## 3 Error Characterization and Simulation Method
### Errors Description
#### 3.1.2 Calibration Errors  
Calibration is important, since it can produce the calibrated limb radiances. These errors in calibration should be avoided in some degree. However, residual errors will also remain after data processing. The two-point calibration is a widely used method for microwave radiometer calibration ([PERSON] et al., 2018; [PERSON] et al., 2018). The schematic of two-point calibration is illustrated in Figure 1.  
The two-point calibration equation can be established as ([PERSON] et al., 2018):  
\[\dot{I}_{A}^{i}=\frac{C_{A}^{i}-C_{C}^{i}}{C_{H}^{i}-C_{C}^{i}}\left(\dot{I}_ {H}^{i}-I_{C}^{i}\right)+\dot{I}_{C}^{i}+\Delta I_{A}^{i}\, \tag{3}\]  
Figure 1: Schematic of two-point calibration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Source** & **Value** \\ \hline Sideband weight & 0.2\% \\ Local oscillator & 0.5 MHz \\ Pointing offset (LOS) & 0.021\({}^{\circ}\) \\ Measurement noise (NEDT) & 2.2 K, 2.2 K, 2.2 K, 5.1 K \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Uncertainties assumed for instrument. \[\Delta I_{A}^{i}=u\Big{(}i_{H}^{i}-I_{C}^{i}\Big{)}^{2}\frac{\Big{(}C_{A}^{i}-C_{H} ^{i}\Big{)}\Big{(}C_{A}^{i}-C_{C}^{i}\Big{)}}{\Big{(}C_{H}^{i}-C_{C}^{i}\Big{)}^ {2}}, \tag{4}\]  
where \(i\) represents the channel, \(C_{A}\), \(C_{H}\), \(C_{C}\) are the counts of Earth, hot and cold targets, respectively. \(I_{H},I_{C}\) are the radiance of the hot target and the cold space obtained by the instrument, respectively. \(\Delta I_{A}\) means the system nonlinearity difference, \(u\) is the nonlinearity coefficient. The possible error sources from calibration are the residual nonlinearity radiance of calibration and the hot target offset, due to the instability of the calibration target. In this study, the hot target temperature is added by a \(-0.5\) K offset to produce the uncertainty in calibration. The residual nonlinearity brightness temperature is set to a standard deviation of 0.3 K, which is a typical value measured by the radiometer test. Another possible source of error is the Rayleigh-Jeans (R-J) approximation used in radiometric calibration [30]. R-J approximation can convert radiance to brightness temperature by using a linear equation, and is widely used in radiometric calibration at the MMW band up to the THz band. However, R-J approximation will lead to a brightness temperature error which varies with frequency and temperature, and it is not easily corrected by adding an offset. In this simulation, \(I\) in Equation (3) is replaced by \(T\) when using the R-J approximation, and \(T_{C}\) and \(T_{H}\) are set to 2.73 K and 300 K, respectively. The variations of calibration errors with temperature and count ratio are shown in Figure 2. The count ratio means \(\frac{C_{A}\sim C_{C}}{C_{H}-C_{C}}\). Figure 2 shows that although the differences \(T_{RJ}-T_{B}\) seem to be a constant error above 50 K, it also varies with count ratio. So, it cannot be corrected easily. TALIS measures atmospheric limb radiances captured by the antenna. The radiance which is accepted by the antenna can be expressed by:  
\[I_{A}^{i}=\int_{band}F(v)\int_{\Omega}I_{B}^{i}(\Omega)W_{A}(\Omega)d\Omega dv\, \tag{5}\]  
where \(i\) represents the channel, \(I_{A}^{i}\) represents the antenna received radiance, \(I_{B}^{i}\) represents the radiance emitted from the atmosphere, \(W_{A}\) describes the normalized antenna response function, \(F(v)\) describes  
Figure 2: The variations of calibration errors with temperature and the count ratio induced by the R–J approximation: **(a)** variations with temperature; **(b)** variations with the count ratio. Here \(T_{B}\) is calculated by Equation (3), and \(T_{RJ}\) is obtained from replacing \(I\) by \(T\) with \(T_{C}\) and \(T_{H}\) set to 2.73 and 300 K, respectively. The count ratio means \(\frac{C_{A}\sim C_{C}}{C_{H}-C_{C}}\). the frequency response of the radiometer, and \(\Omega\) is the solid angle. The integrals are estimated over the full range of frequencies and solid angles where instrument response exists.