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/
Search WWH ::




Custom Search