## 3 TanSat Simulations and Retrieval Assumptions
### Simulation Input  
The forward model used to simulate the TanSat observations consisted of a linearized single-scattering code for spherical particles, a radiative transfer model called VLIDORT that can compute analytical weighting functions [40], and a surface model calculating bidirectional reflectance distribution function (BRDF), as well as modules computing the molecular Rayleigh scattering and trace gas absorption. A similar model was used in the GOSAT and TanSat CO\({}_{2}\) retrieval algorithms [5; 30; 41]. Before conducting a simulation using the forward model, some necessary inputs related to the instrument characteristics, atmospheric states, surface and aerosol types, and observation geometry must be understood. After inputting these parameters, the Stokes vector and their Jacobians with respect to each retrieved parameter (\(K_{ij}\)), were computed, based on the radiative transfer equation from the forward model. For the TanSat, the hyperspectral instrument ACGS measures spectra in three bands: the O\({}_{2}\)-A band, the CO\({}_{2}\) weak absorption band, and the strong NIR absorption band (hereafter expressed as the 1 CO\({}_{2}\) and 2 CO\({}_{2}\) bands). The key characteristics of ACGS are presented in Table 1. The spectral resolution defined by the full width at half maximum (FWHM) and an instrument model are necessary for simulating the spectrum detected by the instrument. After the line-by-line radiative transfer calculations are completed, the original spectrum is convoluted by a Gaussian instrument line shape described by FWHM. The atmospheric temperature, pressure, water vapor and trace gases profiles originate from the US standard atmosphere profile of AFGL (1980). The CO\({}_{2}\) profile is scaled by a factor based on the near real-time CO\({}_{2}\) data from the global CarbonTracker model ([PERSON] et al., 2018), which also serves as the _a priori_ profile. For surface reflectance, we selected the BRDF model of the Moderate Resolution Imaging Spectroradiometer (MODIS) ([PERSON] et al., 2018; [PERSON] et al., 2018) for typical land surfaces, including vegetation and bare soil, instead of Lambertian albedo without angular dependence. As shown in previous studies ([PERSON] et al., 2018), most land surfaces could be regarded by a fraction as a linear combination of vegetation and soil. Therefore, we believe the experiments over these two surface types are adequate for our research over land. The three Ross-Li kernels in the MODIS BRDF describe the isotropic, volumetric, and geometric-optical surface scatterings ([PERSON] et al., 2018; [PERSON] et al., 2018). For our study, we selected one grid from the MODIS observations of vegetation or soil as an example. The coefficients of each kernel were derived from MODIS products and extrapolated to the center wavelength of each ACGS band, on the assumption that surface reflectance underwent little change in each band (Table 2), which is similar to the CAPI research that we have previously conducted ([PERSON] et al., 2018). The corresponding surface reflectivities implied by the BRDF model evaluated in the direct beam, sun-surface-satellite geometries are shown in the Supplemental Material (Figure S5). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instrument Characteristics** & \multicolumn{3}{c}{**Value**} \\ \hline Band name or number & O\({}_{2}\)-A & CO\({}_{2}\) weak & CO\({}_{2}\) strong \\ Spectral range (nm) & 758–778 & 1594–1624 & 2041–2081 \\ Center wavelength (nm) & 768 & 1610 & 2060 \\ FWHM (nm) 1  & 0.04 & 0.12 & 0.16 \\ SNR 1  & 360 & 250 & 180 \\ Spatial resolution (km) & & 1 \(\times\) 2 & \\ Scan range (o) & & \(-\)30–10 & \\ Swath (km) & & 20 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of Atmospheric Carbon Dioxide Grating Spectrometer (ACGS) on board the TanSat.