## 3 Inversion Algorithm of Atmospheric Temperature in MLT
### Peak Relative Intensity Ratio Method  
Through the study of the above forward model, it is found that the correlation between the emission line intensity within O\({}_{2}\)\(A\)-band airglow and temperature is different. Therefore, the indirect retrieval of atmospheric temperature in MLT can be carried out by using the relative relationship of different emission line intensities at a specific temperature. For \(A\)-band airglow, the rotational transition radiation in thermal equilibrium obeys Boltzmann distribution at rotational temperature \(T_{rot}\). For different rotational emission lines in the same band, the rotational partition function \(Q(T_{rot})\), the Einstein coefficient \(A_{762}\), and VER \(\eta\) are the same at a certain temperature \(T_{rot}\). It is worth noting that in the limb mode, the Doppler broadening and the path integral effect in the radiative transfer in MLT, and the convolution effect caused by the instrument system, will change the correlation between the relative intensity of emission line and temperature. Therefore, the traditional peak intensity ratio method needs to be optimized to improve accuracy. When the temperature is \(T_{rot}\), the observation intensity of the central emission line \(v_{c}\) in any field of view is  
\[I(v_{c})=\eta\cdot\frac{A_{m}}{A_{762}}\cdot\frac{g_{m}}{Q(T_{rot})}\cdot \exp\left(-\frac{hcE_{m}}{kT_{rot}}\right)\cdot D(v_{c})\cdot s \tag{16}\]  
where \(A_{m}\), \(g_{m}\) and \(E_{m}\) represent the molecular spectroscopy parameters. \(D(v_{c})\) is Doppler line shape at wavenumber \(v_{c}\). Expand Equation (17) and take logarithms on both sides of the equation and shift terms:  
Figure 12: (**a**) Retrieval of atmospheric temperature in MLT region; (**b**) the bias between retrieved temperature and actual temperature at different tangent heights. \[\ln\left(\frac{I(v_{c})\cdot v_{c}}{A_{m}\cdot g_{m}}\right)=-\frac{hc}{kT_{rot}} \cdot E_{m}+\ln\left(\frac{\eta\cdot s\cdot c}{A_{762}\cdot Q(T_{rot})}\right)- \frac{1}{2}\ln\left(\frac{2k\pi}{m}\right)-\frac{1}{2}\ln(T_{rot}) \tag{17}\]  
The above equation can be considered as the linear equation \(y=k_{0}x+b\) corresponding to the temperature \(T_{rot}\), where, \(y=\ln\left(\frac{I(v_{c})\cdot v_{c}}{A_{m}\cdot g_{m}}\right)\), the slope of the straight line \(k_{0}=-\frac{hc}{kT_{rot}}\), \(x=E_{m}\), \(b=\ln\left(\frac{\eta\cdot s\cdot c}{A_{762}\cdot Q(T)}\right)-\frac{1}{2}\ln \left(\frac{2k\pi}{m}\right)-\frac{1}{2}\ln(T)\) is constant, and where the value of the constant \(b\) has no effect on the slope of the straight line. Therefore, multiple emission lines at a specific temperature \(T_{rot}\) are fitted to a straight line by polynomial fitting, as shown in Figure 13 below. The straight-line slope can be derived from the temperature, \(T_{rot}=-\frac{hc}{k_{0}}\). Based on the observed radiance data, the above peak relative intensity ratio method is conducted for the synchronous retrieval of atmospheric temperature in the whole range of tangent height. The retrieval of atmospheric temperature in MLT, and the bias between retrieved temperature and actual temperature at different tangent heights were shown in Figure 14a,b. Comparing the retrieval of temperature through the above different inversion algorithms, it can be seen that the accuracy of the retrieved temperature based on the optimal estimation method is the highest, and the mean absolute bias is 3.4 K within the range of  
Figure 14: (**a**) Retrieval of atmospheric temperature in MLT region; (**b**) the bias between retrieved temperature and actual temperature at different tangent heights. Figure 13: The straight line fitted by the 6 emission lines at different temperatures. 80-120 km.