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domain 0.003 m
2 m) there exists a strong nugget effect that is not readily
detectable at distances of
h
<
h
<
would be expected to
increase rapidly again, because of measurement errors and the fact that the zinc
have become negligibly small because channel sample length greatly exceeded
crystal dimensions.
The preceding considerations imply that the negative exponential autocor-
relation function previously used (see, e.g., Fig.
2.10
) is too simple for short
distances (
h
2 m. At the microscopic level
ʱ
<
2 m). The true pattern is probably close to that shown in Fig.
6.19
,
which differs from the earlier model in that strong autocorrelation is assumed to
exist over very short distances. It is probably caused by clustering of ore crystals,
although at the microscopic scale there remains rapid decorrelation related to
measurement errors and crystal shapes. The model of Fig.
6.19
is an example of a
nested semi-exponential autocorrelation function as previously used by Mat´rn
(
1981
) and Serra (
1966
). The graph in Fig.
6.19a
satisfies the equation:
c
1
e
a
1
h
c
2
e
a
2
h
ˁ
ðÞ
¼
h
þ
0.1892 as in Fig.
2.10
.
The second term represents the strong autocorrelation due to clustering over very
short distances. The decorrelation at microscopic scale is represented by a small
white noise component (probably augmented with a random measurement error)
coefficient
c
2
in the second term on the right side satisfies
c
2
¼
The coefficients in the first term are
c
1
¼
0.5157 and
a
1
¼
0.4635.
Because of lack of more detailed information on autocorrelation over very short
distances, it is difficult to choose a good value for the coefficient
a
2
. Choosing
a
2
¼
1
c
0
c
1
¼
2 provides a good fit over the entire observed correlogram (Fig.
2.10
). It affects
extrapolation toward the origin with
h
2 m only. Figure
6.19b
shows that
thesecondtermontherightsideofthepreceding equation cannot be detected
in the correlogram for sampling intervals greater than 2 m. Other types of
evidence for existence of strong autocorrelationoververyshortdistancesinthe
Pulacayo orebody have already been discussed (e.g., Fig.
6.15
) and will be
be derived on the basis of self-similarity assumptions. It results in a curve that
resembles the curve in Fig.
6.19
(with
a
2
¼
<
2). For example, for lag distance equal
to 60 cm, the theoretical value according to Fig.
6.19a
is 0.6, while the curve in
Sect.
11.6.3
yields 0.7.
Independent evidence that a model similar to the one shown in Fig.
6.19
also
applies to copper in the Whalesback deposit (see single exponential fit in Fig.
6.16a
)
is as follows. Figure
6.16a
represents an average correlogram based of 24 series of
channel samples taken at 8 ft. intervals perpendicular to drifts on various levels of
the Whalesback Mine. Agterberg (
1966
) had obtained separate results for a series of
111 channel samples along a single drift on the 425-ft. level of this mine. These 8-ft.
long channel samples had been divided into 4-ft. long halves that were separately
analyzed. The correlation coefficient for copper in the two halves amounted to 0.80
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