Geoscience Reference
In-Depth Information
By replacing
R
D ˙
o
ı
C
o
˙
o
ı
(9.49)
C
o
-perturbation in the
in (
9.28
), the first order forecast variation
ıe
induced by a
ı
observation error correlation model is expressed as
@e
@
C
o
˙
o
z
C
o
˙
o
z
˙
o
@e
@
y
;
˙
o
ı
ıe
D
y
;ı
D
(9.50)
R
p
R
p
From (
9.50
), the forecast
C
o
-sensitivity is the rank-one matrix
@e
@
˙
o
@e
@
.
˙
o
z
T
2
R
p
p
/
C
o
D
(9.51)
y
and may be expressed using the elementwise vector product (
9.60
)as
o
ı
@e
@
@e
@
T
2
R
p
p
C
o
D
.
o
ı
z
/
(9.52)
y
o
2
R
p
denotes the vector of values assigned in the DAS to the observation
error standard deviation,
where
˙
o
D
diag.
o
/
.
9.3.3.3
Sensitivity to the Background Error Correlation Specification
The specification of the background error correlations is a key ingredient of the
data assimilation system and ongoing research at NWP centers is focused on
the development of flow-dependent background error covariance models (
Buehner
2005
;
Bannister 2008a
,
b
;
Brousseau et al. 2011
). By replacing
B
D ˙
b
ı
C
b
˙
b
ı
(9.53)
C
b
in the
in (
9.36
), the first order forecast variation
ıe
induced by a perturbation
ı
background error correlation model is expressed as
@e
@
C
b
˙
b
H
T
z
C
b
˙
b
H
T
z
˙
b
@e
@
x
b
;
˙
b
ı
ıe
D
R
n
D
x
b
;ı
(9.54)
R
n
From (
9.54
), the forecast
C
b
-sensitivity is the rank-one matrix
˙
b
H
T
z
T
@e
@
˙
b
@e
@
2
R
n
n
C
b
D
(9.55)
x
b
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