The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 137

The solution of the inverse problem for the irradiance observations in the
atmosphere and its results will be described in Chap. 5.
Before we present the concrete formulas and algorithms of the search for
minimum (4.4), we will mark that the mathematical aspects of the mentioned
problem solving are presented often in a rather abstract manner (Kondratyev
andTimofeyev 1970) (coming fromthe approaches of variationcalculus and the
theory of self conjugate operators in Gilbert space, Elsgolts 1969; Kolmogorov
and Fomin 1989). Sometimes it is complicated in practical applications of
the abstract expressions and they are perceived as formal receipts for the
problem solving without the real physical meaning. Besides, the important
questions of the choice of the mathematical model for the direct problem
solving, the choice of its concrete parameters and their influence are out
of the scope of such a presentation. Our experience of solving the inverse
problems of atmospheric optics demonstrates that the understanding of the
physical meaning of the relations in use plays an important role together
with the formal mathematical approaches. Thus, we will try to present the
indicated mathematical approaches not from the abstract positions but from
the applied ones in the simplestmanner not ignoring even the technical aspects
of the realization. To understand such a presentation knowledge of linear
algebra (Ilyin and Pozdnyak 1978) and mathematical statistics (Cramer 1946)
is enough. We shouldmention that it is very convenient for comprehension and
analysis of the described approaches to consider themapplying to the problems
of the minimal dimensions (one-dimensional and two-dimensional).
The methodology presented below is not the only approach to the search for
minimum (4.4). In fact, the stated problem relates to the class of mathemat-
ical extreme problems, whose solutions are well known nowadays (Vasilyev
F 1988). For example, in practice such elementary manner as a sorting of
a limited quantity of the vector X variants (Kaufman and Tanre 1998) is often
used for the solution search. However, the methodology described below is the
mathematically faultless one and allows for the correct account of the obser-
vational uncertainties that is particularly important. Its application becomes
increasingly popular with the development of the possibilities of computer
techniques.
Wewillbeginthepresentationfromthedefinitionofthedistancebetween
the vectors. Let us use the standard Euclid metrics (Kolmogorov and Fomin
1989) i. e. assume the following:

N

1
N

( y (1 i − y (2) ρ ( Y (1) , Y (2) ) = ) 2 . (4.5) i

= i 1

Thematter of Euclidmetrics (4.5) is the SDof two vectors, i. e. from the physical
point of view we are interested in the closeness between the observational
results and the direct problem solution in average over the entire observational
data set i

= 1,..., N . The choice of this metric is predetermined because only it

succeeds the construction of the real algorithms for the search of the metrics
minimum. For example, if we take not an average difference between the