Mathematical Geosciences

, Volume 41, Issue 6, pp 679–701 | Cite as

Two-dimensional Conditional Simulations Based on the Wavelet Decomposition of Training Images

Special Issue

Abstract

Scale dependency is a critical topic when modeling spatial phenomena of complex geological patterns that interact at different spatial scales. A two-dimensional conditional simulation based on wavelet decomposition is proposed for simulating geological patterns at different scales. The method utilizes the wavelet transform of a training image to decompose it into wavelet coefficients at different scales, and then quantifies their spatial dependence. Joint simulation of the wavelet coefficients is used together with available hard and or soft conditioning data. The conditionally co-simulated wavelet coefficients are back-transformed generating a realization of the attribute under study. Realizations generated using the proposed method reproduce the conditioning data, the wavelet coefficients and their spatial dependence. Two examples using geological images as training images elucidate the different aspects of the method, including hard and soft conditioning, the ability to reproduce some non-linear features and scale dependencies of the training images.

Keywords

Wavelet Conditional simulation Model analogue 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arpat G, Caers J (2004) A multiple-scale, pattern-based approach to sequential simulation. In: Leuangthong O, Deutsch CV (eds) Geostatistics, Banff, 2004. Springer, Dordrecht, pp 255–264 Google Scholar
  2. Chiles J-P, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley series in probability and statistics. Wiley, New York Google Scholar
  3. Choi H, Baraniuk R (2001) Multiscale image segmentation using wavelet domain hidden Markov models. IEEE Trans Image Process 10:1309–1321 CrossRefGoogle Scholar
  4. Crouse MS, Nowak RD, Baraniuk RG (1998) Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans Signal Process 46(4):886–902 CrossRefGoogle Scholar
  5. Daly C (2004) Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong O, Deutsch CV (eds) Geostatistics, Banff 2004. Springer, Dordrecht, pp 215–224 Google Scholar
  6. Fan G, Xia X-G (2003) Wavelet-based texture analysis and synthesis using hidden Markov models. IEEE Trans Circuits Syst J, Fundam Theory Appl 50:106–120 CrossRefGoogle Scholar
  7. Flandrin P (1992) Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans Inf Theory 35:197–199 CrossRefGoogle Scholar
  8. Gloaguen E, Dimitrakopoulos R (2008). Conditional wavelet based simulation of non-stationary geology using geophysical and model analogue information. In: Ortiz JM, Emery X (eds) Geostats2008, Proceedings of the 8th international geostatistics congress, Santiago FCFM—GecaMin, pp 227–236 Google Scholar
  9. Golub GH, Van Loan CF (1983) Matrix computations. Johns Hopkins Press, London Google Scholar
  10. Goovaerts P (1997) Geostatistics for natural resources evaluation. Applied geostatistics series. Oxford University Press, London Google Scholar
  11. Guardiano F, Srivastava RM (1993). Multivariate geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics Troia, 1992. Kluwer Academic, Dordrecht, pp 133–144 Google Scholar
  12. Journel AG (1989) Imaging of spatial uncertainty: a non-Gaussian approach. In: Buxton BE (ed) Geostatistical, sensitivity and uncertainty methods for ground-water flow and radionuclide transport modeling. Batelle Press, Columbus, pp 585–599 Google Scholar
  13. Julesz B (1962) Visual pattern discrimination. IRE Trans Inf Theory 8:84–92 CrossRefGoogle Scholar
  14. Kumar P, Foufoula-Georgiou E (1997) Wavelet analysis for geophysical applications. Rev Geophys 35:385–412 CrossRefGoogle Scholar
  15. Mallat S (1989) Multifrequency channel decompositions of images and wavelet models. IEEE Trans Acous Speech Signal Process 37:2091–2110 CrossRefGoogle Scholar
  16. Portilla J, Simoncelli EP (2000) A parametric texture model based on joint statistics of complex wavelet coefficients. Int J Comput Vis 40:49–71 CrossRefGoogle Scholar
  17. Rubinstein RY (1981) Simulation and the Monte-Carlo method. Wiley, New York CrossRefGoogle Scholar
  18. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34:1–21 CrossRefGoogle Scholar
  19. Tjelmeland H, Eidsvik J (2004) Directional metropolis: Hastings updates for posteriors with non linear likelihood. In: Leuangthong O, Deutsch CV (eds) Geostatistics, Banff 2004. Springer, Dordrecht, pp 95–104 Google Scholar
  20. Tran T, Mueller UA, Bloom LM (2002). Multi-scale conditional simulation of two-dimensional random processes using Haar wavelets. In: Proceedings, Geostatistical Association of Australasia Symposium, Perth, pp 56–78 Google Scholar
  21. Verly GW (1994) Sequential Gaussian cosimulation: a simulation method integrating several types of information. In: Soares A (ed) Geostatistics Troia 1992. Kluwer Academic, Dordrecht, pp 85–94 Google Scholar
  22. Zhang T, Switzer P, Journel AG (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38:63–80 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2009

Authors and Affiliations

  1. 1.QuébecCanada
  2. 2.MontrealCanada

Personalised recommendations