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where
)} represents element concentration value determined on a
neighbourhood size measure
B
x
at point
x
,
ˁ
{
B
x
(
E
)} represents amount of metal,
and
E
is the Euclidean dimension of the sampling space. In general,
ʼ
{
B
x
(
E
)} is an
average value of element concentration values for smaller
B
's at points near
x
with
different local singularities. Consequently, use of the power-law relationship as it
stands would produce biased estimates of
c
(
x
) and
ˁ
{
B
x
(
E
(
x
). How could we obtain
estimates of
c
(
x
) that are non-singular in that they are not affected by the differences
between local singularities within
B
x
? Chen et al. (
2007
) proposed to replace the
original model by:
α
ðÞ
E
α
ðÞE
c
x
ˁ
ðÞ
¼
x
where
α
*(
x
)and
c
*(
x
) are the optimum singularity index and local coefficient,
respectively. In the Chen algorithm the initial crude estimate
c
(
x
)isconsideredto
be obtained at step
k
¼
1 of an iterative procedure. It is refined repeatedly by
using:
ðÞ
E
α
k
ðÞE
c
k
1
¼
c
k
x
This procedure will be explained by application to the 118 zinc concentration
values of the Pulacayo Mine example.
11.6.1 Pulacayo Mine Example
In the 1-dimensional Pulacayo example,
E
1min
two directions from each of the 118 points along the line parallel to the mining drift.
Suppose that average concentration values
¼
1; and, for
E
¼
1,
B
x
extends
/2
¼
E
3, 5, 7
and 9, by enlarging
B
x
on both sides. The yardsticks can be normalized by dividing
the average concentration values by their largest length (
ˁ
{
B
x
(
)} also are obtained for
E
¼
E
9). Reflection of the series
of 118 points around its first and last points can be performed to acquire approximate
average values of
¼
ˁ
{
B
x
(
)} at the first and last 4 points of the series. A straight line can
be fitted by least squares to the five values of log
e
ʼ
E
{
B
x
(
α
(
x
)log
e
E
E
)} against
then
provides estimates of both ln
c
(
x
)and
α
(
x
) at each of the 118 points. Estimates of
c
(
x
)and
tively. These results of ordinary local singularity mapping duplicate estimates previ-
ously obtained by Chen et al. (
2007
) who proposed an iterative algorithm to obtain
improved estimates.
Employing the previous least squares fitting procedure at each step resulted in
process, and for
k
α
¼
1,000 after convergence has been reached. Values for the first
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