The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 154

thus it is mostly used for the inverse problems of atmospheric optics. Note that
the solution dependence of X disappears for the nonlinear problems, where
just the difference between the parameters is considered during the expansions
into Taylor series, i. e. the statistical regularization is equivalent to the adding
of D −1 to the matrix subject to inversion. Parameters X are usually chosen as
a zeroth approximation. Using the following identity:

( G + S − Y G + D −1 ) −1 G + S −1 = DG + ( GDG + + S y ) −1 , (4.47)

Y

which is elementarily tested by multiplying both parts from the left-hand side
by combination G + S − Y G + D −1 and from the right-hand side by combination
GDG + + S y . For some types of problems, it is more appropriate to rewrite
solution (4.46) in the equivalent form not requiring the covariance matrix
inversion:

=

X + DG + ( GDG + + S Y ) −1 ( Y − G 0 − GX ) .

X (4.48)

Compute the uncertainties of obtained parameters X using observational
uncertainties S Y ,i.e.the posterior covariance matrix of the parameters X
uncertainties. According to the definition, the following is correct: S X

=

( X − X )( X − X ) + ,where X is solution (4.48), and X is the random devia-
tion from it caused by the observational uncertainties. Substituting (4.48) to
matrix S X definition, accounting Y

= G 0 + GX , after the elementary manipu-

D − DG + ( GDG + + S Y ) −1 GD .Notethatacertain
positively defined matrix is subtracted from the a priori covariance matrix
in this expression, thus the observations cause the decreasing of the a priori
SD of the parameters, which has a clear physical meaning: the observations
cause precision of the a priori known values of the desired parameters. For the
furthertransformationofmatrix S X , the following relation is to be proved:

= lations we are inferring S X

( D −1 ) −1 −( G + S − Y G + D −1 ) −1 = DG + ( GDG + + S Y ) −1 GD .

Use for that the identity A −1 − B −1

=

B −1 ( B − A ) A −1 with accounting (4.47).

Finally, the following is obtained:

= ( G + S − Y G + D −1 ) −1 . S X (4.49)

It should be emphasized that (4.49) has the same form as (4.44) in spite of
the complicated method of deriving it, namely: the covariance matrix of the
uncertainties of thedesiredparameters is just the inversematrixof the algebraic
equation system subject to solving, i. e. it is directly obtained in the process of
calculation.

As has been mentioned hereinbefore, posterior SD √ ( S X ) kk obtained with

(4.49) are always not exceeded by a priori values √ ( D ) kk . The ratio of these
SD characterizes the information content of the accomplished observations
relative to the parameter in question. The lower this ratio themore information
about the parameter is contained in the observational data. It is curious that