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where it is recalled that
E
fg denotes the
expected value of its argument. Furthermore,
Bennett et al.
(
2000
) notes that the
expected values of parts
m
is the number of observations, and
J
B
and
J
R
of the objective function
J
are
E
f
J
B
.
x
a
/
gD
Tr
HBH
T
D
1
/;
.
(12.20)
and
E
f
J
R
.
x
a
/
gD
Tr
RD
1
/;
.
(12.21)
where
Tr
denotes the trace of the matrix argument
A
. These results may be
further specialized to compute the expected value of subsets of terms in
.
A
/
J
B
and
J
R
…
l
(
Talagrand 1999
;
Desroziers and Ivanov 2001
). Define
as a projection operator
such that
x
l
D …
l
J
l
associated with
x
l
x
, then the expected value of
is given by
Desroziers and Ivanov
(
2001
)
T
E
f
J
l
.
x
a
/
gD
Tr
.
…
l
HBH
T
D
1
…
l
/:
(12.22)
…
k
so that
y
k
D …
k
Likewise, define the projection operator
y
, then the expected
J
k
J
R
is
value for
of
T
E
f
J
k
.
x
a
/
gD
Tr
.
…
k
RD
1
…
k
/:
(12.23)
12.3.2
Validation of Error Variances by Posterior Diagnosis
Desroziers and Ivanov
(
2001
) utilize the above relations (
12.22
)and(
12.23
)
to validate the error variances in the objective function based on the
posterior
diagnosis
of the assimilation system. They demonstrate how to produce realistic
error variances for simulated observations in a cost-effective manner. This approach
was further evaluated and developed by
Chapnik et al.
(
2004
,
2006
)and
Sadiki and
Fischer
(
2005
) for operational data assimilation systems. Following
Chapnik et al.
(
2004
), the objective function (
12.8
) is rewritten as
B
R
X
X
J
l
.
J
k
.
x
/
x
/
J
.
x
/
D
C
;
(12.24)
s
l
s
k
l
D
1
k
D
1
where
s
l
R
components of the
background and the observations, respectively. The analysis
x
a
.
and
s
k
B
and
are scalar tuning parameters for the
s
/
is now a function
s
l
;
s
k
/
of the tuning parameter vector
s
D
.
(
Chapnik et al. 2004
),
x
a
.
/
D
x
b
C
K
y
Hx
b
/;
.
/.
s
s
(12.25)
.
/
where the tuned
Kalman gain
,
K
s
, takes the form
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