JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. D15, PAGES 17,267-17,291, AUGUST 16, 2001
Sensitivity of cirrus bidirectional reflectance to vertical
inhomogeneityof ice crystal habits and size distributions
for two Moderate-Resolution Imaging Spectroradiometer
(MODIS) bands
PingYang,•,2Bo-CaiGao,3BryanA. Baum,
4 WarrenJ.Wiscombe,
2 YongX. Hu,4
Shaima
L. Nasiri,5 PeterF. Soulen,2,
6 AndrewJ.Heymsfield,
7 GregM. McFarquhar,
7
andLarryM. Miloshevich
7
Abstract. A commonassumption
in satelliteimager-based
cirrusretrievalalgorithmsis that the
radiativepropertiesof a cirruscloudmay be represented
by thoseassociated
with a specificice
crystalshape(or habit)anda singleparticlesizedistribution.However,observations
of cirrus
cloudshaveshownthatthe shapesandsizesof ice crystalsmay vary substantially
with height
within the clouds.In this studywe investigatethe sensitivityof the top-of-atmosphere
bidirectionalreflectancesfor two Moderate-Resolution
ImagingSpectroradiometer
(MODIS)
bandscenteredat 0.65 •tm and2.11 •tmto cirrusmodelscomposed
of eithera single
homogeneous
layer or threedistinct,but contiguous,layers.First, we definethe single-and
three-layercirruscloudmodelswith respectto ice crystalhabit andsize distributionson the
basisof in situreplicatordataacquiredduringthe First InternationalSatelliteCloudClimatology
Project(ISCCP) RegionalExperiment(FIRE-II), held in Kansasduringthe fall of 1991.
Subsequently,
fundamentallight-scattering
andradiativetransfertheoryis employedto
determinethe single-scattering
andthe bulk radiativepropertiesof the cirruscloud.For radiative
transfercomputations
we presenta discreteform of the adding/doubling
principlethat is
computationally
straightforward
andefficient. For the 0.65 pm band,at whichabsorption
by ice
is negligible,thereis little differencebetweenthe bidirectionalreflectancescalculatedfor the
one-andthree-layercirrusmodels.This resultsuggests
thatthe verticalinhomogeneity
effectis
relativelyunimportantat 0.65 gm. At 2.1 1 [tm the bidirectionalreflectancescomputedfor both
opticallythin (z = 1) andthick ('c= 10) cirruscloudsshowsignificantdifferencesbetweenthe
resultsfor the one-andthree-layermodels.The reflectances
computedfor the three-layercirrus
modelare substantiallylargerthanthosecomputedfor the single-layercirrus.Furthermore,our
analysisshowsthat the cirrusreflectancesat boththe 0.65 and2.11 [tm bandsarevery sensitive
to the opticalpropertiesof the smallcrystalsthatpredominatein the top layer of the three-layer
cirrusmodel. It is criticalto definethe mostrealisticgeometricshapefor the small"quasispherical"ice crystalsin the top layer for obtainingreliablesingle-scattering
parametersand
bulk radiativepropertiesof cirrus.
1. Introduction
emit infrared radiation to space.The Moderate Resolution
ImagingSpectroradiometer
(MODIS) [Kinget al., 1992] on the
Cirrusclouds
located
in theuppertroposphere
andlower recently
launched
Terraspacecraft
willenhance
thecapability
for
stratosphere
areimportant
to theEarth's
climate
[Liou,1986; monitoring
cirrus
clouds
incomparison
withprevious
generations
Stephens
etal., 1990].Theyreflectsolarradiation,
absorb
the of satelliteinstruments
suchas the advanced
very high
thermal
emission
fromtheground
andthelower
atmosphere,
and resolution
radiometer
(AVHRR).
MODIShasatotalof36bands
for studiesof land, ocean, and the atmosphere,including the
•GoddardEarth Scienceand TechnologyCenter, University of
1.38 gm band for cirrus detectionand correction[Gao and
Kaufman, 1995], the 0.65, 1.66, and 2.11 gm bands for
MarylandBaltimoreCounty,Baltimore,Maryland.
implementing cloud microphysical and optical property
2NASAGoddardSpaceFightCenter,Greenbelt,
Maryland.
retrieval
techniques [King et al., 1997] (available at
3RemoteSensingDivision,Naval ResearchLaboratory,Washington,
D.C.
http://eospso.gsfc.nasa.
gov/atbd/modistables.html),
and the 8.5,
4NASALangleyResearch
Center,Hampton,Virginia.
11, and 12 gm bands for applying infrared cloud property
5Cooperative
Institutefor Meteorological
SatelliteStudies/University
of Wisconsin,Madison, Wisconsin.
6JointCenterfor EarthSystemsTechnology,Universityof Maryland
BaltimoreCounty,Baltimore,Maryland.
7NationalCenterfor Atmospheric
Research,
Boulder,Colorado.
retrievaltechniques[Ackerraanet al., 1990]. MODIS data will
allow improved retrievals of cirrus optical and microphysical
parameters
suchas opticalthicknessof cirrusandmeaneffective
size of the ice crystalsin theseclouds.
Copyright2001 by the AmericanGeophysical
Union
Various algorithmshave been developedto retrieve cirrus
optical and microphysicalpropertiesin the past20 years [e.g.,
Papernumber2000JD900618.
Liou et al., 1990; Ou at al., 1993; Minnis et al., 1993a, 1993b;
0148-0227/01/2000JD900618509.00
Rosrow and Lacis, 1990]. They can be categorizedinto the
17,267
17,268
YANG ET AL.: CIRRUSBIDIRECTIONALREFLECTANCE
AND ICE CRYSTALS
techniques
basedon eitherinfraredemission
or solarreflection. presenta numericallystableradiativetransfermodelbasedon the
The representativealgorithm of the former is the method adding/doubling principle. The adding/doubling model is
developedby Inoue [1985] for determiningthe infrared expressedin a discrete form for calculating reflected and
emissivityof cirrus clouds on the basis of the brightness transmittedintensitiesresulting from multiple scatteringand
temperature
difference
between11 •tmand12 •tmwavelengths absorptionof cirrusclouds.
with an assumption
of implicitmeanparticlesize.An infrared
trispectralalgorithmusing the 8.5, 11, and 12 •tm bands 2. Data and Models
[Ackerman
et al., 1990,1998;Strabalaet al., 1994]witha recent
improvement
[Baumet al., 2000a,2000b]formsthebasisof an 2.1. Data
infraredretrievalalgorithmusingMODIS infraredchannels.
The
The size distributions and ice crystal habit information
representative
retrievalalgorithm
basedonsolarreflection
is that obtainedfrom two casesof replicatotmeasurementscarriedout
developed
by NakajimaandKing[1990],whouseda trispectral in Kansas during the First International Satellite Cloud
(0.75, 1.6, and2.2 •tm) methodto simultaneously
retrievethe Climatology Project (ISCCP) Regional Experiment (FIRE)
opticalthicknessand mean effectiveparticlesize for water [Starr, 1987] phaseII (hereafter,FIRE-II) are usedin this study.
clouds.This approachhasbeenappliedto the retrievalof the The balloon-borneice crystalreplicatorswere launchedat 1337
opticalthickness
andmeanparticlesizeof icecrystals
for cirrus UTC on November 25, 1991, and at 2045 UTC on December 5,
[Wielickiet al., 1990].
To develop
a reliableretrievalalgorithm
forcirrusopticaland
microphysicalproperties,it is critical to generatereliable
precalculated
lookuptablesof bidirectional
reflectance
for cirrus
clouds over a practical range of effective sizes, optical
1991.
The replicator balloons had an ascent rate of
approximately
4 m s-• whilepassing
through
thecloudlayers.
As
measuredby a radiosondeconnectedto the balloonpackage,the
cloud top temperatureon November25 was-57øC, while on
December5 the cloudtop temperature
was-65øC.
thicknesses,and viewing geometry(i.e., solar zenith angle,
The replicatorcollectsparticlesin a liquidplasticsolutionthat
viewingzenithangle,andrelativeazimuthangle).At present, coats a moving, 35-mm-wide transparent leader tape. The
most algorithmsfor retrievingcirrus optical thicknessand particlesbecomeimbeddedin the plasticcoating,and whenthe
effective size assumethat the ice crystalsare of one specific solvent in the solution evaporates, detailed ice crystal
habit,suchas spheres,
hexagonal
plates,hexagonal
columns,or impressionsand size spectraof crystalsare recordeddown to
fractalpolycrystals
[Mackeet al., 1996].In addition,a common crystalsizesof approximately10 pm. The particlesgenerallydo
assumption
is that a singlesize distributionis sufficientto not break up upon impact on the replicatortape becauseof the
determinethe scattering
properties
of the ice crystalswithinthe slow rate of ascentof the balloon.The efficiencywith which the
cirruslayer.However,observations
basedon aircraft-bome
two- replicatorcollectssmallcrystalshasbeenquantifiedtheoretically
dimensionaloptical cloud probe (2D-C) and balloon-borne andexperimentally[MiloshevichandHeymsfield,1997]. Particle
replicator
measurements
[e.g.,Heymsfield
andPlatt, 1984;Arnott sizeconcentrations
usedin this studyare adjustedto accountfor
et al., 1994; Mitchell et al., 1996a, 1996b; McFarquhar and the imperfectcollectionefficienciesof smallparticles.Analysis
Heymsfield,1996, 1997]demonstrate
the widerangeof shapes of the ice crystal data collected during the balloon's ascent
thatthe ice crystalsin cirruscloudsmay have,includingbullet throughthe cirrus provided 28 size spectrain the vertical on
rosettes,
solidandhollowcolumns,
plates,andirregularlyshaped November25 and 33 spectraon December5, with eachspectrum
aggregates.
In addition,Heymsfield
andcolleagues
haveshowed representingthe data collected from approximately100 m of
that ice crystal habits and size distributionsare vertically
inhomogeneous
in cirrusclouds[e.g.,Heymsfield
andIaquinta,
vertical ascent.
2000].
2.2 Single-ScatteringProperties
Sincesatellite-based
retrievaltechniquesessentiallycompare
The rangeof ice crystalsizesin cirruscloudsis predominately
library computationsof bidirectionalreflectancesto actual
measurements
in their implementations,it is necessaryto assess within the applicablesize parameterregime of the geometric
the effect of the vertical inhomogeneityof the ice crystalsizes opticsmethodat visible andnear-infraredwavelengths.Eachray
andshapes
withincirrusontheradiativetransfercalculations
for
generating the reflectance libraries. Our objective is to
can be localized
on the wave
front
of the incident
radiation
(electromagnetic
wave); consequently,
Snell's law andFresnel's
understandthe effect of the vertical inhomogeneityin the formulas can be applied to trace the ray propagationand the
structure of cirrus clouds on their radiative properties. We electricfield magnitude,aswell asthe polarizationconfiguration
employfundamental
scatteringand radiativetransfertheoryto associatedwith the ray. In the conventionalgeometricoptics
investigate
the bidirectionalreflectanceof cirruscloudsfor the approachfor derivingthe scatteringpropertiesof a particle,the
MODIS 0.65 and 2.11 gm bandsusingin situ crystalhabitand scattered field in the radiation zone is regarded as the
superposition of diffracted rays and FresnelJan rays. The
size distribution for a case of midlatitude cirrus on November 25,
contributionfrom ray diffraction can be determinedusing the
1991, further describedin section2.
The outlineof the paperis asfollows.The dataandmodelsare standardFraunhofer theory, whereas the contribution from
providedin section2. Section3 describes
the development
of the FresnelJanrays can be computedby the ray-tracingtechnique.
three-layerandsinglehomogeneous
cirrusmodelsaswell asthe This approachsuffersfrom severalshortcomings,as was noted
single-scattering
properties
associated
withthetwocirrusmodels. by Yang and Liou [1995]. In particular,it is assumedthat the
In section4 we presentthe differencesbetweenbidirectional extinctionefficiencyis 2 regardlessof particlesize.
To overcomethe shortcomings
of the conventionalgeometric
reflectancescomputedfor the three-layercirrusmodel and its
one-layer counterpart. Also presented in this section is a optics, Yang and Liou [1996] developeda geometric-opticssensitivitystudyregardingthe shapeeffect of the small "quasi- integral-equationapproach(hereafter referred to as GOM2).
theory,the scattered
spherically" nonspherical ice crystals on cloud reflectance. Accordingto fundamentalelectrodynamics
Conclusionsare given in section5. Finally, in AppendixA we far field can be obtainedif the tangentialcomponentsof the
YANG ET AL.' CIRRUS BIDIRECTIONAL
electricandmagneticfields on the particlesurfaceare specified.
In principle,GOM2 employsthe ray-tracingtechniqueto solve
the nearfield on the particlesurfaceandthenmapsthe nearfield
to the far field via the following rigorous electromagnetic
relationship:
eikr k2
E (r)[
_0oo
= 42r
REFLECTANCE AND ICE CRYSTALS
17,269
where//= cos0. The derivatives
in equations
(3a) and(3b) are
confinedto the facet. The geometricconfigurationassociated
with thesetwo equations
hasbeenillustratedby Cox and Munk
[1954].Sincethereis little quantitative
experimental
information
regardingthe roughness
of ice crystalsurfacesat present,the
surfaceroughness
is treatedin a similarfashionto thatof a wavy
sea surface, which can be specified by the Gram-Charlie
distribution[Cox and Munk, 1954]. If the tilt distributionof the
is azimuthally
homogeneous
(i.e.,independent
of angle
- •x[fisxH(r,)]}e-ild'r'd2?, (1) roughness
(p),the statisticalprobabilitydensityfunctionfor the condition
wherehsand•: areunitvectors
alongthenormaldirections
of thatthe slopesof a facetalongthetwo axisdirectionsaregiven
particle faces and scattering direction, respectively, and by the first-orderGram-Charlie,or a two-dimensional
(2-D)
k -- 22r/X in which•, isthewavelength
of anincidentwavein a Gaussiandistribution, is as follows:
s
vacuum. With GOM2 it is not necessaryto partition the
diffraction
and FresnelJan
contributions
to the far field.
1
A
simplified algorithm for GOM2 is employed to reduce the
computationalcost.
To computethe extinctionandabsorption
crosssectionsof ice
crystals, we use the ray-tracing technique coupled with the
followingexactelectrodynamic
relationships:
2
2 ry2
P(Zx,Zy
)=zc-•-exp[-(Z$
+Zy)/ ],
(4)
where rxis a parameterdeterminingthe magnitudeof roughness.
Values of oTM0-0.005, 0.005-0.05, 0.05-0.3 correspond
to slight,
moderate,anddeeproughness
in the single-scattering
calculation,
respectively.Furthertechnicaldetailsconcerningthe treatmentof
surfaceroughnessin the GOM2 light scatteringcomputations
were given by Yangand Liou [1998].
iEo2III[e(r')1]E(r')'E2
(r')d3?
' (2a)
v
Cabs
=[Eo-----T
k IIIei(r')E(r')'
E*(r')
d37'
v
2.3.
Radiative
Transfer
Model
(2b) Radiative transfercalculationsfor cirrus are performedusing
method.The adding/doubling
principlehas
where Cext and Cab
s are extinctionand absorptioncross the adding/doubling
sections,
respectively,
E o is theelectricfield associated
with the beenexpressedmathematicallyin a matrix form [ Twomey,1966;
incidentwave, e = er + iei is the complexpermittivity,andthe Hunt and Grant, 1966] and in an integral form [Hansen and
Travis, 1974]. A conciseformulationin a discreteform for the
adding/doubling
methodis providedin AppendixA. In Appendix
A, somenumericalconcernsin the radiativetransfercomputation
are addressed,such as the truncationof the forward scattering
peak in the phasefunctionand a stableexpansionof the phase
function in terms of the renormalizedLegendrefunction. The
1957].
Becauseof the complicatedmechanismsinvolved in ice discrete expression of the adding/doubling principle is
andefficientin numericalrealization.The present
crystalgrowth,suchas sublimation,riming,or the aggregation
of straightforward
adding/doubling
computationalprogramhasbeenvalidatedwith
particles,the surfacesof ice crystalsmay be rough.Roughened
ice crystalsurfaceshavebeenobservedin laboratoryexperiments respectto the variouscasespresentedby Lenoble[1985] and also
[Cross, 1968] and in recentin situ observationsof tropic anvil in comparison with the discrete ordinate method radiative
cirrusclouds(A. J. Heymsfield,private communication,1999). transfer model (DISORT) [Stamnes et al., 1988] (also
gov/pub/wiscombe/Multiple_ScaWDISOR
The scatteringphasefunctionsof roughenedparticlesdisplayless ftp://climate.gsfc.nasa.
variation[Macke et al., 1996; Yangand Liou, 1998] than their T_l.2/DISORTReport.pdf) for a numberof canonicalproblems.
counterpartsfor ice crystals possessingsmooth facets. As
articulatedby Mishchenkoet al. [1996] on the basisof ground- 3. Development of Cirrus Models
basednephelometer
and aircraftradiancemeasurement
of cirrus
clouds[Foot, 1988;Francis,1995;Gayetet al., 1995;Posseand 3.1. Cirrus Three-Layer Model
yonHoyningen-Huene,
1995],the scattering
phasefunctionsfor
The vertically inhomogenousnature of cirrus clouds was
asteriskindicatesthe complexconjugateoperation.The volume
integrals in equations(2a) and (2b) are carried out along
individualray pathsinsidethe particlevia a Monte Carlo/ray-byray algorithm[Yang and Liou, 1997] that is a generalizationof
the well-known anomalous diffraction theory [van de Hulst,
some ice phase clouds can be rather featurelesswith no
observedduring the FIRE-II program.Figures l a and lb show
appreciablehalos.
two different vertical profiles basedon replicatorimages of ice
Based on these studies, we account for surface roughness crystals in cirrus clouds collected on November 25 and
specifically for ice crystal aggregatesin this study. In the December 5, 1991. For thesetwo cases,three distinct regimes of
numericalcomputationthe particle surfaceis regardedas a ice crystalsare evidentfrom the replicatordata.In the uppermost
number of small facets whose normal direction is tilted from that
layer, small nonspherical"quasi-spheres"
are predominant.The
in the smoothcase,specifiedby localzenithandazimuthangles0 middle layer of cirrus is composedprimarily of pristine ice
and q0,respectively.The slopeof a facet alongtwo orthogonal crystalswith well-defined hexagonalshapesor bullet rosettes.
directionsthat are perpendicular
to the local zenithdirection,say, The bottom layer containslarger but irregular aggregates.The
thex andy directions,canbe specifiedby
edgesof theseirregularice crystalsseemto be rounded,perhaps
due
to the effect of sublimation.Roughnesscan also be noted
•)Z
/2
(3a)
COS(•,
from the replicatorimagesof the irregularice crystals.In both
imagesit is apparentthat the particlesincreasein size and the
shapesbecomemore complex from the top to the base of the
gx_• =(//-2
_17)1
sin
rp
Zy 3Z
3y (//-2 1)1/2
(3b)cirrus.
17,270
YANG ET AL.' CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
E
-o
E
-o
0
0
0
u0{•oun=! eseqd
0
YANGETAL.' CIRRUSBIDIRECTIONAL
REFLECTANCE
ANDICECRYSTALS
• • 25 Nov1991Replicator
Profile
17,271
zz-•
+zlz12
/2 J•(O,z'
)n(O,z'
)dz'
•i(O'z)
=' j.z+•/2
n(O,z'
)dz'
Jz-zlz
(5b)
/2
-SSøC
where D is the maximum dimensionof ice crystals. The
precedingaveragingprocedureis alsoappliedto obtainthe mean
size distributionandhabitpercentage
for cirrusby assuming
a
singlesizedistribution
to represent
thecirruscloud.
To illustratethe differences
betweenthe useof a three-layer
model and a single-layer cirrus model, we first confine our
~ 50oc
discussion to the November
25 case where the cirrus cloud
spanneda vertical range of 2.7 km and had a total optical
thicknessof approximately
1. The geometricheightsof the top,
middle, and bottom layers are 0.79, 0.73, and 1.18 km,
respectively.
The sizedistributions
andcrystalhabitpercentages
for the three layers are shownin histogramsA, B, and C of
Figure2a. For the uppermostcirrus layer, "quasi-spherical"
habitsare dominantfor the small-particlemode(D < 100 lam),
whereasbulletrosettesare dominantfor the large-particle
mode
(D > 100 lam). The middle layer is composedprimarily of
columnsandbulletrosettes.In the bottomlayer,ice crystalsare
mainly irregular aggregates.The percentagesof various ice
crystalhabitsintegratedoverthe entirecirruslayerare shownin
histogramD of Figure2a. Two geometries
(sphereandhexagon)
are used to representthe small ice crystals,as is illustratedin
histograms
A andD of Figure2a. The two particlemorphologies
arechosento investigate
the sensitivity
of theradiativeproperties
of cirrusto thepresence
of thesesmallicecrystals
nearcloudtop.
- 45"C
- 40'C
- 35"C
----
5 Dec 1991 Replicatot
Figurela. Replicator
imagesof icecrystals
froma cirruscloud
observed
on November25 1991duringthe FirstInternational
Profile
-65'C
Satellite Cloud ClimatologyProject (ISCCP) Regional
Experiment
(FIRE-U) field experiment
[afterHeymsfieldand
Iaquinta,2000withpermission
of theAmerican
Meteorological
Society]. Note the three-layerstructure
with smallquasispherical
crystals
in thetoplayer,andcolumns
andbulletrosettes
inthesecond
layer.Thethirdlayeriscomposed
mostlyof large
.60øC
aggregatedcrystals.
Thesmallparticles
in theuppermost
layerhavenonspherical
shapes
with an aspectratioapproaching
1. Sometimes
theterm
"quasi-spherical"
is usedin the analysisof observed
dataandin
theoretical
studies.Thistermis oftenmisleading
because
the
opticalproperties
of sphericalandnonspherical
particlesare
significantlydifferenteven if the nonsphericity
of particle
geometryis not substantial.
Analysisof thereplicator
datahasyieldeddetailedinformation
on the dominanthabitsof ice crystalsandsizedistributions
for
the two casesshownin Figuresla and lb. As discussed
in
section
2.1, datawereobtained
for 28 and33 verticallayersat
approximately100 m resolutionin FIRE-II in situ observations
fortheNovember
25 andDecember
5 cases,
respectively.
Using
thesedata,we constructed
the percentages
of the variousice
crystalhabitsand size distributions
for the top, middle,and
bottomlayersof cirrusclouds.For a givenlayercentered
at z
withthickness
of Az,themeansizedistribution
andpercentage
of
a specifichabitaregivenby
] fz+•/2
dz',
h'(D,z)
='•'.•z-•
/2 n(D,z')
- SOOC
.>.
-45øC
-4o'c-.
-e
IIt
.......
:"
'"
$00 .pm
-$5'C
--
Figurelb. SameasFigurela, except
thattheobservation
was
madeonDecember
5, 1991,andthetoplayeris dominated
by
pristinecolumns.Data courtesyof L. Miloshevichand S.
(5a)Aulenbach,
National
Center
forAtmospheric
Research.
17,272
YANGETAL.' CIRRUSBIDIRECTIONAL
REFLECTANCE
ANDICECRYSTALS
0.4
•
0.3-
Z=9.62-10.41
km
D<100pm 9(@)
(bz=0.79
km)
D>100
pm•
to- 0.6øC
0.20.1
I
0
100
,
I
I
300
200
400
500
0.6
600
,
Z=8.89-9.62
'"'• 04
km
--'• (Az=0.73
km
)
,
30ø/'0
(• +705'0
•
0.2
0
.
....
o
o
1 oo
I
....
200
I
....
I
300
....
400
I
....
500
,,
Z=7.71-8.89 km
E
z
600
0.3
C
(Az=1.18
km)
200/,0
• +800/,0
•
0.1
[•.r•_
T=-37.1
to
45.3øC
o IIIII1 I
0
100
200
300
.... , .... , ....
400
,
Averaged
0.2
6OO
500
0.3
,-,(Az=2.7
km)
D
(@)
o,,
- ,zq--rlh•D<100pm22%•+11%u
,+35%
•+32Yo'i•
0.1
0
0
100
200
300
400
500
600
Maximum Dimension(Iam)
Fisure 2a, Size distributionmodelingthe cirrusobservedon November25, 1991, that is shownin Figurela.
HistogramsA, B, andC showthe sizeandhabitdistributions
for the top, middle,andbottomlayer,respectively.
HistogramsD showsthe meansizedistributionaveragedoverheight.
Accordingto Figures2a and 2b, both size distributions
and ice
crystalhabitsvary substantially
with altitude.While insufficient
evidenceexiststo makegeneralizations
regardingtheverticalsize
distributionof ice particlesin cirrusclouds,thereare additional
analysesin thisregardbasedon radarobservations
[Maceet al.,
1997] thattendto supportthe conceptof multiple-layered
cirrus
phenomena[Ohtake, 1970].
Shown in Figure 2b are the size distributionsand habit cloudswith compositemorphologies.
percentages
for thecaseof December
5. Thegeometric
heightsof
the threelayersare 1.24, 1.12, and 1.17 km, respectively.
The
3.2. Radiative Propertiesof Cirrus Layers
December5 casehad a coldercloudtop temperature
thanfor the
We employthe scatteringcomputationalmodel describedin
November 25 case, and many hexagonalcolumnsand small
"quasi-spherical"
droxtals[Thurnan
andRobinson,
1954;Ohtake, section 2 to compute the extinction cross sections, single1970] were observedin the crystalpopulation.The divisionof scatteringalbedos, and phase functions for ice crystals. Ice
largeandsmallmodesfor ice crystalhabitsis at 50 !xmfor the crystalsare assumedto be orientatedrandomlyin the atmosphere.
caseof December5. Again, "quasi-spherical"
ice crystalsseemto First, to characterizethe bulk propertiesof size distribution,we
dominatethe small-particlemode.There are substantially
large definethe meanmaximumdimensionfor a givensizedistribution
numbersof "quasi-spherical"
particlesevenfor themiddlelayer. as follows:
Thesesmall ice crystalsare often misidentifiedas spheresin
observationsbasedon the particle imageswith blurrededges.
Even with the useof an opticalmicroscope,
the shapesof small
ice crystalsare unlikelyto be seenclearlybecause'of
the poor
instrumentalresolving power causedby optical diffraction
YANG ET AL.' CIRRUS BIDIRECTIONAL
REFLECTANCE AND ICE CRYSTALS
r-]"l Z=l1.39-12.63
km
D>50 pm t•
55.2 to -65.4ø(3
0
100
A
D<50pm 9 ('0)
(Az=1.24
km)
17,273
300
200
400
500
600
0.6
•
0.4
_ III
'l'l't'l'
0
Z=
10.27-11.39
km
(Az=1.12km)
I
I
I
I
I
100
I
D<50pm •
('0)
D>50 pm u
½
•
200
300
400
500
600
Z=9.1 - 10.27 km
C
r] (Az:1.17
km)
0.2
30%
u , +70%
•
o.1
o
,
....
0
I
....
I
100
_
H
'l'l'l'l•
0
....
I
200
....
I
300
....
I
400
6OO
500
Averaged
(0 )
3.53km)
D<50pm
90%•+3%u • +7%'•
D>50
pm75%(,½,•+25%
'•
I I'
100
•
,
,
I ....
200
I ....
300
I
400
I
,
,
,
I
'
'
500
i
600
Maximum Dimension (pm)
Figure 2b. SameasFigure 2a exceptfor the caseof December5, 1991.
<D > =
•Dmin
Omax
Dn(D)dD
,
[Drain
Dmax
n(D)dD
whereft(D) is the percentage
of a specifichabitat sizeD. The
(6)
summation
overindex i is carriedoutfor all the ice crystalhabits.
We notethat the precedingdefinitionof effectiveradiusreduces
to that defined by Hansen and Travis [1974] in the case of
whereDmin andDma
x arethecutoffsof sizedistribution
at small sphericalparticles,that is, re =< rø>/<r • >. The mean
and large sizes, respectively.Studiesby Foot [1988], Francis
et al. [1994], Fu [1996], and Wyserand Yang[1998] havefound
that the details of the size distribution are not important to
specifyingthe bulk opticalpropertiesof cirruswith respectto the
effectivesize of ice crystalsif the effectivesize is definedas the
ratio of total volumeto the total projectedarea.This featurehas
alsobeenobservedin the caseof watercloudscomposedof liquid
dropletswhose scatteringpropertiescan be solved using Mie
theory [Hansen and Travis, 1974; Hu and Starnnes,1993].
Followingthesestudies,
we definetheeffectivediameter
D e and
effective radius r e for nonsphericalice crystals with a
combination
of various habits as follows:
Drnax
3 Dmin
•/Vii(D)•(D)n(D)dD
extinction cross section, single-scatteringalbedo, and phase
functionaregivenby
fDmax
ZCext,
i(D)J3(D)n(D)dD
•
l
Cext
= fDmax
,
aDmm
Zf .(D)n(D)dD
Dmax
-- fD•
ZCscat,
i(O)•(O)n(D)dD
I
(0=•D,•
Dm•,,
• Cext,
i(D)•(D)n(D)dD
P(O) =
(8b)
Dmx
D•
• Cscat,
i(D)Pi(e'D)fi(D)n(D)dD
(8c)
ID
Dmax
• • Cscat'i(D)J•(D)n(D)dD
We
=2Fe
='•Jømax
3&(O)3(O)n(D)aO'
(7)
Dmin
(8a)
i
17,274
YANG ET AL.' CIRRUSBIDIRECTIONALREFLECTANCEAND ICE CRYSTALS
11
11
11
•'
10
Top
Layer
,-9 Middle
Layer
••'1; OOo •'
• 1; O
o•
Bottom
Layer
'-
'•
7
O
'''1'''1'''1'''1
40 80 120 160
0
' I ' I ' I ' I ' I
60 62 64 66 68 70
<D>( pm)
11
11
E 10 0
•
De (pm)
o.65
pm
•'10
9
• 9
O
'• 8
O
7
O
'• 8
0.2
0.3
0.4
0.5
0.6
0.7
O
7
.... , .... I .... I .... I .... I .... I .... I
0.1
2.11 pm
O
.... I .... I .... I .... I .... I .... I .... I
0.1
0.8
0.2
0.3
•e (1/km)
11
11
•'10
0.4
0.5
0.6
0.7
0.8
•e (1/krn)
2.11 pm
0.65 IJm
O
9
8
7
....
i
0.99
1
0.88
11
0.89
•o
0.9
0.85
0.9
11
10
O
2.11 pm
0.65urn
9
O
8
o
7
....
I
0.75
....
I
0.8
0.85
g
g
11
11
0
0.65pm
o
9
O
8
•-
0
0
2.11 pm
•'10
o
9
._•
(•
0.04
0.08
0.12
f5
0.16
o
o
8
7
I
0
,
•
j
,
•
j
,
,
0.08
0.04
f5
Figure3a. Themeansizeandsingle-scattering
properties
forthethree-layer
(November
25,FIRE-II) cirrusmodel.
Theverticallinesindicate
theresultscomputed
usingtheone-layer
modelmeansizedistribution
(i.e.,thecloudis
assumedto be verticallyhomogeneous).
where
Cscat,
i isthescattering
cross
section
ofhabit
i, given
bythe thickness
of the layer.The lowertwo rowsprovide
the
difference
of extinction
andabsorption
cross
sections
thatcanbe asymmetry
parameter
of thephasefunctions
andthefraction
of
computed
onthebasis
ofequations
(2a)and(2b).
deltatransmission
[Takano
andLiou,1989a]in scattered
energy.
Figures
3a and3b showthebulkmicrophysical
andopticalNotethatthedeltatransmission
is anartifact
pertaining
to the
properties
forthesizedistributions
shownin Figures
2aand2b. ray-tracing
technique,
whichcanbe circumvented
by usinga
Theupperrowshows
thegeometric
configuration
of thethree moreaccurate
physical
optics
approach
[Mishchenko
andMacke,
layers
of cirrus
andthemeanmaximum
dimension
andeffective1998].In thepresent
GOM2calculation
based
ona simplified
sizeof icecrystals
in theselayers.Thesecond
andthirdrows algorithm
[YangandLiou,1996],we do notaccount
for the
showtheextinction
coefficient
andsingle-scattering
albedo.
Note spreading
of theraysassociated
withdeltatransmission
for size
thattheextinction
coefficient
associated
witha specific
cirrus parameters
largerthan100.Theuseof eithera geometric
optics
layeris givenby the meanextinction
crosssectionandthe methodor a physicalopticsapproach
in dealingwith delta
17,275
YANG
ETAL.'CIRRUS
BIDIRECTIONAL
REFLECTANCE
ANDICECRYSTALS
,-0.65
/zm, Thin Cirrus (-r
½-60
(/)--0¸
60
4.0
6O
•
5O
3.5
T• 40
4O
oO
5O
*
•0
2O
*
•0
10
30 ¸
25
2.0 ¸
3>
o
't 0
20
30
40
50
60
0
10
20
30
40
5O
6O
1.5
½-120 ¸
6o
½-180 ¸
6O
1.0
50
0.5
40
•:n 40
•o
•
N
ß
0.0
2O
20
-0.5
10
10
-1.0
0
0
10 20 50 40 50 60
SolarZenithAngle(o)
0
10 20 .50 40 50 60
SolarZenithAngle( o )
Plate
2.The
pcrccm
r½lati¾½
difference
oœ
bidir½ctio•a!
reflectance
computed
usin•
the
threea•do•½-laycr
models
atMODI$
0.65
pmba•d
fortbi•cirrus
(•!). 1'be
maximum
difference
forthis
case
isabout
5%a•ddepends
mainlyonscattering
angle.
17,276
YANG ET AL.' CIRRUSBIDIRECTIONALREFLECTANCE
AND ICE CRYSTALS
A-0.65
/zm,
Cirrus (-r- 10)
Thick
•--0 ø
- ½__60ø
60
60
3.0
50
2.5
40
•_ 30
50
ß
20
20
m
10
10
o
o
N
2.0
1.5
>.
o
10
20
.50
40
50
60
1.0
o
10
20
30
40
50
60
0.5
½-120 ø
½-180 ø
6o
6o
0.0
5o
•
40
...c ,.50
3O
ß
2O
N
-0.5
4O
20
-1.0
-1.5
10
o
10
20
30
40
50
Solar Zenith Angle ( o )
60
-2.0
o
10
20
50
40
50
Solar Zenith Angle ( o )
Plate 3. SameasPlate2, exceptfor thick cirrus(x=10).
60
YANG ET AL.' CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
/X-2.11
17,277
/zm, Thin Cirrus
(/)--0 ¸
½-60 ¸
6O
6O
5O
5O
12.0
11.0
4O
4O
3O
3O
2O
2O
10
10
10.0
9.0
8.0
20
30
40
50
60
0
10
½-120 ¸
20
30
40
50
6O
½-180 ¸
6O
7.0
6O
5O
6.0
40
4O
,30
30
20
2O
5.0
4.0
10
3.0
10
20
30
40
50
Solar Zenith Angle (o)
60
0
10
20
30
40
50
Solar Zenith Angle (o)
Plate4. SameasPlate2, except
forMODIS2.11gmband.Notetherelative
difference
ismuchhigher(up
to 12%)thanat0.65gmwavelength
dueto absorption
by ice.
6O
17,278
YANG ET AL.'
k-2.1
CIRRUS BIDIRECTIONAL
REFLECTANCE
AND ICE CRYSTALS
1 /zm, Thick Cirrus ('r- 10)
•--0 ¸
½-60 ¸
60
60
50.0
50
40
45.0
•: 50
50
•
20
N
20
40.0
[
10
o
..
0
10
20
.30
40
50
0
60
0
10
20
.30
40
.50
60
35.0
½-120 ¸
½-180 ¸
6O
60
5O
30.0
•
40
4O
,_..50
5O
e
20
2O
m
10
10
25.0
N
0
0
10
20
30
40
50
60
Solar Zenith Angle (o)
20.0
0
10
20
50
40
50
60
Solar Zenith Angle (o)
Plate5. Same
asPlate
3,except
forMODIS2.11pmband.Thedifferences
forlarge
optical
thickness
reach
upto
50%anddepend
alsoonviewingandsolarzenithangles.
YANGET AL.' CIRRUSBIDIRECTIONAL
REFLECTANCE
AND ICECRYSTALS
13
13
13
•' 12
TopLayer
•11
Middle
Layer .c:
._• 11
._o)
•10
-r'
9
I
•' 12
•'12
O
._•
(•10
(•10
9
O
•11
O
-r'
Bottom
Layer
0
O
'1'
0
40
80
120
0
160
40
<D> ( pm )
13
0.65 pm
Ell
._•
2.11pm
•11
O
._•
el0
O
0
9
,
0.3
0.4
0.5
,
,
0.2
0 3
I•e (1/krn)
13
2.11 IJm
•'12
0.65 pm
..•11
Ell
._•
O
(•10
'1-
9
O
9
0.99
1
0.8
13
0.5
O
._•
(•10
0.4
I•e (1/km)
13
•'12
120
De(Pm)
•' 12
0.2
80
13
•'12
'1-
17,279
....
I ....
0.85
I ....
0.9
1
0.95
13
0.65 pm
•'12
•' 12
_•.11
2.11
pm
•11
._•
._•
(•10
el0
0
9
0.75
0.8
0.85
0.8
g
0.85
0.9
g
13
13
•'1• 0.65
pm
•'12
'1-
•10
•11
._•
O
,
0.04
,
,
I
0.08
2.11 pm
'
0.12
0.16
9
0.02
0.06
0.1
Figure3b. SameasFigure3a, exceptfor theDecember5 casethatis shownin Figure2b.
transmissionin the single-scatteringcalculationfor large size
parametersdoesnot make a significantdifferencein the radiative
transfercomputation.
For the November 25 case (Figure 3a), the mean maximum
dimensionof the ice crystalsis 74 pm, 112 pm, and 121 pm for
the top, middle, and bottom cirrus layers. The mean effective
diametersfor the top, middle,andbottomcirruslayersare 65, 64,
and 66 pm, respectively.For the December5 case(Figure 3b),
the mean maximum dimensionof the ice crystalsis 30 pm, 80
pm, and 132 pm for the top, middle, and bottom cirrus layers,
respectively.The corresponding
meaneffectivediametersare47,
92, and 89 pm for the top, middle, and bottom layers,
respectively.We note that the effective diametersfor the middle
andbottomlayersare substantially
smallerfor theNovember25
case than for the December 5 case. The reason for this is that
there are high numbersof bullet rosettesin the November 25
case, and bullet rosettes tend to have a small ratio of volume to
projected area. In addition, the smaller mean effective diameter
may be attributedin partto the porousstructures
of ice crystal
aggregates
whicharepresentin a largernumberconcentration
on
November
25 than on December 5.
The optical propertiesof ice crystalsare computedfor
wavelengthsrepresentative
of two MODIS bandscenteredat 0.65
and 2.11 pm. Thesewavelengthswere chosento representthe
MODIS bandsby integratingover the instrumentalresponse
functionsfollowing Baum et al. [2000a]. For the 0.65 and
17,280
YANG
ET AL.:
CIRRUS
BIDIRECTIONAL
2.11 !xmbands,themaximumextinctioncoefficientis derivedfor
the middle layer for both the November 25 and December5
cases. This occurs because the number concentration
of ice
REFLECTANCE
AND ICE CRYSTALS
single-layer cirrus models for the purpose of comparingthe
radiativefeaturesof eachmodel.To understand
the physicsin the
comparison, one needs to interpret the scattering geometry
involving the Sun and a satellite. For a given solar geometry
crystalsis substantiallyhigher in the middle layer than in the
bottomlayer. The numberconcentrationin the top layer may not specifiedby (Os,rPs)and a viewing geometryspecifiedby
scattering
angleisgivenby
be lower than in the middle layer, but the crosssectionsfor the (0v,rpv), thecorresponding
small ice particlesin the top layer are very smallin comparison
O=cos
-1[-cosOscosO
v+sin0
ssin0vCOS•], (9)
with thoseof the particlesin the middlelayer.
In the November 25 case, the single-scatteringalbedo at where • = rps-rpv is the relativeazimuthanglebetweenSun
2.11 lam for the middle layer is largerthan the onesassociated and satellite.Note that Os and 0v are the inclinationangles
with the top and bottom layers becauseof the small effective measuredfrom zenith. Figure 4 illustratesthe contoursof
particle size of the middle layer. Since the effective size is scatteringanglesversussolar and view zenith anglesfor four
defined as the ratio of volume to projected area, which is casesof azimuthangles.The solarzenithand viewingzenith
proportional
to the meanpathlengthof raysinsidetheparticles,a anglesrangebetween0ø and 60ø. The scatteringanglesfor the
geometry
considered
in thepresent
studyare
smalleffectivesizeimplieslessabsorption
andthusa largervalue regionof view-solar
of the single-scattering
albedo.In contrast,the single-scattering essentiallyconfined to side scatteringand backscattering
The variationalpatternof the scattering
angleversus
albedoat 2.11 lamfor the December5 caseis higher in the top directions.
layer than in the lower layers due to the prevalenceof small solarzenith and viewing zenith anglesdependson the relative
azimuthangle.A similarcontourdiagramof the scattering
angle
crystalswithin the top layer.
by Mishchenko
et
For bothbandsthe asymmetryparameteris smallerin the top versuscos0s and cos0v hasbeenpresented
and bottom layers than in the middle layer for both the al. [1996].
Plate2 showsthe relativedifferencebetweenthe computed
November 25 and December 5 cases. In the top layer the
reflectances
of the three-layerandone-layercirrus
asymmetryparameteris reducedbecausethe particlesare smaller, bidirectional
which tendsto reducethe magnitudeof the forwardpeak of the modelsfor opticallythin cirrus('• = 1) at 0.65 gm. The relative
phase function. In the bottom layer the reducedasymmetry difference is defined as
parameteris causedby the particleroughness.
e(Os,Ov,rp)
= 100[R3(Os,Ov,rP)-R1(Os,Ov,rp)]/R1(Os,Ov,rp
(10)
The vertical variability of delta transmissionis similar to that
of the asymmetryparameter.In the top layer,deltatransmission
is where R3 and R1 are the bidirectional reflection functions
substantially
reducedin the December5 casebecauseof the ray- computedusing the three-layer and one-layer cirrus models,
spreadingeffect associated
with small sizeparameters[Yangand respectively.The maximum differencein this case is about 5%.
Liou, 1996]. In the bottomlayerthe roughness
of particlesurface Whenthe opticalthicknessis small,the photonsoriginatingfrom
alsoreducesthe deltatransmissioneffect.For the sakeof brevity single-scattering events dominate the total radiance. The
of presentation, in the following discussionswe select the contributionof single-scatteringeventsto the radiance in the
December 5 case to investigate the effect of vertical three-layercaseis given by
inhomogeneity on phase function and multiple-scattering
propertiesof cirrus.
Plate 1 showsthe phasefunctionsassociated
with the singlescatteringpropertiesshownin Figure3b. At 0.65 lam,substantial wherethe summation
is carriedfor all threelayersof cirrus.Thus
differencesbetweenthe phasefunctionsfor the bottomlayer and for thin cirrusthe bidirectional
reflectance
functionis linearly
other layers can be noted at scatteringanglesnear 120ø. The proportional
to thephasefunction.Referringto Figures3b and4,
phasefunctionvaluesfor the bottomlayer are much largerthan the contoursshownin Plate2 canbe explainedas follows.For
those for the top and middle layers in the scatteringregion azimuth angles of 0ø and 60ø, the maximum difference is
between 5ø and 20ø. Evidently, the phase function values observed
nearscattering
anglesof 120ø, whichcorresponds
to the
computedby assumingthat cirrus cloudsare a homogeneous phasefunctiondifferenceat thesescattering
angles.For azimuth
mixture of particle shapesand sizesare significantlydifferent anglesof 120ø and 1800, the maximum differencesfor the threefrom the phasefunctionsof the three layersin somespecific layerandone-layer
resultsaremainlynotednearscattering
angles
of 155 ø and 180 ø.
scatteringangleregions.
At 2.11 lam the forward peaks of the phasefunctionsare
Plate3 is similarto Plate2, exceptthattheopticalthickness
of
smallerthan at 0.65 lambecausethe size parameters
are smaller. the cloudis 10. The contribution
of multiplescattering
increases
1 l=1
• [•)lZl•'lPl(Os,Ov,•)],
(11)
r(Os'Ov'rP)
=4cøsOscøsOv
The magnitudeof the phasefunctionfor side scatteringand with increasingoptical thickness.The differencesoccur at
backscattering
anglesis muchlowerat 2.11 lamthanat 0.65 gm. scatteringanglesbetween90 ø and 120ø, between150ø and 160ø,
The differencesbetweenthe phasefunctionsfor the different and between 170ø and 180ø. From Plates 2 and 3 the difference
layers are enhancedat 2.11 lam becauseparticleabsorptionis betweenthe three-layerand one-layermodelsis within a few
stronglysensitiveto the particlesizes.Althoughnotpresented
for percentregardlessof optical thickness.Based on theseresults,
the November25 case,the overallfeaturesof the correspondingmodelingthe cirrusasa singlelayerwouldseemto be sufficient
phase function analysisare similar to that for the December 5 at 0.65 gm.
case.
4. Results
4.1. Comparisonof ReflectanceFeature for Two Cirrus
Models
Plates4 and5 are similarto Plates2 and3, exceptthatthe
calculationsare performedat 2.11 lam.Absorptionby ice at
2.11 gm is muchhigherthan at 0.65 gm. The reflectance
obtained
usingthethree-layer
modelisalwayslargerthanthatfor
the one-layermodelat 2.11 lam. Becauseof ice absorption
at
2.11 lam,the top layerof cirrusdominates
the contribution
to the
The radiativetransfermodeldescribedin AppendixA is used cloudreflectance
becausephotonsthat penetrateintothe lower
to computethe bidirectionalreflectances
for the three-layerand layersarelargelyabsorbed.
The meansizeof the icecrystals
in
YANG ET AL.: CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
17,281
Contours of Scottering Angle
ø
•-•
=60 ø
60
60.•o .-,,,;o
.'-,,g. -%,,
40
N 20
20 %
0
0
0
20
40
60
0
½-120
•
i
20
40
½=180ø
60
ß
.
4o
r • •60
-ß
20
0
0
20
40
Solar Zenith An91e( * )
60
0
20
40
60
Solar Zenith Angle ( ø )
Figure4. The scattering
angleversussolarzenithandviewzenithanglesfor four azimuthalangles.Notethatthe
scatteringanglesareessentiallyfor sidescatteringandbackscattering
directions.
the top layer is muchsmallerthanthat associated
with the general essentiallynonspherical.We wish to clarify whether their
one-layer cirrus model. For a cirrus cloud with a given optical morphologies
can be treatedas spheresin light scatteringand
thickness, the cloud reflectance increases with the decrease of
radiativetransfercalculations.It is expectedthat the answerwill
mean size of ice crystals.With the increaseof optical thickness, dependon the wavelength.
At a far infraredwavelength,say, 15
the difference between the three-layer and one-layer models pm or larger,the nonsphericity
of theseparticlesmay not be
increases.For an opticalthicknessof 10, the differencescan reach significant
because
thesizeparameter
becomes
small.For a small
up to 50%. Becausethe 2.11 gm bandis usedfor the retrieval of particlehavingan effectivediameterof 20 pm, the sizeparameter
the mean size and optical depth of cirrus cloud, it is suggested is approximately4 at a wavelengthof 15 gm. In this case,
that the vertical inhomogeneitymay be importantto developing sphericaland spheroidalgeometrieshave been used in light
more realistic cirrus retrieval algorithms.In comparingPlates4 scattering
computations
[e.g.,SunandShine,1994;Takanoet al.,
and 5, it may be noted that the differencebetweenthe three-layer 1992]. However, for the 0.65 and 2.11 pm bands, the size
and one-layer models dependsmainly on the scatteringangle parameterfor a small particlehavingan effectivediameterof
when the cloud is optically thin. However, for optically thick 20 pm is approximately 100 and 30 at the band centers,
cirrus, the difference dependsnot only on scatteringangle but respectively.For a size parameterof this magnitude,the
also stronglyon the viewing zenith and solarzenith angles.This nonsphericity
effect of a particleon its scatteringpropertiescan
is becausethe radiancepathvarieswith the solarandview angles. be significant.
For large solar zenith or viewing zenith angles,the ray path is
To investigate
the sensitivity
of cirrusopticalproperties
to the
large and the difference betweenthe three-layer and one-layer shapes
assumed
for the small"quasi-spherical"
ice crystals,in
cirrus models, and their associatedsingle-scatteringproperties, thisstudywe compare
the resultsassociated
with spherical
and
becomesmore significant.
hexagonal(with an aspectratio of unity) assumptions
for the
4.2. Sensitivity of Cirrus Reflectanceto Shapesof
Quasi-Spherical Particles
As discussedin section3.1 regardingthe replicatorimagesof
ice crystals,the small, so-called"quasi-spherical"
ice crystalsare
morphologies
of theseparticles.
The "quasi-spherical"
particles
primarilyaffectthe top andmiddlelayersandare largelyabsent
in the bottomlayer. The effectivesizesfor the top and middle
layersare47 pm and92 pm if hexagonalshapesareusedfor the
"quasi-spherical"
particles,whereasthe sizesare 52 pm and
17,282
YANG ET AL.: CIRRUS BIDIRECTIONAL
Table 1. Single-Scattering
Propertiesof the Top and Middle
Layers, Which are Computed in ConjunctionWith Two
Assumptions
for the Shapesof the "Quasi-Spherical"
SmallIce
Crystals
(IceSpheres
andHexagons
WithAspect
Ratioof 1)a
)•=0.65gm
)•=2.11gm
Spheres Hexagons
REFLECTANCE AND ICE CRYSTALS
e(Os,Ov,d))
= 100[Rsph
(Os,Ov,d))
- Rhex(Os,Ov,d))]/Rhex(Os,Ov,d)),(12)
whereRsp
h andRhe
x indicate
thereflection
functions
associated
with sphericaland hexagonalshapes,respectively,which are
Spheres Hexagons assumed for the small "quasi-spherical" ice crystals. The
for
for
for
for"Quasi- maximum differencesshown in Plate 6, which correspondto
"Quasi"QuasiSpherical"
Spherical" Spherical"
Spherical"scatteringanglesbetween 130ø and 140ø, are causedby the
"Quasi-
Particles
Particles
Particles
Particles
Top Layer
/•e(1/km)
0.32207
0.36926
0.33092
•
0.99999
0.99999
0.91208
0.91335
•/
0.83271
0.77965
0.86188
0.80258
0.070293
0.11689
f•
0.038339
0.36005
0.035189
Middle Layer
•e(1/km)
0.41679
0.42553
0.41837
•
0.99998
0.99998
0.85099
0.85187
g
0.81627
0.80842
0.87949
0.86934
0.12765
0.13488
f•
0.087641
0.42446
0.086295
rainbow feature of ice spheres.It can also be noted that the
assumption of ice spheres leads to an overestimation of
reflectancenear 180ø (backscattering).As optical thickness
increases, the contrast decreases for the rainbow feature.
However,the enhancedbackscattering
derivedusingspheresas
the "quasi-spherical"crystal shape is still noticeable. For
opticallythick cirrus,Plate 7 showsthat the assumptionof ice
spheres for the "quasi-spherical" particles leads to an
underestimationof cloud reflection at 2.11 gm except for
scatteringanglesnear 180ø
Plates8 and 9 are similar to Plates6 and 7, exceptthat the
computationshave been performedat 0.65 gm. Again, we see
pronounceddifferencesbetween the results associatedwith the
hexagonaland sphericalassumptions
for the smallice crystalsin
the uppermost layer. The positive maximum near the
aNote
thattheshapes
of icecrystals
withsizelarger
than50lamare backscattering
peaknotedin Plate6, however,
isnotobserved
in
assumed
tobeunchanged.
theresults
shown
inPlate8,because
thephase
function
value
for
the spheres is less than that of hexagons at 0.65 •tm.
Additionally,at 0.65 gm a distinctrainbowfeaturecan be noted
in the case of thin cirrus. For the optically thick cirrus, the
rainbow is blurreddue to multiple scatteringeventsoccurring
95 gm if perfect spherical geometry is assumed for these
particles. Evidently, the assumptionof shapefor the "quasi- within the clouds. Plates 6-9 illustrate that the influence of smallspherical"ice crystalsin cirrus cloudscan lead to a changeof particle shapein the uppermostlayer of cirrus is significantat
effective size by asmuch as 5 gm.
bothvisibleandnear-infraredwavelengths.
Table 1 lists the single-scatteringpropertiesof the top and
middle layers. Substantial differences are noted for delta 5. Conclusions
transmissions
at 0.65 gm becauseof the absenceof parallelfaces
in spheres. The asymmetry factor of the phase function is
In this studywe have defined a three-layercirrusmodel in
substantially different at 2.11 gm, showing clearly the terms of ice crystal habit and size distributionbasedon in situ
dependenceof the scatteringpropertieson the assumptionof replicatordata acquiredduringthe NASA-sponsoredFIRE-II
habits.The differencesof the resultsfor the two shapesare larger field observationprogram.We have describeda fundamental
for the top layer than for the middle layer becausethe formerhas scattering
modelanda numericallystableradiativetransfermodel
a largerpopulationof"quasi-spherical"ice crystals.
for the computation
of the single-scattering
properties
of various
Figure 5 showsthe phasefunctionsassociated
with the single- ice crystalsandthe bidirectionalreflectionof cirrusclouds.
scatteringpropertieslisted in Table 1. Substantialdifferencesof
We havefoundthattheeffectof verticalinhomogeneity
within
the phasefunctionscan be seenfor the top layer at both the 0.65 cirrusis not significant
at 0.65 gm, a wavelength
for whichthe
and 2.11 gm bands. The overall feature is that the spherical absorption
of ice is negligible.However,in comparison
withthe
assumptionleads to larger phasefunction values for scattering one-layercirrusmodel,a verticallyinhomogeneous
cirruscloud
anglesbetween 10ø and 45ø and lower values at side scattering produces substantiallylarger reflectance at 2.11 gm, a
angles.However,the sphericalassumptionleadsto a pronounced wavelengthfor which absorptionby ice is important. The
scatteringmaximum between 130ø and 140ø, which corresponds increase in reflectance occurs because the mean size of ice
to the rainbowfeatureof ice spheres.Additionally,the spherical crystalsin thetop layerin thethree-layermodelis smallerthanin
assumptionleads to larger phase function values near 180ø the caseof the one-layermodelandthe totalreflectedradianceis
(backscattering)
at 2.11 gm. In the middlelayer,the differences dominatedby the contribution
from the top layer.For a given
causedby the assumptionof habit for the small particlesare optical thickness,the reflectanceincreaseswith decreasing
reduceddueto the smallnumberof the smallparticlesin the size particlesize.
distribution.
Evidently,usingthemostrealisticparticleshapes
for
We also investigatedthe sensitivityof reflection of cirrus
the small particlesin the top layer of cirruswill be crucialto cloudsto theparticleshapes
of the"quasi-spherical"
ice crystals
obtainingreliablesingle-scattering
properties
of the cloudat thathavebeenoftenassumed
to be spheres.
For thetwo cirrus
2.11 gm.
Shown in Plates 6 and 7 are the differences
between
cloudcasespresented
in thisstudy,the uppermost
portionof the
cloud tendsto be predominantlycomposedof very small ice
bidirectional
reflectances
computed
usingnonspherical
hexagonal crystals. Numerical results have demonstrated that the
and sphericalgeometriesin the three-layercirrusmodelfor thin bidirectional
reflectionfunctionof cirrusis very sensitive
to the
('•= 1) and thick ('•= 10) cirrus at the 2.11 gm band.The shape of these particles at both visible and near-infrared
differences shown in Plates 6 and 7 are defined as
wavelengths.
YANGET AL.' CIRRUSBIDIRECTIONAL
REFLECTANCE
AND ICECRYSTALS
X-2.11
17,283
/zm, Thin Cirrus
•--0 ø
½-60 ø
6O
60
10.0
5O
Zr• 40
40
x:: 30
3O
5.0
0.0
•
20
20
ß
10
lO
N
-5.0
0
lO
20
30
40
50
60
O
10
20
30
40
6O
-10.0
½-120 ø
½-180 ¸
6O
6O
-15.0
5O
40
% 40-
-20.0
_c .50
3O
ß
20
N
20
-25.0
lO
o
o
0
10
20
50
40
50
Solar Zenith Angle ( o )
60
-30.0
0
10
20
30
40
50
6O
Solar Zenith Angle ( o )
Plate6. Thepercent
relativedifference
of thebidirectional
reflectances
computed
assuming
spherical
and
hexagonal
shapes
for thesmall"quasi-spherical"
icecE/stals.
Thedifference
contours
shownarefor thincirrus
('r=1)atMODIS2.11[tmband.Notethelargedifferences
attheicerainbow
andbackscattering
angles.
17,284
YANG ET AL.' CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
X-2.1 1 /•m, Thick Cirrus (T- 1O)
•--0 ø
•-60 ø
6O
60
5.0
50
•
4o
40
_c 30
30
0.0
-5.0
•
N
20
20
10
-lO.O
10
20
50
40
50
60
0
10
20
50
40
50
6O
-15.0
•-120
ø
•-180 ø
6O
6O
-20.0
5O
•
40
40
-25.0
•_ 30
30
e
20
N
20
-30.0
10
0
•o
20
•o
40
50
Solar ZenithAngle(o)
eo
-35.0
o
•o
2o
•o
40
50
60
Solar ZenithAngle(o)
Plate7. SameasPlate6, exceptforthickcirrus(z=10). Notethesmoothing
of therainbow
maximum.
YANG ET AL.: CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
A-0.65
17,285
/zm, Thin Cirrus
(• --0 ø
½-60 ø
60
60
50
50
40
40
3O
30
20.0
15.0
10.0
20
20
10
lO
5.0
O
10
20
50
40
50
60
•
0
10
20
30
40
50
60
0.0
½-120 ø
½-180 ø
6O
6O
5O
5O
4O
4O
-5.0
-lO.O
5O
5O
2O
2O
-15.0
10
lO
•
10
20
30
40
$0
Solar Zenith Angle (o)
60
0
10
20
30
40
-20.0
50
60
Solar Zenith Angle (o)
Plate 8. SameasPlate6, exceptfor MODIS 0.65 lamband.Note theabsence
of thepositivebackscattering
anglemaximum.
17,286
YANG ET AL.' CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
7,-0.65
/zm,
Thick
Cirrus
•--0 ø
i O)
½-60 ø
60
60
9.0
50
7.0
7:• 4O
40
_c: 30
30
•
20
20
•
10
10
0
o
5.0
N
0
10
20
50
40
50
3.0
1.0
60
o
lO
20
½-120 ø
50
40
50
60 -1.0
½-180 ø
6O
-3.0
6O
5O
•
40
-5.0
40
-7.0
...c:..50
..5O
-9.0
6,
20
2O
6,
10
10
N
-11.o
-13.o
10
20
50
40
50
Solar Zenith Angle ( ø )
60
0
lO
20
50
40
50
Solar Zenith Angle ( o )
Plate9. SameasPlate8, exceptforthickcirrus(x= 10).
60
YANG
ETAL.'CIRRUS
BIDIRECTIONAL
REFLECTANCE
ANDICECRYSTALS
10
3_
10
3
Top
Layer
I
•=0.65
pm
102
100
102•
101-•
[
100
.. :
•c
10-2 ...........
•-
103
=
•=0.65
pm
- ß
..........
spherical
10-1
:•
Middle
Layer _
1
nonspherical
101
17,287
0
6•0
10'1-
.........
1•0
180
I .....
10'2
0
60
• .....
120
180
103
LL
03 102
r-
Top
Layer
t
Middle
Layer
102
X,=2.11
prn
101
•:2.11
pm
101
ß
100
100
10-1
10-1
10-2 ...............
0
6•0
1•0
180
10-2 .....
0
• .....
60
• .....
120
180
ScatteringAngle (deg.)
Figure
5.Comparison
ofthetopand
middle
layer
phase
functions
computed
byassuming
that
thesmall
"quasispherical"
icecrystals
are
either
spheres
ornonspherical
hexagons
with
anaspect
ratio
ofunity.
Note
thepresence
oftheicesphere
rainbow
feature
between
130øand140ø.
Appendix
A' Discrete
Expression
of
To economize
computational
costandmemory
requirements,
weapply
aFourier
expansion
overtheazimuth
angle
forradiance
Adding/Doubling
Principle
andbidirectionalreflectionandtransmission
functions:
M
Theadding/doubling
method
is oneof themostrobust
approaches
tosolve
theradiative
transfer
equation
formultiplescattering
events.
Thestandard
mathematical
expression
ofthis
method
involves
various
tedious
angular
integrals,
although
it can
I i'r't(-T-•,rp)
= • Ii'r't(m)(¾-•)cosmrp,(Ala)
m=0
Mr(rn)(]l,
]1'
(•'),
Ii,r,t(ll,
rp,
ll',rp')=Z
)cosm(rp-
bewritten
symbolically
inaverysimple
form.
Inthissection
we
present
a discrete
formofthemethod
byintroducing
a direct
transmitting
function.
Asa practical
mathematical
expression
froma computational
viewpoint,
thediscrete
adding/doubling
m--0
(Alb)
M
t(•,rp,•',rp')=
• t(m)(•,•')cosm(rp-rP'),
(Alc)
m=0
equations
arestraightforward
andmore
efficient
innumerical superscripts
i, r, andt indicate
incident,
reflected,
and
implementation.
In addition,
the discrete
formof the where
transmitted
intensities,
respectively,
and$t andit' arepositive
adding/doubling
method
ismoresuitable
foraddressing
some withallowable
valuesin [0,1].Themaximum
number
of Fourier
numerical
concerns,
suchas the numerical
singularity
of
for theconvergent
solutions
of equations
adding/doubling
calculation
andtheperformances
ofvariousterms(M) required
quadrature
schemes.
(Ala)-(Alc) depends
ontheincoming
andoutgoing
radiation
17,288
YANG ET AL.: CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
beamsaswell asthemagnitude
of theasymmetry
of thephase Evidently,to guarantee
the equalityin equation(A6) in the
function.
Forexample,
onlythefirsttermisnecessary
fora solar discrete
procedure,
theDiracdeltafunction
should
bereplaced
by
zenithangleof 0ø (overheadSun),whereasmorethan 100 terms Kroneckersymbolin the form of
may be required for a low solar elevation angle (Sun near the
6(/1j-/1i) -• 8ji/Wj = 1/Wjj = 1,
(A7a)
horizon).For solarzenith and viewing zenithanglessmallerthan
60ø, 30 to 40 terms are normally required in the case of cirrus
•J(/1j--/1i)•> •Jji/ •' = 0 j • 1.
(A7b)
clouds if the strong forward peak of the phase function is
Thusthe direct•smission functionin discretefore is givenby
truncated.A comprehensivestudy regardingthe number of the
termsrequiredin the Fourier serieshas beencarriedout by King
0 +6m0)%S
[1983] using the Henyey-Greensteinfunction and the phase
A•)(v)
=
function for a fair weather cumulus.
1
For the discretequ•tities definedwith respectto the set of
According to the definitions of reflection and transmission
discrete
points[•l,•2,'"•n],
functions[Hansen and Travis, 1974], it can be proven that the operator• definedby
Fourier componentsof reflected and transmittedintensitiesare
given by
we introduce
a mathematical
•0' •j•
) BS)=(1+amO)Z
j=l
Ir(m)(it)=(1
+tSmo)l•r(m)(it,
it')li(m)(-it')I
t' dit',(A2a)
x
exp(-v/•j)6ij
.
(AS)
•J •"
(a9a)
cSm)
(g)/•(m)
)j=lZ C(m)
•'jk =(1+(•mO
j B(m)"
jk t•jWj..
• fl,(m)
lt(m)(it)
=(1+t,m0Jj0•
(it,it')li(m)(-it'
)it' dit', (A2b)
(A9b)
whereism0is theKronecker
deltafunction.
It should
bepointed The operator© is similarto an ordinarymatrixmultiplication
out that the radiancesdefined in equations(A2a) and (A2b) are
exceptthat a weight is includedin the former.Thus for one
diffusive
intensities,
thatis,theyoriginate
fromthescattering
of homogeneous
layer,thereflected,
diffusely
transmitted,
andtotal
incidentradiationby the particlesin the scatteringlayer. If the
total (direct + diffuse) transmitted intensity, indicated as
transmitted radiances are related to the incident radiation via the
following relationships:
I' (-/1, rp),isdefinedin thesamemanner
asin thediffusive
case,
I•(m)
=I5.(m)
©r);),
7t(m)
=ij(m)
(g)
[)•n)
I•(m)
=Is(m)
(g)
t3;)''k
'
it follows that
(A10)
•t(m)
(it)=(1+(•mO
)I••(m)
(it,it,)]i(m)
(_It,
)It,dIt'
One of the interestingfeaturesof usingthe operator© is the
=(l+(SmO
)I•[t(m)
(It,It')+A(m)
(It,
It')]li(m)
(-It')It'dIt',(ABa)
variationof subscripts
in the expressions
in equation(A10): The
by subscriptjis redirected
to thedirection
where /(/1,/1') is the mth Fouriercomponent
of the total incidentbe• denoted
transmissionfunction. At"•(/1,/1,) is associatedwith the denoted
by subscript
k afterinteracting
withthesca•eringlayer.
transmission
of incidentradiationandis givenby
Similarly,for two layersindicatedby superscripts
a •d b, we
A(m)
(It,It,)=(1+ am0
1 )It' exp(-•:
/It')8(It
- It'),
in which •(/1-/1')
havethefollowingrelationships:
(A3b)
is the Diracdeltafunction.Evidently,the
quantitiesdefined in equations(A2) and (A3) are continuous
functionsof the arguments/1 and /1' thatrangecontinuously
in
the interval[0,1]. We selecta setof discretepointsin the region
[0,1] for /1 and /1'; that is, the two variablescanonly havethe
valuesof [/11,/12,"'/1n].
Withrespect
to thediscrete
set,we
definethe followingdiscretequantities
with singlesubscript
and
doublesubscripts:
g(m)=
ij.(m)•
[ri(m
) +•i(m)••y)• 7a*(m)
and
jkt(m)
•ln
where
the asterisk
indicates
•nk ]' (Allb)
that the transmission
function
co•espondsto the caseof illuminationcomingfrom below. The
quantities
D and U inequations
(A 11a) •d (A 11b) me givenby
I•'t(m)
=I i't(m)
(--/1j
),I;© =I r(m)
(/1j),
(A4a)
r•.
m)=r(m)
(/1j,
/1k
), tg.
m)=
(A4b)
•.m)
=?(m)(/1j,/1k)
' A•) =zl(m)(/1j,/1k)
' (A4c)
i•m'=•i• '"(n',U("=roO(m'
+i];'* rff
(m',(A12a)
n=l
O•'),(n+l)
=o];),(n)
. o•),(1),
D•Y
)'(1)
=ri?m)• roq*(m).
(A12b)
Sincethe continuous
region[0,1] is discretized
by usinga setof
points,it is requiredthat an integrationof a functionf(#) with Accordingto the sensitivi
w studyby Hansenand Tr•is [1974],
respectto its argumentdefinedin [0,1] be replacedby a discrete we use N=12, 5, •d 3 in the summation involved in the first
summation in the form of
expressionin equation(A12a) for m<10, 10<m<100,andm>100,
respectively. The remaining terms are approximatedby a
geometricseries.The physicsof the adding/doubling
principle
i=1
canbe viewedcle•ly in terns of the variationsof the subscripts
where Wi are the weightsin the summation.For an integral •om left to right in the right-h•d sidesof equations
(A11a) and
involving the Dirac delta function, the definition of the delta
(A 1lb). The reflection and tr•smission functions for the
functionandequation
(AS) leadto thefollowingrelationship:
combinedlayer•e givenby
f(Iti) = fgt)8(It-Iti)dIt-• •f(•j)PlOS(Itj
- Iti). (6)
n
j=l
RJ
7)=ri(m)
+•j(m)
at r(m)
a 7a*(m)
• In
'nk
'
(A13
a)
YANGETAL.' CIRRUSBIDIRECTIONAL
REFLECTANCE
ANDICECRYSTALS
17,289
and
~
~
7b(m) ~
7b(m)
5(k
m)=
tj•(m)
©'lk
+t•(m)
©D•Y
)©'nk
'
(A13b)
Thetransmission
function
givenin equation
(A13b)contains
the
contribution
dueto directtransmission,
whichis implicitly
in the
formof a deltafunction,andit maypotentiallycauseinaccuracy
in numerical
computation.
Thusit is necessary
to separate
the
diffusive
anddirectcomponents
in equation
(A13b).It isnoted
(A17a)
that
=[t•/m)
+A57)(Ta)]©[t•k
(m)
t~fi(m)
©7b(m)
x,b(m)
+4•)(•'b)]
=tf/m)
©t•k(m)
+exp(-•'a/
,a(m)
b/ •k )+g•)(va+vb).
(A14)
'jk exp(_v
4ktikt
j
•2A•.2
[(pi•c(m
)/•k)©(p•m)
/•k)
+32•i•
+(•.•(m)
/•k)©(p•.(m)
/•k)]'
Thusthe diffusivetr•smission functionfor the combinedlayer is
givenby
•a(m)
exp(_z
b/
+
x•b(m) •
+ ©©
(A17b)
Inequmions
(A17a)
•d (A17b),
•f(m)and
•5(m)•e
defined
as
)
follows:
7b(m)
©
2 •iktj
'
(A15)
M
•f(m)
= Z(2_•mO)•l•m(_•i)•m(•j)
l=m
Equations
(A12a),(A12b),(A13a),and(A15)constitute
the
adding/doubling
equations.
It should
bepointed
thatthematrix
M
= Z(2--SmO)•l•m(•i)•m(--•J
)'
(A18a)
l=m
associated
with direct transmission
is diagonal.In numerical
computation
the numerical
efficiency
andaccuracy
canbe
improved
substantially
if the multiplication
of the direct
M
•5(m)
= Z(2_SmO)•l•m(•i)•m(•j),(A18b)
l=m
transmissionfunction with another quantity is evaluated
inwhich•l isgivenby
analytically,suchas
&l=2/+1
2 •
A(/?)(z)
©A3•Y)
=exp(-z/#)A)?,
A);) ©A•.)(z)
--- •(m)exp(-•7/#/c).
•jk
(A16)
2l+1
•
= 2 I0[P(•)
+(_l)lv(_•)]•(•)a•,(• 9)
As mentioned in preceding discussions,the continuous
integration
involvedin the adding/doubling
calculationmustbe whereP(•) is the phasefunctionand •(•) is the Legendre
of/th order.In numerical
computation
we usethe
replacedby a properquadrature
scheme.Mathematically,we polynomial
need
toselect
proper
pairs
of(]2i,
W/)forthedefinition
oftheRadau
quadrature
scheme
fortheintegration
inequation
(A19).
mathematical
operator
©inequations
(A9a)
and
(A9b).
The
mostThus
phase
function
information
attheexact
forward
and
popular
quadrature
sclae•nes
areGauss,
Lobatto,
and
Radau
backward
directions
isaccounted
for.Thefunction
p/m
quadrature
schemes
[Press
etal.,1986;
Hildebrand,
1974].For
in equations
(A18a)
and(A18b)
arethe"renormalized"
the
angular
region
involved,
these
three
schemes
cover
//(or//) (or referred
to as the "normalized"
in DISORT
• (0,1),[0,1
], and(0,1),respectively.
Thatis,Gaussian
(ftp://climate.gsfc.nasa.
gov/pub/wiscombe/Multiple_Scat
quadrature
isopen
atboth
ends,
Labotto
quadrature
isclosed
at RT_l.21DISORTReport.
pdf))
associated
Legendre
polynomials
both
0and
1,and
theRadau
scheme
isopen
at0butclosed
at1. first
introduced
byDave
and
Armstrong
[1970],
defined
as
The radiancedata at nadir view are usuallyrequiredin retrieval
p/m
(l+m)!
m)!
P/
(g)'
~(•)=I(lrn
applications.
Thusan extrapolation
mustbe usedto obtainthe
nadir view radiance if Gaussian quadrature is used. The
(A20)
Legendre
function.
Thenormalized
disadvantage
of usingGaussian
quadrature
in radiativetransfer wherep/mis theordinary
Legendre
functions
canbe calculated
onthebasisof
calculationhas also been noted by Mishchenkoet al. [1999]. associated
the
following
recurrence
relationship'
Althoughthe Labottoschemeis closedat both endsof the
integralregion,theinformation
at # = 0 actuallydoesnotmake
any contribution
to the angularintegration,
as is evidentfrom
equations
(A9a) and(A9b). In addition,including//= 0 will
causea singularityin the initializationof the adding/doubling
calculation. Therefore we use the Radau scheme in the present
study.
To initializethe adding/doubling
process,
we startwith a very
thinhomogeneous
layerwithoptical
depth
A•'(-10-8). The
reflection and transmissiontruncation for this layer can be
obtainedon the basisof the invarianceprinciple [Hansen and
Travis, 1974] as follows:
~m
2/+1
•/rn(]/)
•l+• =4(l+m+1)(lm+1)
- (l+(l+rn)(l-m)
m+1)(/m+1)P/m-1
a)
~(•)' (A21
withthetwo initialvaluesforthepreceding
recurrence
givenby
/(2rn1)!•
bmm(#)=(--1)m•/
•'rt•)•
'(1--#2)
m/2
,
~m
P;;+•(#)
=#d2m
+1bff(#).
(A2
•b)
17,290
YANG ET AL.: CIRRUS BIDIRECTIONAL REFLECTANCE AND ICE CRYSTALS
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commentsand suggestions.
This researchhasbeensupported
by a grant
of NASA's MODIS project and partially by the Office of Naval
Research.This studywas alsosupported
by the Atmospheric
Radiation
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Energy (DOE) undercontractDE-AI02-00ER62901, NASA/EOS grant
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