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because for infinitely slow variations of
L
!1), the normalization factors
must converge to the above expression for the constant diagonal elements
(
.Itis
noteworthy that the formula (
8.44
) is found to be useful even in the case of strong
inhomogeneity
d
. In particular, numerical experiments of
Mirouze and Weaver
(
2010
) have shown that such an approximation provided 10 % errors in a simplified
1d case.
The accuracy of (
8.44
) can formally be increased by considering the next
term in the expansion of the diagonal elements of
B
g;m
. The technique of such
asymptotics has been well developed for the diagonal of the Gaussian kernel (
8.42
)
in Riemannian spaces (e.g.,
Gusynin and Kushnir 1991
;
Avramidi 1999
). More
recently, the approach was considered by
Purser
(
2008a
,
b
) in the atmospheric data
assimilation context. The application of this technique to the diffusion operator (
8.1
)
in flat space yields the following asymptotic expression for the diagonal elements
of
B
g
in the local coordinate system where
1
.
x
/
is equal to the identity matrix, and
D
takes the form of the Laplacian operator:
2
tr
h
Crdiv
h
1
.2/
n=2
1
1
2
tr
h
1
12
C
O.
5
/
B
g
.
;
/
D
x
x
(8.45)
Here
h
is a small (j
h
j
) correction to
within the vicinity of
x
. Note that the
O.
3
/
terms in the parentheses have the order
, because each spatial differentiation
adds an extra power of
.
The asymptotic estimate (
8.45
) involves second derivatives which tend to amplify
errors in practical applications when
may not be small. Therefore, using (
8.45
)
in its original form could be inaccurate even at a moderately small value of
.To
increase the computational efficiency, it is also desirable to formulate the first-order
approximation as a linear operator, which acts on
d
0
.
x
/
. Keeping in mind that j
h
j
, and utilizing the relationships:
/
1
.2/
n=2
tr
h
1
1
2
d
0
.
/
D
.2/
n=2
˝.
x
x
(8.46)
.=2/
I
C
1
exp
2
;
(8.47)
the second term in the parentheses of (
8.45
) can be represented as follows:
r div
h
D
1
tr
h
Crdiv
h
0
n
(8.48)
where
h
0
is the traceless part of
h
. On the other hand, if the divergence of
h
0
is
neglected, the (
8.45
) can be rewritten in the form
1
C
n
2
1
1
2
tr
h
1
.2/
n=2
B
g
.
x
;
x
/
(8.49)
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