Quantitative analysis of detrital modes: statistically rigorous confidence regions in ternary diagrams and their use in sedimentary petrology

Detrital modes are the basic type of quantitative data used in sedimentary petrology. They represent estimates of the petrographic framework composition of sand or sandstone obtained by point counting of thin sections. At present, geological inferences from analysis of these numerical data are semi-quantitative only, because many of the data-processing methods employed in sedimentary petrology lack a firm theoretical basis. A review of existing methods reveals a number of fundamental statistical problems associated with the use of univariate statistics and the construction of so-called hexagonal fields of compositional variation. It is shown that these problems can be overcome by using multivariate methods that honour the non-negativity and unit-sum constraints on compositional data, and incorporate the covariance structure. Spread in compositional data can be modelled on two levels: (1) by regarding the grain as unit of observation, and the total point count as a sample from a multinomial distribution; (2) by regarding the total count as unit of observation, and a set of counts as a sample from an additive logistic normal distribution. Numerical methods are presented for constructing statistically rigorous confidence regions in ternary diagrams by inversion of significance tests for these two distributions. Statistical analyses of the framework composition and heavy-mineral assemblages of modern beach and river sands illustrate their use. The case histories are followed by a brief overview of popular multivariate methods for reconstructing sediment provenance.