"Geostatistical
simulation for realistic mine planning", Mining Príbram
Symposium in Science and Technology,
Dr. Isobel Clark FSS FIMA FIMM FSAIMM CMath CEng
Geostokos Limited, Alloa Business Centre,
Geostatistical
modelling for realistic mine planning
Introduction
This paper is a brief discussion of problems which arose during the evaluation of a Zinc project
in
Problems arose when a routine statistical
analysis of borehole samples within the same fault zone and hosted in the same
rock type were found to exhibit evidence of multiple populations. Narrowing
down the study area failed to remove this behaviour.
Statistical
Analysis
Extensive statistical analysis was carried out to pin down the complexities evident in the
sample data. The data files were broken down by rock type and by fault block.
Histograms were produced for all areas and all rock
types. Where too few samples were available in a single rock type, a histogram was simply drawn for all fault blocks combined. Since the
fault seem to produce only movement between the and do
not (apparently) affect the mineralisation, this is a valid statistical (but
not geostatistical) approach.
Representative histograms for all of the
rock types and all faults were studied and the results
are summarised here. It was apparent from all of the
histograms that there were complexities in the mineralisation
which had not been coded into the geological data. Histograms and
probability plots suggest the presence of at least three major mineralisation
phases – possibly three hydrothermal events.
In an attempt to make sense of the
distribution of sample values, smaller and smaller blocks of ground were inspected within the major fault blocks. The histograms
for all areas revealed two or three components in varying proportions. Blocks
of ground were selected for detailed analysis to
identify:
·
how many components
were actually present and
·
the characteristics of
those component distributions
Logarithmic (natural) values were used throughout the analyses. Only arkose samples were used, to avoid any further geological complexities.
All arkose samples in the central fault
block were selected. The area was
then divided into east/west strips 200m wide in the north/south
direction. Histograms were constructed for each 200m
strip. Multi-component lognormal models were fitted to
the histograms.
It is clear from these
histograms that there are at least three component. However, it is unclear
whether the variations from strip to strip are due to geological changes or
simply the differing coverage of the samples. For example, drilling is
shallower at the north and south of the block and
deepest in the middle.
Of the seven strips analysed, the centre
strip was selected for further detailed analysis. This
volume contains the deepest drilling and most complete coverage of.
To obtain enough samples to produce
histograms, overlapping layers were taken at 40m
intervals within this volume. Towards the surface more
samples are available, so that the top 40m was split into separate 20m layers.
Histograms were produced for each sub-volume and
multi-component models fitted to these graphs.
This analysis showed the presence of two
to four components in all layers. Manipulation of the components reveals the
following pattern:
·
A low grade, extremely
variable, component exists at all levels. The percentage contribution of this
component ranges from over 18% at depth up to 64% in the
centre of the volume. The percentage then falls off to 9% in the upper
20m closest to surface.
·
A medium grade, fairly consistent, component exists at all levels up to the
top 20m. Above 520m the grade begins to drop sharply
to non-economic at surface. The percentage of samples in this component is very
high in the sub-volumes to 520m and drops sharply to below 20% above this
level. It is quite possible that the component exists in the top 20m at such a
low percentage that it has not been identified by the
statistical analysis.
·
A high
grade component is seen when considering samples above 520m. Average
grades at this depth are around very high at depth, dropping swiftly to
moderately low in the top 20m of the deposit. The percentage of samples
contributing to this component rises from 10% at depth to almost 80% close to
the surface.
·
A fourth component is only seen in the top 20m of this volume. This may be (for
example) some sort of supergene enrichment in the
oxide zone. With an extremely high average value, this component accounts for
almost 30% of the samples.
The four components overlap one another to a great extent. It is difficult to pick
"discriminator" values which would give
reliable separation between the components from the histogram graphs. It is
also not possible to use standard geostatistical methods in a deposit in which
the character of the distributions varies so widely between volumes so close
together.
Geostatistical
Analysis
The statistical analysis reveals the
presence of three --- or possibly four --- phases of mineralisation within the
main body of the deposit. It is not possible to separate these component
'populations' statistically. However, there is a recognisable
break point between the bulk of the 'background' low grade
component and the rest of the values, which allows Ordinary Kriging to be used
as a stable estimation technique. The estimation method proposed, therefore,
was a three part approach:
1.
An indicator kriging is used to evaluate the
proportion of the block likely to be above or below the breakpoint value;
2.
ordinary kriging is used to predict the likely value
for the "low grade" mineral;
3.
ordinary kriging is used to predict the likely value
for the "high grade" mineral.
These three values are
combined to produce an estimated grade for each block. There is no
mathematically valid way to evaluate the "kriging standard error" for
the final estimate. It is generally recommended that
the standard error for the most variable of the three components be used as a
conservative measure of reliability for the estimated value. This is most often
the "high grade" component on value.
Three separate geostatistical analyses were carried out. The first exercise was to find indicator
"break-points" which would separate the heterogeneous histograms into
quasi-Normal components so that ordinary kriging could be used for block
estimation.
After detailed investigation, it was deduced that the best "break-point" value was
2%Zn. around half of the "low grade" population lies above this
value. However, the semi-variograms produced for this break-point show that
there is a natural geological cutoff at around 2%Zn. This is borne out by the
geological interpretation. The values above this cutoff consist of samples from
three (possibly four) different components. Whilst the means of these
components differ, the standard deviations tend to lie around 0.4 (in natural
logarithms) for all of the components. It is the variable low
grade component which complicates the geostatistical analysis, with its
very high logarithmic variance.
The data was then broken into two parts
at 2%Zn. Semi-variograms were constructed on the two
data sets, using natural logarithms. Models were fitted
to all three sets of semi-variograms.
Semi-variograms were calculated and
modelled for each of the component exercises.
Cross validation was
carried out for all models and found to be satisfactory. A cross
validation exercise was also carried out for the
combined estimate versus the original value of each sample. This was also found to be satisfactory. Trial runs were carried out using all samples as opposed to restricted
by rock type. These were not found to be satisfactory.
This is a firm indicator that the mineralisations are
strongly controlled by the host rock type.
Summary of study
Statistical and geostatistical analysis
of the borehole data reveals the presence of three phases of mineralisation
throughout the economically mineable area of the project
studied. There is, apparently, geological evidence for multiple events although
current discussions are divided between two totally
different genetic models. The picture is further complicated
by oxidation and apparent enrichment in the 20 metres immediately below
surface.
Estimation could be improved dramatically if efforts were made to identify the component geological populations. Grade control during production will be far more effective if the geological complexities are identified and incorporated into the estimation process.
Figures
Figure 1: Histogram of all borehole cores within the arkose rock type
Figure 2: Example of multi-component lognormal model fitted to arkose samples in the central 200 metres of the central fault block
Figure 3: Averages of lognormal components fitted to subsections of samples in the central band
Figure 4: Backtransfomed average values for lognormal components fitted to subsections of samples in the central band
Figure 5: Mixing percentages of the three/four component lognormals within the central band
Figure 6: cross validation statistics from three part indicator/lognormal kriging exercise --- estimated values versus actual borehole core values.
