Geoscience Reference
In-Depth Information
reflectivity of the liquid surface can be computed using Cox and Munk
1
,
2
empirical formulation together with a normal probability distribution of
water surface facets and applying Fresnel formulae to each surface facet as
shown in Ref. 3. However, this approach do not include the influence of
wave states or any of the physical properties of the surface on the com-
putation of the MSS. In a recent work, Zhao and Toba
4
introduced an
analytical method to calculate the MSS which is consistent with current
empirical algorithms, especially at low wind speeds. The MSS is calculated
by integrating a widely accepted wind-wave spectrum which includes the
gravity-capillary regime into the formulation as well as the physical prop-
erties of the ocean boundary. This new treatment, coupled with linear wave
theory is used here for the computation of the bi-directional reflectivity
of the ocean surface. The radiative transfer equation is then solved using
a modified one-dimensional radiative transfer code
5
based on the widely
known doubling and adding method
6
with the ocean surface treated as a
boundary condition to the problem. For the surface term, We use a method
similar to Deuze
3
with the exception that we compute the bi-directional
reflectivity function directly in azimuth space and then Fourier transform
the results via a FFT. In this way, the azimuth independent part of the
radiative transfer equation is preserved and effectively solved by the dou-
bling and adding algorithm and the interaction principle.
7
2. Radiative Transfer
For a plane-parallel finite atmosphere with a solar source, the uni-
dimensional radiative transfer equation for a single wavelength can be
written as
µ
∂I
(
τ, µ, φ
)
∂τ
2
π
+1
0
4
π
P
(
τ, µ, φ
;
µ
,φ
)
I
(
τ, µ
,φ
)d
µ
d
φ
=
I
(
τ, µ, φ
)
−
0
−
1
+
0
4
π
e
(
τ/µ
s
)
P
(
τ, µ, φ
;
µ
s
,φ
s
)
F
s
,
(1)
where
τ
represents the optical depth,
0
is the single scattering albedo,
(
µ, φ
)and(
µ
,φ
) are the cosine of the zenith and azimuth angle of the
reflected and incident light beams, respectively,
F
s
refers to the total solar
flux at the top of the atmosphere,
P
(
τ, µ, φ
;
µ
,φ
) the scattering matrix,
and
I
represents the intensity of the radiation.
There are several methods to solve the radiative transfer equation for a
plane-parallel atmosphere, several of these methods are described in Ref. 6.
However, to solve Eq. (1), two boundary conditions must be specified: the
