Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 213

µ 0,2 could be
substituted to the second of (6.12). The positive value of the square root
is chosen, owing to the demand of the logarithm argument positiveness.
Consider the observations of reflected radiance

µ 0,1 , where subscript i indicates that any of two values

ρ 1 and ρ 2 at two viewing an-

µ 1 and arccos µ 2 . The first of (2.24) gives difference [ ρ ∞ ( µ µ 0 )− ρ gles: arccos , ],

ρ where the arguments of measured value are omitted. The ratio of differences

ρ ∞ ( µ 1 , µ 0 )− ρ 1 ] | ρ ∞ ( µ 2 , µ 0 )− ρ 2 ] for different µ 1 and µ 2 provides the follow- [ [

τ 0 after the algebraicmanipulations
(Melnikova and Domnin 1997; Melnikova et al. 1998, 2000):

τ = ing expressions for values s and 3(1− g )

ρ 0 ( ϕ µ 1 µ 0 )− ρ 1 ] K 0 ( µ 2 )−[ ρ 0 ( ϕ µ 2, µ 0 )− ρ 2 ] K 0 ( µ 1 ) [ , ,

s 2 =

µ 1 ) K 2 ( µ 1 )

µ 2 ) − R

,

µ 1 ) − K 2 ( µ 2 )

ρ 0 ( ϕ µ 2, µ 0 )− ρ 2 ] K 0 ( [ ,

K 0 ( K 0 (

where specified

(6.13)

µ µ µ 0.955 a 2 (

0 ) K 0 ( 1 ) K 0 ( 2 )

= µ 1 − µ 2 ], R [

q (1 + g )

(2 s ) −1 ln mlK (

ρ 1 + ll

µ i ) K ( µ 0 )

τ =

ρ ∞ ( ϕ µ i , µ 0 )− ,

ϕ where is the viewing azimuth relative to the Sun's direction. It is possible to

use these formulas for processing the multi-directional satellite observational
data of the reflected solar radiance.
The couples of different pixels of the satellite image are characterized with
different solar and viewing angles. Let the cosines of the zenith solar and
viewing angles µ 2 relate to the second

pixel. It is suitable to apply this approach for the one-directional satellite
observations of the reflected solar radiance. Then the following expression of
parameter s 2 is derived from the ratio of the radiances:

µ 0,1 , µ 1 relate to the first pixel and µ 0,2 ,

ρ 0 ( ϕ 1 , µ 1 , µ 0,1 )− ρ 1 ] K 0 ( µ 2 ) K 0 ( µ 0,2 )

[

ρ 0 ( ϕ 2 , µ 2 , µ 0,2 )− ρ 2 ] K 0 ( µ 1 ) K 0 ( µ 0,1 ) −[

s 2 =

µ 1 ) K 0 ( µ 0,1 ) K 0 (

[

ρ 2 ] K 2 ( µ 1 )

µ 0,2 ) + a 2 ( µ 2 ) a 2 ( µ 0,2 )

− R 1

µ

µ 1 ) − K 2 (

2 )

ρ ϕ 2 , µ 2 , µ 0,2 )− × (

12 q

K 0 ( K 0 (

where specified

= µ 2 ) K 0 ( µ 0,2 ) R 1 K 0 ( (6.14)

[

ρ 1 ] K 2 (

+ a 2 (

µ 2 ) µ 1 ) µ 1 ) a 2 ( µ 0,1 )

µ 0,2 ) − K 2 (

ρ ϕ 1 , µ 1 , µ 0,1 )−

× (

µ 1 ) 12 q K 0 ( K 0 (

With the verybigmagnitudes of optical thickness, the atmosphere is considered
as a semi-infinite one. In this case, difference [ ρ ∞ ( µ µ 0 )− ρ , ]tendstozero