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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