Bidirectional reflectance spectroscopy: 1. Theory
An approximate analytic solution to the radiative transfer equation describing the scattering of light from particulate surfaces is derived. Multiple scattering and mutual shadowing are taken into account. Analytic expressions for the following quantities are found: bidirectional reflectance, radiance factor, radiance coefficient, normal, hemispherical, Bond, and physical albedos, integral phase function, phase integral, and limb-darkening profile. Scattering functions for mixtures can be calculated, as well as corrections for comparing experimental laboratory transmission or reflection spectra with observational planetary spectra. An expression for the scattering efficiency of an irregular particle large compared with the wavelength is derived. For closely spaced, nonopaque particles this efficiency is approximated by (1 + αDe)−l, where α is the true absorption coefficient and De is an effective particle diameter of the order of twice the mean particle size. For monomineralic surfaces it is shown that α = ( 1 − w)/wDe, where w is the single-scattering albedo and can be determined from reflectance measurements of a powder, so that α may be calculated from reflectance. This theory should be useful for interpretations of reflectance spectroscopy of laboratory surfaces and photometry of solar system objects. From photometric observations of a body the following may be estimated: average single-scattering albedo, average particle phase function, average macroscopic slope, and porosity.