Validity of soil isoline equation for a system of canopy and soil layers

For retrievals of soil optical properties, relationships between the two reflectances of different wavelengths become a key information. This relationship is known as soil isolines from the fact that the reflectances of two different wavelengths form a contour line (isoline) in a reflectance subspace when the soil properties are set to be constant. In our previous work, a relationship between two reflectances of different wavelengths has been derived under a constant soil optical properties. Although accuracy of the derived relationship has been demonstrated numerically, its validity had not been fully investigated from fundamental and physical points of view. The objective of this study is to provide theoretical validity of the derived soil isoline equation and numerically demonstrate its validity by a radiative transfer model of a canopy-soil system of layers. Specifically, the polynomial fittings employed during the derivation will be validated by conducting an analysis based on a theory of radiative transfer. As a result, a similar form of the previously derived soil isoline equation has been derived successfully based on the analytical form of TOC reflectance.