## 2 Methodology
### Linear Error Analysis
Based on the errors of the forward model parameter \(\mathbf{S}_{\text{b}}\) and the Jacobians with respect to parameter \(\mathbf{K}_{\text{b}}\), we specified the forward model parameter error covariance as follows:  
\[\hat{\mathbf{S}}_{f}=\mathbf{G}_{y}\mathbf{K}_{\text{b}}\mathbf{S}_{\text{b}} \mathbf{K}_{\text{b}}^{T}\mathbf{G}_{y}^{T}. \tag{10}\]  
In fact, in some studies, the a posteriori error only includes the first three errors noted above. The forward model and all the parameter inputs are as accurate as they can be; otherwise, we could improve the forward model. However, for our research regarding aerosol effects, if all the parametersin the aerosol model are not retrieved with CO\({}_{2}\), the uncertainties of those nonretrieved parameters should be considered in the XCO\({}_{2}\) retrieval error. Thus, the forward model parameter error related to the nonretrieved aerosol parameter uncertainties is also included in our analysis. The errors in other aerosol parameters, retrieved synchronously with CO\({}_{2}\), are components of the interference error. Finally, the total covariance matrix is the sum of each component:  
\[\hat{\mathbf{S}}=\hat{\mathbf{S}}_{s}+\hat{\mathbf{S}}_{i}+\hat{\mathbf{S}}_{m} +\hat{\mathbf{S}}_{f}. \tag{11}\]  
The a posteriori error variance in XCO\({}_{2}\) is \(\sigma_{\text{XCO2}}^{2}=h^{\mathsf{T}}\hat{\mathbf{S}}h\), and the corresponding error variance due to each error source can also be calculated in the same way by replacing \(\hat{\mathbf{S}}\) with another covariance matrix, such as the XCO\({}_{2}\) interference error \(\sigma_{i}\), which is estimated by \(\sigma_{i}^{2}=h^{\mathsf{T}}\hat{\mathbf{S}}_{i}h\). In other words, the XCO\({}_{2}\) a posteriori error \(\sigma_{\text{XCO2}}\) is the sum of \(\sigma_{s}\), \(\sigma_{i}\), \(\sigma_{m}\) and \(\sigma_{f}\). Generally, when focusing on the XCO\({}_{2}\) error due to aerosol parameters, those components related to aerosol parameters, both in the interference error and the forward model parameter error, are evaluated and compared among different aerosol models. From the hyperspectrala and Jacobians in the O\({}_{2}\)-A band, and two CO\({}_{2}\) NIR absorption bands of the TanSat, which is simulated by a vector linearized discrete ordinate radiative transfer model (VLIDORT), the DFS values of both CO\({}_{2}\) and aerosol properties in the state vector are calculated. After quantifying 4 components of the XCO\({}_{2}\) retrieval errors, errors related to aerosol parameter uncertainties were brought into focus. When the aerosol model in the CO\({}_{2}\) retrieval algorithm is altered, the corresponding DFS and retrieval errors differ, particularly those associated with aerosol properties. Accordingly, we sought to identify the ensemble of aerosol parameters in the aerosol model that yielded the largest amount of CO\({}_{2}\) information with the fewest aerosol-induced XCO\({}_{2}\) errors. Considering the complex aerosol mixtures, four aerosol scenarios were compared according to different solar zenith angles (SZAs). In adopting this new approach, the aerosol model is optimized to minimize the interference of aerosols in XCO\({}_{2}\) retrieval and could be applied to future retrievals from real data.