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case of the relative discrepancy minimization together with the specifying of
therelativeSDisalmostequivalenttothecaseoftheabsolutediscrepancy
minimization with the specifying of the absolute value of the SD for every
point. The calculation scheme with “weight” has been used for accounting the
observational uncertainties in Sect. 3.2.
Parameters
x
k
desiredwithLSTaretobelinearlyindependent,otherwisethe
matrix of equation system (4.11) would be degenerate and the inverse matrix
would not exist. However, there are the cases, when the linear constraints
between the desired parameters are to be accounted for, this very situation has
been described in Sect. 3.2 during the secondary processing of the sounding
results. We can write the mentioned constraints in a general form as:
K
=
=
c
j
0
+
c
jk
x
k
0,
j
1,...,
J
.
(4.16)
=
k
1
Obviously, conditions (4.16) are to be linearly independent and
J<K
(oth-
erwise, the linear dependent lines should be excluded from system (4.16) by
decreasing value
J
). As per
the theorem of basis minor
(Ilyin and Pozdnyak
1978) there are
J
independent columns in the conditions (4.16). We will as-
sume they are the first ones from the left-hand side (otherwise, components
x
k
should be renumbered). Divide vector
X
into two parts:
X
(
J
)
=
≡
(
x
k
),
k
1,...,
J
and:
X
(
K
−
J
)
=
≡
(
x
k
),
k
J
+1,...,
K
. Then conditions (4.16) are written in the
matrix form as:
C
0
+
C
(
J
)
X
(
J
)
+
C
(
K
−
J
)
X
(
K
−
J
)
=
0 ,
(4.17)
=
=
(
c
j
0
),
C
(
J
)
1,...,
J
,
C
(
K
−
J
)
where
C
0
≡
≡
(
c
jk
),
k
≡
(
c
jk
),
k
J
+1,...,
K
,
=
1,...,
J
.Matrix
C
(
J
)
is non-degenerate, hence system(4.17) is soluble relating
to
X
(
J
)
:
j
X
(
J
)
=
(
C
(
J
)
)
−1
(−
C
0
−
C
(
K
−
J
)
X
(
K
−
J
)
) .
(4.18)
On the basis of (4.18) the expression of vector
X
as a whole through its inde-
pendent part
X
(
K
−
J
)
is inferred:
=
B
0
+
BX
(
K
−
J
)
X
,
(4.19)
where vector
B
0
≡
b
k
0
and matrix
B
≡
b
kl
possess a similar structure:
=
((
C
(
J
)
)
−1
(−
C
0
))
k
=
((
C
(
J
)
)
−1
(−
C
(
K
−
J
)
))
kl
,
b
k
0
and
b
kl
=
=
for
k
1,...,
J
,
l
1,...,
J
;
=
=
=
b
k
0
0,
b
kk
1and
b
kl
=
k
0,
=
=
for
k
J
+1,...,
K
;
l
J
+1,...,
K
.
=
Substituting (4.19) to initial equation system
GX
Y
, writing its solution with
LST for independent variables
X
(
K
−
J
)
and passing again to the whole vector
X
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