8/3/2019 Pocket Geostatistics for Field Geologists
http://slidepdf.com/reader/full/pocket-geostatistics-for-field-geologists 1/4
PREPRINTNUMBER
84 -108
a
C
*
POCKET GEOSTATISTICS FOR FIELD GEOLOGISTS
Jean-Michel M. Rendu
Xewmont Services Limited
Danbury, Connecticut
Permission is hereby given to publish with appropriate acknowledgments,excerpts or summaries not to exceed one-fourth of the entire text of the paper,Permission to print in more extended form subsequent to publication by the Institute
must be obtained from the Executive Director of the Society of Mining Engineersof AIME.
I f and when this paper is published by the Society of Mining Engineers of AIME, i tmay embody certain changes made by agreement between the Technical Publications
Committee and the author, so that the form in which it appears here is not necessarilythat in which i t may be published later.
These preprints are available for sale. Mail orders to PREPRINTS, Society of MiningEngineers, Caller No. D, Littleton, Colorado 80127,
PREPRINT AVAILABILITY LIST IS PUBLISHED PERIODICALLY INMINING ENGINEERING
8/3/2019 Pocket Geostatistics for Field Geologists
http://slidepdf.com/reader/full/pocket-geostatistics-for-field-geologists 2/4
Introduction
Plost programs can process up to approximately
120 data points. The processing time varies
from a few minutes to a number of hours, depend-. .Increasingly, geologists use geostatistical
ing on the number of data points and the pro-
methods to evaluate drilling programs. The
gram being used. Both storage limitations and
complexity of the methods varies with the processing time should be taken into account
nature of the available sample theto decide whether or not the pocket geostatisti-
objective of the statistical- nalysis and thecal package should be used. In most instances
geologist's experience. Occasionally, estimat-analysis of data sets containing more than 50
ing the average sample grade is all that isdata points will be considered impractical.
required. In other instances a semivariogram
calculation may be considered. If sufficient
data is available, the geologist may wish to
estimate blocks or panels of complex shape,
using the kriging method of evaluation. Histor-
ically, the use of geostatistics by geologists
has been limited by a number of factors, most
significant being the difficulty in accessing
and using computer programs needed to complete
the necessary studies.
Geologists are often working in remote areas
from which electronic communication is unreli-
able, if not impossible. Even when access to
computer programs is technically possible, the
costs incurred in accessing and using these
programs are often considered prohibitive. In
many instances using the programs requires an
understanding of computer systems far beyond
that which a field geologist can justifiably
be interested in acquiring and maintaining.
These problems are resolved in part by
developing better software, and using hardware
which can be installed in exploration offices.
h recent trend is towards user-friendly inter-
Areas of Application
The pocket geostatistical software was
developed for the complete geostatistical
study of two dimensional data sets. It is
directly applicable to the detailed analysis
of vein or stratified deposits, where the
variables of interest are the deposit thickness,
the average grade between foot-wall and hang-
ing-wall, and possibly the metal accumulation.The accumulation is a measure of the quantity
of metal per area unit, and is defined as the
product of thickness by grade. The system is
also applicable to early analysis of large
deposits, when the 'information obtained from
the drill holes can be reduced to a thickness
mineralized and the corresponding metal accumula-
tion. The system has also been used successfully
to study blast hole assays, as well as geo-
chemical data.
Possible direct and indirect applications of
the pocket geostatistical software include the
following:
active programs installed on mini or micro com-
puters. This approach is likely to result in a Statistical Analysis:significantly improved working environment for
many geologists. Xevertheless, there will
remain situations where access to even these
d o n scaled geostatistical systems will be
impossible because of cost and limited trans-
portability.
In an effort to demonstrate thac these last
constraints to the use of geostatistics can be
removed, a complete suite of fully interactive
~eostatistical rograms has been developed for
a 'nand held programmsble calculator. Even
though new limitations are introduced by the
pocket size of the computer, these programs
have been used successfully on a number of
deposits. The questions which can be answeredby using pocket geostatistics, and the func-
tions which have been made available to answer
these questions are described below.
Hardware Configuration and Limitations
The pocket geostatistical package was devel-
oped on a Hewlett Packard HP41CV hand held com-
puter, to which were added a card reader, an
extended function module and two extended
memory modules. This hardware configuration was
chosen to produce a system which can be carried
in a coat pocket. If desired, a Hewlett Packard
printer-plotter can be added to the calculator,
with no significant reduction in portability.A11 ?roSrams have been rdritten to operate
either with or without a princer. The ruse of
rechargeable batteries is reconmended to supply
a reliable source of energy.
- Calculation of average grade with
confidence limits, for normally or
lognormally distributed values.- Calculation of proportion and average
grade of samples above cutoff grade.- Analysis of frequency distribution,
with possible determination of outliers
or mixed geologic or statistical en-
vironments.
Semivariogram:- Semivariogram calculation in user
specified directions, for normally or
lognormally distributed values.- Determination of trends, drifts, and
directions of ore chutes.- Determination of distance of influence
and optimization of drill hole location
and spacing.- Determination of nugget effect and
assessment of sampling and assaying
procedures.- Semivariogram modelling, and development
of a mathematical model to be used Sv
rhe reserve calculating programs.
?.eserve Evaluation:
- Evaluation of the averase value o i
blocks located on a regular 2rid, usin;
linear krigin;.- Evaluation of the average value of
panels of irresular shapes, usin? Linea-
krizin:.
8/3/2019 Pocket Geostatistics for Field Geologists
http://slidepdf.com/reader/full/pocket-geostatistics-for-field-geologists 3/4
- Estimation of the precision with which
reserves are estimated.- Estimation of the influence of drillhole location and spacing on the pre-
cision with which the reserves are
estimated.- Optimization of drill hole location.
and spacing.- Calculation of optimal kriging weights
to improve routine evaluation of
blocks using suboptimal weightedaveraging techniques.
Influence of Selectivity on Tonnages and
Grades: - Estimation of the variability in grade
which could be ex~ected f a specified
level of mining selectivity was
achieved.- Estimation oi the tonnage of material
and average grade which could be minedabove a given cut off grade, if a
specified level of mining selectivity
was achieved.
As is usually the case with geostatisticalanalysis of geological data, the programs do
not always give a final answer to the questions
which a geologist may have. Hore commonly, theprograms should be used to assist in the
analysis of the data, and to supply quantita-
tive information which must be interpreted,
and added to the geologist's experience and
knowledge to nake a rational decision.
Available Geostatistical Functions
The normal procedure for data analysis con-
sists of creating a drill hole data base,
statistically analyzing the drill hole values,
developing a semivariogram model and estimating
reserves, either for the deposit as a whole,
within specific panels or withia blocks on a
regular grid.
The geostatistical functions which are avail-
able to perform these analyses can be classified
as the following:- Xormal and lognormal statistical
analysis.- Semivariogram calculation and
modelling.
- alock or panel evaluation.- Calculation of grade - tonnage
relationship.
In addition, necessary support functions
are supplied, including data base management,
and optional printer and plotter displays.The pocket geostatistical system is composed
of 13 main programs, which are stored on
approximately 100 magnetic cards.
Data Base Xanagernent:
Programs are supplied for the creation,listing, and updating of drill hole data
bases. The file names are specified by the
user, as well as the number of samples and the
easting, northing and value of each sample.
If areas of irregular shape are estimated
using the panel kriging program, a polygon
data base must be created by the user contain-ing the easting and northing of the vertices
of the polygonal area defining the panel, The
name of the polygon file is user specified.When blocks are estimated by kriging, a
block file is automaticaliy created, containing
for each block the number of samples used to
estimate the block, the estimated value andthe variance of estimation.
All files can be listed and edited. Statis-
tics can be calculated on both drill hole files
and block files. File backups can be saved
on magnetic cards.
Statistical Analysis:
Normal and lognormal statistics can be
calculated on either drill hole files or
block files. If a normal option is specified,
the following statistics are obtained:- Number of samples.- Mean,variance and standard deviation.- Upper and lower confidence limits for
the mean,- Histogram, frequency distribution and
cumulative frequency distribution.
Iflognormal statistics are requested, a
third parameter additive constant can be speci-
fied. The following statistics are obtained:- Number of samples.- Logarithmic mean and variance.- Sichel's "tl' estimate of the mean.- Upper and lower confidence limits for
the mean.- Histogram, frequency distribution, and
cumulative frequency distribution.
The histogram is obtained by classifying thesample values into grade categories. The grade
limits are calculated on a linear of logarithmicscale, depending on the option specified. The
sample average grade is calculated within each
grade class, which can be used to analyze the
influence of cut off grades. When a printer is
available, che above statistics are printed and
a histogram plot is produced.
Calculation of Experimental Semivariograms:
Absolute, relative or logarithmic semivario-
grams can be calculated on drill hole files. In
addition to the name of the file to be processed,
the user must specify the azimuth of the direct-
ion to be considered, a tolerance angle, and
distance classes within which sample pairs willbe grouped. Within each distance interval the
following statistics are calculated:- Number of pairs of samples.- Average distance between samples.- Semivariogram value.
Other statistics which can be calculated at
the user's request include for each distance
interval:- Relative semivariogram.- Average grade of samples.- Average drift between samples.- Average direction of sample pairs.
If a printer is available, all statistics
will be printed and the ~emi~ariogramsill be
plotted.
Xodelling of Semivariogram:
A two diinensional semivariogram aodel can be
specified as the sum of three submodels:
8/3/2019 Pocket Geostatistics for Field Geologists
http://slidepdf.com/reader/full/pocket-geostatistics-for-field-geologists 4/4
- A nugget effect.- Two spherical submodels, each one hav-
ing its own sill, range, direction,
and ratio of anisotropy.
Once the model has been specified, the
value of the semivariogram can be calculated
in any direction, for any distance. If a
printer is available, the semivariogram will be
plotted. The scale of the plot can be chosen
to facilitate visual coniparison of the experi-
mental and modelled semivariograms.
Estimation of Block Values by Kriging:
A kriging program is available to estimate
blocks on a regular grid. The following para-
meters must be specified by the user:- Name of drill hole file.- hlinimum and maximum easting of the
area to be estimated.- Minimum and maximum northing of the
area to be estimated.- Block size in both the east-west and
north-south directions.- Semivariogram models. This can be the
sum of a nugget effect and two aniso-
tropic spherical submodels.- Xaximum search distance. An aniso-
tropic search can be used.- Xaximum number of samples to be con-
sidered for block evaluation.
In addition, the user must specify the
number of p'oints which should be considered
within each block for the discrete calculation
of block variances and covariances.
An output file is created containing the
number of samples used to estimate each block,the estimated block value and the estimated
variance. The file can be listed, printed or
edited. It can also be accessed by the statis-
tics program to obtain tonnages and average
grades within specified grade intervals.
If desired,additional statistics can be
listed after completion of the kriging program:- Dispersion variance of the true block
values, calculated from the semi-
variogram.- Covariances between the last estimated
block and the samples used to estimate
the block.- Sample values and optimal weight given
to each sample.- Lagrange multiplier.
Estimation of Panels of Irregular Shape:
A kriging program is available to estimate
the average value of panels of irregular shape
from sample values stored in a drill hole file.
The number of samples which can be used to esti-
mate a panel is limited, and analysis of an
entire deposit may require its division into a
number of panels to be estimated independently.
Each panel is specified by a polygon file which
must be created before using the panel kriging
program. The following parameters are request-
ed by the program:- Xame of the drill hole file.- Xame of the polygon file.- Semivariogram model.- A parameter specifying whether all the
samples on the drill hole file should
be considered, or only the samples
within the panel.
In addition, the user must specify a block
size to be used in the east-west and north-
south directions for discrete calculations
of polygon variance and covariances.
The displayed or printed output of the panel
kriging program is similar to that of the block
kriging program.
Calculation of the Grade - Tonnage Relation-
ship:
Provided simple but often satisfied as-
sumptions are made concerning the statistical
properties of selective mining units, it is
possible to estimate what will be the effect
on the reserves of changing mine selectivity
or cutoff grade. A program is available to
assist in this estimation. To specify the
properties of the deposit, the following para-
meters must be specified:- Total tonnage mineralized, assuming no
selectivity.- Average grade of entire deposit.- Whether the block values are normally
or lognormally distributed. If a
distribution is specified, a third
parameter can be added.- Dispersion variance of the selective
mining units.
The dispersion variance is a function of the
block size or the degree of selectivity with
which the deposit will be mined. It can be
estimated using the block kriging program.
Given the above parameters, the following
statistics can be calculated for any userspecified cut off grade:- Tonnage of ore material.- Average grade of ore material.- Metal content of ore material.
Conclusion
Clearly the use of a pocket calculator can-
not be advocated as the ultimate solution to
the geostatistical evaluation of mineral
deposits. The software presented here was
developed in an attempt to alleviate some of
the frustration often expressed by geologists
in their constant effort to use modern tech-
nology in an environment where access to this
technology is often only too limited. Earlyin the development of this software it was
decided that the mathematical integrity of
all calculations should not be compromised.
The cost of this decision is found in the
slow speed of execution of some of the programs.
However, the results obtained are at least as
accurate as those expected from more powerful
software - hardware combinations. Constant
evolution in hardware technology and the develop-
ment of faster processors should result in the
timely elimination of all constraints to the
field utilization of geostatistics.