1.2. Overview of Canopy Reflectance Models 
The anisotropy of reflectance from vegetated surfaces and its remote sensing has led to the development of a 
rich array of mathematical models that describe the surface BRDF. These models can be characterized as fol 
lowing one of two general approaches—physical or empirical. In the empirical approach, a function is fitted 
that describes the shape of the BRDF based on the observations at hand. That is, the BRDF is modeled as an 
empirical function of viewing and illumination angles and azimuths in the hemisphere (e. g., Walthall et al., 
1985, for soil; Barnsley, 1993, for vegetation). For accurate fitting of a complete BRDF, however, this 
approach requires many observations at many combinations of viewing and illumination positions. Although 
simple and direct, empirical models are not very practical for satellite remote-sensing applications, because the 
number of angular observations of a surface typically acquired will be small. Further, the coefficients that fit 
empirical models cannot be readily interpreted in terms of scene or surface properties. Instead, relationships 
between surface properties and empirical functions must be obtained by further empirical techniques, such as 
correlation analyses. 
In the physical approach, a physical scattering model is constructed that explains anisotropic surface 
scattering using physical principles (e. g., Hapke, 1981, 1984, 1986, Hapke and Wells, 1981, for soil; Suits, 1972, 
for vegetation). By inversion, reflectance observations are used to infer the physical parameters that drive the 
model (e. g., Goel, 1988). Once these are known, the BRDF of the surface may be determined for any view or 
illumination position without calibration by further measurements. Moreover, the parameters typically have 
physical interpretations in their own right that are of intrinsic interest beyond simply generating the BRDF. 
Additional advantages accrue to the physical approach. To describe the complete BRDF, fewer 
parameters are typically required. Further, because the physical meaning of the parameters is understood, it is 
often possible to make reasonable a priori choices for their values. In addition, a carefully-drawn physical 
model may be simplified by successive approximations and assumptions. And, considering that some parame 
ters are independent of waveband, fewer total parameters will be required for multiband BRDF inference 
using a single physical model than for a suite of multiband empirical models that must be independently cali 
brated for each band. Thus, the physical approach to bidirectional reflectance modeling is probably the most 
suitable for remote sensing applications. 
A variation on these two approaches, which we may term “semiempirical,” combines physical and 
empirical models (e. g., Roujean, 1992; Deuzé et al., 1993; Rahman et ah, 1993a, 1993b). Here, the BRDF is 
modeled as a weighted sum of a few empirical functions that describe the shape of the BRDF. However, these 
functions are typically derived from physical approximations, and so have some physical meaning. The weight 
to be given to each function is determined empirically by fit to the observations. Thus, it is the weights of the 
physically-based functions that are retrieved, not a set of physical parameters governing the surface scattering. 
Another type of physical model uses computation in lieu of a formally-parameterized mathematical 
description. Usually this type of model is applied to a scattering layer composed of numerous volume scatter 
ing elements—for example, a leaf canopy. An example is a ray tracing model (Kimes and K.rchner, 1982,; 
Goel et ah, 1991; Lewis and Muller, 1992), in which a Monte Carlo model of scattering events is used to char 
acterize the BRDF of a specific scattering layer or surface. Another example is the radiosity model (Borei et 
ah, 1991), in which a sparse matrix of mutual view factors between Lambertian scattering elements within the 
volume layer is computed and used to find the BRDF. 
1.3. Features of Physical Surface Scattering Models 
The angular behavior of land surface reflectance is a function of at least three physical phenomena: coherence, 
volume scattering among scattering elements, and surface scattering effects of self-shadowing and specular 
reflectance according to the three-dimensional arrangement of scattering elements. For any particular surface 
cover, the magnitude of these effects will depend on the positions of both the sensor and source of irradiance 
in the hemisphere. 
Coherence effects can provide a strong backscatter peak (hotspot) to the surface reflectance function, 
and occur when the mean free path length of multiple scattering within the medium is near the wavelength of 
the irradiance. Coherent backscatter is important for lunar soils and seems to explain the opposition effect 
observed for many planetary bodies (Hapke et al., 1993); however, since the mean free path length within a 
leaf canopy is very large when compared to optical wavelengths, current vegetation BRDF models ignore 
coherent backscatter. Volume scattering is quite important for porous media such as vegetation layers or snow, 
and can be described accurately by radiative transfer theory. Surface effects largely involve shadowing, or 
geometric, effects, in which surface projections or volume scattering elements shadow other surface projec 
tions or volume scatterers. These effects are important on scales ranging from soil surface perturbations to 
topographic relief. The distribution of scattering surface normals also conditions specular scattering. Geomet 
ric-optical models have been used to describe these effects. Both volume scattering and geometric effects must 
be accommodated in any realistic physical description of surface reflectance behavior. 
Another complicating factor is the fact that more than one layer of surface scattering material may be 
present. A vegetation cover over soil, snow, or standing water is an example. If the upper layer is thick, as in a 
dense forest cover or closed crop canopy, the scattering behavior of the lower layer may be ignored or perhaps 
approximated as Lambertian—that is, independent of view or illumination angles. However, if the upper layer