Three-Dimensional Representation of Geochemical Data from a Multidimensional Compositional Space
ABSTRACT
Systems described by a wide set of variables, like rock compositions, may be often modeled by a reduced set of components, like minerals, that can be represented in diagrams in two or three dimensions. This paper deals with an original algorithm that allows the representation of compositional data in tetrahedral diagrams, provided that they can be recast on the basis of four end members. The algorithm is based on the orthogonal projection of a given point belonging to Rn to the 3D-space through four Rn points representing the compositions of suitable end members. The algorithm is applied to the assessment of mass balance problems (in weight% or molar basis) as well as to the identification of the geochemical imprint revealed by isotope ratios in igneous rock suites. The fields of possible applications are by far wider, encompassing all problems of comprehensive data representation from a multidimensional space to a bi-dimensional plot.
Systems described by a wide set of variables, like rock compositions, may be often modeled by a reduced set of components, like minerals, that can be represented in diagrams in two or three dimensions. This paper deals with an original algorithm that allows the representation of compositional data in tetrahedral diagrams, provided that they can be recast on the basis of four end members. The algorithm is based on the orthogonal projection of a given point belonging to Rn to the 3D-space through four Rn points representing the compositions of suitable end members. The algorithm is applied to the assessment of mass balance problems (in weight% or molar basis) as well as to the identification of the geochemical imprint revealed by isotope ratios in igneous rock suites. The fields of possible applications are by far wider, encompassing all problems of comprehensive data representation from a multidimensional space to a bi-dimensional plot.
Cite this paper
nullP. Armienti and P. Longo, "Three-Dimensional Representation of Geochemical Data from a Multidimensional Compositional Space," International Journal of Geosciences, Vol. 2 No. 3, 2011, pp. 231-239. doi: 10.4236/ijg.2011.23025.
nullP. Armienti and P. Longo, "Three-Dimensional Representation of Geochemical Data from a Multidimensional Compositional Space," International Journal of Geosciences, Vol. 2 No. 3, 2011, pp. 231-239. doi: 10.4236/ijg.2011.23025.
References
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[1] P. Armienti, “TETRASEZ an Interactive Program in BA- SIC to Perform Tetrahedral Diagrams,” Computer & Geosciences, Vol. 12, No. 2, 1986, pp. 229-241. doi:10.1016/0098-3004(86)90010-5 [2] R. O. Sack, D. C. Walker and I. S. E. Carmichael, “Experimental Petrology of Alkalic Lavas: Constraints on Cotectics of Multiple Saturation in Natural Basic Liquids,” Contributions to Mineralogy and Petrology, Vol. 96, No. 1, 1987, pp. 1-23. doi:10.1007/BF00375521 [3] M. J. O’Hara, “The Bearing of Phase Equilibria Studies on the Origin and Evolution of Basic and Ultrabasic Rocks,” Earth-Science Reviews, Vol. 4, 1968, pp. 69-133. doi:10.1016/0012-8252(68)90147-5 [4] H. S. J. Yoder and C. E. Tilley, “Origin of Basalt Magma: An Experimental Study of Natural and Synthetic Rock Systems,” Journal of Petrology, Vol. 3, No. 3, 1962, pp. 342-532. [5] J. Aitchison, “The Statistical Analysis of Compositional Data,” Chapman and Hall, London, 1986, p. 416. [6] C. J. Allègre, B. Hamelin, A. Provost and B. Dupré, “Topology of Isotopic Multispace and Origin of Mantle Chemical Etherogenities,” Earth and Planetary Sience Letters, Vol. 81, 1986, pp. 319-337. [7] P. Armienti, S. Tonarini, F. Innocenti and M. D’Orazio, “Mount Etna Pyroxene as Tracer of Petrogenetic Processes and Dynamics of the Feeding System,” In: L. Beccaluva, G. Bianchini, M. Wilson, Eds., Volcanism in the Mediterranean and Surrounding Regions, Geological Society of America, Special Paper, Vol. 418, 2007, pp. 265-276. [8] J. F. Schairer and H. S. Yoder Jr., “Crystal and Liquid Trends in Simplified Alkali Basalts,” Carnegie Institution of Washington, Vol. 63, 1964, pp. 65-75. [9] C. Perinelli, P. Armienti and L. Dallai, “Thermal Evolu- tion of the Lithosphere in a Rift Environment as Inferred from the Geochemistry of Mantle Cumulates. Northern Victoria Land, Antarctica,” Journal of Petrology, Vol. 52, No. 4, 2011, pp. 665-690. doi:10.1093/petrology/egq099 [10] S. Hart, E. H. Hauri, L. A. Oschmann and J. A. Whitehead, “Mantle Plumes and Entrainment: Isotopic Evidence,” Science, Vol. 256, No. 5056, 1992, pp. 517-520. doi:10.1126/science.256.5056.517 [11] E. Hauri and S. R. Hart, “Rhenium Abundances and Sys- tematics in Oceanic Basalts,” Chemical Geology, Vol. 13, No. 1-4, 1997, pp. 185-205. doi:10.1016/S0009-2541(97)00035-1 [12] P. Armienti and D. Gasperini, “Do We Really Need Man- tle Components to Define Mantle Composition?” Journal of Petrology, Vol. 48, No. 4, 2007, pp. 693-709. doi:10.1093/petrology/egl078 [13] J. A. Martìn-Fernàndez, C. Barcelò-Vidal and V. Pawlowsky-Glahn, “Dealing with Zeros and Missing Values in Compositional Data Sets Using Nonparametric imputation,” Mathematical Geology, Vol. 35, No. 3. 2003, pp. 45-54. [14] S. S. Sun and W. F. McDonough, “Chemical and Isotopic Systematics of Oceanic Basalts: Implications for Mantle Composition and Processes,” In: A. D. Saunders and M. J. Norry, Eds., Magmatism in Ocean Basins, Geological Society Special Publication, London, 1989, pp. 313-345. [15] A. D. Saunders, M. J. Norry and J. Tarney, “Origin of MORB and Chemically Depleted Mantle Reservoirs: Trace Element Constraints,” Journal of Petrology, Vol. 1, No. 1, 1988, pp. 414-445. [16] D. J. Chaffey, R. A. Cliff and B. M. Wilson, “Characterization of the St. Helena Magma Source,” In: A. D. Saunders and M. J. Norry Eds., Magmatism in Ocean Basins, Geological Society Special Publication, London, Vol. 42, 1989, pp. 257-276.