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Geostatistics with Madagascar or Variograms and Random Fields with FFTs

Geostatistics with Madagascar or Variograms and Random Fields with FFTs. Jim Jennings and Sergey Fomel Carbonate Reservoir Characterization Research Laboratory Bureau of Economic Geology Jackson School of Geosciences The University of Texas at Austin August 31, 2006. Outline.

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Geostatistics with Madagascar or Variograms and Random Fields with FFTs

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  1. Geostatistics with MadagascarorVariograms and Random Fields with FFTs Jim Jennings and Sergey Fomel Carbonate Reservoir Characterization Research Laboratory Bureau of Economic Geology Jackson School of Geosciences The University of Texas at Austin August 31, 2006

  2. Outline • Variograms with Madagascarwhat is a variogram? how to compute a variogram with FFTs implementation in Madagascar examples • Random Fields with Madagascarwhat is stochastic simulation? how to make random field with FFTs implementation in Madagascar examples 2 J. W. Jennings Geostatistics with Madagascar

  3. 3 J. W. Jennings Geostatistics with Madagascar

  4. 4 J. W. Jennings Geostatistics with Madagascar

  5. 5 J. W. Jennings Geostatistics with Madagascar

  6. 6 J. W. Jennings Geostatistics with Madagascar

  7. 7 J. W. Jennings Geostatistics with Madagascar

  8. Variogram Array 8 J. W. Jennings Geostatistics with Madagascar

  9. 9 J. W. Jennings Geostatistics with Madagascar

  10. Variogram Computation with FFTs The trick is to think of an FFT not as an approximation to the Fourier integral transform, but as a tool for exact and efficient computation of the discrete product sum: … for all possible values of the discrete lag vector h. 10 J. W. Jennings Geostatistics with Madagascar

  11. Variogram Computation with FFTs Then, expand the variogram definition into a collection of product sums: 11 J. W. Jennings Geostatistics with Madagascar

  12. 12 J. W. Jennings Geostatistics with Madagascar

  13. Variogram Computation with FFTs Then, expand the variogram definition into a collection of product sums: 13 J. W. Jennings Geostatistics with Madagascar

  14. Variogram Computation with FFTs … that can be computed efficiently with FFTs: 14 J. W. Jennings Geostatistics with Madagascar

  15. Variogram Computation with FFTs … that can be computed efficiently with FFTs: Marcotte, D., 1996, Fast variogram computation with FFT, Computers & Geosciences, v 22, n 10, pp. 1175–1186. 15 J. W. Jennings Geostatistics with Madagascar

  16. Implementation in Madagascar 16 J. W. Jennings Geostatistics with Madagascar

  17. Implementation in Madagascar 17 J. W. Jennings Geostatistics with Madagascar

  18. Implementation in Madagascar 18 J. W. Jennings Geostatistics with Madagascar

  19. Example Application 19 J. W. Jennings Geostatistics with Madagascar

  20. Data Array 20 J. W. Jennings Geostatistics with Madagascar

  21. Indicator Array 21 J. W. Jennings Geostatistics with Madagascar

  22. Pair-Count Array 22 J. W. Jennings Geostatistics with Madagascar

  23. Variogram Array 23 J. W. Jennings Geostatistics with Madagascar

  24. Data Array 24 J. W. Jennings Geostatistics with Madagascar

  25. Data Array, Matrix Only 25 J. W. Jennings Geostatistics with Madagascar

  26. Indicator Array, Matrix Only 26 J. W. Jennings Geostatistics with Madagascar

  27. Pair-Count Array, Matrix Only 27 J. W. Jennings Geostatistics with Madagascar

  28. Variogram Array, Matrix Only 28 J. W. Jennings Geostatistics with Madagascar

  29. Variogram Array 29 J. W. Jennings Geostatistics with Madagascar

  30. Stochastic Simulation • Unconditional stochastic simulation in geostatistics is the process of generating a random field with a specified variogram model. • Unconditional stochastic simulation is like variography in reverse. • Conditional stochastic simulation makes random fields that have a specified variogram and have specified values at given control points. 30 J. W. Jennings Geostatistics with Madagascar

  31. Stochastic Gaussian Simulation 31 J. W. Jennings Geostatistics with Madagascar

  32. Stochastic Gaussian Simulation with FFTs 32 J. W. Jennings Geostatistics with Madagascar

  33. Implementation in Madagascar 33 J. W. Jennings Geostatistics with Madagascar

  34. Implementation in Madagascar 34 J. W. Jennings Geostatistics with Madagascar

  35. Example Application 35 J. W. Jennings Geostatistics with Madagascar

  36. Example Application 36 J. W. Jennings Geostatistics with Madagascar

  37. 1D Random Field 37 J. W. Jennings Geostatistics with Madagascar

  38. 2D Random Field 38 J. W. Jennings Geostatistics with Madagascar

  39. 3D Random Field 39 J. W. Jennings Geostatistics with Madagascar

  40. Deep-Water Channels with Background Noise 40 J. W. Jennings Geostatistics with Madagascar

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