emission and extinction coefficients, we are defining the radiative transfer
equation:
dI
dl
¼ ÀaI þ e
(1.17)
In spite of the simple form, Eq. 1.17 is the general transfer equation that accepts
the coefficients a and e as variable values. This derivation of the radiative transfer
equation is phenomenological. The rigorous derivation must be done using the
Maxwell equations.
Move to the consideration of particular cases of transfer Eq. 1.17 in conformity
with shortwave solar radiation in the Earth atmosphere. Within the shortwave
spectral range we omit the heat atmospheric radiation against the solar one and
seem to have the relation e ¼ 0. However, we are taking into account that the
enhancing of emitted energy within the elementary volume could occur also owing
to the scattering of external radiation coming to the volume along the direction of
the transfer in Eq. 1.17 (i.e. along the direction normal to the side dS). Specify this
direction ~ r 0 and scrutinize radiation scattering from direction ~ r with scattering
angle g (Fig. 1.6). Encircling the similar volume around direction ~ r (it is denoted
as a dashed line), we are obtaining energy scattered to direction ~ r 0 . Then employing
precedent value of energy E 0 , we are obtaining the contribution to the emission
coefficient corresponded to direction ~ r:
deð~ rÞ ¼
s
4p xðgÞIð~ rÞdSdOdldtdOdl
dVdOdldt
¼
s
4p
xðgÞIð~ rÞdO:
Then it is necessary to integrate value deð~ rÞ over all directions and it leads to the
integro-differential transfer equation while taking into account the scattering:
dIð~ r 0 Þ
dl
¼ ÀaIð~ r 0 Þ þ
s
4p
ð
4p
xðgÞIð~ rÞdO:
(1.18)
dl
dS
I
I + dI
r 0
d W
g
(r)
e
dW
r
®
®
®
)
(r
I
®
Fig. 1.6 The derivation of
the radiative transfer equation
1.3 Radiative Transfer in the Atmosphere
13