Geoscience Reference
In-Depth Information
In practical applications, the diffusion operator is not homogeneous, and the analytic
representations ( 8.6 )and( 8.7 ) cannot be obtained. However, the action of B on
a state vector can be computed numerically at a relatively low cost. The major
problem with such modelling is the efficient estimation of the diagonal elements
Z
B n .
B n .
x
;
x
/
x
;
y
/ı.
x y
/d
y
(8.8)
R n
which are necessary to rescale B to have its diagonal elements equal to unity. In
practice, the rescaling factors
are defined as reciprocals of B n .
N n .
.
Computing the integral ( 8.8 ) numerically is expensive, because the convolutions
with the
x
/
x
;
x
/
-functions have to be performed at all points x of the numerical grid.
However, reasonable approximations ( Purser et al. 2003 ; Yaremchuk and Carrier
2012 )for
ı
N n .
can be obtained by using asymptotic expansions of ( 8.8 ) under the
assumption of weak inhomogeneity (see Sect. 8.3 ).
x
/
8.2.2
The Gaussian Model and Its Binomial Approximations
The Gaussian-shaped correlation model is widely used in geophysical applications.
Numerically, it is implemented by approximating exp
.a 2 D
=2/
with the binomial:
I C a 2 D
2m
m
. a 2 D
B g .
D
/ D exp
2 /
;
(8.9)
where
is a large positive integer. This numerical approach is often referred to
as “integration of the diffusion equation” and has been used in practice for several
decades ( Derber and Rosati 1989 ; Egbert et al. 1994 ; Weaver et al. 2003 ; Di Lorenzo
et al. 2007 ). There is, however, a certain disadvantage associated with the numerical
stability of the integration: The number of “integration time steps”
m
has to be large
enough for the eigenvalues of the binomial operator in the rhs of ( 8.9 )tobelessthan
1 in the absolute value. This constraint may limit
m
m
from below by a large value,
which can make the computation rather expensive.
Another option is to use a different approximation in ( 8.9 ):
I a 2 D
2m
m
B m .
D
/ D
:
(8.10)
The eigenvalues of the operator in the rhs of ( 8.10 ) do not exceed 1, and the
“integration procedure” is unconditionally stable. This approach is often referred
to as “implicit integration of the diffusion equation” (see Appendix 2). and has been
used in many practical applications as well ( Ngodock et al. 2000 ; Di Lorenzo et al.
2007 ; Carrier and Ngodock 2010 ).
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