The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 144

Write initial equation (4.22) for iterationnumber n and add product G n ( X n −
X 0 ) to both parts. After accomplishing elementary manipulations and solving
the equation with LST, as has been shown by Timofeyev et al., (1986), the
solutioncouldbeexpressedintheform:

= X 0 +( G n WG n ) −1 G n W ( Y − G ( X n )+ G n ( X n − X 0 )) . X n +1 (4.24)

In algorithm (4.24), all iterations are counted off X 0 that turns out a certain
obstacle for too strong spread of the values. At any rate, according to the
practical application, (4.24) demonstrates a higher effectiveness than (4.23)
does, in spite of the increased calculation volume. Furthermore, we will use
only algorithm (4.24) for solving the problem with LST.
Quite often the additional conditions (constraints and restrictions) are im-
posed to desired parameters x 1 ,..., x k based on the physical reasons, i. e. the
problemof searching not absolute but conditional extremums appears fromthe
mathematical point of view. This problem is more complicated and common
methods of its solution, e. g., the classical method of Lagrange indeterminate
multipliers (Vasilyev F 1988), do not always blend with the LST ideology.
In some separate cases, we succeeded in accounting for the constraints and
restrictions to the desired parameters using special approaches. For example,
in the above-considered case of the linear constraints between parameters
x 1 ,..., x k expressed through vector B 0 and matrix B the following is elemen-
tarily obtained:

= X 0 + B ( B + G n WG n B ) −1 B + G n W ( Y − G ( X n )+ G n ( X n − X 0 )) . X n +1 (4.25)

Algorithm(4.25) is a generalizationof algorithm(4.20) for the case of nonlinear
problems.
Some special difficulties will arise if the restrictions to the possible values of
the parameters are written as inequalities. For example, practically all param-
eters (gases and aerosols contents, surface albedo, etc.) are to be non-negative
proceeding from their physical meaning. In this case, the rather evident way of
theremovalofrestrictionsistheconversionofthevaluestotheirlogarithms
(Virolainen 2000; Potapova 2001). However, strictly speaking, in this case the
values of the logarithms providing the minimum of the discrepancy mustn't
correspond to the values of the parameters providing the same minimum.
That is to say, that taking logs brings an additional uncertainty to the solution
obtained with LST. Hence, in spite of its simplicity and attractiveness it is nec-
essary to use this approach carefully, studying its “pluses and minuses” when
applying it to concrete problem conditions.
At the same time there is a general method allowing the approximate ac-
counting of any complicated constraints and restrictions to the retrieved pa-
rameters - the method of penalty functions (Vasilyev F 1988).
Let J conditions of the constraints be imposed on the desired parameters,
which could be written without breaking off the generality as:

= = c j ( x 1 ,..., x K ) 0, j 1,..., J . (4.26)