Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 73

applicability region of the formulas derived in the work of Yanovitskij (1991)
and Dlugach and Yanovitskij (1983). For the weakly extended phase function
( g

≤

0.5) calculations, the errors are not exceeding 1%.
The accuracy of the calculations of the radiative characteristics with (2.53)-
(2.56) for multilayer cloudiness has been tested for the following cases:

τ i =

= ω 0 i = 5, 7, 10; g i 0.99, 0.995, 0.999 by comparison with

the values calculated with the doubling and adding method. Equations (2.53)-
(2.56) for the irradiance and radiance are accurate for all values of

0.65, 0.75, 0.85 and

ω 0 i and g i

τ i ≥ 7, the errors are less than 1-2%, and when τ i ∼ 5 the error reaches when

10% (Melnikova and Zhanabaeva 1996).

2.5
Conclusion

Specific features of two methods are considered in Chap. 2: the first is the
Monte-Carlo method, one of the most widely used numerical methods for the
calculation of radiative characteristics; the other is a method of asymptotic
formulas from transfer theory applied to the calculation of radiative charac-
teristics in the case of the overcast sky.
The Monte-Carlo method allows for all features of the interaction of radi-
ation with the atmosphere and surface with high accuracy that makes it in-
dispensable for the standard calculations of the radiative characteristics of the
atmosphere. Besides, the Monte-Carlo method makes possible the simulation
of the processes of the real radiation measurements, which is especially impor-
tant for problems of observational data interpretation (Fomin et al. 1994). This
is the main reason for the application of the method in our analysis of airborne
observational data of the solar irradiances that will be considered in Chaps. 3-
5. Finally, we would like to mention that the Monte-Carlo method is rather
simple and flexible, which allows easy realization of computing algorithms on
PC and the application of these to different problems of the theory of radiative
transfer. Further dissemination of the method could be expected in the near
future taking into account the appearance of modern computer systems with
the ability to perform parallel calculations (Sushkevitch et al. 2002). The main
and rather serious disadvantage of this method is the random error contained
in its results (i. e. the method is a full analog of the observations). An increase
in computing time and modern computing systems can lead to a decrease in
this error.
Theapproachforthecalculationofthereflectionfunctionofthesemi-
infinite atmosphere with the analytical formulas is proposed for the Henyey-
Greenstein phase function. We would like to point out that on the one hand the
phase function for real clouds could be more complicated than the Henyey-
Greenstein formula. However, on the other hand Raleigh scattering together
with the influence of multiple scattering could turn out to be rather significant
and smooth the shape of the real cloud phase function. Thus, the proposed
approach can provide less computational error for the real cloudiness than
is to be expected according to the theory. We would like to stress that the