The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 136

It is possible to distinguish two essentially different cases: clear and overcast
sky.
In case of the overcast sky, we succeeded in obtaining the explicit analytical
solution, i. e., to write the components of vector X through the results of
observations Y as explicit analytical expressions. Moreover, these expressions
are not the approximations or empirical formulas, which are often used, but
the consequences of the rigorous relations of the radiative transfer theory.
We should point out that deriving similar relations for the inverse problem
of atmospheric optics is a rather rare case against the backcloth of the recent
mass enthusiasm for the numerical solving of the inverse problems on PC.
Actually, it corresponds to the philosophical traditions of physics according
to which the analytical methods of description of the natural phenomena are
preferable.
As follows from the results of the well-known study by Tikhonov (1943)
concerning the mathematical aspects of the inverse problem theory: if the in-
verse problemsolution is the limited set of continues functions 1 (the analytical
solution is the limited set), this solution will be stable. It has been shown in
the topic by Prasolov (1995) that the analysis of the stability of the inverse
problem solution (robustness) in the limited class of functions is reduced to
the statement of the intervals of the continuity of the functions describing the
solution. It follows from Chebyshev theorems about the solution stability in
the polynomials basis and from the Weierstrass theorem about the existence
of the uniform limit (converging to the solution) in the continuous function
space. In the case of the analytical solution, its analysis for the continuity is not
complicated. Further, the corresponding results will be presented while in de-
tail considering the possibilities of the analytical approaches. The derivations
of the pointed analytical relations will be shown in Chap. 6, and the analysis of
the results of the observational data processing for the cloudy atmosphere will
be considered in Chap. 7.
Regretfully, a similar analytical solution for the clear atmosphere has not
succeeded. It is easy to understand it basis on general principles. The variant of
the overcast sky, when only the diffused radiation is measured, and the variant
of the pure clear atmosphere, when only the direct radiation is accounted for
(the optical thickness is easily obtained from Beer's law) are the limit cases of
very strong diffusion or its absence. The real clear atmosphere is an interme-
diate case from the point of view of the diffuse strength and the intermediate
cases are usually more complicated than the limit ones. So, while processing
theverticalprofilesofthespectralirradiances,(Chap.3)theinverseproblem
has been put as a problem of numerical choice of the parameters satisfying
theabove-formulateddemandoftheminimummin X ∈ T ρ

\

X )). The
search for the minimum (4.4) is not physical but a mathematical problem.
Thus, in this chapter this solution will be considered from the mathematical
side while accounting for the physical conditions and observational errors.

( Y , G ( X , U

1 IntheoriginalwordingbyAndreyTikhonov,theterm“continuesmappinginthecompactspace”
is used. It is more general than that which we are using but these terms coincide in the case of finite
dimensioned space, which we are considering.