at ions Zk and retrieve 
as few parameters S as 
ist exist and implement 
ion if it exists, and (4) 
n). 
imber of measurements 
capability to represent 
proper validation. The 
d Verstraete, 1992). 
any given independent 
riables S and the type 
ariables. For example, 
ieved from the spectral 
ith the way the system 
>f the physical model / 
elevant for the retrieval 
te variables which must 
sses X ; of independent 
nputationsd standpoints 
conditions required for 
a model /(X, S) against 
be inverted with respect 
; acquired while varying 
vational sampling in X 1 
a in the inversion. 
rie, either because large 
Sj varies little, then this 
ersion against a data set 
r specific to the hot spot 
e hot spot and adequate 
also follows that if the 
e retrieval of iP, then no 
us Theorem implies that 
if observations where the 
the inversion. 
I on the particular figure 
e measurements Z, and 
sbust (noise insensitive), 
na) and computationally 
ieved simultaneously. Of 
lion. 
of independent variables 
^dependent of each other, 
d inverted independently 
ersion of the appropriate 
Corollary. If some of the state variables in S j l depend on S ;2 , then there exists a causal model Z — 
f\{Xi 1 , S jl ) describing the observations Z in terms of the state variables S jl and a set of causal models of 
the type Sji = /2(X j2 , S ;2 ) describing the values of the state variables S jl in terms of the state variables 
S j2 . If, in addition, the associated subsets of independent variables X J1 and X-’ 2 are disjoint, then the model 
fi must be repeatedly inverted against Z with respect to X jl for various values of 2 , and the models / 2 
must be inverted against the newly obtained sets of state variables S j l with respect to X j2 to retrieve the 
state variables S j2 . 
In this case, the model describing the measurements could be formally written Z = f\ [X ;1 , S ;1 (X j2 , S j 2 )]. 
This discussion occurs because the remote sensing instruments on board space platforms typically used to 
investigate surface processes acquire data for particular directions of illumination and observation in a number 
of spectral bands. As indicated earlier, this results in data variability with respect to the directional and 
spectral independent variables, and current models associate these disjoint subsets of independent variables 
to distinct subsets of state variables. However, some of the state variables of the radiative transfer problem, 
which describe the directional variance of the signal (typically the reflectance and transmittance of the 
scatterers, which contribute to the anisotropy of the surface), are functions of the state variables which can 
be retrieved by inversion with respect to the spectral independent variable. 
For instance, in the solar spectral range, (i) the inversion of a BRDF model f\ capable of describing 
the physics of the measured signal in terms of the structural and optical properties of the medium must be 
performed before the analysis of the spectral variance, and (ii) the latter, executed through the inversion 
of a spectral model /2, can focus exclusively on those state variables which are spectrally dependent. This 
satisfies Theorem 3 by minimizing the number of model parameters at any one step, and takes advantage 
of the causal relationship between the state variables. From a numerical point of view, a higher confidence 
may be placed in the estimates of the state variables, since the higher the number of model parameters, the 
higher the probability the optimization procedure may find a local minimum. 
This nesting of models no doubt results from the fact that the fields of view of all current and planned 
airborne and space instruments (as well as those of the vast majority of laboratory and field instruments) 
are sensitive to a large number of scatterers at any one time. Remote sensing measurements are therefore 
intrinsically characteristic of the medium under observation, and not only of the individual scatterers. The 
deconvolution of the two effects must be performed before a meaningful spectral analysis can be carried out. 
The new situation resulting from this discussion is shown in Figure 4. Strahler (1994) discussed the full 
range of BRDF models f\ currently available, so that this information does not need to be repeated here. 
Similarly, Jacquemoud and Baret (1991) described a model /2 to estimate the reflectance and transmittance 
of a leaf in terms of the chlorophyll and water content, as well as a leaf structural parameter N. 
Soo 
i' 
Si(r,f,A) Z(f,t,A,n) 
!'■ i’ 
Sj(f,f) Y(r,<) 
Figure 4: Graphical representation of the goal of remote sensing, showing the two step 
inversion with respect to the directional and spectral dimensions. 
The following outstanding issues remain and should be underscored: (i) Models need to be validated before 
they are used extensively. This issue has been addressed in the literature ( e.g ., Pinty and Verstraete, 1992), 
but much more effort should be expended to validate and intercompare models than has been done in the 
Pftri. (ii) Perhaps the major obstacle to the validation of models is the lack of simultaneous remote sensing, 
field, and laboratory data acquisitions. The advent of the PARABOLA2 instrument, described by Deering 
(1994), offers new opportunities, but significant effort will be needed to collect the relevant ancillary data 
sets and make them accessible to the co mmuni ty, (iii) The identification of the minimal number of an gular