Description:

_Journal of Structural Geology 32 (2010) 86–92_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _Visualizing strain and the Rf–F method with an interactive computer program_ _Paul Karabinos a,*, Chris Warren b a Department of Geosciences, Williams College, Williamstown, MA 01267, USA b Office of Information Technology, Williams College, Williamstown, MA 01267, USA_ _article info_ _Article history: Received 9 February 2009 Received in revised form 1 September 2009 Accepted 2 September 2009 Available online 10 September 2009_ _Keywords: Strain analysis Rf–F method Computer program Conglomerate_ _abstract_ _The Rf–F method is a powerful graphical approach for estimating finite strain of deformed elliptical objects, but one that students commonly find difficult to understand. We developed a program that allows users to explore visually how deforming a set of elliptical objects appears on Rf–F plots. A user creates or loads the ellipses and then deforms them by simple shear, pure shear, or rigid rotation. As the ratio of the long to short axis of the ellipses (Rf) and long-axis orientations (F) change in one window, the Rf–F plot continuously and instantaneously updates in another. Users can save snapshots of the deformed elliptical objects and the Rf–F plots to record graphical experiments. The program provides both Rf vs. F and polar ln(Rf) vs. 2(F) plots. The user can ‘undeform’ ellipses quickly and easily, making it possible to inspect the ‘original’ shapes and orientations of objects, and to evaluate the plausibility of the determined strain values. Users can export information about the pebbles' shape and orientation to spreadsheets for rigorous statistical analysis. This program is written in Java and so can run on virtually any operating system. Both the source code and the application will be freely available for academic purposes._ _1. Introduction_ _Deformed ellipsoidal objects, such as pebbles and oolites, are common in rocks, and they offer an intuitive, visually appealing approach for teaching fundamental strain concepts in structural geology. Students can easily grasp the effect of strain on an initially spherical object and, with practice, can visualize the fate of initially ellipsoidal markers. Furthermore, the study of deformed pebble conglomerate and oolitic limestone provides an excellent opportunity for students to gain experience in data acquisition and error analysis. They must also confront a host of important problems that plague all attempts to quantify strain, such as ductility contrast between marker object and matrix, initial shape and distribution of marker objects, area or volume change during deformation, and the relationship between two-dimensional strain measured in planar sections with the three-dimensional strain experienced by rocks._ _Structural geologists commonly exploit elliptical objects in their research to quantify strain in naturally deformed rocks, understand the development of deformation fabrics, and examine strain gradients in folds and fault zones. The importance of this approach to strain measurement has inspired numerous studies to overcome its inherent limitations, or at least to understand them thoroughly._ _* Corresponding author. Tel.: +1 413 597 2079; fax: +1 413 597 4116. E-mail address: pkarabin@williams.edu (P. Karabinos)._ _0191-8141 $ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2009.09.001_ _Through numerical experiments, Lisle (1979) tested several methods for averaging shape and orientation data to determine the most accurate for estimating the strain ellipse, and concluded that the harmonic mean was the most reliable. Hossack (1968) and Treagus and Treagus (2002) also discussed in detail the problems of determining strain from pebble shapes in a conglomerate._ _Ramsay (1967) derived the equations of the Rf–F method for quantifying finite strain, and Dunnet (1969) showed how the Rf–F method can be used as a practical tool for strain determination from elliptical objects. As discussed in detail below, the Rf–F method assumes an initially random distribution of ellipse long-axis orientations, and a range of initial long to short axial ratios, Ri (Table 1). Ramsay and Huber (1983) presented an especially useful, and well-illustrated, discussion of the Rf–F method making the technique more accessible to researchers and students. Lisle (1985) offered a very complete and useful treatment of the method. Our contribution is to provide a program that links deformation of elliptical objects with Rf–F plots, giving students a visual explanation of how the method works, and offering students and researchers a tool to quickly estimate strain from outcrops and samples._ _Our program provides both the familiar Cartesian Rf–F plot and the innovative polar plot of Elliott (1970) for comparison with the ellipse population. Elliott’s (1970) approach employed a novel ‘‘shape factor grid’’ and a polar plot of ln(Rf) vs. 2(F) that should, in theory, allow assessment of the initial distribution of long axes of elliptical objects, and estimate the strain. The greatest limitation of this approach is the apparent complexity of the distribution of undeformed elliptical objects (Boulter, 1976; Paterson and Yu, 1994). Yamaji (2005) developed an inverse method to overcome some of these limitations for the special case of a bivariate normal distribution of sedimentary particles. Another limitation to Elliott’s (1970) approach, and a possible explanation for why the method has been underutilized, is the significant difficulty most users have with visualizing the effect of strain on elliptical objects in polar plots. Our program helps overcome the latter limitation by showing how Cartesian and polar Rf–F plots change during deformation._ _Lisle’s (1985) approach to testing the assumption of an initially random distribution of pebble long axes was to create a set of ‘‘marker deformation grids’’ using Cartesian Rf–F plots, and to examine the distribution pattern of deformed ellipses. Sets of pebbles that were consistent with the assumption should show a symmetrical pattern about both the harmonic mean of Rf and the vector mean of F. De Paor (1988) developed another novel and useful approach to the Rf–F method that uses a hyperbolic net and symmetry principles to estimate strain from ellipsoids, but our program does not include hyperbolic plots._ _Several commercially available drafting programs allow users to create a set of elliptical objects and to simulate deformation with tools that linearly transform the ellipses by pure shear, simple shear, and rigid rotation. These programs are very useful for teaching purposes. Also, several commercial programs are available that permit researchers to determine strain with the Rf–F method using axial ratio and orientation data. Some of these programs incorporate statistical methods to assess the validity of the assumption of initial random distribution of long-axis orientations. We have not duplicated the capabilities of these programs. Instead, we developed a relatively simple program that focuses on visualizing the relationship between strained elliptical objects and plots of axial ratio vs. orientation. Our program is complimentary to the existing software because it is straightforward to examine deformed objects with our program and then, to export information about the ellipses to data files for use in these programs._ _Our goal was to create a simple and easy-to-learn interactive computer program that allows the user to simulate deformation of elliptical objects by pure shear, simple shear, and rigid rotation. Throughout the linear transformations, Cartesian or polar Rf–F plots are continuously and instantaneously updated. The advantage of this program is that the color coding and tracking options make it possible to visualize the distribution and paths of points representing elliptical objects on Rf–F plots. This is especially valuable for the polar plots of ln(Rf) vs. 2(F), and it helps highlight the potential of this neglected approach. We also show here how the program is used to introduce students to the Rf–F method, determine strain in natural samples, and simulate ‘retro-deformation’ of samples to recover the original shapes and orientation of the pebbles for critical evaluation of the method. This program is written in Java, and so can run on virtually any operating system. Both the source code and the finished application will be freely available for academic purposes._ _Fig. 1. Screen captures of the entire program window. A. Individual ellipses can be drawn in editing mode with a wide variety of colors. B. Photographs (up to 700 by 700 pixels) of deformed objects, such as these quartz pebbles, are imported in the display window for tracing ellipses as shown in blue._ _2. Summary of the program_ _The program contains a large display window on the left and a display control window on the right (Fig. 1A). Ellipses are created in the display area by dragging with the mouse in editing mode. The user can import a 700 by 700 pixel photograph or other image as a background, and trace elliptical objects from it (Fig. 1B). Alternatively, text files containing information about the position, shape, and orientation of elliptical objects can be imported._ Ключевые слова: tectonophysics, radial distribution, simple, point centered, image, shear, data, re-sizeable window, deformation fabric, rf–f plot, initial shape, vertical shortening, instantaneously updated, display screen, neglected, graphical, paor, strain ellipse, horizontal shortening, unstrained conguration, outlier, reference, deformed display, yamaji, elliptical, strain ratio, strain estimate, snapshot, pebble outline, strain, user click, ln rf, imported, onasch, geological, undeformed, deformed object, pure shear, short, pattern reects, stretching direction, tool, inverse strain, simulated deformation, rigid rotation, inverse, editing mode, estimate strain, pebble, toggles, rfmin, open, random, elliptical object, highlight, long-axis, structural, rotation, orientation data, developed, horizontal, ind var, rf–f method, button, ratio, ?les, rs, long axis, maximum, geology, strain determination, rigid, rf, method, range, axial, critical, axial ratio, structural geology, student, plot, track, deformation, program, radio button, area, simulate, press, ramsay, ellipse, rf–f, preferentially oriented, area change, simple shear, rfmax, excel workbook, practical, save, polar plot, wa, change, application, lisle, deformed, equation, create, direction, vertical, rff method, long, limitation, entire range, magni?cation, valuable, object, rfmax rfmin, cartesian, photograph, background image, sample, journal structural, position, strain analysis, selected, axis, pure, user, karabinos, number, window, shortening, analysis, ax, conglomerate, point, source code, ellipsis, hossack, click, rimax, initial axial, dunnet, pixel, display window, rff plot, screen capture, form, orientation, initial, long-axis orientation, display area, program doe, polar, background, ?le, shape, boulter, quantify strain, elliott, simulate deformation, determined, ln, initially, journal, shape orientation, open congurations, technique, deformed pebble, estimate, extension, display, approach, initially random, maximum initial, distribution, random distribution