The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 152

nonlinear algorithms, if matrix G is to be taken at the last iteration. Note that
(4.44) relates also to the penalty functions method (4.30) and (4.31). As the
additional yield to discrepancy at the last iteration is zeroth for this method (at
least, theoretically), hence the matrix of the system (4.36) is similar to above
matrix A .
The main stage of the inverse problem solving with LST and of the method
of the maximal likelihood (4.43) is solving a linear equation system, i. e. the
inversion of its matrix. However, in the general case the mentioned matrix
could be very close to a degenerate one. Then, with real computer calculations,
matrix ( G + S − Y G ) −1 is unable to inverse or the operation of the inversion is ac-
companiedwith a significant calculation error. The reason of this phenomenon
is connected with the incorrectness of the majority of the inverse problems of
atmospheric optics (that is a general property of inverse problems). The de-
tailed theoretical analysis of the incorrectness of the inverse problem together
with the numerous examples of the similar problems is presented in the topic
by Tikhonov and Aresnin (1986). The simple enough interpretation was per-
formed in the previous section while discussing the phenomenon of the strong
spread of the desired values during the consequent iterations. Technically, the
incorrectness appears as mentioned difficulties of matrix ( G + S − Y G ) −1 inver-
sion, i. e. its determinant closeness to zero. Note that not all concrete inverse
problems are incorrect, however, the solving methods of the incorrect inverse
problems should always be applied if the correctness does not follow from the
theory. It is necessary because the analysis of the incorrectness is technically
inconvenient, as it needs a large volume of calculations (Tikhonov and Aresnin
1986). Thus, further we will consider the problemof the parameters X retrieval
from observations Y as an incorrect one. Assume for brevity the linear case of
the formulas and then automatically apply the obtained results to the algorithm
recommended for the nonlinear inverse problems.
The method of the incorrect inverse problems solving is their regularization
- the approach (in our concrete case of the linear equation system) of replacing
the initial systemwith another one close to it in a certainmeaning and forwhich
the matrix is always non-degenerate (Tikhonov and Aresnin 1986). Further,
we consider two methods of regularization usually applied for the inverse
problems solving in atmospheric optics.
The simplest approach of regularization is adding a certain a priori non-
degenerate matrix to the matrix of the initial system. Instead of solution (4.43),
consider the following:

= ( G + S − Y G + h 2 I ) −1 G + S − Y ( Y − G 0 ) , X (4.45)

where I is the unit matrix, h is a quantity parameter. It is evident that solution
(4.45) tends to “the real” one (4.43) with h 0. Thus, the simple algorithm

follows: the consequence of solutions (4.45) is obtained while parameter h
decreases and value X with the minimumdiscrepancy is assumed as a solution.
This approach is called “ theregularizationbyTikhonov ”(althoughithadbeen
known for a long time as an empiric method, Andrey Tikhonov gave the
rigorous proof of it (Tikhonov and Aresnin 1986)).

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