Canopy bidirectional reflectance dependence on leaf orientation

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Abstract
The bidirectional reflectance patterns of a complete (dense) canopy are examined as functions of canopy architecture, as specified by azimuth angle δe and zenith angle ψ for a leaf normal. The leaves are assumed to be opaque Lambertian reflectors, all with identical orientation and reflectance properties throughout the canopy, and randomly distributed with respect to the irradiation field and the viewing direction. Multiple reflections are not considered and irradiation is by direct beam only. Simple analytical expressions for the bidirectional reflectance factor are presented and analysed. The nadir reflectance (expressed as a fraction of the leaf reflectance) for canopies whose leaves face the sun, δe = 0, is bounded by cos ψ and 1/2; cos ψ. The nadir reflectance initially increases with increasing ψ, but then decreases when ψ reaches moderate to large values. For a δe = π canopy, on the other hand, the much lower nadir reflectance is bounded by ½ cos ψ and 0, and decreases with increasing ψ throughout the entire range of ψ (0 to ½π). The maximum bidirectional reflectance occurs at large viewing zenith angles (i.e. close to the horizon). The maximum reflectance is always higher for a δe = 0 canopy than for a δe = π canopy, but the differences become small when ψ approaches ½π. The bidirectional reflectance thus depends on the leaf azimuth as well as the zenith angle. Leaf-area azimuthal distributions should be considered when conducting model inversions to infer canopy characteristics and architecture.