Geoscience Reference
In-Depth Information
8.1
Introduction
In recent years, heuristic background error correlation (BEC) modelling has become
an area of active research in geophysical data assimilation. Of particular interest are
the BEC models constructed with positive functions of the diffusion operator,
D
Drr
(8.1)
where
is the spatially varying positive-definite diffusion tensor. This type of BEC
model is attractive for several reasons: (a) it guarantees positive definiteness of the
resulting correlation functions (CFs), (b) it is computationally inexpensive in most
practical applications, and (c) it allows straightforward control of inhomogeneity
and anisotropy via the diffusion tensor. In the traditional approach of correlation
modeling where spatial correlations are specified by prescribed analytical functions,
care should be taken to maintain positive definiteness of the respective correlation
operator, especially in anisotropic and/or inhomogeneous cases (
Gaspari et al. 2006
;
Gregori et al. 2008
).
Among the most popular operators
B
used in practical BEC modeling are those
using the exponential and the inverse binomial functions of
D
:
I
a
2
D
m
m
.a
2
D
B
g
D exp
/
I
B
m
D
(8.2)
where
I
is the identity operator,
a
m
is a positive
integer. Since
D
has a non-positive spectrum whose larger eigenvalues correspond
to the smaller-scale eigenvectors, the operators
B
g
and
B
m
are positive-definite and
suppress small-scale variability. Both types of BEC models (
8.2
)areextensively
used in geophysical applications. Numerically, they are implemented by integration
of the diffusion equation using either explicit (in the case of
B
g
Derber and Rosati
1989
;
Egbert et al. 1994
;
Weaver et al. 2003
) or implicit (in the case of
B
m
Ngodock
et al. 2000
;
Di Lorenzo et al. 2007
) integration schemes.
A disadvantage of the BEC models (
8.2
) is that there is a limited freedom in
the shape of local CFs, which have either the shape of the Gaussian bell (
B
g
)or
provide its
is a scaling parameter and
th-order strictly positive approximations (
B
m
)(
Xu 2005
;
Yaremchuk
and Smith 2011
). In order to allow negative correlations, one has to consider
operators generated by the arbitrary polynomials in
D
. The quadratic polynomial
case was studied recently by
Hristopulos and Elogne
(
2007
,
2009
)and
Yaremchuk
and Smith
(
2011
), who obtained analytic representations of the CFs and derived
relationships between the polynomial coefficients and the spectral parameters of
B
in the homogeneous case.
In a more realistic inhomogeneous setting, the diffusion tensor varies in space,
making analytic methods inapplicable. Nevertheless, they can still give a reasonable
guidance for quick estimation of the diagonal elements of
B
(normalization factors),
whose values are crucial for constructing the BEC models. The importance of
m
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