Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 70

σ i −1 ( τ i −1 , µ 0 , µ n i ). The expressions for the radiance are obtained as:

k

µ 0 )
n 1

K 1 (

) n i f ) n k f σ k =

K i (

µ τ i ) = σ k −1 K k ( η τ k ), ( (

σ σ

(2.55)

=

i 1

) n k f

ρ k = µ µ τ k )] σ k −1 . [ a k ( )− K k ( (

ρ

The subscripts in these expressions are related to the layer with the correspon-
dent number and optical parameters g i , τ i .Ifthereisaconservative

scattering in the layer with number i ,escapefunction K (

ω 0 i and

µ )convertstofunc-

µ

), values n i and a i are equal to unity, n i accepts value 1

|

tion K 0 ( (1 − A i )and

τ i ), defining the dependence upon the optical thickness
are expressed with the formula:

τ i )and f σ ( functions f (

ρ

4(1 − A i )
(1 − A i )[3(1 − g i )

f f

τ i ) = τ i ) =

( ( . (2.56)

ρ σ

τ i +3 q ]+4 A i

For the case of the layers of the arbitrary optical thickness, (2.52) is used for
the derivation of the expressions for multilayer clouds analogous to the thick
layers. The formulas for the irradiances for the upper layer coincide with (2.52).
The irradiances for the lower layers are expressed with:

F i −1 A i +

,

A i +1 V i
1− A i +1 A i

F i −1 A i

F i = =

(2.57)

V i
1− A i +1 A i

F i −1 V i F i −1

F i = = ,

where values A i and V i arecomputedwiththerelations:

1

= F i µ µ = τ τ A i 2 d 1− f i [ p (

i )(ch k i +1)−1],

(2.58)

0

= τ i )(ch k τ i +1)−ch k τ i ]. V i f i [ p (

2.4
Uncertainties and Applicability Ranges of the Asymptotic Formulas

The asymptotic formulas of the transfer theory presented in this chapter are
obtained rigorously. It is necessary to take into consideration that they are
describing the radiation field within the boundaries and at them the more
exact, the bigger the optical thickness is and the less true absorption is. In
addition, there is a strong relationship between the accuracy and the degree
of the scattering anisotropy (the extension of the phase function forward or
the magnitude of parameter g ). Certain mathematical aspects concerning the
estimation of the applicability ranges for the asymptotic formulas of reflection