Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 218

the boundaries between layers i −1and i defines the albedo corresponding to
the boundary of the i -th layer: |σ i −1 (0.67).

Scaled optical thickness of separate layers

ρ i ( − 0.67)

τ i = τ i is described with 3(1 − g i )

the following formulas:
-fortheupperlayer: i = 1

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

1− ,

µ µ 0 ) s 1 (4−9 s 1 ) 1 ) K 0 ( 8 A 1 s 1

1− A 1 a 1

τ 1 =

ρ ∞ − ρ 1 ) (

(6.24)

- for the layer with number i> 1

ln l i 1+ 2 K 0 (

1− .

µ σ i −1 s i (4−9 s i ) 1

2 s i

) 8 A i s i

1− A i a i

τ i =

µ σ i −1 − ρ i ) ( a i ( )

(6.25)

Theobtainedexpressionscouldbeappliedfortheretrievaloftheoptical
parameters of the cloud layer from the observations of solar radiation at the
layer boundaries of the multilayer cloud system.
If the layers are not optically thick, it is possible to use the corresponding
formulas:
-fortheupperlayer: i

=

1

(1 − F 1 ) 2 −(1− A 1 ) 2 F ↓ 2

s 1 =

1

,

16[ u 1 − v 1 ]

r 1 u 1 + v 1 + ( u 1 − v 1 )( r 1 −1)
u 1 + r 1 v 1

(6.26)

τ 1 = s −1 3(1 − g 1 ) ln ,

1

where

1− F 1
(1 − A 1 ) F 1

u 1 − A 1 F 1 ( p 1 −1) and v 1 v 1 + A 1 F 1 p 1 . = = = r 1 , u 1

- for the layer with number i> 1

(1 − F i ) 2 −(1− A i ) 2 F ↓ 2 i

s i = ,

16 F ↓ 2

i −1 [ p i − q i ]

(6.27)

ln r i p i + q i + ( p i − q i )( r i −1)
p i + r i q i

τ i = s −1 3(1 − g i ) ,

i

where

1− F i
(1 − A i ) F i

= = p i − A i F i q i = q i + A i F i p i . r i , p i and q i