Geoscience Reference
In-Depth Information
a
b
2.5
200000
180000
160000
2
140000
1.5
120000
100000
80000
60000
1
0.5
40000
20000
0
0
0.5
1
1.5
2
2.5
α
3
3.5
4
4.5
−
30
−
25
−
20
−
15
−
10
0
Log
2
(x)
−
5
5 0 520
c
6
4
2
0
−
30
−
25
−
20
−
15
−
10
−
5
5 0 520
−
2
−
4
−
6
Z - Value
Fig. 11.9 (a) Histogram method illustrated in Fig.
11.7
applied to concentration values with
d
¼
0.6 and
n
¼
20; (b) Frequency distribution curves corresponding to two multivariate spectra
shown in a; frequencies of limiting form slightly exceed logbinomial frequencies but difference is
zero at center and endpoints; (c) Lognormal
Q
-
Q
plot of upper bound frequency distribution shown
in b; near center, frequency distribution resulting from model of de Wijs is lognormal, and in the
tails it is weaker than lognormal (Source: Agterberg
2001
, Fig. 9)
concentration value. Thus box-counting of a monofractal binary pattern for a
narrow neighborhood centered on any specific value
x
/
E
α
2
created from a
realization such as the one shown in Fig.
10.22
would give a fractal dimension
f
(
α
) provided that the number of cells used for box-counting is sufficiently large.
The theoretical multivariate spectrum
f
(
α
) (for
n
¼
1
) can be used to compute
theoretical frequencies of the concentration values
x
for specific values of
n
and
d
.
These frequencies are not independent of
n
, and the frequency distribution curve
continues to change when
n
is increased. Various scaling and rescaling procedures
have to be applied in order to derive the results shown in Fig.
11.7
and the
frequencies, subsequently, to be derived from the limiting form of
f
(
α
). The
required calculations were given in detail in a FORTRAN program in Agterberg
(
2001
) (Fig.
11.9
).
11.3 Multifractal Spatial Correlation
Cheng (
1994
; also see Cheng and Agterberg
1996
) derived general equations for the
semivariogram, spatial covariance and correlogram of any scale-independent
multifractal including the model of de Wijs. Their model is for sequences of
Search WWH ::
Custom Search