C.-H. Wang, Y.-Y. Feng and K. Yue et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 225 (2019) 166–179
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coupled atmosphere and ocean systems and further considered the
inelastic scattering of the ocean waters, which is of much signifi-
cance to the polarized remote sensing. Wang et al. [42] studied the
time-dependent polarized radiative transfer through a polystyrene-
microsphere medium, which contributed significantly to the un-
derstanding of polarization propagation in biological tissues. The
above mentioned applications of the polarized information demon-
strate the necessity of further investigation of polarized radiative
transfer through a participating medium.
For the medium exposed to an external collimated irradiation
considered in this paper, the Stokes vector I is divided into two
parts
I = Ic + Id,
(4)
where Ic is the Stokes vector of the collimated radiation and Id is
the Stokes vector of the diffused radiation.
The collimated Stokes vector within the medium decreases ex-
ponentially and obeys the Beer’s law
In practical applications, radiative transfer problems usually in-
volve complex media with irregular geometries and inhomoge-
neous medium properties, such as the organic tissues. Some re-
searchers have studied the radiative transfer problems in irregular
media [43–48], but they did not take the polarization effect into
account. Currently, most of the investigations on polarized radia-
tive transfer are limited to plane-parallel or regular-shaped me-
dia. Nowadays, the expanding applications of polarized radiative
transfer to the fields of biomedical and optical imaging make it an
emerging work to study the polarized radiative transfer in irreg-
ular participating media. This paper aims at solving the polarized
radiative transfer problems in complex media exposed to an ex-
ternal illumination. The numerical method used in this work is a
discontinuous finite element method (DFEM). It has been recently
developed for polarized radiative transfer [49,50], but the previ-
ous works only solve the polarized radiative transfer problems in
the square and cubic media with structured grids. In this paper,
by applying the unstructured meshes for spatial discretization, the
DFEM is extended to study the polarized radiative transfer in com-
plex media.
ꢀ · ∇Ic(r, ꢀ) = −β(r)Ic(r, ꢀ),
(5)
then it can be analytically obtained by
ꢁ
ꢂ
ꢀ
s
ꢃ
ꢄ
I
r, ꢀ = I exp
−
β r(s) ds δ ꢀ − ꢀ0
,
(6)
(
)
(
)
c
0
0
where I0 is the Stokes vector of the incident irradiation, s is the
propagating distance of the radiation light when reaching location
r, and ꢀ0 is the incident direction of the external irradiation.
the diffuse radiation can be written as
ꢀ · ∇Id(r, ꢀ) + β(r)Id(r, ꢀ) = Sd(r, ꢀ),
where Sd is the source term for calculating the diffuse radiation,
(7)
expressed as
ꢀ
κs
4π
κs
4π
Sd(r, ꢀ)=
Z(ꢀꢀ → ꢀ)Id(r, ꢀ)dꢀꢀ +
Z(ꢀ0 → ꢀ)Ic(r, ꢀ0 ).
(8)
4π
Considering the specular and diffuse reflection, the boundary
condition is written as
The outline of this paper is as follows. In Section 2, the spe-
cific treatment of the governing equation for polarized radiative
transfer induced by an external collimated illumination, the space
discretization with triangular elements, and the shape function
constructed on each element are presented. In Section 3, the cor-
rectness of the DFEM procedure is validated. Polarized radiative
transfer problems in a complex media by using the DFEM are
solved and discussed in Sections 4, followed by the conclusions in
ꢀ
1
Id(rw, ꢀ)=RsId(rw, ꢀꢀꢀ) +
RdId(rw, ꢀꢀ)|nw · ꢀꢀ|dꢀꢀ,
ꢀ
nw·ꢀn >0
π
(9)
where the subscript ‘w’ denotes variables defined on the wall, n
is the unit outward normal vector, Rs and Rd are the specular and
diffuse reflection matrices, respectively.
2.2. DFEM discretization
2. Governing equation and the DFEM discretization
In this section, the DFEM is applied to solve Eq. (7) to obtain
the diffuse Stokes vector component distributions. Without any
ambiguity, the subscript ‘d’ in Eq. (7) is omitted in the following
expressions for simplicity. The continuous computational domain
(see Fig. 1a) is first divided into a collection of non-overlapped tri-
spect to a weight function w (r, ꢀ) and get
2.1. Governing equation
The polarization state of a radiation bundle is denoted by Stokes
vector I = (I, Q, U, V)T, in which the component I is the intensity,
the other components Q, U, and V describe the polarization state
of the radiation light, and the symbol ‘T’ denotes the matrix trans-
formation. The governing equation of the polarized radiative trans-
fer in an absorbing-scattering medium, corresponding to the Stokes
vector components, is the vector radiative transfer equation (VRTE)
which is written as
ꢀ
ꢀ
w (r, ꢀ)ꢀ · ∇IdAdꢀ
ꢁꢀ Ae
ꢀ
ꢀ
=
w (r, ꢀ)[−β(r)I(r, ꢀ) + S(r, ꢀ)]dAdꢀ,
(10)
ꢀ · ∇I(r, ꢀ) + β(r)I(r, ꢀ) = S(r, ꢀ),
(1)
ꢁꢀ Ae
where ꢀ is the radiation direction, r is the coordinate, β(r) is the
In the DFEM spatial discretization, the discrete elements are
assumed to be separated on the inner boundaries (see Fig. 1c).
The field variables or their derivatives are therefore discontinuous
across the neighboring boundaries. Consider the variable disconti-
nuity and apply Gauss divergence theory twice to Eq. (10), one gets
extinction matrix given by
⎛
⎞
β(r)
0
⎜
⎝
β(r)
⎟
⎠
β(r) =
,
(2)
β(r)
0
β(r)
ꢀ
ꢀ
ꢀ
ꢀ
where β(r) is the extinction coefficient of the medium. The source
w (r, ꢀ)ꢀ · ∇IdAdꢀ+
w(r, ꢀ)–I n · ꢀdꢂdꢀ
ꢂ
ꢁꢀl Ae
ꢀ
ꢁꢀl
ꢀ
κs
4π
Z(ꢀꢀ, ꢀ)I(r, ꢀꢀ)dꢀꢀ,
(3)
=
[−β(r)I + S(r, ꢀ)]w(r, ꢀ)dAdꢀ,
(11)
S(r, ꢀ) =
ꢁꢀl Ae
4π
where κs is the scattering coefficient of the medium, and Z is the
where n denotes the unit outward normal vector of the element
boundary in Fig. 1a, ꢂ denotes the element boundary(see Fig. 1d),
scattering phase matrix which is calculated from the Appendix A.
Wang, Huan
