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The benefit of analytical consideration is its ability to reveal local correlation
structure and therefore provide a reasonable guidance to construction of more
general operators
B
. In addition, as it has been shown recently, good approximations
to diag
B
can be obtained by using analytical results obtained with the homogeneous
versions of
B
(e.g.,
Purser et al. 2003
;
Mirouze and Weaver 2010
;
Yaremchuk
and Carrier 2012
). Therefore, analytical formulas describing homogeneous BEC
operators are of significant practical interest. The analytical results may facilitate
practical design of the cost functions in variational data assimilation problems,
because they give explicit relationships between the shape of the local CFs and the
structure of the corresponding BEC operator.
8.2.1
Correlation Functions and Normalization
R
n
;n
D
1;:::;3
Consider an anisotropic, homogeneous diffusion operator (
8.1
)in
,
with
x
2
R
n
representing points in the physical space. By using the coordinate
transformation
x
0
D
1=2
x
, the problem can be reduced to considering isotropic
operators of the form
B
D
F.
/;
(8.4)
where
is the Laplacian (e.g.,
Xu 2005
;
Hristopulos and Elogne 2007
)and
F
is an
arbitrary positive function. In the case of an inhomogeneous diffusion (
)
the global transformation cannot be found. Transformations of this type, however,
can be used locally for constructing
B
and the normalization factors (Sect.
8.3
).
All of the formulas that are written below are assumed to be in the transformed
coordinates
x
0
with primes omitted to simplify the notation.
The operator (
8.4
) is diagonalized with the Fourier transform, and the diagonal
elements are
ยค
const
k
2
/
where
k
is the Fourier coordinate (wavenumber).
Because of homogeneity, the matrix elements of
B
in the
x
-representation depend
only on the distance
B.
k
/
D
F.
r
Dj
x
j from the diagonal. They can be computed by applying
the inverse Fourier transform to
B.
k
/
:
/
D
.2/
n
Z
B
n
.
x
B.
k
/
exp
.
i
kx
/d
k
:
(8.5)
R
n
R
n
(Appendix 1), (
8.5
) can be reduced to
By integrating over the directions in
Z
B
n
.r/
D
.2/
n=2
B.k/k
n
1
.kr/
s
J
s
.kr/dk
(8.6)
0
where
.
The respective matrix elements of the correlation operator (CFs) are obtained by
normalization:
k
j
k
j,
J
denotes the Bessel function of the first kind, and
s
D
1
n=2
C
n
.r/
D
B
n
.r/=B
n
.0/
(8.7)
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