2. The search space is explored using a family of points. The method exploits the relative order between the 
candidate points to drive the search in a better direction (e.g.. Simplex method). 
3. The search space is explored using a population of points. The method identifies the subdomain in which 
the global minimum is located (e.g.. Genetic Algorithms method). 
Four minimization methods have been tested in this study: Quasi-Newton (QN) and Marquardt (MQ) 
often used in least squares minimiz ation, Simplex (SP), and a coupled method Genetic Algorithms + QN (GQ). 
Very briefly, the QN method (Gill and Murray, 1972) minimiz es, at each iteration, a quadratic approximation 
of the merit function Д 2 . We used here the routine E04JAF of the NAG library. The MQ method (Marquardt, 
1963) combines the best features of the gradient search (steepest descent) with a linearization of the fitting 
function (Taylor's expansion). The HAUS59 routine (Roux and Tomassone, 1973) was used. Instead of starting 
fr om a single point in the p-dimensional search space, the SP method (Nelder and Mead, 1965) considers a 
geometrical figure consisting of p+1 points, the simplex. Through a sequence of elementary geometric 
transformations (reflection, contraction, and extension), the initial simplex progresses in parameter space until 
it surrounds the minimum . We used the routine E04CCF of the NAG library. Finally, the GQ method (Renders 
et al., 1992) combines the explorative qualities of Genetics Algorithms with those of exploitation of the QN 
method; Genetic Algorithms (Goldberg, 1989) is a global search method based on an analogy with the process 
of natural selection and evolutionary genetics; in the coupled method. Genetic Algorithms create generations of 
points while QN drives the selection of individuals. The GQ method used in this paper is the Lamarck-inspired 
of Renders et al. (1992). 
It is not the intention of this paper to describe in detail the mechanism of these optimization 
methods; the reader is referred to the above references for more information. However, several points are 
worthy of note: these algorithms only require function evaluations (no analytical derivative) which makes them 
easy to use. In order to avoid function evaluations at infeasible points, they were bounded according to the 
domain of applicability of the model parameters: 1<N<2.5, l<Cab<100 fig cm -2 , 0.1<LAI<10, 5°<0i<85°, and 
(kSi<l. When required, the initial guesses have been fixed to N=1.75, Cab=50.5 fig cm -2 , LAI=5.05, 01=45°, 
and Si=0.5. Note that one of the difficulties which may arise when inverting the model is that there may be 
more than one local minimum fa- the merit function within a reasonable range of values fa the parameters; 
while the first three strategies (QN, MQ, SP) are likely to provide local minima, GQ is assumed to identify the 
global minimum of the merit function. 
2 - EXPERIMENTATION 
2.1. Synthetic Data 
In order to compare the different methods, a number of inversion procedures were performed using reflectances 
generated with the PROSPECT+SAIL model. Five surfaces representing five different vegetation canopies have 
been defined by varying the model parameters (Table 1). 
Surface 
N 
Cab 
LAI 
01 
Si 
A 
1.28 
36.6 
3.46 
27.7 
0.77 
В 
2.14 
5.7 
2.00 
54.9 
0.48 
C 
1.65 
62.2 
0.54 
79.9 
0.24 
D 
1.41 
25.5 
5.62 
10.3 
0.36 
E 
1.07 
16.6 
1.10 
64.3 
0.16