Geoscience Reference
In-Depth Information
incident radiation at the top of the atmosphere
I
(0
,µ,φ
) and surface reflec-
tion at the bottom of the atmosphere
I
(
τ,
µ, φ
). For near-UV to near-IR,
there is no source other than direct sunlight at the top of the atmosphere.
There is, in general, a source at the bottom of the atmosphere from reflec-
tion of the surface. The surface reflected radiation is a function of the down-
ward radiation incident upon it. If the surface is a pure specular reflector,
then the full reflection matrix can be written in terms of Fresnel equations.
In reality however, the rough sea surface cannot assumed to behave like a
pure specular reflector and as such a bi-directional reflectance distribution
function (BDRF) is required to model sea surface reflectivity.
−
2.1.
Rough sea surface reflection: Cox and Munk MSS
Computing the scattering of solar radiation for a smooth, specular liquid
surface requires only the direct application of Fresnel equations. However,
for the case of a rough liquid surface, these equations are no longer directly
applicable since specular reflection will only exist for suitable tilted liquid
surface facets. According to Cox and Munk
1
,
2
study of the Sun glitter, the
probability distribution of water surface facet normals under the influence
of the wind is given by the following Gaussian distribution (isotropic rough
surface, independent of wind-direction):
exp
.
µ
n
δ
2
µ
n
1
πσ
2
1
−
F
(
µ
n
,φ
n
)=
−
(2)
The term
δ
2
is called the mean-square slope (MSS) of the sea surface.
Cox and Munk,
1
,
2
based on their photographs of the Sun glint, formulated
a linear relation between the MSS and the wind speed
V
w
(measured at
12.5 m height) given by the following equation,
δ
2
=0
.
003 + 0
.
00512
V
w
.
(3)
This form of the MSS has been widely used by the scientific community
over the years to describe specular reflection from sea surface wave slopes.
2.2.
Rough sea surface reflection: Zhao and Toba MSS
In a recent work, Zhao and Toba
4
proposed a spectral approach to deter-
mine the MSS from radar altimeter data. The proposal includes the integra-
tion for a wind-wave spectrum but extended to include the gravity-capillary
