analytical approaches on scale of roughness or
analytical approaches, more information on the
severity of slopes means that one can work with
distribution of the medium is generally required
truly realistic surface profiles. This is accom-
than just rms variation and correlation length.
plished, however, at impressive computational
The specific character of the overall correlation
cost: two-dimensionally rough surfaces are rarely
function is needed or assumed; in volume scatter-
attempted, though we expect progress here in the
ing density information is needed, and other
near future. True Monte Carlo treatment of vol-
higher-order spectral or geometrical information
ume scattering as is done for surfaces is currently
may be required for complete analysis. This is
out of the question. The sheer detail required to
particularly so when the limits of the approxi-
describe the internal geometry of a realistic vol-
mate theories are exceeded.
ume of environmental material is insurmount-
It has been noted that a number of different
able; the computational requirements will also be
models may produce reasonable agreement with
invincible for the near future for solution of elec-
observations even though they use contrasting
tromagnetic equations over a truly life-like varia-
versions of the underlying physics, even when
tion of internal medium geometry.
the input data have been constrained by ground-
Monte Carlo volume-scattering treatments in
truth measurements (Carsey 1992). Among other
a certain limited sense have been carried out in
things, this serves to re-emphasize the fact that
the construction and analysis of collections of very
rms variation and correlation length do not
simple shapes. For example, Ding et al. (1992)
uniquely define the system. For more extreme yet
have used Monte Carlo procedures in achieving
realistic media, we will have to recognize and
arrangements of spheres to determine the statisti-
implement additional parameters that catch the
cal pair distribution functions needed for their
particular character of the variability, beyond what
dense media volume-scattering calculations.
is incorporated in the traditional quantities. This
Tsang et al. (1992b) have solved Maxwell's equa-
makes sense and is even desirable: it says that we
tions rigorously over similar dense media, for
need to focus on the identification of new, spe-
cases with as many as 4000 spheres, presumably
cific, distinguishing medium characteristics that
with volume fraction limits imposed by the con-
can be linked to some distinguishing statistical
vergence of the iterative solution procedure. Good
character in the received data. Only by doing this
solutions were obtained at densities of 25% by
will we recognize realistically distinct media in
volume. Alternatively, Chuah and Tan (1992a,b,c)
our sensed data. While overall this is likely to
have used Monte Carlo procedures to calculate a
mean that we must consider additional param-
cascade of idealized contributing scattering events
eters, such as higher statistical moments, in some
in vegetation. This statistical treatment of the con-
instances it may mean that we discard altogether
tributing scattering events is fundamentally dif-
one or more of the traditionally important me-
ferent from a rigorous deterministic analysis of
dium quantities. One might even say that we re-
events on a precisely realistic geometry for later
quire new physically based, probably case-depen-
ensemble averaging, i.e., for Monte Carlo treat-
dent parameters and signal characterization
ment as meant here. In addition, because they
specifically to simplify our task to the point of
proceed on the basis of a photon transport equa-
tractability.
tion and not Maxwell's equation, coherent inter-
actions as needed for dense media are absent; this
may suffice for some vegetation applications.
APPLICATIONS
Tsang and his co-workers (1992b) are also evi-
dently applying Monte Carlo WT analyses to ob-
The following is suggested as a general scheme
tain coherent and incoherent effects for vegeta-
of the media and physical features for which mod-
tion that, though not dense, is clustered.
els are sought. In each case both active and pas-
In summary, we should not look for volume-
sive polarimetric systems are ultimately desired.
scattering progress in the near future from the
ensemble averaging of numerical solutions of
Cultural features
Maxwell's equations over truly realistic geom-
Buildings and transportation constructions,
etries in the same manner as is occurring for sur-
urban/suburban environments, edges or
faces. Beyond this we note that pushing rough-
linear structures such as roads and wires.
ness limits using numerical solutions brings
Atmospheric applications
dangers with its opportunities. Even in the more
Active propagation and emission horizon-
8
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