ABSTRACT

Monte Carlo simulation of large replications of 100,000 was used to evaluate the final total propagated error represented by standard deviation or percentage of relative standard deviation (%RSD) in conventional ternary diagrams, as well as in bivariate diagrams based on statistically coherent additive, centered, and isometric log-ratio transformations of three compositional variables. The error distortion, amplification and reduction, evidenced in a ternary diagram, were not observed in any of the bivariate diagrams based on log-ratio transformations, in which equal %RSD errors in the initial compositional variables showed equal standard deviation values in the transformed space. As an example of these findings, the three types of log-ratio transformations were evaluated from a geochemical database of basaltic rocks from arc (IAB), within-plate (WPB), and mid-ocean ridge (MORB) tectonic settings, as better, statistically coherent alternatives to the highly used Zr-3Y-Ti/100 ternary diagram. This diagram showed relatively low (<25%), acceptable (86%), and non-existent (0%) correct discrimination for the arc, within-plate, and mid-ocean ridge settings, respectively. The exact same bivariate diagram, with correct discrimination of 46.2%, 89.0%, and 60.6%, respectively, for IAB, WPB, and MORB was obtained from linear discriminant analysis performed on the additive, centered, and isometric log-ratio transformations of the same database. Nevertheless, the fulfillment of the basic assumption of statistical samples drawn from binormal distributions before linear discriminant analysis increased the correct discrimination in the new bivariate diagrams by about 0.5% to 1.5% for IAB, 1.1% to 7.3% for WPB, and –7.7% to +0.9% for MORB. The performance of the new bivariate diagrams was significantly better than that of the ternary discrimination diagram. Thus, this work highlights the successful replacement of ternary diagrams by bivariate diagrams based on log-ratio transformations. Therefore, the bivariate diagram based on normally distributed isometric log-ratio data can be recommended to replace the widely used Zr-3Y-Ti/100 ternary diagram.