Geoscience Reference
In-Depth Information
three types of permanent frequency distributions (lognormal, probnormal and
asymmetrical bivariate binomial distributions) will be exemplified in this section.
Other types of orthogonal polynomials are required for other types of permanent
frequency distributions. Topics covering theory of orthogonal polynomials include
Beckmann (
1973
) and Szeg
¨
(
1975
).
Theory will be summarized and applied in case history studies dealing with the
areal distribution of acidic volcanics in the vicinity of Bathurst, New Brunswick,
and in the Abitibi area on the Canadian Shield. Both examples have been used
extensively in previous chapters in order to exemplify a variety of statistical
techniques and methods of mathematical morphology. These rocks constitute
favorable environments for the occurrence of volcanogenic massive sulphide
deposits. It should be kept in mind that the information used as input is 2-D only.
In reality these rock formations have 3-D structures that are only known to a limited
extent although during the past 10 years significant progress has been made in 3-D
12.8.1 Permanent Frequency Distributions
Suppose that a rock or geological environment is sampled by randomly
superimposing on it a large block with average value
X
2
, and that a small block
with value
X
1
is sampled at random from within the large block. Then the expected
value of
X
1
is
X
2
,or
E
(
X
1
|
X
2
)
X
2
. Let
f
(
x
1
) and
f
(
x
2
) represent the frequency
density functions of the random variables
X
1
and
X
2
, respectively. Suppose that
X
1
can be transformed into
Z
1
by
X
1
¼
ˈ
1
(
Z
1
), and
X
2
into
Z
2
by
X
2
¼
ˈ
2
(
Z
2
) so that the
random variables
Z
1
with marginal density function
f
(
z
1
), and
Z
2
with
f
(
z
2
), satisfy
a bivariate density function of the type
¼
"
#
1
j
¼1
ˁ
j
Q
j
zðÞ
fz
1
;
ð
z
2
Þ
¼
f
1
zðÞ
f
2
zðÞ
1
þ
S
j
zð =
h
1
j
h
2
j
represents the product-moment correlation coefficient of
Z
1
and
Z
2
.
Q
j
(
z
1
) and
S
j
(
z
2
) are orthogonal polynomials with
f
1
(
z
1
) and
f
2
(
z
2
)as
weighting functions, and with norms
h
1
j
and
h
2j
, respectively. It is implied that
In this equation,
ˁ
¼
ˁ
j
S
j
zðÞ
EQ
j
Zð j
Z
2
h
1
j
=
h
2
j
In most applications,
f
(
z
1
,
z
2
) is symmetric with
f
1
(
z
i
)
¼
f
2
(
z
i
) for
i
¼
1, 2. For
example, if
f
1
(
z
1
) and
f
2
(
z
2
) are standard normal,
Q
j
(
z
1
)
¼
H
j
(
z
2
) are Hermite polynomials with squared norms
j
! for both
Q
j
(
z
1
) and
S
j
(
z
2
).
Then the preceding expression for the bivariate density function becomes the well-
known Mehler identity.
¼
H
j
(
z
1
) and
S
j
(
z
2
)
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