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a
b
B matrix in DAS−I
B matrix in DAS−II
40
40
0.08
0.08
30
0.06
30
0.06
0.04
20
20
0.04
0.02
10
10
0.02
0
10
20
30
40
10
20
30
40
c
d
B−sensitivity in DAS−I
B−sensitivity in DAS−II
40
40
4
0.06
2
0.04
30
30
0.02
0
0
20
20
−2
−0.02
−4
10
10
−0.04
−6
−0.06
10
20
30
40
10
20
30
40
Fig. 9.3
(
a
) Time-averaged background error covariance in DAS-I. (
b
) The specification of the
background error covariance in DAS-II. (
c
) Time-averaged forecast error
B
-sensitivity in DAS-I.
(
d
) Time-averaged forecast error
B
-sensitivity in DAS-II. Displayed is the symmetric part of the
B
-sensitivity matrices
suboptimal system DAS-II, the
B
-sensitivity matrix distinguishes a pronounced
off-diagonal structure from the noisy entries and provides an indication that the
background error correlations are misspecified in the model
B
.
The time-averaged structure of the background error correlation matrix
C
b
in
DAS-I is shown in Fig.
9.4
a. The time-averaged structure of the forecast error
sensitivity to the background error correlation model in DAS-II (
C
b
-sensitivity)
is shown in Fig.
9.4
b. The diagonal entries of the correlation model
C
b
are
constrained to
C
ii
D
1;i
D
1
W
n
. For visual display purposes only and
to emphasize the cross-correlation structure, the diagonal entries of the
C
b
and
C
b
-sensitivity matrices in Fig.
9.4
a, b respectively, were set to zero. Negative values
of the entries in the
C
b
-sensitivity matrix in DAS-II are associated with positive
values of the background error correlations in DAS-I (and vice versa) which is in
agreement with the guidance provided by the steepest descent direction (
9.57
). The
C
b
-sensitivity in Fig.
9.4
b provides a first order guidance on the
C
b
update that is
necessary to correct the correlation model in DAS-II from its current specification
of
C
b
D
I
toward the correlation structure of the background errors in DAS-I
(Fig.
9.4
a).
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