Geoscience Reference
In-Depth Information
In the Fourier representation both models (
8.9
)and(
8.10
) approximate the same
Gaussian function of
k
:
m
1
a
2
k
2
2m
.
a
2
k
2
2
B
e
.k/
D
exp
/
(8.11)
m
exp
1
C
a
2
k
2
2m
.
a
2
k
2
2
B
m
.k/
D
/
(8.12)
Since the value of
in (
8.11
) is fairly large in practice, the resulting CF is hardly
distinguishable from a Gaussian-shaped curve with a half-width
m
a
.
Substituting (
8.12
)into(
8.6
), integrating over
k
, and normalizing the result by
B
m
.0/
yields the CFs of t
he
Matern family (
Stein 1999
) enumerated by s D
m
n=2
and scaled by
p
a
D
a=
2m
:
C
m
./
D
s
K
s
./
/
;
(8.13)
2
s
1
.
s
where
stands for the modified
Bessel function of the second kind (
Abramowitz and Stegun 1972
). The respective
normalization factors are
D
r=a
,
is the gamma-function and
K
p
.m/
.m
1=2/
!
n
a
n
N
m
D
(8.14)
where
!
1
D
2
,
!
2
D
2
,and
!
3
D
4
. In the limiting case of
m
!1,theCFs
(
8.13
) take the Gaussian form:
C
1
D exp
.
r
2
=2a
2
/
I
n
D
1;::
(8.15)
Consecutive approximations of the Gaussian CF by (
8.13
)areshowninFig.
8.1
.It
is remarkable that when
in both
two and three dimensions (see also Table
8.1
). This means that
in the continuous
case
the first-order approximations become invalid when
m
D
1
,theCFs(
8.13
) have singularities at
D
0
n>1
. Numerically,
however, the CFs do exist for
n>1
, but their decorrelation scale is limited by
the grid size
(the corresponding CF is shown by the dotted line in the left panel
of Fig.
8.1
). This occurs because the numerical analogue of the
ı
-function is never
singular, but has a finite amplitude inversely proportional to the volume of a grid
cell, therefore, resulting in a finite value of the convolution (
8.8
) even if it is infinite
in the continuous case. After normalization by that finite value, the CF is 1 at
ı
r
D
0
,
but its effective decorrelation scale remains proportional to the local grid size.
The left panel in Fig.
8.1
shows that low-order binomial approximations (
8.13
)
underestimate the decorrelation scale
of the target Gaussian function. This
unpleasant property can be corrected by optimizing the value of
a
in (
8.10
) to obtain
the best fit with the Gaussian CF. Since the Gaussian and its approximating functions
a
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