Geoscience Reference
In-Depth Information
computerized nearest-neighbor (NN) method (Sinclair and
Blackwell
2002
).
8.1.6
Classic Polygonal Method
The polygonal estimate is also based on assigning areas of
influence around drill hole intercepts. The drawing of poly-
gons around drill hole data is based on the perpendicular bi-
sectors between a sample and all its neighbors, see Fig.
8.3
.
The perpendicular bisector of a line segment is a line for
which points are at the same distance from either side of the
line segment. This concept can be extended to three-dimen-
sions, although polygons are typically drawn in two-dimen-
sions and the volume of influence is defined perpendicular
to the polygonal plane.
The polygon of influence is such that each point within
the polygon is closest to the central sample than any other
sample. Special care must be taken with samples on the
outer edges. These samples are not completely surrounded
by other samples, so bounding the polygon is important.
There are several alternatives, including the use of geologic
boundary (if available), or, more commonly, a fixed maxi-
mum distance from the sample. In any event, the closing of
the polygons on the outer edges may be arbitrary and can
have a significant impact on the final results.
The polygonal method corresponds with the intuitive idea
that the amount of information provided by each sample
is proportional to its area (or volume) of influence. In this
sense, the method has found modern application as a spatial
declustering tool, calculating weights to avoid biased statis-
tics based on spatial drill hole data aggregation (Chap. 2).
One example is that it can be used to provide a deposit-wide
estimate of the average grade (Isaaks and Srivastava
1989
).
Fig. 8.2
A sketch showing estimation in one dimension
deposit, such as those discussed in Chap. 2, where methods
were proposed to remove bias resulting from clustering.
8.1.4
Weighted Linear Estimation
Estimates are often made as weighted linear estimators. A
common approach is to estimate the values as deviations
from a mean or trend surface, see Fig.
8.2
. The estimate
reverts to the mean some distance away from the data, see
the far right edge. The deviations from the mean surface
are estimated at unsampled locations with some method of
interpolation. The most common interpolation scheme is a
weighted linear estimate:
n
∑
z
*
( )
u
−
m
( )
u
=
λ
·
[()
z
u
−
m
( ]
u
0
0
i
i
i
i
=
1
where the
*
denotes an estimate,
u
0
denotes the unsampled
location being estimated,
z
(·) denotes the variables value,
m
(∙) denotes the mean or trend value and
i = 1,…, n
is the
index of data values.
Estimation then becomes an exercise in determining the
weights
λ
i
using certain criteria. Factors considered when as-
signing weights may include the closeness to the location
being estimated; the redundancy between data values; the
anisotropic continuity (preferential direction); and the mag-
nitude of continuity/variability.
8.1.7
Nearest-Neighbor Method
This is a variant of the polygonal method, but in this case the
grades or attributes are assigned directly to a block model.
This is the most common computerized polygonal estima-
tion method and it has evolved into two common uses.
The first and more traditional use is the calculation of
mineral resources. A grid of blocks is assigned grades by the
closest drill hole data sample or composite. The method does
not average values from different samples. The original vari-
ance of the data is maintained. There is no smoothing and
grades from one block to the next change abruptly producing
artificial discontinuities.
The nearest-neighbor method is not as good as Inverse
Distance methods and Kriging at estimating grades. From
theory and practice (see for example Knudsen et al.
1978
;
Baafi and Kim
1982
; Readdy et al.
1982
; Knudsen
1990
),
8.1.5
Traditional Estimation Methods
Simple (traditional) estimation techniques can be used to
assign values to blocks. Polygonal methods and Inverse
Distance (ID) methods are often applied at early stages of a
mining project or for checking. These methods are not par-
ticularly accurate, but can provide an order-of-magnitude re-
source estimate. They can also be used to check the results of
more sophisticated geostatistical estimation methods.
According to Popoff (
1966
) polygonal methods have
been used since the early 1900s. Variants of the polygo-
nal method include the sectional estimation method (Stone
and Dunn
1996
), the classic polygonal method, and the
