Revisiting a few popular discrimination diagrams

In this section, a few historically important and popular tectonic discrimination diagrams will be discussed. They are:

• Ti-V (Shervais, 1982)
• Ti-Zr (Pearce and Cann, 1973)
• Ti-Zr-Y (Pearce and Cann, 1973)
• Zr-Y-Nb (Meschede, 1986)
• Th-Ta-Hf (Wood, 1980)
• SiO$_2$-Al$_2$O$_3$-TiO$_2$-CaO-MgO-MnO-K$_2$O-Na$_2$O (Pearce, 1976, but without FeO)
• Ti, Zr, Y and Sr (Butler and Woronow, 1986)

The word ``discrimination diagram'' is used instead of ``discriminanant analysis'', because most of these diagrams are only loosely based on the principles of discriminant analysis outlined in Section 2 and the decision boundaries were drawn by eye. This section will revisit the combinations of elements used in these discrimination diagrams. An extensive dataset of 756 samples (Figure 10) was compiled from the PETDB and GEOROC databases (Lehnert et al., 2000). It contains:

• 256 Island arc basalts (IAB) from the Aeolian, Izu-Bonin, Kermadec, Kurile, Lesser Antilles, Mariana, Scotia and Tonga arcs.
• 241 Mid-ocean ridge (MORB) samples from the East Pacific rise, Mid Atlantic ridge, Indian ocean and Juan de Fuca ridge.
• 259 Ocean-island (OIB) samples from St. Helena, the Canary, Cape Verde, Caroline, Crozet, Hawaii-Emperor, Juan Fernandez, Marquesas, Mascarene, Samoan and Society islands.

All the training data had SiO$_2$ concentrations between 45 and 53%. Duplicate analyses were excluded from the database to avoid potential bias towards overrepresented samples. From this database, two sets of training data were generated:

• 11 major oxides (in weight percent): SiO$_2$, TiO$_2$, Al$_2$O$_3$, Fe$_2$O$_3$, FeO, CaO, MgO, MnO, K$_2$O, Na$_2$O and P$_2$O$_5$.
• 45 major, minor and trace elements (in ppm): Si, Ti, Al, Fe(III), Fe(II), Ca, Mg, Mn, K, Na, P, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Sc, V, Cr, Co, Ni, Cu, Zn, Ga, Rb, Sr, Y, Zr, Nb, Cs, Ba, Hf, Ta, Pb, Th and U.

The datasets are available as an electronic appendix. Not all samples were analysed for all the components. The dataset of major oxides is redundant, but a rescaling from % to ppm is avoided by treating it separately. Being admitted to the GEOROC and PETDB databases, it was assumed that the training data are reliable. Each datapoint in the electronic appendix is associated with a unique ID that allows the user to recover the original publication source. Different normalization procedures were used for different datasets, but this is unlikely to have major consequences for the discriminant analysis. So many data sources are mixed that at most, this mixing of normalization and laboratory procedures would have induced some additional random uncertainty, with only minor effects on the actual decision boundaries. Mixing different data sources and normalization procedures in the training data has the positive side-effect that the user is more or less free to use whichever normalization procedure (s)he wishes.

First, two simple bivariate discrimination diagrams will be discussed: the Ti-V diagram of Shervais (1982) and the Ti-Zr diagram of Pearce and Cann (1973). Many of the problems that plague the study of compositional data and were discussed in Section 3 are far less serious in the bivariate than the ternary case. Of course, Ti and V, or Ti and Zr are still subject to the (physical) constant-sum constraint, but considering they typically consitute less than a few percent of the total rock composition, a change in one element will have little effect on the remaining two when the raw measurement units are used on the axes of the bivariate discrimination diagrams. In contrast with this, all popular ternary discrimination diagrams have been rescaled to a (diagrammatic) constant sum of 100%, thus magnifying the effects of closure. For all of the following discriminant analyses, a uniform prior was used. Statistical analysis was done with a combination of Matlab $^{\copyright}$ and R (http://www.r-project.org).