## 3 Retrieval Method  
### Optimal Estimation Theory and Retrieval Error Analysis Method  
The retrieval is done using optimal estimation theory. The DFS is used to represent the number of independently-retrievable quantities that can be derived from the inversion. Here, we summarize the main formulas used in the analysis. The DFS is defined as the trace of the averaging kernel matrix, which describes the sensitivity of the retrieval to the true state vector:  
\[\mathbf{A}=\frac{\partial\mathbf{\hat{x}}}{\partial\mathbf{x}}=\mathbf{G} \mathbf{K}. \tag{3}\]  
Here, \(\mathbf{K}\) is a Jacobian matrix (derivatives of the simulated measurements with respect to elements of the state vector), and \(\mathbf{G}\) is the contribution function matrix, defined as:  
\[\mathbf{G}=\left(\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{-1}\mathbf{K}+\mathbf{S }_{a}^{-1}\right)^{-1}\mathbf{K}^{T}\mathbf{S}_{\epsilon}^{-1}, \tag{4}\]  
in which \(\mathbf{S}_{\epsilon}\) is the observation error covariance matrix and \(\mathbf{S}_{a}\) represents the a priori error covariance matrix. The averaging kernel matrix relates the retrieval state \(\mathbf{\hat{x}}\) to the a priori \(\mathbf{x}_{\text{a}}\) and the true state vector \(\mathbf{x}\) as follows:  
\[\mathbf{\hat{x}}=(\mathbf{I}_{n}-\mathbf{A})\mathbf{x}_{a}+\mathbf{A}\mathbf{ x}+\mathbf{G}_{y}\boldsymbol{\varepsilon}_{y}, \tag{5}\]in which \(\mathbf{I}_{n}\) is a unit matrix with dimension n (the rank of the state vector) and \(\epsilon_{y}\) represents random error in the measurements. In general, the averaging kernel quantifies the ability to infer the a posteriori state \(\mathbf{\hat{x}}\) for specified observation noise and a priori characterization. The closer \(\mathbf{A}\) is to the unit matrix, the greater is the information that can be obtained from observation and the less it depends on the a priori characterization. We will discuss the DFS for a number of simulation scenarios in the following section. From Equation (5), an expression for the retrieval error can be derived:  
\[\mathbf{\hat{x}}-\mathbf{x}=(\mathbf{A}-\mathbf{I}_{n})(\mathbf{x}-\mathbf{x }_{a})+\mathbf{G_{y}}\mathbf{K_{b}}\Big{(}\mathbf{b}-\mathbf{\hat{b}}\Big{)}+ \mathbf{G_{y}}\Delta\mathbf{f}\big{(}\mathbf{x},\mathbf{b},\mathbf{b}^{\prime} \big{)}+\mathbf{G_{y}}\epsilon, \tag{6}\]  
The retrieval error comprises four sources: the smoothing error, the model parameter error, the forward model error and the measurement noise. In this equation, the vector \(\mathbf{b}\) is the ancillary model parameters that are not included in the state vector. The smoothing error is the component related to a priori uncertainties. Thus, the smoothing error covariance matrix \(\mathbf{S_{s}}\) corresponding to the a priori error \(\mathbf{S_{a}}\) is:  
\[\mathbf{S_{s}}=(\mathbf{A}-\mathbf{I}_{n})\mathbf{S_{a}}(\mathbf{A}-\mathbf{ I}_{n})^{T}. \tag{7}\]  
Similarly, measurement noise \(\mathbf{S_{m}}\) and the model parameter error covariance \(\mathbf{S_{f}}\) can be expressed as Equations (8) and (9), respectively (assuming these are distributed normally):  
\[\mathbf{S_{m}}=\mathbf{G}\mathbf{S_{\epsilon}}\mathbf{G}^{T}. \tag{8}\]  
\[\mathbf{S_{f}}=\mathbf{G}\mathbf{K_{b}}\mathbf{S_{b}}\mathbf{K_{b}^{T}} \mathbf{G}^{T}. \tag{9}\]  
The error covariance matrix \(\mathbf{S_{\epsilon}}\) indicates instrument noise, which is usually random and uncorrelated between channels, and is therefore defined by the SNR in each channel of CAPI in our study. The evaluation of forward model error cannot be completed easily by some matrix calculation, so in this work, we do not take this error into consideration and only estimate the uncertainty introduced by ancillary parameters in the forward model. In addition, we ignore the uncertainties of other parameters, with auxiliary vector \(\mathbf{b}\) only including the aerosol profile shape parameters.