Mathematical Geology

, Volume 34, Issue 5, pp 479–503 | Cite as

Kriging of Regionalized Directions, Axes, and Orientations I. Directions and Axes

  • K. Gerald van den Boogaart
  • Helmut Schaeben


The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector space is presented. Its explicit justification in terms of mathematical assumptions concerning stationarity/homogeneity and isotropy is included. The data are modelled by a stationary random field, and the spatial correlation is represented by modified multivariate variograms and covariance functions. Various types of isotropy assumptions concerning invariance under translation/rotation of the data locations, the measurements, or a combination of both, can be distinguished and lead to different simplifications of the general cross-covariance function. Beyond spatial prediction a measure of confidence in the estimates is provided.

geostatistic manifolds isotropy assumptions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitchison, J., 1986, The statistical analysis of compositional data: Chapman and Hall, London, 416 p.Google Scholar
  2. Chilès, J.-P., and Delfiner, P., 1999, Geostatistics: Modeling spatial uncertainty: Wiley, New York, 695 p.Google Scholar
  3. Cressie, N. A. C., 1993, Statistics for spatial data, Rev. edn.: Wiley, New York, 900 p.Google Scholar
  4. Freeden, W., Gervens, T., and Schreiner, M., 1998, Constructive approximation on the sphere with applications in geomathematics: Oxford Science Publications, Oxford, 427 p.Google Scholar
  5. Hoschek, J., and Seemann, G., 1992, Spherical splines: Math. Modelling Num. Anal., v. 26, no. 1, p. 1–22.Google Scholar
  6. Jupp, P. E., and Kent, J. T., 1987, Fitting smooth paths to spherical data: J. R. Stat. Soc., Ser. C, v. 36, no. 1, p. 34–46.Google Scholar
  7. Lee, J.-C., and Angelier, J., 1994, Paleostress trajectory maps based on the results of local determinations: The “lissage” program: Comput. Geosci., v. 20, no. 2, p. 161–191.Google Scholar
  8. Mendoza, C. E., 1986, Smoothing unit vector fields: Math. Geol., v. 18, no. 3, p. 307–322.Google Scholar
  9. Parker, R. L., and Denham, C. R., 1979, Interpolation of unit vectors: Geophys. J. R. Astr. Soc., v. 58, no. 3, p. 685–687.Google Scholar
  10. Pawlowsky, V., Olea, R. A., and Davis, J. C., 1995, Estimation of regionalized compositions: A comparison of three methods: Math. Geol., v. 27, no. 1, p. 105–127.Google Scholar
  11. Prentice, M. J., 1987, Fitting smooth paths to rotation data: J. R. Stat. Soc., Ser. C, v. 36, no. 3, p. 325–331.Google Scholar
  12. Schumaker, L. L., and Traas, C., 1991, Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines: Num. Math., v. 60, no. 2, p. 133–144.Google Scholar
  13. Traas, C., Siemes, H., and Schaeben, H., 1994, Smoothing pole figures using tensor products of trigonometric and polynomial splines, in Bunge, H. J., ed., Proceedings of the 10th international conference on textures of materials (ICOTOM10): Materials Science Forum, Aedermannsdor, Vol. 157-162, p. 453–458.Google Scholar
  14. van den Boogaart, K. G., 1999, Kriging von Richtungen, Achsen und Orientierungen: Terra Nostra, Schriften der Alfred-Wegener-Stiftung 99/1, Old crusts new problems: Geodynamics and utilisation, 89th annual meeting of the Geologische Vereinigung e.V. Abstracts und Programme, Freiberg, February 22-26, 1999, K¨oln, p. 62-63.Google Scholar
  15. van den Boogaart, K. G., and Schaeben, H., 2000, Estimation of regionalized polar and axial unit vectors, in Proceedings of the 20th gOcad meeting, Nancy, June 19-23, 2000, 9 p.Google Scholar
  16. Wackernagel, H., 1998, Multivariate geostatistics, 2nd completely revised version: Springer, Berlin, 291 p.Google Scholar
  17. Watson, G. S., 1985, Interpolation and smoothing of directed and undirected line data, in Krishnaiah, P. R., ed., Multivariate analysis VI: Elsevier, Amsterdam, p. 613–625.Google Scholar
  18. Young, D. S., 1987a, Random vectors and spatial analysis by geostatistics for geotechnical applications: Math. Geol., v. 19, no. 6, p. 467–480.Google Scholar
  19. Young, D. S., 1987b, Indicator kriging for unit vectors: Rock joint orientations: Math. Geol., v. 19, no. 6, p. 481–502.Google Scholar

Copyright information

© International Association for Mathematical Geology 2002

Authors and Affiliations

  • K. Gerald van den Boogaart
    • 1
  • Helmut Schaeben
    • 1
  1. 1.Mathematical Geology and Computer Sciences in GeologyFreiberg University of Mining and TechnologyFreibergGermany

Personalised recommendations