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On the other hand, ensemble-based forecast error covariance has a potential to
capture all inter-variable correlations, and is also inherently time-dependent. How-
ever, even with error covariance localization, the low-rank limitation of ensemble
error covariance does not allow accurate representation of all cross-correlations
between variables. Fortunately, there are still inter-variable correlations that can
be represented well in ensemble-based methods. Typically, a correlation between
variables at near-by points is well represented even by a limited size ensemble. This
property essentially allows ensemble forecast error covariance to include all terms
defined by (
19.4
)-(
19.7
), however with reduced accuracy. The practical problem
is how to distinguish between “good” and “bad” correlations, and the answer
is not yet clear. More aggressive localization may have an appearance of better
controlling cross-variable correlations, but it does potentially impact dynamical
balance of the analysis and would prevent some important correlations in vertical for
well-developed cloud systems. Alternatively, one could introduce other localizing
functions that would selectively apply localization to off-diagonal matrix blocks
depending on the variable.
In addition to algebraic representation of all-sky assimilation issues with respect
to forecast error covariance described above it is also instructive to visually examine
its structure. The structure can be inspected by plotting columns of the forecast error
covariance that is also related to “single-observation” data assimilation experiments.
Let define a vector with all zero elements except for the
i
-th element with the value
one
z
i
D
0
1
0
i
1
1
i
0
i
C
1
0
Ns
T
(19.10)
where the index refers to a grid point and a variable (e.g., index of the state vector),
and
N
S
is the dimension of state vector. After multiplying vector
z
i
by matrix
P
f
one obtains the
i
-th column of the forecast error covariance matrix
c
i
D
P
f
z
i
D
f
1
•
f
i
i
C
1
•
f
Ns
T
i
1
f
i
i
f
i
(19.11)
f
j
with
. Note that location refers to
a grid point and variable. Following
Thepaut et al.
(
1996
)and
Huang et al.
(
2009
),
one can derive the analysis increment for single observation at
representing the
i
-th column value at location
j
i
-th point
/
P
f
y
h.x
f
/
x
a
x
f
(19.12)
i
Applying the matrix-vector product in (
19.12
), and using (
19.11
)
/
y
h.x
f
/
x
a
x
f
i
c
i
(19.13)
i.e. the analysis increment is simply the
-th column of forecast error covariance
scaled by the observation increment. This result, also expected on the basis of (
19.3
),
allows us to interpret a column of the forecast error covariance as analysis response,
and thus give a physical meaning to the structure of forecast error covariance.
i
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