## 3 Retrieval
### Inversion Experiments
#### 3.2.3 Results
As a result the direct photon path often dominates (if not observing parallel polarization at large solar zenith angles) and an imperfect parameterization of scattering becomes less important. The domination of the direct photon path also results in a larger total radiance and, correspondingly, smaller stochastic errors in perpendicular polarization (Figures 7 and 10, right). The larger systematic XCO\({}_{2}\) errors of about 4 ppm at 60\({}^{\circ}\) and parallel polarization are a result of the poor surface reflectivity in this observation geometry and associated with large stochastic errors of about 8 ppm and little error reduction (analog for XH\({}_{2}\)O). This means, applied to real measurements, such retrievals would most certainly be filtered during post processing. Note that due to the non-Lambertian surface, the retrieved albedo may have values larger than one. Figure 7 (right) shows that the shape of the CO\({}_{2}\) a priori error covariance matrix can considerably influence the stochastic XCO\({}_{2}\) a posteriori uncertainty, even though the a priori XCO\({}_{2}\) uncertainty has not been changed. Stiffening the covariance matrix by enhancing the layer-to-layer correlations as done for the _synth_ and _stiff_ setups (Figures 11 and 12), reduces the stochastic XCO\({}_{2}\) uncertainty from typically about 1 ppm to 0.7 ppm (_synth_) and 0.4 ppm to 0.6 ppm (_stiff_), which does not necessarily mean that results actually improve. Figure 10: As Figure 7 but for XH\({}_{2}\)O. Except for the _XCO\({}_{2}\)_+6 _ppm_ scenario, the systematic errors of the _3-Scat_, _3-Scat-synth_, and _3-Scat-stiff_ setups are very similar. This is not the case for the _0-Scat_, _0-Scat-synth_, and _0-Scat-stiff_ setups for which the systematic errors increase with stiffness of the CO\({}_{2}\) a priori error covariance matrix. Apparently, the (loose) profile retrieval of the _0-Scat_ scenario happens to somewhat compensate light path related errors. In the case of the _3-Scat_ setups, the scattering parameters are doing this job. Figure 13 shows, that the largest deviations of the retrieved profiles from the true profile (a priori) indeed occur for the _0-Scat_ setup. Figure 11: As Figure 4 but for a synthetic a priori error covariance matrix as proposed by [PERSON] et al. (2010) but with a correlation length of 1.0 \(p_{0}\). Figure 12: Same as Figure 4 but for a synthetic a priori error correlation matrix as proposed by [PERSON] et al. (2010) but with a correlation length of 100 \(p_{0}\). The degree of freedom for the CO\({}_{2}\) profile is about 2.2 for the _3-Scat_ setup and reduces to 1.8 for the _3-Scat-synth_ and 1.0 for the _3-Scat-stiff_ setup. The degree of freedom for the H\({}_{2}\)O profile reduces from 2.2 for the _3-Scat_ setup to 1.0 for the _3-Scat-stiff_ setup. Additionally, the column averaging kernels (AKs) change and show larger deviations from unity; specifically, as illustrated in Figure 14, the XCO\({}_{2}\) AK increases to about 1.2 in the boundary layer and reduces to 0.6 in the stratosphere. As a result, the systematic error (in this particular case, the smoothing error) increases for the _stiff_ setups to about 1.6 ppm (Figure 7, left). As illustrated in Figure 10, scattering related systematic XH\({}_{2}\)O errors are usually negative and larger for the _0-Scat_ setups. Stiffening the H\({}_{2}\)O a priori error covariance matrix has little influence on the systematic or stochastic error which is usually about 10 ppm. SIF is almost not influenced by the mostly low scattering optical thicknesses of the tested scenarios and the stochastic a posteriori error is usually between 0.2 and 0.3 mW/m\({}^{2}\)/sr/nm (Figure 15). All tested retrieval setups do not have the ability to change the number of dry-air particles in the atmospheric column, e.g., by fitting the surface pressure, or a shift of the temperature profile.