996 
for the potentially very large number of state variables used to describe the various aspects of the real world, 
while /, stands for the economically invertible physically-based models. 
See 
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z 
/A 
S 1’ 
\ ’ 
Y 
Figure 3: Graphical representation of the goal of remote sensing, showing two kinds of 
physical models, those that are used mostly in direct mode, and those that can be routinely 
inverted to retrieve the values of the state variables. 
The goals of remote sensing must be updated accordingly. If the objective is to promote a more fundamental 
understanding of the processes controlling the measurements, then complex models must be developed to 
represent the current state of the theory of radiation transfer. Recent advances in this direction have 
occurred with models capable of describing the transport of photons in three-dimensional structured media, 
using techniques such as discrete ordinates, ray tracing, or radiosity. However, if the prime objective is to 
routinely characterize the state of the system under observation, then somewhat simpler, and less demanding, 
models must be derived, and these models must be invertible at a reasonable computational cost. 
In this context, we use the phrase ‘model inversion against a data set’ to designate the numerical 
process whereby the parameters of the model are estimated on the basis of the information contained in the 
observations, using an optimization procedure and a figure of merit function. In the case of the physical 
models described here, the model parameters are the state variables of the radiative transfer problem, so 
they are measurable quantities, and this feature allows the validation of the model as explained in Pinty 
and Verstraete (1992). The inversion procedure will be investigated further below; for the time being, it 
is sufficient to know that this approach implies that the number of model parameters must be as small as 
feasible while maintaining both a realistic representation of the physical processes and an accurate description 
of the measurements. 
After such a model /, has been inverted against the observations and the model parameters S have been 
retrieved, a further scientific goal is to establish quantitative relations g, between the variables of interest Y 
and the state variables S just obtained, although it will be seen that this falls outside the scope of remote 
sensing per se. The development of these models Y = ÿj(S) contributes directly to a better understanding 
of the basic processes that control the environment, justifies the use of remote sensing as a source of data 
and of the radiation transfer models as the tools to assess the state variables of the system, and permits 
to objectively identify the variables of interest which can realistically be retrieved from remote sensing with 
sufficient reliability and accuracy, as will be seen below. 
INVERTING PHYSICAL MODELS 
Because of the importance of the inversion procedure to estimate the state variables and ultimately to 
characterise the system under observation, we investigate further some of the requirements and implications 
of this approach. As long as the model Z — f(S) requires only one state variable S to estimate the 
measurement Z, this equation can be analytically or numerically inverted to yield the value S — 
However, if the model / requires more than one state variable to describe the observation Z, the model 
cannot be inverted (or, equivalently, the values of S cannot be uniquely estimated) because the problem is 
underdetermined: we have multiple unknowns and only one equation. The standard approach consists in