Geoscience Reference
In-Depth Information
a
b
4.5
4
0
−
0.5
3.5
−
1.
−
1
3
2.5
2.
−
2
2
−
−
3.
−
3
1.5
1
−
4.
−
4
0.5
0
−
5
0
2
4
6
8
10
0
0.5
1
1.5
2
LAG
LOG LAG
Fig. 6.13 Comparison of covariance models fitted to 118 Pulacayo zinc values under finite and
infinite variance assumption. (a) Exponential function (straight line with slope -
a
¼ 0.1892).
Nearly half of the variance of the zinc values can be attributed to white noise; (b) Semivariogram
mic distance scale.
Straight line
has unit slope. This is in agreement with exponential covariance
model applied to original zinc values of Table
2.4
(Source: Agterberg
1994
, Fig. 2)
6.2.2 Correlograms and Variograms:
Pulacayo Mine Example
De Wijs (1951) assumed that, if a block of ore is divided into halves, the ratio of
average element concentration values for the halves is equal to the same constant
regardless of the size of the block that is divided into halves. If greater value is
divided by lesser value, this ratio can be written as
1. Matheron (
1962
)
generalized this original model by introducing the concept of “absolute dispersion”
written as
ʷ >
)
2
/log
e
16. This approach is equivalent to what is now better
known as scale invariance. It leads to the more general equation
ʱ
¼
(log
e
ʷ
2
˃
(log
e
x
)
¼
2
(log
e
x
) represents logarithmic variance of element concen-
tration values
x
in smaller blocks with volume
v
contained within a larger block of
ore with volume
V
. The corresponding semivariogram along a line then satisfies:
ʳ
h
¼
A
log
e
V
/
v
where
˃
3
A
· log
e
h
. This model does not have a sill but is useful for modeling spatial
correlation over very short distances.
Matheron (
1989
) pointed out that in rock-sampling there are two possible infinities
if number of samples is increased indefinitely: either the sampling interval is kept
constant so that more rock is covered, or size of study area is kept constant whereas
sampling interval is decreased. These two possible sampling schemes provide addi-
tional information on sample neighbourhood, for sill and nugget effect, respectively.
In practice, the exact form of the nugget effect usually remains unknown because
extensive sampling would be needed at a scale that exceeds microscopic scale but is
less than scale of sampling space commonly used for ore deposits or other geological
bodies. Nevertheless, there are now several methods by means of which the nugget
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