3 Journal of Quantitative Spectroscopy & Radiative Transferatmos.ucla.edu/~liougst/Group_Papers/Liou_JQSRT_2013b.pdf · 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

Q1

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

www.elsevier.com/locate/jqsrthttp://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010mailto:[email protected]://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

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.


http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

K.N. Liou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 3

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.


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



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61


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.


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



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

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.)


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


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



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

−90

αs = 0.07

αs = 0.4

θ0 = 60o, φ − φ0 =0/180o

Bid

irect

iona

l Ref

lect

ance

Zenith Angle θ (deg.)

τa = 0.1

τa = 0.25

τa = 0.5

Deg

ree

of L

inea

r P

olar

izat

ion

−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.


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


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),



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

Model -A12 for Lichen

φ - φ0

θ0 30 60 90

60

240

30

210

90

180

330

150

300

120

270

0

0

−0.0

4

0.02

0.04

−0.0

2

30

210

60

240

90

270

120

300

150

330

1800

−0.0

9

−0.0

5

−0.0

1

0.01

0.09

0.05

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.


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


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



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

53

55

57

59

61

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

αs = 0.07

αs = 0.4

Bid

irect

iona

l Ref

lect

ance

τa = 0.1

τa = 0.25

τa = 0.5

Deg

ree

of L

inea

r P

olar

izat

ion

−Q/I


0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

−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

αs = 0.07

αs = 0.4

Bid

irect

iona

l Ref

lect

ance

τa = 0.1

τa = 0.25

τa = 0.5

Deg

ree

of L

inea

r P

olar

izat

ion

−Q/I


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.


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.


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



1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

Q2


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.

References

[1] Lyot B. Recherches sur la polarisation de la lumière des planètes etde quelques substances terrestres. Ann Obs Paris (Meudon) 1929;8.161 p. [available in English as NASA TTF-187, 1964].

[2] Hansen JE, Hovenier JW. Interpretation of the polarization of Venus.J Atmos Sci 1974;31:1137–60.

[3] Mishchenko MI, Cairns B, Kopp G, Schueler CF, Fafaul BA, Hansen JE,et al. Accurate monitoring of terrestrial aerosols and total solarirradiance: introducing the Glory mission. Bull Am Meteorol Soc2007;88:677–91.

[4] Waquet F, Goloub P, Deuzé J-L, Léon J-F, Auriol F, Verwaerde C, et al.Aerosol retrieval over land using a multiband polarimeter andcomparison with path radiance method. J Geophys Res 2007;112:D11214. http://dx.doi.org/10.1029/2006JD008029.

[5] Waquet F, Cairns B, Knobelspiesse K, Chowdhary J, Travis LD, SchmidB, et al. Polarimetric remote sensing of aerosols over land. J GeophysRes 2009;114:D01206. http://dx.doi.org/10.1029/2008JD010619.

[6] Diner DJ, Xu F, Martonchik JV, Rheingans BE, Geier S, Jovanovic VM,et al. Exploration of a polarized surface bidirectional reflectancemodel using the ground-based multiangle spectropolarimetric ima-ger. Atmosphere 2012;3:591–619.

[7] Meng Z, Yang P, Kattawar GW, Bi L, Liou KN, Laszlo I. Single-scattering properties of tri-axial ellipsoidal mineral dust aerosols:a database for application to radiative transfer calculations. J AerosolSci 2010;41:501–12.

[8] Bi L, Yang P, Kattawar GW, Kahn R. Single-scattering properties oftriaxial ellipsoidal particles for a size parameter range from theRayleigh to geometric-optics regimes. Appl Opt 2009;48:114–26.


[9] Mishchenko MI, Travis LD, Mackowski DW. T-matrix method and itsapplications to electromagnetic scattering by particles: a currentperspective. J Quant Spectrosc Radiat Transfer 2010;115:1700–3.

[10] Draine BT, Flatau PJ. Discrete-dipole approximation for scatteringcalculations. J Opt Soc Am A 1994;11:1491–9.

[11] Yang P, Liou KN. A geometric-optics/integral-equation method forlight scattering by nonspherical ice crystals. Appl Opt 1966;35:6568–84.

[12] Yang P, Liou KN. Light scattering by hexagonal ice crystals: solutionsby a ray-by-ray integration algorithm. J Opt Soc Am A 1997;14:2278–89.

[13] van de Hulst HC. Light scattering by small particles. New York:Wiley; 1957.

[14] Liou KN. An introduction to atmospheric radiation. 2nd ed.SanDiego: Academic Press; 2002.

[15] Stokes GG. On the composition and resolution of streams ofpolarized light from different sources. Trans Cambridge Philos Soc1852;9:339–416.

[16] Liou KN. Applications of the discrete-ordinates method for radiativetransfer to inhomogeneous aerosol atmospheres. J Geophys Res1975;80:3434–40.

[17] Takano Y, Liou KN. Solar radiative transfer in cirrus clouds. Part II.Theory and computation of multiple scattering in an anisotropicmedium. J Atmos Sci 1989;46:20–36.

[18] Liou KN, Takano Y. Interpretation of cirrus cloud polarizationmeasurements from radiative transfer theory. Geophys Res Lett2002;29. http://dx.doi.org/10.1029/2001GL014613. p. 27-1–27-4.

[19] Perrin P. Polarization of light scattered by isotropic opalescentmedia. J Chem Phys 1942;10:415–26.

[20] Kattawar GW, Adams CN. Stokes vector calculations of the sub-marine light field in an atmosphere-ocean with scattering accordingto a Rayleigh phase matrix: effect of interface refractive index onradiance and polarization. Limnol Oceanogr 1989;34:1453–72.

[21] Hansen JE, Travis LD. Light scattering in planetary atmosphere. SpaceSci Rev 1974;16:527–610.

[22] d'Almeida GA, Koepke P, Shettle EP. Atmospheric aerosols.Hampton:Deepack; 1991.

[23] Ginoux P. Effects of nonsphericity on mineral dust modeling. J GeophysRes 2003;108:4052. http://dx.doi.org/10.1029/2002JD002516.

[24] Breon RM, Tanre D, Lecomte P, Herman M. Polarized reflectance ofbare soils and vegetation: measurements and models. IEEE TransGeosci Remote Sensing 1995;33:487–99.

[25] Wu T, Zhao Y. The bidirectional polarized reflectance model of soil.IEEE Trans Geosci Remote Sensing 2005;43:2854–9.

[26] Suomalainen J, Hakala T, Puttonen E, Peltoniemi J. Polarised bidirec-tional reflectance factor measurements from vegetated land sur-faces. J Quant Spectrosc Radiat Transfer 2009;110:1044–56.

[27] Peltoniemi J, Hakala T, Suomalainen J, Puttonen E. Polarised bidirec-tional reflectance factor measurements from soil, stones, and snow. JQuant Spectrosc Radiat Transfer 2009;110:1940–53.

[28] Litvinov P, Hasekamp O, Cairns B, Mishchenko MI. Reflection modelsfor soil and vegetation surfaces from multiple-viewing angle photo-polarimetric measurements. J Quant Spectrosc Radiat Transfer2010;111:529–39.

[29] Litvinov P, Hasekamp O, Cairns B. Models for surface reflection ofradiance and polarized radiance: comparison with airborne multi-angle photopolarimetric measurements and implications for mod-eling top-of-atmosphere measurements. Remote Sensing Environ2011;115:781–92.

[30] Litvinov P, Hasekamp O, Dubovik O, Cairns B. Model for land surfacereflectance treatment: physical derivation, application for bare soiland evaluation on airborne and satellite measurements. J QuantSpectrosc Radiat Transfer 2012;113:2023–39.


http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref1http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref1http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref1http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref2http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref2http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref3http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref3http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref3http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref3http://dx.doi.org/10.1029/2006JD008029http://dx.doi.org/10.1029/2006JD008029http://dx.doi.org/10.1029/2006JD008029http://dx.doi.org/10.1029/2008JD010619http://dx.doi.org/10.1029/2008JD010619http://dx.doi.org/10.1029/2008JD010619http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref6http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref6http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref6http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref6http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref7http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref7http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref7http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref7http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref8http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref8http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref8http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref9http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref9http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref9http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref10http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref10http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref11http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref11http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref11http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref12http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref12http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref12http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref13http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref13http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref14http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref14http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref15http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref15http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref15http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref16http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref16http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref16http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref17http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref17http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref17http://dx.doi.org/10.1029/2001GL014613http://dx.doi.org/10.1029/2001GL014613http://dx.doi.org/10.1029/2001GL014613http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref19http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref19http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref20http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref20http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref20http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref20http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref21http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref21http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref22http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref22http://dx.doi.org/10.1029/2002JD002516http://dx.doi.org/10.1029/2002JD002516http://dx.doi.org/10.1029/2002JD002516http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref24http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref24http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref24http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref25http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref25http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref26http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref26http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref26http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref27http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref27http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref27http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref28http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref28http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref28http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref28http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref29http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref29http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref29http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref29http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref29http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref30http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref30http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref30http://refhub.elsevier.com/S0022-4073(13)00212-4/sbref30http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010http://dx.doi.org/10.1016/j.jqsrt.2013.05.010

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

Documents

3 Journal of Quantitative Spectroscopy & Radiative Transferatmos.ucla.edu/~liougst/Group_Papers/Liou_JQSRT_2013b.pdf · 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43