## 3 Retrieval
### Inversion Experiments
#### 3.2.3 Results  
Primarily, we are interested in XCO\({}_{2}\) retrieval results of high quality; the correct retrieval of other state vector elements is less important as long as the XCO\({}_{2}\) quality is not affected. Figure 7 summarizes the systematic errors and stochastic uncertainties of the retrieved XCO\({}_{2}\) for all retrieval setups and geophysical scenarios. The _baseline_ scenario is mainly to ensure consistency of the RT used to simulate the measurements (SCIATRAN) and the RT of the retrieval (FOCAL). Additionally, the _baseline_ scenario allows estimates of the retrieval's noise error. With SCIATRAN, it is not simply possible to simulate FOCAL's scattering approximations, that is why this scenario excludes scattering. The systematic errors of the _baseline_ scenario are always very small (0.03 ppm at maximum), which confirms the RT consistency in the absorption only case and ensures that, e.g., the number of particles is basically identical in the SCIATRAN and the FOCAL \"world\". Figure 7: Error characteristics of nine retrieval setups and twelve geophysical scenarios. Each box includes six sub-boxes representing polarization parallel (left) and perpendicular (right) to the SPP as well as three solar zenith angles (20\({}^{\circ}\), 40\({}^{\circ}\), and 60\({}^{\circ}\), from bottom to top). Gray boxes represent not converging retrievals. **Left**: Systematic error (retrieved minus true XCO\({}_{2}\)). **Right**: Stochastic uncertainty as reported by the optimal estimation retrieval. The systematic errors of the _SIF_ scenario are not larger than for the _baseline_ scenario, because i) SIF is solely determined from the SIF fit window and ii) there is no SIF flux emitted in the CO\({}_{2}\) fit windows. A more complex case for FOCAL is the _Rayleigh_ scenario, because Rayleigh scattering takes place in the entire atmospheric column with a peanut-shaped SPF. This means, it cannot be expected that FOCAL is able to perfectly fit the simulated measurement. Figure 8 (top) shows a spectral fit in all fit windows but with a state vector not including any scattering parameter, so that the geophysical results (e.g., XCO\({}_{2}\)) become identical with those of the _0-Scat_ setup. Not surprisingly, the residual in the O\({}_{2}\) fit window becomes large compared to the simulated measurement noise (\(\chi_{\text{O}_{2}}=6.825\)). The residuals in the CO\({}_{2}\) fit windows are already small compared to the instrumental noise even without fitting scattering parameters (\(\chi_{\text{WO}_{2}}=0.026\), \(\chi_{\text{SCO}_{2}}=0.049\)). This is only partly explained by Rayleigh scattering having an Angstrom exponent of four and, therefore, a much smaller scattering optical thickness at longer wavelengths. It also indicates that disentangling scattering parameters and CO\({}_{2}\) concentration from measurements in the CO\({}_{2}\) fit windows may be difficult. In other words, most of the scattering information must be imprinted in the residual of the O\({}_{2}\) fit window. This is also why the results of the _3-Scat_-O\({}_{2}\) setup are similar to the _3-Scat_ setup and why the _3-Scat_-CO\({}_{2}\) retrievals are often not converging (Figure 7). Allowing the _3-Scat_ retrieval setup to fit the scattering parameters \(p_{s}\), \(\tau_{s}\), and \(\dot{A}\), reduces the O\({}_{2}\) residual to become typically four times smaller than expected from instrumental noise (\(\chi_{\text{O}_{2}}=0.250\), Figure 8, middle). Simultaneously, the XCO\({}_{2}\) error reduces from \(-0.43\,\text{ppm}\) to \(0.10\,\text{ppm}\) (\(-0.89\,\text{ppm}\) and \(0.16\,\text{ppm}\) for perpendicular polarization). All other scattering related scenarios are even more \"complicated\" for FOCAL because different particles contribute to scattering. For example, cloud particles have different properties like height or Angstrom exponent as aerosol particles, but FOCAL can only retrieve one effective height and one effective Angstrom exponent. Additionally, the SPFs of aerosols and clouds are less isotropic.