## 3 TanSat Simulations and Retrieval Assumptions
### The A Priori and Measurement Error Assumptions  
In addition to the simulated spectra, error analysis requires characterization of the _a priori_ error covariance matrix \(\mathbf{S_{a}}\) and the measurement error covariance matrix \(\mathbf{S_{\varepsilon}}\). The _a priori_ uncertainty of each retrieved parameter, described as a standard deviation (1\(\sigma\)), is summarized in Table 4. To prevent any unreasonable perturbation of the retrieved CO\({}_{2}\) profile, we imposed smoothness constraints on the CO\({}_{2}\) profile via nonzero off-diagonal elements in the _a priori_ error covariance matrix, \(\mathbf{S_{\varepsilon_{\text{CO2}}}}\). The square roots of the diagonal elements of \(\mathbf{S_{\varepsilon_{\text{AC02}}}}\) from the surface to the top of atmosphere (TOA) are presented in Table 4 and expressed as a vector \(\mathbf{\sigma_{\text{AC02}}}\). Figure 4 presents the correlation coefficient matrix (\(\mathbf{R}\)) of the CO\({}_{2}\) concentrations in 15 layers, similar to ([PERSON] et al., 2018). In other words, \(\mathbf{S_{\varepsilon_{\text{AC02}}}}\) is calculated by \(\mathbf{R\sigma_{\text{AC02}}}\mathbf{\sigma_{\text{AC02}}}^{\text{T}}\). Overall, the total _a priori_ uncertainty of XCO\({}_{2}\) is scaled to approximately 5.6 ppm. The _a priori_ error of water vapor column concentration is assumed to be 50% to describe its large uncertainty. The higher \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameter** & **Number of Parameters** & _a priori_ Uncertainty (1\(\sigma\)) \\ \hline CO\({}_{2}\) (ppm) & Mixing ratio profile & 15 & 7.09, 6.5, 6.0, 53.4, 79.3, 87, \\  & & & 2.75, 1.96, 1.84, 3.72) \\ H\({}_{2}\)O (ppm) & Scaling factor of mixing ratio profile & 1 & 50\% \\ Surface pressure (hPa) & Pressure at surface layer & 1 & 1\% \\  & Aerosol optical depth (AOD) & 1 & 100\% \\  & The effective radius (\(\tau_{\text{eff}}\)) of particle size & \(1\times 2\) & From AERONET \\  & distribution (PSD) for each mode & \(1\times 2\) & From AERONET \\ Aerosol & The effective variance (\(\tau_{\text{eff}}\)) of PSD for each mode & \(1\times 2\) & From AERONET \\  & each mode & \(1\times 2\) & From AERONET \\  & The peak height of aerosol profile (H\({}_{\text{fp}}\)) & 1 & 100\% \\  & The half width of aerosol profile (H\({}_{\text{w}}\)) & 1 & 100\% \\  & Fine mode fraction (fmf) & 1 & 100\% \\  & Parameters for real part of refractive & \(2\times 2\) & From AERONET \\  & index for each mode (\(a_{\text{r}}\), b\({}_{\text{r}}\)) 1 & \(2\times 2\) & From AERONET \\  & Parameters for imaginary part of & \(2\times 2\) & From AERONET \\  & refractive index for each mode (\(a_{\text{i}}\), b\({}_{\text{i}}\)) 1 & \(2\times 3\) & 20\% \\ Surface & BRDF coefficient of Lambertian kernel in each band & \(1\times 3\) & 20\% \\ \hline \hline \end{tabular}  
* a\({}_{r}\), b\({}_{r}\), a\({}_{\text{i}}\) and b\({}_{\text{i}}\) correspond to those parameters in Equations (12) and (13).  
\end{table} 
Table 4: The state vector for TanSat CO\({}_{2}\) retrieval and the corresponding _a priori_ uncertainty. accuracy of the surface pressure results in a smaller uncertainty. While the _a priori_ uncertainties of AOD, aerosol profile parameters, and fine mode fraction (fmf) (100%) for different aerosol types were assumed to be equivalent, and the standard deviation of the inversion products from four AERONET sites (Table 4) were used as the _a priori_ uncertainties of the refractive index coefficients (\(a_{r}\), \(b_{r}\), \(a_{i}\) and \(b_{i}\)) and PSD parameters (r\({}_{\text{eff}}\) and v\({}_{\text{eff}}\)). Considering the crucial role of the Lambertian kernel in the three-kernel BRDF surface model, we only retrieved the BRDF coefficient for the Lambertian kernel at each band, and the _a priori_ uncertainty was assumed to be 20%. Further analysis of the influences of different surface reflectances on aerosol interference are shown in the Supplemental section and not presented in the main text. Similar to the assumptions in [PERSON], et al.