Vegetation isoline equations for an atmosphere-canopy-soil system

The relationship between the reflectances at two different wavelengths, over a three-layer system comprising atmosphere, canopy, and soil layers is derived. This work is an extension of the isoline equation previously derived in a red and near-infrared (NIR) reflectance space for a canopy-soil system. As a result of retaining only the zeroth and first-order interaction terms between the layers, the relationship has a linear form in which the slope and offset are functions of the optical properties of each layer. Numerical examples of isolines based on the derived expressions are obtained under various conditions and are shown to demonstrate some of the known behaviors of isolines. Since the derived expression relates a pair of reflectances, it is expected to be useful for the analysis of satellite data products involving algebraic manipulations of spectral reflectances, such as spectral vegetation indexes.