Contributions to Mineralogy and Petrology

, Volume 102, Issue 1, pp 69–77 | Cite as

Matrix analysis of metamorphic mineral assemblages and reactions

  • George W. Fisher
Article

Abstract

Assemblage diagrams are widely used in interpreting metamorphic mineral assemblages. In simple systems, they can help to identify assemblages which may represent equilibrium states; to determine whether differences between assemblages reflect changes in metamorphic grade or variations in bulk composition; and to characterize isograd reactions. In multicomponent assemblages these questions can be approached by investigating the rank, composition space (range) and reaction space (null-space) of a matrix representing the compositions of the phases involved. Singular value decomposition (SVD) provides an elegant way of (1) finding the rank of a matrix and detemining orthonormal bases for both the composition space and the reaction space needed to represent an assemblage or pair of assemblages; and (2) finding a model matrix of specified rank which is closest in a least squares sense to an observed assemblage. Although closely related to least squares techniques, the SVD approach has the advantages that it tolerates errors in all observations and is computationally simpler and more stable than non-linear least squares algorithms. Models of this sort can be used to interpret multicomponent mineral assemblages by straightforward generalizations of the methods used to interpret assemblage diagrams in simpler systems. SVD analysis of mineral assemblages described by Lang and Rice (1985) demonstrates the utility of the approach.

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References

  1. Albarede F, Provost A (1977) Petrological and geochemical mass balance equations: an algorithm for least-square fitting and general error analysis. Comput Geosci 3:309–326Google Scholar
  2. Fletcher CJN, Greenwood HJ (1979) Metamorphism and structure of Penfold Creek area, near Quesnel Lake, British Columbia. J Petrol 20:743–790Google Scholar
  3. Golub GH, Van Loan CF (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17:883–893Google Scholar
  4. Golub GH, Van Loan CF (1983) Matrix computations. Johns Hopkins Univ Press, Baltimore, 476 pGoogle Scholar
  5. Greenwood HJ (1967) The N-dimensional tie-line problem. Geochim Cosmochim Acta 31:467–490Google Scholar
  6. Greenwood HJ (1968) Matrix methods and the phase rule in petrology. XXIII Int Geol Cong Proc 6:267–279Google Scholar
  7. Korzhinskii DS (1959) Physico-chemical basis of the analysis of the paragenesis of minerals. Consultants Bureau, New York, 142 pGoogle Scholar
  8. Lang HM, Rice JM (1985) Regression modeling of metamorphic reactions in metapelites, Snow Peak, northern Idaho. J Petrol 26:857–887Google Scholar
  9. Pigage LC (1976) Metamorphism of the Settler Schist, southeast of Yale, British Columbia. Can J Earth Sci 13:405–421Google Scholar
  10. Press WH, Flannery BP, Teukolsky SA, Vettering WT (1986) Numerical recipes. Cambridge Univ Press, Cambridge, 818 pGoogle Scholar
  11. Reid MJ, Gancarz AJ, Albee AL (1973) Constrained least-squares analysis of petrologic problems with an application to lunar sample 12040; Earth Planet Sci Lett 17:433–445Google Scholar
  12. Spear FS, Rumble D, Ferry JM (1982) Linear algebraic manipulation of n-dimensional composition space. Min Soc Am Rev Mineral 10:53–104Google Scholar
  13. Stewart GW (1973) Introduction to matrix computations. Academic Press, New York, 441 pGoogle Scholar
  14. Thompson JB (1957) The graphical analysis of mineral assemblages in pelitic schists. Am Mineral 42:842–858Google Scholar
  15. Thompson JB (1982) Reaction space: an algebraic and geometric approach. Mineral Soc Am Rev Mineral 10:33–52Google Scholar
  16. Thompson JB (1988) Paul Niggli and petrology; order out of chaos. Min Schweiz Pet (in press)Google Scholar
  17. Van Huffel S (1987) Analysis of the total least squares problem and its use in parameter estimation; doctoral thesis submitted to the Department of Electrical Engineering, Catholic Univ of Leuven, Belgium, 370 pGoogle Scholar
  18. Van Huffel S, Vandewalle J (1988) Analysis and properties of the generalized total least squares problem AX=B when some or all columns in A are subject to error. SIAM J Matrix Anal Appl (in press)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • George W. Fisher
    • 1
  1. 1.Department of Earth and Planetary SciencesJohns Hopkins UniversityBaltimoreUSA

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