Correction of Cloud Optical Thickness Retrievals from Nadir Reflectances in the Presence of 3D Radiative Effects. Part II: Theory

Cloud optical thickness (COT) retrieved from airborne and satellite nadir-view measurements can be seriously underestimated if 3D radiative effects are not taken into account. This happens when retrievals for isolated or broken clouds are based on 1D radiative transfer computations (such as the widely used bispectral technique). In our previous work, we introduced and validated a linear correction technique for retrieved COT, which is based on the cloud’s geometric aspect ratio. A heuristic theoretical framework was developed to show that this technique is consistent with the process of radiation escape from cloud sides in 3D geometry. However, the theory suggests that the correction factor is to be applied to reflectance, while in our technique, it is applied to COT. In this study, we resolve this inconsistency by introducing general renormalization theory based on a nonlinear renormalization function for reflectances, which must satisfy certain conditions. The link between the renormalization function and COT is established within this theory by using COT-dependent model reflectance functions. In these terms, the linear correction for COT corresponds to a specific form of the renormalization function for reflectance, which satisfies the required conditions. This proves the validity of the linear COT correction technique within the framework of general renormalization theory. We tested our conclusions on two specific model reflectance functions based on the two-stream diffuse approximation (power law) and backscattering approximation (that utilizes an exponential function).