Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 226

theaveragevalueoftheopticalthicknessiscalculatedforeverypixel.The
relative deviations of the optical thickness obtained for every direction from
the average one could be taken as a measure of the deviation of the cloud top
from the plane. It is necessary to have in mind that parameter r also includes
the influence of the radiation scattering by the above atmospheric layers and
thin semitransparent above clouds. Then the following is proposed for the
evaluation of parameter r :

N

1
N¯

= τ τ i | ,

1 |

r ¯ − (6.39)

τ

= i

τ

where N is the number of viewing directions for every pixel and ¯ is the average

optical thickness over viewing directions. This methodology was applied to
POLDER (Polarization and Directionality of the Earth's Reflectance) Level-2
data containing the reflected radiance at 14 directions (Melnikova and Naka-
jima 2000).

6.3
Analysis of Correctness and Stability of the Inverse Problem Solution

The above-proposed set of formulas is the solution of the inverse problem of
atmospheric optics for the accepted cloud model. According to the topic by
Prasolov (1995) the range of the continuality of the obtained functions is to be
analyzed for testing the solution correctness.
In the case of (6.1) the analysis of continuality and positiveness of function
s 2 ( F ↓ , F ↑ ,

µ 0 ) taking into account evident condition F (0) τ 0 )yieldsthe

≥ F (

following inequalities:
- For cosine of solar incident angle µ

0 > 0.3

1 > 2[ F ↑ 2 (0) + F ↓ 2 (

τ 0 )− F ↓ ( τ 0 ) F ↑ (

τ 0 )] + F (0) ,
s [8.0 + 0.2(1 − A )] > 0.54(1 − A ) 2 +0.3(1− A ),

(6.40)

µ 0 > 0.9 - For cosine of solar incident angle

1 > 0.7[ F ↑ 2 (0) + F ↓ 2 ( τ 0 )] + 1.1 F (0) ,

s [8.0 − 0.2(1 − A )] > 0.54(1 − A ) 2 −0.2(1− A ).

τ 0 )−0.7 F ↓ ( τ 0 ) F ↑ (

(6.41)

The concrete numerical magnitudes of the parameters providing continuality
and positiveness of function s 2 ( F ↓ , F ↑ ,

µ 0 ) are different for every observed pair
of upwelling and downwelling irradiances at the single level and wavelength.
Thus, the experimental data have to be tested for satisfying these inequalities
before applying (6.1) to the observational results. Corresponding procedures
are provided in the algorithms of the observational data processing. The anal-
ogous inequality could be easily derived for all cases considered hereinbefore
andthecorrespondinganalysisisincludedtotheprocessingalgorithms.