Page 257 - Introduction to Mineral Exploration

240 M.K.G. WHATELEY & B. SCOTT

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Estimation variance (σ e ) variance is a linear function of the weights
No matter which method is used to estimate (see “Kriging” above) established by kriging
the properties of a deposit from sample data, (Journel & Huijbregts 1978). Kriging deter-
whether it is manual contouring or computer mines the optimal set of weights that minim-
methods, some errors will always be intro- ize the variance. These weighting factors are
duced. The advantage of geostatistical methods also used to estimate the value of a point or
over conventional contouring methods is their block. As it is rare to know the actual value (Z)
ability to quantify the potential size of such for a point or block it is not possible to calcu-
errors. The size of the error depends on factors late Z–Z*(c.f. cross-validation above).
such as the size of the block being estimated In order to minimize the estimation vari-
(Whitchurch et al. 1987), the continuity of the ance, it is essential that only the nearest
data, and the distance of the point (block) being samples to the block being estimated are be
estimated from the sample. The value to be used in the estimation process.
estimated (Z*) generally differs from its esti-
mator (Z) because there is an implicit error Kriging variance (σ k )
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of estimation in Z–Z* (Journel & Huijbregts Kriging variance is a function of the distance
1978). The expected squared difference be- of the samples used to estimate the block
tween Z and Z* is known as the estimation value. A block, which is estimated from a near
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variance (σ e ): set of samples, will have a lower σ k than one
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that is estimated using samples some distance
2
2
σ e = E([Z − Z*] ) away. Once the kriging weights for each block
have been calculated (see “Kriging” above) the
2
variances can be calculated. Once σ k is deter-
with E = expected value. mined it is possible to calculate the precision
The estimation variance can be derived from with which we know the various properties of
the semi-variogram: the deposit we are investigating, by obtaining
confidence limits (σ e ) for critical parameters.
2γ (h) = E(z (i) − z (i+h) ) 2 Knudsen and Kim (1978) have shown that the
errors have a normal distribution. This allows
the 95% confidence limit, ±2(σ e ), to be calcu-
By rewriting this equation in terms of the lated (sections 8.3.1 & 10.1):
semi-variogram, as described by Journel and
Huijbregts (1978), the estimation variance can 2
then be calculated. σ = σ e
e
Knudsen (1988) explained that this calcula-
tion takes into account the main factors affect- 95% confidence limit = Z* ± 2(σ e ).
ing the reliability of an estimate. It measures
the closeness of the samples to the grid node Estimation variance and confidence limit are
being estimated. As the samples get farther extremely useful for estimating the reliability
from the grid node, this term increases, and of a series of block estimates or a set of con-
the magnitude of the estimation variance will tours (Whateley 1991). An acceptable confid-
increase. It also takes into account the size of ence limit is set and areas that fall outside this
the block being estimated. As the size of the parameter are considered unreliable and should
block increases the estimation variance will have additional sample data collected. Thus
decrease. In addition it measures the spatial the optimal location of additional holes is
relationship of the samples to each other. If determined. The confidence limits for critical
the samples are clustered then the reliability parameters such as Au grade or ash content
of the estimate will decrease. The variability of coal can be obtained. Areas which have
of the data also affects the reliability of the unacceptable variations may require additional
estimate, but since the continuity (C) is meas- drilling, or in a mining situation it may lead
ured by the semi-variogram this factor is to a change in the mine plan or a change in the
already taken into account. The estimation storage and blending strategy.