Chapter 14
Calculating Radiative Characteristics
with the Single Scattering Approximation
Abstract The approximation of the single scattering is considered for case of
homogeneous plane atmosphere. The surface reflection is taken into account. The
description of the practice is presented.
14.1 Expressing the Radiative Intensity (Radiance) in Terms
of the Source Function
Let us return to the transfer equation Eq. 1.25
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
(14.1)
The diffuse radiation is treated as additional sources in the atmosphere, that
gives the second and third items in the right side. Introduce the source function:
Bðt; m; m 0 ; ’Þ ¼
oðtÞ
4p
ð
2p
0
d’
0
ð 1
0
Iðt; m
0
; m 0 ; ’
0
Þxðg; tÞdm
0
þ
oðtÞ
4p
F 0 xðg 0 ; tÞ expðÀt=m 0 Þ
(14.2)
Then the transfer equation develops to
m
dIðt; m; m 0 ; ’Þ
dt
¼ ÀIðt; m; m 0 ; ’Þ þ Bðt; m; m 0 ; ’Þ
(14.3)
I. Melnikova et al., Remote Sensing of the Environment and Radiation Transfer,
DOI 10.1007/978-3-642-14899-6_14, # Springer-Verlag Berlin Heidelberg 2012
139