Geoscience Reference
In-Depth Information
1
1
1
b=2
b=1
b=2
b=4
b=2
b=2
b=10
b=4
b=2
0.8
0.8
0.8
0.6
0.6
0.6
n=1
n=2
n=3
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
0
2
4
6
0
2
4
0
1
2
3
Fig. 8.2
Two-parameter CFs corresponding to the inverse BEC (
8.21
) with
a
D
1;M
D
1
.The
horizontal axis
is scaled by
a
.
Dotted lines
show CFs corresponding to the special case with two
k
1
D
a; k
2
D
b
negative roots
not described by the spectral polynomial (
8.21
)
C
n
.r/
The corresponding correlation functions
are obtained through normalizing
B
n
.0/
(
8.28
)by
. The first three values at
r
D
0
are
B
1
.0/
D
X
m
h
q
m
z
m
ij
z
m
j
2
(8.29)
X
B
2
.0/
D
1
h
q
m
log
z
m
i
(8.30)
m
X
B
3
.0/
D
1
2
h
q
m
z
m
i
(8.31)
m
C
n
.r/
R
n
:
The normalization factors can be found by integrating
over
X
h
q
m
z
2
m
i
j
z
m
j
4
2
B
n
.0/
N
n
D
(8.32)
m
Relationships (
8.28
)-(
8.32
) provide analytical expressions for the CFs and the
normalization factors.
In the important case of the quadratic polynomial (
M
D
1
) the BEC model
is defined by two parameters
(Fig.
8.2
). Expressions for the respective CFs in
1- and 3-dimensional cases can be rewritten in terms of the elementary functions
(
Yaremchuk and Smith 2011
;
Yaremchuk and Sentchev 2012
)
a;b
p
a
2
C
b
2
b
.br
arctan
a
C
1
.a;b;r/
D
.
ar/
b
/
exp
cos
(8.33)
sin
.br/
br
C
3
.a;b;r/
D exp
.
ar/
(8.34)
and the normalization factors are given by
4a
a
2
C
b
2
I
N
2
D
8ab
2.a
2
C
b
2
/
2
arctan
8a
.a
2
C
b
2
/
2
N
1
D
I
N
3
D
(8.35)
.b=a/
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