Bidirectional transmittance and reflectance models for soil signature analysis.

The propagation of radiation in an absorbing-scattering soil with constant or spatial variation of the refractive index is investigated. The soil consists of a plane parallel with Fresnel reflection at the boundaries and is exposed at one boundary to a diffuse or collimated incident radiation. The discrete spherical harmonics method using Marshak boundary conditions is introduced to approximate the directional hemispherical reflectance and transmittance as well as the bidirectional reflectance. The effect in spatial variation of the refractive index on the reflectance and transmittance predictions is examined. A comparison of the directional transmittance and reflectance with the literature results demonstrates that the present method gives accurate results for optically thin and thick soil with a maximum relative error in all cases less than 1%. The bidirectional radiance for variable refractive index soils also shows excellent agreement as compared to the literature results. The results demonstrated that the anisotropic soil interfaces cause a significant decrease of energy reflected and transmitted as well as the bidirectional reflectance.