Geoscience Reference
In-Depth Information
Equation (
9.43
) provides the forecast sensitivity to each observation error covari-
ance weight coefficient,
@e
@
@e
@s
i
x
b
/
H
x
a
x
b
/
i
y
i
;Œ
.
.
D
y
h
R
p
i
i
@e
x
b
/
H
x
a
x
b
/
T
D
Œ
.
.
y
i
;i
2
I
y
h
(9.44)
@
By replacing (
9.42
)in(
9.36
) and with the aid of the analysis equation (
9.7
), the first
order variation in the forecast aspect
x
a
/
ıs
b
as
e.
is expressed in terms of
ıe
D
ıs
b
@e
@
x
a
x
b
x
b
;
(9.45)
R
n
Equation (
9.45
) provides the forecast sensitivity to the background error covariance
weight coefficient,
@e
@
x
a
x
b
@e
@s
b
D
x
b
;
(9.46)
R
n
s
b
-sensitivity
and the identity (
9.21
) allows the observation-space evaluation of the
as
@e
@s
b
D
Œ
@e
@
x
b
/
H
x
a
x
b
/
T
y
h
.
.
(9.47)
y
From (
9.44
)and(
9.47
) it is noticed that the identity (
9.21
) is formally equivalent to
@s
b
C
X
i
2
I
@e
@e
@s
i
D
0
(9.48)
and reflects an intrinsic property of the optimization problem (
9.1
) in variational
data assimilation: multiplication of both
R
and
B
matrices by the same positive
constant has no impact on the analysis.
9.3.3.2
Sensitivity to the Observation Error Correlation Specification
The standard practice in operational data assimilation and forecast systems is to
neglect the statistical correlation of the observation errors and tuning of the assigned
observation error variance parameters is used to compensate for unrepresented error
correlations. Recent diagnostic studies have shown evidence of both spatial and
inter-channel error correlations in the radiance data provided by the atmospheric
sounders (
Garand et al. 2007
;
Bormann et al. 2010
,
2011
) and research to assess
the potential gain that may be achieved from modeling the observation error corre-
lations is becoming increasingly important as the next generation of hyperspectral
instruments will further increase the data density.
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