tering should normally be significant in dense
fects. Hopes are even more dim for development
media. Thus, computational maneuvers that rely
of an "equivalent" CRT formulation that lacks
on weak or insignificant multiple scattering are
detailed realism in the mechanisms incorporated
unlikely to apply. Using the DMRT approach,
but that also somehow reliably produces correct
Wen et al. (1990) show the convex relation be-
behavior.
tween extinction and particle density that has been
In principle, there is no problem in including
observed in measurements (e.g., Ishimaru and
different species as well as different sizes of par-
Kuga 1982, Gibbs and Fung 1990).
ticles in DMRT formulations. This suggests that
DMRT simulations show a relation between
water inclusions might be modeled as small ring-
interparticle correlation and size distribution ef-
shaped or otherwise nonspherical inclusions at
fects. While Wen et al. (1990) show reasonable
points of particle contact. However nonspherical
agreement with snow backscatter measurements,
particles interfere with the established strategies
they assume a uniform grain size. More reason-
for the necessary PDF-related computations. Be-
able physical parameters produce good agreement
cause the liquid water inclusions are small, their
when a grain size distribution is employed (West
shape may not be so important and they might be
et al. 1993). Most simulations up through the work
modeled as many very small spheres concentrated
of West et al. assume Rayleigh scattering, i.e., ka <
somehow in the smaller crevices of the medium.
1, which limit applicability to the bottom of the
Ding et al. (1993) have produced "sticky" particle
MMW frequency range. (This limitation is not
formulations, by which specified degrees of par-
inherent in the basis of DMRT.) In this regime, for
ticle clustering can be achieved in the numerical
independent scattering, the scattering cross sec-
construction of the medium. Beyond its possible
tion increases as the sixth power of particle ra-
relevance for the inclusion of liquid water, this
dius a. Thus, it should not be surprising that
sort of formulation has shown promise in pro-
DMRT calculations based on a Rayleigh distribu-
ducing more realistic morphology and scattering
tion of sizes produces backscatter on the order of
behavior (Shih et al. 1995). Extreme degrees of
15 dB greater than that from a uniform distribu-
anisotropic "stickiness" might successfully treat
tion, with size equal to the mode radius in the
some of the anisotropic particle effects mentioned
distribution. A relatively small number of larger
above. In any case, ongoing work at MIT seeks
particles can produce a disproportionately great
ways to generate dense distributions of spheroi-
effect. At the same time, the increase in scattering
dal particles, together with their correlation/PDF
by the larger particles in a nonuniform distribu-
functions. As in nature, some degree of average
tion is not nearly as great as one would expect
particle orientation is forced by the density of the
from the sixth-power law. The point is that the
medium.
close proximity of the smaller particles to the
The particle basis for DMRT formulations here-
larger particles reduces the overall correlation
tofore imposes considerable computational bur-
length of the medium, which is expressed through
den as the particles become large relative to the
the PDF.
wavelength. Restricting particles to a spherical
These results suggest that both dense-medium
shape facilitates both PDF and scattering deter-
and particle-size-distribution effects must be
mination as dimensions increase, inasmuch as
treated, and must be treated together, for high-
Mie scattering schemes may be employed. How-
fidelity simulations. For a particular application,
ever, to date most formulations and computa-
i.e., to snow, one might hope that it could be
tions have been performed assuming particles that
possible to develop an "equivalent" uniform par-
are small enough to warrant the assumption of
ticle size. That is, if snow tends to come in various
simpler Rayleigh scattering. When nonspherical
particular micromorphologies with particular at-
grains are treated, however, the Mie scattering
tendant particle distributions, it might be pos-
formulations are inapplicable. When particles are
sible to develop systematically a corresponding
relatively large, the details of their shape have
fictional uniform particle size that would give the
more effect on overall scattering behavior and
correct results in simulations. This particle size
one has less liberty to just assume a somehow
would in general be different from the average
equivalent sphere. In response to this, work at
particle size. It is questionable that such a quan-
MIT has also been directed into correlation func-
tity could be implemented so as to portray accu-
tions for "bicontinuous" nongranular geometries.
rately all effects of interest, e.g., polarization ef-
In the end, this may be both simpler and more
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