total, albedo αt is often a quantity of interest,
Stamnes 1992, and Smith et al. 1992a) and can
have a deleterious impact on living organisms
since it is a measure of the total solar energy
(Smith 1989, Smith et al. 1992b). The familiar spec-
absorbed by the ice and ocean (Maykut and
Untersteiner 1971, Maykut and Perovich 1987). It
trum of visible light is also shown in Figure 1
can be expressed in terms of the spectral albedo
from violet (400 nm) to blue (450 nm) to green
and the spectral incident irradiance as
(550 nm) to yellow (600 nm) to red (650 nm).
"Properties" refers to the parameters that are
∫ α (λ) Fd (0, λ) dλ .
used to describe the reflection, absorption and
αt =
(1)
∫ Fd (0, λ) dλ
transmission of solar radiation by sea ice. The
terminology of radiative transfer is intricate and
The total albedo depends on the spectral distri-
voluminous. It also has the unfortunate attribute
bution of the incident irradiance as well as on the
that the same physical quantity may have a dif-
spectral albedo of the surface. Thus a change in
ferent name, depending on whether an oceanog-
cloud conditions, and thereby the incident spec-
rapher, an astrophysicist or a biologist is speak-
tral irradiance, can result in changes in the total
ing. To avoid a Babel of jargon we shall limit
albedo (Grenfell and Maykut 1977).
ourselves to the terms needed for a basic under-
For some problems a knowledge of the angu-
standing of the optical properties and shall fol-
lar distribution of the reflected radiance is needed.
low the terminology conventions of the sea ice
For example, in climate studies it would be use-
literature.
ful to derive large-scale ice albedos from satellite
The spectral radiance I(θ,φ,λ) is the power in a
data. However, satellite sensors have narrow fields
ray of light in a particular direction, where θ is
of view and measure reflected radiance. The key
the zenith angle (0 pointing downward, π point-
then is to relate the radiance reflected at the view-
ing upward), φ is the azimuth angle and λ is the
ing angle of the instrument to the albedo of the
wavelength. The spectral radiance is defined as
ice. In order to do this the angular distribution of
the radiant flux/nanometer per unit area per unit
reflected radiance, characterized by the bidirec-
solid angle in a particular direction and has units
tional reflectance distribution function (BRDF),
of W m2 sr1 nm1. The spectral irradiance F(λ) is
must be known. The formal definition of the BRDF
simply the radiance projected onto a plane sur-
is (Nicodemus et al. 1977, Warren 1982, Perovich
face and integrated over a hemisphere. Because
1994)
of this projection the radiance is scaled by cos θ.
dI (θ, φ, λ)
The downwelling irradiance Fd(λ) is the radiance
R (θ0 , θ, φ0 , φ, λ) =
cos (θ0 ) dF (θ0 , φ0 , λ)
integrated over downward directions (e.g., from
the sky), and the upwelling irradiance Fu(λ) is the
where θ0 and φ0 are the solar zenith and azimuth
radiance integrated over upward directions (e.g.,
angle, F(θ0, φ0, λ) is the incident spectral irradi-
from the surface). This can be expressed formally
as:
ance, and R has units of steradians1. Formally R
is a derivative quantity, similar to a probability
2π π / 2
∫ I (θ, φ, λ) cos θ sinθ dθdφ
Fd (λ) =
density function, defined in terms of infinitesi-
∫
mal angles. In practice, the definition is extended
φ=0 θ=0
to finite, measurable angles, so that dI→∆I and
dF→∆F.
2π
π
I (θ, φ, λ) cos θ sinθ dθdφ .
Fu (λ) =
∫
∫
Light transmission through the ice is charac-
terized by the transmittance T(λ), which is simi-
φ=0 θ= π/2
lar to the albedo in that it is the fraction of the
The most studied, and most used, optical prop-
incident irradiance that is transmitted through
erty of sea ice is the albedo (α). The spectral al-
the ice. Light attenuation in the ice is often repre-
bedo is simply defined as the fraction of the inci-
sented using an irradiance extinction coefficient
dent irradiance that is reflected:
-1 dFd (z, λ)
κ (z, λ) =
Fu (0, λ)
Fd (z, λ)
α (λ) =
dz
Fd (0, λ)
where Fd(z, λ) is the downwelling spectral irradi-
where the 0 designates the surface. In sea ice ther-
ance at depth z in the ice.
modynamic studies the wavelength-integrated, or
Let us now examine the difficulties in deter-
2
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