A Few Words About Geostatistics - Geovariances

A Few Words About Geostatistics(Article published in the Mining Equipment Review)J. Deraisme (Geovariances)Nowadays geostatistics can't be ignored when dealing with the reserves evaluation. It is nowof common use in a large number of companies from the exploration to the production stages.The time when only variograms and kriging were accessible is over. Practitioners requiremore to assess reserves in relation with cut-off grades. The so called effects of support, andconstraints are no longer mysterious, and the knowledge of geostatistical solutions is notshared by the scientific community only.Support EffectAny mining engineer knows that recovered grades are lower when selectivity is poor, in otherwords the bigger the mining units the lower the grades.The average grade of a huge block of several thousands of cubic meters can be considered asthe average of the grades of smaller blocks of a few cubic meters contained in the big block.The distribution of grades of huge blocks is obviously less scattered than that of the smallsamples.The only grades that are known experimentally are those of the samples; in order to forecastthe distribution of grades for blocks of different dimensions, geostatistics provides models ofchange of support, which are based on the experimental histogram of sample grades as well astheir spatial correlation through the variogram.Fig. 1: Support effect

Information EffectDuring the exploitation of the mine, the true grade of the mining units is still unknown.Therefore the decision to send the mining unit to the ore plant or to the waste dump is stilltaken from the estimated grade and not from the true grade. As a consequence it is notpossible to avoid sending blocks to the wrong destination: rich mining units will end up on thewaste dump, because they are estimated as "poor", while poor mining units will feed the oreplant, as illustrated in figure 2.Fig. 2: This cross-plot of estimated grades vs true grades of the mining units illustratesthe information effectNon linear geostatistical methods such as Disjunctive Kriging can quantify the amount of lossof selectivity due to the support and effect. These techniques allow to dimension an adequatesampling mesh for pre-mining boreholes.Technical constraints imposed by the exploitation method add even more penalties to therecovery of the ore material. It may prove necessary to leave some rich parts of the orebody inplace, if the technical infrastructures involved would cost too much -e.g in undergroundmining- or if the stripping ratio is much too high -e.g in open pit mining-.

Moreover the minimum opening width of stopes as well as blasts are responsible for dilutionthat again lower the grades and the selectivity.When technical constraints are simple, e.g stability slopes in open pit, footwall or hangingwall so that the whole seam is extracted without further vertical selectivity, a technicalparameterization of reserves can be applied to optimize the ore recovery.When constraints are more complex, it is necessary to use a realistic numerical model of theorebody that reproduces a spatial distribution of grades similar to the true distribution. Thismodel built through geostatistical conditional simulations can then be used to estimaterecoverable reserves.With the development of computer technology, these techniques become accessible with areduction of the effort to be made by the user. The difficulty that remains relies essentially onbasic notions to be examined with common sense: What are the most appropriate variables,how to introduce geological parameters in the geostatistical approach, what are the hypothesesthat can be reasonably made ?It is a great advantage to access in a package to detailed spatial analysis together with thegeostatistical processing from the simplest one like kriging to the most advanced one likenon-linear techniques or simulations.