Geoscience Reference
In-Depth Information
and the nudged dynamics (
2.7
) takes the form
x.k
C
1/
D
Mx.k/
C
GŒ
z
.k/
Hx.k/
(2.36)
We can rewrite (
2.36
)as
x.k
C
1/
D
Ax.k/
C
G
z
.k/
(2.37)
A
D
.M
GH/
Iterating (
2.37
), it follows that the solution is given by
k
X
A
j
GH M
k
1
j
x.0/
x.k/
D
A
k
x.0/
C
j
D
0
k
X
A
j
GV.k
1
j/
(2.38)
j
D
0
where we have used the fact that the true state
x.k/
dynamics is given by
x.k
C
1/
D
M x.k/
(2.39)
with
x.0/
as the initial condition and
k
x.k/
D
M
x.0/
(2.40)
Hence substituting (
2.35
)and(
2.39
)in
e.k/
D
z
.k/
x.k/
and simplifying we readily obtain a decomposition into deterministic and stochastic
parts as follows:
e.k/
D
F.k/x.0/
A
k
x.0/
C
.k/
(2.41)
where
2
3
k
X
k
k
1
j
4
HM
5
A
j
GHM
F.k/
D
(2.42)
j
D
0
and
k
X
A
j
GV.k
1
j/
.k/
D
V.k/
j
D
0
D
V.k/
GV.k
1/
C
AGV.k
2/
C
A
2
GV.k
3/
C
:::A
k
2
GV.1/
(2.43)
Search WWH ::
Custom Search