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adjustments that are necessary to improve the DAS performance. A summary and
further research perspectives are in Sect.
9.5
. The notational convenience adopted in
this work and some useful elements of matrix calculus are in the appendix.
9.2
The Analysis Equation
Variational data assimilation (
Kalnay 2002
) provides an analysis
x
a
2
R
n
to the
true state
x
t
of the atmosphere by minimizing the cost functional
/
D
1
x
x
b
/
C
1
x
x
b
/
T
B
1
.
T
R
1
Œ
J.
x
2
.
2
Œ
h
.
x
/
y
h
.
x
/
y
(9.1)
where
x
b
2
R
n
is a prior (background) state estimate,
y
2
R
p
is the vector
of observational data, and
h
W
R
n
!
R
p
is the observation operator that maps
the state into observations. In a four-dimensional variational (4D-Var) DAS the
operator
h
incorporates the nonlinear forecast model and evolves the initial state to
the observation time. Statistical information on the background error
b
D
x
b
x
t
o
D
x
t
/
and observational error
is used to specify the weighting matrices
B
2
R
n
n
and
R
2
R
p
p
that are representations in the DAS of the background
and observation error covariances
B
t
D
E.
b
b
y
h
.
T
and
R
t
D
E.
o
o
T
/
/
respectively,
where
denotes the statistical expectation operator.
In practice, an approximate solution to the nonlinear minimization problem (
9.1
)
is obtained using a linearization of the observation operator (
Courtier et al. 1994
),
E.
/
x
b
/
C
H
x
x
b
/
h
.
x
/
h
.
.
(9.2)
where
@
h
@
j
x
D
x
b
2
R
p
n
H
D
(9.3)
x
is the Jacobian matrix of the observation operator
h
evaluated at
x
b
. In this study we
consider a single outer loop iteration such that the analysis state is expressed as
x
a
D
x
b
C
K
x
b
/
Œ
y
h
.
(9.4)
where the gain matrix
K
is defined as
K
D
B
1
C
H
T
R
1
H
1
H
T
R
1
D
BH
T
HBH
T
C
R
1
(9.5)
The observation-space evaluation of the analysis (
9.4
) is a two-stage process
consisting of solving the linear system
HBH
T
C
R
z
D
y
h
x
b
/
.
(9.6)
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