Geoscience Reference
In-Depth Information
"
#
Þ
¼
ˆ
ð
z
1
;
z
2
Þ
ð
z
2
ˁ
z
2
Þ
ˆ
ð
z
2
j
z
1
zðÞ
¼
ˆ
p
1
ˆ
2
ˀ
ˁ
2
As discussed in more detail before, the Mehler identity is:
"
#
1
j
¼1
ˁ
j
H
j
zðÞ
ˆ
ð
z
1
;
z
2
Þ
¼
ˆ
zðÞˆ
zðÞ
1
þ
H
j
zð =
j
!
where
H
j
(
z
1
) and
H
j
(
z
2
) are Hermite polynomials with the property:
Z
z
2
H
j
zð ˆ
zðÞ
dz
2
¼
H
j
1
zð ˆ
zðÞ
1
Use of this property during integration with respect to
z
2
, replacement of
z
1
by
the standard normal random variable
Z
, and replacing
z
2
by the constant
ʲ
yields the
random variable
"
#
1
j
¼1
ˁ
ˁ
Z
ʲ
1
j
H
j
Z
X
¼
ʦ
p
¼ 1
ʦðÞˆðÞ
ðÞ
H
j
1
ðÞ=
j
!
ˁ
2
This is equivalent to assuming that the “probit” of
X
has normal distribution
(Fig.
12.38
). The binary random variable
X
1
for points randomly located within
cells with values assumed by
X
can be defined by letting
can
be interpreted as the correlation coefficient between
X
1
and
X
. The mean of
ˁ
tend to one. Then
ˁ
1
2
2
j
H
j
1
ðÞ
ðÞ
1
.By
2
2
X
satisfies
ʼ
¼
1
ʦ
(
ʲ
) and its variance is
σ
¼
ˆ
ðÞ
1
ˁ
j
j
¼
ʲ
ˁ
means of these two equations it is possible to obtain estimates of
and
from
2
as is shown graphically in Fig.
12.39
. Of course, it also is
possible to estimate these parameters directly by fitting a straight line to a
Q
-
Q
plot
of cell value percentage values and their observed frequencies using prob-prob
paper.
ʼ
σ
estimates of
and
12.8.3 Bathurst Area Acidic Volcanics Example
Occurrences of volcanogenic massive sulphides and their relationship with acidic
volcanics in the Bathurst area, New Brunswick were taken for example in Sect.
1.5.2
to illustrate various Minkowski operations. In this section the same example is
probnormal model.
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