Geoscience Reference
In-Depth Information
Observational data
Y
contain the random errors characterized with the SD
of components
y
i
,
i
=
1,...,
N
. In general, the errors could correlate, i. e. they
are interconnected (although everybody aims to avoid this correlation with all
possible means in practice). Thus, the observational errors are described with
symmetric covariance matrix
S
Y
of dimension
N
N
,whichcanbeobtained
conveniently by writing schematically according to Anderson (1971) as:
×
(
Y
−
Y
)(
Y
−
Y
)
+
,
=
S
Y
(4.37)
where
Y
is the exact (unknown) value of the measured vector,
Y
is the observed
value of the vector (distinguishing from the exact value owing to the observa-
tional errors), the summation is understood as an averaging over all statistical
realizations of the observations of the random vector (over the general set).
The relation for covariance matrix of the errors
S
X
of parameters
X
,of
dimension
K
K
written in the same way as (4.37). Then, substituting relation
(4.36) to it, the following is obtained:
×
A
(
Y
−
Y
)(
Y
−
Y
)
+
A
+
,
(
AY
−
AY
)(
AY
−
AY
)
+
=
=
S
X
(4.38)
=
AS
Y
A
+
.
S
X
A set of important consequences directly follows from (4.38)
Consequence 1.
Equation (4.38) expresses the relationship between the co-
variance matrices of observational errors
Y
and parameters
X
linearly linked
with them through (4.36), i. e. allows the finding of errors of the calculated
parameters from the known observational errors. Namely, values
√
(
S
X
)
kk
are
the SD of parameters
x
k
,values(
S
X
)
kj
|
(
S
X
)
kk
(
S
X
)
jj
are the coefficients of the
correlation between the uncertainties of parameters
x
k
and
x
j
.Intheparticular
case of non-correlated observational errors that is often met in practice, (4.38)
converts to the explicit formula convenient for calculations:
N
a
ki
a
ji
s
i
=
=
=
(
S
X
)
kj
,
k
1,...,
K
,
j
1,...,
K
,
(4.39)
=
i
1
where
a
ki
are the elements of matrix
A
,
s
i
is the SD of parameter
y
i
.Inthe
case of the equally accurate measurements, i. e.
s
=
=
=
s
N
,thedirect
proportionality of the SD of the observations and parameters follows from
(4.39):
s
1
...
s
2
N
=
(
S
X
)
kj
a
ki
a
ji
.
=
i
1
Consequence 2.
From the derivation of (4.38) the general set could be evi-
dently replaced with a finite sample from
M
measurements
Y
(
m
)
,
m
=
1,...,
M
,
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