Geoscience Reference
In-Depth Information
Equation (4.7) in the case of the linear dependence is written as follows:
K
=
y
i
g
i
0
+
g
ik
x
k
.
(4.9)
=
k
1
Coefficients
g
i
0
,
g
ik
are not to be the identical constants at all. They can be
rather complicated functions of vector
U
X.
It should only be noted that the
coefficients are constants fromthe sense of the considered relationshipbetween
the observations and desired parameters because all parameters of vector
U
\
X
are known and fixed within the range of the concrete inverse problem. The
substitution of (4.9) to equation system (4.8) leads to the system of
K
linear
algebraic equations with
K
unknowns:
\
x
j
N
g
ij
g
ik
K
N
=
=
(
y
i
−
g
i
0
)
g
ik
,
k
1,...,
K
.
(4.10)
=
=
=
j
1
i
1
i
1
Rewrite (4.10) in the matrix form using above-defined vectors
X
≡
(
x
k
),
Y
≡
(
y
i
) and introducing vector
G
0
≡
(
g
i
0
) together with matrix
G
≡
(
g
ik
),
=
=
i
1,...,
N
,
k
1,...,
K
:
(
G
+
G
)
X
G
+
(
Y
−
G
0
)
=
(4.11)
where the sign “+” specifies the matrix transposition. The vectors are assumed
as columns; the first indices of the matrix are assumed as indices of a line while
writing system (4.11), and we will stick to this order. Multiplying both parts of
(4.11) from the left-hand side to combination (
G
+
G
)
−1
the desired solution is
obtained:
=
(
G
+
G
)
−1
G
+
(
Y
−
G
0
) ,
X
(4.12)
We s h ou l d me n t i on t h a t ma t r i x (
G
+
G
) of equation system (4.11) is symmetric
(
i
=
i
=
1
g
ik
g
ij
) and positive defined (as per Sylvester criterion (Ilyin
and Pozdnyak 1978)). Hence, solution (4.12) exists, it is unique (because the
determinant of the positive defined matrix exceeds zero) and corresponds to
the discrepancy minimum (because the positive defined matrix (
G
+
G
)isits
second-order derivative). Equation (4.12) is called
a solution of the system of
linear equations
G
0
+
GX
1
g
ij
g
ik
=
=
Y
with LST. Further, we will use this terminology.
The following standard normalizing approach (Box and Jenkins 1970) is
recommended here and further to diminish the possible uncertainty con-
necting with accumulation of the computer errors of the rounding-off dur-
ing the practical calculations with (4.12). Specify system (4.11) as
AX
=
B
=
√
a
kk
,
k
=
for a brevity and introduce operator
d
k
1,...,
K
.Passtosystem
A
X
=
B
,where
a
jk
=
|
(
d
j
d
k
),
b
k
=
|
d
k
and after its solution
X
=
(
A
)
−1
B
a
jk
b
k
=
x
k
|
obtain final results
x
k
d
k
. The effective square root technique (Kalinkin
1978) is appropriate for the matrix
A
inversion owing to its symmetry and
positive definiteness. The computing of the factors in (4.12) is to be accom-
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