Discussion - Grade Estimation And Its Precision In Mineral Resources: The Jackknife Approach - Technical Papers, Mining Engineering Vol. 48, No.2, pp. 84-88 - Adisoma, G.S., Hester, M.G.

Discussion by D.G. Krige I read the paper "The jackknife approach" with interest. There appears to be a widespread misunderstanding of the kriging variance as a measure of the error variance of the kriged grade estimate of an ore block. Virtually all mineral deposits display skew frequency distributions and are subject to the well-known proportional effect. Thus, it is usual to find in an ore body, when subdivided a really into subpopulations, that the variances of the grades (or the total sills of the variograms) for the subpopulations will be correlated with the corresponding mean grades (usually linearly with the squares of the mean grades). Thus, the variogram will be station. When this feature is evident, the relative variograms for the subpopulations (i.e., with variogram values divided by the relevant mean squared) will be similar, or they could even be accepted at an averaged model for the whole ore body with stationarity. Using this approach, the kriging variance will reflect the relative error variance that is to be multiplied by the kriged block grade estimate squared to yield the usual kriging variance for purposes of confidential limits. The objection that the kriging variance does not take into account the local grades used for a block valuation then falls away. In the example given by the authors, the global mean grade is not recorded. But the global variance, taken at the value of the total sill, is 0.005, whereas the variance of the 11 values used in the example is 0.000152. The two variances differ by a factor of nearly six. If the relative variogram approach is applicable in this case, the need for a jackknife technique will fall away. Also note that, for skew distributions, the confidence limits for ore-block estimates will be skew as well, and symmetrical normal limits will not apply. RepIy by G.S. Adisoma and M.G. Hester There are a few interesting ways to utilize the block-by-block kriging variance that the authors have encountered. One of them is to classify a particular block into a proven-probable reserve or a possible-resource category, using a somewhat subjectively determined kriging variance value. Here, the kriging variance still represents the drilling density, i.e., data configuration more than the data values. Professor Krige mentions another ingenious approach through the use of relative variograms. This will yield a block kriging variance that takes into account local (block) grade, as he correctly points out, for purposes of confidence limits. However, we are more interested in quantifying the uncertainties associated with grade estimation in a much larger scale than a block. e.g., for a mining phase or the minable reserve. The jackknife, through the block kriging shortcut, provides a simple solution to obtain a single estimation variance for a general shape. The authors' main reference discusses different ways to handle skewed distributions typical in mineral resource estimation, e.g., the weighted jackknife approach or data transformation. Also, as with any technique, we can always improve our estimate by subdividing into smaller populations that are geologically and statistically separate domains. As we have mentioned; the jackknife is but a tool in our collection of tools to solve estimation problems. The jackknife's appeals are its simplicity and versatility. It is handy when no other solution is available, or when the available solution is not practical. It is used with other methods, not to replace them altogether. ?