A comparison of methods used in measuring finite strains from ellipsoidal objects

The available methods for measuring strains in two and three dimensions from passive ellipsoidal objects are compared in an attempt to determine the most useful and precise procedure for the structural geologist. A comparison of data collection techniques showed that the use of thin orthogonally cut slabs or mylar overlays used with a monocomparater provides the most reproducible data. Lisle's theta-curve and Shimamoto and Ikeda's algebraic methods provided the most precise, and probably most accurate, two-dimensional data while Miller and Oertel's procedure and possibly Dunnet's PHASE 5 program gave the best three-dimensional results.An examination of errors encountered in strain analyses suggests that all of the available methods give accurate orientations of the finite strain ellipsoid. However, the magnitudes of strain ratios show large variations that are dependent on the sample size and procedure used. Shimamoto and Ikeda's method again proved to be the most precise, giving reproducible results with as few as ten elliptical objects.The samples used in the above comparison are part of a larger analysis of strains occurring in southeastern Maine. Structural elements observed in four selected areas of Avalonian belt rocks and the strain data collected suggest that the region has undergone at least three non-coaxial deformations with D₁ >D₂ >D₃.     

The Bi-Gaussian approach: A simple method for recovery estimation

A new and simple method is proposed to obtain estimates of recovery functions: the Bi-Gaussian approach. Existing methods estimate recovery functions with conditional distributions where the conditioning set is all the data available. Here instead the simple kriging estimate of the Gaussian transform is proposed to be used. Results in the point recovery case are identical to the multi-Gaussian approach of Verly (1983, 1984), whereas in the non-point-support situation, an approximation is derived which saves computer time as compared to employing the strict multi-Gaussian hypothesis. Two examples compare favorably with the well-established disjunctive kriging method (discrete Gaussian model).     

A comparative error analysis of manual versus automated methods of data acquisition for algebraic strain estimation

This paper presents a statistical analysis of the algebraic strain estimation algorithm of Shimamoto and Ikeda [Shimamoto, T., Ikeda, Y., 1976. A simple algebraic method for strain estimation from deformed eillipsoidal objects: 1. Basic theory. Tectonophysics 36, 315–337]. It is argued that the error in their strain estimation procedure can be quantified using an expected discrepancy measure. An analysis of this measure demonstrates that the error is inversely proportional to the number of clasts used. The paper also examines the role of measurement error, in particular that incurred under (i) a moment based and (ii) manual data acquisition methods. Detailed analysis of these two acquisition methods shows that in both cases, the effect of measurement error on the expected discrepancy is small relative to the effect of the sample size (number of objects). Given their relative speed advantage, this result favours the use of automated measurement methods even if they incur more measurement error on individual objects. Validation of these results is carried out by means of a simulation study, as well as by reference to studies appearing in previous literature. The results are also applied to obtain an upper bound on the error of strain estimation for various studies published in the literature on strain analysis.     

The effect of sample size on geological strain estimation from passively deformed clastic sedimentary rocks

The role of sample size in the estimation of geological strain, both finite strain (R_s) and that of the orientation of the finite strain ellipse (φ_s), is investigated for clastic sedimentary rocks. This study looks at four strain methods, the Robin method, the linearization method, the Mulchrone and Meere method and the mean radial length method that are initially tested using simulated strained data sets and subsequently by applying the methods to real data. It is found that the optimum strain analysis sample size for a clastic sedimentary rock is primarily dependant on the intensity of strain suffered by that rock because of the error behavior associated with R_s estimates. An iterative process is therefore recommended starting with a minimum sample size of 150, which can be maintained or reduced based on the initial R_s estimates.     

The estimation of finite strain from the orientation distribution of passively deformed linear markers: Eigenvalue relationships

The eigenvalues and eigenvectors of a direction cosine dispersion matrix may be used to characterise sets of orientation measurements in either two or three dimensions. General equations are presented that relate ratios of the eigenvalues of such a measured set to the finite strain ratios of the set assuming that the original orientation distribution, before a deformation, was uniform. The measured linear features in the rock are assumed to have behaved passively during a homogeneous deformation. The application of these relationships to strain estimation is discussed and tables for the three-dimensional case are given.     

Effect of grain packing tightness on strain estimation from the Fry method

How tightly should a sample be packed for strain estimation by the Fry method? We address this issue using synthetic simulations of 900 images such that each image contains 200 randomly distributed grains, but differs from other images with respect to the packing tightness and the degree of sorting. Each image is coaxially distorted by various known strain ratios and the strain estimates from distorted images are obtained by the Fry method. The statistical errors in the strain estimates are found to grow larger with the decrease in the packing tightness irrespective of the level of distortion. We demonstrate that a progressive decrease in the packing tightness results in an increasingly clustered nature of grain center distribution and hence the larger errors. These results, obtained from the synthetic images, are corroborated by two natural examples of sandstone, one loosely packed and the other tightly packed. Based on the results of tests on synthetic and natural examples, we recommend that the Fry method should be used only on those samples that have \({>}30\%\) packing density.     

A method for quantitatively analysing dynamic recrystallization in deformed quartzitic rocks

Quantification of the microstructural changes brought about by dynamic recrystallization is essential for the interpretation of deformation mechanism histories and for the understanding of recrystallization as a syn-kinematic process. A method is presented for analysing the degree of dynamic recrystallization and for reconstructing the original grain-size distribution from that measured in the deformed specimen. This is based on size distribution measurements and comparative volume calculations between subsets of grains which contain rutile inclusions and subsets which do not. Application of the method to some quartzites from the Kilmory Bay Syncline, S. W. Highlands, Scotland, demonstrates that up to 25% of the apparent matrix grains are new grains produced by the dynamic recrystallization of porphyroclasts, and that the new grains alone compose more than 12% of the total rock volume. These figures are 2–3 times larger than estimates made by normal petrographic inspection. It is also shown that grain-size distributions alone convey little information about the microstructural changes and that grain-size vs volume fraction graphs are more meaningful.     

A simple method for estimating excess pressure over horsts from seismic sections

Observations in the North Sea Basin max indicate significant overpressure in sediments over horst blocks but not over grabens at the same submudline depth. The purpose is to show that over a horst, of top width W, with grabens on either side of top widths G₁ and G₂, respectively, the equivalent mud density. _r can be estimated approximately from the simple equation.

A linear algebraic method for the calculation of pyroxene endmember components

Summary A linear algebraic model for the sequence-independent calculation of pyroxene components of microprobe analyses is presented. Assuming ideal stoichiometry, the theoretical Fe³⁺ is obtained by using charge balance criteria. The calculation of 11 linearly-independent endmember parameters (Jd, Ac, Ur, TiTs, CaTs, FeTs, CrTs, Pm, Fs, En, Wo) in conjunction with a rigorous mass balance enables the appreciation of the stoichiometry and the endmember definition. The presented method offers a normalized comparison base for a wide spectrum of pyroxene analyses. On the basis of the 11 linearly-independent endmember parameters other common pyroxene components (e.g. Di, Hd, Jh, FeATs, CrATs, MgTs, Fe²⁺Ts) may be calculated using exchange reactions.

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Die Berechnung von Pyroxen-Endgliedern mit Methoden der Linearen Algebra

Zusammenfassung Unter Verwendung von Methoden der Linearen Algebra wird eine reihenfolgenunabhängige Berechnung von Pyroxen-Endgliedern aus Mikrosondenanalysen vorgestellt. Die Annahme idealer Stöchiometrie ermöglicht die Bestimmung des theoretischen Fe³⁺ aus einer Ladungsbilanz. Die Berechnung von 11 linear-unabhängigen Pyroxen-Endgliedern (Jd, Ac, Ur, TiTs, CaTs, FeTs, CrTs, Pm, Fs, En, Wo) erfolgt unter einer strengen Massenbilanz und gewährleistet eine theoretische Auswertung der auferlegten Bedingungen über die Stöchiometrie und die Endgliederdefinition. Darüber hinaus stellt diese Methode eine normierte Vergleichsbasis für ein breites Spektrum von Pyroxenanalysen dar. Ausgehend von den 11 linear-unabhängigen Endgliedern können weitere in der Natur auftretende Pyroxen-Endglieder wie z.B. Di, Hd, Jh, FeATs, CrATs, MgTs, Fe²⁺ Ts mittels Austauschreaktionen berechnet werden.

     

A simple estimation model for 3D braced excavation wall deflection

A series of two-dimensional (2D) and three-dimensional (3D) finite element analyses using the Hardening Soil (HS) model were carried out to investigate the influences of soil properties, wall stiffness, excavation length, excavation depth, clay thickness at the base of the excavation and wall embedment depth, on the maximum wall deflection induced by braced-excavation. The results show that the 3D maximum wall deflections are generally much smaller than those for 2D. Comparisons were also made with other commonly used semi-empirical charts. Based on the finite element results in this study, a simple wall deflection equation was developed for estimating the maximum wall deflection that takes the 3D effects into consideration through different ratios of excavation length over excavation width.     

Model-free estimation from spatial samples: A reappraisal of classical sampling theory

A commonly held view among geostatisticians is that classical sampling theory is inapplicable to spatial sampling because spatial data are dependent, whereas classical sampling theory requires them to be independent. By comparing the assumptions and use of classical sampling theory with those of geostatistical theory, we conclude that this view is both false and unfortunate. In particular, estimates of spatial means based on classical sampling designs require fewer assumptions for their validity, and are therefore more robust, than those based on a geostatistical model.     

Strain analysis from objects with a random distribution: A generalized center-to-center method

Existing methods of strain analysis such as the center-to-center method and the Fry method estimate strain from the spatial relationship between point objects in the deformed state. They assume a truncated Poisson distribution of point objects in the pre-deformed state. Significant deviations occur in nature and diffuse the central vacancy in a Fry plot, limiting the its effectiveness as a strain gauge. Therefore, a generalized center-to-center method is proposed to deal with point objects with the more general Poisson distribution, where the method outcomes do not depend on an analysis of a graphical central vacancy. This new method relies upon the probability mass function for the Poisson distribution, and adopts the maximum likelihood function method to solve for strain. The feasibility of the method is demonstrated by applying it to artificial data sets generated for known strains. Further analysis of these sets by use of the bootstrap method shows that the accuracy of the strain estimate has a strong tendency to increase either with point number or with the inclusion of more pre-deformation nearest neighbors. A poorly sorted, well packed, deformed conglomerate is analyzed, yielding strain estimate similar to the vector mean of the major axis directions of pebbles and the harmonic mean of their axial ratios from a shape-based strain determination method. These outcomes support the applicability of the new method to the analysis of deformed rocks with appropriate strain markers.     

A method for the determination of the initial shapes of deformed xenoliths in granitoids

Measuring strain from deformed xenoliths is problematic due mainly to the large initial shape variations of these markers. A method is described which allows mean initial shape to be determined for a number of xenolith populations by displaying their logarithmic ranges (log R_fmax — log R_fmax) on a Range diagram. The diagram contains a check on validity and allows bad samples to be recognised. Xenolith data from a deformed granite in northwest Ireland is analysed using the method.     

A direct boundary element method for plane strain poroelasticity

This paper presents a direct boundary element method (BEM), formulated in the Laplace transform space, for plane strain poroelasticity. The paper expands on work by Cheng and Liggett by recasting the theoretical foundation of BEM within the framework of Rice and Cleary's formulation of the Biot theory of poroelasticity. Furthermore, the numerical algorithm is generalized to deal with both interior and exterior domain problems, and a method for indirectly calculating the Cauchy principal value of the singular integrals is presented. Formulae for the stress and flux inside the domain are also derived. Finally, the algorithm is validated by comparing the numerical results with the analytical solution of a borehole subject to a far-field deviatoric stress (exterior domain) and with the solution of Mandel's problem (interior domain). These two examples provide a critical test of the algorithm.     

In-Situ strain measurement by photoelastic bar gages, 1. The method

A technique is described for measuring recoverable strain in situ using bridged photoelastic bar gages and a simple reflecting polariscope. Rosettes of 0.2 × 0.4 × 4-cm bars are bonded to the rock for 0.6 cm at each end. Change in strain is measured following overcoring with a 10-cm bit. Correction for strains in the order of 30 · 10^?6/°C, due to differential thermal expansion between rock and gage, are made by monitoring dummy gages mounted on free blocks. Bar gages have been satisfactorily tested against foil resistance gages. Rosettes mounted closely adjacent on a free granite block showed a reasonable homogeneity of response.     

A simple theory for the dispersion of real and apparent finite-strain lineations

In some metamorphic terrains, lineations in folded surfaces are coaxial to the folds at their hinges, but show a systematic dispersion on the limbs. A simple theoretical model is presented, based on two assumptions: (1) the layering is folded according to two idealised models, “ideal compression folding” and “ideal shear folding”, which assume that the rock material is homogeneous and the layering passive; (2) the lineation is a manifestation of the total product of the pre-folding and folding strains. In an ideal compression fold, only apparent lineations can be dispersed away from the fold-axial trend; in an ideal shear fold, however, both real and apparent lineations are dispersed in a similar way, the degree of similarity depending on the X/Y ratio of the pre-fold strain. The lineation loci of the two models are sufficiently distinct for them to be used, together with other features of the fabric, to distinguish between folds produced by dominantly vertical movements, and those produced by dominantly horizontal movements.