Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 38

Direct radiation is necessary to take into consideration also for the description
of the reflection because:

= µ 0 T ↓ ( µ µ , ϕ ϕ ) σ

0 (

µ µ 0 ) + exp(− τ 0 |µ 0 )] . , , [ ,

π

µ ϕ ) the anal-
ogous expressions are obviously deriving. Further, according to the definition
of the operator (1.58) and with a subject to equality I 0 (

For the case of the illumination from the bottom to direction (

S (due to the
orthotropy of the reflection, the link between the radiance and irradiance (1.4)
and equality of the irradiances in definitions (1.77)) we finally obtain for the
bottom of the atmosphere:

µ , ϕ ) =

1

T ↑ (

µ µ , ϕ ϕ ) = µ ˜ ρ

0 (

µ , µ µ , , , 2 ) d

0

i. e. the T ↑ depends only on

µ .

The analogous expression is obtained for the top of the atmosphere

1

T (

µ µ , ϕ ϕ ) = σ

0 (

µ µ ) + exp(− τ 0 |µ µ d µ , , , 2 [ ˜ , )]

0

where direct radiation and condition T (

T ↑ (

µ µ , ϕ ϕ ) = µ µ , ϕ ϕ )aretaken , , , ,

into account.
The product of the integral operators is found by definitions (1.58) and
(1.73)

π

2 1

1
π

R T ↑ ( µ µ , ϕ ϕ ) = ϕ µ ϕ µ , ϕ ) T ↑ ( µ , µ , ϕ , ϕ ) µ d µ , , , d R ( , ,

0 0

that yields after substituting the above-obtained expressions, in particular
R = A ,constant AC for R T ↑ ,where

1 1

= ρ

0 (

µ , µ ) µ µ d µ d µ C 4 ˜ . (1.80)

0 0

Absolutely analogously RT ↓ is found as A µ 0 φ µ 0 ), where (

1

µ , µ d µ + exp(−

φ µ 0 ) = σ

0 (

µ 0 ) τ 0 |µ 0 ). ( 2

0