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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 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
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
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Variogram Array 8 J. W. Jennings Geostatistics with Madagascar
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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
Variogram Computation with FFTs Then, expand the variogram definition into a collection of product sums: 11 J. W. Jennings Geostatistics with Madagascar
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Variogram Computation with FFTs Then, expand the variogram definition into a collection of product sums: 13 J. W. Jennings Geostatistics with Madagascar
Variogram Computation with FFTs … that can be computed efficiently with FFTs: 14 J. W. Jennings Geostatistics with Madagascar
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
Implementation in Madagascar 16 J. W. Jennings Geostatistics with Madagascar
Implementation in Madagascar 17 J. W. Jennings Geostatistics with Madagascar
Implementation in Madagascar 18 J. W. Jennings Geostatistics with Madagascar
Example Application 19 J. W. Jennings Geostatistics with Madagascar
Data Array 20 J. W. Jennings Geostatistics with Madagascar
Indicator Array 21 J. W. Jennings Geostatistics with Madagascar
Pair-Count Array 22 J. W. Jennings Geostatistics with Madagascar
Variogram Array 23 J. W. Jennings Geostatistics with Madagascar
Data Array 24 J. W. Jennings Geostatistics with Madagascar
Data Array, Matrix Only 25 J. W. Jennings Geostatistics with Madagascar
Indicator Array, Matrix Only 26 J. W. Jennings Geostatistics with Madagascar
Pair-Count Array, Matrix Only 27 J. W. Jennings Geostatistics with Madagascar
Variogram Array, Matrix Only 28 J. W. Jennings Geostatistics with Madagascar
Variogram Array 29 J. W. Jennings Geostatistics with Madagascar
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
Stochastic Gaussian Simulation 31 J. W. Jennings Geostatistics with Madagascar
Stochastic Gaussian Simulation with FFTs 32 J. W. Jennings Geostatistics with Madagascar
Implementation in Madagascar 33 J. W. Jennings Geostatistics with Madagascar
Implementation in Madagascar 34 J. W. Jennings Geostatistics with Madagascar
Example Application 35 J. W. Jennings Geostatistics with Madagascar
Example Application 36 J. W. Jennings Geostatistics with Madagascar
1D Random Field 37 J. W. Jennings Geostatistics with Madagascar
2D Random Field 38 J. W. Jennings Geostatistics with Madagascar
3D Random Field 39 J. W. Jennings Geostatistics with Madagascar
Deep-Water Channels with Background Noise 40 J. W. Jennings Geostatistics with Madagascar