Geoscience Reference
In-Depth Information
Tabl e 8. 1
Correlation functions associated with the power approximations (
8.10
) of the Gaussian
CF in
and 3 are rewritten in terms of elementary functions
for convenience. The correlation radius adjustment coefficients
n
dimensions. The CFs for
n
D
1
m
are shown below the formulas
e
m
in approximation of the Gaussian CF (bold
together with the corresponding relative errors
numbers)
n
D
1
n
D
2
n
D
3
m
D
1
.
/
K
0
./
.
/=
exp
exp
p
0.33
—
—
m
D
2
.1
C
/
.
/
K
1
./
.
/
exp
exp
p
2
p
=2
p
8=
0.13
0.19
0.33
.1
C
C
2
=3/
exp
.
/
2
K
2
./=2
m
D
3
.1
C
/
exp
.
/
p
p
16=3
p
3=4
27=8
0.08
0.10
0.13
8.2.3
The Inverse Polynomial Model
A certain disadvantage of the binomial models (
8.9
)and(
8.10
) is their inability to
represent oscillating CFs whose spectra may have multiple maxima. This issue can
be overcome by considering the BEC models of the form:
2
3
1
J
X
4
I
C
a
j
D
j
5
B
D
(8.19)
j
D
1
Here
a
j
are the real numbers, constrained by the positive definiteness requirement
of
B
. In the Fourier representation, the operator (
8.19
) acts as multiplication by the
inverse of the polynomial in
k
2
, and the positive-definiteness property translates into
the requirement that the spectral polynomial
J
X
B
1
.k
2
/
D
1
C
a
j
.
k
2
/
j
(8.20)
j
D
1
k
2
>0
should be positive for all
. This constraint is equivalent to the statement that
the rhs of (
8.20
) must not have real positive roots. Therefore,
B
1
.k
2
/
can also be
represented in the form
Y
B
1
.k
2
/
D
1
Z
.k
2
C
z
2
m
/.k
2
C
z
2
m
/;
(8.21)
m
D
1
where
M
D
J=2
,
Z
D
Y
m
j
z
2
m
j
2
;
(8.22)
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