Approaches to the Simulation of Compositional Data: A Nickel-Laterite comparative case study

An accurate prediction of benefit for some types of ore may require not just the ore grade, but a whole compositional characterization of the estimates, that is, its full mineral composition, including waste composition and nuisance elements. An example is Nickel laterite ores, where Ni/Co content must be complemented by estimates of many other elements, as these might have a notable impact on ore processing.
Estimates are often obtained with (co)kriging or co-simulation. However, geostatistics applied to compositional data may yield spurious and inconsistent results because of the constant sum constraint. The log-ratio approach avoids such problems. Like anamorphosis, it proposes a three-step procedure: (1) data are mapped to a set of log ratios of components, for example the additive log-ratio transformation (alr); (2) transformed scores are modelled with an appropriate fully multivariate geostatistical toolbox (e.g. direct/cross-variograms and cokriging or co-simulation); and (3) results are back-transformed to the original units. Some practical aspects may make the application of this technique difficult: the assumption of Gaussianity and the high dimensionality of the compositions combined with the need for using multivariate methods. This contribution compares several ways of treating compositional data (combining anamorphosisnormal score transformation, logratio transformations, and minimum maximummax autocorrelation decomposition) with respect to their ability to generate sensible simulations. The various approaches considered are illustrated with a Nickel laterite data set for which 10 variables are available. Both theoretical considerations and illustration results suggest that the best combination is (log)ratio-anamorphosisnormal score transform-MAF, in this order. The factors so obtained are closer to Gaussian, approximately spatially decorrelated and can be simulated independently. Simulations can later be recombined to alr-simulations, which in turn may be converted back to point compositions. The resulting simulated compositions are compared with a full Gaussian co-simulation of the raw data. The logratio methodsy show a reasonable to good reproduction of mean values and distribution of the data set and by construction honour the total sum constraint, in contrast to the simulations based on the raw data for which the total sums fluctuate strongly.