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smoothing parameter
compatible with the one given by the relationship (
8.50
)
derived from the asymptotic approximation of the Gaussian kernel diagonal. In
deriving the ansatz (
8.51
) for the LH1 model, we followed the approach of
Purser
et al.
(
2003
), who proposed to smooth the zeroth-order diagonal by the square-root
of the BEC operator in the one-dimensional case. Using the asymptotic technique
for the heat kernel expansion, we obtained a formula for higher dimensions, and
tested its validity by numerical experimentation.
It should be noted that the formal asymptotic expansion (
8.45
) is local by
nature and tends to diverge in practical applications, where spatial variations of the
diffusion tensor may occur at distances
L
comparable with the typical decorrelation
N
scale
. To effectively immunize the expansion from the ill-effects of the abrupt
changes in
, we utilized a non-local empirical modification, still fully consistent
with the original expansion in the limit
N
=L
!
0
, but sufficiently robust with
respect to the numerical errors related to the high-order derivatives of
. A similar
technique was developed by Purser (
Purser et al. 2003
;
Purser 2008a
), who used
empirical saturation functions to stabilize higher-order approximations of the
B
g
.
In general, results of our experiments show high computational efficiency of the
LH1 scheme, whose total CPU requirements is just a fraction of the CPU time
required by the convolution with the BEC operator - a negligible amount compared
to the cost of a 3dVar analysis. Therefore, LH1 approximations to the BEC diagonal
may serve as an efficient tool for renormalization of the correlation operators in
variational data assimilation, as they are capable of reducing the error to 3-10 % in
realistically inhomogeneous BEC models.
A separate question, that requires further investigation, is the accurate treatment
of the boundary conditions. In the present study we assumed that boundaries affect
only the magnitude of the corresponding columns of
B
, but not their structure. This
approximation is only partly consistent with the zero normal flux conditions for
D
,
but can be avoided if one uses “transparent” boundary conditions (e.g.
Mirouze and
Weaver 2010
) which do not require computation of the adjustment factors. On the
other hand, it might be beneficial to keep physical (no-flux) boundary conditions in
the formulation of
D
, as they are likely to bring more realism to the dynamics of the
BE field.
Another important issue is parameterization of
using the background fields
and their statistics. In the simple diffusion tensor model used in the experiments,
anisotropic BE propagation is governed by the background velocity field and
superimposed on the small-scale isotropic BE diffusion, which takes place at
scales that are not well resolved by the grid (less than
.
x
/
3ı
). More sophisticated
parameterizations of
are surely possible and require further studies. In par-
ticular, recent studies have shown that since
.
x
/
independent
components, it can be estimated from ensembles of moderate (
100n
.
x
/
has only
n.n
C
1/=2
) size with
reasonable accuracy (
Belo-Pereira and Berre 2006
;
Pannekoucke and Massart 2008
;
Pannekoucke et al. 2008
;
Berre and Desroziers 2010
). Finally, the considered
BEC models could also be effectively used for adaptive/flow-dependent covariance
localization (
Bishop and Hodyss 2007
,
2011
;
Yaremchuk and Nechaev 2013
), which
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