Abstract
Finite strain estimation is a widely used technique for the study of rock deformation in structural geology. One particular algorithm proposed by Shimamoto and Ikeda uses the ‘average shape matrix’ of deformed markers. This paper provides a detailed error analysis for resulting strain estimates in two dimensions. When the number of markers exceeds 100, estimators of components of the strain tensor are shown to have an approximately Gaussian distribution with variances that increase with their mean. Equal variance estimators are obtained by applying a log transform for the elongation and an arcsin transformation for the orientation estimates. Confidence interval formulae for strain tensor components are proposed. Lithology specific constants arising in these formulae are estimated from undeformed samples. The results are validated by application to simulated data as well as observational data from thin sections of sandstone sampled from SE Ireland.
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References
Anderson TW (1984) An introduction to multivariate statistical analysis, 2nd edn. Wiley, New York, 752 p
Dunnet D (1969) A technique of finite strain analysis using elliptical particles. Tectonophysics 7:117–136
Dunnet D, Siddans AWB (1971) Non-random sedimentary fabrics and their modification by strain. Tectonophysics 12:307–325
Flinn D (1979) The deformation matrix and the deformation ellipsoid. J Struct Geol 1(4):299–307
Fry N. (1979) Random point distribution and strain measurements in rocks. Tectonophysics 60:89–105
Heilbronner R (2000) Automatic grain boundary detection and grain size analysis using polarization micrographs or orientation images. J Struct Geol 22:969–981
Matthews PE, Bond RAB, Van Den Berg JJ (1974) An algebraic method of strain analysis using elliptical markers. Tectonophysics 24:31–67
McNaught MA (2002) Estimating uncertainty in normalized Fry plots using a bootstrap approach. J Struct Geol 24(2):311–322
Means WD (1976) Stress and strain: basic concepts of continuum mechanics for geologists. Springer, New York, 339 p
Meere PA, Mulchrone KF (2003) The effect of sample size on geological strain estimation from passively deformed clastic sedimentary rocks. J Struct Geol 25:1587–1595
Mulchrone KF (2005) An analytical error for the mean radial length method of strain analysis. J Struct Geol 27:1658–1665
Mulchrone KF, Choudhury KR (2004) Fitting an ellipse to an arbitrary shape: implications for strain analysis. J Struct Geol 26:143–153
Mulchrone KF, Walsh K (2006) The motion of a non-rigid ellipse in a general 2D deformation. J Struct Geol 28:392–407
Mulchrone KF, O’Sullivan F, Meere PA (2003) Finite strain estimation using the mean radial length of elliptical objects with bootstrap confidence intervals. J Struct Geol 25:529–539
Ramsay JG, Huber MI (1983) Strain analysis. The techniques of modern structural geology, vol 1. Academic Press, London, 307 p
Roy Choudhury K, Mulchrone KF (2006) A comparative error analysis of manual versus automated methods of data acquisition for algebraic strain estimation. Tectonophysics 421(3):209–230
Roy Choudhury K, Meere PA, Mulchrone KF (2006) Automated grain boundary detection by CASRG. J Struct Geol 28(3):363–375
Serfling R (1980) Approximation theorems in mathematical statistics. Wiley, New York, 400 p
Shimamoto T, Ikeda Y (1976) A simple algebraic method for strain estimation from deformed ellipsoidal objects, 1. Basic theory. Tectonophysics 36:315–337
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Choudhury, K.R. Analysis of the Variability of a Two-Dimensional Finite Strain Estimate. Math Geosci 41, 535–553 (2009). https://doi.org/10.1007/s11004-008-9185-1
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DOI: https://doi.org/10.1007/s11004-008-9185-1