m
Iðt; m; m 0 ; ’Þ
dt
¼ ÀIðt; m; m 0 ; ’Þ þ
o 0 ðtÞ
4p
ð
2p
0
d’
0
ð 1
À1
xðt; gÞIðt; m
0
; m 0 ; ’
0
Þdm
0
þ
o 0 ðtÞ
4p
F 0 xðt; g 0 Þe
Àt=m 0
(1.25)
Point out that Eq. 1.25 is written only for the diffuse radiation. The boundary
conditions are taking into account by the third term in the right part of Eq. 1.25.
The sense of this term is the yield of the first order of scattering to the radiance and
the integral term describes the contribution of multiple scattering.
The ground surface at the bottom of the atmosphere is usually called the
underlying surface or the surface. Solar radiation interacts with the surface
reflecting from it. Hence, the laws of the reflection as a boundary condition at
the bottom of the atmosphere should be taken into account. However, it is done
otherwise in the radiative transfer theory. As it will be shown in the following
section, there are comparatively simple methods of calculating the reflection by the
surface if it is obtained the solution of the transfer equation for the atmosphere
without the interaction between radiation and surface. Thus, neither direct nor
reflected radiation is included to Eq. 1.25. As there is no diffused radiation at the
atmospheric top and bottom, the boundary conditions are as follows:
Ið0; m; m 0 ; ’Þ ¼ 0 m > 0; Iðt 0 ; m; m 0 ; ’Þ ¼ 0 m < 0:
Transfer Eq. 1.25 together with boundary conditions defines the problem of the
solar diffused radiance in the plane parallel atmosphere. Nowadays different
methods both analytical and numerical are elaborated. Some of them will be
considered in the following sections.
1.4 Transformation of the Radiation Transfer Equation
Return to the transfer Eq. 1.25 and transform it. Introduce the following noting in
the form:the average intensity multiplied to 4p:
Iðt; m 0 Þ ¼
ð
2p
0
d’
ð 1
À1
Iðt; m; m 0 ; ’Þdm
(1.26)
the diffuse irradiance:
Hðt; m 0 Þ ¼
ð
2p
0
d’
ð 1
À1
Iðt; m; m 0 ; ’Þmdm
(1.27)
1.4 Transformation of the Radiation Transfer Equation
17
Iðt; m; m 0 ; ’Þ
dt
¼ ÀIðt; m; m 0 ; ’Þ þ
o 0 ðtÞ
4p
ð
2p
0
d’
0
ð 1
À1
xðt; gÞIðt; m
0
; m 0 ; ’
0
Þdm
0
þ
o 0 ðtÞ
4p
F 0 xðt; g 0 Þe
Àt=m 0
(1.25)
Point out that Eq. 1.25 is written only for the diffuse radiation. The boundary
conditions are taking into account by the third term in the right part of Eq. 1.25.
The sense of this term is the yield of the first order of scattering to the radiance and
the integral term describes the contribution of multiple scattering.
The ground surface at the bottom of the atmosphere is usually called the
underlying surface or the surface. Solar radiation interacts with the surface
reflecting from it. Hence, the laws of the reflection as a boundary condition at
the bottom of the atmosphere should be taken into account. However, it is done
otherwise in the radiative transfer theory. As it will be shown in the following
section, there are comparatively simple methods of calculating the reflection by the
surface if it is obtained the solution of the transfer equation for the atmosphere
without the interaction between radiation and surface. Thus, neither direct nor
reflected radiation is included to Eq. 1.25. As there is no diffused radiation at the
atmospheric top and bottom, the boundary conditions are as follows:
Ið0; m; m 0 ; ’Þ ¼ 0 m > 0; Iðt 0 ; m; m 0 ; ’Þ ¼ 0 m < 0:
Transfer Eq. 1.25 together with boundary conditions defines the problem of the
solar diffused radiance in the plane parallel atmosphere. Nowadays different
methods both analytical and numerical are elaborated. Some of them will be
considered in the following sections.
1.4 Transformation of the Radiation Transfer Equation
Return to the transfer Eq. 1.25 and transform it. Introduce the following noting in
the form:the average intensity multiplied to 4p:
Iðt; m 0 Þ ¼
ð
2p
0
d’
ð 1
À1
Iðt; m; m 0 ; ’Þdm
(1.26)
the diffuse irradiance:
Hðt; m 0 Þ ¼
ð
2p
0
d’
ð 1
À1
Iðt; m; m 0 ; ’Þmdm
(1.27)
1.4 Transformation of the Radiation Transfer Equation
17