A study on kriging small blocks

"In order to define the ore/waste boundaries as finely as possible, mine planners often insist on kriging very small blocks (relative to the spacing of the available sample grades). These kriged estimates for very small blocks are over-smoothed. For several simple sampling patterns the relation between the block size and the variogram range, and the kriging results were studied. Meaningful estimates of individual blocks are obtained when the variogram range is large compared to the block size and the sample spacing. For a variogram range less than half the sample spacing, the kriged block estimates were found to be uncorrelated with the actual grades. The over-smoothed estimates should not be used for calculating recoverable reserves IntroductionNowadays most mining companies know that the geostatistical estimation method, kriging, gives the best estimates of individual block grades. As most mining engineers and geologists are now familiar with the basics of geostatistics, the definitions of the variogram and of kriging will not be repeated here. An appendix containing definitions and mathematical formulae is given at the end of the article for the sake of completeness.Even though it is well known that the kriged estimates for very small blocks are over-smoothed, mine planners who want to define the ore/waste limits as finely as possible (especially for selective underground mines), are often tempted to krige very small blocks. So we ask how small is ""too small""? Clearly ""smallness"" cannot be expressed in yards or metres. It depends on the nature of the orebody under study and this is reflected in the variogram model chosen to describe it.To answer this question, the changes in four key parameters are studied:• the kriging variance,• the variance of the estimated grade,• the slope of the linear regression of the true but unknown grade on its estimated value, and• the coefficient of correlation between the actual grade and the estimate;as a function of the block size and the variogram range. (Definitions of these four parameters are given in the appendix). Ideally the kriging variance should be low while the variance of the estimated grade should be equal to the dispersion variance of the blocks. The slope should be l.0, and so should the correlation coefficient.These four parameters have been calculated for several simple 2-D examples. The effect of using kriged block estimates that are too heavily smoothed are shown, and the dangers of trying to use these estimates for predicting recoverable reserves for small blocks (i.e. selective mining units) are pointed out."