## 2 Cold-Sky Calibration Simulator  
Synthetic aperture radiometers can obtain high-sensitivity and large-spatial-domain measurements of Earth's surface simultaneously. The different interference baselines are performed by cross-correlating pairs of antenna elements. The visibility function can be formed based on the interference patterns from reasonable spatially distributed antennas. A Fourier synthesis process of the cross correlations is performed through Equation (1), where \(\xi\),\(\eta\) represent the direction cosines and are defined with respect to the angles in the antenna reference frame, \(\Omega\) is the solid angle of the antennas, \(T_{B}\) is the simulated brightness temperature of the scene, \(T_{rec}\) is the estimated equivalent noise temperature of the receivers, and \(F(\xi\eta)\) is the normalized antenna pattern [8].  
\[\begin{split}&\mathrm{V}_{i,j}(u,v)=\frac{1}{\sqrt{\Omega_{1} \Omega_{2}}}\iint\limits_{\xi^{2}+\eta^{2}\leq 1}\frac{T_{B}(\xi\eta)-T_{rec}}{ \sqrt{1-\xi^{2}-\eta^{2}}}\\ &\cdot F_{1}(\xi,\eta)F_{2}{}^{*}(\xi,\eta)\tilde{r}_{12}\cdot \exp(-j2\pi(u\xi+v\eta))d\xi d\eta\end{split} \tag{1}\]  
The brightness temperature image reconstruction of the synthetic aperture radiometer can be approximately computed using the inverse Fourier transform of the visibility function, as shown below:  
\[\tilde{T}_{B}^{sim}(\xi,\eta)\propto F^{-1}(V(u,v)-\mathrm{V}_{R}(u,v)) \tag{2}\]  
The simulator is a forward algorithm [9], which can compute the brightness temperature of the antenna, given the parameters of the surface and surrounding environment (e.g., the ocean, land, sun, moon, and celestial sky) [10; 11]. The effects of propagation, attenuation, its own emitted radiation into the atmosphere, and the radiation coming from the Earth's surface are considered in the simulator. The brightness temperature of the sources, which, considered here, are the ocean and land surfaces over Earth, the sun/moon, and the celestial sky background, are modeled using real historical data or empirical data. The effects of the troposphere and Faraday rotation of L-band signals are modeled in the simulator, too. The antenna pattern of each feed horn or reflector antenna with a common feed horn is used to select the appropriate area that meets the requirements for cold-skycalibration. The optimum conditions are based on those described in detail in [10] and some new considerations. The brightness temperature of the target's scene at each antenna is first to be calculated, and then the synthetic images can be calculated. Earth is assumed to be spherical and ellipsoid in shape, respectively, to obtain the radiation contribution from Earth's surface because the cost of calculation of the Earth ellipsoid model is much more than the spherical model when simulating satellite operation on board. Even a whole year's worth of orbital data need to be looked through to find a suitable cold-sky calibration time and location. It is necessary to trade off efficiency and accuracy for the run time of the simulator. The angle pointing error brought by the spherical model is about 0.2 degrees against the World Geodetic System of 1984 (WGS-84) Earth ellipsoid model [12]. The approximation error of spherical and ellipsoid models is calculated and evaluated, and the inversed brightness temperature of the synthetic aperture radiometer can be given to show the effect of cold-sky calibration.