Geoscience Reference
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for the vector
z
2
R
p
and followed by a post-multiplication operation
x
a
D
x
b
C
BH
T
z
(9.7)
In NAVDAS-AR the computational steps (
9.6
)and(
9.7
) are performed using a
matrix-free implementation (
Xu et al. 2005
;
Rosmond and Xu 2006
).
9.2.1
Adjoint-DAS Observation Impact Estimation
Adjoint techniques are currently implemented as an effective approach (all-at-once)
to estimate the impact of any data subset in the DAS on reducing the forecast errors.
The forecast score is typically defined as a short-range forecast error measure
x
f
x
v
T
E
x
f
x
v
e.
x
/
D
.
f
/
.
f
/
(9.8)
where
x
f
D
M
t
0
;t
f
.
x
/
is the model forecast at verification time
t
f
initiated at
t
0
from
x
,
x
v
f
t
f
and serves as a proxy to the true state
x
t
f
,
and
E
is a diagonal matrix of weights that gives (
9.8
) units of energy per unit mass.
The adjoint approach to observation impact (OBSI) estimation relies on the
adjoint-DAS
operator
K
T
to obtain an observation-space estimation of the change
in the model forecast due to the assimilation of all data in the DAS
is the verifying analysis at
x
b
/
˝
g
x
a
x
b
˛
R
n
D
˝
K
T
g
x
b
/
˛
x
a
/
e.
e.
;
;
y
h
.
(9.9)
R
p
The order of the approximation (
9.9
) is determined by the specification of the vector
g
2
R
n
(
Gelaro et al. 2007
;
Daescu and Todling 2009
).
In NAVDAS-AR the OBSI assessment is performed based on a second-order
accurate approximation introduced by
Langland and Baker
(
2004
) with
g
defined
as the average of two forecast gradients that are evaluated with adjoint model
integrations along the analysis and background trajectories
1
2
@e
@
x
a
/
C
1
2
@e
@
x
b
/
M
t
o
;t
f
T
E
x
f
x
v
M
t
o
;t
f
T
E
x
f
x
v
g
D
x
.
x
.
D
Œ
.
f
/
C
Œ
.
f
/
(9.10)
where
M
t
0
;t
f
t
f
. A measure of the
contribution of individual data components in the assimilation scheme to the
forecast error reduction, per observation type, instrument type, and data location,
is obtained as
denotes the tangent linear model from
t
0
to
y
i
/
D
˝
f
K
T
g
g
i
;
f
y
h
x
b
/
g
i
˛
(9.11)
where
y
i
2
R
p
i
is the data component whose impact is being evaluated. Data
components for which
OBSI.
.
R
p
i
contribute to the forecast error reduction
(improve the forecast), whereas data components with
OBSI.
y
i
/<0
increase
the forecast error (degrade the forecast). The second-order approximation (
9.9
)and
(
9.10
) has been found to provide satisfactory results for OBSI estimates associated
OBSI.
y
i
/>0
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