## 3 Error Characterization and Simulation Method
### Model and Retrieval Method  
In order to retrieve the atmospheric information, an accurate forward model which can simulate the radiative transfer, spectroscopy and instrumental characteristics, are necessary. In limb sounding, scattering can usually be neglected above the upper troposphere, since there are almost no clouds at these altitudes. While polar stratospheric clouds can be present, the particle sizes are smaller than the TALIS observation wavelengths. The cloud-free radiative transfer equation can be described as:  
\[I_{B}(S_{2})=I_{B}(S_{1})e^{-\tau_{(S_{1},S_{2})}}+\int_{S_{1}}^{S_{2}}\alpha~ {}(s)B~{}(T)e^{-\tau_{(}S_{2})}ds\,, \tag{8}\]  
where \(I_{B}\) is the radiance, \(\alpha\) is the absorption coefficient, \(S_{1}\) and \(S_{2}\) means the two position of the radiance, and \(\tau\) is the opacity or optical thickness. \(B(T)\) represents the atmospheric emission which is given by the Planck function. The optimal estimation method (OEM) is the most widely used retrieval method in atmospheric sounding [34]. The cost function in the retrieval method can be defined as:  
\[\chi^{2}=\left[\mathbf{y}-\mathbf{F}(\mathbf{x},\mathbf{b})\right]^{T}\!S_{y}^{-1}\!\left[\mathbf{y }-\mathbf{F}(\mathbf{x},\mathbf{b})\right]+\left[\mathbf{x}-\mathbf{x}_{a}\right]^{T}\!S_{a}^{-1} \!\left[\mathbf{x}-\mathbf{x}_{a}\right]\,, \tag{9}\]  
where \(x_{a}\) is a priori state vector, \(\mathbf{S}_{a}\) and \(\mathbf{S}_{y}\) are the covariance matrices representing the variability of the state vector and the measurement error vector, respectively. The core theory of OEM is to compare \begin{table}
\begin{tabular}{c c} \hline \hline
**Source** & **Value** \\ \hline Hot target offset & \(-0.5\) K \\ Nonlinearity & \(0.3\) K \\ R–J approximation & / \\ Antenna pattern used & 3 dB FWHM \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Errors assumed for calibration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Source** & **Value** \\ \hline Temperature a priori & 5 K \\ Pressure a priori & 5\% \\ Species a priori & 20\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Uncertainties assumed for external errors. the simulated radiance \(F(x,b)\) with the observed radiance \(y\) until the state which minimizes the \(\chi^{2}\) could be found. The Levenberg-Marquardt method, which is the modification of the Gauss-Newton iterative, is used to solve the nonlinear problem [37]. The solution is given by:  
\[\mathbf{x}_{i+1}=\mathbf{x}_{i}+\left[(\mathbf{I}+\gamma)\mathbf{S}_{a}^{-1}+\mathbf{K}_{xi}{}^{T} \mathbf{S}_{y}^{-1}\mathbf{K}_{i}\right]^{-1}\left(\mathbf{K}_{xi}{}^{T}\mathbf{S}_{y}^{-1}[ \mathbf{y}-\mathbf{F}(\mathbf{x}_{i})]-\mathbf{S}_{a}^{-1}(\mathbf{x}_{i}-\mathbf{x}_{a})\right)\,, \tag{10}\]  
where \(\gamma\) denotes the Levenberg-Marquardt parameter, and \(\mathbf{K}_{xi}\) represents the weighting function matrix (Jacobian). The error estimates induced by random and systematic uncertainties are given by:  
\[\varepsilon=|\hat{x}-\hat{x}_{0}|\,, \tag{11}\]  
where \(\varepsilon\) is the error induced by perturbing the parameters and \(\hat{x}\) is the state vector retrieved after changing the parameter according to its uncertainty, while \(\hat{x}_{0}\) is the state vector retrieved before changing the parameter. In the following simulations, the Atmospheric Radiative Transfer Simulator (ARTS) is used to calculate the measured radiance [38]. The retrieval tool is Qpack, which is the OEM tool developed for interaction with ARTS [39]. Temperature, H\({}_{2}\)O, O\({}_{3}\) and HCl are treated as main products of the 118 GHz, 190 GHz, 240 GHz and 643 GHz radiometers, respectively. A mid-latitude summer atmospheric condition extracted from FASCOD, which is provided by ARTS, is chosen to perform the simulation, and local spherical homogeneity is assumed. A vertical retrieval grid of 2.5 km resolution is selected. The choice of a retrieval grid is closely related to the antenna field of view (FOV). Generally speaking, the resolution of the retrieval grid should match the antenna FOV.