Geoscience Reference
In-Depth Information
8.4
Block Kriging
based on practical considerations, and will be different for
different geologic and estimation domains.
Some of the factors that influence the decision on neigh-
borhood size and geometry include: (1) The influence of far
away data is screened or diminished by the closer data; (2)
the variogram values at long distances are derived from the
model γ ( h ), not from the data; (3) if the number of data n
chosen is large, the covariance matrices used to obtain the
kriging weights are very large, and the weights themselves
will be small and very similar for most data values; and (4)
there are some cases when solving the kriging system may
pose computational problems. This may be the case if clus-
tering is present, the variogram model is very continuous,
and the nugget effect is very small or 0; or when kriging on
a string (Deutsch 1994 , 1996 ).
The maximum search radii should be the limit of reli-
ability and effectiveness of the variogram model, not just
its range; recall that zonal anisotropy models can have very
long ranges. It should also be related to data density. The
distance may be anisotropic, following the anisotropy ob-
served in the variogram model. If each nested structure of the
variogram model is allowed to have different anisotropies,
then search neighborhoods can be customized for different
kriging passes.
Often the neighborhood is split into sectors (quadrants
or octants), with only the nearest data being retained within
each sector. This reduces the effect of clustered data, aiding
the effect obtained from the term
Most variables estimated or simulated in mining average
linearly. This allows to use the linear kriging estimator to
obtain directly linear averages of the variable z ( u ). Some of
the most common examples are kriged block estimates of
grades. Section 8.2 above already discussed estimated values
on a different support than the original samples.
Consider the estimation of a block average, defined as:
N
1
1
u uu u
'
'
'
z
()
=
z
( )
d
z
( )
v
j
|
V
|
N
j
=
1
Vu
()
where V ( u ) represent the block or panel centered at u , and
the
'
u 's are the N points used to discretize the volume V .
Block kriging then amounts to estimating the individual
discretization points z ( u j ') and then average them to obtain
the block value z v ( u ). The “block kriging” system applies a
different set of covariance values: the right-hand-side point-
to-point covariance values are replaced by averages (point-
to-block) covariance values of the form:
CV
(
( ),
uu
)
α
N
1
1
'
'
'
=
d
u uuu
C
(
)
d
C
(
u u
)
j
α
|
v
||
v
|
N
α
β
j
=
1
vu
()
vu
()
α
β
Cv v
(, )
i
in the covari-
j
ance matrix.
Another important decision is the minimum and maxi-
mum number of samples to be used in the estimation. Also,
the related decisions regarding the minimum of drill holes
and the number of samples per quadrant or octant to use.
There is a direct relation between the number of samples
used and the conditional bias vs. recoverable resource ac-
curacy debate.
Other implementation decisions include the use of hard
and soft boundaries among estimation domains, the use of
multiple kriging passes or not, the number of discretization
nodes if doing block kriging, and the use of different data
sources, including secondary data.
All these parameters can be modified to a certain extent to
obtain a resource model that achieves specific goals. Ideally,
the process of setting up the kriging plans becomes iterative
because some kind of calibration procedure is being used.
The type of calibration that can be used depends on whether
the mineral deposit being estimated is in production or not.
Comparisons with prior production are best, assuming
that the data that reflects production is indeed reliable. These
may be blast hole data in open pits; stope definition or grade
control data in underground mines; or grades and tonnages
fed to the mill. But in all cases, the reliability of production
data should be demonstrable, since the resource estimation
Two major mistakes are sometimes made with respect to
block kriging. The first is to estimate using block kriging
variables that do not average linearly. Most of these are ei-
ther geometallurgical or geotechnical variables (Chap. 5), for
which a different estimation strategy is needed.
The second is to incorrectly apply non-linear transforms.
One common example is log-normal kriging: the average of
log-transforms is not the log-transform of the average of the
z ( u ')'s. Thus, the anti-log of a block estimate is not a kriged
estimate, and is a biased estimate of the block value z v ( u ).
Another example is indicator kriging (Chap 9).
8.5
Kriging Plans
The kriging plan mostly determines the quality of the grade
estimate. The strategy used in the estimation of the variables
has a very significant impact on the final estimates. Often
is more significant than the actual kriging method chosen,
which is the case for those kriging variants that use neigh-
borhoods for local estimation (Boyle 2010 ; Rivoirard 1987 ;
Vann et al. 2003 ).
There are several variables and parameters that constitute
the kriging plan. The kriging neighborhood itself is chosen
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