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_Journal of Structural Geology 32 (2010) 975e981_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg Strain analysis from point fabric patterns: An objective variant of the Fry method Richard J. Lisle School of Earth and Ocean Sciences, Cardiff University, Cardiff CF10 3YE, United Kingdom Article info Article history: Received 25 August 2009 Received in revised form 2 June 2010 Accepted 16 June 2010 Available online 23 June 2010 Keywords: Strain analysis Fry method Statistical tests Point patterns Abstract A simple technique is devised for obtaining finite strain estimates from deformed patterns of points possessing anticlustered properties. As an alternative to the Fry approach where analysis is carried out in the deformed state, the new method performs a large number of trial de-strainings of the point pattern. For each de-strained dataset, a statistical analysis tests for the assumed properties of the point fabric in the pre-deformational state, that the points come from an isotropic but non-Poisson distribution. This computer-based procedure usually yields a number of acceptable solutions for the possible strain ellipse in the given geological situation. The variability of the strain estimates for a given dataset provides an assessment of the precision of the strain results. Testing of the method with both synthetic and real datasets suggests that a best estimate for the strain ellipse is one corresponding to the centre of gravity on the Elliott strain plot. The performance of the new method compares favourably with existing Fry methods. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Geological strain analysis requires geometrical information relating to the rock in its pre-deformational and deformed states. The entities that yield this information are referred to as strain markers. Some strain markers consist of individual objects with specific shape, e.g., fossils, ooids, and sedimentary clast outlines. Other strain markers may consist of collections of objects with particular orientation patterns, e.g., preferred orientation patterns of muscovites in a sample of slate. However, a problem arises where neither type of strain marker is present in the rock being analyzed. To overcome this, Ramsay (1967) recognized a third type of strain marker where the necessary geometrical information is provided by mutual spatial arrangements of objects in the rock. This led to the development of strain analysis techniques based on the distances of separation of point objects in sedimentary, metamorphic and igneous rocks (Ramsay, 1967, p. 195; Fry, 1979; Hanna and Fry, 1979). The most popular of these inter-object distance techniques is the all-object separation method devised by Fry (1979). This is a strain method based on a polar plot of vectors depicting the separation of the centres of all possible pairs of objects. Assuming that pairs of adjacent objects were no closer than a certain threshold distance in the pre-deformation state, the plotted vectors of the inter-object distances in the deformed state will surround an elliptical region devoid of data points, the so-called vacancy field. If the threshold distance applies strictly, the elliptical region portrays directly the shape of the finite strain ellipse. The Fry method has been used to determine the finite strain in deformed clastic sedimentary rocks (Treagus and Treagus, 2002), in gneisses (Lacassin and van der Driessche, 1983), igneous rocks (Schwerdtner et al., 1983), etc. where grain centres provide the required object data. In addition, the method has been applied to deformed patterns of points not consisting of the centres of grains. Examples of such non-granular data used for Fry analysis are sand volcanoes (Waldron and Jensen, 1985). For the majority of applications of the Fry method, the choice of the ellipse that best describes the periphery of the vacancy field is normally a source of ambiguity. This uncertainty arises where a strict inter-point threshold distance does not exist because neighbouring points show a weaker tendency for constant separation, i.e., a lesser degree of anticlustering. This spatial variability arises from the stochastic nature of the underlying processes that typically form the point patterns studied, which has motivated several refinements of the Fry method that attempt to compute the best-fit ellipse as an alternative to visual selection (Waldron and Wallace, 2007). After an estimate of the finite strain has been obtained by selection of the best-fitting ellipse, a problem remains of assessing the robustness of this estimate (Crespi, 1986). This is because different pre-deformation point distributions show different degrees of anticlustering. In cases of strong anticlustering (Fig. 1a), the shape of the vacancy field should approximate closely to that of the strain ellipse (Fig. 1b). However, in cases of very weak anticlustering, where the undeformed distribution of points approaches that resulting from a random (Poisson) distribution 976 R.J. Lisle Journal of Structural Geology 32 (2010) 975e981 heterogeneities in strain, such as those arising from the competence contrasts between grains and matrix, need not invalidate the method. 2) The strain history is coaxial, i.e., infinitesimal strains during the deformation are parallel to finite strains. This assumption is a cautionary one and is made to avoid special situations where strain may leave the original point configuration unaltered, e.g., the ‘invisible’ simple shear deformation of regular lattice patterns of points (Genier and Epard, 2007). Fig. 1. Anticlustering and Fry plots. A, Grains in deformed rock with strongly anticlustered centres; B, corresponding Fry plot with well defined ellipse; C, Weakly anticlustered grain centres; D, corresponding Fry plot with poorly defined ellipse. (Fig. 1c), the vacancy field in the deformed point pattern will have a size influenced by the number of points in the dataset and a shape that is largely independent of the applied strain (Fig. 1d). Although several methods have been suggested for determining the strain ellipse from point data, with the exception of McNaught (2002) for granular point data, none of these allows assessment of confidence limits for the obtained result. The aim of this paper is to propose a new and simple method of strain analysis from point datasets of granular and non-granular types. The proposed method addresses both problems of subjectivity in obtaining a strain estimate, and the issue of qualifying the robustness of that estimate. 2. The assumptions of the method The Fry plot has in the past been used as a display for a variety of types of spatial point patterns without necessarily a strain connotation. Examples are spatial distributions of mineralization (Vearncombe and Vearncombe, 1999) and of star clusters (Cartwright and Whitworth, 2008). Even when the point data are grain centres in a tectonite, the resulting elliptical Fry plot is not guaranteed to have a direct relationship to the finite strain ellipse. For this reason it seems advisable to generally refer to the Fry pattern as a descriptor of a point fabric ellipse rather than as the strain ellipse (Erslev, 1988). Only when the user is willing to make certain fundamental assumptions about the nature of the starting point patterns and of the imposed deformation, may the Fry plot be used to deduce the strain ellipse. In this context, the proposed method determines the strain as an inverse problem; it uses the data from a deformed point distribution to constrain the principal parameters that had an influence on the final configuration of the deformed points. To estimate the magnitude and orientation of the finite strain ellipse, assumptions are made regarding the character of the strain and of the initial point distribution. The assumptions concerning the strain are: Concerning the initial point configuration, the pattern is assumed to be isotropic in the sense that distances between pair of points, whether immediate neighbours or not, do not vary systematically as a function of the orientation of the tie line. An example of such a population of point pairs is illustrated in a polar graph of the ends of vectors representing the lengths and orientations of the tie-lines drawn between all possible pairs of points in the sample (Fig. 2). Fry (1979) used this plot for the visual estimation of the strain ellipse. For the method, the plot is divided into three concentric hoop-shaped sampling cells. The assumption of isotropy means that a successful de-straining of the point data produces a uniform distribution of directions within each hoop-shaped sampling cell. 3. The method Strain analysis using the proposed method involves a three-stage process. Firstly, data consisting of the coordinates of points in the deformed array are acquired and the inter-point vector (Fry) diagram constructed (Fig. 3a). Secondly, this dataset is subjected to a series of different geometrical transformations, each equivalent to a different retrodeformation of the point array (Fig. 3b). This stage effectively subjects the dataset to a series of trial retrodeformations, each corresponding to the imposition of a strain ellipse with a particular axial ratio and orientation. In the final stage, the characteristics of each of the transformed datasets on the Fry plot are compared to those assumed for the pre-deformation configuration of the points (Fig. 3c, d). Where sufficient similarity exists, the corresponding trial retrodeformation is used to derive an acceptable solution of the finite strain parameters (Fig. 3d). The three stages of the method are performed by a FORTRAN 95 program. 1) The strain is statistically homogeneous on scales ranging from that of the inter-point distances upward to the size of the ov' Ключевые слова: vector, complete revolution, simple, set, cell, shear, data, fry diagram, waldron wallace, grid, separation, circle, rmist, elliott plot, paor, strain ellipse, frys method, cheeney, point distribution, problem, trial de-strainings, strain estimate, isotropy, successful de-straining, strain, waldron, deformed pattern, geological, distance, solution eld, point fabric, poisson, successful, fabric, trial retrodeformation, hoop, assumption, statistical test, centre, size, datasets, hoop diagram, trial, ?nite, strain plot, type, rock, applied, degree, pair, point data, pre-deformation, diagram, structural, mardia, test statistic, solution, fry plot, uniform grid, ratio, rs, initial isotropy, geology, expected, acceptable, acceptable strain, strain determination, vector diagram, method, expected number, range, mis?t, axial, axial ratio, uniform, structural geology, plot, vearncombe, deformation, examples, test, appendix, area, ramsay, ellipse, mcnaught, table, simple shear, erslev, acceptable solution, real, point encountered, de-strainings, lisle, schwerdtner, result, mis?t factor, deformed, ha, shear strain, direction, produce, object, isotropic, assumed, scale, sample, dened ellipse, determination, crespi, journal structural, de?ned, strain analysis, spatial distribution, anticlustering, number, grain, vacancy, mist factor, analysis, statistical, point, explanation, point pattern, wallace, ellipsis, marker, grid node, pattern, ?nite strain, anticlustered, dataset, vacancy eld, orientation, proposed, grain centre, fry method, shape, fry, elliott, society, factor, uniform distribution, nite strain, strain marker, journal structural geology, sample size, journal, strain ellipsis, technique, total number, difference, poisson distribution, estimate, ?eld, distribution