Geoscience Reference
In-Depth Information
Fig. 12.45 Theoretical frequency distributions of amount of acidic volcanics per cell of variable
size. Includes two frequency distributions of Fig.
12.44
(Source: Agterberg and Fabbri
1978
,
Fig. 12)
random cell of variable size. In order to model the frequency distribution of
X
, the
probnormal model can be used. Suppose that the random cell
i
with value
x
i
¼
h
2
(
z
i
)
contains a much smaller cell with value
h
1
(
u
i
) which occurs at random within the
larger cell
i
;
h
1
(
u
i
) is either one or zero. The pair of values (
z
i
,
u
i
) is assumed to be a
realization of a standard bivariate normal distribution with correlation coefficient
r
.
Thus
z
i
and
u
i
are realizations of random variables
Z
and
U
with standard normal
distributions. The transformed random variables
h
1
(
U
) and
X
¼
h
2
(
Z
) can be
expanded in terms of Hermite polynomials.
Figure
12.45
shows a sequence of frequency distributions arising from the
random cell with variable side
a
. For any value of
a
, the frequency distribution of
X
is determined by the parameters
m
,
c
, and
p
because
C
0
(
a
),
b
, and
r
are derived
from these three basic parameters. The five examples in Fig.
12.45
are for
a
¼
0,
4, 10, 20, and 40 km, respectively. When
a
tends to zero (
a
!
0),
r
!
1 and in the
limit (
a
¼
0) the density function of
X
consists of separate spikes at
x
¼
0 (with
Search WWH ::
Custom Search