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is an issue of crucial importance in improving the forecast skill of the state-of-the-art
data assimilation systems.
Acknowledgements
This study was supported by the Office of Naval Research (Program element
0602435N) as part of the project “Exploring error covariances in variational data assimilation”.
Appendix 1
R
n
and
Let
be the angle between
x
and
k
in
n>2
. Then the integral (
8.5
) can be
rewritten in spherical coordinates as
Z
Z
B
n
.r/
D
.2/
n
/k
n
1
dkd˝
n
1
;
B.k/
exp
.
ikr
cos
(8.53)
0
˝
n
1
where
changes symmetrically within the limits of integration, the imaginary part of the
exponent vanishes. Furthermore, using the identity
d˝
n
1
is the element of the surface area of the unit sphere. Since cos
d˝
n
1
D
d˝
n
2
sin
n
2
d
,
the integral (
8.53
) can be rewritten as
Z
Z
Z
B
n
.r/
D
.2/
n
B.k/k
n
1
dk
sin
n
2
d
d˝
n
2
.kr
/
cos
cos
(8.54)
0
˝
n
2
0
Integration over
and substitution of the formula for the surface of
.n
2/
-
dimensional unit sphere into (
8.54
) yields (
8.6
).
The general relationship (
8.6
) also holds for
n
D
1;2
although these cases require
a special (less complicated) treatment.
Appendix 2
In practice, the matrix elements of the operator (
8.10
) are never calculated explicitly
due to the immense cost of such a computation. Instead, the result
x
m
.
/
x
of the
x
0
.
action by
B
on a (discrete) model state vector
x
/
is calculated by solving the
linear system of equations
I
m
D
x
m
D
x
0
;
=2m
(8.55)
D
denotes the discretized diffusion operator. If
x
0
.
where
x
/
represents the “initial
state” and the “time step”
ıt
is prescribed such that the “integration time” is
mıt
D
1
, then action of the operator (
8.55
) can be identified as a result of a discrete-time
integration of the diffusion equation
@
t
x
D
D
=2
x
with the implicit scheme
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