Extension of Chandrasekhars formula to a homogeneous non-Lambertian surface and comparison with the 6S formulation

The classical Chandrasekharʹs formula relating the surface reflectance to the top of the atmosphere radiance rigorously applies to a Lambertian surface. For a homogeneous non-Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been recently implemented in the second simulation of the satellite signal in the solar spectrum (6S) algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal is derived in a plane-parallel atmosphere bounded by a homogeneous non-Lambertian surface. Then the 6S algorithm extension is compared with the exact formula and approximations and their validity are pointed out. The methods used for the derivation of the exact formula are classical. They are based on the separation of direct and diffuse components of the radiation fields, on the introduction of the Greenʹs function of the problem, and on integrations of boundary values of the radiation fields with the Greenʹs function.