Analytical model for radiative transfer including the effects of a rough material interface

The reflected and transmitted radiance due to a source located above a water surface is computed based on models for radiative transfer in continuous optical media separated by a discontinuous air-water interface with random surface roughness. The air-water interface is described as the superposition of random, unresolved roughness on a deterministic realization of a stochastic wave surface at resolved scales. Under the geometric optics assumption, the bidirectional reflection and transmission functions for the air-water interface are approximated by applying regular perturbation methods to Snell's law and including the effects of a random surface roughness component. Formal analytical solutions to the radiative transfer problem under the small-angle scattering approximation account for the effects of scattering and absorption as light propagates through the atmosphere and water and also capture the diffusive effects due to the interaction of light with the rough material interface that separates the two optical media. Results of the analytical models are validated against Monte Carlo simulations, and the approximation to the bidirectional reflection function is also compared to another well-known analytical model.