Geoscience Reference
In-Depth Information
Box 6.7 (continued)
Z
1
C
(
y
) and semivariogram
ʳ
(
y
)as
Cy
ðÞ
¼
V
½
fx
ðÞ
M
½
fx
ðÞ
M
dx
Z
V
2
dx
. The covariance of the averages
ʳ
ðÞ
¼
y
2
V
1
fx
ð
þ
y
Þ
fx
ðÞ
and
½
V
m
1
and
m
2
for two volumes
v
1
and
v
2
can be determined as:
˃
ð
m
1
;
m
2
Þ
¼
Z
Z
dx
1
Z
v
2
dx
2
Z
1
Vv
1
v
2
½
fx
ð
þ
x
2
Þ
M
dx
:
½
m
1
M
½
m
2
M
dx
¼
½
fx
ð
þ
x
1
Þ
M
V
v
1
V
When
y
represents distance between all possible pairs of points, it follows
Z
dx
1
Z
1
v
1
v
2
that :
˃
ð
m
1
;
m
2
Þ
¼
Cy
ðÞ
dx
2
. When
v
1
and
v
2
coincide it reduces
v
2
Z
dx
1
Z
v
1
v
2
2
m
1
to
dx
2
representing the variance of average con-
centration
m
for a block with volume
v
. By defining an auxiliary function
˃
ðÞ
¼
Cy
ðÞ
v
v
v
2
Z
dx
1
Z
1
2
(
m
)
Fm
ðÞ
¼
v
ʳ
ðÞ
y
dx
2
this variance can be rewritten as
˃
¼
Z
dx
1
Z
v
F
(
M
)
F
(
m
). Definition of
Gm
1
;
ð
m
2
Þ
¼
1
v
1
v
2
v
2
ʳ
ðÞ
y
dx
2
results in
v
1
˃
G
(
m
1
,
m
2
). Using
m
1
of
v
1
to estimate
m
2
of
v
2
results
in the “extension” variance
(
m
1
,
m
2
)
¼
F
(
M
)
2
(
m
1
E
˃
¼
˃
m
2
) that can also be written as
2
(
m
1
)
2
(
m
2
)or
E
E
˃
¼
˃
2
˃
(
m
1
,
m
2
)+
˃
˃
¼
2
G
(
m
1
,
m
2
)
F
(
m
1
)
F
(
m
2
).
Matheron (1964) has shown that the average of
n
adjoining channel sample
concentration values has variance equal to
2
E
/
n
. This is another important result:
The extension variance
˃
E
0.0622 is for logarithmically transformed zinc con-
centration values. As discussed in Sect.
2.2
, it can be assumed that the zinc values
(
X
i
with
i
˃
¼
, 118) for the original channel samples systematically underes-
timate zinc values for the massive sulphide (Fig.
2.9
). By setting
¼
1,
...
2
E
and
˃
¼
˃
2
(
X
i
) of the original zinc values.
These variances can then be used to calculate approximate 95 % confidence limits
for zinc concentration values of 1.3 m
μ
(
X
)
¼
X
i
, it is possible to estimate the variances
˃
2 m plates formed by extending the 1.3 m
long channel samples by 1 m on both sides. Table
6.2
shows
(
X
i
) error bars
for 11 original zinc values and for averages of adjacent values for wider plates at the
same locations. These sets of overlapping plates, that are 20 m apart, were selected
for example so that both low and high zinc concentrations are represented. The error
bars in Table
6.2
for plates wider than 2 m are relatively narrow. Uncertainty is
greatest for the 1.3 m
1.96
˃
ʱ
¼
0.0165 is
underestimated over very short distances resulting in error bars that are too wide.
2 m plates but this is probably because
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