Geoscience Reference
In-Depth Information
Fig. 2.1
Illustration of the
cost functions from
Example
2.2
with
m
D
m
D
1:1.ı
D
0/
Figure: Plot of J Vs. g
8
,
7
2
D
0:01
x
0
D
1:1
,
h
D
1
,
,
and
x
0
D
0:9:J
3
.g/
and
6
J
2
.g/
are represented by
“xxx” and “——”,
respectively
5
4
3
2
1
0
0.5
1
1.5
g
e.1
W
4/
D
.e.1/;e.2/;e.3/;e.4//
T
2
R
4
and the analog of the covariance
where
matrix
V
in (
2.45
)isgivenby
2
4
3
5
1
g
ag
a
2
g
g
C
g
2
g
C
ag
2
ag
C
a
2
g
2
ag
g
C
ag
2
1
C
g
2
C
a
2
g
2
g
C
g
2
a
C
g
2
a
3
a
2
g
ag
C
a
2
g
2
g
C
g
2
a
C
g
2
a
3
1
C
g
2
C
a
2
g
2
C
a
4
g
2
V
D
2
a
D
.m
g/
where
.
A comparison of the plots of
J
2
.g/
J
3
.g/
and
in (
2.53
)and(
2.54
) for the case
m
D
m
D
1:1.ı
D
0/
x
0
D
1:1
h
D
1
2
D
0:01
x
0
D
0:9
when
,
,
,
,and
is given in
J
3
.g/
Fig.
2.1
. It is easily seen that the minimum of
is to the right of the minimum
J
2
.g/
of
.
2.3.2
Estimation of G Using Kalman-Like Nudging Scheme
This approach is due to
Vidard et al.
(
2003
) which is a nice hybrid scheme that
combines the Kalman filter like predictive part and the conventional nudging scheme
to combine the innovation or the prediction error [
Kalman
(
1960b
)].
Let
x.0/
D
x
b
.0/
C
ıx.0/
be the initial state for the nudged dynamics where
x
b
.0/
is the background/prior information about
x.0/
and
ıx.0/
is the perturbation
x
b
.0/
added to the background. Let
B
be the covariance of the background state
.
A two step nudging scheme is then given by
x
f
.k/
D
M.x.k
1//
(2.55)
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