597 
and view angles. Added to it is a weighted geometric scattering component, based on a physical model of 
Lambertian protrusions on a flat plain; and a weighted volume scattering component derived from a simple 
single-scattering radiative transfer model for randomly-oriented isotropic scatterers. A comparison with direc 
tional reflectance data for a range of vegetated surfaces shows a reasonably good fit for most continuous 
canopy covers. Forest BRDFs, however, appear to be an exception. 
2.5 Geometric-Optical Models 
New developments have also occurred in geometric-optical modeling of vegetation canopy reflectance. Li and 
Strahler (1992) modified their previous model (1986) to more properly accommodate the effects of mutual 
shadowing of individual plant crowns by one another. Their model is driven by the shape and spacing of indi 
vidual plant crowns, which are taken as geometric objects that cast shadows on the background and on other 
crowns. The surface reflectance is modeled as a function of four scene components—sunlit crown, shadowed 
crown, sunlit background, and shadowed background—that are viewed in varying proportions, depending on 
illumination and viewing positions. The new mutual shadowing model accounts for the effect that when plant 
crowns are closely spaced and of similar size, the shadow of one crown tends to fall preferentially on the base 
of an adjacent crown. Thus, when the canopy is viewed from a low angle, only the sunlit tops of crowns are 
seen. This effect gives the BRDF a typical bowl-shape when plotted in hemispherical projection. The model 
fits the directional reflectance of conifer forests in Oregon (Abuelgasim and Strahler, 1993) and at Howland, 
Maine (Schaaf and Strahler, 1994), with good accuracy. 
Most recently, Li et al. (1994) have developed a hybrid model combining geometric optics with prin 
ciples of radiative transfer. It relies on gap probabilities and path length distributions to model the penetration 
of irradiance into the canopy and its single and multiple scattering in the direction of view. Within a plant 
crown, the probability of scattering is a negative exponential function of path length. Within-crown scattering 
provides the source for single scattering radiance, which exits with probabilities proportional to further path- 
length distributions in the direction of exitance (including the hotspot effect). Single scattering provides the 
source for double scattering, and then higher orders of scattering are solved successively by a convolution 
function. The model is parameterized by a per-meter scattering coefficient, within-crown pro jected leaf area as 
a function of angle, and statistical variables describing crown shape, count density, and the height of the 
canopy layer. Early validation using data from a conifer stand at Howland, Maine, shows good agreement 
between modeled and observed reflectance. 
2.6. Computational Models 
Mention has already been made of the computational models for vegetation scattering recently developed by 
Borei et al. (1991) and Goel et al. (1991, 1992). In the radiosity approach of the Borei et al., Lambertian scat 
tering by leaves is described by a sparse matrix of view factors between leaf surfaces. The approach of Goel et 
al. uses ray tracing on a realistic plant model parameterized by L-systems. Lewis and Muller (1992) have 
recently provided ARARAT, an advanced radiometric ray tracing program, which allows an arbitrary BRDF 
model for scattering by surface elements. It can be used to simulate a wide range of scenes, from crop canopies 
to vegetation-covered topographic landscapes. Further, the program can utilize a sky radiance model for 
downwelling irradiance, such as that of Zibordi and Voss (1989), and thus compute bidirectional reflectance 
factors (BRFs) as well as the BRDF. 
3-DISCUSSION 
3.1. Validation Needs 
Studies of the anisotropic reflectance of the earth’s vegetated surface conducted over the last decade or so 
have not lacked for models. There are literally dozens of models in the literature, and undoubtedly still more 
are lurking within the fertile brains of scientists and applied mathematicians who work in this field. These 
models vary widely in their abstractions of the physics of the interaction of light with the canopy, yet most 
seem to do a reasonable job of approximating the anisotropic reflectance of at least some type of vegetation 
cover. Except possibly for the coherent backscattering of leaf surfaces (Hapke et al., 1993), it would appear 
that no new physical mechanisms for the interaction of light with vegetated surfaces will be discovered or 
applied in the near future. We may expect, however, that models will continue to add degrees of complexity, 
such as accounting for stems and branches and accommodating more soil scattering parameters. Perhaps these 
models will eventually merge with microwave scattering models, providing a “unified theory” for the interac 
tion of electromagnetic radiation with the plant cover. 
What has been lacking, however, are ample quantities of directional reflectance measurements of 
vegetated surfaces that are sufficiently well characterized to validate the physical abstractions of their models. 
For example, every radiative transfer canopy model requires some parameterization of leaf angle distribution, 
yet there are probably less than a dozen sets of angular radiance measurements of canopies for which the leaf 
a ngle distribution has been accurately determined. Leaf scattering functions (leaf BRDFs) have been mea