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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
R
Œ
C R
y h
.
R p D
y
(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 .
/
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 /
o /
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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