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b
K
X
oT
P
a
1
g
X
0
b
K
and consequently j
X
o
b
K
j D
0:
D
0;
(16.6)
N
Second, note that if
,
respectively, we may deduce that the perturbation ensemble sample covariance
matrix
P
e
ii
and
-
ii
denote the diagonal elements of
ƒ
and —
ƒ
associated with the transformed ensemble perturbations is given by
X
r
X
rT
K
1
X
o
TT
T
X
oT
K
1
=
X
o
B
—
ƒ
1
B
T
X
oT
K
1
P
e
D
D
X
K
X
1
K
1
x
i
b
i
b
i
x
oT
i
1
K
1
x
i
b
i
b
i
x
oT
i
D
D
(16.7)
1=2
ii
1=2
ii
—
i
D
1
i
D
1
where
b
i
is the ith column of
B
. Equation
16.7
shows that because j
X
o
b
K
j D
0
,
P
e
is entirely independent of the value assigned to Kth eigenvalue. Throughout this
discussion we will assume that every ensemble contains K1 linearly independent
ensemble perturbations.
16.2.2
Signal Variance and Forecast Error Variance Reduction
for Adaptive Observation with the ETKF Technique
If the true analysis error covariance at the observation time after assimilating all
routine observations was given by
P
e
D
X
r
X
rT
K
1
then the posterior analysis error
covariance
P
i
after assimilating the ith feasible deployment of adaptive observations
y
i
in addition to the routine observations is given by
H
i
i
C
I
1
H
i
P
i
D
P
r
P
r
H
aT
P
r
H
aT
P
r
(16.8)
i
H
i
where
describes the mapping from the model state vector to the observation
vector normalized by the inverse square root of the observation error covariance
R
1=2
i
associated with the ith feasible deployment; in other words,
H
i
x
t
D
R
1=2
y
t
i
(16.9)
i
where
x
t
denotes the true model state and
y
t
i
denoted the true value of the observed
variable. As shown in
Bishop et al.
(
2001
), if
X
i
X
aT
i
K
1
P
i
D
(16.10)
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