## 3 Inversion Algorithm of Atmospheric Temperature in MLT
### Optimal Estimation Method  
The optimal estimation method is used to retrieve atmospheric state parameters from indirect measurements, thus determining the statistical optimal solution of the state from the set of possible states [9]. If the atmospheric state parameter is a vector \(x\in R^{n}\) and the forward model is \(F\): \(R^{n}\to R^{m}\), then the data vector of the limb observed radiance \(y\in R^{m}\) is  
\[y=F(x) \tag{4}\]  
In reality, the observed limb radiance \(y_{\varepsilon}\) is affected by measurement noise \(\varepsilon\). Equation (4) is modified as follows:  
\[y_{\varepsilon}=F(x)+\varepsilon \tag{5}\]  
Figure 8: Limb observed radiance vertical profiles of O\({}_{2}\)\(A\)-band night glow with random noise. Figure 7: Comparison of a priori information and actual temperature information. The inverse problem is defined as the target state vector \(x\) retrieved from the measurement \(y_{\epsilon}\), as follows:  
\[x=F^{-1}(y_{\epsilon}) \tag{6}\]  
In the case of a linear relationship between \(x\) and \(y\), the measurement vector \(y_{\epsilon}\) with noise \(\epsilon\) can be related to the actual state vector \(x\) through the weighting function matrix\(K\):  
\[y_{\epsilon}=Kx+\epsilon \tag{7}\]  
where the weighting function \(K=\frac{\partial y}{\partial x}\). For ill-fitting non-linear systems, it is usually not feasible to directly inverse the forward model \(F^{-1}\). The inverse problem can be approximated as a minimization problem. In a sense, the optimal solution is defined as the minimization of the difference between the forward model calculation and the actual observation, as follows:  
\[\min_{x}\parallel F(x)-y_{\epsilon}\parallel^{2} \tag{8}\]  
The inherent measurement error results in an infinite number of possible solutions within the measurement error. In order to improve the solution, a priori knowledge \(x_{a}\) of state parameters can be introduced as constraints. Meanwhile, in order to reduce the impact of measurement errors and ensure the uniqueness of solutions, regular terms can be introduced as constraints. Equation (8) can be expressed by the cost function \(J(x)\):  
\[J(x)=(F(x)-y_{\epsilon})^{T}S_{\epsilon}{}^{-1}(F(x)-y_{\epsilon})+(x-x_{a})^ {T}S_{a}{}^{-1}(x-x_{a}) \tag{9}\]  
where \(S_{\epsilon}\in R^{m\times m}\) is the covariance matrix for measuring noise. \(x_{a}\) is a priori state vector from the atmospheric state parameters, and \(S_{a}\in R^{n\times n}\) is the corresponding covariance matrix. In the traditional optimal estimation method, this value is the square of a priori value multiplied by an 'uncertainty factor', which generally takes a value of 10. However, in practical applications, the variance with large values will lead to the non-convergence of some inversion problems, so the 'uncertainty factor' can also be reduced to between 5 and 0.1 until convergence is satisfied. In this paper, the Tikhonov regularization method is used to construct the a priori covariance matrix. Tikhonov regularization [10] is one of the most widely used regularization methods. The Tikhonov regularization matrix \(S_{a}{}^{-1}\) is defined as  
\[S_{a}{}^{-1}=\alpha_{0}{}^{2}L_{0}{}^{T}L_{0}+\alpha_{1}{}^{2}L_{1}{}^{T}L_{1 }+\alpha_{2}{}^{2}L_{2}{}^{T}L_{2}+\cdots+\alpha_{n}{}^{2}L_{n}{}^{T}L_{n} \tag{10}\]  
where \(L_{0}\) to \(L_{n}\) are the 0 to n-order Tikhonov regularization matrices, respectively, and where \(\alpha_{0}\) to \(\alpha_{n}\) are the corresponding weight factors, which are specified according to the comparison between the contribution degree of the total regularization term in the cost function and the contribution degree of the measurement results. Ideally, the information content in the inversion results should come from the measurements themselves. The cost function minimization problem can be solved adequately by an iterative optimization algorithm. The Gauss-Newton iterative algorithm is used in this paper. The Gauss-Newton iteration method is shown below [11].