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and the relationship between
ıe
and
ı
R
is obtained by replacing (
9.26
)in(
9.23
)
@e
@
HBH
T
1
Œ
x
b
/
ıe
D
x
a
;
K
ı
R
Œ
C
R
y
h
.
(9.27)
R
n
An observation-space formulation to (
9.27
) is obtained by using the adjoint-DAS
operator
K
T
and Eqs. (
9.6
)and(
9.19
)
K
T
@e
@
@e
@
Rz
HBH
T
1
Œ
x
b
/
ıe
D
x
a
;ı
R
Œ
C
R
y
h
.
R
p
D
y
;ı
(9.28)
R
p
From (
9.28
)and(
9.62
), the forecast
R
-sensitivity is the rank-one matrix
@e
@
R
D
@e
y
z
T
2
R
p
p
(9.29)
@
In an observation-space DAS, the evaluation of the vector
z
is performed in
the intermediate stage (
9.6
) of the analysis. An
R
-sensitivity formulation that is
equivalent to (
9.29
) and may be used in both observation-space and analysis-space
data assimilation systems is obtained by expressing
z
from (
9.6
)and(
9.7
)as
z
D
R
1
Œ
x
b
/
H
x
a
x
b
/
y
h
.
.
(9.30)
If the observation error correlations are not modeled in the DAS then
R
is a diagonal
matrix,
R
D
diag.
o
/
, and the forecast sensitivity to the specification of the
observation error variance is expressed from (
9.29
)as
@e
@
o;i
D
@e
@y
i
z
i
;i
D
1
W
p
(9.31)
R
,
A first order assessment of the forecast performance of a new covariance model
R
D
R
R
in (
9.28
) and requires only the additional ability to provide the matrix/vector product
Œı
as compared with the model
R
in the DAS, may be obtained by setting
ı
R
z
,
T
@e
@
x
a
.
R
x
a
.
eŒ
/
eŒ
R
/
f
Œı
R
z
g
(9.32)
y
R
are specified as
In particular, if the observation error covariance models
R
and
R
D
diag.
o;i
/
D
diag.
O
o;i
/
diagonal matrices,
R
and
,then(
9.32
) is expressed
as
ı
o;i
@e
p
X
x
a
.
O
x
a
.
o
/
eŒ
o
/
eŒ
z
i
(9.33)
@y
i
i
D
1
The right side of (
9.33
) provides an
all-at-once
first order assessment to the forecast
impact of each individual variation
ı
o;i
DO
o;i
o;i
. The impact estimates (
9.32
)
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