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for fixed values of these hidden variables, their applicability is even more reduced. This is typically the case 
when a relation is deemed applicable only to a certain type of ecosystem, for instance. 
Another difficulty arises because of the role of the state variables Sj that affect the measured signal 
but are unrelated to the variable of interest Y: Although the relation g may still be useful in practical 
applications, its interpretation may be difficult or impossible. The ‘contaminating’ effects of the atmosphere 
and soils on the value of vegetation indices, and therefore on the variables of interest estimated from them, 
provide typical examples of such difficulties. 
Vegetation indices (VI) have been used extensively for a wide variety of applications, and probably 
constitute the most common approach to the direct estimation of the variables of interest. Initially, vegetation 
indices were designed as linear combinations of sensor spectral channels (for the most part, one in the red 
and one in the near-infrared), in such a way as to maximise dVI/dY. Only minor attention was given to the 
perturbing factors dVI/dSj. New vegetation indices (e.g., Huete, 1988; Kaufman and Tanre, 1992; Pinty 
and Verstraete, 1992) or procedures (such as the maximum value composite proposed by Holben, 1986) were 
introduced when a better compromise was found between maximising dVI/dY, and minimising dVI/dSj. 
Significant further developments can be expected in this direction over the next few years. 
Of course, any number of empirical relations of the type Y — g( Z) may be established, one for each 
variable of interest Y . In fact, the literature abounds in correlations between vegetation indices and variables 
of interest, from plant properties to ecosystem structure and functioning, and from precipitations to large 
mammals density. Lest we forget many of these relations are based on the same small number of independent 
data items, the following Theorem underscores the simplistic view of the world subtended by this practice: 
Theorem 8. If more than one variable of interest Y is correlated to a given vegetation index, or if more than 
n variables of interest are correlated with n independent remote sensing data items Zk, these variables are 
correlated between themselves and no additional information is gained on the system, unless these relations 
also depend explicitly on additional and independent hidden variables. 
Two more issues will be mentioned, although they would require a more detailed treatment than will 
be possible here. First, the value of vegetation indices is different for each remote sensing instrument, even 
if they observe the same location at the same time. This is due to differences in the spectral responses of 
the sensor, as well as in spatial resolution. Furthermore, most of the empirical relations Y = g( Z) already 
derived, say on the basis of AVHRR data, will not be transferrable to the new sensors currently under 
development. Vegetation indices therefore appear to be specific to the sensor, or class of similar sensors, for 
which they were designed. 
The other (and perhaps more important) issue is expressed in the following Theorem: 
Theorem 9. Vegetation indices compress remote sensing data by a factor equal to the number of channels 
used, but also significantly reduce the information contained in the original data set. 
This point has been addressed elsewhere (Verstraete, 1994), but can easily be demonstrated by pointing 
out that the mathematical process of computing a vegetation index is not reversible: From two wide-band 
spectral measurements, we can compute their sum and their difference. The latter is a vegetation index, the 
former could be an estimator of the albedo of the target. It is not possible to estimate this albedo from the 
vegetation index alone. 
EMPIRICAL BRDF MODELS 
A final category of bidirectional reflectance models should be mentioned for completeness, namely those that 
aim to describe the anisotropy of the surface without imposing any constraint, other than being able to 
fit the given data Z as well as possible, on the analytical form of these models. Their parameters can be 
adjusted to fit a wide range of conditions, but have no particular physical meaning. Formally, these models 
can be written Z = h(X, P). This approach is schematically represented in Figure 6: