## 2 Methodology
### Linear Error Analysis  
For XCO\({}_{2}\) retrieval, the state vector \(\mathbf{x}\) consists of the CO\({}_{2}\) profile \(\mathbf{u}\) and non-CO\({}_{2}\) elements \(\mathbf{e}\), including the surface reflectance, water vapor, surface pressure and aerosol properties, which are retrieved simultaneously:  
\[\mathbf{x}=\left( \begin{array}{c}\mathbf{u}\\ \mathbf{e}\end{array} \right). \tag{4}\]  
XCO\({}_{2}\) is defined as the retrieved CO\({}_{2}\) profile weighted by the pressure weighting function \(\mathbf{h}\), such that XCO\({}_{2}=h^{T}\hat{\mathbf{x}}\). In fact, those elements of \(\mathbf{h}\) corresponding to the non-CO\({}_{2}\) profile (\(\hat{\mathbf{e}}\)) are zero. As shown in [16; 17], the a posteriori error of XCO\({}_{2}\) results from a list of sources, including the _a priori_ uncertainties of the CO\({}_{2}\) profile, interference of the non-CO\({}_{2}\) retrieved parameters, and inaccurate observation processes. Thus, the uncertainty in XCO\({}_{2}\) is expressed as follows:  
\[\Delta\mathbf{X}_{\text{CO2}}=\mathbf{h}^{T}(\hat{\mathbf{x}}-\mathbf{x})= \mathbf{h}^{T}(\mathbf{A}_{uu}-\mathbf{I})(\mathbf{u}-\mathbf{u}_{a})+ \mathbf{h}^{T}\mathbf{A}_{ue}(\mathbf{e}-\mathbf{e}_{a})+\varepsilon_{u}. \tag{5}\]  
Here, \(\mathbf{A}_{\text{uu}}\) and \(\mathbf{A}_{\text{ue}}\) are submatrices of \(\mathbf{A}\), denoting the CO\({}_{2}\)-only components, and components characterized by the relationship between the CO\({}_{2}\) profile \(\mathbf{u}\) and non-CO\({}_{2}\) elements \(\mathbf{e}\), respectively. \(\mathbf{I}\) is the identity matrix, and the subscript \(a\) represents the _a priori_ uncertainty. The first term on the righthand side of Equation (5) is the smoothing error derived from the CO\({}_{2}\) profile uncertainty, and the second term represents the interference error caused by non-CO\({}_{2}\) components in the state vector. \(\varepsilon_{u}\) includes all other errors. The corresponding covariance matrix for each error source can be specified as follows. The smoothing error covariance matrix is as follows:  
\[\hat{\mathbf{S}}_{s}=(\mathbf{A}-\mathbf{I})\mathbf{S}_{\text{a}}(\mathbf{A} -\mathbf{I})^{T}. \tag{6}\]  
The interference error is calculated using the following:  
\[\hat{\mathbf{S}}_{i}=\mathbf{A}_{ue}\mathbf{S}_{\text{ec}}\mathbf{A}_{ue}^{T}, \tag{7}\]  
where \(\mathbf{S}_{\text{ec}}\) describes the ensemble covariance for the non-CO\({}_{2}\) elements, which is not necessarily equal to the _a priori_ error, \(\mathbf{S}_{\text{a}}\). If considering the error in XCO\({}_{2}\) due to each state vector element (\(\sigma_{XCO2e}\)), the expression is as follows:  
\[\sigma_{XCO2e,j}=\left\{ \begin{array}{l}\mathbf{h}^{T}(\mathbf{A}-\mathbf{I}) \sigma_{j},j=1,\;q\\ \mathbf{h}^{T}\mathbf{A}\sigma_{j},j=q+1,\;n\end{array} \right., \tag{8}\]  
where \(\sigma_{j}\) is the _a priori_ uncertainty (or the square root of \(\mathbf{S}_{\text{ec}}\) for non-CO\({}_{2}\) elements) of each state vector element. The first \(q\) elements represent uncertainties related to the CO\({}_{2}\) profile (i.e., the smoothing errors) while the others are the interference error components. Two further errors include measurement noise and the forward model parameter uncertainties, which are not retrieved in the state vector. For measurement errors, the covariance matrix \(\hat{\mathbf{S}}_{m}\) is described by the following:  
\[\hat{\mathbf{S}}_{m}=\mathbf{G}_{y}\mathbf{S}_{\epsilon}\mathbf{G}_{y}^{T}, \tag{9}\]  
where \(\mathbf{S}_{\epsilon}\) is an estimation of the actual measurement errors. In our study, we only considered the measurement noise based on the instrument signal-to-noise ratio (SNR) and the simulated spectra of TanSat.