Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 29

of the scattering is necessary to account in the series. Mark that, according
to (1.56), source function B linearly depends on q .Hence,sourcefunction
B (and the desired radiance) is directly proportional to value S ,i.e.tothe
extraterrestrial solar flux. So it is often assumed S = 1 and finally the obtained

= |π

radiance multiplied by the real value S F 0 .

= µ 0 BI 0 ,where I 0 = I (0, µ µ 0 , ϕ = πδ µ µ 0 ) δ ϕ As per (1.55) q , ) ( − ( )isthe

= τ µ µ 0 , ϕ extraterrestrial radiance. Consequently the desired radiance I I ( , , )

also linearly depends on I 0 and it is possible to formally write the following:
I = TI 0 , (1.57)

where T is the linear operator and the problem of calculating the radiance is
reduced to the finding of the operator. As function I 0 is the delta-function of
direction ( ϕ 0 ) (where the azimuth of extraterrestrial radiation is assumed

arbitrary) the radiance could be calculated for no matter how complicated an
incident radiation field I 0 (

µ 0 ,

µ 0 , ϕ 0 ) after obtaining the operator T as a function

µ 0 , ϕ 0 ) due to the linearity of (1.57). The following of all possible directions T (

relation is used for that:

π

2 1

= ϕ 0 µ 0 , ϕ 0 ) I 0 ( µ 0 , ϕ 0 ) d µ 0 . I d T ( (1.58)

0 0

The linearity of (1.57) is widely used in the modern radiative transfer theory
including the applied calculations. It is especially convenient for describing the
reflection from the surface that will be considered in the following section.
The presentation of the solution of the differential and integral equation as
a series expansion over the orthogonal functions is the standard mathematical
method. Certain simplification is succeededafter expanding the phase function
over the series of Legendre Polynomials in the case of the radiative transfer
equation. Legendre Polynomials are defined, e. g. (Kolmogorov and Fomin
1999) as,

d ( z 2 −1) n
dz

1
2 n n !

= P n ( z ) .

However, during the practical calculation the following recurrent formula is
more appropriated:

2 n −1
n

zP n −1 ( z )− n −1
n

= P n ( z ) P n −2 ( z ) (1.59)

= = where P 0 ( z ) z .

With (1.59) the relations P 1 ( z )

1, P 1 ( z )

= = | 2(3 z 2 −1)etc.areobtained.

Legendre Polynomials constitute the orthogonal function system within the
interval [−1, 1]:
1

z , P 2 ( z ) 1

1

2
2 n +1

= = P n ( z ) dz = P n ( z ) P m ( z ) dz 0, for n m and

−1 −1