Geoscience Reference
In-Depth Information
From the properties of
V.k/
it follows that
EŒe.k/
D
F.k/x.0/
A
k
x.0/
EŒ.k/
D
0
and
(2.44)
The covariance matrix
V
D
ŒV
ij
is now given by
V
ii
D
E
.i/
T
.i/
(2.45)
and
D
E
.i/
T
.j/
for
V
jj
i
¤
j
As an example, consider the case when
N
D
4
. Then from (
2.43
),
.1/
D
V.1/
.2/
D
V.2/
GV.1/
.3/
D
V.3/
ŒGV.2/
C
AGV.1/
.4/
D
V.4/
GV.3/
C
AGV.2/
C
A
2
GV.1/
Hence,
V
11
D
E
.1/
T
.1/
D
E
V.1/V
T
.1/
D R
V
22
D
E
.2/
T
.2/
D R C
G
T
R
G
V
33
D
E
.3/
T
.3/
D R C
G
T
T
T
R
G
C
AG
R
G
A
V
12
D
E
.1/
T
.2/
DR
T
G
D
V
21
V
13
D
E
.1/
T
.3/
DR
T
A
T
D
V
31
G
V
14
D
E
.1/
T
.4/
DR
T
.A
2
/
T
D
V
41
G
V
23
D
E
.2/
T
.3/
DR
T
T
T
G
C
GRG
A
D
V
32
V
24
D
E
.2/
T
.4/
DR
T
A
T
C R
T
.A
2
/
T
D
V
42
G
G
V
34
D
E
.3/
T
.4/
DR
T
T
T
.A
2
/
T
D
V
43
G
C
G
R
G
A
C
AG
R
G
Thus, the elements of
V
are polynomial matrices in
G;A;
and
R
.
Hence the shape of
J
3
.G/
in (
2.33
) in general depends on the (unknown) model
error,
x.0/
,
x.0/
,
R
, random realizations of the observational errors, and
G
. Clearly
the problem of determining the optimal
is much more involved than implied in
the literature.
To
simplify matters, we generally assume that the model is perfect,
that is,
G
.
The deterministic part
M
D
M
F.k/
in (
2.42
) of the error
e.k/
in (
2.41
) takes a much
simpler form when the model is
per
fect; that is,
M
D
M
and
H
D
I
in which case
z
.k/
D
x.k/
. Substituting
M
D
M
and
H
D
I
in (
2.34
), we obtain
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