Geoscience Reference
In-Depth Information
The practical problem of gold mining in this example consists of predicting the
value of a panel that will be 30 ft. ahead of the panel whose value is known at a
given time. In order to solve this problem, one must estimate, as Krige (
1962
) has
done:
E
(
Y
β
1
x
using the means, standard deviations and correlation
coefficient (see Box
4.1
). This linear regression results in the straight line (A) in
Fig.
4.1
. Krige (
1962
) pointed out that, in this manner, one avoids making the
erroneous assumption that a given amount of gold per panel in a mine is represen-
tative for a larger region surrounding it. For example, if a panel values is
x
j
x
)
¼
β
0
+
1,102,
the value predicted from it for a panel that is 30 ft. ahead is not 1,102 but 688 which
is considerably less. On the other hand, if
x
¼
x
) is estimated to be
158 which is significantly more. Consequently, relatively high values overestimate
amount of gold per unit volume in their immediate surroundings, but relatively low
values tend to underestimate it. Suppose that each sample value is representative of
a larger volume of ore around it and mining would be restricted to only those blocks
for which the sample value is relatively large, exceeding a cut-off value and that the
remainder of the orebody would be left unmined. In the past, mining engineers
found to their surprise that the average for a volume blocked out in this manner
systematically overestimates the average amount of metal that is actually mined out
later. Problems of this type can be avoided by the use of “kriging”. Figure
4.1
provides a simple example of this: if a cut-off grade it set, it should be based on the
predicted values (curve A) rather than on the actual values (curve B).
In general, the aim of the various methods of kriging that have been developed in
geostatistics is to provide unbiased estimates of values at points with arbitrary
tration value for a small block would be available, it could be used as an estimate of
a larger block surrounding the small block. The variance of the single value for the
larger block then would be a so-called extension variance (Sect.
6.2.5
).
¼92,
E
(
Y
j
Box 4.1: Basic Elements of Covariance and Correlation Coefficient
If
X
and
Y
are two random variables, then:
E
(
XY
)
¼
RR
xy
·
f
(
x
,
y
)
dxdy
where
f
(
x
,
y
) is the bivariate frequency distribution. Suppose that the double integral
is approximated by a double sum, then:
E
(
XY
)
¼
∑
i
∑
j
x
i
y
j
P
(
x
i
,
y
j
). Mutual
independence of
two random variables can be defined as:
P
(
x
i
,
y
j
)
¼
P
(
x
i
)
P
(
y
i
). Hence:
E
(
XY
)
(
EX
)·(
EY
) but only for
independent random variables. The covariance of
X
and
Y
is:
¼
∑
i
∑
j
[
x
i
P
(
x
i
)]
y
i
P
(
y
i
)]
¼
˃
(
X
,
Y
)
¼
E
(
X
ʼ
x
ʼ
y
. The correlation coefficient is the covariance
of two standardized random variables, or
ʼ
x
)(
Y
ʼ
y
)
¼
E
(
XY
)
Þ
¼
˃
X
;ðÞ
˃
ˁ
¼
ˁ
ð
X
;
Y
. Conditional
ðÞ ˃
X
ðÞ
Y
dependence can be defined as:
P
(
Y
¼
y
j
X
¼
x
)
¼
P
(
X
¼
x
,
Y
¼
y
)or
P
(
y
j
x
)
¼
P
(
x
,
y
)/
P
(
x
). Consequently,
P
(
x
,
y
)
x
)·
P
(
x
). Suppose
X
and
Y
are
normal. Standardization to Z
1
and Z
2
, then results in the bivariate frequency
density:
¼
P
(
y
j
h
p
1
1
1
21
ˁ
z
1
ˆ
ð
z
1
;
z
2
Þ
¼
exp
2
ˁ
z
1
z
2
. After some manip-
ð
2
Þ
2
ˀ
ˁ
2
. This is the
z
2
z
1
¼
ˆ
z
1
;
z
ð Þ
p
1
ˁ
p
z
2
ˁz
1
1
ˁ
ulation, it follows that:
ˆ
ˆ zðÞ
¼
1
ˆ
2
2
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