Spectral theory for photon transport in dense vegetation media: Caseology for the Canopy equation

In this paper, the spectral theory for the transport of photons in dense vegetation structure is developed. The one-dimensional, one-angle, time-independent canopy equation, which models the passive response of green canopies to incident sun-light, is analyzed with spectral methods borrowed from conventional transport theory. The nonrotational invariance characteristic of canopy architectures precludes the use of Legendre polynomials to expand the scattering kernel. Nevertheless, it is shown that Case's method can be applied with proper modification to area canopy equations with a finite rank kernel. "Caseology" is developed to characterize the complete spectrum (discrete and continuous) as well as to determine the eigenfunctions. Orthogonality and full-range completeness are shown to be inherent properties of the full set of modes. The spectral theory was used to derive the two integral equations that form the backbone of the Fn method. Caseology provides a natural bridge between numerical computation and remote sensing applications that finally justifies the need for such theoretical analysis.