diffuse, as in an aerosol, but the denser the me-
Whatever the current (evolving) limitations on
dia, the more interactions between the elements
density (volume fraction) and particle size rela-
must be included. In the limit of very dense me-
tive to wavelength, we note that important links
dia, with a great deal of contact between elements
have been made between simulations based on
or with continuous, tortuous intermixed phases,
dense media theory and Monte Carlo calculations,
it may no longer be a promising approach to re-
in effect verifying each. The latter have the ad-
gard the medium in terms of contributions from
vantage that true solutions of Maxwell's equa-
discrete elements.
tions may be obtained, including coherent interac-
The distinction between continuous medium
tions, and no adjustable parameters are intro-
and discrete scatterer theory is somewhat mis-
duced. Dense media theory calculations have
leading, in that the former can generally deal with
agreed well with Monte Carlo calculations of ex-
discrete scattering elements as well. One merely
tinction rate for volume fractions up to 25% (Tsang
specifies a discrete permittivity distribution and
et al. 1992b); they have also agreed well with
performs much the same computational motions
laboratory data where input parameters were de-
as for a continuous distribution, ending up with
rived from measurement (Wen et al. 1990, Ishi-
essentially the same statistical descriptors for the
maru and Kuga 1982 ).
medium as under the discrete scatterer approach.
In the radiative transport (RT) approach it is
In either case, the statistical medium description
assumed that interacting field components are
must be integrated into a set of equations cover-
sufficiently uncorrelated so that addition of power,
ing the medium as a whole. The principal ap-
as opposed to addition of fields, holds. Thus, an
proaches have been via analytical wave theory
energy balance equation is formulated in terms of
and radiative transport theory.
specific intensity, in scalar form for completely
The perceived difficulty of applying wave
unpolarized fields or in vector form in the sense
theory (WT) to an extremely complex mixture
that the (modified) Stokes vector is the depen-
has appalled some authors, to the extent that the
dent variable when polarization is important.
WT approach is sometimes dismissed out of hand.
Maxwellian underpinnings may reside in dif-
Nevertheless, the brave beginnings of its recent
fraction and interference effects expressed for the
incarnations in the early '80s have been followed
behavior of constituent elements in the medium.
by the undaunted into formulations of ever greater
However, these are ultimately swallowed up
sophistication (e.g., Nghiem et al. 1990, Nghiem
through the phase and extinction functions (ma-
1991). In principle, WT can include all multiple
trices) in the intensity-based governing equation.
scattering, diffraction, and interference effects. In
Comprehensive introductions to RT are provided
practice, it is limited by the difficulty of formulat-
by a number of recognized sources (Chandra-
ing the characteristics, behavior, and distribution
sekhar 1960, Ishimaru 1978, Tsang et al. 1985,
of constituent elements and then solving equa-
Ulaby et al. 1986). As in the WT approaches, re-
tions incorporating their interactions. Green's
searchers have built increasingly complex and di-
function-based integral equations are generally
verse scattering elements into the medium for-
used. Perhaps the simplest method for making
mulations, treating increasingly complex shapes
the integral equations tractable is the Born ap-
and distributions of scattering elements and me-
proximation, which has been used for increas-
dia. Although complex media descriptions may
ingly complex configurations in recent years, e.g.,
be built into RT formulations more easily than for
an isotropicanisotropic three-layer system with
WT, numerical solution can still be daunting. It-
rough surface (Borgeaud et al. 1986). The Rytov
erative methods are sometimes used, though they
and Foldy approximations are also applied to ex-
are suited primarily to sparse or weakly scatter-
tend the range of validity or tractability. The dis-
ing media and are rarely carried beyond the sec-
torted Born approximation extends the WT den-
ond order. Kuga (1991) reports a system that pro-
sity and fluctuation strength range further,
vided third- and fourth-order iterations and
particularly in conjunction with strong fluctua-
compared well with "exact" numerical solutions
tion theory. Recently, dense media formulations
obtained with the discrete ordinate method. While
have been advanced for media with particle-size
the method requires low density and spherical
distributions, employing the quasi-crystalline ap-
particles, it can deal with large particles. Rough
proximation (QCA) and QCA with coherent po-
surfaces have been included in WT approaches
tential (QCACP) and pair distribution functions
simply by incoherently adding the independently
(Ding and Tsang 1991, Tsang and Kong 1992).
calculated volume and surface scatter (e.g.,
3
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