Quantitative analysis of detrital modes: statistically rigorous confidence regions in ternary diagrams and their use in sedimentary petrology
Introduction
Quantitative estimation of detrital modes of sand and sandstone is traditionally performed by point counting. 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. Most sands and sandstones contain many different types of grains, indicating that their compositional variation can only be properly characterised by multivariate statistical methods. However, sedimentary petrologists have traditionally limited themselves to the analysis of ternary (sub)compositions that can be displayed in ternary diagrams. The prime example of the use of ternary diagrams in sedimentary petrology is the inference of plate-tectonic setting of sedimentary basins from sandstone composition Dickinson, 1985, Dickinson, 1988.
The introductory section of this article discusses the procedures by which detrital modes are obtained. The main body of this review is devoted to theory and application of statistical analysis of ternary compositions. It concludes with a brief section on the use of multivariate methods of compositional data analysis in sedimentary petrology.
Section snippets
Classification schemes
Following standard practice in sedimentary petrology, the framework composition of sandstone is estimated by counting a certain number of points in thin section (usually between 300 and 600) according to the Glagolev–Chayes method Chayes, 1949, Chayes, 1956, Galehouse, 1971. Each grain beneath the crosshair is assigned to a category within the petrographic classification system used. The petrographic modal composition of sands and sandstones may be regarded as a mineralogical mode augmented by
Compositional data
Compositions are the principal data used in sedimentary petrology. Such data are commonly expressed as proportions, percentages, or parts per million, and thus sum to a constant value C (equal to 1, 100, or 106, respectively):Because all component abundances xi are non-negative by definition, the value of the k-th component in a composition of k parts is automatically fixed by the sum of the other k−1 values. This inevitable physical limitation implies that compositional data
Concluding remarks
The fundamental problems associated with the use of univariate statistics and the construction of ‘hexagonal fields of variation’ can be summarised as follows:
⋅The assumption that the distribution of component proportions can be adequately approximated by a normal distribution is not generally valid, especially in cases of small samples and/or (average) proportions close to zero or unity, because it leads to predictions extending beyond 〈0;1〉, the physical limits on proportions.
⋅Hexagonal
Model A: the grain as unit of observation
During point counting, each point or observation is classified into one of a number of mutually exclusive categories. All categories together should represent an exhaustive
General properties
The construction of confidence regions for population parameters is complementary to common significance tests. The purpose of such tests is to determine if parameter values estimated from a sample are consistent with known population parameters. A well-known univariate test of this type is the one-sample Student's t-test (e.g., Davis, 1986, Rock, 1988, Press et al., 1994). Student's t, the test statistic, provides a measure of discrepancy between population and sample means that can be
A goodness-of-fit measure for the multinomial distribution
A classical goodness-of-fit measure for data that are assumed to follow a multinomial distribution is Pearson's chi-squared statistic (PXS). This measure of discrepancy between data and model is a member of the class of so-called power-divergence statistics, which encompasses all commonly used goodness-of-fit statistics for categorical data. The behaviour of all power-divergence statistics has been shown to converge to that of the PXS statistic in the case of large samples Cressie and Read, 1984
The additive logistic normal model
Many patterns of compositional variation can be described by additive logistic normal distributions (Aitchison, 1986). This class of statistical models is based on the logratio transformation, which removes the non-negativity and constant-sum constraints on compositional variables (Eq. (1)). Let xi represent the relative abundances of components in a composition made up of k constituents (1≤i≤k). The k-th component xk, whose value is fully specified by the sum of the other k−1 values, is used
Data
Compositional data acquired in the course of a systematic sampling programme of modern beach sands (Aiello et al., 1978) will be used to illustrate the above techniques. Two sets of specimens were collected from sections dug on the berms of Tyrrhenian beaches near the Cecina River mouth (about 30 km south of Livorno, Italy). Sections X and Y are located 7.5 km north and 1.0 km south of the river mouth, respectively. Four specimens were sampled in each section at different depths below the
Data
Point-count data of sands from the Sagavanirktok river, Alaska (Robinson and Johnsson, 1997) will be used to illustrate the flexibility of the additive logistic normal distribution (Model B) for characterising compositional variability in a system dominated by chemical and mechanical weathering. The results obtained by applying Model B to the data will be compared to the use of hexagons constructed from univariate normal approximations. The data consist of 19 specimens of medium-sized sand,
Compositional hierarchy
Compositional data used in sedimentary petrology represent different levels of information within a compositional hierarchy. Compositional elements at a given level may be expressed as linear combinations (mixtures) of elements at a lower level. For detrital modes of sands and sandstones, this hierarchy can be represented by:
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Grain assemblages (sediment sources),
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Mineral assemblages (polymineralic grains),
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Minerals (monomineralic grains),
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Chemical elements.
Acknowledgements
Raymond V. Ingersoll and Jeffrey L. Howard reviewed an earlier version of this manuscript and made many suggestions for improvement. The final version also benefited from the comments of ESR reviewers Alex Woronow and Gerald M. Friedman. I have many people to thank for sharpening my view on the nature of compositional variation in sediments, but above all I am grateful to John Aitchison for his unwavering encouragement and stimulating interest in all matters compositional. Computer programmes
Gert Jan Weltje (1962) is assistant professor of applied clastic sedimentology and mathematical geology at Delft University of Technology. He obtained a MSc in natural sciences (1988) and a PhD in earth sciences (1994) at Utrecht University, The Netherlands. Between 1994 and 1997 he carried out postdoctoral research in applied nuclear physics and sedimentary geology, and worked as a statistical-sedimentological consultant for the oil industry. From 1997 to 2000 he was employed as a researcher
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Gert Jan Weltje (1962) is assistant professor of applied clastic sedimentology and mathematical geology at Delft University of Technology. He obtained a MSc in natural sciences (1988) and a PhD in earth sciences (1994) at Utrecht University, The Netherlands. Between 1994 and 1997 he carried out postdoctoral research in applied nuclear physics and sedimentary geology, and worked as a statistical-sedimentological consultant for the oil industry. From 1997 to 2000 he was employed as a researcher at TNO-NITG, the National Geological Survey. He received the 1997 Vistelius Research award of the International Association for Mathematical Geology, of which he is presently a council member. His research focuses on process-response simulation for predictive reservoir modelling, and on inversion of geological data.