The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 133

together with simplifications and approximations are inevitable, so any model
is simpler than the reality is, so it is inadequate when compared to the reality
to a certain degree. Hence, the choice of the concrete model together with its
parameters is always ambiguous and it is defined either with the physical pro-
cesses put to the model, or with the degree of approximation of the description
of these processes. For example, if we are considering only the radiative trans-
fer, the parameters of the model will be the following: optical thickness, single
scattering albedo, and phase function (see Sect. 1.3). Then we could account
for the processes of the radiation-media interaction defining the mentioned
values (see Sect. 1.2), and the parameters of the model will be: vertical profiles
of the pressure, temperature, concentrations of the atmospheric gases, and
volume coefficients of the aerosol scattering and absorption.
Secondly, the number of parameters describing the mentioned processes
is always finite in the range of the chosen model .Itisreadilyseenfromthe
technical point of view and needs no comment. However, from the other side
the number of the measured characteristics is finite too. Actually, if even the
continuous spectrum of the irradiance or radiance is registered, really it is
representing as a finite array of the measured characteristics (see Sect. 3.1).
The opposite case is impossible because of digitations of the output signal.
Thus,itissafetosaywithoutthegeneralitylossthatwhilesolvingthedirect
problem we realize an algorithm allowing the calculation of a strictly limited
set of values through a strictly limited set of parameters. This statement is
expressed with the mathematically formal relation:

Y

= G ( U ), (4.1)

where Y = 1,..., N is the set, i. e. the vector, of the calculated val-

ues, corresponding to real N measurements; G is the operator of the direct
problem solving, i. e. the realization of a certain (concretely chosen as has
been pointed out above) mathematical model of the observational process;
U

≡ ( y i ), i

= = 1,..., M is the vector of parameters of the model in question.

In general the components of vectors Y and U could be inhomogeneous, i. e.
could have different meaning and different units (it is always so for vector U ).
We s h ou l d me n t i on t h a t v e c t o r U includes all necessary parameters for solving
the direct problem (not only parameters characterizing the atmosphere and
surface but also the solar zenith angle, value of the incident flux at the top of the
atmosphere, spectroscopic parameters for computing the volume coefficient
of the molecular absorption - Sect. 1.2 etc.), and vector Y contains only the
observational results.
The formal statement of the inverse problem is determination (in atmo-
spheric optics it is accepted to say retrieval )ofthecomponentsofparameter
vector U with the specified concrete values of observational result vector Y .
However, there is no sense in retrieving all parameters included in vector U .
Actually, some parameters of vector U , for example the solar zenith angle, are
known (exacter: are supposed to be known). Therefore, from the components
of vector U let us select vector X

( u j ), j

= 1..., K , which has to be retrieved.

The concrete variants of this selection are considered in the study by Timofeyev

≡ ( x k ), k