The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 155

proper observational results are not needed for the calculation of posterior SD
(4.49) in the linear case; it is enough to know the algorithmof the only solution
of the direct problem (matrix G ). Thus, calculating the possible accuracy of
the parameters retrieval and the information content estimation could be done
evenattheinitialstageofthesolvingprocessbeforetheaccomplishmentof
the observations. Strictly speaking, this confirmation is not correct for the
nonlinear case, when the matrix of the derivatives G depends on solution X ;
nevertheless, even in this case (4.49) is often used for analyzing the information
content of the problem before the observations.
The choice of a priori covariance matrix D causes some difficulties while
using the statistical regularization method. If there are sufficient statistics
of the direct observations of the desired parameters then matrix D will be
easily calculated. Otherwise, we need to use different physical and empirical
estimations andmodels. The a priorimodels will be discussed inChap. 5 for the
concrete problem of the processing of sounding results. Note that in the case of
thenecessityofmatrix D interpolation it is elementarily recalculatedwith (4.38)
as per consequence 5. It should be mentioned that the results of the covariance
matrix calculation have to be presented with a rather high accuracy without
rounding off the correlation coefficients. Otherwise, the errors of rounding
cause the distortions of the matrix structure (according to consequence 4),
those, in turn, lead to difficulties in the use of the matrix. In particular, all
reference data about the correlation coefficients of the atmospheric parameters
are presented with accuracy up to 2-3 signs, hence, these matrices are not to
inverse while using them. However, the difficulties with matrix D inversion
could be principal, as thismatrixwould be degenerate if the desired parameters
stronglycorrelatetoeachother.
To overcome the mentioned difficulties and to optimize the algorithm it is
necessary to transform the desired parameters to independent ones for those
there are no correlations for and the matrix D is diagonal. This transformation
is provided by matrix P consisting of the eigenvectors of matrix D ,inciden-
tally matrix D converts to diagonal matrix L with the known formulas of
the coordinates conversion L PDP −1 (Ilyin and Pozdnyak 1978). The inverse

transformation to the desired parameters P −1 S X P is to be realized after the cal-
culation of the posterior covariance matrix and we infer the following solution
of (4.46) with accounting for eigenvectors orthogonality ( P −1

=

=

P + ):

=

X + P + ( PG + S − Y GP + + L −1 ) −1 PG + S −1

( Y − G 0 − GX ) . X (4.50)

Y

The method of the revolution (Ilyin and Pozdnyak 1978) should be used for
calculating the eigenvectors and eigenvalues of matrix D .Althoughitisslow,it
works successfully for the close (multiple) eigenvalues. To prevent the accuracy
lost during the eigenvalue calculations the following approach of normalizing
is recommended. The a priori SD of parameter x k is assumed as a unit of
measurement, i. e. introduce vector d k = √ ( D ) kk and pass to the values:

x k = | x k = x k | g 0 k = g ik = g ik d , D ) ik = | x k d k , d k , g 0 k , ( D ) ik ( d i d k ),