Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 217

τ i = τ i (1 − g i )looklike The expressions for 3

2 s 1 ln l 1 1+ 2 K 0 (

1− ,

µ 0 ) s 1 (4−9 s 1 )

1 8 A 1 s 1

1− A 1 a 1

τ 1 = = i 1,

µ 0 )− F ↑ (0) a (

ln l i 1+ 2 s i (4−9 s i )

1− ,

1
2 s i

8 A i s i
1− A i a i

τ i =

a i − A i −1

(6.21)

µ 0 )and a ∞ are the plane and spherical albedo of the upper layer and
a i is the spherical albedo of the i -th layer.
For the data of the radiance observations the expressions for parameter s 2
are the following:
-fortheupperlayer( i

where a (

= 1)

K 0 (

µ ) 2 ( ρ 0 − ρ 1 ) 2 − K 0 ( µ ) 2 σ

1

s 2 =

) 2 K 0 (

1− A 1 2 − J

, (6.22)

1 A 1

µ µ µ 0 ) 2 − σ

) 2 K 0 (

16 K 0 (

where J is specified as following

2 A 1
1− A 1 [

= µ µ

K 0 (

µ ρ 0 − ρ 1 ) 2

]

J a 2 ( )+ n 2 (1 − w ( )) )(

µ 0 ) K 0 (

µ µ

) 2 (

ρ 0 − ρ 1 )

+ a 2 (

) a 2 (

6 q

) K 0 (

1

A 1
1− A 1 K 0 (

A 1
1− A 1 K 0 (

−24 q µ µ ρ 0 − ρ 1 ) 2 − µ σ )( )

- for the layer with number i> 1

K 0 (

µ ) 2 ( σ i −1 − ρ i ) 2 − K 0 ( µ ) 2 σ

i

s i =

) 2 K 0 (

1− A i 2 − J

,

i A i

µ µ σ σ

i −1 −

16 K 0 ( ) 2

2 A i
1− A i [

= µ µ

K 0 (

µ σ i −1 − ρ i ) 2 J a i ( )+ n 2 (1 − w ( ))

]

)(

(6.23)

µ ) K 0 ( µ ) 2 ( σ i − ρ i ) +2 a 2 (

) K 0 (

i .

A i
1− A i K 0 (

A i
1− A i K 0 (

−24 q

ρ i ) 2 −

µ µ σ i − µ σ

)( )

µ µ µ Functions a 2 ( )andvalue n 2 are calculated for phase function

parameter g i corresponding to the properties of the i -th layer. The subscripts
are omitted in the formula for brevity.
Remember here the above conclusion concerning the definition of albedo A i .
The ratio of the radiances observed at viewing angles

), K 0 ( )and w (

ϑ 1,2 = arccos( ± 0.67) at