Computers & Geosciences 28 (2002) 81–85
Short Note
STEREOFIT: A program for fitting directional data to multiple planes and some applications in structural geology$ Domingo Aerdena,*, Javier Sanchez San Romanb b
a Departamento de Geodinamica, Universidad de Granada, c/Fuentenueva s/n, Granada 18071, Spain Area de Geodinamica, Departamento de Geologia, Universidad de Salamanca, Salamanca 37008, Spain
1. Introduction Most programs for plotting and analyzing structural data (Allmendinger et al., ftp://silver.geo.cornell.edu/ pub/rwa programs; De Paor, 1996; Duyster, http: // homepage.ruhr-uni-bochum.de / Johannes.P.Duyster/ stereo/stereo1.htm; Mancktelow http://www.erdw. ethz.ch/Bneil/stereoplot.html; van Everdingen http:// www.geomem.co.uk/geomem /products/quikplot.html.) allow a best-fit plane to be determined for a set of lines. This can be done algebraically by minimizing the sum of the squares of deviations between poles and the best-fit plane (Ramsay, 1967), or numerically by calculating the average angle of data with a finite number of differently oriented test planes, and finding the one yielding the smallest value. In structural geology a best-fit plane is frequently calculated to determine the average orientation of fold axes in a region. If the fold structure is approximately cylindrical, lines perpendicular to the folded surface (‘‘poles’’) will be contained in a plane normal to the fold axes (Ramsay and Huber, 1987; Fig. 1A). However, natural fold shapes often deviate from ideal cylindrical. In particular, superposed fold generations can produce complex geometries, and lead to pole distributions in two or more planes (e.g. Ragan, 1973; Mulchrone, 1991; Fig. 1B). We have written a program in FORTRAN that allows lines to be fitted to up to three planes. Possible applications are described and illustrated with two examples. Appendix sections referred to in this paper and the complete computer code can be viewed or downloaded from http://iamg.org/ $
Code available from server at http://www.iamg.org/ CGEditor/index htm. *Corresponding author. Tel.: +349-5827-2883; fax: +3495824-8527. E-mail address:
[email protected] (D. Aerden).
CGEditor/index.htm. An executable Macintosh application can be requested from the first author. The problem in calculating multiple best-fit planes resides in the large number of combinations that can be formed with a small number of elements. In spherical coordinates, 32,401 planes are defined for a coordinate spacing of 11. From these, 5.1 108 different plane pairs can be formed, and over 5.4 1012 plane triplets. Testing all these for a data set of, let us say, 1000 lines is not practical even with modern computers. Consequently, our program starts with an initial set of only 43 regularly spaced test planes (Step 1), calculates a preliminary best-fit solution with these planes, and subsequently refines it in an iterative loop (Step 2) until a resolution of 11 is reached.
2. Program description Step 1: Calculation of a preliminary best-fit solution. The 43 initial test planes are defined by superposing a square grid on an equal-area stereonet with radius conveniently chosen as 13. Grid-line intersections define the poles of test planes (‘‘test poles’’), and two poles are added ‘‘manually’’ on the primitive circle (Fig. 2). Grid coordinates are converted to spherical coordinates using the formulas for equal-area projection (Appendix-I on server), and the user is asked whether to make a best fit to one, two or three planes. Via a dialogue box a data file is selected containing two columns for plunge and plunge direction (azimuth) of a set of lines. The program reads the data and calculates a best-fit solution by minimizing the average deviation of the data with a single plane, a plane pair or a plane triplet. For this purpose, the angle of each data with each of the 43 test poles is calculated and stored in a two-dimensional array called ‘‘ANGLE(i, j)’’, where i and j label data and test planes,
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 4 6 - 2
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respectively. Thus, for 1000 data, the array contains 1000 43 values. From this point onward, the program follows different paths for best fits to one, two or three
Fig. 1. (A) Cylindrical fold with poles of folded surface contained in one plane whose normal is fold axis. (B) Refolded fold with poles of folded surface contained mainly in two planes perpendicular to two predominant fold axis directions.
Fig. 2. Definition of regularly distributed test poles by superposing a rectangular grid on an equal-area stereonet. Two additional points on perimeter are defined separately to fill gaps left by grid points. Figure corresponds to Appendix-I (see http://iamg.org/CGEditor/index.htm).
planes, but as these paths are essentially analogous, only the case of three planes will be described here. For three planes, the heart of the program is an algorithm composed of four nested loops (Appendix-II on server), designed to test all possible plane triplets (10,660 total) that can be made with the initial 43. For each triplet, the following procedure is executed: the angles between the first data and the three poles of the first plane triplet (already stored in ‘‘ANGLE(i, j)’’) are compared and the largest value is retained. This is repeated for the second data, the third, and so on until the last data line. The sum of all retained angles is stored in a three-dimensional array called SCORE(i, j, k), where i, j and k identify the tested plane triplet. After this has been completed for all 10,660 plane triplets, the largest value in SCORE(i, j, k) is searched and the corresponding plane triplet (i, j, k) selected as the preliminary best-fit solution. Step 2: Iterative refinement. A new set of 27 test poles is defined in three square clusters centered about the three preliminary best poles of Step 1 (Fig. 3; Appendix-III on server), whereby the grid spacing is reduced by half. New plane triplets are now formed by combining one pole out of each cluster (9 9 9=729 triplets), and they are tested in a similar way as described for Step 1. The newly obtained best triplet is returned to the start of Step 2 for a second iteration with again half the grid spacing. During each subsequent iteration, grid spacing is again halved until, after four iterations, a resolution of about 11 is reached. When pole clusters are redefined close to the perimeter of the stereonet, some poles fall outside it and obtain a negative plunge. Such poles are transferred to the opposite side of the stereonet by summing 1801 to
Fig. 3. Hypothetical example of test-pole definition during Step 1, and during first two iterations of Step 2. Poles falling outside primitive circle are transferred to opposite side of stereonet. Figure corresponds to Appendix-III (see http://iamg.org/ CGEditor/index.htm).
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their azimuth and changing the sign of their plunge. In this way, clusters are always adequately defined, although not necessarily aligned along grid lines (Fig. 3). Step 3: Listing and evaluation of results. Once the final best-plane triplet is identified, the average- and standard deviation of the data are calculated and printed (Appendix IV on server). Data are listed in three blocks for each best plane, which can be copied to the user’s favorite plotting program to visualize results (e.g. Figs. 4 and 5). Also the deviation of individual data and their list number in the input file are printed. In order to compare best fits to one, two and three planes for the same data set, and to select one as the most adequate solution, the average deviation of data is calibrated against uniform data. That is, the average deviation is divided by the average deviation of pseudo-uniform data from the same best-fit planes. This ratio is printed as
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‘‘0.641 the average deviation of uniform data’’ (see Appendix IV on server), and gives an indication of the ‘‘tightness’’ or quality of a solution. For example, an average deviation of 101 of data from two planes and a tightness ratio of 0.25 is better as an average deviation of 81 from three planes but a tightness ratio of 0.5.
3. Application Application to the analysis of superposed folds is illustrated with an example of a km-scale cross-fold developed in low-grade Palaeozoic sediments of the Variscan belt in NW-Spain (Fig. 4A). Poles to bedding exhibit a wide spread, but are separated in two groups according to two best-fit planes (Fig. 4B). The poles of the latter match fold-axis orientations estimated
Fig. 4. (A) Simplified geological map showing traces of bedding in Armada-Pallide tectonic klippen in Cantabrian Zone of Variscan Belt in NW-Spain outlining cross-fold structure. (B) Best fit of 66 bedding orientations to two planes yields tight average deviation of 8.91 (0.466 average deviation for uniform data). Poles to best-fit planes (triangles) match fold axis directions estimated from geological map (solid arrows). Results were plotted with Mancktelow’s Stereoplot program. (C) Best fit of same data to small circle (cone) performed using Mancktelow’s Stereoplot gives slightly higher standard deviation.
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Fig. 5. (A) Artificial example of area with polyphase folding modified after Ragan (1973). Solid symbols are Ragan’s original data. Remainder of data have been added to obtain an even aerial distribution. (B) Best fit of poles corresponding to orientation data in (A) to three planes. Poles of calculated best-fit planes are represented as arrows in (A) to show that they match fold axis trends. (C) Best fit of same data to two planes results in merging of two best-fit planes in (B) whereas third remains practically unchanged. (D) Best fit to cone (small circle) using Mancktelow’s Stereoplot is clearly unfavorable with respect to best fit to three planes.
from the geological map. Thus, the example can be considered equivalent to Fig. 1B. A second example is an artificial map constructed by Ragan (1973), in which we added additional orientation symbols to simulate an even aerial distribution of data (Figs. 5A–C). Statistical output parameters indicate that a best fit to three planes is favored over two planes, and the poles of the three planes match fold-axis trends observable on the map. The data from both examples were also fitted to cones (small circles) using Mancktelow’s ‘‘Stereoplot’’ (Figs. 4C and 5D), as this is favored when the style of folding is close to concentric. Indeed, superposed concentric folds of similar wavelength will produce a dome-and-basin structure with a perfect distribution of poles in a cone. At the other extreme, however, two generations of Chevron-type folds (angular fold hinges connected by long, straight limbs) will yield a
perfect match of poles to two planes. A continuous range of intermediate situations can exist in nature, but the poles of our examples are better fitted to planes (Figs. 4 and 5). Other possible applications include the quantification of double-girdle distributions of crystallographic axes in tectonites (cf. Sander, 1970), the identification of axial foliation sets preserved in different porphyroblast generations in a rock (Aerden, 2001), or the analysis of joint systems. It should be borne in mind, however, that the results of any statistical analysis must be interpreted with care. In this situation, conclusions should be supported by independent evidence to justify a particular best-fit solution. It should also be realized that, although the minimum number of data necessary to calculate one, two or three best-fit planes is 3, 5 and 8, respectively, many more data are required for a meaningful result.
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Acknowledgements We appreciate and have benefited from the comments of three anonymous referees. D.A. gratefully acknowledges a 1 year visiting lectureship at the University of Salamanca.
References Aerden, D.G.A., 2001. Preferred orientation of porphyroblast inclusion trails in Variscan NW-Iberia: a record of subduction, collision and orocline development. Journal of structural Geology, submitted. Allmendinger, R., et al., ‘‘Stereonet’’. ftp://silver.geo.cornell. edu/pub/rwa programs. De Paor, D.G. (Ed.), 1996. Structural Geology and Personal Computers. Pergamon, Oxford, 542pp.
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Duyster, J., ‘‘Stereonett’’. http://homepage.ruhr-uni-bochum. de/Johannes.P.Duyster/stereo/stereo1.htm. Mancktelow, N., ‘‘Stereoplot’’. http://www.erdw.ethz.ch/ Bneil/stereoplot.html. Mulchrone, K.F., 1991. The interpretation of fold axial data from regions of polyphase folding. Journal of Structural Geology 13, 275–280. Ragan, D.M., 1973. Structural Geology; An Outline to Geometrical Techniques. Wiley, New York, 208pp. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, 568pp. Ramsay, J.G., Huber, M.I., 1987. The Techniques of Modern Structural Geology, Vol. 2: Folds and Fractures. Academic Press, London, 700pp. Sander, B., 1970. An Introduction to the Study of Fabrics of Geological Bodies. Pergamon Press, Oxford, 641pp. van Everdingen, http://www.geomem.co.uk/geomem/products/ quikplot.html.
