## 3 Inversion Algorithm of Atmospheric Temperature in MLT
### Pre-Selection of Observation Band and Construction of Observation Vector  
A total of 113 spectral lines within the O\({}_{2}\)\(A\)-band airglow full spectrum are numbered and segmented, and three representative bands including weak, medium, and strong bands are selected, respectively. Table 1 below shows the observed spectra of each band in the field of view of 92 km in Figure 5. The sensitivity of emission line integral intensity to temperature on each band is studied, respectively, which is used as the basis to match the best band for temperature inversion. A certain perturbation is made to the input atmospheric background temperature, and defining the integral observation radiance change ratio \(\Delta I(\
u)/I(\
u)\) of each target band as the response of this band to the temperature perturbance. Based on the above forward model, the temperature response of each target band is shown in Figure 6 below. From the figure above, we can see that the temperature response of each band is slightly different, and the absolute change ratio of Band 2 is the largest. The summary of the pre-selection of the three bands for temperature inversion is shown in Table 2 below, which shows that Band 2 may have a significant advantage in the inversion of atmospheric temperature in MLT. At the same time, the pre-selected band can avoid the need for the instrument to design too wide a channel to detect the full \(A\)-band airglow spectra and avoid too large a quantity of data to improve the inversion efficiency, and can obtain higher precision, as described in Section 4 below. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Band 1** & **Band 2** & **Band 3** \\ \hline
13,059.5 cm\({}^{-1}\) & 13,084.2 cm\({}^{-1}\) & 13,136.2 cm\({}^{-1}\) \\
13,061.3 cm\({}^{-1}\) & 13,086.1 cm\({}^{-1}\) & 13,138.2 cm\({}^{-1}\) \\
13,068.1 cm\({}^{-1}\) & 13,091.7 cm\({}^{-1}\) & 13,140.6 cm\({}^{-1}\) \\
13,070.0 cm\({}^{-1}\) & 13,093.7 cm\({}^{-1}\) & 13,142.6 cm\({}^{-1}\) \\
13,076.3 cm\({}^{-1}\) & 13,098.8 cm\({}^{-1}\) & 13,144.5 cm\({}^{-1}\) \\
13,078.2 cm\({}^{-1}\) & 13,100.8 cm\({}^{-1}\) & 13,146.6 cm\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Pre-selection of observation band and containing emission lines. Figure 4: Simulation configurations for the simulated observed spectra. Subgraph (**a**) corresponds to the number density of oxygen, atomic oxygen, nitrogen, and (**b**) corresponds to the temperature, respectively. The vertical profiles in Figure 7 are a priori information on temperature taken from the atmospheric model database and the actual temperature information, respectively. According to the actual temperature information and the above forward model, the O\({}_{2}\)\(A\)-band night glow limb observed radiance spectrum is simulated. Considering the inevitable measurement error in the actual observation process, the measurement noise is added, as shown in the following equation  
\[y_{\epsilon}=F(x)+\epsilon \tag{2}\]  
where \(y_{\epsilon}\) represents the actual observed spectrum; \(F(x)\) represents the simulated ideal observed spectrum, and \(\epsilon\) represents the measurement noise; and where the mean value of noise is 0, and the variance is equal to the value of \(F(x)\). The concept of signal-to-noise ratio (\(SNR\)) is introduced to evaluate the noise level. \(SNR\) is calculated by the following formula:  
\[SNR=ave./t \tag{3}\]  
where \(ave\) is mean value of radiance \(Bt\), \(t\) is the standard deviation of radiance \(Bt\), and \(Bt\) is the multiple detected radiance at a wavenumber. Assuming the \(SNR\) = 100, an array of observation spectra in all the vertical fields of view is constructed, and the corresponding