convenient for some problems where the phase function needs an analytical
approximation. The one of the widely used approximations is a Henyey-Greenstein
phase function:
xðgÞ ¼
1 À g
2
ð1 þ g 2 À 2g cos gÞ
3=2
;
(1.15)
where g is the approximation parameter (0 g < 1),
g ¼
1
3
x 1 ¼
1
4p
ð
2p
0
d’
ð 1
À1
xðgÞgdg;
(1.16)
it coincides with the mean cosine of the scattering angle, changes in the ranges
[0,1], and is called often the asymmetry factor because it governs the degree of the
phase function forward extension.
The function describes the main property of the aerosol phase functions –
the forward peak – (the prevalence of the scattering to the forward hemisphere
0 g p/2 over the scattering to the back hemisphere p/2 g p) and it is very
suitable for the theoretical consideration, as it will be shown further.
1.3 Radiative Transfer in the Atmosphere
Within the elementary volume, the enhancing of energy along the length dl could
occur in addition to the extinction of the radiation considered above. Heat radiation
of the atmosphere within the infrared range is an evident example of this process,
though as it will be shown further the accounting of energy enhancing is really
important in the short-wave range either. Value dE – the enhancing of energy – is
proportional to the spectral dl and time dt intervals, to the arc of solid angle
dO encircled around the incident direction and to the value of emitting volume
dV ¼ dSdl. Specify the volume emission coefficient e as a coefficient of this
proportionality e ¼
dEr
dVdOdldt .
Consider now the elementary volume of medium within the radiation field. In
general case both the extinction and the enhancing of energy of radiation passing
through this volume are taking place (Fig. 1.6). Let I be the radiance incoming to
the volume perpendicular to the side dS and I + dI be the radiance after passing the
volume along the same direction. According to energy definition in Eq. 1.1 incoming energy is equal to E 0 ¼ IdSdOdldt then the change of energy after passing the
volume is equal to dE ¼ dIdSdOdldt. According to the law of the conservation of
energy, this change is equal to the difference between enhancing dE r and extincting
dE e energies. Then, taking into account the above definitions of the volume
12
1 Radiation in the Earth Atmosphere
approximation. The one of the widely used approximations is a Henyey-Greenstein
phase function:
xðgÞ ¼
1 À g
2
ð1 þ g 2 À 2g cos gÞ
3=2
;
(1.15)
where g is the approximation parameter (0 g < 1),
g ¼
1
3
x 1 ¼
1
4p
ð
2p
0
d’
ð 1
À1
xðgÞgdg;
(1.16)
it coincides with the mean cosine of the scattering angle, changes in the ranges
[0,1], and is called often the asymmetry factor because it governs the degree of the
phase function forward extension.
The function describes the main property of the aerosol phase functions –
the forward peak – (the prevalence of the scattering to the forward hemisphere
0 g p/2 over the scattering to the back hemisphere p/2 g p) and it is very
suitable for the theoretical consideration, as it will be shown further.
1.3 Radiative Transfer in the Atmosphere
Within the elementary volume, the enhancing of energy along the length dl could
occur in addition to the extinction of the radiation considered above. Heat radiation
of the atmosphere within the infrared range is an evident example of this process,
though as it will be shown further the accounting of energy enhancing is really
important in the short-wave range either. Value dE – the enhancing of energy – is
proportional to the spectral dl and time dt intervals, to the arc of solid angle
dO encircled around the incident direction and to the value of emitting volume
dV ¼ dSdl. Specify the volume emission coefficient e as a coefficient of this
proportionality e ¼
dEr
dVdOdldt .
Consider now the elementary volume of medium within the radiation field. In
general case both the extinction and the enhancing of energy of radiation passing
through this volume are taking place (Fig. 1.6). Let I be the radiance incoming to
the volume perpendicular to the side dS and I + dI be the radiance after passing the
volume along the same direction. According to energy definition in Eq. 1.1 incoming energy is equal to E 0 ¼ IdSdOdldt then the change of energy after passing the
volume is equal to dE ¼ dIdSdOdldt. According to the law of the conservation of
energy, this change is equal to the difference between enhancing dE r and extincting
dE e energies. Then, taking into account the above definitions of the volume
12
1 Radiation in the Earth Atmosphere