## 4 Conclusions  
We presented the fast atmospheric trace gas retrieval FOCAL including a RT model which approximates multiple scattering effects at an optically thin isotropic scattering layer and assessed the  
Figure 14: XCO\({}_{2}\) and XH\({}_{2}\)O column averaging kernels for the 3-_Scat_ and 3-_Scat-stiff_ retrieval setups and the geophysical _Rayleigh_ scenario (\(\theta_{0}=40^{o}\), perpendicular polarization). Figure 15: Retrieved solar induced chlorophyll fluorescence for the 3-_Scat_ retrieval setup and the geophysical _baseline_, _SIF_, and all scattering related scenarios. The error bars represent the \(1\sigma\) a posteriori uncertainty. potential performance of various XCO\({}_{2}\), XH\({}_{2}\)O, and SIF retrieval setups for an OCO-2 like instrument. FOCAL accounts for scattering by splitting up the top of atmosphere (TOA) radiance into parts originating from direct reflection at the scattering layer or the surface and parts originating from multiple scattering of the diffuse radiant flux between scattering layer and surface. FOCAL's relatively simple approximation of the RT problem allows unphysical inputs such as negative scattering optical thicknesses or albedos. This can be an advantage when analyzing measurements including noise and assuming Gaussian a priori error statistics. FOCAL accounts for polarization only implicitly by the retrieval of a variable scattering optical thickness. The PPDF method (e.g., 19,20) gains its computational efficiency by applying the theorem of equivalence to replace computationally expensive multiple scattering RT computations with a set of fast transmission computations. This is conceptually similar to FOCAL which uses an effective transmission function for the diffuse flux. However, different from the PPDF method, FOCAL accounts for multiple scattering by solving the geometric series of successive (flux) scattering events. In principle, the PPDF method can simulate arbitrary SPFs. This is not possible for FOCAL which can only simulate an isotropic scattering layer. However, splitting the radiance into direct and diffuse parts can be interpreted as a SPF with a sharp forward peak and which is isotropic otherwise. This still represents typical Mie SPFs not very well but much better than an entirely isotropic SPF. Strictly, the theorem of equivalence only applies for spectral regions with constant scattering and reflection properties ([PERSON], 2017) making the PPDF shape, e.g., depending on surface albedo. This can make it complicated to transfer scattering information from one fit window into another. Reflection and scattering properties of FOCAL are allowed to vary within the fit windows and can be used to transfer information between fit windows, e.g., via the Angstrom exponent. We confronted several different retrieval setups with simulated OCO-2 radiance measurements of a set of different geophysical scenarios, solar zenith angles, and polarization directions. Due to often relatively low systematic XCO\({}_{2}\) and XH\({}_{2}\)O errors with low polarization dependency, well controlled retrieved profiles, lowest CO\({}_{2}\) smoothing errors, a relatively realistic a priori error correlation matrix for CO\({}_{2}\), and advantageous AKs, we conclude that the _3-Sat_ setup is a promising candidate for further studies with measured OCO-2 data. The _3-Sat_ setup fits the OCO-2 measured radiance in four fit windows by simultaneously retrieving the following geophysical parameters: five layered CO\({}_{2}\) and H\({}_{2}\)O concentration profiles, the pressure (i.e., height), scattering optical thickness at 760 nm, and the Angstrom exponent of a scattering layer, SIF, and polynomial coefficients describing the spectral albedo in each fit window. As accurate XCO\({}_{2}\) retrievals will probably always require a rigorous cloud and aerosol screening, we concentrated on scenarios with scattering optical thicknesses in the range of about 0.03 and 0.70.