## 3 Retrieval
### Inversion Experiments
#### 3.2.3 Results
Therefore, the residuals (Figure 8, bottom) and more importantly, the systematic errors typically increase for these scenarios (Figure 7, left). Figure 9 shows the retrieved scattering parameters for the _3-Scat_ setup and a set of scattering related plus the baseline scenario. As the baseline scenario does not include any scattering, the retrieved \(p_{s}\) and \(\dot{A}\) are close to their a prior values and have a large a posteriori uncertainty. Consistent with the expectations, the retrieved effective Angstrom exponent is close to four (about 3.8) for the Rayleigh scenario and reduces to 2.8-3.6 for the aerosol and 2.1-2.6 for cloud scenarios. This means the scattering optical thickness at longer wavelengths increases relative to the shorter wavelengths. Rayleigh scattered light is unpolarized in forward and backward scattering direction but polarized perpendicular to the incident beam for scattering angles of 90\({}^{\circ}\). For this reason, the retrieved \(\tau_{s}\) is always larger for the polarization direction perpendicular to the SPP. As expected, this effect is more/less pronounced for larger/smaller solar zenith angles (not shown). In contrast to the _3-Scat_ and _4-Scat_ setups, the _0-Scat_ retrievals cannot fit \(\tau_{s}\) which results in a larger polarization dependency of the resulting systematic errors (Figure 7, left). As shown in Figure 9, the highest scattering optical thicknesses at 760 nm are obtained for the urban aerosol and the cloud scenarios. However, the quantitative interpretation of the retrieved values of \(\tau_{s}\) and \(p_{s}\) is difficult because they are effective values representing all kinds of scattering in the atmospheric column. Additionally, \(\tau_{s}\) and \(p_{s}\) may not be perfectly independent because light path modifications are expected to become larger when enhancing the height of the scattering layer. It can be observed that the retrieved values of \(\tau_{s}\) are generally smaller than the scattering optical thicknesses computed by SCIATRAN (Section 3.2.2). This is expected because of the different SPFs assumed by SCIATRAN and FOCAL. Especially for Mie scattering of cloud and aerosol particles, the SCIATRAN simulations use SPFs with a distinct forward peak contributing to the total scattering optical thickness. FOCAL, however, interprets scattering in forward direction as transmission (not contributing to \(\tau_{s}\)). This means, \(\tau_{s}\) is best comparable for the _Rayleigh_ scenario with a SPF without forward peak. Figure 8: As Figure 3 (bottom) but for the _Rayleigh_ scenario and the _0-Scat_ setup (**top**), the _Rayleigh_ scenario and the _3-Scat_ setup (**middle**), and the _Rayleigh+Aerosol BG+Water cloud_ scenario and the _3-Scat_ setup (**bottom**). Figure 9: Retrieved scattering optical thickness, Ångström exponent, and height of the scattering layer for the _3-Scat_ setup and the scenarios (\(\theta_{0}\) = 40\({}^{\circ}\)): _baseline_, _Rayleigh_, _Rayleigh_+_Dark surface_, _Rayleigh_+_Bright surface_, _Rayleigh_+_Aerosol BG_, _Rayleigh_+_Aerosol _out_, _Rayleigh_+_Aerosol _urban_, _Rayleigh_+_Aerosol BG_+_Water cloud_, and _Rayleigh_+_Aerosol BG_+_Ice cloud_. The scenarios _Rayleigh_, _Rayleigh+Dark surface_, and _Rayleigh+Bright surface_ differ by their surface albedo. However, the retrieved scattering parameters show little differences because in FOCAL these parameters represent (within the limits of the made assumptions) approximations of real geophysical quantities. Applying FOCAL to the _Rayleigh+Ocean glint_ scenario with a highly non-Lambertian surface BRDF results in systematic XCO\({}_{2}\) and XH\({}_{2}\)O errors usually comparable to the _Rayleigh_ scenario (Figures 7 and 10, left) except for solar zenith angles of 60\({}^{\circ}\) and polarization parallel to the SPP. In near-glint geometry, specular reflectance dominates the radiation field but with increasing solar zenith angle the reflected radiation becomes more and more polarized.