Energy conservation in soil surface scattering in the optical region

Most of the studies in the past used empirical phase functions that were not directly related to the physical and electromagnetic properties of the medium. These phase functions do not conserve energy. Although one can normalize these functions to realize conservation, no meaningful albedo or extinction properties can be derived from them, and energy transfer is not correctly represented. In fact, the albedo and extinction coefficient have to be determined separately and empirically. Furthermore, when these phase functions are incorporated into a scattering model, attempts to retrieve soil parameters from such models sometimes result in unstable conditions. The purpose of this study is to derive a phase function for soil based upon physical principles using a ray-tracing technique that allows the summation of multiply scattered rays. The final result is expressed in terms of the electromagnetic properties of soil. It is shown that such a phase function conserves energy and includes a proper amount of absorption. Then, by incorporating this phase function into a radiative transfer formulation, the authors arrive at a scattering model for the soil medium. The inputs to this model are particle and background permittivity, volume fraction, particle size relative to wavelength, and individual particle and soil surface roughness. The scattering model can be used for soil parameter retrieval without the problem of instability