Scattering by Lambertian-leaves canopy: dependence on leaf-area projections

Abstract

A single-scattering model is constructed for a canopy with Lamber-tian leaves. The azimuthal distribution of the leaves is represented by fractional abundances of the leaf-area in the cardinal directions with respect to the Sun. The canopy bi-directional reflectances are found to be controlled by the projections of the leaf-areas onto horizontal and vertical planes. The sum of the leaf-area projections onto the horizontal plane determines the reflectance to the zenith when the Sun is at the zenith. For a complete canopy this reflectance is where w_xh is the fractional projection onto the horizontal plane of leaf-area of leaf-category x, g_x is the leaf reflectance (assumed equal to the leaf transmittance), and ψ_x, is the zenith angle of the leaf normal for this category. As the view and solar zenith angles deviate from the nadir, the change in the reflectance in the principal plane of the Sun is controlled by ihe difference in the leaf-area projections onto the vertical plane of the leaves with leaf-normals in opposite quadrants in the principal plane. When these two leaf-categories are identical (other than in their azimuths), a large region around the zenith exhibits the Lambertian viewing property, that is, the reflectance does not change with the view or illumination directions. Forward scattering and backscattering, which become intense when both the illumination zenith angle θ₀ and the view zenith angle θ_v are large (approach 90°) while ψ is not small (or when ψ is large while θ₀ and θ_v are not small), are controlled by the sum of these two leaf-area projections. The reflectance has then the limiting value g sinψ(cot θ₀ + cotθ_v), where g and ψ characterize the two leaf categories with normals in the principal plane. This expression represents a generalization of a result obtained for a field of thin vertical cylinders