## 2 Analysis of Antenna Pattern Errors
### Antenna Pattern Differences  
Under ideal conditions, the antenna patterns of all the elements in Equation (1) are assumed to be identical. The relationship between the visibility function and the reconstructed BT satisfies the Fourier transform \(V(u,v)=F\{T(\xi,\eta)\}\). The G-matrix is analogous to an inverse Fourier transform. When backlobe and cross-polarization effects are disregarded, the reconstructed BT can be expressed as an inverse Fourier transform IF of the visibility function:  
\[\mathbf{invT}=IF(\mathbf{V})=\mathbf{G}^{-1}\mathbf{V} \tag{4}\]  
The elements of the G-matrix \(G_{kj}^{0}(\xi,\eta)\) are expressed as follows:  
\[G_{kj}^{0}(\xi,\eta)=\frac{\left|F_{n0}(\xi,\eta)\right|^{2}}{\Omega_{0}\sqrt {1-\xi^{2}-\eta^{2}}}e^{-j\pi(u\xi+\eta)}\Delta\xi\Delta\eta \tag{5}\]  
where \(F_{n0}(\xi,\eta)\) represents the identical ideal antenna pattern applicable to all elements in the array, and \(\Omega_{0}\) is the corresponding solid angle of the antenna. For an FPASR system, the limitations in machining accuracy and the variations in the electromagnetic environments surrounding each antenna result in disparities in electrical performance among different antennas. It is reflected in the antenna pattern differences of each element. In addition, the processing and assembly of the feed network and antenna often introduce amplitude and phase errors into their respective patterns. For the antenna array, these errors are not the same for each element, which is also the reason for the antenna pattern differences. At this point, the visibility function equation deviates from the simple Fourier transform. The elements in G-matrix are expressed as Equation (2), where \(F_{nk}(\xi,\eta)\
eq F_{nj}(\xi,\eta)\) and \(\Omega_{k}\
eq\Omega_{j}\). The antenna pattern differences can be characterized as an additive error to the average antenna pattern, which is different for each antenna within the array. Then, the antenna pattern can be expressed as follows:  
\[ \begin{array}{l}\left|F_{nk}(\xi,\eta)\right|=\left|\overline{F_{n}}(\xi, \eta)\right|+\left|D_{k}(\xi,\eta)\right|,\varphi_{k}(\xi,\eta)=\overline{ \varphi}(\xi,\eta)+\varphi_{Dk}(\xi,\eta)\\ \left|F_{nj}(\xi,\eta)\right|=\left|\overline{F_{n}}(\xi,\eta)\right|+\left|D _{j}(\xi,\eta)\right|,\varphi_{j}(\xi,\eta)=\overline{\varphi}(\xi,\eta)+ \varphi_{Dj}(\xi,\eta)\end{array}  \tag{6}\]  
where \(\left|F_{nk,nj}(\xi,\eta)\right|\) represents the amplitudes of the antenna pattern. \(\varphi_{k,j}(\xi,\eta)\) represents the phases of the antenna pattern. \(\overline{F_{n}}(\xi,\eta)\) is the average antenna pattern, \(\overline{F_{n}}(\xi,\eta)=\frac{F_{nl}(\xi,\eta)+F_{n2}(\xi,\eta)+...+F_{nM }(\xi,\eta)}{M}\). \(M\) indicates the number of antenna elements in the array. \(\left|\overline{F_{n}}(\xi,\eta)\right|\) represents the amplitude of the average pattern. \(\overline{\varphi}(\xi,\eta)\) represents the phase of the average pattern. \(D_{k,j}(\xi,\eta)\) represents the deviation from the array average pattern. \(\left|D_{k,j}(\xi,\eta)\right|\) represents the amplitude of the deviation, and \(\varphi_{Dk,Dj}(\xi,\eta)\) represents the phase of the deviation. The impact of antenna pattern differences on BT reconstruction is highly complex, and there is currently no universally accepted metric for quantifying the extent of these differences. In ref. [11], the SMOS pattern is used as a reference to quantify the level of antenna pattern differences. However, this method lacks general applicability. This paper adopts the average antenna pattern as a reference standard.