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ADAPTATION OF A BIDIRECTIONAL REFLECTANCE MODEL 
INCLUDING THE HOT-SPOT TO AN OPTICALLY THIN CANOPY 
J. Iaquinta & B. Pinty 
Université Blaise-Pascal, Laboratoire de Météorologie Physique, URA 267/CNRS, 
F-63177 Aubière Cedex, France. 
ABSTRACT 
This model presents an improvement of the original model by Verstraete et al. [1990] in order to account for the 
effects due to an underlying soil below the vegetation canopy. The singly scattered component is solved exactly 
using an analytical hot-spot description. The multiply scattered component is approximated on the basis of a 
Discrete Ordinates Method reduced to a one-angle problem. This model is then compared to others existing 
reference models and it is also used in an inversion procedure to examine its performances for real remote sensing 
applications. 
KEYWORDS: Bidirectional Reflectance Factor, Hot-spot, Inversion. 
INTRODUCTION 
The global monitoring of the state and evolution of the terrestrial surfaces requires a quantitative interpretation of 
satellite measurements at a very high spatial and temporal resolution. This interpretation of remotely sensed data 
should rely on the inversion of physically based models representing the radiance field emerging from the 
medium under study. The highly anisotropic behavior of the bidirectional reflectance over a vegetation canopy, in 
the optical domain, requires a careful and accurate modeling of radiation transfer. This implies that we are able to 
link the biological and physical properties characterizing the surface to parametric radiative quantities including 
the phase function (which describes the angular distribution of the energy scattered by a particle), the single 
scattering albedo (ratio of the scattered energy to the total energy either scattered or absorbed) and the leaf area 
index (area of one-side leaf surface projected onto a horizontal plane). In general, it is not possible to find an 
analytical solution for the total radiance field, except for the single scattering contribution, and consequently 
numerical schemes have to be used for solving exactly the integro-differential equation with appropriate boundary 
conditions. In this paper we present a new model where the single scattering component, including the hot-spot 
and the underlying soil effects, is given explicitly through an analytical solution and where the multiply scattered 
radiance, which is more difficult to handle, is approximated using a Discrete Ordinates Method (DOM), reduced 
to a one-angle problem. This modeling approach is clearly based on a compromise (and hence some 
approximations) between purely analytical (fast) and numerical (accurate, but very time consuming) methods. 
Therefore, the model we propose appears as a good candidate for inversion purposes to the extent that any 
inversion technique will require computationally cheap formulations. 
In the first section of this paper, we present the model and describe the modelling strategy. Test examples are 
shown in the second section, together with a comparison between the results obtained from our model and those 
given by (1) Nilson & Kuusk’s model [1989], in which the multiple scattering is calculated according to the 
Schwarzschild approximation, and (2) the more accurate results given by the resolution of the transport equation 
using a complete 3-D DOM [Myneni R. B. and Asrar G., 1993]. In the third section, we apply an inversion 
technique against synthetic data generated by our model and the 3-D DOM code. The results are discussed in light 
of the ability of retrieving accurately and simultaneously the leaf area index and the optical properties of the 
canopy. 
1 • DESCRIPTION OF THE BIDIRECTIONAL REFLECTANCE FACTOR MODEL 
Fig-1: Schematic decomposition of the radiation field. Fig-2: Geometry of illumination and observation.