Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A mathematical model for orientation data from macroscopic elliptical conical folds

Abstract

An iterative least-squares technique to fit circular and elliptical conical surfaces to orientation data from folds is presented. A statistical model is used which assumes that each data point is an observation from a Fisher distribution. The mean of this distribution is assumed to lie on the curve to be fitted. Estimates of variances and covariances for the fitting parameters are calculated, and confidence intervals for the cone axis and half apical angle are estimated from variances and covariances. A normal test with null hypothesis that the cone angle is 90° determines if a conical model fits the data better than a cylindrical model. AnF test is used to determine whether an elliptical cone is a better model than a circular cone. In this fashion, macroscopic folds are classified into cylindrical, circular conical, or elliptical conical folds. Examples of these three types of fold are given. The Wynd Syncline near Jasper, Alberta is the first natural elliptical conical fold described as such.

This is a preview of subscription content, log in to check access.

References

  1. Bevington, P. R., 1969, Data reduction and error analysis for the physical sciences: McGraw-Hill, New York, 336 p.

  2. Charlesworth, H. A. K.; Langenberg. C. W.; and Ramsden J., 1976, Determining axes, axial planes, and sections of macroscopic folds: Can. J. Earth Sci., v. 13, n. 1, p. 54–65.

  3. Cruden, D. M. and Charlesworth, H. A. K., 1972, Observations on the numerical determination of axes of cylindrical and conical folds: Geol. Soc. Amer. Bull., v. 83, n. 7, p. 2019–2024.

  4. Cruden, D. M. and Kelker, D., 1978, Simple graphical methods for estimating the confidence region about the orientation of the intersection of two planes: Can. J. Earth Sci., v. 15, n. 10, p. 1598–1604.

  5. Draper, N. and Smith, H., 1981, Applied Regression Analysis, 2nd ed. John Wiley & Sons, New York, 708 p.

  6. Haman, P. J., 1961, Manual of sterographic projection: West Can. Res. Publ., Ser. 1, n. 1, 67 p.

  7. Hobbs, B. E.; Means, W. D.; and Williams, P. F., 1976, An outline of structural geology: John Wiley & Sons, New York, 571 p.

  8. Johansen, S., 1984, Functional relations, random coefficients and nonlinear regression with application to kinetic data: Lecture Notes in Statistics, v. 22, 126 p.

  9. Johnson, R. A. and Wichern, D. W., 1982, Applied multivariate statistical analysis: Prentice-Hall, Englewood Cliffs, New Jersey, 594 p.

  10. Kelker, D. and Langenberg, C. W., 1982, A mathematical model for orientation data from macroscopic conical folds: Math. Geol., v. 14, p. 289–307.

  11. Kendall, M. G. and Stuart, A., 1973, The advanced theory of statistics, v. 2: Hafner Publishing Co., New York, 723 p.

  12. Langenberg, C. W. 1983, Polyphase deformation in the Canadian shield, NE Alberta: Alberta Research Council, Bull. 45, 33 p.

  13. Langenberg, C. W., 1985, The geometry of folded and thrusted rocks in the Rocky Mountain Foothills near Grande Cache, Alberta: Can. J. Earth Sci., v. 22, p. 1711–1719.

  14. MacKay, B. R. 1929, Cadomin, Alberta: Geological Survey of Canada, MAP 209A.

  15. Marquardt, D. W., 1963, An algorithm for least-squares estimation of non-linear parameters: J. Soc. Ind. Appl. Math., v. 11, n. 2, p. 431–441.

  16. Stauffer, M. R., 1964, The geometry of conical folds: New Zealand J. Geol. Geophys., v. 7, p. 340–347.

  17. Stauffer, M. R., 1967, The problem of conical folding around the Barrack creek Adamellite, Queanbeyan, New South Wales: J. Geol. Soc. Australia, v. 14, n. 1, p. 45–56.

  18. Stesky, R. M., 1985, Least squares fitting of a non-circular cone, Comput. Geosci., v. 11, p. 357–368.

  19. Stockmal, G. S. and Spang, J. H., 1982, A method for the distinction of circular conical from cylindrical folds: Can. J. Earth Sciences, v. 19, p. 1101–1105.

  20. Watson, G. S., 1965, Equatorial distributions on a sphere: Biometrika, v. 52, p. 193–201.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kelker, D., Langenberg, C.W. A mathematical model for orientation data from macroscopic elliptical conical folds. Math Geol 19, 729–743 (1987). https://doi.org/10.1007/BF00893011

Download citation

Key words

  • orientation data
  • circular conical folds
  • elliptical conical folds