Distribution law for mineral and chemical constituent fractions in rocks and ores

Abstract

Mineral and chemical constituents are multiple in numbers (say, C-dimension with C > 3) and are estimated as positive random fractions in open interval (0, 1) of total weight/volume of measurement. Such data suffer from several drawbacks as regards their statistical analysis and inference and subsequent geological/mining applications. These defects can be listed as: (i) data are closed with spurious negative correlations among the constituents(that is data are not FULLRANK lacking unique Inverse Matrix), (ii) data are Binomial/Poisson for major/trace components where means are highly correlated to corresponding variances and Not Independent as is required for Gaussian density that is needed for conventional LINEAR statistical estimations, tests of hypotheses etc. and mean values are highly positively skewed for constituent fractions less than 0. 5 (1/C = 0. 33), (iii) variances are Not Homoscedastic over mean values as is required for Gaussian density for regression(correlation) studies, and (iv) data belong only to the interior points of C-dimensional complex excluding apices, hyper-edges, hyper-planes etc. These drawbacks can be eliminated by suitable pre-analysis nonlinear transformation, such as log (c (i)/ 1- c(i)) where c(i) is the fractional value of the ith constituent in the rock/ore sample that was petrographically /chemically measured. The advantages of Gaussian density of constituents are LINEAR additions of random variables remain Gaussian (or, data are CLOSED or have CONSTANT SUM under summations) and hence, their second order statistics (mean, covatiance/correlation) are the only Two Non-ZERO Parameters needed for statistical characterization, as all the higher cumulants are zeros. Since rocks/ores are heterogeneous solids, a suitable volume/weight, representative elementary volume (REV), must be sampled and chemically analyses so that the resulting data are consistent, stable, unbiased and reliable. The author and his students have been using a log(c/1-c)), or log (odds) having multiplicative errors(NOT additive errors as given in conventional probability theory), pretransformation for Gaussianising the geochemical data and their subsequent statistical inference since 1980s and have obtained very good geological inferences for several ores including, Au, BIF, Pb-Zn-Cu- Ag, Cu-Ag, P and associated single and multi-element mineralisations. A measure-theorectic proof of the applicability of this logarithmic pre-transform, (log (c/(1-c)), to obtaining Gaussian density is given. Applications of the pre-transform for Gaussianisation of fractional geochemical data for optimal grinding and beneficiation of ores, blending operations and processing to Marketable Grades, for exploration, anomaly detection, development & mine planning, and for obtaining Pathfinders of Single and Multipleelement ores are included here.