assessment of the accuracy of snow surface direct beam spectral albedo under a variety of overcast...

Assessment of the accuracy of snow surfacedirect beam spectral albedo under a variety ofovercast skies derived by a reciprocal approachthrough radiative transfer simulation

Shusun Li and Xiaobing Zhou

With radiative transfer simulations it is suggested that stable estimates of the highly anisotropic directbeam spectral albedo of snow surface can be derived reciprocally under a variety of overcast skies. Anaccuracy of �0.008 is achieved over a solar zenith angle range of �0 � 74° for visible wavelengths and upto �0 � 63° at the near-infrared wavelength � � 862 nm. This new method helps expand the databaseof snow surface albedo for the polar regions where direct measurement of clear-sky surface albedo islimited to large �0’s only. The enhancement will assist in the validation of snow surface albedo modelsand improve the representation of polar surface albedo in global circulation models. © 2003 OpticalSociety of America

OCIS codes: 240.6490, 120.5700, 030.5620, 160.1190, 290.1090, 350.6050.

1. Introduction

Albedo is the ratio of the reflected shortwave irradi-ance at the surface to the incident shortwave irradi-ance. Because surface albedo is an importantgeophysical parameter that determines shortwaveenergy exchange at the Earth’s surface, accurate al-bedo estimates are required to determine the surfaceenergy balance in global circulation models �GCMs�.1Snow surface albedo is particularly important be-cause snow cover—having a high reflectance, occu-pying �46% of the land in the Northern Hemispherewinter2 as well as vast areas of winter sea ice,3–5 andexerting a strong positive feedback effect on the sur-face energy budget6–11—exhibits a notorious varia-tion of surface albedo with the aging of the snow pack,variation of solar zenith angle ��0�, and changes in thespectral composition and angular pattern of the in-

S. Li �[email protected]� is with the Geophysical Institute, Uni-versity of Alaska Fairbanks, 903 Koyukuk Drive, Fairbanks,Alaska 99775. X. Zhou �[email protected]� is with the Departmentof Earth and Environmental Science, New Mexico Institute ofMining and Technology, 801 Leroy Place, Socorro, New Mexico87801.

Received 21 January 2003; revised manuscript received 12 June2003.

0003-6935�03�275427-15$15.00�0© 2003 Optical Society of America

2

coming radiation under various cloud coverconditions.12–16

Under clear skies, the snow surface albedo in-creases with �0 because of the strong forward scat-tering from the snow grains in the snow cover.12,17,18

Cloud cover can change the effective incidence an-gle.12 Clouds also absorb more strongly in the infra-red than in the visible: Therefore under cloudyconditions a greater portion of the incident irradianceis at visible wavelengths where the snow and icealbedo is larger. Consequently, the snow and icesurface all-wave albedo under cloudy skies is typi-cally 4–11% larger than clear-sky values when thesolar incidence angle is not extremely large.16,19–21

To reduce the uncertainty of snow surface albedo inGCM models and snow and ice physical and biologicalprocess models, it is more appropriate to evaluatesnow surface all-wave albedo through integration ofdirect beam spectral albedo over the full range ofincidence angles with the pattern of directional down-welling spectral radiance as a weighting function.The advantage of such an approach is that the directbeam spectral albedo is intrinsic only to the surfacecondition.

With radiative transfer �RT� models we are able toproduce a surface direct beam spectral albedo over afull range of incidence angles for certain snow-covered surface types.12,22 However, a thorough val-idation of such models is hampered because of the

0 September 2003 � Vol. 42, No. 27 � APPLIED OPTICS 5427

difficulty in measuring the surface direct beam spec-tral albedo for a wide range of incident angles. Thisdifficulty is caused by two facts. One is the lack ofclear-sky conditions for most areas. For example,we had only five clear-sky sea ice stations during our45-day cruise in the Southern Ocean in February andMarch 2000. The other is the narrow range of solarincidence angles that occur during measurement.This is especially true when the measurements aremade at high latitudes and within a few hours duringa day. In the field of snow and sea ice investigation,there are few direct beam spectral albedo data avail-able that cover a wide range of snow and ice types anda full range of incident angles.

To expand the snow surface albedo database for thesake of improving snow surface albedo representa-tion in GCMs and other physical and biological pro-cess models, we experimented with a new method toderive a surface direct beam spectral albedo for awide range of solar incidence angles using surfacedirectional spectral reflectance measurements madeunder overcast conditions. The derivation is basedon the reciprocity between the direct beam spectralalbedo and the hemispherical-directional spectral re-flectance values. In this study we first provide atheoretical error analysis of the reciprocity betweenthe hemispherical-directional spectral reflectancemeasured under overcast conditions and the directbeam spectral albedo for clear skies. We then use amultilayer zenith- and azimuth-dependent RT modelto simulate both snow surface direct beam spectralalbedo under clear skies and directional spectral re-flectance under various overcast conditions. Fi-nally, we assess the feasibility of the reciprocalmethod in practical use by comparing the two sets ofsimulation results.

2. Theoretical Error Analysis of the Reciprocal Method

In optical reflectometry, there is reciprocity betweenthe hemispherical-directional spectral reflectivityR��v, �v� and the clear-sky direct beam spectral al-bedo b,��0, �0� �Fig. 1�:

R��v, �v� � b,��0, �0�, for �v � �0, �v � �0

(1)

when the incident radiance from the hemisphericalillumination is uniform in all directions.23 Here thesubscript � represents wavelength dependence and bdenotes beam incidence. Hereafter �x and �x repre-sent zenith and azimuth angles with the subscript xreplaceable by v for viewing direction, 0 for beamincidence, and i for downwelling diffuse incidence ingeneral. However, these distinctions blur when thebidirectional reflectance quantities originally definedunder clear skies are adopted for overcast conditionsand some basic reciprocal relations23 are pursued.To avoid confusion of the subscripts of the directionalvariables, we follow the convention for the bidirec-tional quantities: The directional variables in pa-rentheses are related to incidence if they appearbefore the semicolon, and those after it are related toreflection.

Note that there is an azimuthal dependency of thetwo quantities in Eq. �1�. This implies that Eq. �1�holds even for surfaces that exhibit a variation ofdirect beam spectral albedo with the azimuth of thebeam incidence if the isotropic condition of the inci-dent radiance is met. For snow surface, such a vari-ation can be caused by the formation of snow dunes,barchans, and sastrugi at the snow surface.15

Although the ground-level sky diffuse light underovercast skies seems to be isotropic on the basis ofvisual observation, it may actually be isotropic.Therefore whether we can produce accurate esti-mates of snow surface direct beam spectral albedousing the reciprocal method for typical overcast con-ditions is a question we must answer to assess thefeasibility of the reciprocal method in practical use.To answer this question, we first conduct a theoreti-cal analysis of errors in the derived direct beam spec-tral albedo when the sky diffuse light is arbitrarilyanisotropic.

According to Siegel and Howell,23 thehemispherical-directional spectral reflectance un-der overcast conditions can be expressed as

R��v, �v� �

�� L�2��i, �i� fr,��i, �i; �v, �v�cos �i sin �id�id�i

Edif,�2 ,

(2)

where fr,� is the bidirectional reflectance distributionfunction �BRDF�.24 The downwelling diffuse irradi-ance Edif,�

2 equals the hemispherical integral of down-welling diffuse radiance L�

2:

Edif,�2 � �� L�

2��i, �i�cos �i sin �id�id�i. (3)

R��v, �v� can be directly measured in the field witha spectroradiometer and a reference Lambertian sur-face with known spectral reflectance values. Be-cause R��v, �v� is independent of azimuth for thickstratus cloud and most surfaces, R��v, �v� � R��v� �

Fig. 1. Reciprocity between the clear-sky direct beam spectralalbedo and the hemispherical-directional spectral reflectancewhen the incident radiance is uniform.

5428 APPLIED OPTICS � Vol. 42, No. 27 � 20 September 2003

R��v�. We can evaluate R��v� by averaging R��v,�v� over the �vterms:

R��v� � R��v� �

�0

2

R��v, �v�d�v

2

�

2Edif,�2 �� L�

2��i, �i� fr,��i, �i; �v, �v�

� cos �i sin �id�id�id�v. (4)

This expression is especially useful for measure-ment because the averaging can significantly reducethe impact of random errors. Because direct beamspectral albedo also does not depend on azimuth an-gle for most homogeneous natural surfaces, we useR��v� to compare with b,��0 � �v� throughout therest of the paper.

Similarly, BRDF depends on the relative azimuthangle between incidence and viewing directions in-stead of individual azimuth angles for most homoge-neous natural surfaces. Hereafter, we primarily usebidirectional reflectance function �BRF� instead ofBRDF to describe the anisotropic nature of surfacereflectance. BRF is a dimensionless quantity result-ing from when the BRDF is multiplied by a scalingfactor . The advantage of our using BRF is thatBRF values generally have a magnitude similar tothe surface spectral albedo. We can introduce theazimuthally averaged BRF and derive the reciprocityof the azimuthally averaged BRFs �R��i; �v� versusR��v; �i� when the viewing and incidence zeniths areinterchanged on the basis of the underlying nonre-strictive reciprocity between BRDFs23:

R��i; �v� �

�0

2

fr,��i, �i; �v, �v�d��v � �i�

2

�

�0

2

fr,��i, �i; �v, �v�d�v

2

�

�0

2

fr,��v, �v; �i, �i�d�i

2

� R��v; �i�. (5)

Also, we introduce azimuthally averaged sky dif-fuse spectral radiance,

L�2��i� �

�0

2

L�2��i, �i�d�i

2(6)

and express the direct beam spectral albedo b,� interms of the BRDF23:

b,��0 � �v� � �� fr,��v, �v; �i, �i�

� cos �i sin �id�id�i. (7)

Using Eqs. �4�–�7� we obtain the discrepancy be-tween azimuthally averaged hemispherical-directional reflectance and the direct beam albedo:

�b,��0 � �v� � R��v� � b,��0 � �v�

�2

Edif,� � L�2��i� R��i; �v�cos �i sin �id�i

� �� fr,��v, �v; �i, �i�cos �i sin �id�id�i,

�2

Edif,�2 � L�

2��i� R��v; �i�cos �i sin �id�i

� 2 � R��v; �i�cos �i sin �id�i. (8)

Introducing L�2� Edif,�

2 �, �L�2��i� � L�

2��i� � L�2,

and �R��v; �i� � R��v; �i� � b,��v�, we can furtherexpress Eq. �8� as follows:

�b,��0 � �v� �2

Edif,�2 �

0

�2

�L�2 � �L�

2��i�

� R��v; �i�cos �i sin �id�i

� 2 �0

�2

R��v; �i�cos �i sin �id�i

�2

Edif,�2 �

0

�2

�L�2��i� R��v; �i�cos�i sin �id�i

�2

Edif,�2 �

0

�2

�L�2��i��R��v; �i�cos�i sin �id�i

�2b,��v�

Edif,�2 �

0

�2

�L�2��i�cos�i sin �id�i.

(9)

Based on the definition of �L�2��i� and Eqs. �3� and �6�,

�0

�2

�L�2��i�cos �i sin �id�i

� �0

�2

L�2��i�cos �i sin �id�i

� �0

�2

L�2 cos �i sin �id�i

�Edif,�2

2�

Edif,�2

2� 0. (10)


Thus we finally obtain the following relation:

�b,��0 � �v�

�2

Edif,� �0

2�L�

2��i��R��v; �i�cos �i sin �id�i.

(11)

This means that the error �b,��0 � �v� depends onthe anisotropy of both sky diffuse light and surfaceBRF. When sky diffuse radiation is uniform, all�L�2��i� values are zero, and �b,��0 � �v� must be

zero. This is exactly the condition for the reciprocityin Eq. �1� to hold. When R��v; �i� is constant, thesurface is Lambertian. �b,��0 � �v� must be zerotoo. From the viewpoint of statistics,25 �b,��0 ��v� is obviously proportional to the covariance of thezenith-dependent and azimuthally averaged sky dif-fuse radiance and surface BRF. Consequently, theerror would be small when the sky diffuse light andsurface BRF vary randomly. For general overcastconditions, therefore, we focus on the relations among�b,��0 � �v�, R��v; �i�, and L�

2��i� only when skydiffuse light and surface BRF exhibit nonrandomvariations.

3. Simulation of Snow Surface Clear-Sky Direct BeamSpectral Albedo and Diffuse Sky DirectionalSpectral Reflectance

To study nonrandom variation of diffuse sky light andsnow surface BRF patterns, we use a plane-parallellayered model to represent stratus clouds and snowcover. We use a RT model to simulate both directbeam spectral albedo under clear skies and surfacehemispherical-directional spectral reflectance undervarious overcast conditions.

A. Models Used in Simulation

An azimuth- and zenith-dependent plane-parallel RTcode26 is used in simulation. The code is based ondoubling and adding methods27–29 to handle the cou-pling of multiple layers, each of which can be opticallythick. The code takes the single-scattering albedo,asymmetry factor, and optical thickness values of in-dividual layers as input. Given the solar incidenceangle above the top layer and the albedo of an under-lying surface beneath the lowest layer, the modelproduces a full range of angular patterns of the up-welling and downwelling spectral radiance at the topand bottom of the individual layers. The handling ofzenith and azimuth variation of the radiation field isrealized through Gaussian quadrature-based divi-sion in the zenith domain and evenly distributed sec-ond divisions in the azimuth domain. Fourierdecomposition of the RT equation in the azimuth isperformed by fast cosine expansion30 of the single-scattering phase function in the azimuth domain tosave computational time. Specifically, we subdivideeach of the up and down hemispheres into 16 �zenith�by 64 �azimuth� divisions to study the patterns ofsnow surface BRF under clear skies and downwellingdiffuse radiance and hemispherical-directional reflec-

tance at the snow surface under overcast skies. Wealso use this model to calculate the snow surfacedirect beam spectral albedo through hemisphericalintegration of BRF.

1. Snow ModelTo simplify the simulation, we use a one-layer modelto simulate snow surface spectral albedo �Fig. 2�a� .To investigate the relation between the error in thedirect beam spectral albedo estimation and the an-isotropy of surface reflectance, the same model isused to simulate the snow surface BRF and azimuth-ally averaged BRF patterns. The specifications ofthe model are listed in Table 1. The selection ofsnow grain radii �rs� is conventional because 0.2 and1.0 mm are traditionally considered as the fine-grained old snow and old snow near the meltingpoint.12 Hereafter we call them medium and largesnow grain sizes for simplicity. A snow opticalthickness ��s� of 100 is also adequate because snowpack may be semi-infinite although it is often opti-cally thick.

The selection of a snow particle model is a challenge.Recently, Mishchenko et al. compared three basicshape models, i.e., irregular, spherical, and hexagonal,for snow particles to study the impact of nonsphericityof snow particles on their scattering properties.31 Theresulting differences in the asymmetry factor arelarge. Field observations of snow pack suggest thatsnow particles tend to become rounded because ofmetamorphism as they age.32,33 Under near-meltingconditions, they are further sintered together to formcomposite grains because of melt and refreeze condi-tions.34 These facts imply that a spherical model, ir-regular model, or something in between is appropriatefor medium and large snow grains.

To simplify the simulation, we use a spherical modelin this investigation for several reasons. First, simu-lation of single-scattering properties of spherical par-ticles is well established, and the computer codes areavailable to us.35,36 Second, our focus in this study isthe snow surface direct beam spectral albedo. Thespherical model that has been proven appropriate forsimulation of snow surface albedo14,37 is thus used inthis study. Third, our particular interest in this re-search is the accuracy of the reciprocally derived al-bedo. As indicated by Kokhanovsky and Macke,38 aspherical-shape model generates the largest asymme-try factor among all convex large nonspherical parti-

Fig. 2. �a� Snow model and �b� coupled cloud–snow model used inthe RT simulation.


cles. Our trial simulations suggest that larger albedoerrors occur when larger asymmetry factors are usedin the reciprocal approach and when the single-scattering albedo values are kept the same in all cases.Therefore selection of the spherical-shape model helpsset an upper boundary of the errors of the albedo de-rived from the reciprocal method.

With the assumption of sphericity, the single-scattering properties such as the single-scattering al-bedo and the asymmetry factor are computed with aMie code with the complex refractive index of ice andthe selected effective snow grain radius as input.35

Once the asymmetry factor is obtained, the scatteringphase function is calculated as a function of scatter-ing angle by use of the Henyey–Greenstein �H-G�approximation.39 Aoki et al. compared field mea-surements of snow surface BRF with both Mie andH-G phase functions and found that the H-G phasefunction was more suitable than the Mie phase func-tion.37 Their investigation and the simulation byMishchenko et al.31 both indicate that adoption of theH-G phase function will have only a negligible impacton the accuracy of the albedo simulation.

Solar incidence angles used in the simulation areselected from the abscissas of Gaussian quadraturewith 16 zenith divisions.30 They are selected for di-rect comparison with hemispherical-directional spec-tral reflectance at the snow surface calculated from thecloud–snow coupled model described in Subsection3.A.2. In addition, solar incidence angles of 0°, 30°,65°, and 80° are used to represent vertical incidenceand incidence with small, moderately large and extralarge solar zenith angles in the investigation of BRFpatterns as a function of solar zenith angle.

2. Diffuse Sky ModelWe use the same RT model for a two-layered stratuscloud and snow-coupled model �Fig. 2�b� . The de-tails of the two-layer model are also listed in Table 1.

To study the impact of various sky conditions on theaccuracy of the derived direct beam spectral albedofrom the reciprocal method, 72 sky conditions areformed from combinations of eight cloud optical thick-nesses ��c�, three equivalent radii of cloud droplets �rc�,and three solar incidence angles above clouds ��0c�.The selection of �c is based on a ground-based remotesensing study of clouds in the Arctic,40 as well as a25-month database of stratus cloud properties at aSouthern Great Plains site.41 The mean �c in bothinvestigations is around 25–30 with a standard devi-ation of 13–20. Therefore �c values that range from 5to 60 are used in the simulation. The selection of rc’s�rc � 4, 8, 14 �m�, the size distribution parameter �� 8�, and the wavelengths are identical to those in Le-ontieva and Stamnes40 so that the single-scatteringalbedo and asymmetry factor of the cloud layer at thegiven wavelengths can be directly obtained from theirresults. Three above-cloud solar incidence angles��0c � 30°, 65°, and 80°� are applied to the model tostudy the �0c effect on the accuracy of the estimates.

The bottom snow layer is identical to that used inthe simulation of the clear-sky snow surface albedoand the BRF.

The output of the simulation includes thehemispherical-directional reflectance pattern and thedownwelling sky diffuse spectral radiance pattern atthe cloud–snow interface. The former is comparedwith the direct beam spectral albedo from the clear-sky snow model to assess the accuracy of the esti-mates from the reciprocal method. The latter isused to analyze the relation between the error in thedirect beam albedo estimation and the departure ofsky diffuse light from isotropy.

B. Simulation Results

1. Clear-Sky Snow Surface AlbedoFor a homogeneous snow layer with an optical thick-ness ��s � 100� underlain with a moderately dark

Table 1. Specifications of Snow and Cloud–Snow Coupled Models Used in the Simulation

Snow Model Cloud–Snow Coupled Model

Model Used One-layer RT Model Model Used Two-layer RT Model

Snow grain radius �rs� 0.2, 1.0 mm Equivalent radius of cloud droplets�rc�

4.0, 8.0, 14.0 �m

Cloud droplet size distribution pa-rameter ��

8

Snow optical thickness ��s� 100 Cloud optical thickness ��c� 5, 10, 15, 20, 25, 30, 40, 60Wavelengths �� 415, 500, 610, 665,

862, 2250 nmWavelengths �� 415, 500, 610, 665, 862, 2250 nm

Solar incidence angles �for BRFcomparison with results fromreciprocity� ��0�

5.9°, 13.5°, 21.1°,28.6°, 36.0°, 43.2°,50.1°, 56.8°, 63.1°,68.9°, 74.3°, 79.0°,83.0°, 86.1°, 88.4°,89.7°

Solar incidence angles above cloud��0c�

30°, 65°, 80°

Solar incidence angles �for BRFinvestigation� ��0�

0°, 30°, 65°, 80°

Bottom albedo �Lambertian� 0.2 Bottom snow layer See snow model to the left except forsolar incidence angles


Lambertian surface �albedo of 0.2�, the snow surfacedirect beam spectral albedo �b,�� at the six selectedwavelengths were calculated for the solar zenith an-gles ��0� listed in Table 1. Figure 3 shows the pat-terns of the direct beam spectral albedo as a functionof �0 with snow grain size �rs� as a parameter. Forall cases, snow surface b,� increases with �0. Therate of increase of b,� increases with �0. This non-linear feature becomes more and more conspicuous asthe wavelength increases. The b,� also increaseswith decreasing rs with a greater sensitivity at near-infrared �NIR� and shortwave-infrared �SWIR� wave-lengths �� 862 and 2250 nm�. Figure 4 shows thesame simulation with the wavelength used as a pa-rameter. For both medium- and large-sized snowgrains �rs � 0.2 and 1.0 mm�, b,� values are all highat visible wavelengths �� 415–665 nm�, and differ-ences among them are barely discernable. As thewavelength further increases, the spectral albedo atthe NIR and SWIR wavelength regions have muchlower values. Because we do not focus on variationof snow surface albedo with wavelength and snowgrain size in this paper, we do not discuss the relationamong them in detail. Nevertheless, we point outthat the selected cases cover a wide range of snowsurface direct beam spectral albedo and are represen-tative when used to assess the accuracy of the snow

surface spectral albedo derived from the reciprocalmethod.

2. Snow Surface Bidirectional ReflectanceFunction PatternWe also derive snow surface BRF patterns for theselected cases for an intuitive analysis of the errors ofthe derived albedo in relation to the anisotropy ofsurface reflectance, although the theoretical back-ground was discussed in Section 2.

Figure 5 shows representative BRF patterns forvarious cases. Figures 5�a� and 5�b� provide BRFpatterns with vertical solar incidence for rs at 0.2 and1.0 mm, respectively. For both snow grain sizes, theBRF patterns are symmetric with the vertical axisand exhibit a limb-darkening pattern at the visible�500-nm� and NIR �862-nm� wavelengths and a trendof general limb brightening at SWIR �2250 nm�. Atvisible and NIR wavelengths, the strongest differen-tial scattering intensity does not occur at the near-surface sublayer because of multiple scatteringamong nonabsorbing snow grains. Consequently,viewing along a vertical path will see a brighter scat-tering layer deep in the snow pack than along a slantpath with larger viewing zenith angles ��v�. Thisleads to limb darkening. At the SWIR wavelengthregion, stronger absorption of radiation by snowgrains causes a rapid decrease of scattering intensitywith snow depth. This effect favors a generally en-hanced BRF with a large �v because more brightscatterers near the surface are seen along such apath. Figures 5�c� and 5�d� present BRF patterns

Fig. 3. Direct beam spectral albedo as a function of solar inci-dence angle at wavelength �a� 415, �b� 500, �c� 862, and �d� 2250 nm.The snow grain size is given as a parameter.

Fig. 4. Direct beam spectral albedo as a function of solar inci-dence angle with snow grain sizes �a� 0.2 and �b� 1.0 mm. Wave-length is given as a parameter.

Fig. 5. Representative BRF patterns, including patterns withvertical incidence for �a� rs � 0.2 mm and �b� rs � 1.0 mm at threewavelengths, patterns along the principal plane for �c� � � 500 nmand rs � 0.2 mm and �d� � � 500 nm and rs � 1 mm under fourdifferent solar incidence �inc� angles, and full BRF patterns for �e�� 500 nm for �0 � 30° and rs � 1.0 mm and �f � �0 � 65° and rs �1.0 mm at six individual viewing zenith angles.


along the principal plane at � � 500 nm for mediumand large rs terms, respectively. The principal planeis the plane defined by the vertical axis and the vectorof the solar beam. Figures 5�c� and 5�d� reveal limbdarkening for vertical and small �0’s �0°–30°� andstrong forward scattering for large �0’s �65°–80°�.These patterns are similar to those presented in Li42

and Han et al.,43 except that the snow optical thick-ness is not semi-infinite in our study. Figures 5�e�and 5�f � provide full BRF patterns at � � 500 nm witha snow grain size of 1.0 mm for small �0 �30°� andlarge �0 �65°�, respectively. The x axis representsdifferent viewing azimuth angles ��v�, and differentcurves represent different viewing zenith angles ��v�.For small incidence angles, limb darkening is re-vealed by the generally lower position of curves forlarger �v terms than those for smaller �v terms.However, slant incidence causes a skewed BRF pat-tern that exhibits less limb darkening at the forwarddirection ��v � 0°�. For large solar incidence angles,a strong scattering peak appears at large �v terms atthe forward direction ��v � 0°�.

For an analysis of the accuracy of the reciprocallyderived direct beam spectral albedo, we further lookinto the anisotropy of the azimuthally averaged snowsurface spectral BRF �R��0; �v� based on Eq. �11�.Figure 6 shows representative snow surface R��0; �v�patterns in relation to �0, �v, �, and rs. Note thatR��0; �v� is on a logarithmic scale. Figures 6�a� and6�b� provide R��0; �v� patterns for small �30°� andextra large �80°� incidence angles, respectively. Fig-ures 6�c� and 6�d� give a three-dimensional view ofthe R��0; �v� patterns at visible �� 500-nm� andSWIR �� 2250-nm� wavelengths, respectively.Their symmetric patterns also reveal the reciprocitybetween R��0; �v� and R��v; �0� �Eq. �5� when theincidence and viewing zenith angles are inter-changed. For small �0 �Figs. 6�a� and 6�c� , the limbdarkening at visible and NIR wavelengths and the

general limb brightening at the SWIR wavelengthare similar to those for vertical incidence �see Fig.5�a� . When �0 is large, the R��0; �v� patterns ex-hibit a generally increasing trend with increasing �v�Figs. 6�c� and 6�d� because of the forward-scatteringnature of snow grains. When �0 becomes extralarge, the R��0; �v� patterns are extremely anisotro-pic �Figs. 6�b�–6�d� . It is obvious �Fig. 6�b� that, forthe extra large �0 �80°�, the R��0; �v� patterns at � �2250 nm are much more anisotropic than those atshorter wavelengths. This contrast can also be seenby a comparison of Figs. 6�c� and 6�d�.

On the basis of these simulation results and Eq.�11�, we expect that the errors of direct beam spectralalbedo derived by the reciprocal approach would belarger for extra large �0 terms than those for smalland intermediate �0 terms if the sky diffuse lightexhibits a monotonously increasing or decreasingpattern. Also, the errors for extra �0 terms would beeven larger at � � 2250 nm because the strongerabsorption of snow particles and the consequent lessmultiple scattering in the snow pack at this wave-length lead to stronger anisotropy in the surface bi-directional reflectance pattern.

3. Patterns of Sky Diffuse LightUsing the coupled cloud and snow model described inSubsection 3.A.2 and the specifications listed in Table1, we simulated the patterns of sky diffuse radiancefor 72 sky conditions and at the six wavelengths de-fined in Subsection 3.A.2 over snow packs with dif-ferent snow grain sizes �rs � 0.2 and 1.0 mm�. Tofacilitate a comparison among different cases, theradiance values in this subsection are normalizedwith respect to the case-specific averaged spectralradiance value at the cloud–snow interface.

Figure 7 shows examples of normalized L�2��i, �i�

at � � 862 nm for various �c’s and �0c’s with rc � 8 �mand rs � 1.0 mm. The wavelength � � 862 nm ischosen because the absorption of cloud droplets at the

Fig. 6. Representative R��0; �v� patterns for �a� �0 � 30°, �b� �0 �80°, �c� three-dimensional view for � � 500 nm and rs � 1.0 mm,and �d� three-dimensional view for � � 2250 nm and rs � 0.2 mm.In �a� and �b�, the symbols are given with the numerical values for� and the letters M and L for medium and large snow grain sizes.

Fig. 7. Patterns of normalized L�2��i, �i� as a function of �i with

�i �expressed as Zi in the figures� as a parameter at � � 862 nm for�a� �c � 5 and �0c � 30°, �b� �c � 5 and �0c � 80°, �c� �c � 10 and�0c � 30°, �d� �c � 20 and �0c � 30°. The cloud droplet size is 8 �mand the snow grain size is 1.0 mm.


wavelength is intermediate and the general relationof L�

2��i, �i� patterns with �c’s and �0c’s is represen-tative for other cases. Figures 7�a� and 7�b� are theresult of the same thin clouds ��c � 5� but with dif-ferent �0c’s �30° and 80°�. The L�

2��i, �i� patterns forboth cases vary not only with diffuse incoming zenithangles ��i� shown as different curves �with symbols Zi� xx.x� but also with azimuth angles ��i�. The dif-ferences between the two cases are �1� the L�

2��i, �i�pattern for large �0c is less anisotropic �the ratio ofmaximum over minimum is 2.0� than that for small�0c �the ratio of maximum over minimum is 3.5�, and�2� there is strong peak of L�

2��i, �i� at �v � 30° thatis close to the direct beam direction ��0 � 30°�whereas there is no such coincidence for �0 � 80°.These differences are due to the difference in scatter-ing path lengths when the direct beam passesthrough the cloud layer with different incidence an-gles. The larger solar incidence angle results in alonger scattering path, more attenuation, and lessanisotropic diffuse light. As cloud optical thicknessincreases �Figs. 7�c� and 7�d� , the dependence ofL�2��i, �i� on �i disappears. However, the sky dif-

fuse light still exhibits a limb-darkening pattern,which is shown in Figs. 7�c� and 7�d� where curves ofL�2��i, �i� with larger �i’s have smaller radiance val-

ues than those with smaller �i’s. These examplesindicate that, when the clouds are sufficiently thick,azimuthal dependence of sky diffuse light disappears.

For the purpose of an accurate assessment of thedirect beam spectral albedo derived from the recipro-

cal approach, we focus on the anisotropy of the azi-muthally averaged sky diffuse radiance L�

2��i� as wedid for snow surface BRF.

Figure 8 provides the normalized L�2��i� patterns

as a function of �i with �c and �0c as parameters, andwavelength �� varies among the graphs and is spec-ified in the figure caption. Only patterns for � �500, 862, and 2250 nm and rs � 1 mm are given asexamples because patterns of other combinations areeither similar to one of them or something betweenthem. In the legends, t05 and t10 denote �c � 5 and�c � 10; and inc30, inc65, and inc80 represent �0c �30°, 65°, and 80°, respectively. For moderate tolarge cloud optical thicknesses ��c � 15, 20, 25, 30, 40,and 60�, the differences among the normalized L�

2��i�patterns are extremely small. Therefore only therange and mean values are presented as the maxi-mum, minimum, and mean.

Figures 8�a�, 8�b�, and 8�c� illustrate L�2��i� pat-

terns for a wide cloud droplet size range �rc � 4, 8,and 14 �m� with � � 500, 862, and 2250 nm, respec-tively. For each of these graphs all 72 sky conditionsdefined in Subsection 3.A.2 are involved, includingcombinations from 6 �c’s, 3 rc’s, and 3 �0c’s. Compar-ing Fig. 8�a� with Fig. 8�b�, it is obvious that a vari-ation of cloud droplet size causes only slightvariations of L�

2��i� patterns for thin clouds ��c � 5�and exerts almost no impact on the L�

2��i� patternsfor thicker clouds ��c � 10� at visible and NIR wave-lengths �� 500 and 862 nm�. For thin clouds ��c �5�, L�

2��i� patterns �blue curves� vary with �0c signif-

Fig. 8. Patterns of normalized L�2��i� as a function of �i at �a� � � 500, �b� � � 862, �c� � � 2250, �d� � � 2250 nm for various sky conditions

��c � 5–60, rc � 4–14 �m, and �0c � 30°–80°� over snow pack with rs � 1.0 mm. The continuous curves represent mean values. Thedifferent colors represent different cloud optical thicknesses with blue for thin ��c � 5�, green for moderately thin ��c � 10�, and red forothers ��c � 15–60�. Different shades of blue and green curves denote different �0cs in the thin and moderately thin cloud cases. Theexception is �d� where only a �c range from 15 to 60 is involved and different colors represent different rc terms. inc, incidence; cr, clouddroplet radius.


icantly. They do not exhibit a pattern of monoto-nous increase or decrease with �i. For small �0c�30°�, the maximum of L�

2��i� appears where �i � �0c.For large �0c �65° and 80°�, the L�

2��i� maxima arelocated at the middle range of �i’s. As �c increases,the variation of the L�

2��i� pattern with �0c decreases.When �c � 15, all L�

2��i� curves appear as a singleline and reveal a monotonously decreasing pattern�limb darkening�. The rate of limb darkening is in-dependent to rc and �0c. The rate is 8.5% � 0.4% for� � 500 nm and rs � 1 mm and is 27.9% � 0.4% for� � 862 nm and rs � 1 mm.

At SWIR �� 2250 nm�, L�2��i� patterns also vary

more with �0c for thin clouds ��c � 5� than forthicker clouds �Figs. 8�c� and 8�d� . Contrary toshorter wavelengths, the cloud droplet size �rc� notonly results in a variation of the L�

2��i� pattern forthin clouds �Fig. 8�c� , but also causes a variation ofthe degree of limb darkening for moderate to thickclouds ��c � 15� �Fig. 8�d� . Clouds with a smallerrc �4 �m� appear less anisotropic than clouds with alarger rc �14 �m�. The limb-darkening factor is3.682 � 0.003 for the former and is 4.517 � 0.028for the latter.

For snow grain size rs � 0.2 mm, we do not presentfigures because the general patterns are similar tothose for rs � 1.0 mm. For �c � 15, rc � 4 � 14 �mand �0c � 30°, 65°, and 80°, the limb-darkening ratesfor � � 500 and 862 nm are 8.1% � 0.3% and 13.7%� 0.3%, respectively. At � � 2250 nm, the limb-darkening factor is 3.056 � 0.003 for rc � 4 �m andis 3.656 � 0.022 for rc � 14 �m. At � � 862 and 2250nm, the difference in the limb-darkening rate of theL�2��i� patterns above snow packs with different snow

grain sizes is caused by the difference in the snowsurface BRF patterns that contribute to multiplescattering between snow and clouds. The strongeranisotropy of the snow surface BRF with larger snowgrain size results in stronger anisotropy in the L�

2��i�patterns.

In short, the L�2��i� patterns under thin clouds

��c � 5� diverge significantly with the variation of�0c and exhibit the maximum at �i � �0c for small �0cand at the middle range of �i’s for large �0c’s. Asclouds become moderately thin ��c � 10�, the vari-ation of L�

2��i� with �0c decreases significantly.For moderate to thick stratus clouds ��c � 15�, theL�2��i� pattern has a stable limb-darkening pattern.

At visible and NIR wavelengths, the rate of slight tomoderate limb darkening ��8% for � � 500 nm and14–28% for � � 862 nm� is independent of skyconditions, including variations of �c, rc, and �0c, butdependent on rs for � � 862 nm. At SWIR �� 2250 nm�, the high rate of limb darkening �a factorof 3.06–4.52� is independent of �c and �0c, yet isdependent on both rc and rs. For clouds betweenthin and moderately thin thicknesses ��c � 5–10�under large �0c’s ��0c � 65° and 80°�, the overallL�2��i� patterns appear less anisotropic than those

under thicker clouds ��c � 15�, although the formeris less stable and more sensitive to a slight changeof sky conditions than the latter.

4. Patterns of Diffuse Sky Snow SurfaceHemispherical-Directional Spectral ReflectanceR��v�Using the same coupled cloud and snow model de-scribed in Subsection 3.A.2, we also simulated thepatterns of the diffuse sky snow surfacehemispherical-directional spectral reflectance R��v�under various sky and surface conditions. Figure 9shows patterns of azimuthally averagedhemispherical-directional spectral reflectance as afunction of viewing zenith angle �v for various skyconditions. The same patterns can also be viewed asthe reciprocally derived direct beam spectral albedoestimates as a function of solar incidence angle �0with �0 � �v. Note that in each graph the R��v�values are given for all 72 sky conditions defined inSubsection 3.A.2. The dark blue and red curves pro-vide mean values of R��v� for the cloud optical thick-ness ranges �c � 5–60 and �c � 10–60, respectively.The two curves are so close to each other that theyform a single curve except for � � 415–610 nm wherethe two curves slightly diverge only at large �v’s �e.g.,Fig. 9�a� . The upper and lower boundaries of R��v�are also given by discrete symbols of the same colorsfor these two given �c ranges. It is obvious that theR��v� values are almost independent of rc and �0c formoderate to thick clouds ��c � 10–60� although in-clusion of thin clouds ��c � 5� leads to much widedivergence because of the wide variation of the L�

2��i�patterns with �0c above thin clouds �see Subsection3.B.3�. The simulated R��v� patterns at all the se-lected combinations of � and rs show a monotonousincreasing trend with increasing �v �Figs. 9�a�–9�f � .These results suggest that we can obtain consistentand stable estimates of the direct beam spectral al-bedo from the reciprocal approach for moderate tothick clouds ��c � 10–60�. In Figs. 9�a�–9�f � the di-rectly simulated spectral albedo b,��0 � �v� valuesare also given as continuous green curves for com-parison. In addition, the cases with the minimumroot-mean-square �rms� errors with respect tob,��0 � �v� are presented as light blue curves in thegraphs with the curve for Best.

4. Comparison and Error Patterns

From Figs. 9�a�–9�c� it is obvious that R��v� matchesb,��0 � �v� well for small to moderate �v’s �or �0’s�and then departs from b,��0 � �v� more and more as�0 further increases. To evaluate the overall perfor-mance of the reciprocal method, we plot in Fig. 10 thereciprocally derived R��v� against b,��0 � �v� for allcombinations of 16 �0’s, 6 �’s, 2 rs’s, 8 �c’s, 3 rc’s, and3 �0c’s listed in Table 1 except for the worst sky con-dition cases for small �0c �30°� above thin clouds ��c �5�. The diagonal line represents a one-to-one rela-tion expected for the two sets of albedos. Large de-partures from the line mean a large error in theestimation. It is obvious that most points fall on ornear the one-to-one diagonal line, indicating that theestimates from the reciprocal approach are generallygood approximations of the direct beam spectral al-


bedos. The overall correlation coefficient r2 �0.9962. Note that there are still points that are faraway from the one-to-one line. A close inspection ofthe data reveals that those points are cases for ex-tremely large �0’s �86.1°, 88.4°, and 89.7°�. Whenthe points in the rectangles are omitted for thoseextremely large �0’s, the two sets of albedos matchbetter with an improved overall correlation coeffi-cient r2 � 0.9996.

To evaluate the accuracy of the reciprocal method,we further look into the patterns of the error�b,��0 � �v� for various wavelengths and snow

grain sizes under various overcast sky conditions.Figure 11 is produced based on Eq. �8�. The mean-ings of the symbols in Fig. 11 are the same as those inFig. 9. When �0’s are not extremely large, the errorsare small to moderate. More specific informationabout error patterns is given in Figs. 12 and 13, in-cluding rms errors in relation with sky conditions forthe full �0 range �Fig. 12�a� and the reduced ��0 �83°� range �Fig. 12�b� , enlargements of the errors forthe solar zenith angle range ��0 � 75°� under typicalcloud conditions ��c � 10–60� for medium �Fig. 13�a� and large �Fig. 13�b� snow grain sizes, and the max-

Fig. 9. Comparison of direct beam spectral albedos derived from direct simulation �dir simu� and from a reciprocal approach for variouscases: �a� � � 500 nm, rs � 0.2 mm; �b� � � 610 nm, rs � 1.0 mm; �c� � � 862 nm, rs � 0.2 mm; �d� � � 862 nm, rs � 1.0 mm; �e� � �2250 nm, rs � 0.2 mm; �f � � � 2250 nm, rs � 1.0 mm. The green curve provides the direct beam spectral albedo from direct simulation.The other curves are from the reciprocal approach with different sky conditions.


imum absolute �Fig. 13�c� and the maximum relative�Fig. 13�d� errors for individual wavelengths.

For visible and NIR wavelengths �Figs. 11�a�–11�d� , small positive errors are found for small tomedium zenith angles ��0 � 45°� under typical over-cast conditions ��c � 10–60�. The errors are positivebecause limb darkening in both L�

2��i� �Figs. 8�a� and8�b� and R��0; �v� �Figs. 6�a� and 6�c� for small tomedium �0’s ��45°� results in positive covariance �seeEq. �11� . For large and extremely large �0’s, theerrors become negative because the limb darkening ofL�2��i� �Figs. 8�a� and 8�b� and the limb brightening

in snow surface R��0; �v� �Figs. 6�b� and 6�c� resultin negative covariance �Eq. �11� . The magnitude ofthe errors for large �0’s �60°–68°� is moderate. Forextremely large �0’s, the magnitude of the errors in-creases rapidly with �0. At medium zenith angles�35° � �0 � 56°�, the errors are minimum because ofthe transition of the error signs from positive to neg-ative.

For SWIR �� 2250 nm�, the errors are alwaysnegative because the snow surface R��0; �v� patternsalways show limb-brightening patterns �Figs. 6�a�and 6�d� regardless of the solar incidence anglewhereas L�

2��i� exhibits a stable limb-darkening pat-tern. The increase of limb brightening of R��0; �v�with �0 leads to an increase in magnitude of the errorwith �0 �Figs. 11�e�, 11�f �, 13�a�–13�d� . At ex-tremely large �0’s, the errors are extremely large.

The best cases presented in Fig. 11 are the best skyconditions identified by their lowest rms error valuesamong all sky conditions for individual �–rs combi-nations and are denoted by encircled solid trianglesin Fig. 12�a�. Note that the best conditions are as-sociated with thin clouds ��c � 5� with large or extralarge �0c’s. Looking back to Fig. 8, we find that thosesky conditions have the maximum L�

2��i� valueswithin the middle range of �i. When such a patternis combined with either limb-darkening or limb-brightening patterns of the R��0; �v� patterns, errorvalues tend to be small to moderate based on Eq. �11�.The signs of the individual errors depend on the signsof the covariance given in Eq. �11� and are not so

straightforward as those for the typical cloud thick-ness range ��c � 10–60�.

Figures 12�a� and 12�b� both present rms of�b,��0 � �v� under various sky conditions. Out of12 possible combinations of � and rs, only five areselected. Because the patterns in the visible wave-length region �� 415–665 nm� are similar to eachother, only a combination of � � 500 nm and rs � 0.2mm is chosen. Similarly, many moderate to thickcloud optical thickness cases ��c � 15, 20, 30, and 40�were dropped, and only 36 sky conditions are selectedfrom the 72 experimented sky conditions so that Figs.12�a� and �b� not too busy. The symbols under thehorizontal axis are organized in a �0c–rc–�c interleav-ing manner to reflect variations of �0c �S for 30°, L for65°, and X for 80°�, rc �4, 8, 14 �m�, and �c �5, 10, 25,and 60�. The difference between Figs. 12�a� and12�b� is that the former provides patterns of the rmsof �b,��0 � �v� for the full �0 range and the lattergives a pattern for the reduced range ��0 � 83°� withthe errors for the three largest �0’s dropped. Be-cause �b,��0 � �v� values for the largest �0’s are thelargest �Fig. 11�, omission of the largest errors leadsto smaller rms for the reduced �0 range �Fig. 12�b� .The best sky conditions are identified by their lowestrms error values for individual �–rs combinations.Among all �c’s, the thin clouds almost always giveboth the best overall results with large or extra large�0c’s and the worst ones �the largest rms� with small�0c’s. As �c increases, the difference in rms errorsamong different �0c’s decreases rapidly. At visibleand NIR wavelengths, the rms of �b,��0 � �v� be-comes independent of sky conditions as �c � 10. AtSWIR �� 2250 nm�, the rms of �b,��0 � �v� is moresensitive to �rc� than to �c for �c � 10. This differencecan be explained by the fact that L�

2��i� patterns areindependent of rc at shorter wavelengths whereas theanisotropy of the L�

2��i� patterns at � � 2250 nm aresensitive to rc �see Subsection 3.B.3�. The strongeranisotropy in L�

2��i� resulting from larger rc leads tolarger rms errors based on Eq. �11�.

For all combinations of visible wavelengths �� 415, 500, 610, and 665 nm� and snow grain sizes andfor all moderate to thick overcast conditions �rc � 4,8, and 14 �m; �0c � 30°, 65°, and 80°; and �c � 10, 15,20, 25, 30, 40, and 60�, the average of the rms of�b,��0 � �v� over the full �0 range is 0.0099 with astandard deviation of 0.0011. When the reduced �0range ��0 � 83°� is considered, the average and stan-dard deviation values are reduced to 0.0044 and0.0005, respectively.

At NIR wavelength �� 862 nm�, the rms errorsare also sensitive to snow grain size �Figs. 12�a� and12�b� . Under moderate to thick overcast conditions��c � 10–60�, the statistics of the rms of �b,��0 � �v�for the full �0 range are 0.014 � 0.00039 and 0.028 �0.00045 for rs � 0.2 mm and rs � 1.0 mm, respec-tively. The statistics for the reduced �0 range ��0 �83°� are reduced to 0.0066 � 0.00025 and 0.013 �0.00030, respectively. Over both full and reduced �0ranges, the averages of the rms of �b,��0 � �v� forlarge snow grains �rs � 1.0 mm� are twice as large as

Fig. 10. Reciprocally derived direct beam spectral albedos versusthe directly simulated albedo for all experimental cases except forthin clouds ��c � 5� with small �0c �30°�. The rectangles encirclepoints for extremely large �0’s �86.1°, 88.4°, and 89.7°�.


those for medium snow grains �rs � 0.2 mm�. Mean-while, the standard deviations are all much smallerthan the averages and do not vary much with snowgrain size. These statistics again indicate that theerrors of ��0 � �v� at this wavelength are moder-ate. They are larger than those in the visible wave-lengths. In addition, errors are more sensitive tosnow grain size than sky conditions.

At SWIR �� 2250 nm�, the corresponding statis-tics for the medium and large snow grain sizes are0.078 � 0.0037 and 0.049 � 0.0018 for the full �0range and are 0.026 � 0.0015 and 0.0087 � 0.00043for the reduced �0 range, respectively.

Figures 13�a� and 13�b� provide details of the pat-

terns of �b,��0 � �v� under the typical cloud con-ditions ��c � 10–60, rc � 4–14 �m, �0c � 30°, 65°,and 80°� as a function of �0 at individual wave-lengths for two snow grain sizes �rs � 0.2 and 1.0mm�. The curves are successively displaced down-ward by an amount of 0.02 except for � � 415 nm.Over the typical cloud optical thickness range ��c �10–60�, the mean, maximum, and minimum valuesof �b,��0 � �v� at individual �0’s are given as solidcurves and the discrete symbols close to the curves.The best sky conditions for individual �–rs combi-nations for the full �0 range are also provided. Fig-ures 13�c� and 13�d� present the maximumdeviations and the maximum relative errors in the

Fig. 11. Errors of direct beam spectral albedos ��b,��0 � �v� for various cases: �a� � � 500 nm, rs � 0.2 mm; �b� � � 610 nm, rs � 1.0mm; �c� � � 862 nm, rs � 0.2 mm; �d� � � 862 nm, rs � 1.0 mm; �e� � � 2250 nm, rs � 0.2 mm; �f � � � 2250 nm, rs � 1.0 mm.


b,��0 � �v� for both snow conditions �rs � 0.2 and1.0 mm�. The spectral curves are successivelyshifted by an amount of 0.01% and 1%, respectively.

It is obvious that under the typical cloud condi-tions the maximum deviations of reciprocally esti-mated b,��0 � �v� at visible and NIR wavelengths�� 415–862 nm� are within �0.0035 over therange �0 � 57°. The corresponding relative errorsare within �0.5%. The errors become larger as �0increases. For visible wavelengths �� 415–665nm�, the errors are still within �0.008 ��0.9%� upto �0 � 74°. For the NIR wavelength �� 862 nm�,the same accuracy can be achieved up to �0 � 63°.The maximum deviation reaches �0.017 when �0approaches 74° and the snow grain size is large,although the maximum deviation is smaller �within�0.009% and 1%� for the smaller snow grain size�rs � 0.2 mm� �Fig. 13�a� . The albedo estimates atthe SWIR wavelength �� 2250 nm� are accuratewithin �0.009 over the range �0 � 57° with themaximum deviations occurring with medium snowgrain size �rs � 0.2 mm�. But the maximum rela-tive errors �from �5% to �13%� for the same �0range are associated with large snow grain size�rs � 1.0 mm� because of the even smaller b,��0 ��v� values for the larger snow grains. As �0 furtherincreases, the errors become larger.

5. Discussion and Conclusions

The spherical snow particle model is selected in ourRT simulation based on our field observations of sum-mer snow cover on sea ice in the Southern Ocean. Itis well known that the spherical particle model re-sults in the largest asymmetry factor in single scat-

tering. Our trial simulations suggest that we canobtain smaller albedo errors with the reciprocal ap-proach using smaller asymmetry factor values whilekeeping the single-scattering albedo unchanged invarious cases. Therefore the assessment based onthe spherical snow particle model tends to set anupper boundary of the albedo errors that can beachieved by the reciprocal approach for actual snowcover.

The simulated snow surface reflectance isstrongly anisotropic when the solar zenith angle islarge. The anisotropy is even stronger in theSWIR spectral region because stronger absorptionreduces multiple scattering of solar radiation in thesnow pack.

Under typical stratus clouds ��c � 10–60� the skydiffuse light over snow surface exhibits slight limbdarkening at visible wavelengths, but shows a stronglimb darkening at the SWIR. The anisotropy of skydiffuse light is independent of sky conditions, includ-ing variations of �c, rc, and �0c, for all the wavelengthsinvestigated except for � � 2250 nm where sky dif-fuse light slightly depends on rc.

Consequently, stable estimates of the snow surfacedirect beam spectral albedo can be derived by thereciprocal approach over a wide range of overcast skyconditions. The results are accurate to within�0.008 up to a solar zenith angle ��0� around 74° atvisible wavelengths. The same accuracy can beachieved up to �0 � 63° for the NIR wavelength. Thealbedo estimates at � � 2250 nm are within �0.009over the range �0 � 57°, although the relative errorsare large �5–13%�. As �0 further increases, the er-rors become larger. Overall better estimates can beobtained from thin clouds ��c � 5� when �0c is large�65° and 80°�. However, the estimates can becomeworse than those from the thicker cloud cases when�0c is small.

The new method helps expand the database ofsnow surface albedo for the polar regions where di-rect measurement of clear-sky surface albedo is lim-

Fig. 12. rms of errors under selected sky conditions for �a� the full�0 range and �b� reduced �0 range ��0 � 83°�. L, large; S, small; X,extra large; WV, wavelength; r, snow grain radius.

Fig. 13. Details and statistics of �b,��0 � �v� patterns for �c � 10for �a� rs � 0.2 mm, �b� rs � 1.0 mm, �c� maximum deviation for bothrs’s, �d� maximum relative deviation for both rs’s. Wavelength isgiven as a parameter.


ited to large �0’s only. The enhancement will assistus in validating snow surface albedo models and im-proving the representation of polar surface albedo inGCMs.

Because the theoretical derivation is general forany surfaces, the method is likely to be used at visibleand NIR wavelengths for any type of Earth surfacewhere sky is not obscured by terrain, trees, and build-ings.

This research is supported by the National Aero-nautics and Space Administration under grantNAG5-6338 to the University of Alaska Fairbanks.We are grateful to the two anonymous reviewers whoprovided many constructive suggestions that contrib-uted to a significant improvement of the manuscript.We also thank the topical editor, M. I. Mishchenko,for his prompt response to our concerns and encour-agement during our modification to the manuscript.

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assessment of the accuracy of snow surface direct beam spectral albedo under a variety of overcast...

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