Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 27

and the transfer equation is rewritten as follows:

τ µ µ 0 , ϕ dI ( , , )

µ = τ µ µ 0 , ϕ τ µ µ 0 , ϕ − I ( , , )+ B ( , , ) . (1.51)

τ d

Equation (1.51) is the linear inhomogeneous differential equation of type
dy ( x ) | = dx ay ( x )+ b ( x ). Its solution is well known:

x

= b ( x )exp( a ( x − x )) dx . y ( x ) y ( x 0 )exp( a ( x − x 0 )) +

x 0

Applying it to (1.51) under boundary conditions (1.49), it is obtained:

τ

)exp − τ

d

τ 1

µ

−

τ µ µ 0 , ϕ = τ , µ µ 0 , ϕ τ µ I ( , , ) B ( , > 0,

µ

0

(1.52)

τ

)exp − τ

d

0

τ

− 1

−

τ µ µ 0 , ϕ = τ , µ µ 0 , ϕ τ µ < 0. I ( , , ) B ( ,

µ µ

τ

Certainly(1.52)arenottheproblem'ssolutionbecausesourcefunction
B ( τ µ µ 0 , ϕ ) itself is expressed through the desired radiance. However, (1.52)

allows the calculation of the radiance if the source function is known, for exam-
ple in the case of the first order scattering approximation when only the second
term exists in the definition of function B (

, ,

τ µ µ 0 , ϕ ) (1.50). The expressions

forthereflectedandtransmittedscatteredradianceofthefirstorderscattering
in the homogeneous atmosphere (where the single scattering albedo does not
depend on altitude) have been obtained (Minin 1988):

, ,

χ 0 ) 1−exp −

0 )

τ ( 1 + 1

µ 0 ω 0
4

S µ µ

τ µ µ 0 , ϕ = µ I 1 ( , , ) x ( < 0,

µ µ 0 +

χ 0 ) exp( − µ

) − exp( − τ

µ 0 )

µ 0 ω 0
4

S

τ µ µ 0 , ϕ = µ I 1 ( , , ) x ( > 0.

µ µ 0

−

Return to general expressions for the radiance (1.21), substitute them to source
function definition (1.19), and deduce the following:

π τ

ϕ

2 1

= ω 0 (

τ µ

µ

)

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

π 4

0 0 0

τ

exp − τ

d

ϕ )exp − τ

d

τ

0 0

τ µ

µ

τ −

) d

−

τ − τ χ τ , µ , µ 0 , × x ( , B (

µ µ

τ −1

+ ω 0 (

τ )

τ χ 0 ) exp(− τ|ζ Sx ( , ).

4

(1.53)