. In the real case,due to the non-negligible differences in antenna patterns, it is necessary to measure all antenna patterns in the array accurately. Before the radiometers are put into operation, each antenna element within the system undergoes measurement in a grounded anechoic chamber. The limitations of antenna machining accuracy, the insufficiency of the test system's precision, and the changes in the in-orbit environment may lead to inaccurate antenna pattern measurements and antenna pattern differences. These antenna pattern errors may introduce distortions in the reconstructed images. Furthermore, non-negligible backlobe antenna patterns and the cross-polar antenna patterns will bring additional reconstruction errors. Some studies have found that the antenna pattern has an important impact on the aperture synthesis radiometer. After taking into account the coupling effect between antennas, a new formulation of the visibility function equation (called the Corbella equation) was derived in [8; 9]. [PERSON] et al. analyzed the impact of antenna errors on radiometric accuracy and emphasized that antenna patterns are critical and need to be calibrated [10]. [PERSON], [PERSON], et al. investigated the relationship between the reconstruction error and antenna pattern differences. They took the SMOS antenna as the reference and analyzed the influence of the pattern differences and the element spacing on the reconstruction error [11]. At present, some methods have been proposed to mitigate antenna pattern error. The flat target transformation (FTT) method proposed in reference [12] can be used to correct the antenna pattern error. This method reduces the antenna pattern error by measuring a \"flat\" target such as the cold sky near the galactic poles. Refs. [13; 14] proposed to observe a reference stable ocean scene instead of the cold sky in [12], thereby further reducing the image errors caused by antenna pattern error. However, it is not easy to find an ideal reference target for observation, and the quality of its correction effect depends on the flat target being observed. In addition, these methods only correct part of the antenna pattern errors. In this paper, antenna pattern errors are analyzed for FPASR and a two-step correction method is proposed to correct the antenna pattern errors.