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accurately computing diag
B
is evident from the fact that the operators
B
under
consideration are formulated numerically as multiplication algorithms by the
matrices, whose elements are not explicitly known. On the other hand, since the
BEC operator
C
is represented numerically by the correlation matrix, it must have
a unit diagonal and, therefore, knowledge of the diagonal elements of
B
is required
for renormalization:
/
1=2
B
/
1=2
C
D
.
diag
B
.
diag
B
(8.3)
Equation (
8.3
) shows that the considered BEC models involve two separate
algorithms: one for computing the action of
B
and another for estimating the
normalization factors
.
diag
B
/
that are necessary for computing the action of
/
1=2
.
Purser with coauthors (
Purser et al. 2003
;
Purser 2008a
,
b
)wereamongthefirstto
employ analytic methods for estimating the normalization factors for the Gaussian
operator
B
g
in geophysical applications. Somewhat earlier, an asymptotic technique
was developed for estimating the diagonal of the Gaussian kernel in Riemannian
spaces to study quantum effects in general relativity (e.g.,
Gusynin and Kushnir
1991
;
Avramidi 1999
). These ideas can be utilized to derive a useful algorithm for
estimating the normalization factors.
In this chapter, we first give an overview of the recent developments in construct-
ing the
D
-operator BEC models, and illustrate their major features with the examples
in the homogeneous case
.
diag
B
D
const
. In particular, in Sect.
8.2.2
, the relationships
between the scaling parameters for the Gaussian model and its
th-order approxi-
mation (
8.2
) are obtained and the respective CFs are given. In Sect.
8.2.3
the inverse
binomial model is extended to an arbitrary polynomial of
D
: Expressions for the
CFs and normalization factors are derived, and relationships are established between
the structure of the BEC spectrum and the polynomial coefficients. In Sect.
8.3
,
after a brief overview of the diagonal estimation methods, a heuristic formula for
computing diag
B
g
is derived (Sect.
8.3.2
) and then tested numerically against other
methods in a set of realistic oceanographic applications (Sects.
8.3.3
-
8.3.5
). Results
of similar tests with the
B
m
model are also presented. Summary and discussion of
the prospects for the
D
-operator BEC modeling complete the chapter.
m
8.2
Diffusion Operator and Covariance Modeling
The convenience of the diffusion operator (
8.1
) for constructing the BEC models
can be explained by the non-negative spectrum of
D
: An operator that is generated
by a positive rational function
of
D
whose eigenvalues tend to zero at large
wavenumbers, is positive-definite and has a smoothing property, i.e. tends to
suppress high-frequency components of the solution. In this section we consider
two types of such functions: Those that are generated by the
F
th-order binomials
(Sect.
8.2.2
) and the others by the inverse of a positive polynomial (Sect.
8.2.3
). To
allow analytical treatment, anisotropic homogeneous case in the boundless domain
is considered.
m
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