## 2 Materials and Methods  
### Optimal Estimation Theory  
Following the optimal estimation (OE) theory [26], ALH retrieval is the process of obtaining the best matching ALH parameter based on satellite observations. The forward model can be described as  
\[\mathbf{y}=F(\mathbf{x},\mathbf{b})+\epsilon \tag{1}\]  
where \(\mathbf{x}\) is a state vector, \(\mathbf{b}\) comprises parameters that are not present in \(\mathbf{x}\) but have a quantitative impact on TOA measurements, \(\mathbf{y}\) is an observation vector, \(F\) is a forward model, and \(\epsilon\) represents an experimental error term that integrates observation noise and forward model uncertainty. Meanwhile, the maximum likelihood analysis of \(\mathbf{x}\) can be described as  
\[\hat{\mathbf{x}}=\mathbf{x}_{a}+\left(\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{-1}\mathbf{K}+\mathbf{S}_{ a}^{-1}\right)^{-1}\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{-1}(\mathbf{Y}-\mathbf{K}\mathbf{x}_{a}) \tag{2}\]  
where the superscript \"\(-1\)\" is the inverse operation of the matrix, \(\mathbf{S}_{a}\) is the error covariance matrix of the a priori estimate \(\mathbf{x}_{a}\), and \(\mathbf{K}\) is the Jacobian matrix, which consists of the partial derivatives of \(F(\mathbf{x})\) with respect to \(\mathbf{x}\) and describes the sensitivity of the forward model to the state vector. \(\mathbf{S}_{\epsilon}\) is the covariance matrix of the error. The a priori estimate provides constraints for the retrieval of the optimal estimation algorithm to prevent the generation of unreasonable results. In Equation (2), \(\hat{\mathbf{x}}\) corresponds to the minimum value of the scalar-valued cost function after convergence, and the scalar-valued cost function \(\mathbf{J}\) is used as the form of  
\[\mathbf{J}=\left(\mathbf{y}-\mathbf{K}\mathbf{x}\right)^{T}\mathbf{S}_{\epsilon}^{-1}(\mathbf{y}-\mathbf{ K}\mathbf{x})+\left(\mathbf{x}-\mathbf{x}_{a}\right)^{T}\mathbf{S}_{a}^{-1}(\mathbf{x}-\mathbf{x}_{a}) \tag{3}\]  
The posteriori error is also known as the theoretical retrieval error. It can be calculated by combining the prior error with error propagation, which can be written as  
\[\hat{\mathbf{S}}=\left(\mathbf{S}_{a}^{-1}+\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{-1}\mathbf{K} \right)^{-1} \tag{4}\]  
This indicates that the statistical uncertainties in retrieved \(\hat{\mathbf{x}}\) due to noise in measurements and error propagation, as well as the posterior errors (absolute errors), are equal to the square root of the diagonal elements. Besides the posterior error covariance matrix, another key parameter that measures the amount of information about the parameters recovered from observations is the averaging kernel function \(\mathbf{A}\):  
\[\mathbf{A}=\frac{\partial\hat{\mathbf{x}}}{\partial\mathbf{x}}=\left(\mathbf{K}^{T}\mathbf{S}_{ \epsilon}^{-1}\mathbf{K}+\mathbf{S}_{a}^{-1}\right)^{-1}\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{ -1}\mathbf{K} \tag{5}\]  
The degree of freedom for signal (DFS), also known as the trace of the averaging kernel matrix \(\mathbf{A}\), denotes the amount of information about the retrieved parameters that can be explored from the satellite data. When the matrix is the identity matrix, the retrieved state can fully reflect the true state; Typically, we may assume that the parameter can be retrieved when the DFS result is \(>\)0.5 [27; 28]. Each diagonal element, in turn, describes the information contained in each retrieved parameter, given that the averaging kernel specifies the relative weight of information about the state vector contained in satellite measurements as well as from a priori measurements [29]. Considering that the averaging kernel describes the relative weight of information about the state vector from the satellite measurements compared with that from the a priori measurements, the DFS value depends on the a priori uncertainty assumption. In other words, a higher \"degree of freedom for signal\" corresponds to a lower posterior error with the same a priori measurement.