Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 36

thetopoftheatmosphere.Introduceoperator T ↓ ,sothat I ↓ = T ↓ I 0 specifying

that operator T ↓ istodescribethetransferofbothdiffusedand direct radiation
throughout the atmosphere. The latter has been excluded from the radiative
transfer equation and itmust be taken into account whenwe are considering the
reflection from the surface. The solution of the radiative transfer problemwith
the illumination from the bottom is I

TI 1 where I 1 describes the radiation
field coming from the bottom to the lower boundary, which is taken into
account according to (1.58). Besides, operator T hastodescribealsothetransfer
of direct reflected radiation (i. e. the radiation transferring from the surface
without scattering). Operator T ↑ describes the radiance incoming from above
tothelowerboundaryilluminatedfromthebottomwithradiance I 1 so that
I ↑

=

T ↑ I 1 .Value I ↑ means the radiance reflected fromthe surface then scattered
to the atmosphere and after all returned to the surface. Mathematically, the
problem of constructing all operators T , T ↓ ,

=

T ↑ is uniform as it follows from T ,

the previous section.
The radiance with a subject to the surface reflection is evidently obtained
as a sum of the following components. Firstly, it is the radiance of direct solar
radiation diffused to the atmosphere TI 0 . Secondly, it is the radiance of direct
anddiffusedradiationreflectedfromthesurface TI 1 that is the combination
TRT ↓ I 0 with taking into account (1.74). Further, it follows a subject to sec-
ondary reflected radiation TI 2

T ( r T ↑ RT ↓ I 0 ), etc. Finally, for the
desired radiance calculation we are obtaining the following:

T ( r T ↑ I 1 )

= =

TI 0 + T ( 1 + RT ↑ +( RT ↑ ) 2 +( RT ↑ ) 3 +...) RT ↓ I 0 . (1.75)
Expression (1.75) is known as a radiance expansion over the reflection order.
It is widely used in the algorithms of the numerical methods where it allows
organizing the recurrent calculations of the desired radiance. Note that the
series converges faster if the reflection is weaker. The operator approach is
presented in particular in the topics by Hulst (1980) and Lenoble (1985).
Consider a particular problem concerned with radiative transfer and re-
flection from the surface. Let us consider only the radiance at the boundaries
I (0,

= 2 I

µ µ 0 , ϕ µ < 0) and I ( τ 0 , µ µ 0 , ϕ µ > 0) without consideration of it

between the boundaries. The obvious examples are the problems of the inter-
pretation of the satellite and ground-based observations of the diffused solar
radiance. In these problems, the viewing angles are assumed to be in the range
[0,

, )( , )(

π| µ

is assumed positive. Then the desired values of the
radiance are written as I (0, −

2],i.e.thevalueof

µ µ 0 , ϕ )and I ( τ 0 , µ µ 0 , ϕ , , ) according to transfer

µ 0 > 0).
Specify thereflectionandtransmissionfunctions in accordance with Sobolev
(1972) are shown as
I (0, −

geometry (In any case it is

µ µ 0 , ϕ = µ 0 ρ µ µ 0 , ϕ τ 0 , µ µ 0 , ϕ = µ 0 σ µ µ 0 , ϕ

) , (1.76)
where the reflection from the surface is not taken into account. Specify the
analogous function for the case of illumination from the bottom:
I (0, −

, ) S ( , ), I ( , ) S ( ,

µ µ , ϕ = µ ˜ ρ µ µ , ϕ τ 0 , µ µ , ϕ = µ ˜ σ µ µ , ϕ

S

), I (

S

, ) ( , , ) ( , ) , (1.77)