Surface albedo A is assumed by formulas:
K 0 ðmÞ ¼ K 0 ðmÞ þ A 1 À A
ð
Þ
=
;
K 2 ðmÞ ¼ K 2 ðmÞ þ
A
1 À A
3K 0 ðmÞ
3:8m À 2:7
1 þ g
þ n 2
;
(11.11)
11.3 The Analytical Presentation of the Reflection Function
The following group of the formulas is the approximations obtained from the
analysis of the numerical values of the reflection function. As it is usually done,
let us describe the reflection function over the azimuth angle cosine to separate the
item independent of the azimuth angle:
rð’; m; m 0 Þ ¼ r
0
ðm; m 0 Þ þ 2
X 1
m¼1
r
m
ðm; m 0 Þ cos m’;
(11.12)
where functions r
m
(m,m 0 ) are the harmonics of the reflection function of order m.
Superscripts specify here the number of the azimuthal harmonics. As it has been
mentioned above, we are using here the phase function described by HenyeyGreenstein formula (1.15).
The analysis of numerical calculations shows that for the accurate description of
function r(’,m,m 0 ) it is enough to know the zeroth and first six harmonics if either of
cosines m and m 0 are greater than 0.15 even for value g ¼ 0.9 unfavorable for the
computing accuracy. This limitation does not restrict our consideration because it is
also necessary to use a complicated model of the spherical atmosphere and to take
into account the refraction of solar rays for the small cosines of zenith solar and
viewing angles. These cases are not studied here.
The values of r
m (m,m 0 ) for m ¼ 0,. . .,6 have been analyzed and the following
expression, is used for the description of high harmonics r
m
(m,m 0 ) :
r
m
ðm; m 0 Þ ¼ a
m mm 0 þ b
m m þ m 0
ð
Þþc
m
½
m þ m 0
ð
Þ
=
(11.13)
This presentation provides the reciprocity of the reflection function relatively to
the both zenith viewing and zenith solar angles.
Table 11.2 Values of the second coefficient a 2 (m) of the plane albedo expansion
m
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
g
0.75 1.310 2.220 3.118 4.078 5.126 6.256 7.475 8.786 10.19 11.70 13.29
0.80 1.267 2.236 3.151 4.117 5.163 6.289 7.494 8.796 10.18 11.66 13.23
0.85 1.201 2.242 3.181 4.148 5.198 6.320 7.512 8.798 10.17 11.63 13.18
0.90 1.092 2.244 3.208 4.193 5.237 6.350 7.529 8.808 10.16 11.60 13.12
110
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic
K 0 ðmÞ ¼ K 0 ðmÞ þ A 1 À A
ð
Þ
=
;
K 2 ðmÞ ¼ K 2 ðmÞ þ
A
1 À A
3K 0 ðmÞ
3:8m À 2:7
1 þ g
þ n 2
;
(11.11)
11.3 The Analytical Presentation of the Reflection Function
The following group of the formulas is the approximations obtained from the
analysis of the numerical values of the reflection function. As it is usually done,
let us describe the reflection function over the azimuth angle cosine to separate the
item independent of the azimuth angle:
rð’; m; m 0 Þ ¼ r
0
ðm; m 0 Þ þ 2
X 1
m¼1
r
m
ðm; m 0 Þ cos m’;
(11.12)
where functions r
m
(m,m 0 ) are the harmonics of the reflection function of order m.
Superscripts specify here the number of the azimuthal harmonics. As it has been
mentioned above, we are using here the phase function described by HenyeyGreenstein formula (1.15).
The analysis of numerical calculations shows that for the accurate description of
function r(’,m,m 0 ) it is enough to know the zeroth and first six harmonics if either of
cosines m and m 0 are greater than 0.15 even for value g ¼ 0.9 unfavorable for the
computing accuracy. This limitation does not restrict our consideration because it is
also necessary to use a complicated model of the spherical atmosphere and to take
into account the refraction of solar rays for the small cosines of zenith solar and
viewing angles. These cases are not studied here.
The values of r
m (m,m 0 ) for m ¼ 0,. . .,6 have been analyzed and the following
expression, is used for the description of high harmonics r
m
(m,m 0 ) :
r
m
ðm; m 0 Þ ¼ a
m mm 0 þ b
m m þ m 0
ð
Þþc
m
½
m þ m 0
ð
Þ
=
(11.13)
This presentation provides the reciprocity of the reflection function relatively to
the both zenith viewing and zenith solar angles.
Table 11.2 Values of the second coefficient a 2 (m) of the plane albedo expansion
m
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
g
0.75 1.310 2.220 3.118 4.078 5.126 6.256 7.475 8.786 10.19 11.70 13.29
0.80 1.267 2.236 3.151 4.117 5.163 6.289 7.494 8.796 10.18 11.66 13.23
0.85 1.201 2.242 3.181 4.148 5.198 6.320 7.512 8.798 10.17 11.63 13.18
0.90 1.092 2.244 3.208 4.193 5.237 6.350 7.529 8.808 10.16 11.60 13.12
110
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic