1001 
.DF models, and of the 
ilea remains a research 
>ly still take time, (iv) 
ticular with the use of 
sired state variable. In 
of the impact of noise 
s from remote sensing 
etween the parameters 
rt is, strictly speaking, 
these relations justifies 
round but scientifically 
ssess the values of the 
The answer appears to 
l on terrestrial surfaces 
the analysis of images 
an), or the exploitation 
he variables of interest 
>y the numerical value 
Terent nature than the 
relations is also clearly 
landing of the physical 
> the estimation of Y. 
that are state variables 
se that can be entirely 
one other variable, not 
or biochemical (causal) 
lels /1 and /3 described 
is of the type Y = g( Z) 
iriables S are necessary 
t perfect) estimation of 
lels against the data Z, 
.able provides a perfect 
empirical, even if there 
even if the variable of 
Y = g( Z) is reliable or 
regetation indices when 
the leaves from remote 
tion can reasonably be 
Lnce and transmittance 
s the measurements are 
e of the canopy, we can 
li min e the chlorophyll 
and exclusively on the 
5 state variables Sj are 
o establishing multiple 
{the latter exists. The 
is gj, even if the model 
g, is perfect. Again, there may be sound theoretical justication for seeking such a relation, but the final 
relation g will also be empirical. 
In the vast majority of cases where relations of the type Y = g( Z) are developed and used, the variable 
of interest Y is an environmental or economic variable whose value is determined only partly by the state 
variables S. We will call ‘hidden variables’ H all the variables which affect the value of Y, but which are not 
state variables of the radiative transfer problem, and do not depend on them. These hidden variables include, 
for instance, all underground characteristics not accessible by optical methods (e.g., deep soil moisture, root 
depth), all physical, chemical or biological properties unrelated to the transfer of radiation in the spectral 
range considered here (e.g., the parameters that describe the competition of species), and all parameters 
describing human activities (e.g., amount of fertiliser and irrigation). Since hidden variables directly affect 
the variable of interest, but have no relation whatsoever with the remote sensing measurements, the latter 
cannot include any information on the former and cannot provide the basis for a deterministic determination 
of the variable of interest. In fact, the more the variable of interest Y depends on hidden variables H and 
the less it is controlled by the state variables S, the less reliable the relation Y = g( Z) will be. 
Figure 5 exhibits the place of these relations and the role of the hidden variables in the graphical 
scheme used before, and the following Theorem identifies which variables of interest may in fact reasonably 
be expected to be estimated from remote sensing data on the basis of these empirical relations: 
Soo 
1' 
Si(r,f, A) Z(f,t,A,n) 
J/» | 
S 2 (f,t) Y(r,t) H 
Figure 5: Graphical representation of the goal of remote sensing, showing the role of 
hidden variables and the place of empirical relations and vegetation indices. 
Theorem 6. A meaningful relation of the type Y = g( Z) can be established only if the variable of interest 
Y is in fact one of the state variables S which control the measurements Z, or a function of these variables. 
In the latter case, these state variables must signiGcantly affect the variable of interest. 
It is important to note that this Theorem places a necessary but not a sufficient condition for such a 
relation to be meaningful. We must therefore question and precisely document its range of applicability. 
Most of the relations Y = g(Z) are traditionally established by correlating remote sensing measurements to 
field observations of the variable of interest. The fact that these in situ measurements take place at spatial 
and temporal scales different by one or more orders of magnitude from those of space observations should 
he of major concern to anyone involved in the design or use of such relations. Strictly speaking, the validity 
of such statistical relations is limited to the set of conditions which are accurately sampled by these remote 
sensing measurements and field observations. The field of statistical inference defines the degree of confidence 
that can be placed in various statistical statements about a population, when information is available only 
about a sample. Nevertheless, the establishment of these empirical relations hinges on the availability of 
field measurements of the variable of interest Y, and their verification can only be achieved through the 
collection of more field data. The following Theorem expresses this paradox between the conflicting needs 
to verify and to apply these statistical relations: 
Theorem 7. A relation Y = g( Z), obtained by statistically correlating remote sensing measurements and 
Geld observations, is potentially useful only where and when it is not verified, and remote sensing does not 
provide additional information whenever this relation is verified. 
The reliability of these relations may be increased by allowing a dependency on hidden variables: Y = 
H), provided the values of these variables are available independently. If these relations are established