Mathematical Geosciences

, Volume 41, Issue 6, pp 611–641 | Cite as

Directional Wavelets and a Wavelet Variogram for Two-Dimensional Data

  • E. H. Bosch
  • A. P. González
  • J. G. Vivas
  • G. R. Easley
Special Issue


This paper describes two new approaches that can be used to compute the two-dimensional experimental wavelet variogram. They are based on an extension from earlier work in one dimension. The methods are powerful 2D generalizations of the 1D variogram that use one- and two-dimensional filters to remove different types of trend present in the data and to provide information on the underlying variation simultaneously. In particular, the two-dimensional filtering method is effective in removing polynomial trend with filters having a simple structure. These methods are tested with simulated fields and microrelief data, and generate results similar to those of the ordinary method of moments variogram. Furthermore, from a filtering point of view, the variogram can be viewed in terms of a convolution of the data with a filter, which is computed fast in O(NLogN) number of operations in the frequency domain. We can also generate images of the filtered data corresponding to the nugget effect, sill and range of the variogram. This in turn provides additional tools to analyze the data further.


Directional wavelets Wavelet analysis Trend analysis Variogram 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Antoine J-P, Murenzi R, Vandergheynst P (1999) Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns. Appl Comput Harmon Anal 6(3):314–345 CrossRefGoogle Scholar
  2. Boogaart KG, Brenning A (2001) Why is universal kriging better than IRFk-kriging: estimation of variograms in the presence of trend. In: Ross G (ed) Proceedings of 2001 annual conference of the international association for mathematical geology, CD-ROM, September 6–12, 2001, Cancún, Mexico Google Scholar
  3. Bosch EH, Oliver MA, Webster R (2004) Wavelets and the generalization of the variogram. Math Geol 36(2):147–186 CrossRefGoogle Scholar
  4. Bruce LM, Koger CH, Li J (2002) Dimensionality reduction of hyperspectral data using discrete wavelet transform feature extraction. IEEE Trans Geosci and Remote Sens 40(10):2331–2338 CrossRefGoogle Scholar
  5. Candès EJ, Donoho DL (2002) New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities. Commun Pure Appl Math 57(2):219–266 CrossRefGoogle Scholar
  6. Chiles J-P, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York Google Scholar
  7. Coifman RR, Wickerhauser MV (1995) Adapted waveform “de-Noising” for medical signals and images. IEEE Eng Med Biol 14(5):578–586 CrossRefGoogle Scholar
  8. Coifman RR, Lafon S (2006) Diffusion maps. Appl Comput Harmon Anal 21:5–30 CrossRefGoogle Scholar
  9. Daubechies I (1988) Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41(7):909–996 CrossRefGoogle Scholar
  10. Dimitrakopoulos R (1990) Conditional simulation of intrinsic random functions of order k. Math Geol 22(3):361–380 CrossRefGoogle Scholar
  11. Donoho DL (2001) Sparse components of images and optimal atomic decomposition. Constr Approx 17(3):353–382 CrossRefGoogle Scholar
  12. Easley GR, Labate D, Lim W (2006) Optimally sparse image representations using shearlets. In: Proceedings of IEEE 40th asilomar conference on signals, systems, and computers, CA, pp 974–978. doi: 10.1109/ACSSC.2006.354897
  13. Etemad K, Chellapa R (1995) Dimensionality reduction of multi-scale feature spaces using a separability criterion. IEEE Proc Int Conf Acoust Speech Signal Proc 4:2547–2550 Google Scholar
  14. Jasper WJ, Garnier SJ, Potlapalli H (1996) Texture characterization and defect detection using adaptive wavelets. Opt Eng 35(11):3140–3149 CrossRefGoogle Scholar
  15. Lark RM, Webster R (1999) The analysis and elucidation of soil variation using wavelets. Eur J Soil Sci 50(2):185–206 CrossRefGoogle Scholar
  16. Lark RM (2006) The representation of complex soil variation on wavelet packet bases. Eur J of Soil Sci 57(6):868–882 CrossRefGoogle Scholar
  17. Lark RM, Webster R (2001) Changes in variance and correlation of soil properties with scale and location: Analysis using an adapted maximal discrete wavelet transform. Eur J Soil Sci 52(3):547–562 CrossRefGoogle Scholar
  18. Lark RM, Cullis BR (2004) Model-based analysis using REML for inference from systematically sampled data on soil. Eur J Soil Sci 55:799–813 CrossRefGoogle Scholar
  19. Li L, Qian W, Clarke LP (1996) X-ray medical image processing using directional wavelet transform. IEEE Proc Int Conf Acoust Speech Signal Proc 4:2251–2254 Google Scholar
  20. Lin C, Lee Y, Ou H (2000) Satellite sensor image classification using cascaded architecture of neural fuzzy network. IEEE Trans Geosci Remote Sens 38(2):1033–1043 CrossRefGoogle Scholar
  21. Maengseok N, Youngjo L (2007) REML estimation for binary data in GLMMs. J Multivar Anal 98(5):896–915 CrossRefGoogle Scholar
  22. Mallat S (1998) A wavelet tour of signal processing. Academic Press, San Diego Google Scholar
  23. Matheron G (1965) Les variables régionalisées et leur estimation. Masson, Paris Google Scholar
  24. Mohan A, Sapiro G, Bosch E (2007) Spatially coherent nonlinear dimensionality reduction and segmentation of hyperspectral images. IEEE Geosci Remote Sens Lett 4(2):206–210 CrossRefGoogle Scholar
  25. Oliver MA, Bosch EH, Slocum K (2000) Wavelets and kriging for filtering and data reconstruction. In: Kleingeld, WJ, Krige, DG (eds) Geostatistics 2000, Cape Town, vol 2, pp 571–580 Google Scholar
  26. Walker JS (1999) A primer on wavelets. Chapman and Hall/CRC, Boca Raton Google Scholar
  27. Walnut DF (2002) An introduction to wavelet analysis. Birkhauser, Boston Google Scholar
  28. Walter GG (1994) Wavelets and other orthogonal systems with applications. CRC Press, Boca Raton Google Scholar
  29. Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New York CrossRefGoogle Scholar
  30. Whitcher B, Guttorp P, Percival DB (2000) Wavelet analysis of covariance with application to atmospheric time series. J Geophys Res 105(D11):14941–14962 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2009

Authors and Affiliations

  • E. H. Bosch
    • 1
  • A. P. González
    • 2
  • J. G. Vivas
    • 3
  • G. R. Easley
    • 4
  1. 1.National Geospatial-Intelligence AgencyRestonUSA
  2. 2.Universidad de La CoruñaCoruñaSpain
  3. 3.Universidade Federal da BahiaSalvadorBrazil
  4. 4.University of MarylandCollege ParkUSA

Personalised recommendations