b
m and c
m in Tables 11.3 and 11.4 gives an acceptable presentation for all the
harmonics of the reflection function considered here. The errors of this approximation have been shown to depend on the values of the zenith solar and viewing angles
cosines, on number of the harmonic m, and on phase function parameter g. Some
details of the error analysis will be presented in Sect. 11.4.
The presented asymptotic formulas (11.1), (11.4) and (11.5), expansions (11.6),
(11.7) and approximations (11.8)–(11.11), (11.13) and (11.15) allow computing the
reflected and transmitted radiance and irradiance together with the radiative divergence for cloud layer if the layer properties and the geometry of the problem are
known. The considered model has to satisfy the applicability ranges of the
presented formulas: big optical thickness and weak true absorption. These ranges
will be analyzed in Sect. 11.4 in detail. However, it is necessary to point out that for
the application of Eqs. 11.1–11.5 the big optical thickness with known asymptotic
functions and constants is enough. The use of the expansions (11.6) and (11.7)
needs the weak absorption.
11.4 Diffused Radiation Field Within the Cloud Layer
Radiation within the cloud layer (in the diffusion domain: t 0 À t N–1 ) 1 and
t 1 ) 1) is described with formulas different from those presented above. Here
we are offering the results useful for the further consideration.
The diffused radiance in energetic units in the diffusion domain at optical depth
t satisfied conditions t> > 1, t 0 À t> > 1 is expressed with the equation:
Iðt; m; m 0 ; t 0 Þ ¼
F 0
p
Kðm 0 Þm 0 e
Àkt0 iðmÞe
kðt0ÀtÞ
À iðÀmÞ
le
Àkðt0ÀtÞ
1 À l
le À2kt0
;
(11.16)
where F 0 is the solar incident flux, function i(m) characterizes the angle dependence
of the radiance in deep levels of the semi-infinite atmosphere. The behavior of
function i(m) relatively the phase function shape and absorption in the medium has
been studied. The expansion for function i(m) has been derived in case of the weak
true absorption, which is presented here in terms of parameter s:
iðmÞ ¼ 1 þ 3sm þ 3
1 À g
2
þ 2P 2 ðmÞ
1 þ g
s
2
þ 9ð1 À 1:5gÞm þ
10:8P 3 ðmÞ
ð1 þ gÞð1 þ g þ g 2 Þ
þ
3:6m
1 þ g
s
3
(11.17)
Functions P i (m) for i ¼ 1,2, . . . are Legendre polynomials of power i.
The diffused irradiance in relative units of F 0 within the optically thick cloud
layer is described with the following:
112
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic
m and c
m in Tables 11.3 and 11.4 gives an acceptable presentation for all the
harmonics of the reflection function considered here. The errors of this approximation have been shown to depend on the values of the zenith solar and viewing angles
cosines, on number of the harmonic m, and on phase function parameter g. Some
details of the error analysis will be presented in Sect. 11.4.
The presented asymptotic formulas (11.1), (11.4) and (11.5), expansions (11.6),
(11.7) and approximations (11.8)–(11.11), (11.13) and (11.15) allow computing the
reflected and transmitted radiance and irradiance together with the radiative divergence for cloud layer if the layer properties and the geometry of the problem are
known. The considered model has to satisfy the applicability ranges of the
presented formulas: big optical thickness and weak true absorption. These ranges
will be analyzed in Sect. 11.4 in detail. However, it is necessary to point out that for
the application of Eqs. 11.1–11.5 the big optical thickness with known asymptotic
functions and constants is enough. The use of the expansions (11.6) and (11.7)
needs the weak absorption.
11.4 Diffused Radiation Field Within the Cloud Layer
Radiation within the cloud layer (in the diffusion domain: t 0 À t N–1 ) 1 and
t 1 ) 1) is described with formulas different from those presented above. Here
we are offering the results useful for the further consideration.
The diffused radiance in energetic units in the diffusion domain at optical depth
t satisfied conditions t> > 1, t 0 À t> > 1 is expressed with the equation:
Iðt; m; m 0 ; t 0 Þ ¼
F 0
p
Kðm 0 Þm 0 e
Àkt0 iðmÞe
kðt0ÀtÞ
À iðÀmÞ
le
Àkðt0ÀtÞ
1 À l
le À2kt0
;
(11.16)
where F 0 is the solar incident flux, function i(m) characterizes the angle dependence
of the radiance in deep levels of the semi-infinite atmosphere. The behavior of
function i(m) relatively the phase function shape and absorption in the medium has
been studied. The expansion for function i(m) has been derived in case of the weak
true absorption, which is presented here in terms of parameter s:
iðmÞ ¼ 1 þ 3sm þ 3
1 À g
2
þ 2P 2 ðmÞ
1 þ g
s
2
þ 9ð1 À 1:5gÞm þ
10:8P 3 ðmÞ
ð1 þ gÞð1 þ g þ g 2 Þ
þ
3:6m
1 þ g
s
3
(11.17)
Functions P i (m) for i ¼ 1,2, . . . are Legendre polynomials of power i.
The diffused irradiance in relative units of F 0 within the optically thick cloud
layer is described with the following:
112
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic