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JOURNAL OF CLIMATEVOLUME 17

Transmission of Solar Radiation by Clouds over Snow and Ice Surfaces: A
Parameterization in Terms of Optical Depth, Solar Zenith Angle, and Surface Albedo
MELANIE F. FITZPATRICK
Department of Earth and Space Sciences, University of Washington, Seattle, Washington

RICHARD E. BRANDTDepartment of Atmospheric Sciences, University of Washington, Seattle, Washington

STEPHEN G. WARREN
Department of Earth and Space Sciences, and Department of Atmospheric Sciences, University of Washington, Seattle, Washington(Manuscript received 21 January 2003, in final form 25 June 2003)
ABSTRACT
A multilevel spectral radiative transfer model is used to develop simple but accurate parameterizations forcloud transmittance as a function of cloud optical depth, solar zenith angle, and surface albedo, for use oversnow, ice, and water surfaces. The same functional form is used for broadband and spectral transmittances, butwith different coefficients for each spectral interval. When the parameterization is applied to measurements of‘‘raw’’ cloud transmittance (the ratio of downward irradiance under cloud to downward irradiance measured
under clear sky at the same zenith angle), an ‘‘effective’’ optical depth t is inferred for the cloud field, which
may be inhomogeneous and even patchy. This effective optical depth is only a convenient intermediate quantity,
not an end in itself. It can then be used to compute what the transmittance of this same cloud field would beunder different conditions of solar illumination and surface albedo, to obtain diurnal and seasonal cycles ofcloud radiative forcing. The parameterization faithfully mimics the radiative transfer model, with rms errors of
1%–2%. Lack of knowledge of cloud droplet sizes causes little error in the inference of cloud radiative properties.
The parameterization is applied to pyranometer measurements from a ship in the Antarctic sea ice zone; thelargest source of error in inference of inherent cloud properties is uncertainty in surface albedo.

1. Introduction
Clouds have large effects on the earth’s radiation budget, both longwave and shortwave. Lack of knowledgeof cloud distributions and cloud properties, and of the
behavior of clouds during climatic change, limits theaccuracy of climate-prediction models (Cess et al.
1990). Cloud distributions and radiative properties are
now being monitored from space by the International
Satellite Cloud Climatology Project (ISCCP; Rossow
and Schiffer 1999). There is also a need for surface
measurements of cloud properties to obtain the effects
of clouds on the radiation budget at the surface as well
as at the top of the atmosphere, and to infer cloud properties in regions where satellite cloud detection is difficult, particularly over snow and ice surfaces.
Field experiments [e.g., the Atmospheric Radiation
Measurement (ARM) Program; Stokes and Schwartz
Corresponding author address: Melanie F. Fitzpatrick, Departmentof Earth and Space Sciences, University of Washington, Box 351310,
Seattle, WA 98195.
E-mail: fitz@atmos.washington.edu

q 2004 American Meteorological Society

1994] can obtain cloud properties with fine spectral detail at a few geographic locations. However, climate
research would benefit if the routine solar radiation measurements made at weather stations worldwide, and on
many research ships, using broadband pyranometers,
could be used to determine the cloud radiative forcing(CRF) at the surface. Here we investigate what information about clouds can be obtained from broadband
measurements, with particular attention to the effect of
surface albedo on the measured downward shortwave
irradiance.
The number of weather stations measuring solar radiation is a small fraction of the number making visualcloud observations, and the number of research ships ismuch smaller than the total number of merchant ships
making routine weather observations (including cloud
observations) at sea. In order that the isolated surface
measurements of CRF can be extended to larger areasusing a cloud climatology (e.g., Warren et al. 1986,
1988), inherent cloud radiative properties are needed.
A value for cloud transmittance can be obtained just bycomparing the measured downward solar irradiance be-15 JANUARY 2004

FITZPATRICK ET AL.

neath the cloud to that obtained at the same solar zenith
angle, u, on a clear day in the same month. However,this same cloud would have a different measured transmittance if illuminated at a different zenith angle or if
positioned over a surface with different albedo, a. To
use a cloud climatology to compute surface CRF therefore requires an intermediate quantity that is inherent
to the cloud field (which may be inhomogeneous andnonovercast), which can be used to compute transmittance given u and a. This intermediate quantity is theeffective optical depth t. It is ‘‘effective’’ in two senses:(a) it is computed assuming a horizontally homogeneous
overcast cloud, and (b) the cloud is assumed to consistof liquid water droplets with a standard effective radius(reff 5 8.6 mm, as explained later). The effective cloudoptical depth is the same as the true optical depth if theobserved cloud actually is horizontally homogeneousand has reff 5 8.6 mm. In all other cases the effective
optical depth is just a convenient intermediate quantity
whose utility is determined by its ability to predict thetransmittance of the observed cloud field over surfaces
and under illumination different from those observed.
Such an effective cloud optical depth has been usedpreviously by Barker et al. (1998) and is used by ISCCP(Hahn et al. 2001; Rossow and Schiffer 1999). Boerset al. (2000) showed that errors in true average opticaldepth retrieved from ground-based pyranometers can beas large as 50% due to cloud inhomogeneities. For thisreason we are careful to note our use of optical depth
only as an intermediate quantity. It is designed to givethe correct transmittance but cannot be expected to givethe correct cloud water content.
Several previous studies have determined cloud optical properties in polar atmospheres from ground-based
measurements of solar irradiance. In polar regions withsnow-covered ground, Shine (1984) and Gardiner(1987) noted that multiple reflection between the snow
and the underside of the cloud can increase the radiation
received at the surface by a factor of 2. In characterizing
clouds, this multiple reflection must be considered.
However, Wiscombe (1975) showed that because of thestrong spectral variation of cloud albedo and surface
albedo, the difference between cloud-top reflectance andcloud-bottom reflectance (spectrally averaged) depends
on cloud optical depth, so that simple spectrally averaged multiple-bounce models like Gardiner’s cannot accurately relate transmittance of radiation through a cloud
to cloud optical depth.
Previous work on inferring cloud optical depth using
only broadband measurements was carried out by Leontyeva and Stamnes (1994). They used broadband measurements of downwelling irradiance and surface albedo
together with a multilayer radiative transfer model toiteratively determine cloud optical depth. A further
study by Leontyeva and Stamnes (1996) determinedcloud optical depth from spectral transmittance using
lookup tables that had been derived from the radiativetransfer model. In both cases good characterization of267

surface albedo was found to be necessary. Lubin andSimpson (1997) determined cloud scattering opticaldepth at visible wavelengths using a radiative transfer
model to determine the clear sky irradiance. They theniterated with a discrete ordinates model, adjusting theestimated cloud optical depth until the calculated irradiance matched the measured irradiance. All these methods rely on an iterative solution to the radiative transfer
equation for each irradiance measurement, a time-consuming process if large quantities of data are to be processed.
To simplify this step, a simple parameterization ispresented here that relates cloud transmittance to cloudoptical depth, surface albedo, and solar zenith angle.
This parameterization can mimic the results of a multilayer, multiwavelength radiative transfer model. We
apply this parameterization to obtain cloud properties
from radiation measurements made during a ship voyage
in the Southern Ocean.
2. Parameterization
We define a quantity, ‘‘raw’’ cloud transmittance (trc),that can be obtained from the measurements alone without any model. It is simply the ratio of downward irradiance measured under cloud to that measured underclear sky at the same solar zenith angle. It is this quantity, trc, for which we will derive a parameterization.
Using a two stream multiple-reflection model, Wiscombe (1975) motivated the form of a parameterization for
Arctic summer stratus clouds, relating cloud transmittance to cloud droplet number density for a grey underlying surface. We have rederived this parameterization using a spectrally varying surface albedo anddifferent cloud drop-size distributions. In the original
paper on the two stream method, Schuster (1905)
showed that the monochromatic transmittance througha nonabsorbing, isotropically scattering atmosphere of
optical depth t is just 1/(1 1 t). Wiscombe (1975, his
Eq. 24) generalized this functional form as 1/(c 1 dn c ),where c and d are empirical coefficients and n c is cloud
droplet number density. We replace cloud droplet number density with cloud optical depth t, in the geometrics–optics limit, as discussed later. Cloud transmittanceis greater if the sun is higher, and Wiscombe (1975)
found that a linear function of cosu was adequate todescribe this effect. Wiscombe accounted for multiple
reflections between cloud and ground by including aspectrally averaged albedo in the denominator as a multiplier of cloud droplet number density. However, Shine(1984) showed that neglecting the spectral variation ofhigh surface albedos leads to an underestimate of surface irradiance. In the radiative transfer model used todevelop the parameterization we therefore use a spectrally varying surface albedo, with spectral shapes characteristic of snow and ice surfaces. The broadband parameterization itself uses the spectrally averaged albedo,
so it is valid only for surfaces of snow, sea ice, and268

JOURNAL OF CLIMATEVOLUME 17

FIG. 1. Schematic diagram of the physical basis of the parameterization. The cloud is underlain by two different surface types: a dark
surface on the right and a bright surface, which supports multiple
reflections, on the left. The measured raw cloud transmittance, trc, is(i) proportional to the cosine of the solar zenith angle u, (ii) inversely
proportional to cloud optical depth t, and (iii) increased by multiple
reflection between the cloud base and the surface with albedo a.

water; however, the narrowband parameterizations derived next should be generally valid.
Our parameterization isa(t ) 1 b(t ) cosu
trc 5.
1 1 (c 2 da)t(1)

Figure 1 shows the physical reasoning behind this parameterization. The coefficients a and b were constantsin Wiscombe’s (1975) parameterization; however, tomake the parameterization accurate in the limit of t →0 we found it necessary to make these coefficients functions of t :a(t ) 5 a1 1 (1 2 a1 ) exp(2k1 t )andb(t ) 5 b1 [1 1 b2 exp(2k2 t ) 1 b3 exp(2k3 t )].

E

r 2 n(r) dr,

where h is cloud thickness, r is cloud droplet radius,and n(r) is the cloud drop-size distribution. Effective
radius is defined as the ratio of the third moment to thesecond moment of the particle radius distribution function, and is given by

reff 5(2)(3)

Figure 2 shows the coefficients a(t) and b(t): they asymptote to constant values for large t. At small t, aapproaches 1 and b approaches 0, removing the solar
zenith angle dependence of cloud transmittance. We
tried a variety of t dependencies of a and b, including
lnt and t 2 , but the best relationship found is an exponential dependence. The coefficients were determinedusing a nonlinear least squares algorithm (Press et al.
1992, 678–683) separately for each of three different
drop-size distributions, covering a range of solar zenith
angles, surface albedos, and cloud optical depths.
The optical depth used in (1), and which we attempt
to retrieve from measurements of trc, is the geometric–
optics optical depth, t g ; that is, with extinction efficiency Qext 5 2. We use t g to describe the clouds because
it is a quantity that is independent of wavelength. Allforms of the parameterization given later, whether
broadband or spectral, have coefficients chosen to givet g . The relationship between cloud optical depth anddrop-size distribution is given byt g 5 2phFIG. 2. The functional forms of a(t) and b(t) used in the broadband
parameterization (for reff 5 8.6 mm) to ensure correct asymptotic
behavior for both thin clouds and thick clouds.

(4)

E
E

r 3 n(r) dr.

(5)r n(r) dr2

A representative drop-size spectrum for Arctic stratus
clouds was given by Jayaweera and Ohtake (1973). It
has reff 5 8.6 mm. Observed effective radii for springtime cloud droplets in the Southern Ocean range from
4.5 to 16.6 mm (Boers and Krummel 1998). They were
measured between 408 and 558S during the First Aerosol
Characterization Experiment (ACE-1), in November
and December 1995. No trend was observed in reff within this latitude range, so these values may also be representative for other parts of the Southern Ocean (R.
Boers 2000, personal communication). However reff isprobably smaller in summer than in winter because there
are more cloud condensation nuclei in summer (Ayers
and Gras 1991).
3. ModelTo derive the coefficients of the parameterization we
use a multilayer atmospheric radiative transfer model,
ATRAD (Wiscombe et al. 1984), with 118 spectral intervals in the UV, visible, and near-infrared. The model
assumes a plane-parallel, horizontally homogeneous atmosphere, and neglects polarization. The adding–doubling method is used to compute radiative transfer. Ab-

15 JANUARY 2004

FITZPATRICK ET AL.

FIG. 3. Spectral albedos used to develop the parameterization for
12 different surfaces, ranging from open water to fine-grained snow
for a solar zenith angle of 608 under a clear sky. The labels indicate
the spectrally averaged albedo for each surface type under these conditions.

sorption and scattering by aerosols, clouds, and all natural atmospheric gases are included. Exponential sum
fitting of transmittance (ESFT) is used to computegaseous absorption. Mie theory is used to compute scattering and absorption by cloud droplets, with exact calculations (Wiscombe 1980) performed for size parameters below 100, and approximate calculations (Nussenzveig and Wiscombe 1980) for larger-size parameters.
We used the top-of-atmosphere solar spectrum as given by Labs and Neckel (1970) and Neckel and Labs
(1984). For atmospheric profiles of temperature, ozone,
and water vapor we used the subarctic summer standard
atmosphere (McClatchey et al. 1972) modified to force
a relative humidity of 100% (with respect to liquid water) within the cloud. For all optical depths used, thecloud base was placed at 0.6 km and the cloud top at
1.3 km; the optical depth was varied by varying thedroplet number density, keeping the cloud thickness
fixed. This assumption, as well as the assumption thatreff is independent of height within the cloud, has little
effect on the computed trc.
The underlying surface is described by a spectral albedo, 1 of 12 choices shown in Fig. 3; the reflectedradiation is distributed with angle according to an azimuthally averaged bidirectional reflectance function
given by Warren (1982, his Fig. 15). The 12 spectral
albedos are meant to represent surfaces ranging fromfine-grained new snow (uppermost curve), throughcoarse-grained old melting snow, to granular sea ice of
decreasing scattering coefficient. The nearly horizontal
curve at the bottom is that of open water from Briegleb
and Ramanathan (1982), but it more closely resembles
the spectral albedo measured for thin nilas by Brandt etal. (1999). These spectral albedos for hypothetical (mod-269

FIG. 4. Three different drop-size distributions used in developing
the parameterization. The distribution with reff 5 8.6 mm is an observed distribution measured in Arctic stratus clouds (Jayaweera andOhtake 1973). The other two are lognormal distributions (Reist 1993,
chapter 2).

eled) surfaces types were used because they are available in the ATRAD model. However, they do resemble
spectral albedos measured for snow and for various
types of sea ice (Grenfell et al. 1994; Grenfell and Perovich 1984; Brandt et al. 1999).
The spectral and broadband cloud transmittance wascomputed for three zenith angles (cosu 5 0.2, 0.4, 0.6),
14 cloud optical depths (t from 0.02 to 88 with geometric spacing, and t 5 0 for clear sky), and the 12
surface albedos with broadband albedos from 0.09 (openwater) to 0.87 (fine-grained snow); a total of 504 different conditions. They were then repeated for the threedifferent drop-size distributions shown in Fig. 4. The
standard drop-size distribution is that measured for water droplets in an Arctic stratus cloud by Jayaweera andOhtake (1973); it has an effective radius of 8.6 mm. The
other two drop-size distributions are lognormal distributions with effective radii of 6 and 20 mm.
The parameterization proved difficult to fit to ATRAD
results for surface albedos a $ 0.83. Furthermore, theinference of t becomes increasingly indeterminate as a
→ 1. We therefore fitted the coefficients for the parameterization only to model results for a # 0.83. This is
not a severe restriction, as sea ice scenes usually containsome open water or thin ice that lowers the area-averaged albedo.
4. Resultsa. Broadband parameterization
A broadband parameterization, covering the solar
spectrum from 0.3 to 3.0 mm, so as to cover the spectral
range measured by pyranometers, is derived for each of

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JOURNAL OF CLIMATEVOLUME 17

TABLE 1. Values of coefficients in Eqs. (1), (2), and (3) for the broadband parameterization (0.3–3.0 mm) determined using three different
drop-size distributions. The last column gives the root-mean-square difference between values of cloud transmittance obtained from thespectral radiative transfer model and those obtained using the parameterization.
reff

a1

b1

b2

b3

k1

k2

k3

c

d

Rms (%)

6.0 mm
8.6 mm
20 mm

0.58
0.58
0.58

0.75
0.74
0.71

20.2053
20.161220.742020.7935
20.8343
20.26612.0506
1.97850.56000.19680.28280.28812.47902.3042
3.12930.14090.13650.12060.13290.1291
0.1133

1.2
1.3
2.4

the three different drop-size distributions. The values of
the coefficients in Eqs. (1), (2), and (3) are listed inTable 1. Cloud transmittance values derived using theparameterization with an effective radius of 8.6 mmshow an rms error of 1.3% when compared with values
obtained from the radiative transfer model (Fig. 5). A
synthetic dataset is used to determine the difference in
optical depths obtained from the three different dropsize distributions, using values of trc, cosu, and a toderive t. Figure 6 shows the inferred optical depths that
result from different assumptions for the effective radius, at a solar zenith angle of 608 and a surface albedo
of 0.55. The comparison was performed for other valuesof solar zenith angle and surface albedo and a similar
percentage variation in t is obtained. For a given cloudoptical depth, a cloud with large particles has greater
forward scattering and hence greater cloud transmittance. For the range of observed effective radii in theSouthern Ocean, incorrect assumptions of effective radius in the parameterization introduce a 10%–15% error
in the inferred effective optical depth.
However, for climatological applications, effective
optical depth is simply an intermediate quantity thatallows us to find the transmittance of the same cloud

FIG. 5. Comparison of transmittance obtained from the radiativetransfer model to that obtained from the parameterization, using a
synthetic dataset, an rms error of 1.3% is shown.

field over different surfaces and at different solar zenith
angles. The errors introduced in predicted cloud transmittance, caused by assuming an incorrect drop-size distribution, are shown for one example in Fig. 7. Using
an initial measured transmittance trc over a low-albedo
surface (a 5 0.1), an effective optical depth t is derived
for each of two drop-size distributions. This intermediate quantity t is then used to predict the transmittanceof the observed cloud field over a different surface (a5 0.6). The error in effective optical depth is typically
10%–15%, but the error in predicted transmittance «trc
is usually less than 2%. The procedure of Fig. 7 was
carried out for numerous cases of different zenith angles
and surface albedos, and the results were similar to thoseshown in Fig. 7.
In all our computations, the model clouds consistedof liquid water droplets. However, we think our results
are valid for ice clouds as well, by the following argument. The spectral absorption coefficient of ice (Warren 1984) is very similar to that of liquid water (Haleand Querry 1973) across the solar spectrum (they areplotted together for comparison by Dozier 1989, his Fig.
2b). Therefore the main reason that optical properties

FIG. 6. Effect of assumed cloud droplet effective radius, reff , on
the effective optical depth t determined from the parameterization.
To enable comparison with Fig. 7, t (the dependent variable) is usedas the abscissa. A larger effective radius of cloud droplets produces
stronger forward scattering (larger asymmetry parameter) and hence
a greater cloud transmittance than a cloud of the same optical depthwith smaller effective radius.

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FIG. 7. Example of the error in derivation of cloud radiative properties due to lack of knowledge of the drop-size distribution. The
cloud was observed over a surface of albedo 0.1, and the measured
transmittance of solar radiation (trc) was 0.35. If we assume reff 5
8.6 mm we obtain t 5 13.8. We estimate using the parameterization[Eq. (1)] that if this cloud were placed over a surface of albedo 0.6,
it would have trc 5 0.528. If we instead had larger drops (reff 5 20mm), we infer t 5 15.2, and estimate that if this cloud were placed
over the high-albedo surface it would have trc 5 0.512, very close
to that estimated using the smaller drops. Lack of knowledge of reffin this case causes errors of 10%–15% in the intermediate quantity
t but only 2% in the prediction of solar transmittance in a different
environment.

of an ice cloud would differ from those of a water cloudof the same optical depth is that the ice particles arelikely to be larger. Representing a nonspherical ice crystal by a collection of spheres having the same total
volume and the same total surface area results in sphereswith the same volume-to-area ratio (V/A) as the crystal.
Using this representation, advocated by Grenfell and
Warren (1999) and Fu (1996), any ice crystal shape canbe described by a single effective radius for computationof absorption and scattering of radiation. This prescription results in little error for the shapes tested so far(hexagonal columns and plates; Neshyba et al. 2003).
Using the equal-V/A description, the effective radii ofcrystals in ice clouds are typically 10–30 mm (Mahesh
et al. 2001; Fu 1996; C. Schmidt and A. Heymsfield
2002, personal communication). Their effect is therefore
represented by the example shown in Fig. 7.
Uncertainties in surface albedo also affect the valuesof cloud optical depth inferred using the parameterization. The albedo to be used in Eq. (1) is the actual
albedo under the observed cloud, not what the albedo
would be under clear sky. The effect on the parameterization of specifying an incorrect surface albedo isshown in Fig. 8 for a case where the estimated surface
albedo was 0.6 but the true surface albedo was different
by as much as 10%. An increase in surface albedo results
in an increase in multiple reflections of incoming radiation. For a given measured cloud transmittance, if

271

FIG. 8. Effect of errors in surface albedo estimates on the inferenceof effective cloud optical depth. This case is for a solar zenith angleof 608 and an erroneous assumed surface albedo of 0.6, with true
surface albedos in the range 0.54–0.66.

the estimated surface albedo entered into Eq. (1) is erroneously low, the inferred cloud optical depth will be
too large. The errors introduced by uncertainties in surface albedo are greater for higher surface albedos. If thetrue albedo is 0.06 greater than the estimated albedo,
the effective cloud optical depth is 15% too high, whereas if the true albedo is 0.06 lower than the estimatedalbedo, the effective cloud optical depth is only 11%
too low. (The percent difference in derived t is nearly
independent of trc, but the absolute value of t decreases
as trc increases. For trc 5 1.0, t 5 0, so the absolutedifference vanishes.) Visual reports of ice-type distributions (as discussed later) are estimated to be accurate
to 60.1 of ice concentration (A. Worby 2001, personalcommunication). We estimate that surface albedos derived from these reports are typically uncertain to 60.06
in albedo.
b. Spectral parameterization
The same form of the parameterization can also be
used for narrow spectral intervals, with different coefficients for each interval. To do this we repeat the procedure described earlier, using spectral results fromATRAD, and derive coefficients for nine different wavelengths across the solar spectrum. These values areshown in Table 2. Using the parameterization insteadof iterating through a radiative transfer model introduces
an error of less than 1% at each of the specified wavelengths. Values are applicable at each specified wavelength, over the range of albedos, optical depths, andsolar zenith angles used in the radiative transfer model.
Attempts were made to fit the coefficients as functionsof wavelength and hence generalize the parameterization across the whole solar spectrum. However, the spectral variation of downwelling irradiance, the spectral
dependence of surface albedo of snow and ice, and the272

JOURNAL OF CLIMATEVOLUME 17

TABLE 2. Coefficients in Eqs. (1), (2), and (3) found for nine different wavelengths l, assuming reff 5 8.6 mm. The last column gives the
root-mean-square difference between values of cloud transmittance obtained from the spectral radiative transfer model and those obtainedusing the parameterization.

l (nm)

a1

b1

b2

b3

k1

k2

k3

c

d

Rms (%)

370
440
560
650
720
810
925
1050
1660

0.78
0.65
0.53
0.49
0.51
0.49
0.53
0.48
0.49

0.31
0.59
0.79
0.88
0.82
0.85
0.74
0.99
1.02

20.0782
20.121220.1054
20.1276
20.105420.0775
20.0442
20.1975
20.0973

20.9218
20.8788
20.8946
20.8724
20.8946
20.922220.9559
20.8025
20.9026

1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3

0.20000.02290.04090.0436
0.05750.05310.20000.11240.22090.80001.17541.2436
1.18361.36371.01010.80001.42461.75440.08350.10080.11530.12150.13070.13240.1426
0.15600.23180.07590.09540.11260.11920.12180.12630.12350.14820.1768

0.8
0.6
0.4
0.4
0.4
0.5
0.9
0.7
0.8

absorption characteristics of liquid water in clouds makesuch a generalization impractical. Downwelling irradiance varies by a factor of 100 over the region from 0.3
to 3.0 mm. Over the same spectral range the albedo ofsnow and ice surfaces decreases with wavelength andexhibits pronounced sharp minima in the infrared. Liquid water in clouds absorbs more strongly in the infrared. All these factors make it difficult to generalize theparameterization as a function of wavelength. We therefore offer parameterizations only for the nine discrete
wavelengths in Table 2.
5. Application to field data

FIG. 9. Frequency distribution of cloud optical depth in the East
Antarctic sea ice zone during a single ship voyage, derived using thebroadband parameterization. An areal concentration of greater than
two-tenths of sea ice is defined as sea ice; less than two-tenths is
considered open water. The bin size is 4 units of optical depth. The
bar at t 5 0 represents clear sky.

The broadband parameterization is applied to measurements in the east Antarctic sea-ice zone during a 2month springtime voyage in 1996. The data were collected on the research vessel Aurora Australis as partof the 49th Australian National Antarctic Research Expeditions (ANARE) between the latitudes of 448 and
698S. Eppley pyranometers covering the spectral range
from 0.3 to 2.8 mm measured downwelling irradianceat 10-s intervals for the duration of the voyage. Spectral
irradiance was measured at infrequent intervals using aspectral radiometer covering the wavelength range 320–
1060 nm. From these measurements raw cloud transmittances (trc) are obtained. An advantage of using measurements rather than model calculations to determine
the clear-sky irradiance is that it does not require absolute calibration of the pyranometer. Furthermore,modeling errors are avoided.
Spectral surface albedos of different ice types in Antarctic sea ice have been measured in situ by Allison etal. (1993) and Brandt et al. (1999). Together with hourly
visual observations, which include estimates of the fractional coverage of the ocean surface, the ice thickness,and the snow cover thickness for each of several ice
types within 2 km of the ship (the relevant area for
multiple reflections between the surface and low
clouds), these albedos are used to determine the areaaveraged surface albedo at a particular location. Using
the broadband parameterization, effective cloud opticaldepth is derived. Figure 9 shows the frequency distribution of effective cloud optical depths obtained for thisone voyage, using only those hourly observations performed while in the sea ice zone. The data are separated
into two classes of surface type: ‘‘open water,’’ where
the areal concentration of ice is at most 0.2, and ‘‘sea

15 JANUARY 2004

FITZPATRICK ET AL.

273

FIG. 11. Spectral surface albedos used to determine effective cloudoptical depth in Fig. 10. The dotted line is based on the visual observation of ice types (one-tenth snow-covered ice, nine-tenths water), whereas the circles are the values of spectral albedo that cause
the optical depth values to agree at each wavelength [Fig. 10(b)].
Some types of sea ice do have spectral albedos corresponding to thecircles (Grenfell and Perovich 1984) but those types were not reported
in the vicinity of the ship during the time of observation.

FIG. 10. Effective cloud optical depth (in the geometrics–optics
limit) for a 15-min period with completely overcast sky, and a surface
composed of one-tenth snow-covered first-year sea ice and ninetenths open water. Values obtained using the broadband parameterization for a single observation at 0500 UTC are compared with thosefor five of the spectral parameterizations in (a) by assuming a weighted average of snow and water and in (b) by adjusting the spectral
albedo (within 60.05) so that all spectral channels give the same
optical depth (in the geometric-optics limit).

ice,’’ where the areal concentration of ice is greater than
0.2. The figure shows that thin cloud is more prevalent
in areas of open water and clear sky is more common
over sea ice. The leftmost bin, 22 , t , 2, contains
some derived optical depths that are negative (alsoshown later in Fig. 12). These are obtained from Eq.
(1) when trc . 1. This may occur under a clear sky due
to imperfect leveling of the pyranometers, and under apartly cloudy sky when the solar beam passes between
clouds causing the downward solar irradiance to exceed
the clear-sky irradiance.
We now investigate whether spectral and broadband
measurements are consistent. To compare the spectral
and broadband values of effective cloud optical depth
a 15-min period was chosen when low concentrationsof sea ice were present (and hence the area-averaged
surface albedo was most accurate), and while the sky
was completely overcast. The surface was nine-tenths
water and one-tenth snow-covered ice. Figure 10 showsthe effective cloud optical depth at five spectral wavelengths and compares them with the broadband value(measured at only one time, 0500 UTC). This comparison is done in two ways. In Fig. 10a, at each wavelength
we use the visual observation of the fractional area covered by each ice type in the vicinity of the ship; at eachof five wavelengths the albedo is computed as a weighted average of the measured spectral albedos of the different types (weighted by their area fractions). The albedos at these five wavelengths are used to computefive values of cloud optical depth. For the mixture of
ice types reported present, the area-averaged albedo
should be nearly constant with wavelength between 370
and 930 nm (dotted line in Fig. 11). The resulting computed optical depths (Fig. 10a), however, did not agreewith each other. In Fig. 10b, we instead determine thespectral variation in surface albedo (also shown in Fig.
11) that would be necessary to calculate the same effective optical depth at all wavelengths.
The improvement in the comparison of effective optical depth in Fig. 10b results from using these latter
spectral albedos that do correspond to some types ofsea ice (Grenfell and Perovich 1984) but not to those
ice types evident near the ship at this time. A possible,
but speculative, explanation for the discrepancy is thatthe effective area of sea surface affecting the pyranometer measurement is greater for visible wavelengths
than for near-infrared wavelengths. Because the cloudis more absorptive at longer wavelengths, photons at

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JOURNAL OF CLIMATEVOLUME 17

of parameterization could be used for various land surfaces, but with different coefficients derived for theirspectral shapes.
The parameterization is intended for use in the analysis of measured surface irradiance to obtain an effective cloud optical depth t from measurements of trc.
The inferred value of t is an inherent property of thecloud field, allowing computation of cloud transmittancefor other conditions of solar illumination and surface
albedo, thus allowing instantaneous point measurements
to be used for computation of diurnal-average and seasonal-average cloud radiative forcing. The parameterization may also find use in climate modeling.

FIG. 12. Latitudinal variation of derived cloud optical depth during
austral spring 1996 on a voyage from Australia to east Antarctica
and return. Horizontal bars are averages of effective cloud opticaldepth over 58 of latitude, excluding cases of clear sky. The approximate edge of the sea ice was at 628S. Points are plotted at hourlyintervals, for daytime hours when the sun was at least 108 above thehorizon. There are 514 points total.

these wavelengths cannot survive many bounces between cloud and surface. The near-IR downward irradiance may therefore be more dominated by the open
water in the lead surrounding the ship, whereas for thevisible wavelengths the relevant albedo is that of a larger
area, extending beyond the lead into snow-covered floes.
We have not attempted to model this effect. However,
we emphasize that, for the typically mixed scenes in thesea ice zone, surface albedo is the least well-characterized of the variables required for use in both the broadband and spectral parameterization.
The latitudinal dependence of effective cloud opticaldepth for the duration of the voyage is shown in Fig.
12. One measurement per hour is plotted, for the entire
voyage from 448S to the edge of the Antarctic continent
at 698S. Observations at night or with the sun less than
108 above the horizon are excluded, leaving a total of
514 points. This figure shows the contrast between cloudoptical depth over the open ocean and over the sea icezone. The approximate edge of the sea ice was at 628S.
It is evident that there are far more instances of clear
sky over the sea ice than over the ocean, probably due
in part to the decrease in ocean–atmosphere moistureexchange in the sea ice zone. However the mean valuesof cloud optical depth show little latitudinal variation.
6. Conclusions
A simple but accurate parameterization has been developed relating raw cloud transmittance trc to cloudoptical depth t, surface albedo a, and solar zenith angle
u. The parameterization was derived using the measured
spectral albedos of snow, ice, and water; the same form

Acknowledgments. We thank Istvan Laszlo (University of Maryland) for helping us implement the ATRAD
model. We thank Ian Allison of the Australian Antarctic
Division for sponsoring the fieldwork for this project,
and Tom Grenfell, Tony Worby, Thomas Ackerman,
Charles Long, and Warren Wiscombe for discussions.
This work was supported by NSF grants OPP-95-27244
and OPP-98-15156, and by a NASA Earth System Science Fellowship to Melanie Fitzpatrick.
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