Open and closed compositional data in petrology

The nature of chemical and modal closed and open compositional data for igneous petrology is reviewed. Chemical analyses of Ben Nevis (Ontario) Archaean volcanic rocks support earlier assertions that major- and trace-element g/100 cc igneous-rock data usually comprise open compositional variables which, unlike traditional closed constant-sum weight percentages, are not constrained by closure.     

Reducing the dimensionality of compositional data sets

The high-dimensionality of many compositional data sets has caused geologists to look for insights into the observed patterns of variability through two dimension-reducing procedures: (i)the selection of a few subcompositions for particular study, and (ii)principal component analysis. After a brief critical review of the unsatisfactory state of current statistical methodology for these two procedures, this paper takes as a starting point for the resolution of persisting difficulties a recent approach to principal component analysis through a new definition of the covariance structure of a composition. This approach is first applied for expository purposes to a small illustrative compositional data set and then to a number of larger published geochemical data sets. The new approach then leads naturally to a method of measuring the extent to which a subcomposition retains the pattern of variability of the whole composition and so provides a criterion for the selection of suitable subcompositions. Such a selection process is illustrated by application to geochemical data sets.     

Quantifying and testing differences among means of compositional data suites

Commonly, geological studies compare mean values of two or more compositional data suites in order to determine if, how, and by how much they differ. Simple approaches for evaluating and statistically testing differences in mean values for open data fail for compositional (closed) data. A new parameter, an f-value, therefore has been developed, which correctly quantifies the differences among compositional mean values and allows testing those differences for statistical significance. In general, this parameter quantifies only therelative factor by which compositional variables differ across data suites; however for situations where, arguably, at least one component has neither increased nor decreased, anabsolute f-value can be computed. In situations where the compositional variables have undergone many perturbations, arguments based upon thef-values and the central limit theorem indicate that logratios of compositional variables should be normally distributed.     

Textural and compositional evidence for magma mixing and its mechanism,Abu volcano group,southwestern Japan

Quaternary basalts, andesites and dacites from the Abu monogenetic volcano group, SW Japan, (composed of more than 40 monogenetic volcanoes) show two distinct chemical trends especially on the FeO^*/MgO vs SiO₂ diagram. One trend is characterized by FeO^*/MgO-enrichment with a slight increase in SiO₂ content (Fe-type trend), whereas the other shows a marked SiO₂-enrichment with relatively constant FeO^*/MgO ratios (Si-type trend). The Fe-type trend is explained by fractional crystallization with subtraction of olivine and augite from a primitive alkali basalt magma. Rocks of the Si-type trend are characterized by partially melted or resorbed quartz and sodic plagioclase phenocrysts and/or fine-grained basaltic inclusions. They are most likely products of mixing of a primitive alkali basalt magma containing olivine phenocrysts with a dacite magma containing quartz, sodic plagioclase and hornblende phenocrysts. Petrographic variation as well as chemical variation from basalt to dacite of the Si-type trend is accounted for by various mixing ratios of basalt and dacite magmas. Pargasitic hornblende and clinopyroxene phenocrysts in andesite and dacite may have crystallized from basaltic magma during magma mixing. Olivine and spinel, and quartz, sodic plagioclase and common hornblende had crystallized in basaltic and dacitic magmas, respectively, before the mixing. Within a lava flow, the abundance of basaltic inclusions decreases from the area near the eruptive vent towards the perimeter of the flow, and the number of resorbed phenocrysts varies inversely, suggesting zonation in the magma chamber.The mode of mixing changes depending on the mixing ratio. In the mafic mixture, basalt and dacite magmas can mix in the liquid state (liquid-liquid mixing). In the silicic mixture, on the other hand, the basalt magma was quenched and formed inclusions (liquid-solid mixing). During mixing, the disaggregated basalt magma and the host dacite magma soon reached thermal equilibrium. Compositional homogenization of the mixed magma can occur only when the equilibrium temperature is sufficiently above the solidus of the basalt magma. The Si-type trend is chemically and petrographically similar to the calc-alkalic trend. Therefore, a calc-alkalic trend which is distinguished from a fractional crystallization trend (e.g. Fe-type trend) may be a product of magma mixing.     

Regional and local variations in the composition of the wolframite series from SW Japan and possible factors controlling compositional variations

The regional and local variations in the composition of the wolframite series associated with the ilmeniteseries and magnetite-series granitoids in the Inner Zone of SW Japan were investigated using X-ray diffraction and microprobe analyses. It is concluded that the variation in Mn/Fe ratios in the source materials (granitic magmas in most cases) is the dominant factor controlling the regional variation in the wolframite compositions, while, as exemplified by the Kaneuchi mine, Kyoto Prefecture, the wallrock chemistry and pH of ore fluids are responsible for the local variation within a single vein system. Within a fresh euhedral crystal from the mine, only slight compositional variation of less than 3 mole% MnWO₄ is found in the form of oscillatory zoning. This study, combined with the fluid inclusion research, indicates that the wolframite compositions (Mn/Fe ratios) cannot be used as a geothermometer.     

Some aspects of transformations of compositional data and the identification of outliers

The statistical analysis of compositional data is based on determining an appropriate transformation from the simplex to real space. Possible transfonnations and outliers strongly interact: parameters of transformations may be influenced particularly by outliers, and the result of goodness-of-fit tests will reflect their presence. Thus, the identification of outliers in compositional datasets and the selection of an appropriate transformation of the same data, are problems that cannot be separated. A robust method for outlier detection together with the likelihood of transformed data is presented as a first approach to solve those problems when the additive-logratio and multivariate Box-Cox transformations are used. Three examples illustrate the proposed methodology.     

Seismological evidence for compositional variations at the base of the mantle transition zone under Japan Islands

Temperature and water content anomalies just above the 660-km discontinuity under the Japan Islands are estimated from seismological observations. Two sets of seismological observations of P-wave velocity perturbations and depth variations of the 660-km discontinuity are used, which are (1) long-wavelength (~ 500 km) variations from seismic tomography based on a grid parameterization and waveform analysis of ScS reverberations and (2) moderate-wavelength (~ 150 km) variations from seismic tomography with a block parameterization and receiver-function analysis. To estimate temperature and water content anomalies, partial derivatives of velocity and depth variations with respect to temperature and water content determined by mineral physics studies are used. Under Southwest Japan, low temperature and high water-content anomalies are obtained from both sets of seismological observations, which have already been found by a previous study. Under Northeast Japan, however, there are discrepancies between the results estimated from the two data sets, possibly due to the different resolution scales. This discrepancy is dismissed when examining distributions of the temperature and water content anomalies. These anomalies can be grouped into two geographical locations, one group under Southwest Japan, the other under Northeast Japan. The two groups are clearly separated and exhibit common features in both sets of the seismological observations. The grouping is interpreted by a compositional difference between the subducting Pacific slab under Southwest Japan and a normal mantle under Northeast Japan.     

Crustal structure of Japan as derived from explosion seismic data

Crustal structures of Japan were investigated under the Upper Mantle Project in three profiles, Kurayosi-Hanabusa, western Japan; Atumi-Noto, central Japan; Kesennuma-Oga, northeastern Japan. These investigations have revealed that the crust of Japan is of continental type. The variation of the crustal structure reflects the topography, especially the water depth; so the thinning of the crust occurs near the shore where the water depth increases rapidly. The velocity below the Mohorovi?i? discontinuity is smaller than 8.0 km/sec, but it is possible that a deeper layer with a velocity of over 8.0 km/sec may exist. The basaltic layer in central Japan, if existing at all, must be thin.     

The statistical study of hydraulic geometry: A new direction for compositional data analysis

Hydraulic exponents and unit hydraulic exponents are unit-sum constrained, which requires that they be analyzed by statistical methods designed for compositional data. Though uncertainties remain regarding selection of the best constraining operation and method of handling departures from the unit-sum constraint, neither category of uncertainty should be an impediment to the selection of the appropriate statistical methodology. In a small sample study, the hydraulic geometry of different types of streams were compared: (1) semi-arid: perennial vs. ephemeral; (2) tropical: Puerto Rico vs. West Malaysia; and (3) semi-arid vs. tropical (by pooling the previous data sets). All three comparisons revealed statistically significant differences in either logratio mean vectorsor logratio covariance matrices but not both. All six categories of data had logistic normal distributions. Because the derivatives at a given discharge of curvilinear hydraulic geometry relationships and hydraulic exponents on either side of the breakpoints of piecewise linear relationships are unit-sum constrained, they also can be studied by compositional methods. However, the compositional approach is limited in cases where distributions have large departures from logistic normality and for streams that have negative hydraulic exponents.     

End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem

Linear mixing models of compositional data have been developed in various branches of the earth sciences (e.g., geochemistry, petrology, mineralogy, sedimentology) for the purpose of summarizing variation among a series of observations in terms of proportional contributions of (theoretical) end members. Methods of parameter estimation range from relatively straightforward normative partitioning by (nonnegative) least squares, to more sophisticated bilinear inversion techniques. Solving the bilinear mixing problem involves the estimation of both mixing proportions and end-member compositions from the data. Normative partitioning, also known as linear unmixing, thus can be regarded as a special situation of bilinear unmixing with (supposedly) known end members. Previous attempts to model linear mixing processes are reviewed briefly, and a new iterative strategy for solving the bilinear problem is developed. This end-member modeling algorithm is more robust and has better convergence properties than previously proposed numerical schemes. The bilinear unmixing solution is intrinsically nonunique, unless additional constraints on the model parameters are introduced. In situations where no a priori knowledge is available, the concept of an “ optimal ” solution may be used. This concept is based on the trade-off between mathematical and geological feasibility, two seemingly contradictory but equally desirable requirements of the unmixing solution.     

New data for Late Pleistocene Pinedale alpine glaciation from southwestern Colorado

New cosmogenic surface-exposure ages of moraine-crest boulders from southwestern Colorado are compared with published surface-exposure ages of boulders from moraine complexes in north-central Colorado and in west-central (Fremont Lake basin) Wyoming. ¹⁰Be data sets from the three areas were scaled to a single ¹⁰Be production rate of 5.4 at/g/yr at sea level and high latitude (SLHL), which represents the average ¹⁰Be production rate for two high-altitude, mid-latitude sites in the western United States (US) and Austria. Multiple nuclide ages on single boulders indicate that this ¹⁰Be production rate yields ages comparable to those calculated with a commonly used ³⁶Cl production scheme. The average age and age range of moraine-crest boulders on terminal moraines at the southwestern Colorado and Wyoming sites are similar, indicating a retreat from their positions ∼16.8 ³⁶Cl ka (Cosmogenic ages in this paper are labeled ¹⁰Be or ³⁶Cl ka or just ka when both ¹⁰Be or ³⁶Cl ages are being discussed; radiocarbon ages are labeled ¹⁴C ka, calibrated radiocarbon are labeled cal ka, and calendar ages are labeled calendar ka. Errors (±1σ) associated with ages are shown in tables. Radiocarbon ages were calibrated using the data of Hughen et al. (Science 303 (2004) 202). This suggests a near-synchronous retreat of Pinedale glaciers across a 470-km latitudinal range in the Middle and Southern Rocky Mountains. Hypothetical corrections for snow shielding and rock-surface erosion shifts the time of retreat to between 17.2 and 17.5 ¹⁰Be ka at Pinedale, Wyoming, and between 16.3 and 17.3 ³⁶Cl ka at Hogback Mountain, Colorado.     

Temporal evolution and compositional signatures of two supervolcanic systems recorded in zircons from Mangakino volcanic centre,New Zealand

Mangakino, the oldest rhyolitic caldera centre delineated in the Taupo Volcanic Zone of New Zealand, generated two very large (super-sized) ignimbrite eruptions, the 1.21 ± 0.04 Ma >500 km³ Ongatiti and ~1.0 Ma ~1,200 km³ Kidnappers events, the latter of which was followed after a short period of erosion by the ~200 km³ Rocky Hill eruption. We present U/Pb ages and trace-element analyses on zircons from pumice clasts from these three eruptions by Secondary Ion Mass Spectrometry (SIMS) using SHRIMP-RG instruments to illustrate the evolution of the respective magmatic systems. U–Pb age spectra from the Ongatiti imply growth of the magmatic system over ~250 kyr, with a peak of crystallisation around 1.32 Ma, ~100 kyr prior to eruption. The zircons are inferred to have then remained stable in a mush with little crystallisation and/or dissolution before later rejuvenation of the system at the lead-in to eruption. The paired Kidnappers and Rocky Hill eruptions have U–Pb zircon ages and geochemical signatures that suggest they were products of a common system grown over ~200 kyr. The Kidnappers and Rocky Hill samples show similar weakly bimodal age spectra, with peaks at 1.1 and 1.0 Ma, suggesting that an inherited antecrystic population was augmented by crystals grown at ages within uncertainty of the eruption age. In the Kidnappers, this younger age peak is dominantly seen in needle-shaped low U grains with aspect ratios of up to 18. In all three deposits, zircon cores show larger ranges and higher absolute concentrations of trace elements than zircon rims, consistent with zircon crystallisation from evolving melts undergoing crystal fractionation involving plagioclase and amphibole. Abundances and ratios of many trace elements frequently show variations between different sectors within single grains, even where there is no visible sector zoning in cathodoluminescence (CL) imaging. Substitution mechanisms, as reflected in the molar (Sc + Y + REE³⁺)/P ratio, differ in the same growth zone between the sides (along a-axis and b-axis: values approaching 1.0) and tips (c-axis: values between 1.5 and 5.0) of single crystals. These observations have implications for the use of zircons for tracking magmatic processes, particularly in techniques where CL zonation within crystals is not assessed and small analytical spot sizes cannot be achieved. These observations also limit applicability of the widely used Ti-in-zircon thermometer. The age spectra for the Ongatiti and Kidnappers/Rocky Hill samples indicate that both magmatic systems were newly built in the time-breaks after respective previous large eruptions from Mangakino. Trace element variations defining three-component mixing suggest that zircons, sourced from multiple melts, contributed to the population in each system.     

Deep structures in the Maritime Territory-Sea of Japan continental margin from seismic data

This paper discusses the results of interpreting seismic profiles on the Earth’s crust of the Maritime Territory and Sea of Japan performed during the 20th century by the Sakhalin Integrated Research Institute and by the Schmidt Joint Institute of Physics of the Earth, Russian Academy of Sciences. The seismic profiles confirmed the presence of structural features under the Maritime Territory and the Sea of Japan that were revealed previously from geological data, such as spreading zones, rifts, deep-seated faults, overthrusts, and subduction zones, suggesting an active type of continental margin in the Far East region. We assumed that a high occurrence of the asthenospheric layer enclosing magmatic chambers explains the high activity of tectonic processes in the Far Eastern continental margin. The identified system of rifts and spreading centers supports this assumption.     

On criteria for measures of compositional difference

Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.