Geoscience Reference
In-Depth Information
plished from the right-hand side to the left-hand side; hence, all operations
willbereducedtothemultiplyingofthevectorbythematrix.
Hereinbeforewe have assumed that the yield to the discrepancy of all squares
of the differences between the observational and calculation results is the same.
However, it is often desirable to account for the individual specific of these
yields. In this case, we use the generalization of
the least-squares technique -
the least-squares technique “with weights”
(Kalinkin 1978). Write the equation
for the discrepancy (4.6) as:
w
i
(
y
i
−
y
i
)
2
N
N
R
2
=
w
i
,
(4.13)
=
=
i
1
i
1
where
w
i
>
0 is a certain “weight”, attributed to point
i
.Thenforlinear
dependence (4.9) system (4.10) transforms to:
x
j
N
w
i
g
ij
g
ik
K
N
=
=
(
y
i
−
g
i
0
)
w
i
g
ik
,
k
1,...,
K
.
(4.14)
=
=
=
i
1
i
1
j
1
=
=
Not a vector but the diagonal weight matrix
W
≡
(
w
ij
),
w
ii
w
i
,
w
i
,
j
=
0,
i
=
=
i
1,...,
N
, is necessary to introduce for writing equation system
(4.14) and for solving it in the matrix form. Then the matrix of system (4.14) is
written as (
G
+
WG
), the free term is written as
G
+
W
(
Y
−
G
0
)andthesolution
is written as:
1,...,
N
,
j
=
(
G
+
WG
)
−1
G
+
W
(
Y
−
G
0
) .
X
(4.15)
It is important to mention that explicit expressions (4.14) are more convenient
to use during the practical calculations of the matrix and free term. The
meaning of the introduced weight matrix
W
will become clear in the following
section. Mention here, that the solution of the problem with LST does not
depend on the absolute magnitudes of the weights, i. e. the multiplying of all
values
w
i
by the constant does not change the values of desired parameters
X
.
In particular, if all
w
i
are equal, then solution (4.15) will coincide with the case
of the solution “without weights” (4.12).
In principle, weights
w
i
could be chosen from different views. The situation
when the inverse square of the mean square uncertainty of the observations
is taken as a weight is rather usual, i. e.
w
i
s
i
,where
s
i
is the SD of the
y
i
observation. The theoretical reasons for this choice will be presented in the
following section. Now we should mention its obvious meaning: the greater
uncertainty the less its yield to the discrepancy and the demand to the closeness
of corresponding values
y
i
and
y
i
is weaker. The other important case of using
the weights is passing to the relative value of the discrepancy, i. e. summarizing
of the squares of not absolute but relative deviations
y
i
from
y
i
in (4.13).
Equality
w
i
=
|
1
y
i
is evidently valid in this case. If the relative value of the
discrepancy is calculated and the relative SD of series points
=
|
1
δ
i
is fixed then
=
|
δ
i
y
i
)
=
|σ
i
the following will be inferred:
w
i
1
(
1
. That is to say, that the
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