Reflection of light from a rough water surface in numerical methods for solving the radiative transfer equation

The well-known expression of Cox and Munk for the bi-directional reflectivity of a rough water surface tends to infinity for reflection angles close to the horizontal direction. This creates evident difficulties in applying it in numerical radiative transfer schemes. Detailed analysis of the process of reflection from a non-flat surface shows that the singularity arises from some contradictory implicit assumptions adopted in the theoretical derivation of the bi-directional reflectivity expression. In the present paper, these assumptions are modified in order to provide for consistency and a bi-directional reflectivity expression applicable for all possible reflection angles is derived. An analytical form of Fourier expansion of bi-directional reflectivity in azimuthal variables is also obtained.