The boarder conditions for the Eqs. 14.1 and 14.3 are
Iðt ¼ 0; m; m 0 ; ’Þ ¼ 0; for m > 0
Iðt ¼ t 0 ; m; m 0 ; ’Þ ¼ 0, for m < 0
(14.4)
It corresponds to the absence of the incoming diffuse radiation at the
atmosphere’s top.
The Eq. 14.3 is an ordinary differential equation of the first order. (of type
dy
dx ¼ aðxÞy þ bðxÞ). It has a known general solution
ðyðxÞ ¼ yðx 0 Þ exp
ð x
x0
aðx
0
Þdx
0
0
@
1
A þ
ð x
x0
bðx
0
Þ exp
ð x
x 0
aðx
00
Þdx
00
0
@
1
A dx
0
Þ:
Applying it to the Eq. 14.3 (x t, y I, aðxÞ À1=m, bðxÞ B=m) by taking
into account for the boarder conditions (14.4) (x 0 0 for m > 0, x 0 t 0 for m < 0,
for both cases y(x 0 ) ¼ 0) yields:
Iðt; m; m 0 ; ’Þ ¼
1
m
ð t
0
Bðt; m; m 0 ; ’Þ exp À
t À t
0
m
dt
0 , for m > 0
Iðt; m; m 0 ; ’Þ ¼ À
1
m
ð
t0
t
Bðt; m; m 0 ; ’Þ exp À
t À t
0
m
dt
0 , for m < 0
8
> > > > > > <
> > > > > > :
(14.5)
Let’s point out that expressions (14.5) are not the solution of the transfer equation
(14.1) because the function Bðt; m 0 ; m; ’Þdepends on desired intensity according to
Eq. 14.2. Nevertheless they are convenient for consideration in certain cases.
14.2 The Approximation of the Single Scattering –
The General Case
Consider the Eq. 14.2 for the source function. Two items have transparent physical
sense. The first one expresses sources of the multiple scattered light in the atmosphere, i.e. scattered more than one time. The second item characterises sources of
the single scattered light that arises by substituting the direct radiation to the
transfer equation. The contribution of single scattered light to the total intensity
significantly (several times) exceeds the contribution of multiple scattering in the
clear Earth atmosphere. Thus it is possible to restrict the single scattering approximation and the first item in the expression for the source function to assume zeroth
in many problems, where the high exactness is not needed. Then Eq. 14.5 give a real
solution for the intensity calculation;
140
14 Calculating Radiative Characteristics with the Single Scattering Approximation
Iðt ¼ 0; m; m 0 ; ’Þ ¼ 0; for m > 0
Iðt ¼ t 0 ; m; m 0 ; ’Þ ¼ 0, for m < 0
(14.4)
It corresponds to the absence of the incoming diffuse radiation at the
atmosphere’s top.
The Eq. 14.3 is an ordinary differential equation of the first order. (of type
dy
dx ¼ aðxÞy þ bðxÞ). It has a known general solution
ðyðxÞ ¼ yðx 0 Þ exp
ð x
x0
aðx
0
Þdx
0
0
@
1
A þ
ð x
x0
bðx
0
Þ exp
ð x
x 0
aðx
00
Þdx
00
0
@
1
A dx
0
Þ:
Applying it to the Eq. 14.3 (x t, y I, aðxÞ À1=m, bðxÞ B=m) by taking
into account for the boarder conditions (14.4) (x 0 0 for m > 0, x 0 t 0 for m < 0,
for both cases y(x 0 ) ¼ 0) yields:
Iðt; m; m 0 ; ’Þ ¼
1
m
ð t
0
Bðt; m; m 0 ; ’Þ exp À
t À t
0
m
dt
0 , for m > 0
Iðt; m; m 0 ; ’Þ ¼ À
1
m
ð
t0
t
Bðt; m; m 0 ; ’Þ exp À
t À t
0
m
dt
0 , for m < 0
8
> > > > > > <
> > > > > > :
(14.5)
Let’s point out that expressions (14.5) are not the solution of the transfer equation
(14.1) because the function Bðt; m 0 ; m; ’Þdepends on desired intensity according to
Eq. 14.2. Nevertheless they are convenient for consideration in certain cases.
14.2 The Approximation of the Single Scattering –
The General Case
Consider the Eq. 14.2 for the source function. Two items have transparent physical
sense. The first one expresses sources of the multiple scattered light in the atmosphere, i.e. scattered more than one time. The second item characterises sources of
the single scattered light that arises by substituting the direct radiation to the
transfer equation. The contribution of single scattered light to the total intensity
significantly (several times) exceeds the contribution of multiple scattering in the
clear Earth atmosphere. Thus it is possible to restrict the single scattering approximation and the first item in the expression for the source function to assume zeroth
in many problems, where the high exactness is not needed. Then Eq. 14.5 give a real
solution for the intensity calculation;
140
14 Calculating Radiative Characteristics with the Single Scattering Approximation