F
#
ðm 0 ; t; t 0 Þ ¼
Kðm 0 Þe
Àkt 0
1 À l
le À2kt 0
i
# e
kðt0ÀtÞ
À i
"
le
Àkðt0ÀtÞ
h
i
;
F
"
ðm 0 ; t; t 0 Þ ¼
Kðm 0 Þe
Àkt 0
1 À l
le À2kt0 i
" e
kðt0ÀtÞ
À i
#
le
Àkðt0ÀtÞ
h
i
;
(11.18)
where i
#
¼ 2
Ð 1
0
iðmÞmdm; i
"
¼ 2
Ð 1
0
iðÀmÞmdm:
Expansions for values i
# and i
" are derived after integrating Eq. 11.17:
i
#"
¼ 1 Æ 2s þ 3s
2 1:5 À g
2
1 þ g
Æ 3s
3 2 À 3g þ
0:8
1 þ g
(11.19)
It is also convenient to describe the internal radiation field with the values of
internal albedo b(t i ) ¼ F
"
(t i )/F
#
(t i ) and net flux F(t i ) ¼ F
#
(t i ) À F
" (t i ):
Fðt; m 0 Þ ¼ F
#
ðt; m 0 Þ À F
"
ðt; m 0 Þ ¼
4sKðm 0 Þe
Àkt
1 À l
le À2kt0 1 þ
le
À2kðt0ÀtÞ
h
i
F
"
ðt; m 0 Þ
F # ðt; m 0 Þ
¼ bðtÞ ¼
b
1
À
le
À2kðt 0 ÀtÞ
1 À b 1
le À2kðt0ÀtÞ
(11.20)
Value b
1 and function b(t) are called the internal albedo of the infinite atmosphere and the internal albedo of the atmosphere of the big optical thickness
respectively, moreover b
1
¼ 1À4s + 8s
2 and the values of function b(t) could
be obtained from the observations or from the calculations of the semispherical
irradiances at level t.
11.5 Case of the Conservative Scattering
In the true absorption absence, according the definition, we have o 0 ¼ 1 and the
expressions for the radiation characteristics are particularly simple.
For the reflection and transmission functions:
rð0; m; m 0 ; ’Þ ¼ r 0 ðm; m 0 ; ’Þ À
4K 0 ðm 0 ÞK 0 ðmÞ
3 ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
sðt 0 ; m; m 0 Þ ¼
4K 0 ðm 0 Þ
K 0 ðmÞ
3 ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
(11.21)
for the semispherical fluxes in relative units of F 0
11.5 Case of the Conservative Scattering
113