Hi all
First off, to all our American colleagues, my condolences following Tuesdays
events.
Sorry this is so late, however I hope that it will be of some use. Note: I
have precede all of the replies, I can send the entire text if so desired
(some of the older messages are no longer available).
Some background, I work on a Witwatersrand gold mine, which has been in
operation for approximately 20 years. We mine a tabular ore body, which is
remarkably continuos and flat (max. dip of 25 degrees). Gold is the primary
mineral that is exploited, though uranium is also present (though in sub
economic concentrations).
I have a sampling data set of + 100 000 points. If I include the data from
the adjoining mines the size of the data set increases to + 300 000 points.
For comparison the size of the Carbon Leader data set (from all the gold
mines in the region) will be almost 1 000 000.
With this wealth of data, I was wondering why most mines sample the primary
development (on reef) at intervals of 2.5m and stopes on a regular 5x5m
grid. The information I turned up suggests that these are historical
hangovers (like the anecdote that explains why the dimensions used by the
vehicles used to transport the Space Shuttle, were determined by the width
of a horses ass back in Roman times).
It would appear that the panels in a stope average a 10m advance per month
and are 30m long (on average) suggesting a sampling grid that would be a
multiple of this mining unit. The next logical step would be to sample the
primary, on reef development at a sub division of the sampling grid used on
a day to day basis.
Obviously population variability (reef thickness and gold value) and
continuity are not taken into account in this practice. The question of what
the sample spacing should be, rather than what it currently is, is what I
wanted to know.
Here are the responses that I had to my questions. I am afraid this may be a
little long, so copying the text into e.g. Word and reformatting may be an
option.
Thanks a lot to Marcel and Jan-Willem, who gave me a lot of leads and things
to think about.
The questions that I had were as follows:
1. What is the optimum number of samples in a block of any particular
size?
2. Does anyone know of a good "idiots guide" to GSLIB?
3. Is there a good declustering program out there?
4. Does anyone know where I can get a good introductory text on
Probability kriging, and is there any software that has been designed to run
this?
I will list the responses in reverse order to the questions, as 1.
Elicited the most response.
Question 4.
5. Does anyone know where I can get a good introductory text on
Probability kriging, and is there any software that has been designed to run
this?
As Isobel pointed out, probability Kriging is covered in GSLIB (and
is referenced in the GSLIB manual on page 87. It is commonly known as
Multiple Indicator Kriging (MIK).
(Ref: Deutsch and Journel, GSLIB: Geostatistical Software Library
and User's guide, Oxford University Press, 1998)
Apcom 87 has an interesting paper by:
Kim, Zhao and Roditis, "Performance Comparison of Local Recoverable
Reserve Estimates Using Different Kriging Techniques, p 65-82.
They also have a useful set of references, esp.
Verly, The Multigaussian approach and its application to the
estimate of local reserves. J math. Geol. V 15, 1983.
The GSLIB Guide has a good reference section and includes e.g.
Isaaks, MSc, 1984, Stanford, " Risk qualified mappings for hazardous waste
sites", which apparently covers some of the aspects of MIK.
I also asked about software, GSLIB is the obvious one, Surpac
apparently does MIK as well (I have not investigated this. Datamine does not
have it as an option, though you could probably write a macro to do so.
Question 3
Is there a good declustering program out there?
Being obtuse, declustering is another term for regularization. GSLIB
does it using the "declus" sub routine.
This is the start of the whole "change of support issue". In my
case, for example, sampling is done every 2.5 m in primary development and
every 5m (5x5m regular grid) when stoping commences. It is a historical fact
that raise sampling on the VCR is treated with some degree of caution, 30%
of sub economic raises are found to be economically viable once ledging has
been commenced.
Thus in this case regularizing the data into a 5x5m grid would
suggest itself.
As Deutsch and Journel state "without a declustering program most
geostatistical studies would be flawed from the beginning." GSLIB p. 5
Question 2
Does anyone know of a good "idiots guide" to GSLIB?
This question may have been interpreted by some as being insulting;
it was not intended to be. There is a good introductory guide on the GSLIB
web site (can't remember it off hand, a search for GSLIB on Gopher.com will
take you there).
Question 1
6. What is the optimum number of samples in a block of any particular
size?
This question got the most response. It would appear that this issue is not
one that is covered in much detail by earth/ life scientists.
A search of the Internet turned up quite a few sites, however they were all
under applied mathematics and statistics. (I will post these later).
Apcom 87 has two interesting papers specifically related to Witwatersrand
gold mining:
Krige, Estimating Grade differentials from drilling results on new gold
properties. PP 31- 41.
Gershon, Comparisons of Geostatistical Approaches for Drillhole Site
Selection. Pp 93-100.
Sampling:
Don Meyers replied
1. If nothing is known about the population distribution, then the
sampling grid/ method will need to be based on other criteria (e.g. cost)
2
. Even if there were no spatial correlation and one would only want to
estimate an average value for the block, you would still need the variance.
3. Sample locations, compute the sample variance and use this to
"predict" the sample size.
4. We need to remember that the data are being used for two different
things: a. to calculate the covariogram/ variogram. B to Krige. An optimal
sampling plan for one will not generally be the optimum for the other.
Jan Willen van Groenigen replied:
1. Sampling in geostatistics is paradoxical, the more you know about a
parameter, the better you can optimize a sampling scheme for it.
2. Webster and McBratey (refs follow) describe an algorithm for
calculating the optimum grid spacing for a sampling scheme, given the
maximum allowed krige variance and a variogram.
3. Jan Willem has developed a simulated annealing based algorithm that
optimizes the optimal location of individual sampling points rather than
optimal grid spacing. A semivariogram needs to be assumed.
4. The use of Krige variance as a measure of interpolation error is
controversial, as it does not take into account the value of a sample; just
it's location. This can give problems if the intrinsic hypothesis does not
hold.
5. What is the sampling objective... to describe spatial variability, to
optimize spatial interpolation and to detect "hot spots"?
Evan Englund replied
Optimum depends on your objective function and the distribution you are
sampling. (Ref to Englund paper under references).
Benjamin Warr replied
"Optimum" depends on the objective; clearly exhaustive sampling would
provide the best estimate, but nay not be the most cost effective.
Generally the size of the block/ grid is fixed, and the number of samples
that may be taken is also fixed (cost)
Often the best sampling method is systematic random or stratified random
sampling.
Beware of regular grids because of possible periodicity artifacts.
Graphs to plot: after Webster and Oliver:
Average sample spacing v's Krige error
Sample density vs krigging error for various block sizes.
The pay off between sample density and estimation precision/accuracy is a
function of the variogram. If the nugget is small you can expect greater
accuracy for the same sample density. If the range of the variogram is long
then the predictive capacity of each sample will be greater than if outside
of the range of spatial dependence.
Marcel Vallee replied:
Note: I have used almost all of Marcell's replies as is, as he replies from
a mining perspective. I have placed all references in the reference spot.
In my opinion, the final purpose of sampling is
Estimation leading to mining extraction that achieves
optimal metal recovery and minimal dilution. Estimating/
modeling of the variogram and kriging are interim
objectives. Geological mapping, interpretation and modeling
are also essential steps with whom sampling methods,
sampling grid dimensions and geostatistical aspects should
be integrated in the determination of global and local
continuity (Sinclair and Vallée, 1994).
The basic problem of sampling grid size should be viewed in
three dimensions. Our usual sampling grids are planned for
efficiency, using a stratified drilling pattern
perpendicular to the plane of apparent structural /
geological continuity. So far, so good!
However, once we have achieved a first delineation and can
calculate a semivariogram along the drill hole axis,
too often we neglect or forget to verify if these
continuity parameters apply are present in the other two
perpendicular directions.
This problem has been detected by Michel David and described
in a sampling paper titled "What happens if?" given at a
sampling symposium in Australia in 1976 where he describes
the problem and recommends sampling specifically designed
to verify continuity parameters in the second and third
dimensions. Sadly, there is only one sentence in
"Geostatistical Ore Reserve Estimation (which was already
in print at the time this paper was prepared and presented)
that refers to this problem (around page 200).
Journel and Huijbregts in "Mining Geostatistics" also
describe this problem (1978) and recommend a few simple
tools, for instance to lay out a cross of more closely
spaced drill holes within the main grid.
I consider, based on my mining experience, that sampling
of rock in place should be targeted stratified sampling,
not random sampling. Regionalized variables require
regionalized sampling
When the exploration or mining geologist halves the sampling
grid dimension, he/she is basically using a similar
strategy. I know the results of this effort are viewed by
geostatisticians as the "clustering" problem! For the mining
geologist, the objective is local estimation and close
determination of ore limits, whether they are grade contacts
or sharp contacts both for planning and extraction.
What is the ideal sampling grid dimension for the mining
geologist and the mining engineer? My answer: the one
that allows to plan and develop and extract the ore (from
stopes or open pit) efficiently as described above. I
understand the student/researcher cannot rely on similar
amounts of funds for drilling.
Another important consideration in the Davil paper and in
Geosatistical Ore Estimaion is that of sample preparation
and assaying quality control. Quoting from the paper"
" ... (frequently) it is the sanple preparation procedure
whick generates the nugget effect rather than the real
mineralization which generates the nugget effect"
[underlined in the text]. This subject makes up almost a
chapter of Geostatistical Ore Reserve Estimation.
A higher "induced" nugget effect, that is a higher sampling/
assaying variance of the values used for a selectin decision
will reduce the accuracy of LOCAL estimation and mine
selection.
What is the final grid dimension (the one we call measured
resource, proven reserve). This will vary depending on
orebody configuration and mining method selected. In a open
pit, with a fairly sizeable, not too complex and not too
nuggetty ore body, you generally can get away with a wider
grid for proven ore than yo can in the majority of cases
underground,
In an open pit you can sample more systematically,
from bench to bench than you can in most underground mine.
Also one has more flexibility for selection by adjusting
blast limits based on test drilling on the pit floor and
sampling of blast holes.
The problem of sampling and geostatistics is that
geostatistics, except for a few exceptions, has taken over
the indifferent, if I may say gently, sampling methods of a
majority of geologists and their frequent lack of concern
for the "measurable reliability" of sampling (W.E. Deming).
Here is a recap of my evolution, also quoting events of the
period and repeating some material from my first note.
Geostatistical Perspectives and Experiments
## In 1973, when drifing a drift for bulk sampling at
the Niobec deposit, 10 kg control samples were taken from
drift
rounds, 3-5 kg channels from drift walls, and diamond drill
holes placed in the walls in slashes to get samples all
along the drift (carbonative with disseminated pyrochlore,
20 to 50 mesh, 0.2% to 1.0% Nb2O5. I had this data
reviewed by Michel David and he found out that diamond
drilling semi-variograms were close to those of rounds,
but those from channel samples were more erratic. I don't
haveexact figures, as all the files were transferred to the
mine office in 1975 and they lost tract of them.
Doing the same now, I would be prescribing larger samples.
# The sampling contents of Geostatistical Ore Reserve
Estimation was based on verifications and tests I did in
1975 at SOQUEM, spurred by questions from Michel David
(as our geostat consultant), regarding possible causes
of 'anomalies' on the histograms of gold assay results
from the gold deposit that became the Doyon Mine
(east of Royn-Noranda). Incidentally this paper, given at
the 1976 CIM AGM, was reviewed by the Geological Society
for CIM Bulletin and turned down, "because all this is in
books on statistics."
# David 1976 sampling paper touched both sampling
methods and sampling strategy; I only found this
paper a year ago, after Michel David's death when reading
his full CV when preparing a formal submission by CIM for
his induction into the Canadian Mining Hall of Fame (he did
not get inducted in 2001, but we tried again this year for
the 2002 induction.
# Journel and Huijbrecths 1978 commented on sampling
strategy, pointed out the usefulness, or need in some cases,
of getting variograms in the three main dimensions of a
deposit; I noticed these comments about 5 or 6 years ago.
# Burn, R.G. 1979, Data reliability in ore reserve
asesessments. Mining Magazine, Oct. p. 289-299.
I think this is a major sampling paper that should be read
by everybody involved in mining, environment, agriculture,
and geostatistics, despite the fact it is not very
quantitative.
# A 1986 case study by Podolsky (INCO's Exploration V.P.)
was based on the underground and development and pilot plant
sampling at the Casa Berardi (Québec) gold deposit; this was
only published in the Proceedings of the CIM Symposium on
"Ore Reserve Estimation: Methods, Models and Reality."
# A 1986 sampling case study by Magri and McKeena (S.A.IMM)
of channel sampling and diamond saw sampling: "conventional
channel samples and <strictly supervised channels> showed
virtually the same semivariograms." However, "to achieve
the same estimation error with conventional channels as with
rock saw sampling at a sampling interval of 5 m, it would be
necessary to have a spacing interval of 2 m to 2.5 m with
chip sampling."
Similar observations were made by Podolsky (1986)when
comparing channel samples with round averages obtained from
35kg rock samples from all trucks load (Vallée 1992, 1998
Sampling QC).
I used a tungsten carbide saw to sample a deposit of
titaniferous magnetite near Chibougamau in 1982 (too hard
to chip); 20 years or so later, most exploration teams in
Québec use such a saw instead channel sampling.
More Recent Practical Applications
1992 - The Guide to the Evaluation of Gold Deposits has an
18 p. chapter on sampling that recommends more efforts for
representivity, for quality control and ends by reminding
the geologists and engineers involved of their professional
and
legal responsibilities.
1993 - I was involved, with Al Sinclair from UBC, in two
papers, at the Michel David Forum (Improved Sampling and
Data
Gathering for Improved Mineral Inventory and Production
Control)and at the APCOM XXIV meeting (Quality Management
Methods for reliable Estimations of Deposits and Reserves).
The second paper proposed a reference the quality plan
framework of the ISO 9000 quality assurance system:
1) targeted objectives,
2) responsibility and authority,
3) activities, methods and procedures,
4) verifications and audits,
5) & 6) feedback to quality system for continuous
improvement.
1997-1998 - I mentioned the CIM Symposium in my first note.
The paper Sampling Quality Control is an update of the
1992-1993 work also quoting from concurrent papers. It
presented a synthesis of what we should could call the
sampling process sensu lato:
sampling stragety
sample collection
sub-sampling/sample preparation
assaying
verification and feedback at all steps.
Among the other papers at the Symposium, I would
particularly recommend
Long's (Practical Quality Control Procedures . ..),
Sketchley's (Gold Deposits: Establishing Sampling Protocols
and Quality control),
François-Bongarçon's(Extensions to the demonstration of Gy's
formula) and,
Sinclair and Benzen (Evaluation of Errors in paired
analytical data by a linear model).
1997-2001 / Draft Standards
Recently, I have been expanding and rewriting the parts
dealing with geological interpretation, sampling,
interpretation (I have further developed the material in
Sampling Quality Control).
There are also updates to make regarding inventory methods
and feasibility requirements from another paper:
Vallée, 2000, Mineral resources + engineering, economic and
legal feasibility = Ore reserve. CIM Bulletin, V. 93, 1038
March,p.53-61.
To conclude, I do not always write such long emails, but
this is a core subject and I do not encounter frequently
people who take these problems at heart.
References
In no specific order!
(Thanks esp. to Marcel and Jan Willem for all the references)
Armstrong and Champigny (1989) " A study in mining small blocks", CIM
Bulletin, V82, 923, pp 128-133.
Van Groenigen, PhD.
Englund and Heravi, 1992, Conditional Simulation: Practical Application for
Sampling design optimization. In Gerostatistics Trioia '92. Soares ed.
Kluwer Academic Publishers, Dordreeht, pp 613-624. (Thanks to Wevan for
sending through a pdf copy of the paper).
Webster and McBratney (1980's) in the Journal of Soil Science
Yfantis et. Al. (1987) Efficeincy of Kriging Estimation for square,
triangular and Hexagonal grids. Mathematical Geology 19 (3), 183:205.
-Burgess, T.M. and Webster, R., 1984. Optimal sampling strategies for
mapping soil types I. Distribution of boundary spacings. Journal of Soil
Science, 35: 641-654.
-Burgess, T.M. and Webster, R., 1984. Optimal sampling strategies for
mapping soil types II. Risk functions and sampling intervals. Journal of
Soil Science, 35: 655-665.
-Burgess, T.M., Webster, R. and McBratney, A.B., 1981. Optimal
interpolation and isarithmic mapping of soil properties IV. Sampling
strategy. Journal of Soil Science, 32: 643-659.
-McBratney, A.B. and Webster, R., 1981. The design of optimal sampling
schemes for local estimation and mapping of regionalized variables II.
Program and examples. Computers and Geosciences, 7(4): 335-365.
-McBratney, A.B. and Webster, R., 1983. Optimal interpolation and
isarithmic mapping of soil properties V. Co-regionalization and multiple
sampling strategy. Journal of Soil Science, 34: 137-162.
-McBratney, A.B., Webster, R. and Burgess, T.M., 1981. The design of
optimal sampling schemes for local estimation and mapping of regionalized
variables I. Theory and methods. Computers and Geosciences, 7(4): 331-334.
-Webster, R. and Burgess, T.M., 1984. Sampling and bulking strategies for
estimating soil properties in small regions. Journal of Soil Science, 35:
127-140.
van Groenigen and Stein 1988, Constraining optomization of saptial sampling
using continious simulated annealing, J of Enviromnental Quality, 27 (5),
1078-1086
van Groenigen, Siderius and Stein, 1999, Constrained optimisation of soil
sampling for minimising the kriging varience, Geoderma, 87: 239-259
van Groenigen, Pieters and Stein, 2000, Optimising spatial sampling for
multivariate contamination in urban areas, Environmetrics, 11, 227-244
Journel and Huijbregts "Mining Geostatistics"
David, "Geostatistical Ore Reserve Estimation"
David M., 1976, What Happens If? A few remarks on Useful
Geostatistical Concepts for the Design of Sampling Patterns.
The Aus. I.M.M Melbourne Branch, Sampling Symposium, Sept.
1976, preprint of proceedings, 16 p.
Postolski, T. A., Sinclair, A. J. (1998) Geology as a
Basis for Refining Semivariogram Models for Porphyry-Type
Deposits. Exploration and Mining Geology, Vol. 7, Nº 1-2,
p. 45-50.
Vallée, M, Dagbert, M, & Côte, D. (1993) Quality control
requirements for more reliable mineral deposit and reserve
estimates. CIM Bulletin, vol. 86. No 969, p. 65-74.
Sinclair, A.J. and Vallée, M. (1994) Reviewing continuity:
An essential element of quality control for deposit and
reserve estimation. Exploration and Mining Geology, Vol.3,
Nº 2, pp. 95-108.
Vallée, M. (1998) Sampling Quality Control. Exploration and
Mining Geology, Vol, 7, Nº 1-2, p. 107-116. (
Vallée, M. (1992) Guide to the evaluation of gold deposits.
CIM Special Volume 45, Canadian Institute of Mining,
Metallurgy and Petroleum, Montréal, Canada, 299 p.
Vallée M. Draft Standards for Exploration and
Resource/Reserve Estimation.
The Volume 7, No 1-2 issue of Exploration and Mining Geology
is a special issue on "Quality Assurance, Continuous
Improvement and Standards in Resource Estimation.
Jeff Myer "Geostatiscal Error Management,"
Vallée, 2000, Mineral resources + engineering, economic and
legal feasibility = Ore reserve. CIM Bulletin, V. 93, 1038
March,p.53-61.
Websites:
http://agronomy.ucdavis.edu/groenigen
http://www.u.arizona.edu/~donaldm
Isobel's Kriging Game:
(as well as a lot of other interesting stuff )
http://uk.geocities.com/drisobelclarke/briefcase.html
In conclusion, thanks again to all those people who have replied to my
questions. Hope the replies posted here help.
To those whom I still owe stuff, I will get there eventually...
Mark Burnett
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