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coupling, most of the coupled models are of this type. Coupling can be used to model surface bidirectional
reflectance factors (BRFs), in which case it is the downwelling distribution of irradiance that is of concern. It
can also used to analyze the effects of canopy and atmospheric parameters on radiance measured at the top of
the atmosphere.
Myneni and coworkers have coupled both one- and three-dimensional radiative transfer canopy mod
els to atmospheric radiative transfer models. In one-dimensional studies, Myneni et al. (1993) showed that the
atmosphere acts to add significant path radiance to the surface radiance at red wavelengths, while the atmo
sphere significantly attenuates surface radiance at infrared wavelengths. A factor that converts top-of-atmo-
sphere directional radiance measurements to (hemispherical) fluxes varies significantly with sun and view
angle. In a further application (Asrar and Myneni, 1993), surface albedo is always reduced by a clear atmo
sphere, and the fraction of photosynthetically active radiation absorbed by the canopy is well predicted by the
atmospherically-resistant vegetation index (ARVI; Kaufman and Tanré, 1992). Exercising the coupled 3-D
model, Myneni and Asrar (1993) reproduced the adjacency effect well as compared to a Monte Carlo simula
tion, and simulated soybean reflectance with good agreement to measured data.
In the radiative transfer formulation of Liang and Strahler (1993a), the coupled atmosphere-canopy
system consists of two plane-parallel layers with a non-Lambertian lower (soil) boundary. The atmosphere is
parameterized by a single scattering albedo and one-term Henyey-Greenstein phase function; each is a
weighted combination of values for Rayleigh and aerosol particles. The leaf canopy is described by a leaf nor
mal distribution function, bi-Lambertian leaf scattering, and a specular reflectance parameter. The flux field is
separated into unscattered radiance, singly-scattered radiance, and multiply-scattered radiance. The unscat
tered radiance field consists of uncollided downwelling irradiance and radiance upwelling from the soil surface.
Within the canopy, the single-scattering radiance field includes the hotspot effect, as parameterized by Nilson
and Kuusk (1989), while the multiple scattering field does not. The total radiance field is solved by Gauss-Sei-
del iteration at finite increments of optical depth, with numerical integrals evaluated using Gauss-Legendre
quadrature.
Although the Liang-Strahler Gauss-Seidel model provides accurate solutions for a realistic parameter
ization of the radiative transfer equation of the coupled atmosphere-canopy medium, it is too cumbersome for
inversion by forward iteration. Accordingly, Liang and Strahler (1993b) provide a simplified model also rely
ing on decomposition into a three-component flux field. Atmospheric multiple scattering is approximated by a
ô two-stream model, which preserves the anisotropic distribution of skylight. In the canopy, multiple scattering
is approximated using asymptotic theory. An inversion from reflectance measurements of a soybean canopy
(Ranson et al., 1984) retrieved leaf area index with good accuracy. Leaf angle distribution parameters were not
estimated as accurately, probably due to lack of measurements near the hotspot. Later work by Liang and
Strahler (in preparation) has applied a four-stream approximation to the coupled atmosphere-canopy model.
This approach yields very good accuracy at useful angles with a calculation speed sufficiently rapid to extend
inversion through forward iteration to large volumes of directional radiance imagery.
2.3. Other Coupled Models
A coupled atmosphere-canopy model is also presented by Rahman et al. (1993a). They utilize a partitioning of
atmospheric radiation into direct and diffuse fields (following Tanre et al., 1983) and couple the atmosphere
and canopy using a multiple reflectance parameter that depends on the proportions of direct and diffuse irra
diance. The canopy is modeled following Verstraete et al. (1990) and Pinty et al. (1990), utilizing parameters
describing leaf angle distribution, single-scattering albedo, phase function for the leaf, and sunfleck geometry.
In a series of simulations oriented toward sensing with the NOAA AVHRR (Advanced Very High Resolution
Radiometer) instrument, they show that canopy optical properties should be retrieved with good accuracy in
most cases. Structural properties can also be well retrieved, if the shape of the hotspot is sampled well.
In a more practical application, Rahman et al. (1993b) simplify the canopy portion of their coupled
model to a semiempirical form that includes terms representing forward-backward scattering and a hotspot.
Three empirical constants calibrate the surface BRDF function. In a validation against the observed direc
tional reflectance of a number of canopy covers, the model showed very good accuracy. For some test datasets,
the authors adjusted observed reflectances for the smoothing of the BRDF that is produced by diffuse illumi
nation, confirming the importance of coupling atmosphere and canopy. In application to AVHRR data, the
coupled model retrieved reasonable values for average optical depth, water vapor content, and surface param
eters for an annual sequence of measurements obtained from two North African desert sites.
Liang and Strahler (submitted) coupled an atmospheric radiative transfer model to a simple six-
parameter empirical model for surface BRDF that is derived by combining the limaçon model of Walthall et
al. (1985) with a two-parameter negative exponential hotspot model. This formulation fits soyoean, shinnery
oak, and conifer forest BRDFs well, with accuracies in the range of 3-10 percent. The results emphasize the
importance of including a non-Lambertian lower boundary for proper modeling of path radiance.
2.4. Semiempirical Models
Roujean et al. (1992) have recently proposed a three-parameter semiempirical model for surface reflectance.
The model expresses BRDF as a sum of three terms. The first term represents reflectance at nadir illumination
