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where matrix
D
is the correlation one. After solving the inverse problem
with the primed variables pass to the initial units of measurements
x
k
=
x
k
d
k
,
(
S
X
)
ik
d
i
d
k
. In addition, note that the eigenvalues of the covariancema-
trix could become negative owing to the above-mentioned distortions while
rounding. The regularization by Tikhonov is recommended against this phe-
nomenonwhenmatrix
D
+
h
2
I
is used instead of matrix
D
with the consequent
increasing of value
h
up to the negative eigenvalues disappearing.
Only several maximal eigenvalues of matrix
D
differ from zero in the strong
correlation between the desired parameters oftenmet in practice. Specify their
number as
m
. Then all calculations would be accelerated if only
m
pointed
eigenvalues remain in matrix
L
(it becomes of the dimension
m
=
(
S
K
)
ik
×
m
)and
matrix
P
contains only
m
corresponding columns (dimension is
m
K
). This
approach is the kernel of the known
method of themain components
.Specifying
the obtained matrices as
L
m
and
P
m
the following is obtained from (4.50):
×
=
X
+
P
m
(
P
m
G
+
S
−
Y
GP
m
+
L
−
m
)
−1
P
m
G
+
S
−
Y
(
Y
−
G
0
−
GX
) .
X
(4.51)
Sometimes we can succeed in reducing the volume of calculations by an order
of magnitude and more using (4.51) instead of (4.50).
The criteria of selection of value
m
in (4.51) could be different. The math-
ematical criteria are based on the comparison of initial matrix
D
and matrix
P
m
L
m
P
m
, which have to coincide for
m
=
K
in theory. Correspondingly, value
m
is selected proceeding fromthe permitted value of their noncoincidence. The
comparison of every element of the mentioned matrices is needless. Usually
the comparison of the diagonal elements (dispersions) or of the sums of these
elements (the invariant under the coordinates conversion (Ilyin and Pozdnyak
1978)) is enough. The objective physical selection of value
m
is proposed in the
informatic approach by Vladimir Kozlov (Kozlov 2000), though it is not conve-
nient for all types of inverse problems because of very awkward calculations.
According to Consequence 2 from (4.38), the variation of the observations
caused by the a priori variations of the parameters is
GDG
+
.Wewillusethe
eigenbasis of this matrix, i. e. the independent variations of the observations.
Then eigenvalues of matrix
GDG
+
are the “valid signal” that is to be compared
with the noise, i. e. with the SD of the observations. If the observations are
of equal accuracy and don't correlate with SD equal to
s
then number
m
is
a number of the eigenvalues exceeding
s
2
. The case of non-equal accuracy and
correlated observations (just that is realized in the sounding data processing)
is more complicated. In this case the observations are preliminary to reduce
to the independency and to the unified accuracy
s
=
1. This transformation is
based on the theorem about
the simultaneous reducing of two quadratic forms
to the diagonal form
(Ilyin and Pozdnyak 1978) and is provided with matrix
P
Y
L
−1
|
Y
,where
P
Y
is the matrix of eigenvectors
S
Y
,and
L
Y
is the diagonal
matrix from eigenvalues
S
Y
corresponded to them. Thus, according to (4.38)
the selection of number
m
is determined by the number of the eigenvalues of
matrix
P
Y
L
−1
|
|
2
GDG
+
L
−1
2
Y
P
Y
, which exceed unity. Note that matrix
G
varies
from iteration to iteration in the nonlinear case, but such awkward calculations
are unreal to be accomplished. That's why it is preliminarily calculated using
Y
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