479 
Models based on implicit functions or explicit functions (IF-models and EF-models) are theoretical 
models which do not require prior adjustments of parameters. Models based on a discrete description 
of the vegetation layer are good examples of EF-models (Karam et al., 1992; Ulaby et al., 1990; 
Wigneron et al, 1993b). Vegetation canopy is treated as a multi-layered medium containing randomly 
distributed discrete scatterers (discs and cylinders corresponding to leaves, branches and trunks). Input 
parameters of discrete models describe accurately the vegetation structure in terms of distribution of 
orientation, size, shape and density of vegetation scatterers. Recently, spatial location, size and shape 
of the individual tree crowns have also been taken into account to model discontinuous tree canopy 
backscattering (McDonald and Ulaby, 1993; Wang et al., 1993). 
2,2 Method of Inversion 
Several statistical techniques have been implemented to retrieve geophysical parameters from remote 
sensing measurements: use of Backus-Gilbert inversion (Westwater and Cohen, 1973), covariances 
matrices associated to principal component analysis (Smith and Woolf, 1976) or to Monte Carlo 
method (Jin and Isaacs, 1987), multiple linear regression schemes (Waters et al., 1975; Jackson et al., 
1990), to name but a few. Yet, most of these approaches require linear relationships between remote 
sensing measurements and geophysical parameters. 
For non-linear problems, when a simple analylitic solution to the inverse problem is not available, a 
very common algorithm to inverse a forward model is the statistical inversion approach (SIA). The 
principle is to search for input parameters (Xj,.^), which include the geophysical parameters of 
interest, that minimize the squared error between the spectral signatures as measured from space (T B ) i; 
and the actual outputs of the model g^Xj,..^). So the inversion problem is (Pulliainen et al., 1993): 
m n 
Minimize G(x ] ,..,xJ = X 1/2 O; 2 {g i (x 1 ,..,x n )-(T B ) i } 2 + X 1/2 ^¡ 2 (xj - x’j) 2 (1) 
i=7 j=l 
where: 
G^,..^,,) = cost function 
and (a priori information): 
x’j = average value of the j* model parameter 
A.j = standard deviation of the model parameter value 
Oj = standard deviation of measurement noise of the i* channel 
Depending on the forward model, several minimization algorithms are available to minimize 
G(x 1 ,..,xJ. To check whether correct values of geophysical parameters (xj,..^) are retrieved when the 
observed data (T B ) f are contaminated by noise (due to the sensor or to the conditions of observation), 
random noise can be added to each value of (T B ),. The sensitivity of the retrieval algorithm to this 
simulated noise can be estimated, by repeating many times the inversion procedure and by calculating 
the standard deviation of the retrieved parameters (Pinty et al., 1989). Pulliainen et al., compared the 
SIA with conventional inversion algorithms for different geophysical parameters (sea ice 
concentration, ocean temperature...) from passive microwave measurements. They conclude that 
(1) SIA gives accurate estimates (better than conventional algorithms) when (i) appropriate a priori 
information is available and when (ii) the number of input parameters contributing to the emission 
behavior is small 
(2) For vegetated land applications, the SIA is the only available method, but both criteria (i) and (ii) 
are not met.