Geoscience Reference
In-Depth Information
Table 6.2 Zinc concentration values (in %) with 95 % confidence intervals for thin plates in the
direction of the mining drift with channel samples at their centers
Plate size
1.3 m
2 m
1.3 m
6 m
1.3 m
10 m
1.3 m
14 m
1.3 m
18 m
24.1
12.2
19.9
5.6
19.4
4.2
17.5
3.2
17.0
2.8
#10
#20
15.1
7.7
13.8
3.9
14.0
3.1
13.3
2.4
13.2
2.1
#30
9.5
4.8
12.1
3.4
15.2
3.3
13.2
2.4
14.7
2.4
#40
10.6
5.4
15.6
4.4
17.0
3.7
15.5
2.9
14.2
2.3
#50
27.4
13.9
18.6
5.3
17.4
3.8
17.4
3.2
17.2
2.8
#60
4.7
2.4
9.0
2.5
8.7
1.9
8.1
1.5
9.0
1.5
#70
9.7
4.9
9.2
2.6
10.5
2.3
10.3
1.9
10.2
1.6
#80
10.6
5.4
11.1
3.2
10.8
2.3
9.3
1.7
9.6
1.6
#90
30.8
15.6
31.6
9.0
30.8
6.7
30.7
5.7
29.2
4.7
#100
22.6
11.5
16.4
4.6
18.6
4.1
20.8
3.8
21.4
3.5
#110
7.9
4.0
17.8
5.0
17.2
3.8
15.9
2.9
14.6
2.4
From Agterberg (
2012
, Table 2)
Results are shown for every 10th value in the original series of 118 values. Error bars for
1.3 m
2 m are too wide because small-scale spatial correlations are not being considered
The problem of overestimation of extension variances of average element
concentration values for small plates due to local strong autocorrelation was
previously considered by Matheron (
1989
, pp. 73-75) as follows. Ten professional
geostatisticians were provided with a set of variogram values with unit of lag
distance equal to 180 m. Independently the participants in this experiment were
asked to (a) fit a variogram, and (b) calculate the corresponding extension variance
for a square plate measuring 180 m on a side. Each variogram fitted by a participant
had a nugget effect, with, in addition, an exponential or (third-order polynomial)
“spherical” variogram curve. The corresponding average of ten estimated extension
variances was 0.4019
0.0127 indicating excellent agreement between participants.
Next, the same ten people were provided with additional variogram values for shorter
unit lag distance interval of 20 m. Again they were asked (a) fit a variogram, and
(b) calculate the corresponding extension of the 180 m
180 m square plate.
The variogram models used during the second stage of the experiment were “richer”
becoming either: nugget + spherical + spherical, or nugget + exponential + spherical,
or nugget + exponential + exponential. A few other answers were given as well.
The revised average extension variance became 0.3686
0.0062. Clearly this revised
estimate of the extension variance is less than the first estimate and outside the 95 %
confidence of the first estimate. Similar results were obtained during a third stage of
this experiment using an even shorter unit lag distance. Although Matheron (
1989
)
did not report the equation of the model used to generate the variograms for longer lag
distances, this model obviously had no or very small nugget effect that is
overestimated by extrapolation toward the origin by means of the standard models.
It is noted that geostatisticians often use the spherical semivariogram with
2
(
h
/
a
)
3
/2 for 0
ʳ
(
h
)/
˃
¼
3
h
/2
a
h
<
a
where
a
is a constant called the range;
2
for
h
and
a
. This model also arises in the following situation. Suppose
that in 3-D two identical copies of a sphere with radius
a
and volume equal to
ʳ
(
h
)
¼
˃
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