where the nomination is introduced: q
0
¼ 2
Ð 1
0
K 0 ðzÞz
2 dz ffi 0:714
In these expansions functions r 0 (m,m 0 ) and K 0 (m) are functions r 1 (m,m 0 ) and
K(m) for the conservative scattering (o 0 ¼ 1) correspondingly, functions a 2 (m) and
K 2 (m) are the coefficients by the item s
2
. They are presented either in analytical or
in table form. Asymptotic expansions (11.6) and (11.7) have been mathematically
rigorously derived, their errors are defined by items ~ s
3 or ~ s
4 omitted in the
series.
The coefficients of items s
2 and s
3 in the expansion for reflection function
r 1 (m,m 0 ) are looking as:
r 2 ðm; m 0 Þ ¼
a 2 ðmÞa 2 ðm 0 Þ
a 2
; r 3 ðm; m 0 Þ ¼
a 3 ðmÞa 3 ðm 0 Þ
a 3
;
(11.8)
where a 2 , a 3 , a 2 (m) and a 3 (m) are the coefficients of s
2 and s
3 in the series for
spherical a
1 albedo as per Eq. 11.6 and in series for plane a(m) albedo as per
Eq. 11.7 respectively. Values of the conservative escape function K 0 (m 0 ) are
presented in the following table (Table 11.1):
The approximation for function K 0 (m) with the error 3% for m > 0.4 has been
proposed as: K 0 (m) ¼ 0.5 þ 0.75 m. The analysis of numerical results yields the
following approximation for coefficients K 0 (m) and K 2 (m) with taking into account
the phase function dependence:
K 0 ðmÞ ¼ ð0:7678 þ 0:0875gÞm þ 0:5020 À 0:0840g:
(11.9)
K 2 ðmÞ ¼ n 2 K 0 ðmÞwðmÞ ¼ 1:667n 2 ðm
2
þ 0:1Þ
The correlation coefficient of the formulas (11.9) and numerical calculations is
about 0.99 – 0.93 depending on parameter g.
Expressions for functions a 2 (m), and a 3 (m) are follows:
a 2 ðmÞ ¼ 3K 0 ðmÞ
3
1 þ g
ð1:271m À 0:9Þ þ 4q
0
a 3 ðmÞ ¼ 4K 0 ðmÞ 4:5g À
1:6
1 þ g
À 3 À n 2 wðmÞ
:
(11.10)
The values of function a 2 (m) are presented in Table 11.2 for four values of
parameter g.
Table 11.1 The escape function K 0 (m 0 ) for clouds (0.65 g 0.9)
m 0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
K 0 (m 0 ) 1.271 1.193 1.114 1.034 0.952 0.869 0.782 0.690 0.591 0.476
11.2 Weak True Absorption of Solar Radiation
109
0
¼ 2
Ð 1
0
K 0 ðzÞz
2 dz ffi 0:714
In these expansions functions r 0 (m,m 0 ) and K 0 (m) are functions r 1 (m,m 0 ) and
K(m) for the conservative scattering (o 0 ¼ 1) correspondingly, functions a 2 (m) and
K 2 (m) are the coefficients by the item s
2
. They are presented either in analytical or
in table form. Asymptotic expansions (11.6) and (11.7) have been mathematically
rigorously derived, their errors are defined by items ~ s
3 or ~ s
4 omitted in the
series.
The coefficients of items s
2 and s
3 in the expansion for reflection function
r 1 (m,m 0 ) are looking as:
r 2 ðm; m 0 Þ ¼
a 2 ðmÞa 2 ðm 0 Þ
a 2
; r 3 ðm; m 0 Þ ¼
a 3 ðmÞa 3 ðm 0 Þ
a 3
;
(11.8)
where a 2 , a 3 , a 2 (m) and a 3 (m) are the coefficients of s
2 and s
3 in the series for
spherical a
1 albedo as per Eq. 11.6 and in series for plane a(m) albedo as per
Eq. 11.7 respectively. Values of the conservative escape function K 0 (m 0 ) are
presented in the following table (Table 11.1):
The approximation for function K 0 (m) with the error 3% for m > 0.4 has been
proposed as: K 0 (m) ¼ 0.5 þ 0.75 m. The analysis of numerical results yields the
following approximation for coefficients K 0 (m) and K 2 (m) with taking into account
the phase function dependence:
K 0 ðmÞ ¼ ð0:7678 þ 0:0875gÞm þ 0:5020 À 0:0840g:
(11.9)
K 2 ðmÞ ¼ n 2 K 0 ðmÞwðmÞ ¼ 1:667n 2 ðm
2
þ 0:1Þ
The correlation coefficient of the formulas (11.9) and numerical calculations is
about 0.99 – 0.93 depending on parameter g.
Expressions for functions a 2 (m), and a 3 (m) are follows:
a 2 ðmÞ ¼ 3K 0 ðmÞ
3
1 þ g
ð1:271m À 0:9Þ þ 4q
0
a 3 ðmÞ ¼ 4K 0 ðmÞ 4:5g À
1:6
1 þ g
À 3 À n 2 wðmÞ
:
(11.10)
The values of function a 2 (m) are presented in Table 11.2 for four values of
parameter g.
Table 11.1 The escape function K 0 (m 0 ) for clouds (0.65 g 0.9)
m 0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
K 0 (m 0 ) 1.271 1.193 1.114 1.034 0.952 0.869 0.782 0.690 0.591 0.476
11.2 Weak True Absorption of Solar Radiation
109