Geoscience Reference
In-Depth Information
2.5.2
Analysis of Backward Nudging
Iterating (
2.108
), it can be verified that the backward solution, at any time k, is given
by
X
k
jD1
.1
˛/
k
j
Z
k
W
W
.
N k
/
D
.
1
˛/
.
N
/
C
˛
.
N j
/
(2.118)
where W(N) is the final condition from which the backward nudging starts.
Substituting (
2.112
)in(
2.118
) and simplifying we get
W
.0/
D DPB C SPB
(2.119)
where the deterministic part, DPB is given by
X
k
j
N
W
1
.1
˛/
N
j
u
T
DPB D
.
1
˛/
.
N
/
C
˛
.0/
(2.120)
D
N
W
.0/
D u
T
/
u
T
.0/.
1
˛/
.
N
The stochastic part, SPB is given by
X
N
jD
1
.1
˛/
Nj
V
SPB D
˛
.
N j
/
(2.121)
whose mean is zero and the variance is given
/
D
˛
2
1
.1
˛/
2N
/
1
.1
˛/
2
1
2
Va r
.
SPB
g
2
.1
C
g/
2
1
1
1
.1
C
g/
2N
D
(2.122)
Thus, W(0) is a Gaussian random variable whose mean is given by DPB and
variance is equal to Var(SPB).
2.5.3
Back and Forth Nudging Scheme
Against this background, we now close the loop between the forward and the
backward steps to get the so called back and forth nudging scheme.
Let u
.
j
/
.0/
be the starting initial condition for the jth forward run of the model
that leads to the sequence of rates given by fu
.
j
/
.0/;
/
g
obtained by running the forward model (
2.107
)whereu
.
j
/
(N) is the final state. In
the jth backward run of the model, the starting final state W
.
j
/
.
u
.
j
/
.1/;
u
.
j
/
.2/;:::::::
u
.
j
/
.
N
N
/
is set to be equal
to the final state u
.
j
/
.
N
/
of the jth forward run just completed.
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