Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 210

The expressions in the numerators of the first relations in (6.3)-(6.5) are
the differences of squares of the net fluxes, measured at corresponding alti-
tude levels. The following nominations are used for brevity: F ↓↑ = F ↓↑ (0, µ 0 ),

0

F ↓↑

µ 0 ). It is important that the dependence upon ground albedo
appears in the case of the bottom sublayer only (through value F N )andthe
dependence upon incident solar angle (through functions K 0 (

= F ↓↑ ( τ i ,

i

µ 0 ))
appears at the upper sublayer only. Meanwhile, the results for the internal
sublayers are independent of these parameters.
Expressions (2.24) for the reflection and transmission functions together
with (2.40) and expansion (2.41) serve the base for inferring the inverse for-
mulas allowing the retrieval of values s 2 and

µ 0 )and a 2 (

τ from radiance observations
within clouds. The formulas are divided into three groups in the case of obser-
vations of the radiance analogously to the case of the irradiance dependence
upon sublayer location.
The following modification of (2.40) is also used similar to (2.44):

µ 0 ) 1+ l e −2 k ( τ 0 − τ )

1
2 mK 0 (

µ = µ µ =

J ( ) I ( )− I (− ) ,

1− ll e −2 k

τ

0

(6.6)

)− l e −2 k (

τ τ

µ b ∞ ( µ

0 − )

I (− )

= .

µ ) l e −2 k (

τ τ

µ

I ( ) 1− b ∞ (

0 − )

where the nomination is specified analogously to value b ∞

- for the cloud top sublayer (level numbers are 0, 1)

ρ 0 − ρ

) 2 − J (

τ 1 ) 2

(

s 2 =

µ 0 ) 2 K 0 ( µ τ 1 ) 2 [2 Q 2 w ( µ µ µ µ Ξ

) 2 + J ( )−9 q 2 ]−9

2 [ I (

)] 2 −

16 K 0 ( )+ I (−

µ µ 0 )

) a 2 (

) a 2 (

Ξ = ρ 0 − ρ where specified ( ,

6 q

ln l

(6.7)

τ 1 )−6 µ µ µ 1

2 s

[ J ( I ( ) s (1 − 3 s )]

τ =

τ 1 )+6 µ µ µ [ J ( I (− ) s (1−3 s )]

⎡ ⎤

µ µ

) s + a 2 (

0 ) a 2 (

)

ρ ∞ − ρ µ 0 ) K 0 ( µ ( )+4 K 0 (

12 q

⎣ ⎦ .

×

µ µ

) s + a 2 (

0 ) a 2 ( )

ρ ∞ − ρ µ 0 ) K 0 ( µ ( )−4 K 0 (

12 q

- for the internal sublayers (level numbers are i −1, i )

τ i −1 ) 2 − J ( τ i ) 2 ] J ( τ i −1 ) 2 [ J (

s 2 = , (6.8)

µ τ i ) 2 I i −1 ( µ µ τ i −1 ) 2 I i ( µ µ

36

2 [ J (

) I i −1 (− )− J ( ) I i (− )]

2 s ln [ J (

,

τ i −1 )−6 I i −1 ( µ µ µ τ i )+6 I i (− µ µ µ

1 ) s (1−3 s )][ J ( ) s (1−3 s )]

τ i − τ i −1 =

τ i −1 )+6 I i −1 (− µ µ µ τ i )−6 I i ( µ µ µ [ J ( ) s (1−3 s )][ J ( ) s (1−3 s )]