Angular distribution of scattered radiation under broken cloud arrays: An approximation of successive orders of scattering

A theoretical model simulating the radiance field below inhomogeneous cloudy skies is derived. The model is applicable to any cloud configuration, i.e. positions of individual clouds can be arranged randomly as regularly occurs in nature. The solution follows the modified theory of successive orders of scattering in which single-scattering radiances from broken clouds and a cloud-free atmosphere are well separated, while their non-trivial superposition is used to determine a second-order approximation to the radiative field. The contributions of both single and secondary scattered beams form the total radiation field below the cloud level. Numerical simulations are performed for a cloud array with regular spacing between single clouds to document the correct model behavior for various cloud fractions, including extreme cases of cloudless and overcast skies. Despite the complexity of the derived equations, the numerical computations can be accomplished rapidly using an inexpensive Intel Pentium processor. The main strengths of the model presented here are its easy parameterization and its ability to predict more realistic radiance patterns for inhomogeneous cloudy skies. The theoretical foundation of the documented solution implies its applicability to a wide range of sky states