Geoscience Reference
In-Depth Information
fW
.
j
/
.0/;
W
.
j
/
.1/;
W
.
j
/
.2/; :::::::
W
.
j
/
.
u
.
j
/
.
/
g be the sequence of
backward states obtained by running the backward model (
2.108
).
The new initial condition, u
.
jC
1/
.0/
Let
N
/
D
N
)th run of the forward model is
set to be equal to the initial rate, W
.
j
/
(0) of the backward run just completed.
To start the overall iterative process at the 0th run of the forward model, the initial
condition u
.0/
.0/
D u
for the (j C
1
, an arbitrary choice.
Our goal is to characterize the limiting behavior of the sequence
fu
.0/
u
.p/
.0/ :::::::
g of initial state of
the forward run induced by the feed-back process between the forward and the
backward runs described above.
We consider two cases.
CASE A: Observations are Noise - free
Under this assumption, V
u
.0/
.0/;
u
.1/
.0/;
u
.2/
.0/; :::::::
.0/
D
in (
2.112
). Consequently, the stochastic part
SPF in (
2.116
) and SPB in (
2.121
) are identically zero.
We now derive a recurrence relation that relates the evolution of the required
initial conditions u
.
j
/
.0/
.
k
/
0
.Thefinalrateu
.
j
/
(N) starting from u
.
j
/
(0) is given by
(
2.115
)as
N
u
.
j
/
.0/
u
T
.0/
u
.
j
/
.
/
D u
T
N
.0/
C
.
1 g
/
(2.123)
Similarly, referring to (
2.120
) the initial rate W
.
j
/
.0/
of the jth backward run is given
by
N
W
.
j
/
.
.0/
W
.
j
/
.0/
D u
T
/
u
T
.0/
C
.
1
˛/
N
(2.124)
Since W
.
j
/
.
/
D u
.
j
/
.
N
N
/
, substituting (
2.123
)into(
2.124
) and simplifying we get,
u
.
jC1
/
.0/
D W
.
j
/
.0/
(2.125)
N
u
.
j
/
.0/
u
T
.0/
D u
T
N
.0/
C
.
1
˛/
.
1 g
/
That is,
N
u
.
j
/
.0/
u
T
.0/
u
.
jC1
/
.0/
u
T
N
.0/
D
.
1
˛/
.
/
1 g
(2.126)
0<
<1
˛
Thus, if
g
,thensois
and (
2.126
) becomes
ju
.
jC
1/
.0/
u
.0/
jD
ˇ
ju
.
j
/
u
T
.0/
j
(2.127)
when
ˇ
D
.1
˛/
N
.1
g/
N
and
0<ˇ<1
for any fixed numbers N
.> 0/
of
observations.
Iterating (
2.127
), we obtain
ju
.
p
/
.0/
u
p
ju
.0/
u
.0/
jD
ˇ
.0/
j
(2.128)
That is, u
.
p
/
.0/
converges to the true but unknown initial state exponentially, That is,
u
.
p
/
.0/
D u
.
T
/
.0/
lim
p
!1
(2.129)
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