The Eddington method is the kind of two-stream approximations. These
approximations of the transfer theory are based on different analytical formulas
approximately representing angle distribution of upward and downward intensity in
the plane media. The substituting these formulas to the integro-differential transfer
equation provides the system of differential equations for upward and downward
irradiances:
dF
"
ðt; m 0 Þ
dt
¼ g 1 F
"
ðt; m 0 Þ À g 2 F
#
ðt; m 0 Þ À F 0 og 3 exp( À
t
m 0
Þ
dF
#
ðt; m 0 Þ
dt
¼ g 2 F
"
ðt; m 0 Þ À g 1 F
#
ðt; m 0 Þ þ F 0 og 4 exp( À
t
m 0
Þ
(12.3)
where F
"#
ðt; m 0 Þ ¼ 2p
R 1
0
Iðt; m 0 ; ÆmÞmdm are solar upward and downward
irradiances in the atmosphere.
In the case of absorption absence or conservative scattering it is true: o 0 ¼1.0,
and the illumination of the surface with the albedo A (the transmitted irradiance at
the atmosphere base) is expressed as:
F
#
ðt 0 ; m 0 Þ ¼
4
4 þ ð3 À x 1 Þð1 À AÞt 0
1
2
þ
3
4
m 0
þ
1
2
À
3
4
m 0
exp À
t 0
m 0
!
(12.4)
In case of vertically homogeneous absorptive atmosphere the expressions for
reflected and transmitted irradiances are:
F
"
ð0; m 0 Þ ¼ C 1 þ C 2 þ D;
F
#
ðt 0 ; m 0 Þ ¼ C 1 expðÀkt 0 Þ þ C 2 expðkt 0 Þ þ D expð
t 0
m 0
Þ þ expð
t 0
m 0
Þ;
(12.5)
Expressions for constants in Eq. 12.5 are:
C 1 ¼
o 0
2
1
1 À k 2 m 0
2
1
D
Á
2 þ 3m 0 þ ð1 À oÞx 1 ð1 þ 2m 0 Þ
½
Š expðktÞð1 þ bÞþ 2 À 3m 0 Àð1 À oÞx 1 ð1 À 2m 0 Þ
½
Š expð
t 0
m 0
Þð1 À bÞ
&
'
;
C 2 ¼ À
o 0
2
1
1 À k 2 m 0
2
1
D
Á
2 þ 3m 0 þð1 À o 0 Þx 1 ð1 þ 2m 0 Þ
½
Š expðÀktÞð1 À bÞþ 2 À 3m 0 Àð1 À oÞx 1 ð1 À 2m 0 Þ
½
Š expðÀ
t 0
m 0
Þð1 þ bÞ
&
'
;
D ¼ À
3 þ ð1 À o 0 Þx 1
1 À k 2 m 0
2
m 0
o 0
2
;
120
12 Calculating Solar Irradiance with Eddington Method