Geoscience Reference
In-Depth Information
k
X
e.k/
D
A
k
e.0/
C
A
j
GV.k
1
j/
(2.83)
j
D
0
is Hurwitz, it follows that the spectral norm
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
A
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
Since
A
of
A
is less than 1 and
k
!1
A
k
D
0
lim
and
k
X
A
j
G
D
k
!1
.I
C
A
C
A
2
C
:::
C
A
k
1
/G
lim
k
!1
lim
j
D
0
D
.I
A/
1
G
(2.84)
.1
x/
1
when j
x
j
<1
in analogy with the expansion of
. Hence
EŒe.k/
0
for
all
k
0
and
8
<
T
9
2
3
2
3
=
k
X
k
X
4
5
4
5
A
j
GV.k
1
j/
A
j
GV.k
1
j/
Co
v
.e.k//
D
E
:
;
j
D
0
j
D
0
(2.85)
k
X
A
j
G
G
T
.A
j
/
T
D
R
j
D
0
Hence,
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
G
G
T
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
A
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
X
2j
cov
.e.k/
R
(2.86)
2
j
D
0
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
G
G
T
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
.1
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
A
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
/
1
D
R
In the special case when the observations are noise free, the second term on the r. h.
s. of (
2.83
) vanishes identically and in this case
k
!1
e.k/
D
0
lim
or
k
!1
x.k/
D
x.k/
lim
(2.87)
. If the norm
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
A
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
Clearly, the rate of convergence is controlled by the choice of
G
is close to zero, the convergence is way too fast which should be avoided.
In the following, we illustrate the choice of
G
using a simple example.
Example 2.4.
Let
n
D
2
and
m
D
1
with
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