Geoscience Reference
In-Depth Information
k
X
F.k/
D
M
k
.M
G/
j
GHM
k
1
j
(2.46)
j
D
0
F.1/
D
.M
G/
D
A
F.2/
D
.M
G/
2
D
A
2
.Itisasimple
It can be verified
,
exercise to prove by induction that
F.k/
D
.M
G/
k
D
A
k
(2.47)
Substituting (
2.47
)in(
2.41
) and simplifying
e.k/
D
A
k
.x.0/
x.0//
C
.k/
(2.48)
Thus, in this case the deterministic part of the error has a simple from and is
essentially controlled by the error in the initial condition.
We now illustrate the impact of model error, error in the initial condition and the
variance of the observation noise on the optimal estimate of the nudging coefficient
for a simple scalar, linear, discrete time model.
Example 2.2.
Numerical Experiment
Consider a scalar linear nudged model given by
x.k
C
1/
D
mx.k/
C
g.
z
.k/
x.k//
(2.49)
starting from the initial condition
x.0/
and
g
2
R
is a nudging parameter. The
observation
z
.k/
D
x.k/
C
V.k/
(2.50)
That is,
h.x/
D
x
,where
V.k/
is a white Gaussian noise, namely
V.k/
N.0;
2
/
,and
x.k/
is the state of the dynamics given by
x.k
C
1/
D
m
x.k/
(2.51)
with
x.0/
as
th
e initial condition.
Let
m
D
m
C
ı
where
ı
denotes the model error and
.x.0/
x.0//
is the error
in the initial condition. Let
e.k/
D
z
.k/
x.k/
(2.52)
Consider the scalar analogs of (
2.22
)and(
2.33
)givenby
1
2
2
e
T
.1
W
4/e.1
W
4/
J
2
.g/
D
(2.53)
and
J
3
.g/
D
1
2
e
T
.1
W
4/V
1
e.1
W
4/
(2.54)
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