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distribution, in which probability density depends only on the above-defined
Y
,
S
Y
and is equal to:
2
exp
−
1
2
(
Y
−
Y
)
+
S
−
Y
(
Y
−
Y
)
.
1
ρ
=
(
Y
)
π
|
|
)
N
2
1
(2
|
S
Y
|
Abstract from the above-discussed non-adequacy of the operator of the in-
verse problem solution and assume that the difference of real observational
results
Y
and calculated values
Y
is caused only by the random error. Then
vector
X
,whichtruevalue
Y
corresponds to (i. e.
Y
Y
), is to be selected as
an inverse problem solution. Substituting this condition to the formula for the
probability density, we obtain it as a function of both the observational and
desired parameters:
=
ρ
(
Y
,
X
). Then use the known
Fisher's scoring method in the
maximum likelihood estimation
according to which the maximum of the com-
binedprobabilitydensityistocorrespondtothedesiredparameters.Writing
explicitly the argument of the exponent through parameter
x
k
the maximum
is found from equation
∂ρ
|∂
=
(
Y
,
X
)
x
k
0 that gives the system of the linear
equations:
x
j
N
)
il
g
lk
K
N
N
N
g
ij
(
S
−1
=
(
y
i
−
g
i
0
)(
S
−1
=
)
il
g
lk
k
1,...,
K
.
Y
Y
=
=
=
=
=
j
1
i
1
l
1
i
1
l
1
(4.42)
The problem solution is obtained after writing (4.42) in matrix form:
=
(
G
+
S
−
Y
G
)
−1
G
+
S
−1
X
(
Y
−
G
0
) .
(4.43)
Y
S
−
Y
is assumed then (4.43) will
almost coincide with solution (4.15) for LST with weights. In particular, for
the case of non-correlated observational random uncertainties obeying Gauss
distribution, matrix
S
Y
is the diagonal one and solution with LST (4.15) is
an estimation of maximal likelihood (4.43). This statement is a kernel of the
known Gauss-Markov theorem (see for example Anderson 1971) - a severe
ground of selecting the inverse squares of the observational SD as weights of
the LST. It is evident that relation
W
=
It is to be pointed out that if equality
W
S
−
Y
is directly applied to all further
algorithms of LST described by (4.20), (4.23)-(4.25), (4.28), (4.30) and (4.32).
As (4.43) has linear constraint form (4.36) between
Y
and
X
, the covariance
matrix of the uncertainties of the retrieval parameters
S
X
is obtained with
(4.36). Substituting the expression
A
=
(
G
+
S
−
Y
G
)
−1
G
+
S
−
Y
from (4.43) to (4.38)
and accounting the symmetry of matrix (
G
+
S
−
Y
G
)
−1
the following relation is
inferred:
=
=
(
G
+
S
−
Y
G
)
−1
.
S
X
(4.44)
Equation (4.44) allows finding estimations of the uncertainty of the retrieved
parameters through the known observational uncertainty, i. e. it almost solves
the problem of their accounting. Equation (4.44) evidently keeps its form for
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