Theorem 3. In order to invert a model /(X, S) against a data set of M observations Z* and retrieve 
the m values of S, (1) the mode] must be mathematically well-behaved and require as few parameters S as 
possible, (2) a figure of merit function must be deSned, (3) an inversion procedure must exist and implement 
an algorithm capable of finding the absolute minimum of the Ggure of merit function if it exists, and (4) 
more observations must be collected than there are parameters in the model (M > m). 
The smaller the number m of parameters a model needs, and the larger the number of measurements 
M can be explained by the model, the more confidence we can place in this model’s capability to represent 
the variance of the measurements, although this verification is not equivalent to a proper validation. The 
latter would require the simultaneous measurement of the actual values S (Pinty and Verstraete, 1992). 
Not all state variables S can be retrieved from an inversion with respect to any given independent 
variable. In fact, there is a strong association between the nature of the state variables S and the type 
of independent variables X that are appropriate for the retrieval of these state variables. For example, 
the chemical composition of the materials in the observed medium can only be retrieved from the spectral 
variations of the measurements. Since the independent variables have only to do with the way the system 
is being observed, not with the state of the system itself, the mathematical form of the physical model / 
that describes the measurements Z determines which independent variables X are relevant for the retrieval 
of each of the state variables S. 
For this reason, it is useful to partition the set S into disjoint classes S’ of state variables which must 
be retrieved simultaneously, i.e., which are physically associated with disjoint classes X’ of independent 
variables, and it is advantageous, both from the observational, mathematical and computational standpoints 
to keep these classes X 1 as small as possible. The following Theorem specifies the conditions required for 
the successful retrieval of such a class of state variables: 
Theorem 4. To reliably and accurately retrieve the state variables S’ by inverting a model /(X, S) against 
a data set Z (1) the model must be sensitive to the values of S’ , (2) the model must be inverted with respect 
to the appropriate class of independent variables X’, (3) the measurements must be acquired while varying 
only the independent variables in X’ while maintaining S = cst, and (4) the observational sampling in X 1 
must be extensive and dense enough to generate sufEcient variance in Z and constrain the inversion. 
In effect, this Theorem implies that if a model parameter Sj plays a minor role, either because large 
variations of Sj produce only minimal changes in the value of /, or because in fact Sj varies little, then this 
parameter should not be included in the model. For example, for the purpose of inversion against a data set 
of directional reflectances, a BRDF model should not include a structural parameter specific to the hot spot 
feature of the reflectance field unless the observed medium does exhibit a noticeable hot spot and adequate 
sampling of the reflectance in the appropriate angular region can be obtained. It also follows that if the 
data Z are acquired while varying only the independent variables appropriate for the retrieval of S’, then no 
other state variable could be retrieved by inversion against these data. Intuitively, this Theorem implies that 
the observational sampling strategy should be designed to provide higher densities of observations where the 
model / is most variable with respect to the independent variables appropriate for the inversion. 
The numerical values of the model parameters S’ retrieved by inversion depend on the particular figure 
of merit function, the noise level in both the independent variables X’ and the measurements Z, and 
the performance of the optimization procedure. Specifically, the latter should be robust (noise insensitive), 
reliable (capable of finding the absolute minimum despite the presence of local minima) and computationally 
cheap if this approach is to be used operationally. 
The last Theorem also implies that all model parameters in S’ should be retrieved simultaneously. Of 
course, these parameters must a priori be independent of the X ; used in the inversion. 
Theorem 5. If the data set Z has been acquired by varying two disjoint classes of independent variables 
(X’ 1 and X’ 2 ), and if the corresponding classes of state variables (S’ 1 and S’ 2 ) are independent of each other, 
then two separate models Z — /i(X j1 , S’ 1 ) and Z — /j(X’ 2 , S’ 2 ) can be defined and inverted independently 
on the same data set. 
In other words, each class of state variables can be retrieved directly by inversion of the appropriate 
model with respect to the relevant independent variables.