spects of the real world, 
taking M measurements, and leads to the following system of equations: 
; two kin de of 
n be routinely 
ite a more fundamental 
i must be developed to 
in this direction have 
ional structured media, 
e prime objective is to 
er, and less demanding, 
tational cost, 
esignate the numerical 
nation contained in the 
de case of the physical 
re transfer problem, so 
as explained in Pinty 
for the time being, it 
rs must be as small as 
an accurate description 
arameters S have been 
variables of interest Y 
ie the scope of remote 
i better understanding 
ng as a source of data 
e system, and permits 
m remote sensing with 
>les and ultimately to 
nents and implications 
ble 5 to estimate the 
he value 5 = f~ 1 (Z). 
:rvat ion Z, the model 
>ecause the problem is 
i approach consists in 
Zi =f(S\ 1 Sl...,S 1 m ) 
z 2 = f(slsl,...,sl) 
( 2 ) 
z M = f(s?,s?,...,s%) 
where it has been assumed that the same model / is applicable to all measurements. This does not help, 
however, because either all Z k are identical (within the errors of measurements), and no additional infor 
mation is gained by multiplying them, or they are different, but then these differences in the observations 
must result from differences in the state variables S. This means that the system has changed between 
observations, and there is no way its state can be uniquely characterised on the basis of these measurements 
alone. This difficulty cannot be resolved without introducing yet another set of variables: 
Theorem 2. A model f describing remote sensing measurements Z in terms of more than one state 
variable S cannot be inverted unless it expresses the variability of Z as a function of at least one mesurable 
independent variable X. 
Four sets of independent variables have been found useful in remote sensing, corresponding to the 
spatial (f), temporal ( t ), spectral (A), and directional (il) dimensions. Many models use more than one 
independent variable simultaneously: For instance, bidirectional reflectance distribution function (BRDF) 
models describe the variability of the measured reflectances in terms of the senith and asimuth angles 
characterizing the directions of illumination and observation. 
The physical model describing the observed signal is now formally expressed as Z = /(X, S), and to 
avoid acquiring the same observation over and over again even though the system does not change (S = cat), 
we adopt an observing strategy designed to change one or more of these independent variables between each 
observation. The system of equations (2) becomes 
Zi = /(*!!,Xai X nl ;Si,S a ,...,Sm) 
Z 2 = f(X 12 , Xu, . . ., X n 2', Si, Si, . . ., S m ) 
(3) 
Zm — f{X im , Xim 1 • • • 1 X n M ; Si, S?,, S m ) 
where Xu, stands for the value of the independent variable X, (* = 1,2,..., n) corresponding to the conditions 
of observations prevailing when the measurement Zk, (Jfc = 1,2,..., M) was acquired, and where the state 
variables Sj, (j = 1,2,..., m), are kept constant for all measurements. 
As before, if M < m, there is no solution because the system is underdetermined. When M = m, there 
is usually no solution, because of errors of measurements in Z or in X, or because of limitations of the model 
/• And if M > m, there is no solution because the system is overdetermined. We are once more forced to 
change objective: rather than searching for the unique true solution of this system, we want to find the ‘best’ 
solution, 1. e., the best estimate of the values of the state variables: those that, together with the model /, 
optimally describe the observations Z. 
This implies that a criterion must be selected to evaluate the ‘goodness’ of a particular solution. A 
figure of merit function is used for this purpose, and the least mean square estimate is usually chosen: 
M 
6 2 = £ Wt [Z k - f{Xi k ,Xi k ,...,X nk -,Si,Si,...,S m )}* (4) 
1 
where W k is the weight given to observation Z k . The problem is solved by searching, among the set of 
Possible solution vectors S, the one that minimises the figure of merit function 6 3 . Standard numerical 
algorithms exist to solve this optimisation problem, provided the function / is sufficiently well-behaved and 
the data set Z provides enough variance to constrain the inversion procedure. The following two Theorems 
explicit these requirements: