Geoscience Reference
In-Depth Information
Referring to Chap.
10
,
Lewis et al.
(
2006
) we can restate that u
.
p
/
(0) converges to
u
T
(0) asymptotically at a linear rate.
CASE B: Noisy Observations
In this case, from (
2.114
,
2.115
,
2.116
and
2.117
) it follows that
N
u
.
j
/
.0/
u
T
.0/
C
.
j
/
.
u
.
j
/
.
/
D u
.
T
/
.0/
C
.
/
/
N
1 g
N
(2.130)
where
.
j
/
.
/
N
.0;
.
//
N
Va r
SPF
(2.131)
Similarly, from (
2.118
)-(
2.112
), we get
N
W
.
j
/
.
C
"
.
j
/
.0/
W
.
j
/
.0/
D u
.
T
/
.0/
C
.
/
u
T
1
˛/
N
.0/
(2.132)
where
©
.
j
/
.0/
N
.0;
Va r
.
SPB
//
(2.133)
with W
.
j
/
.
. Substituting (
2.130
)into(
2.132
) and using the feed—back
law of back and forth nudging, we get,
N
/
D U
.
j
/
.
N
/
u
.
jC1
/
.0/
D W
.
j
/
.0/
N
h
u
.
j
/
.0/
U
T
i
C
.
1
˛/
D u
.
T
/
.0/
C
.
1
˛/
N
N
.
j
/
.
N
/
C
"
.
j
/
.0/
(2.134)
.
1 g
/
.0/
which on rewriting becomes
u
.
jC1
/
.0/
u
T
/
.0/
D
ˇ
u
.
j
/
.0/
u
.
T
/
.0/
C
.
j
/
.0/
(2.135)
where
§
.
j
/
.0/
D
.
N
.
j
/
.
/
C
"
.
j
/
.0/
1
˛/
N
(2.136)
˜
.
j
/
.
©
.
j
/
.0/
D SPM in (
2.121
)in(
2.136
),
Substituting for
N
/
D SPF in (
2.116
)and
§
.
j
/
.0/
it can be verified that
is a mean—zero Gaussian random variable whose
variance is given by
Va r
.
j
/
.0/
D
.
1
2
2N
Va r
1
˛/
.
SPB
/
C Va r
.
SPF
/
(2.137)
Now, iterating (
2.135
), we get, for any integer p
,
u
.
p
/
u
.
T
/
.0/
D D C S
>0
(2.138)
where
p
u
.0/
.0/
u
T
.0/
D D
ˇ
(2.139)
which reads to zero as p grows since
0<“<1
, and S is given by
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