Kubelka-Munk equations in vector–matrix forms and the solution for bidirectional vegetative canopy reflectance

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Abstract
The radiation from different directions can be specified by upward and downward radiation vectors, and the interactions of the radiation with a leaf or with a vegetative canopy can be specified by matrices. The Kubelka-Munk equations, which are applicable only to a canopy with horizontal and Lambertian leaves, can then be extended to describe the directional transfer of radiation in a canopy with nonhorizontal, non-Lambertian leaves. In the extended Kubelka-Munk equations, variables are upward and downward radiation vectors, and the coefficients are matrices. The solutions are found from which the bidirectional vegetative canopy reflectance, including azimuthal variations, can be obtained. Simplified and approximate methods are presented for a canopy with leaves without azimuthal preference in order to reduce the execution time.