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horizontal visibility using the 5S atmospheric correction model (Tanré et al., 1990) which is a single parameter 
no more wavelength dependent. The horizontal visibility was inferred from a look up table built from the 5S 
model, where the entry was the fraction of diffuse radiation measured in few wavebands during the 
experiments. For a given configuration, the PROSPECT+SAIL model computes canopy reflectance spectra 
from canopy structural variables such as the leaf area index LAI, the average leaf inclination 0, and the hot-spot 
parameter s. It also requires variables describing the leaf optical properties such as C ab (pg cm -2 ), water 
equivalent thickness C w (cm), leaf mesophyll structure N, for the leaves, and soil reflectance p s (X.) which was 
assumed lambertian. Figure 1 presents the models and their input and output variables. 
Figure l .The models used to compute canopy reflectance spectra from canopy biophysical characteristics, soil 
reflectance and the measurement configuration. 
2.3. The inversion procedure 
In most cases, the complexity of canopy reflectance models prevents an analytical inversion so that numerical 
methods of optimization are required. Let us consider a model M that relates the vector X of input variables to 
the vector Y of output variables so that Y = M(x). The goal of the inversion process is to estimate some of the 
input variables X e from measurements of the output variables Y and knowledge of complementary input 
variables X c . If n is the number of observations, the inversion procedure consists in determining X e by 
minimizing the merit function S(A' e ) defined by: 
s <*<> = Zb' - M < x «- x c>i ] 2 W 
(=1 
This generally non-linear equation is solved by iterations. Knowledge of the initial parameter set X e is often 
critical to speed up the convergence of the solution and the validity of the convergence itself. Numerical 
instability phenomena may lead to local minima, which means that the uniqueness of the solution is never 
guaranteed. As above mentioned, there are several optimization techniques such as the Gauss-Newton 
algorithm, the Gauss-Marquardt algorithm (Marquardt, 1963) or the simplex method (Nelder and Mead, 1965). 
The choice of one of these algorithms mainly depends on the model, the degree of linearity, the number of input 
variables to be estimated. Jacquemoud et al. (1994) showed that the quasi Newton algorithm (Gill and Murray, 
1972) gave accurate results in most cases, and was quite computationally efficient. In the frame of this work, 
we implemented the qasi-Newton algorithm using the NAG (Numerical Algorithm Group) routine E04JAF. 
This routine allows to impose constrainsts on the lower and upper bounds of the input variables to be estimated.