optics, it conforms to the direction of the direct radiation spreading (ϑ 0 , ’ 0 ).
Actually in the cloudless atmosphere, the intensity of solar direct radiation is
essentially greater than the intensity of scattered radiation. In this case, the direction
of solar radiation is only one, the intensity depends only on the altitude, and the
transfer Eq. 1.19 transforms to the following
dIðzÞ
dz cos # 0 ¼ aðzÞIðzÞ and it is always
sinϑ 0 > 0 here. Differential equation together with boundary condition I ¼ I(z 1 ),
where z 1 is the altitude of the top of the atmosphere (the level above which it
is possible to neglect the interaction between solar radiation and atmosphere) is
elementary solved that leads to:
IðzÞ ¼ Iðz 1 Þ exp À
1
cos # 0
ð
z 1
z
aðz
0
Þdz
0
0
@
1
A :
(1.20)
This relation illustrates the exponential decrease of the intensity in the extinct
medium and it is called Beer’s law.
Introduce the dimensionless value:
tðzÞ ¼
ð
z1
z
aðz
0
Þdz
0
;
(1.21)
that is called the optical depth of the atmosphere at altitude z. Its important
particular case is the optical thickness of the atmosphere in whole t 0 ¼
Ð
z 1
0
aðz
0
Þdz
0 .
Then Beer’s law is written as:
IðzÞ ¼ Iðz 1 Þ expðÀtðzÞ= cos # 0 Þ:
(1.22)
As it follows from definitions (1.20) and (1.22) and from “summation rules”
(1.12), the analogous rules are correct for the optical deepness and for the optical
thickness: tðzÞ ¼
P M
i¼1
t i ðzÞ; t 0 ¼
P M
i¼1
t 0;i .
Therefore, it is possible to specify the optical thickness of the molecular scattering, the optical thickness of the aerosol absorption etc.
According to the accepted in Sect. 1.1 condition we are considering solar
radiation incoming to the plane atmosphere top as an incident solar parallel flux
F 0 from direction (ϑ 0 , ’ 0 ). Then, deducing the intensity through delta-function
(1.10) and substituting it to the formula of the link between the flux and intensity
(1.5) it is possible to obtain Beer’s Law for the solar irradiance incoming to the
horizontal surface at the level t:
F d ðzÞ ¼ F 0 cos # 0 expðÀtðzÞ= cos # 0 Þ:
(1.23)
1.3 Radiative Transfer in the Atmosphere
15
Actually in the cloudless atmosphere, the intensity of solar direct radiation is
essentially greater than the intensity of scattered radiation. In this case, the direction
of solar radiation is only one, the intensity depends only on the altitude, and the
transfer Eq. 1.19 transforms to the following
dIðzÞ
dz cos # 0 ¼ aðzÞIðzÞ and it is always
sinϑ 0 > 0 here. Differential equation together with boundary condition I ¼ I(z 1 ),
where z 1 is the altitude of the top of the atmosphere (the level above which it
is possible to neglect the interaction between solar radiation and atmosphere) is
elementary solved that leads to:
IðzÞ ¼ Iðz 1 Þ exp À
1
cos # 0
ð
z 1
z
aðz
0
Þdz
0
0
@
1
A :
(1.20)
This relation illustrates the exponential decrease of the intensity in the extinct
medium and it is called Beer’s law.
Introduce the dimensionless value:
tðzÞ ¼
ð
z1
z
aðz
0
Þdz
0
;
(1.21)
that is called the optical depth of the atmosphere at altitude z. Its important
particular case is the optical thickness of the atmosphere in whole t 0 ¼
Ð
z 1
0
aðz
0
Þdz
0 .
Then Beer’s law is written as:
IðzÞ ¼ Iðz 1 Þ expðÀtðzÞ= cos # 0 Þ:
(1.22)
As it follows from definitions (1.20) and (1.22) and from “summation rules”
(1.12), the analogous rules are correct for the optical deepness and for the optical
thickness: tðzÞ ¼
P M
i¼1
t i ðzÞ; t 0 ¼
P M
i¼1
t 0;i .
Therefore, it is possible to specify the optical thickness of the molecular scattering, the optical thickness of the aerosol absorption etc.
According to the accepted in Sect. 1.1 condition we are considering solar
radiation incoming to the plane atmosphere top as an incident solar parallel flux
F 0 from direction (ϑ 0 , ’ 0 ). Then, deducing the intensity through delta-function
(1.10) and substituting it to the formula of the link between the flux and intensity
(1.5) it is possible to obtain Beer’s Law for the solar irradiance incoming to the
horizontal surface at the level t:
F d ðzÞ ¼ F 0 cos # 0 expðÀtðzÞ= cos # 0 Þ:
(1.23)
1.3 Radiative Transfer in the Atmosphere
15