Geoscience Reference
In-Depth Information
(20
40 km) cells in the Abitibi area on the Canadian Shield,
similar to those discussed in Sect.
4.3.2
, are approximately lognormally distributed
with logarithmic variances equal to 2.150 and 0.920, respectively. Consequently,
the correlation coefficient between values for (20
20 km) and (40
20 km) and (40
40 km) cells
is estimated to be equal to 0.654.
Box 12.5: Correlation between Original and Standardized Cell
Composition Data
Suppose the mean value of the two random variables
X
1
and
X
2
(for smaller
and larger cells) is written as
ʼ
¼
E
(
X
1
)
¼
E
(
X
2
). In general, the transformations
ˈ
1
and
ˈ
2
can be expanded by using the orthogonal polynomials
Q
j
and
S
j
with
x
1
¼
ˈ
1
(
z
1
)
¼
∑
j
¼ 1
c
j
S
j
(
z
2
) where
x
1
and
x
2
are
ordinary continuous variables. The coefficients differ from one another except
¼
∑
j
¼1
c
j
Q
j
(
z
1
);
x
2
¼
ˈ
2
(
z
2
)
.Ingeneral,
Z
1
1
c
0
*
h
1
j
where
for
c
0
¼
¼
ʼ
f
1
zðÞ
Q
i
zðÞ
Q
j
zðÞ
dz
1
¼
ʴ
ij
ʴ
ij
¼
0
Z
1
if
i
6
¼
j
, d
ʴ
ij
¼
1if
i
¼
j
.Aso,
x
2
¼
ˈ
1
zðÞ
½
fz
1
;
ð
z
2
Þ=
f
2
zðÞ
dz
1
1
X
j
¼
0
ˁ
j
Q
j
zð S
j
zðÞ
h
1
j
h
2
j
"
#
dz
1
¼
ˈ
2
(
z
2
) ¼
∑
j
¼ 0
ˁ
Z
1
j
c
j
S
j
(
z
2
)
h
1
j
/
h
2
j
or
c
j
¼
ˈ
1
zð f
1
zðÞ
1
Z
1
Z
1
j
c
j
h
1
j
/
h
2
j
. For the larger cells,
EX
2
¼
¼
ˁ
x
2
f
X
2
xð dx
2
¼
2
2
zð f
2
zðÞdz
2
ˈ
1
1
Z
1
h
i
dz
2
¼
X
j
¼0
ˁ
X
j
¼0
ˁ
j
c
j
S
j
zðÞ
2
j
c
j
h
1
j
f
2
zðÞ
h
1
j
=
h
2
j
¼
1
2
(
X
2
)
2
2
j
c
j
h
1
j
E
(
X
2
)
¼
∑
j
¼ 1
ˁ
so that their variance satisfies
σ
¼
ʼ
2
2
(
X
2
)
¼
∑
j
¼ 1
ˁ
2
j
c
j
h
1
j
.
or
ˁ
x
σ
12.8.2 The Probnormal Distribution
In an application of the discrete Gaussian model approach, Matheron
(1974
,
pp. 57-61) considered the following cell-value distribution model for binary patterns:
.Inthisequation,
X
is a randomvariable for cell values generated from
ˁ
Z
ʲ
p
1
ˁ
X
¼
ʦ
2
a binary variable
X
0
,
Z
is a standard normal random variable, while
ʲ
and
ˁ
are
constants and
represents the quantile of a normal distribution function.
Cell values are compositional data in that they represent proportions of cell area
underlain by the binary pattern (e.g., for a rock type on a geological map) that is
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