## 2 Cold-Sky Calibration Simulator
### Moon and Sun's Radiation during Calibration  
The estimation of the angle of the sun or moon with respect to the array boresight is critical for calibration. The total antenna temperature and the observed standard deviation error of the celestial sky measurements are affected by the sun's position in SMOS [5]. The brightness temperature drifts, although the receiver temperature does not change much. Therefore, the influence of the sun/moon on the synthetic aperture radiometer is one of the factors that must be considered in external calibration. In a given epoch, the position of the spacecraft in an Earth-fixed coordinate system is \(\mathbf{P_{s}}\) (\(x\), \(y\), \(z\)), and the velocity is \(\mathbf{V_{s}}\) (\(V_{x}\), \(V_{y}\), \(V_{z}\)), as shown in Figure 3. The + \(Z\) axis of the antenna of the radiometer points to the ground, and the +\(X\) axis is oriented towards the track direction when the satellite is flying forward. According to the sun/moon vectors \(\mathbf{SM}\)(\(x\), \(y\), \(z\)), the angle \(\theta_{a}=f\)(\(\mathbf{P_{oz}}\), \(\mathbf{SM}\)) can be obtained, that is, the angle between the sun/moon vector and + \(Z\) axis [6]. \(\mathbf{P_{oz}}\) is the +\(Z_{s}\) axis direction of the spacecraft body frame in the Earth-fixed coordinate system, and it can be obtained by the attitude variations (roll, pitch, or yaw) from spacecraft data. The SM vector originates from where \(\mathbf{P_{oz}}\) originates. The angle between the sun/moon vector and the + \(X\) axis of the antenna coordinate system is \(\varphi_{a}=f\)(\(\mathbf{P_{oz}}\), \(\mathbf{Proj_{oxy\_plane}}\)(\(\mathbf{SM}\))) in the spherical coordinate system, where \(\mathbf{Proj_{oxy\_plane}}\)(\(\cdot\)) is the vector projection function to the XOY plane and \(f\)(\(\mathbf{X}\), \(\mathbf{Y}\)) = \(\tan^{-1}(|\mathbf{X}\times\mathbf{Y}|/\mathbf{X}\cdot\mathbf{Y})\). The sun/moon reflection location on the ground is described in the following equation, assuming that Earth's semi-major axis is \(a_{e}\), the semi-minor axis is \(b_{e}\), and the position of the reflection point \(\mathbf{P_{ref}}\) can be written in the form of the following equations. This situation calls for a more realistic elliptical model for Earth.  
\[\left({{p_{ref}}_{2}}^{2}+{p_{ref}}_{y}^{2}\right)/{a_{e}}^{2}+{p_{ref}}_{z}^{ 2}/{b_{e}}^{2}=1 \tag{7}\]  
\[\left(\frac{\mathbf{SM}-\mathbf{P_{ref}}}{|\mathbf{SM}-\mathbf{P_{ref}}|}- \frac{\mathbf{P_{s}}-\mathbf{P_{ref}}}{|\mathbf{P_{s}}-\mathbf{P_{ref}}|} \right)\cdot\frac{\mathbf{P_{ref}}}{|\mathbf{P_{ref}}|}=0 \tag{8}\]  
\[\left((\mathbf{SM}-\mathbf{P_{ref}})\times(\mathbf{P_{s}}-\mathbf{P_{ref}}) \right)\cdot\mathbf{P_{ref}}/|(\mathbf{SM}-\mathbf{P_{ref}})\times(\mathbf{P_{ s}}-\mathbf{P_{ref}})|=0 \tag{9}\]  
The angle between the reflected sun/moon direction and the antenna boresight can be calculated by solving the position of the sun/moon reflection at the intersection of the grid points with the WGS-84 Earth ellipsoid.  
Figure 4 shows the direct sun/moon and reflected sun angles entering into the antenna's field of view in a complete year for a typical sun-synchronous orbit. The Y axis of the antenna reference frame is perpendicular to the orbital plane, and the X axis is tilted 30 degrees upward from the velocity vector of the satellite. Figure 5 shows the brightness temperature map of the sun with an antenna pattern like SMOS antennas [13]. The brightness temperature of the geometric location of the satellite position is calculated by one antenna element.  
Figure 4: _Cont._  
Figure 3: The geometry of the satellite, Earth, sun/moon, and antenna coordinates. A spherical coordinate system is used to define the elevation and azimuth angles of the antenna coordinate. A typical position of the reflection of the sun/moon on Earth is illustrated in the figure. This figure is not to scale.