Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 26

χ χ 0 the following expression is correct where value is defined by (1.46) and for

according to the same equation:
χ 0 = µµ 0 + 1−

2 1−

µ µ

0 cos(

ϕ ) (1.48)

Point out that (1.47) is written only for the diffuse radiation . The boundary
conditions are taking into account by the third term in the right part of (1.47).
The sense of this term is the yield of the first order of the scattering to the
radiance and the integral term describes the yield of the multiple scattering .
The ground surface at the bottom of the atmosphere is usually called the
underlying surface or the surface . Solar radiation interacts with the surface
reflecting from it. Hence, the laws of the reflection as a boundary condition at
the bottom of the atmosphere should be taken into account. However, it is done
otherwise in the radiative transfer theory. As will be shown in the following
section, there are comparatively simplemethods of calculating the reflection by
the surface if the solution of the transfer equation for the atmosphere without
the interaction between radiation and surface is obtained. Thus, neither direct
nor reflected radiation is included in (1.47). As there is no diffused radiation
at the atmospheric top and bottom, the boundary conditions are following

µ µ 0 , ϕ = µ

I (0,

, ) 0 > 0,

(1.49)

τ µ µ ϕ = µ I (

0 ,

,

0 ,

) 0 < 0.

Transfer equation (1.47) together with (1.46), (1.48) and boundary conditions
(1.49) defines the problem of the solar diffused radiance in the plane parallel
atmosphere. Nowadays different methods both analytical (Sobolev 1972; Hulst
1980; Minin 1988; Yanovitskij 1997) and numerical (Lenoble 1985; Marchuk
1988) are elaborated. Our interest to the transfer equation is concerning the
processing and interpretation of the observational data of the semispherical
solar irradiance in the clear and overcast sky conditions. The specific numerical
methods used for these cases will be exposed in Chap. 2. Now continue the
analysis of the transfer equation to introduce some notions and relations,
which will be used further.
The diffused radiation within the elementary volume could be interpreted
as a source of radiation. It follows from the derivation of the volume emission
coefficient through the diffused radiance in (1.34) if the increasing of the
radiance is linked with the existence of the radiation sources. Then introduce
the source function :

π

2 1

= ω 0 (

τ )

τ µ µ 0 , ϕ ϕ τ χ τ µ , µ 0 , ϕ ) d µ B ( , , ) d x ( , ) I ( ,

π 4

(1.50)

0 −1

+ ω 0 (

τ )

τ χ 0 ) exp(− τ|µ 0 ), Sx ( ,

4