14 Three-Dimensional Radiative Transfer in Vegetation Canopies

Interaction of photons with a host medium is described by a linear transport equation. This equation has a very simple physical interpretation; it is a mathematical statement of the energy conservation law. In spite of the different physics behind radiation transfer in clouds and vegetation, these media have certain macro and micro-scale features in common. First, both are characterized by strong horizontal and vertical variations, and thus their three-dimensionality is important to correctly describe the photon transport. Second, the radiation regime is substantially influenced by the sizes of scattering centers that constitute the medium. Drop and leaf size distribution functions are the most important variables characterizing the micro-scale structure of clouds and vegetation canopies, respectively. Third, the independent (or incoherent) scattering concept underlies the derivation of the extinction coefficient and scattering phase function in both theories (van de Hulst, 1980, p. 4-5); (Ross, 1981, p. 144). This allows the transport equation to relate micro-scale properties of the medium to the photon distribution in the entire medium. From a mathematical point of view, these three features determine common properties of radiative transfer in clouds and vegetation. However, the governing radiative transfer equation for leaf canopies, in both three-dimensional (3D) and one-dimensional (1D) geometries, has certain unique features. The extinction coefficient is a function of the direction of photon travel. Also, the differential scattering cross-section is not, as a rule, rotationally invariant, i.e., it generally depends on the absolute directions of photon travel and , and not just the scattering angle arccos •. Finally, the single scattering albedo is also a function of spatial and directional variables. These properties make solving of the radiative transfer equation more complicated; for example, the expansion of the differential scattering cross-section in spherical harmonics (see, Chap. 4) cannot be used. In contrast to radiative transfer in clouds, the extinction coefficient in vegetation canopies introduced by Ross (1981) is wavelength independent, considering the size of scattering elements (leaves, branches, twigs, etc.) relative to the wavelength of solar radiation. Although the scattering and absorption processes are different at different wavelengths, the optical distance between two arbitrary points within the vegetation canopy does not depend on the wavelength. This spectral invariance 0528