tivity for volumes and of surface height for an
and the diffuse incoherent field or average power.
interface, or the equivalent, plus some measure of
These theoretical average quantities presuppose
the spatial scale and geometry of this variation,
a certain kind of averaging, typically averaging
most commonly through correlation length.
over an ensemble of surfaces, conceptually sub-
Loosely speaking, the correlation length indicates
jected to the same illumination regime and devel-
the general size of a significant permittivity varia-
oped as realizations of the same statistical pro-
tion or of a discrete scatterer in a volume or the
cess. In this context, the correlation function, for
lateral extent of a contour variation on a surface.
example, means the correlation value of quanti-
As such, correlation lengths may be anisotropic
ties (e.g., surface heights at a given separation)
in the sense that different lengths may apply in
averaged over the ensemble of surfaces, not averaged
different directions, as for elongated brine pock-
spatially over a single domain. The latter sense of
ets in saline ice or narrow striations in a surface.
averaging, however, is often the one employed in
The fact that directional information can reside
practice when a limited domain is characterized
in the correlation length underscores the signifi-
experimentally to provide model input. While the
cance of this quantity for polarimetric modeling.
two averaging processes bear some relation, ques-
We also note that the correlation length does not
tions remain in any given application as to the
entirely define the correlation function itself, given
correspondences of the quantities obtained.
that the former is typically only defined as the
Lastly, we mention Monte Carlo techniques as
distance over which the function decays by 1/e.
an approach to calculation of responses of ran-
The entire function itself will depend on the case
dom media. Such techniques have received a great
considered; a Gaussian function is often used.
deal of attention as computing power has in-
Note that the indication of a "Gaussian surface"
creased. The attraction of the non-Monte-Carlo
usually means that the correlation function or,
techniques resides in their ability to proceed di-
equivalently, the power spectrum of a surface has
rectly in relating statistical and spectral features
the form of a negative exponential of distance
of the modeled medium to the statistical proper-
squared. This does not mean that the variation of
ties of the received field or power. In such ap-
surface height is assumed to follow a Gaussian
proaches, ensemble averaging is carried out ana-
lytically in the course of the formulation pro-
the case.
cedures; this is a result of the analytical simplicity
The appearance of statistical quantities both in
of the expressions used and the introduction of
remote sensing data and the modeling implies
limiting simplifications. By contrast, in Monte
some particular kind of averaging procedure. In
Carlo simulation one calculates the response of
physical measurements this is performed, in ef-
specific, geometrically defined (deterministic) me-
fect, by the sensor. One footprint presumably
dium samples, perhaps analogous to the "patches"
spans many patches or subregions, in each of
mentioned above. This is typically done numeri-
which a sufficient sample is present for statistical
cally. Each of the samples is constructed to be an
quantities to apply meaningfully. Thus, integra-
individual realization of the hypothesized under-
tion is performed over each subregion patch, and
lying statistical formation process. Once solutions
the ensemble of subregions is summed as well.
have been obtained for a sizable number of such
Quite significant questions remain as to the corre-
instances, they are added together to obtain effec-
spondence between the kind of averaging im-
tively an ensemble average. Thus by "brute force"
plied in the theoretical bases of our models and
one mimics the supposed process by which a sen-
that performed by any likely or conceivable sen-
sor might accomplish its averaging integration
sor. Among other things, received data are likely
and also accomplishes a faithful rendering of the
to contain effects due to the variability of me-
kind of averaging assumed in the more approxi-
dium properties at a number of different scales,
mate analytical models.
e.g., at both the subregion scale and at the scale of
Monte Carlo techniques have the attraction that
the ensemble of subregions.
many of the limiting assumptions needed to ob-
In model formulations, the averaging process
tain results in the more analytical approaches can
is typically carried out analytically, at a single
be avoided. In addition, because the details of the
scale, in the process of constructing governing
modeled media are known, there are no unknown
relations. This means that the formulations are
tunable parameters, the estimation of which can
ultimately stated in terms of the statistical quanti-
render data-matching exercises suspect. In sur-
ties themselves, such as expected or coherent field
face scattering, going beyond the constraints of
7
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