Geoscience Reference
In-Depth Information
be snow, ice or water). Spectral downwelling and upwelling planar irradiances at the
depth z are given, respectively, by:
Z
2
p
d
u
Z
p=
2
E
d
ð
z
; kÞ
¼
L
ð
z
;
Z
; u; kÞ
cos
Z
sin
ZdZ
ð
3
:
16a
Þ
0
0
Z
2
p
d
u
Z
p
E
u
ð
z
; kÞ
¼
L
ð
z
;
Z
; u; kÞ
cos
Z
sin
ZdZ
ð
3
:
16b
Þ
0
p=
2
where (Z,
/2)
and the azimuth
ˆ
(0
≤ ˆ ≤
2
ˀ
). Integrating the solar irradiances with respect to the
wavelength, the downwelling and upwelling solar energy fluxes are obtained. At the
surface these are also called the incoming and outgoing solar radiation Q
s
and Q
r
,
respectively.
The ratio of upwelling to downwelling irradiance is called spectral re
ˆ
) are the local spherical direction coordinates, the zenith angle (
−ˀ
/2
≤
Z
≤ ˀ
fl
ectance r(z,
ʻ
),
while albedo
ʱ
, literally
'
whiteness
'
, is the ratio of these irradiances integrated over the
wavelength at the surface,
7
respectively:
R
0
E
u
0
; k
rz
; ðÞ
¼
E
u
z
; ðÞ
E
d
z
; ð
; a
¼
Q
r
ð
Þ
dz
Q
s
¼
ð
3
:
17
Þ
R
1
0
E
d
0
; k
ð
Þ
dz
where the notation 0
−
stands for
'
just above the surface
'
. Attenuation of irradiance is
normally modelled with a
first order equation analogous to the Beer-Lambert absorption
law. For the downwelling irradiance, the transfer equation and the solution are:
dE
d
ð
z
;
k
Þ
dz
¼
K
ð
z
; kÞ
E
d
ð
z
; kÞ
ð
:
Þ
3
18
where K is the diffuse attenuation coef
cient. Sometimes (e.g., Warren 1982) K is termed
as the
. It is clear from Eq. (
3.13a
,
b
) that irradiance depends on the
angular distribution of the incoming radiance, and therefore so do also optical properties
de
'
extinction coef
cient
'
ned from the irradiance. These optical properties, which include the albedo, re
fl
ectance
and diffuse attenuation coef
cient, are therefore apparent optical properties (see, e.g.,
Dera 1990; Arst 2003). The angular distribution of the incoming solar radiation depends
primarily on the solar altitude and cloudiness.
Attenuation of irradiance tells how much solar power is available for heating and
primary production. Upwelling irradiance just above the surface consists of the surface
re
fl
ection and scattering and the volume backscatter from beneath the surface. In snow and
7
Sometimes the expression spectral albedo is used instead of surface reflectance.
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