688 
The inversion technique, which can be linked to the above development, aims at inferring the model parameters 
from a set of 'measured' bidirectional reflectances. In this application, the inversion problem is solved following 
the method proposed by Pinty et al [1989], which minimizes the non-linear merit function defined by: 
82= i>j*- r j> 2 
j=l 
where rj is the jth measurement (for a given geometry) and rj is the model-estimation for the same geometry of 
illumination and observation. The optimization routine we used in this study implements a quasi-Newton 
algorithm (subroutine E04JAF) provided by the Numerical Algorithms Group library; the performances of this 
particular routine have been extensively tested elsewhere. It requires initial guesses (taken about the middle of the 
range) for all the model parameters which are also subjected to fixed upper and lower bounds. 
We applied this inversion scheme against synthetic data generated by (1) the model itself where the five 
independent parameters are leaf area index (Lj), leaf hemispherical reflectance (r^) and transmittance (tL), soil 
reflectance (Rs) and hot-spot parameter (2rA), and (2) the model of Myneni & Asrar [1993] with the same 
independent parameters. 
The Root Mean Square of the fit (RMS^ = 8^/nf), where nj represents the number of degrees of freedom (np is 
equal to the number of observations minus the number of fitting parameters), gives an indication of the quality 
of the optimization. 
3.1 - Data-set from Iaquinta & Pinty’s model 
Table-2a indicates the selected set of values for the canopy parameters which was used to simulate the 
bidirectional reflectance field. These values are representative of a green vegetated canopy in the red and near- 
infrared bands. The initial guesses and the upper and lower bounds taken for each individual parameter are 
indicated in Table-2b. The sun zenith angle is 30°, and the leaf angle distribution is uniform (reflectances values 
are computed each 3.2° in the principal plane, and 10° each 30° in azimuth, with 4 digits). The results of the 
inversion are given in Table-2c for various values of the leaf area index representing thin to very thick canopy 
optical conditions. 
Table-2a 
(red) 
(near-infrared) 
l L 
Rs 
2rA 
0.0607 
0.0429 
0.2 
0.2 
0.4357 
0.5089 
0.25 
0.2 
(True values) 
Table-2b 
L T 
«L 
tL 
Rs 
2rA 
2.5 
0.25 
0.25 
0.2 
0.1 
l.E-06 
l.E-06 
l.E-06 
l.E-06 
l.E-06 
15. 
0.99 
0.99 
0.45 
2. 
(Initial guess) 
(Lower bound) 
(Upper bound) 
trueLj 
retrieved Lp 
tL 
tL 
Rs 
2rA 
RMS 
1. 
0.9999 
0.9998 
0.0607 
0.4357 
0.0429 
0.5089 
0.2000 
0.2500 
0.1999 
0.2001 
0.31E-04 
0.30E-04 
3. 
3.0054 
3.0734 
0.0607 
0.4346 
0.0429 
0.5076 
0.2006 
0.2546 
0.1990 
0.2008 
0.28E-04 
0.56E-04 
5. 
5.1011 
4.9450 
0.0607 
0.4359 
0.0429 
0.5092 
0.2183 
0.2493 
0.1993 
0.1998 
0.29E-04 
0.29E-05 
7. 
5.7671 
7.8001 
0.0607 
0.4361 
0.0430 
0.5092 
0.0621 
0.0150 
0.2018 
0.2000 
0.30E-04 
0.82E-04 
(red) 
(near-infrared) 
Table-2c: Retrieved values for the model parameters. 
Clearly the leaf optical properties (r^ and t^) as well as the hot-spot parameter (2rA) are always very well 
retrieved by the procedure. At the same time, it appears that the inference of the leaf area index and soil albedo is 
very sensitive to the leaf area index value. This feature is not surprising since the larger the leaf area index, the 
larger the multiple scattering and the weaker the soil contribution (for example, the soil contribution is less than 
2 % of the total signal when Lj=7.). To illustrate this point, Table-3 gives the percentage of the upward flux