g 3 ¼
1
4
ð2 À 3g
0 zÞ; g 4 ¼ 1 À g 3 ; 3g ¼ x 1
(12.9)
The above set of formulas is the realization of the Eddington method. The phase
function considering leads to delta-Eddington method, where optical parameters o
0
and t
0 , transform in according to expressions:
o
0
¼ o 0
1 À g
2
1 À o 0 g 2 ; t 0 ¼ t
0
¼ tð1 À o 0 g
2
Þ; g
0
¼
g
1 þ g
;
(12.10)
12.2 Considering the Surface Reflection
The taking into account the surface albedo A > 0 is done according to known
relations
F
" ð0; zÞ ¼ F
"
ð0; m 0 Þ þ A s Vðt
0
Þ
F
# ðt 0 ; m 0 Þ
F
# ðt 0 ; m 0 Þ ¼
F
#
ðt 0 ; m 0 Þ
1 À A s Aðt 0 Þ
(12.11)
where A(t ¼ 0) and V(t 0 ) are spherical albedo and the transmittance and defined by
expressions:
Aðt ¼ 0Þ ¼ 2
ð 1
0
F
"
ð0; m 0 Þm 0 dm 0
Vðt 0 Þ ¼ 2
ð 1
0
F
#
ðt 0 ; m 0 Þm 0 dm 0
(12.12)
It seems easy to calculate the result that after integrating. But integrals of
Eq. 12.7 do not lead analytical expressions directly. For diffuse radiation the result
was obtained for reflected radiation A(0) and for transmittance V(t 0 ) without
considering the direct radiation:
Að0Þ ¼
g 2 1 À e
À2kt0
ð
Þ
k þ g 1 þ k À g 1
ð
Þe À2kt0
Vðt 0 Þ ¼
2ke
Àkt 0
k þ g 1 þ k À g 1
ð
Þe À2kt 0
;
(12.13)
122
12 Calculating Solar Irradiance with Eddington Method