Fundamental scattering quantities for the determination of reflectance and transmittance

The bidirectional reflectance distribution function (BRDF) and the bidirectional scattering - surface reflectance distribution function (BSSRDF), which relate radiance at the surface to irradiance and radiant flux, respectively, are regarded as the most fundamental scattering quantities used to determine the reflectance of objects. However, for materials where the optical radiation is transmitted under the surface, this radiance depends not only on irradiance and radiant flux, but also on the size of the irradiated area of the surface. This article provides insight into such dependence under the special condition in which the radiance is evaluated within the irradiated area and, consequently, is produced by both the insurface reflection and the subsurface scattering, in contrast to the situation in which the radiance is evaluated at non-irradiated areas and only subsurface scattering contributes. By explicitly considering both contributions, two other scattering quantities are defined: one that accounts exclusively for the insurface reflection and the other that accounts for subsurface scattering. In this regard, these quantities might be considered more fundamental than the BRDF and the BSSRDF, although they are coincident with these two functions apart from the above-mentioned special condition and for materials with negligible subsurface scattering. In this work, the relevance of the proposed scattering quantities is supported by experimental data, practical considerations are given for measuring them, and their relation to the bidirectional transmittance distribution function (BTDF) is discussed.