Kateri Guertin

A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.

Estimation and Mapping of Rank Related Uniform Transforms of Ion Deposition from Acid Precipitation

In the context of estimation of the spatial distribution of ion deposition from acid precipitation, it is interesting to compare patterns of deposition of various pollutant ions over different periods of time. A simple approach to produce comparable maps of such deposition patterns consists in estimating and mapping the dimensionless rank related uniform transform U of each studied ion deposition phenomenon D. In this paper, the geostatistics of such uniform transforms are presented and a uniformly distributed estimator U** is defined and characterized in terms of estimation variance. The performance of U** as an estimator of U is then illustrated through an application based on conditionally simulated deposition values and involving a regularly spaced data network. Two additional examples show how U** maps allow the identification and comparison of zones of extreme deposition for various pollutant ions and over time.

The choice of working variables in the geostatistical estimation of the spatial distribution of ion concentration from acid precipitation

Abstract Ordinary kriging presents an optimal tool to estimate the spatial distribution of ion concentration from acid precipitation and produce risk-qualified maps of the phenomenon over the studied area. However, being expressed as a linear combination of data values, the kriging estimator should not be calculated directly from ion concentration values unless their corresponding precipitation volumes are essentially constant over the area. In theory, spatial linear combinations of volume-weighted concentrations should be considered, meaning that concentration estimates should be obtained by dividing kriged depositions by their corresponding kriged precipitation totals. In practice, the choice of appropriate working variables for the estimation of ion concentration will depend on the nature of the studied phenomena and on the objective of the study. In this paper, the effect of this choice on estimation results is illustrated through a comparison of experimental results obtained from the direct kriging of concentration and from the estimation of concentration by quotient. It appears that even with fairly constant precipitation totals with respect to ion concentrations and given a small correlation between both variables, discrepancies between results from the two approaches can be significant, especially at a local scale: the spatial distribution of estimated values varies locally according to each procedure, and the level of uncertainty is systematically better predicted through an estimation by quotient, even though it is based on an approximate formula. In order to avoid misleading results, the direct kriging of ion concentration should be applied only for global estimation purposes after its practical equivalence with the estimation by quotient has been appraised.

Practice of geostatistics in the estimation of ion deposition from acid precipitation in the presence of sparse data networks

The estimation of the spatial variability of ion deposition from acid precipitation is commonly faced with the problem of sparse data networks sampled over a short period of time. In such cases, geostatistical techniques are pertinent as long as they are properly applied. According to some practical examples, the use of ordinary kriging 1 given an experimental semi-variogram function based on scattered data values can be misleading in terms of predicted estimation variances; however, the choice of additional sampling stations based on such a semi-variogram remains valid. It is also preferable to work with regularly spaced data values that allow the identification of preferential directional variabilities even from a small number of data points. In order to predict the performance of kriging, the use of semi-variogram cross-validation techniques in the presence of small data sets can be misleading and is not recommended. Finally, the integration of additional information from denser precipitation networks through the cokriging technique is questionable when based on a very small number of concomitant deposition and precipitation data values.