Discussion - Blasthole Sample - A Source Of Bias? - Knudsen, H. Peter

Discussion by G.F. Raymond Knudsen's study presents two curious conclusions: • The kriging of blasthole assays can systematically overstate mill head grades by as much as 21% as a result of unbiased sample variance. • No estimation method is able to reduce this bias to a small margin. The study is based on a simulation using real data and what are presumed to be actual variogram parameters from a real deposit. Although I have no doubt that the first conclusion (21% overestimation) is correct for the author's simulation, I do not believe this represents a realistic mining situation. Over the past 15 years I have done extensive comparisons between exploration drill-hole assays, blasthole assays and mill head grades on seven major open-pit mines, including some very erratic gold deposits. Commonly, nugget effects on blasthole variograms were 10% - 20% higher than on exploration variograms. And in one extreme case, the difference was 50%. Even in the extreme case, ordinary kriging on blastholes agreed well with the mill head grade over the long term. In Knudsen's simulation, the blasthole nugget effect is assumed to be 200% higher than exploration data's. He supports this by variogram plots from each. My guess is that the apparent, large difference between these variograms results from a failure to account for the proportional effect (blasthole assays are likely from a higher-grade area). A simple check would be a comparison of the variance of exploration samples nearest blastholes. As for a nearly conditionally-unbiased estimator of a large random error, the arithmetic mean of all of the data certainly qualifies, provided there is an even data spacing. As a corollary, so does simple kriging, which would include, in this case, a large weighting to the arithmetic mean. Similarly, using a large number of samples with ordinary kriging or indicator kriging would significantly reduce the bias in the case of a large nugget effect for the variogram.[ ] Reply by H. Peter Knudsen Raymond questions two conclusions in the paper. First, he wonders whether a 21 % overestimation represents a realistic mining situation. I agree that 21 % is high, but overestimation in the range of 10% to 15% is certainly common in my experience. Furthermore, several years ago I consulted on a gold mine that was experiencing a 45% overestimation due, predominantly, to poor blasthole samples: In further questioning of the 21 % value, Raymond wonders whether the nugget effect of the blastholes is really so much larger than the nugget values of the exploration data. It is consistently larger throughout the deposit. In my experience with six Nevada gold mines, the high nugget value is not unusual. In fact, for some reason, nugget values for blasthole samples are typically about 0.0005 (opt squared). I am of the opinion that this high nugget effect observed at many gold mines is predominantly due to the inherent inadequacies of the blasthole sample and subsequent sample preparation. The second conclusion Raymond questions is the inability of the estimators tested to reduce the conditional bias. In fact, the conditional bias is extreme with the polygon estimator and greatly reduced by ordinary kriging. However, it was not eliminated. Raymond suggests using simple kriging, or perhaps a larger number of samples, to reduce the conditional bias. The technique of simple kriging may be less affected by the random errors in the data, but I did not test the technique. Using a larger amount of data presupposes that too few samples were used initially. In ordinary kriging and indicator kriging, the screen effect comes into play and ensures that samples beyond the second screen are given zero weight. Hence, increasing the sample size does not change the estimates nor the conditional bias. The main point of my paper is that the random unbiased errors (a fact of life in blasthole samples) cause a conditional bias in our estimates. The mechanics by which the conditional bias is introduced are nicely explained by Springett. My paper simply shows that the bias is also present when working with linear estimators, such as ordinary kriging, and even with nonlinear estimators, such as indicator kriging.[ ]