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Contents lists available at SciVerse ScienceDirect
Journal of Quantitative Spectroscopy &Radiative Transfer
Journal of Quantitative Spectroscopy & Radiative Transfer ]
(]]]]) ]]]–]]]
0022-40http://d
n CorrE-m
PleasQuan
journal homepage: www.elsevier.com/locate/jqsrt
Intensity and polarization of dust aerosols over
polarizedanisotropic surfaces
K.N. Liou a, Y. Takano a,n, P. Yang b
a Joint Institute for Earth System Science and Engineering,
Department of Atmospheric and Oceanic Sciences, University of
California,Los Angeles, CA 90095, USAb Department of Atmospheric
Sciences, Texas A&M University, College Station, TX 77845,
USA
a r t i c l e i n f o
Article history:Received 1 February 2013Received in revised
form10 May 2013Accepted 13 May 2013
Keywords:Surface albedo and polarizationDust aerosolsPhase
matrix elementsDegree of linear polarizationBidirectional
reflectanceRemote sensing
73/$ - see front matter & 2013 Elsevier
Ltd.x.doi.org/10.1016/j.jqsrt.2013.05.010
esponding author. Tel.: +1 310 794 9832; faail address:
[email protected] (Y. Taka
e cite this article as: Liou KN, et al.t Spectrosc Radiat
Transfer (2013),
a b s t r a c t
The effect of surface polarization on the intensity and linear
polarization patterns ofsunlight in an atmosphere containing a dust
aerosol layer is investigated by means of theadding principle for
vector radiative transfer in which the surface is treated as a
layerwithout transmission. We present a number of computational
results and analysis forthree cases: Lambertian (unpolarized
isotropic), polarized isotropic, and polarizedanisotropic surfaces.
An approach has been developed to reconstruct anisotropic 2�2phase
matrix elements on the basis of bidirectional-reflectance and
linear-polarizationpatterns that have been measured from
polarimeters over various land surfaces. The effectof surface
polarization on the simulated intensity patterns over a dust layer
is shown to benegligible. However, the differences in the simulated
linear polarization patterns betweencommonly assumed Lambertian and
polarized anisotropic cases are substantial for dustoptical depths
between 0.1 and 0.5 and for surface albedos of 0.07 and 0.4,
particularly inbackward directions.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Intensity and polarization of sunlight reflected fromaerosols
and clouds have been shown to bear a strongimprint of their size,
shape, optical depth, and otheroptical properties. Perhaps the most
intriguing resultsassociated with the use of polarization data for
inferringparticle size and optical properties have been found in
thestudy of Venus' cloud deck by the French astronomer Lyot[1]. In
a subsequent work, Hansen and Hovenier [2]performed an extensive
multiple scattering investigationand determined from the linear
polarization data that theVenus cloud layer is composed of
spherical particles(rainbow feature) having a mean radius of �1.05
μm anda real refractive index of �1.44 at a wavelength of 0.55
μm.
All rights reserved.
x: +1 310 794 9796.no).
Intensity and polarizathttp://dx.doi.org/10.10
The NASA Glory mission (Mishchenko et al. [3]) had thesimilar
idea of using the reflected spectral polarization ofsunlight from
the Earth's atmosphere without clouds todetermine the optical and
thermodynamic properties ofaerosols, including absorbing black
carbon and dustparticles.
The NASA Glory spacecraft unfortunately failed to reachorbit
after liftoff on March 4, 2011. An attempt, however,has been
initiated to reengage the polarization instrument,referred to as
Aerosol Polarimetric Sensor (APS), to collectpolarimetric
measurements along satellite ground track. Inorder to determine the
physical and chemical properties aswell as the spatial and temporal
distributions of aerosols,APS will measure polarized reflected
sunlight in thewavelength range of 0.4–2.4 μm. Because aerosol
opticaldepths are usually small (τo�1), it appears that the
effectof surface reflection cannot be neglected, especially
overbright land surfaces. Furthermore, the effect of
surfacepolarization properties with reference to the nature of
ion of dust aerosols over polarized anisotropic surfaces.
J16/j.jqsrt.2013.05.010i
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K.N. Liou et al. / Journal of Quantitative Spectroscopy &
Radiative Transfer ] (]]]]) ]]]–]]]2
anisotropy on polarization signals at the top of the atmo-sphere
has not been carefully considered in radiativetransfer simulations
involving aerosols and thin cirrusclouds, although a number of
recent studies haveaddressed the effect of surface polarization
[4–6].
The objective of this paper is to explore the
surfacepolarization effect on the simulated polarization patternsin
an atmosphere containing dust particles. Specifically,we have
extended the surface polarization model bytaking into consideration
recent observations of the polar-ized bidirectional reflectance
from land surfaces. More-over, the database for the
single-scattering of dust aerosolsdeveloped by Meng et al. [7] has
been used, along with theadding method for radiative transfer of
polarized light tocompute the reflected intensity and polarization
patternsat the top of a dust layer to understand the significance
ofsurface polarization.
This paper is organized as follows. First, we present
thesingle-scattering properties of randomly oriented dustparticles
using a tri-axial ellipsoidal model in Section 2.This is followed
by a discussion in Section 3 on the Stokesparameters and phase
matrix in the content of vectorradiative transfer using the
adding/doubling approach andthe definition of surface reflection
matrix and its elementsunder various approximations. We have also
presented anumber of computational results and analysis for
casesinvolving Lambertian (unpolarized isotropic),
polarizedisotropic, and polarized anisotropic surfaces. For the
lastcase, an approach has been developed to reconstruct 2�2phase
matrix elements on the basis of the bidirectional-reflectance and
linear-polarization patterns that have beenmeasured over various
land surfaces.
2. Optical properties of dust aerosols
Airborne mineral dust originates primarily in desertand
semi-arid regions and is globally distributed.
Accuratedetermination of its single-scattering properties is
funda-mental to quantifying aerosol radiative forcing and
criticalto developing appropriate remote sensing techniques forthe
detection of its size, shape, and composition. Electronmicroscopic
images reveal that mineral dust particles arealmost exclusively
nonspherical and have irregular shapeswith no specific habits. Due
to technical difficulties,experimental determinations of the
extinction efficiency,single-scattering albedo and scattering phase
matricesaround the forward and backward scattering directionshave
not been solely determined from measurements.Also, measurements are
usually conducted at visiblewavelengths with a small number of dust
samples. Theapplicability of experimental approaches to the study
ofthe single-scattering properties of dust particles through-out
the entire solar and thermal infrared spectra cannot becarried out
in practical terms.
Bi et al. [8] investigated the single-scattering propertiesof a
tri-axial ellipsoidal model by introducing an additionaldegree of
morphological freedom to reduce the symmetry ofspheroids. They
demonstrated that the optical propertiescomputed from the
ellipsoidal model with optimallyselected particle shapes and their
weightings more closelymatched laboratory measurements.
Additionally, the results
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
computed from the ellipsoidal model fit measurementsbetter than
the spheroidal model, particularly in the caseof the phase matrix
elements associated with polarization.Meng et al. [7] developed a
database for the single-scattering properties and phase matrix
elements for dustparticles assuming tri-axial ellipsoids based on
the compu-tational results from a combination of the T-matrix
method[9], the discrete dipole approximation [10], and an
improvedgeometric optics method [11,12]. The database covers
var-ious aspect ratios and size parameters ranging from Rayleighto
geometric optics regimes.
In the following, we discuss the scattering phase matrixfor dust
aerosols. If no assumption is made about particleorientation, the
“scattering phase matrix” for a set ofnonspherical particles
contains 16 elements, Pij (i, j¼1–4), and can be expressed as
follows:
P¼
P11 P12 P13 P14P21 P22 P23 P24P31 P32 P33 P34P41 P42 P43 P44
266664
377775: ð1Þ
However, if nonspherical particles are assumed to berandomly
oriented in space and their mirror images haveequal numbers such
that the reciprocity principle can beapplied for incoming and
outgoing light beams, the scat-tering phase matrix can be reduced
to six independentelements in the form [13,14]
PðΘÞ ¼
P11 P12 0 0P12 P22 0 00 0 P33 −P430 0 P43 P44
266664
377775: ð2Þ
In this case, the six elements are a function of thescattering
angle Θ, defined by the directions of theincoming and outgoing
light beams.
In order to use the database presented in [7] for dustparticles
assumed ellipsoid in shape, we must specifythree semi-axis lengths
(a, b, and c) to define the shape,where a and b are two semi-minor
and semi-major axes ofthe equatorial ellipse, and c is the polar
radius. For a givencomplex refractive index m for dust, its
single-scatteringproperties are interpolated from the database for
compu-tational purposes. Also, the database includes the
kernellook-up tables Ksca/ext/abs and Kij for dust size
distributions,where the notations sca, ext, and abs denote the
scattering,extinction, and absorption, respectively, and ij
denotesscattering phase matrix elements. For a given size
dis-tribution dNx=dln x, the averaged scattering properties
aregiven by
cscaPijðΘÞ ¼ ∑m
dNxðxmÞdln x
KijðΘ; xmÞ; ð3aÞ
csca=ext=abs ¼∑m
dNxðxmÞdln x
Ksca=ext=abs; ð3bÞ
where xm is the center of the mth bin and the sizeparameter x is
defined by 2πc/λ, where λ is the wavelengthof incident light.
ion of dust aerosols over polarized anisotropic surfaces.
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3. The adding method for vector radiative transferincluding
surface polarization
3.1. Background
For the transfer of a light beam including
polarizationcontribution, we must consider the full Stokes
parametersor Stokes vector [15]. A set of equations governing
thediffuse reflection and transmission matrices R(μ, μ0, ϕ–ϕ0)and
T(μ, μ0, ϕ–ϕ0) for the adding of two homogeneouslayers with
vertical optical depths τa and τb have beenpresented in prior
literature [16–18]. The combined reflec-tion and transmission
matrices can be expressed as fol-lows:
Rab ¼ Ra þ expð−τa=μÞUþ TnaU; ð4aÞ
Tab ¼ expð−τb=μÞDþ Tb expð−τa=μ0Þ þ TbD; ð4bÞ
where D and U correspond to the downward and upwardStokes
parameters (I, Q, U, V) at an interface between thetwo layers, μ
and μ0 are the zenith angles for outgoing andincoming light beams,
Tb is the transmission matrix forlayer b, and Tna denotes the
transmission matrix for layer awhen the light beam comes from
below. Rnab and T
n
ab can becomputed from a scheme analogous to Eqs. (4a) and
(4b).In adding equations, the product of two functions
impliesintegration over an appropriate solid angle so that
allpossible multiple-scattering contributions can beaccounted for.
Moreover, for efficient computations, it isadvantageous to expand
the phase matrix and Stokesparameters in the Fourier series in
terms of the azimuthalangle ϕ−ϕ0. From Rab results, we can
determine thereflected intensity and polarization at the top of a
scatter-ing layer. To speed up the computations, we may set
τa¼τb,referred to as the doubling method. Doubling of two layersis
repeated to build up a desired optical thickness startingfrom a
thin initial layer, say on the order of 10−8. R and Tfor a thin
layer can be approximated by using the single-scattering
approximation involving the “phase matrix”given by
Zðμ; μ0;ϕ−ϕ0Þ ¼ Lðπ−i2ÞPðΘÞLð−i1Þ; ð5Þ
where P(Θ) is the “scattering phase matrix” defined inEq. (2), Θ
is the scattering angle, and i1 and i2 denote theangles between
meridian planes for the incoming andoutgoing light beams,
respectively, and the plane ofscattering. The transformation matrix
for the Stokes vectoris given by
LðχÞ ¼
1 0 0 00 cos 2χ sin 2χ 00 − sin 2χ cos 2χ 00 0 0 1
26664
37775; ð6Þ
where χ is either π−i2 or −i1. These two angles allow
thetransformation of the direction of the incident light beamto
that of the scattered light beam. In general then, thephase matrix
Z defined in Eq. (5) contains 16 elements.With the preceding
introduction, we should now defineand discuss various types of
surface reflection matrices.
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
3.2. Surface reflection matrix
In the context of the adding principle for radiativetransfer,
surface can be considered as a single layer definedby a reflection
matrix but without transmission so thatR¼A, the surface albedo
matrix, and T¼0. The 16 ele-ments of the general scattering phase
matrix P defined inEq. (1) for a surface without any assumption can
bewritten in the form [10,11]
P11 ¼12ðM2 þM4Þ þ ðM3 þM1Þ½ �;
P12 ¼12ðM2 þM4Þ−ðM3 þM1Þ½ �;
P13 ¼ S23 þ S41; P14 ¼−D23−D41;
P21 ¼12ðM2−M4Þ−ðM1−M3Þ½ �;
P22 ¼12ðM2−M4Þ þ ðM1−M3Þ½ �;
P23 ¼ S23− S41; P24 ¼−D23 þ D41;P31 ¼ S24 þ S31; P32 ¼
S24−S31;P33 ¼ S21 þ S34; P34 ¼−D21 þ D34;P41 ¼D24 þ D31; P42
¼D24−D31;P43 ¼D21 þ D34; P44 ¼ S21−S34; ð7Þ
where the terms on the right-hand side of Eq. (7) aredefined
by
Mk ¼ jSkj2; k¼ 1−4;
Skj ¼ Sjk ¼12ðSjSnk þ SkSnj Þ ¼ ReðSkSnj Þ; j; k¼ 1−4; ð8Þ
−Dkj ¼Djk ¼i2ðSjSnk−SkSnj Þ ¼ ImðSkSnj Þ; j; k¼ 1−4;
where * denotes the complex conjugate value. The terms Sj(j¼1,
2, 3, 4) are the amplitude functions which transformthe incident
electric field (El0, Er0) to the scattered electricfield (El, Er)
defined by
ElEr
" #∝
S2 S3S4 S1
" #El0Er0
" #: ð9Þ
We have presented detailed definitions of the scatter-ing phase
matrix elements which are required foranalysis below.
If the normal direction of a point on the surface withrespect to
the tangent plane of that point defined by anangle, say ξ, is
randomly distributed, and the incident andreflected light beams can
be reversed [19], we may expressthe phase matrix similar to Eq. (2)
such that it containssix independent elements, i.e. P21 ¼ P12 ¼
ðM2−M1Þ=2 andP43 ¼−P34 ¼D21:
Consider an ocean surface in which only specularreflection
occurs. We should have El¼S2El0 and Er¼S1Er0so that S3¼S4¼0. The
following phase matrix elements aredefined by
P11 ¼ P22 ¼ ðM2 þM1Þ=2; P33 ¼ P44 ¼ S21; P43 ¼ −P34
¼D21:ð10Þ
ion of dust aerosols over polarized anisotropic surfaces.
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Thus, the scattering phase matrix can be expressed as
P¼
P11 P12 0 0P12 P11 0 00 0 P33 −P430 0 P43 P33
266664
377775: ð11Þ
It follows that there are four independent phase matrixelements
in this case. Furthermore, if absorption does notoccur, the term
D21¼0 (see the last line in Eq. (8)), i.e.P43¼0 so that only three
independent elements remain,which is the same as those presented by
Kattawar andAdams [20]. Similar to Eq. (5), the scattering
phasematrices associated with the preceding conditions mustbe
transformed from a frame of reference fixed to an oceansurface to
one fixed to space in the form
A¼ Lðπ−ηrÞPLð−ηiÞ; ð12Þ
where the angles ηi and ηr correspond to i1 and i2 asdefined in
Eq. (6). Thus, the matrix A is equivalent to thematrix Z defined in
Eq. (5).
For a surface which reflects the incoming irradianceaccording to
Lambert's law, its reflection matrix is given by
A¼ αs
1 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775; ð13Þ
where αs (0–1) is the surface albedo, conventionally definedas
the normalized upward flux (W/m2). This is the case forthe
Lambertian (“unpolarized isotropic”) surface.
In terms of the amplitude coefficients, we can setS1 ¼ Rr ; S2 ¼
Rl ; where Rl and Rr are the Fresnel reflectioncoefficients given
by Eq. (5.3,23a) in Liou [11]. The matrixelements in Eq. (11) can
then be expressed in the form
M1 ¼ jRr j2; ð14aÞ
M2 ¼ jRlj2; ð14bÞ
S21 ¼ ðRlRnr þ RrRnl Þ=2¼ ReðRlRnr Þ; ð14cÞ
D21 ¼ iðRlRnr−RrRnl Þ=2¼ −ImðRlRnr Þ: ð14dÞAs an example, in
order to see the surface polarization
effect, we may set Rl ¼ffiffiffiffiffiffiffi0:9
pand Rr ¼−
ffiffiffiffiffiffiffi1:1
p. It follows that
M1¼1.1 and M2¼0.9 such that (M1+M2)/2¼1, (M2−M1)/2¼−0.1, S21
¼−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:9�
1:1
p≅−0:995, and D21¼0. In this
manner, we have –P12/P11¼0.1. The reflection matrix
cansubsequently be expressed in the form
A¼ αs
1 0 −0:1 0−0:1 1 0 00 0 −0:995 00 0 0 −0:995
26664
37775: ð15Þ
In comparison with Eq. (13), the conventional defini-tion of
unpolarized isotropic surface albedo, five additionalelements are
included in the phase matrix. We refer to thiscase as “polarized
isotropic surface”. In the following wepresent a number of
computational results and analysis.
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
In Fig. 1 we first show the scattering phase matrixelements,
which were determined by interpolation fromthe database presented
in [7]. We selected an ellipsoidshape of a/c¼0.5, b/c¼0.75,
x¼2πc/λ¼4.0 and m¼1.4−i0.001 for dust aerosols. Additionally, we
have alsoemployed an equal-volume elongated ellipsoid of a/c¼0.35,
b/c¼0.4 and x¼2πc/λ¼5.56 for comparison. If weconsider the APS
1.61-μm channel, c is �1 and 1.4 μm forthe two shapes. For the APS
0.672-μm channel, c¼�0.4and 0.6 μm. These values appear to be
comparable to atypical size of dust aerosols (Ref. [22]), which
shows thepeak of the dust size distribution at �0.5 μm. Also,
thereal part of the refractive index 1.4 is comparable to thevalue
of 1.398 for mineral dust at the 1.61-μm wavelengthbased on
interpolation of the values listed in Table 4.3 ofRef. [22]. For
the calculations involving 0.672 μm, we haveused a refractive index
of 1.53−i4.28�10−3. In terms ofshape, we have used a/c¼0.5 and
0.35, and b/c¼0.75 and0.4, resulting in an approximate aspect ratio
of �1.67 and2.68. The former value is comparable to a typical
aspectratio of �1.5 for mineral dust aerosols compiled inRef. [23].
The phase function varies mildly as a functionof the scattering
angle at backscattering directions atwhich weakly polarizing effect
of dust is shown. Thedegree of linear polarization (DLP), which is
equivalentto −P12/P11, has positive values over all scattering
angles.The term P22/P11 is close to 1, while P33/P11 is close to
P44/P11. These elements vary from 1 to −1 in the scatteringangle
domain. The term P43/P11 has positive and negativevalues at small
and large scattering angles, respectively.
Fig. 2 depicts bidirectional reflectance (BR) and DLP −Q/I for
dust aerosols over an unpolarized isotropic (Lamber-tian) surface
and the artificially modified surface matrixaccording to Eq. (15),
which is polarized but isotropic. Inthe calculations, we used a
solar incident zenith angle θ0 of601 and three aerosol optical
depths τa. In the principalplane, ϕ−ϕ0¼01/1801, U and V are zero.
The top-left panelshows that BR values for dust aerosols over
desert arelarger than those over ocean for all three optical
depths. Atzenith angles θ≳301, BR is close to surface albedo (0.07
or0.4); however, for θ≲−301, its values increase substantiallydue
to grazing reflection. For the optical depth of 0.5,shown in the
bottom-left panel, the effect of grazingreflection on BR at θ≲−301
is enhanced by multiplescattering. In the case of isotropic ocean
surface, DLPdisplays peaks at θ≈−601 and 01, as illustrated in the
top-right panel. These peaks correspond to peaks of −P12/P11
atscattering angles Θ of 601 and 1201 denoted in Fig. 1. In
thebottom-right panel, positive polarization at forward direc-tions
(θ≲01, relative to the incident light beam) is strength-ened,
whereas negative polarization occurs at backwarddirections (θ ≳01)
associated with multiple scattering. Asshown in the top-right
panel, positive polarizationincreases by �10% due to the addition
of 10% positivesurface polarization in the case of optically thin
dust layer(τa¼0.1). In the bottom-right panel, positive surface
polar-ization effect is seen to be suppressed by a thicker
dustlayer (τa¼0.5), leading to reduced differences in DLPbetween
unpolarized and polarized isotropic surfaces overboth desert and
ocean. Note that in order to highlightsurface polarization effect,
other potential effects such as
ion of dust aerosols over polarized anisotropic surfaces.
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10−2
10−1
100
101
0 60 120 180
Pha
se F
unct
ion
P11
−1.0
−0.5
0.0
0.5
1.0
0 60 120 180
P33
/P11
0.6
0.8
1.0
0 60 120 180
P22
/P11
−0.5
0.0
0.5
1.0
0 60 120 180
−P12
/P11
−1.0
−0.5
0.0
0.5
1.0
0 60 120 180
P44
/P11
−0.6
−0.3
0.0
0.3
0.6
0 60 120 180P
43/P
11
Scattering Angle (deg.)
Fig. 1. NonzeroQ3 elements of the phase matrix for an
ellipsoidal aerosol with a/c¼0.5, b/c¼0.75, x¼2πc/λ¼4 (blue line)
and a/c¼0.35, b/c¼0.4, x¼2πc/λ¼5.56 (red line), m¼1.4−i0.001, where
a, b, and c are three semi-axes, λ is the wavelength, and m is the
index of refraction. (For interpretation ofreferences to color in
this figure legend, the reader is referred to the web version of
this article.)
K.N. Liou et al. / Journal of Quantitative Spectroscopy &
Radiative Transfer ] (]]]]) ]]]–]]] 5
Rayleigh scattering and water vapor absorption have notbeen
included in the analysis and will be a subject forfurther
study.
Now, if we neglect the P33 and P44 elements in
multiplescattering computations, the 4�4 surface reflectionmatrix
defined in Eq. (15) reduces to a 2�2 matrix inthe form
A¼ αs1 −0:1
−0:1 1
� �: ð16Þ
Using this form, the results of BR and DLP differ onlyslightly
from those presented in Fig. 2, due primarily to thefact that U and
V are zero in the ϕ−ϕ0¼0/1801 plane.
In terms of surface polarization observations, onlylinear
polarization has been measured. For example, Bréonet al. [24]
measured and modeled polarized reflectance ofbar soil and
vegetation. However, their model was unableto reproduce negative
polarization at backward directions.Wu and Zhao [25] measured
polarized BR of soil andshowed that all measured values are
positive. Morerecently, Suomalainen et al. [26] and Peltoniemie et
al.[27] conducted polarized BR measurements from vege-tated land
surfaces, soil, stones, and snow. These authorsdisplayed observed
DLP (−Q/I) patterns in two-dimensional space which can be used for
present polarized
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
radiative transfer computations. Fig. 3a and b showsexamples of
measured −Q/I of the red light reflected froma vegetated surface
[26] and from sand with dead needles[27]. The results are displayed
in a two-dimensionaldiagram in terms of two set of relevant angles
with respectto the incident solar zenith angle.
In order to use these measurements in conjunctionwith radiative
transfer in the atmosphere, we need toreconstruct phase matrix
elements from linear polariza-tion measurements to couple with the
adding method forvector radiative transfer for dust layers.
Consider theintensity Is and linear polarization −Qs measured from
apolarimeter. Because diffuse light was removed from Is andQs [26],
we should have the following matrix operation asfollows:
IsQs
" #¼
A11 A12A12 A22
" #I00
� �; ð17Þ
where I0 ðμ0;ϕ0Þ is the incident solar intensity definedby the
direction ðμ0;ϕ0Þ, so that the two phase matrixelements can be
obtained from two measurements asfollows:
A11ðμ; μ0;ϕ−ϕ0Þ ¼ Isðμ; μ0;ϕ−ϕ0Þ=I0ðμ0;ϕ0Þ; ð18aÞ
ion of dust aerosols over polarized anisotropic surfaces.
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αs = 0.07
αs = 0.4
θ0 = 60o, φ − φ0 =0/180o
Bid
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τa = 0.1
τa = 0.25
τa = 0.5
Deg
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of L
inea
r P
olar
izat
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−Q/I
polarized anisotropicunpolarized isotropic
-60 -30 0 30 60 90−90 -60 -30 0 30 60 90
Fig. 2. Bidirectional reflectance and the degree of linear
polarization −Q/I as a function of zenith angle for a layer of dust
aerosols overlying an unpolarizedLambertian (isotropic) surface and
the polarized isotropic surface given by Eq. (15) for surface
albedos of 0.07 and 0.4.
K.N. Liou et al. / Journal of Quantitative Spectroscopy &
Radiative Transfer ] (]]]]) ]]]–]]]6
A12ðμ; μ0;ϕ−ϕ0Þ ¼Qsðμ; μ0;ϕ−ϕ0Þ=I0 ðμ0;ϕ0Þ: ð18bÞ
Note that the direction of the reflected light beam
isrepresented by the two-dimensional angular functionðμ; μ0;ϕ−ϕ0Þ
for the two phase matrix elements. This isthe case which is
referred to as “polarized anisotropic.”The conventional surface
albedo cannot be defined in thiscase. Also, the third element A22
cannot be determinedfrom two measurements (BR and DLP); however, to
a goodapproximation, we may set A22� A11 (see the curve P22/P11in
Fig. 1).
By definition, A11 is equivalent to BR and −A12 repre-sents DLP
measurements displayed in Fig. 3 based onwhich we have developed an
analytical equation to
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
parameterize A12 values in the form
A12 ¼−C2θπ
� �cos ðϕ−ϕ0Þ; ð19Þ
where C is a constant, θ is zenith angle confined to [0,
π/2],and ϕ−ϕ0 is the azimuthal angle in the range of [−π, π].From
Fig. 3a and b, C is about 0.05 and 0.1. Results of
theparameterization are depicted on the right-hand side ofFig. 3a
and b, representing the two-dimensional domainconsisting of the
reflected zenith angle θ and relativeazimuthal angle ϕ−ϕ0. In the
principal plane ϕ−ϕ0¼01/1801 in the left panel of Fig. 3a, the
observed −Qs/Is ispositive, with values smaller than 0.05, at θo01
(i.e., in theplane ϕ−ϕ0¼01), almost zero at θ¼01 (i.e., at the
zenith),
ion of dust aerosols over polarized anisotropic surfaces.
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Model -A12 for Lichen
φ - φ0
θ0 30 60 90
60
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Model -A12 for Needles_Sand
φ - φ0
θ0 30 60 90
Fig. 3. Linear polarization (−Qs/Is) patterns of the red light
reflected from a vegetated surface (a) measured by Suomalainen et
al. [26] in which the solarzenith angle was from 451 to 471 and
from a sand surface with dead needles (b) measured by Peltoniemi et
al. [27] in which the incident solar zenith anglewas 551. The right
diagrams illustrate the results determined from parameterization
[Eq. (19)] in the zenith and azimuthal angle domain.
K.N. Liou et al. / Journal of Quantitative Spectroscopy &
Radiative Transfer ] (]]]]) ]]]–]]] 7
and negative, also with absolute values smaller than 0.05but
with more variability, at θ401 (i.e., in the planeϕ−ϕ0¼1801), which
were shown at the top-right panel inFig. 3 of Ref. [26]. This
behavior is approximately repro-duced in the right panel in Fig. 3
based on an analyticalmodel developed in this paper with an overall
accuracy ofabout 10% without capturing detailed features inthe
measured data. See also Figs. 3–10 in Ref. [26] andFigs. 8–11 in
Ref. [27], which show the same behavior.The parameterized Eq. (19)
is useful for the interpretationof –Q/I in the principal plane
ϕ−ϕ0¼01/1801 presented inFig. 4. Also, dependence of the observed
–Qs/Is on wave-length is relatively weak [26,27]. For this reason,
thesurface polarization (–Qs/Is) pattern at 0.67 μm shown inFig. 3
can be applied to near infrared wavelengths as well.We also note
that the BR measurements depicted in
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
Suomalainen et al. [26] and Peltoniemie et al. [27]
arecomplicated three-dimensional functions. However, theirvalues
are all close to a constant value.
Fig. 4a shows comparison of BR and DLP at the top ofdust layers
between unpolarized isotropic surface and thepolarized anisotropic
surface defined by Eq. (19) withC¼0.1, corresponding to the results
presented in Fig. 3b.Anisotropic surface polarization does not
affect BR, similarto the results shown in Fig. 2. For an optically
thin dustlayer (τa¼0.1) depicted in the top-right panel,
strongerpositive and negative polarization patterns are
shown,respectively, at forward and backward directions in
com-parison to the isotropic unpolarized surface case.
Negativepolarization at backscattering directions results from
thesurface polarization effect shown in Fig. 3b. For
τa¼0.5,displayed in the bottom-right panel, negative
polarization
ion of dust aerosols over polarized anisotropic surfaces.
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olar
izat
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−90 −60 −30 0 30 60 90−90 −60 −30 0 30 60 90−90 −60 −30 0 30 60
90−90 −60 −30 0 30 60 90
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αs = 0.4
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irect
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τa = 0.25
τa = 0.5
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inea
r P
olar
izat
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−Q/I
polarized anisotropicunpolarized isotropic
Zenith Angle θ (deg.) Zenith Angle θ (deg.)
θ0 = 60 , φ − φ0 =0/180°°θ0 = 60 , φ − φ0 =0/180°°
Fig. 4. (a) Bidirectional reflectance and the degree of linear
polarization −Q/I as a function of zenith angle for a dust layer
overlying an unpolarized isotropicsurface and the polarized
anisotropic surface defined by Eq. (19) with C¼0.1 for two surface
albedos of 0.07 and 0.4 at 1.61 μm wavelength. Rayleighscattering
contributions are included, but are negligible in this case. (b)
Bidirectional reflectance and the degree of linear polarization
−Q/I as a function ofzenith angle for a dust layer overlying an
unpolarized isotropic surface and the polarized anisotropic surface
defined by Eq. (19) with C¼0.1 for two surfacealbedos of 0.07 and
0.4 at 0.672 μm wavelength. Rayleigh scattering contributions are
included.
K.N. Liou et al. / Journal of Quantitative Spectroscopy &
Radiative Transfer ] (]]]]) ]]]–]]]8
at backward directions is reduced due to multiple scatter-ing,
which differs from those presented in Fig. 2. When C isset to 0.05
in Eq. (19), DLP values are between C¼0.1 andunpolarized isotropic
surface cases. Despite subtle differ-ences in the phase function
and −P12/P11 between the twoellipsoids displayed in Fig. 1, the
simulated BR and DLPpatterns for elongated ellipsoid are
substantially similarto those presented in Fig. 4a, particularly at
backwarddirections with reference to aerosol optical depthfor an
anisotropically polarized surface in comparison toan isotropic
unpolarized (Lambertian) surface. For the1.61-μm wavelength, the
Rayleigh optical depth is 0.0013,which is much smaller than aerosol
optical depths ≥0.1employed in this paper. As a result, the
simulated BR andDLP in Fig. 4a are essentially the same as in the
case inwhich Rayleigh scattering is neglected. For the
0.672-μmwavelength shown in Fig. 4b, the Rayleigh optical depth
is0.043. The simulated BR patterns remain approximatelythe same as
those depicted in Fig. 4a; however, the DLPincreases substantially
in the zenith angle region of −30–301 in the case of an albedo of
0.07. In the region of 30–601(backscattering directions), the
results are similar to thoseshown in Fig. 4a. For a higher albedo
of 0.4, the resultsdisplayed in Fig. 4a and b are similar.
Please cite this article as: Liou KN, et al. Intensity and
polarizatQuant Spectrosc Radiat Transfer (2013),
http://dx.doi.org/10.10
4. Concluding remarks
We investigate surface polarization effect on the simu-lation of
intensity and linear polarization patterns abovea dust aerosol
layer using the adding principle for vectorradiative transfer,
which represents a first step to under-stand the anisotropic
surface effects on polarization pat-terns of dusty atmospheres. In
the formulation, surface isconsidered to be a layer defined by a
reflection matrix butwithout transmission. Dust particles are
assumed to berandomly-oriented tri-axial ellipsoids based on which
thescattering phase matrix consists of six independent ele-ments.
In the analysis of surface reflection matrix, weconsider three
surface types: Lambertian (unpolarizedisotropic, commonly assumed),
polarized isotropic, andpolarized anisotropic. We focus on linear
polarization thathas been observed over various land surfaces, and
developan approach to invert BR and DLP measured values toobtain
2�2 phase matrix elements that are required forvector radiative
transfer calculations.
The surface polarization effect on the simulated inten-sity
patterns over a dust layer is shown to be negligible.However,
differences in the simulated linear polarizationpatterns between
commonly assumed Lambertian and
ion of dust aerosols over polarized anisotropic surfaces.
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polarized anisotropic cases are substantial for dust
opticaldepths between 0.1 and 0.5 and covering surface albedofrom
0.07 to 0.4, particularly in backscattering directionsrelative to
the Sun's position. In order to make precisespectral polarization
calculations to interpret measure-ments from space in the UV,
visible, and near infraredwavelengths for the purposes of
determining the size,shape, and single-scattering albedo of various
types ofaerosols, including black carbon, it is critically
importantto account for the surface polarization effect: a
conclusionin line with the recent work on the polarimetric
modelingof surface properties and the polarimetric retrievals
ofaerosol properties over land [28–30].
Uncited reference
[21].
Acknowledgments
This research was supported by Subcontract S100097from the Texas
A&M Research Foundation, which is spon-sored by the NASA under
Grant NNX11AK39G and by theNational Science Foundation under Grant
AGS-0946315.
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ion of dust aerosols over polarized anisotropic surfaces.
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Intensity and polarization of dust aerosols over polarized
anisotropic surfacesIntroductionOptical properties of dust
aerosolsThe adding method for vector radiative transfer including
surface polarizationBackgroundSurface reflection matrix
Concluding remarksUncited referenceAcknowledgmentsReferences