## 2 Cold-Sky Calibration Simulator
### The Antenna Temperature (Ta) of Each Antenna  
The salinity of the sea surface can be monitored from space by a microwave radiometer in the L-band, which requires higher sensitivity and accuracy than other radiometers. The main lobe of the antenna looks toward a uniform cold-sky scene with a low brightness temperature while cold-sky calibration is performed, which is used for the radiometer's absolute calibration and instrument stability test. The scene below the satellite can have an effect on the accuracy of the calibration reference temperature because the brightness temperature of Earth's surface is different, as shown in Figure 1. The brightness temperature of the land or ocean scene is much higher than the cold-sky scene, which impacts the Ta of the antenna through the back lobes [9]. The antenna temperatures of the radiometers due to emissions from all sources are calculated using the equation described below [9; 10]:  
\[\mathrm{T}_{a}=\frac{\iint G(\theta,\varphi)\cdot\overset{\rightarrow}{T}_{ap} (\theta,\varphi)d\Omega}{\iint G(\theta,\varphi)d\Omega} \tag{3}\]  
where Ta is the brightness temperature of the antenna, \(G\) is the antenna directivity pattern, (\(\theta\), \(\varphi\)) is the incident direction, and \(\overset{\rightarrow}{T}_{ap}\) is the effective brightness temperature of the directions. The integral is performed over all incident directions of the domain \(\Omega\). An imaging geometry of the look vector from a given antenna and its rotation for cold-sky calibration are given in Figure 2. The 'target' can be the ocean, the sky, the land, and the sun or moon. A local coordinate system is represented by (\(\hat{x},\hat{y},\hat{z}\)) in the figure. The look vector from a given antenna is defined by:  
\[\left( \begin{array}{c}\overset{\rightarrow}{x}\\ \overset{\rightarrow}{y}\\ \overset{\rightarrow}{z}\end{array} \right)=\left( \begin{array}{c}\sin\theta \cos\varphi\\ \sin\theta\sin\varphi\\ \cos\theta\end{array} \right)\overset{\rightarrow}{l} \tag{4}\]  
where \(\theta\) is the elevation angle, and \(\varphi\) is the azimuth angle of a spherical coordinate system. (\(\overset{\rightarrow}{x}\overset{\rightarrow}{y}\overset{\rightarrow}{z}\)) is a rectangular coordinate system, used to describe the antenna orientation, whose relationship with the spacecraft body coordinate system can be given by:  
\[\mathbf{F}_{x,y,z\rightarrow\overset{\rightarrow}{x},\overset{\rightarrow} {y},\overset{\rightarrow}{z}}=\mathbf{A}_{\mathrm{T}}\cdot\mathbf{M}_{e} \tag{5}\]  
where \(\mathbf{A}_{\mathrm{T}}=(R_{1}(\text{roll})\cdot R_{2}(\text{pitch})\cdot R_{3}( \text{yaw}))^{-1}\) is made of the attitude angle of the satellite, which is the rotational angle value, as shown in Figure 2. The attitude angle of the spacecraft can be measured by the Flight Control System. \(R(\cdot)\) is the Euler rotation matrix about the corresponding axis, and yaw, roll, and pitch stand for the rigid body S/C roll, pitch, and yaw angle, respectively. \(\mathbf{M}_{e}\) is the installation matrix of the antenna position on board. Figure 1: A typical relative orientation of the sun, moon, and satellite when the satellite operates on a sun-synchronous orbit. The white arrow indicates the direction of the moon; the red stands for the direction of the sun; the brown arrow is the forward-moving direction of the satellite; and the black curve shows the orbital plane of the satellite. All the data are instantaneous, which can be marked by the ephemeris. Figure 2: Imaging geometry of the look vector from a given antenna and its projection on the target. In order to obtain the brightness temperature when the satellite is flying forward, the scene into the radiometer's field of view (FOV) is divided into four parts: the sun, the moon, the celestial sky, and Earth. A typical relative orientation of the sun, moon, and satellite when the satellite operates on a sun-synchronous orbit is shown in Figure 1. The antenna brightness temperature is shown in Equation (6):  
\[