Geoscience Reference
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Subtracting (
2.88
) from (
2.97
), we obtain
e.k
C
1/
D
.M
GH/e.k/
C
F.x.k//
F.x.k//
(2.98)
Using the Lipschitz property of
F.x/
in (
2.98
), it becomes
e.k
C
1/
.M
GH
C
L
F
I
n
/e.k/
(2.99)
where
I
n
is the identity matrix. Taking the norms of both sides, we obtain
jj
e.k
C
1/
jj jj
.M
C
L
F
I
n
/
GH
jj jj
e.k/
jj
(2.100)
Thus,
if
..M
C
L
F
I
n
/;H/
is
observable,
then
there
exists
G
such
that
Œ.M
C
L
F
I
n
/
GH
is a Hurwitz matrix.
Clearly, if
F.x/
0
, then and we obtain the results of Sect.
2.4.2
.
2.5
Back and Forth Nudging Scheme
Recently
Auroux
(
2011
) and his collaborators have introduced a nudging scheme
wherein the same set of observations are inserted into the model that runs forward
in time and then backward in time. Starting from an arbitrary initial condition, say
x
.0/
0
D
x
0
,let
x
.0/
N
be the nudged model state at the final forecast time
N
.The
nudged forecast is made using observations f
z
0
;
z
N
1
g. Then the model is run
backwards starting from the final state which is now denoted by
z
1
;:::;
.Let
x
.0/
0
x
N
.
D
x
N
/
be the state at time
resulting from the backward run. Then a new forward run
is initiated from the initial condition
k
D
0
.1/
0
.
D
x
.0/
0
/
x
, the initial state computed by the
backward run just completed. This cycle is repeated. It is shown by
Auroux
(
2011
)
that the sequence of initial states for the forward run:
x
.0
0
;x
.1
0
;x
.2
0
;:::
converges
to the true initial state—that is, the initial state used to create the observations in the
numerical experiment.
In the following we illustrate the power of this idea using a simple dynamics for
both the cases of observations being noiseless and noisy.
Consider a linear advection equation
u
t
C
c
u
x
D
0
(2.101)
where
u
t
and
u
x
are the first partial derivatives of
u
D
u
.x;t/
with respect to the
time variable
t
and the space variable
x
where it is assumed that
x
2
Œ
1;1
and
t
0
.Itisalsoassumedthat
.x;0/
D sin
.x/ .
/
u
initial condition
(2.102)
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