Mathematical Geology

, Volume 31, Issue 6, pp 627–649 | Cite as

Improvement of Fourier-Based Unconditional and Conditional Simulations for Band Limited Fractal (von Kármán) Statistical Models

  • John A. Goff
  • James W. JenningsJr.


We evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation. The method is applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization (the von Kármán model). The simple Fourier unconditional simulation is conducted by Fourier transform of the amplitude spectrum model, sampled on a discrete grid, multiplied by a random phase spectrum. Although computationally efficient, this method failed to adequately match the intended statistical model at small scales because of sinc-function convolution. Attempts to alleviate this problem through the “covariance” method (computing the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function) created artifacts and spurious high wavenumber content. A modified Fourier method, consisting of pre-aliasing the wavenumber spectrum, satisfactorily remedies sinc smoothing. Conditional simulations using Fourier-based methods require several processing stages, including a smooth interpolation of the differential between conditioning data and an unconditional simulation. Although kriging is the ideal method for this step, it can take prohibitively long where the number of conditions is large. Here we develop a fast, approximate kriging methodology, consisting of coarse kriging followed by faster methods of interpolation. Though less accurate than full kriging, this fast kriging does not produce visually evident artifacts or adversely affect the a posteriori statistics of the Fourier conditional simulation.

simulation conditional simulation fourier methods band-limited fractal variogram fast kriging 


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  1. Bracewell, R., 1978, The fourier transform and its applications: McGraw-Hill Inc., New York, 444 p.Google Scholar
  2. Christakos, G., 1992, Random field models in earth sciences: Academic Press, San Diego, CA, 474 p.Google Scholar
  3. Deutsch, C. V., and Journel, A. G., 1992, GSLIB geostatistical software library and user's guide: Oxford University Press, New York, 340 p.Google Scholar
  4. Fisk, M. D., Charrette, E. E., and McCartor, G. D. 1992, A comparison of phase screen and finite difference calculations for elastic waves in random media: Jour. Geophys. Res., v. 97, no. B9, p. 12,409–12,423.Google Scholar
  5. Fouquet, C., 1994, Reminders on the conditional Kriging, in Armstrong, M., and Dowd, P. A., eds., Geostatistical simulations: Kluwer Academic Publishers, Netherlands, p. 131–145.Google Scholar
  6. Frankel, A., and Clayton, R. W., 1986, Finite difference simulations of seismic scattering: Implications for the propagation of short-period seismic waves in the crust and models of crustal heterogeneity: Jour. Geophys. Res., v. 91, no. B6, p. 6,465–6,489.Google Scholar
  7. Gee, L. S., and Jordan, T. H., 1988, Polarization anisotropy and fine-scale structure of the Eurasian upper mantle: Geophys. Res. Lett., v. 15, no. 8, p. 824–827.Google Scholar
  8. Goff, J. A., 1995, Quantitative analysis of sea ice draft 1. Methods for stochastic modeling: Jour. Geophys. Res., v. 100, no. C4, p. 6,993–7,004.Google Scholar
  9. Goff, J. A., and Jordan, T. H., 1988, Stochastic modeling of seafloor morphology: Inversion of sea beam data for second-order statistics: Jour. Geophys. Res., v. 93, no. B11, p. 13,589–13,608.Google Scholar
  10. Goff, J. A., and Levander, A., 1996, Incorporating “sinuous connectivity” into stochastic models of crustal heterogeneity: Examples from the Lewisian gneiss complex, Scotland, the Franciscan formation, California, and the Hafafit gneiss complex, Egypt: Jour. Geophys. Res., v. 101, no. B4, p. 8,489–8,501.Google Scholar
  11. Gutjahr, A., Bullard, B., and Hatch, S., 1997, General joint conditional simulations using a fast Fourier transform method: Math. Geology, v. 29, no. 3, p. 361–389.Google Scholar
  12. Holliger, K., 1996, Upper-crustal seismic velocity heterogeneity as derived from a variety of P-wave sonic logs: Geophys. J. Int., v. 125, no. 3, p. 813–829.Google Scholar
  13. Holliger, K., Levander, A., and Goff, J. A., 1994, Stochastic modeling of the reflective lower crust: petrophysical and geological evidence from the Ivrea Zone (Northern Italy): Jour. Geophys. Res., v. 98, no. B7, p. 11,967–11,980.Google Scholar
  14. Holliger, K., and Levander, A., 1994, Seismic structure of gneissic/granitic upper crust: Geological and petrophysical evidence from the Strona-Ceneri zone (northern Italy) and implications for crustal seismic exploration: Geophys. J. Int., v. 119, no. 2, p. 497–510.Google Scholar
  15. Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.Google Scholar
  16. King, M. J., Blunt, M. J., Mansfield, M., and Christie, M. A., 1995, Rapid evaluation of the impact of heterogeneity on miscible gas injection, in DeHaan, H. J., ed., New developments in improved oil recovery, Geological Society Special Publication No. 84. Geologic Society of London, London, p. 133–142.Google Scholar
  17. Levander, A., England, R. W., Smith, S. K., Hobbs, R. W., Goff, J. A., and Holliger, K., 1994, Stochastic characterization and seismic response of upper and middle crustal rocks based on the Lewisian gneiss complex, Scotland: Geophys. J. Int., v. 119, no. 1, p. 243–259.Google Scholar
  18. Pardo-Igúzquiza, E., and Chica-Olmo, M., 1993, The Fourier integral method: An efficient spectral method for simulation of random fields: Math. Geology, v. 25, no. 4, p. 177–217.Google Scholar
  19. Pardo-Igúzquiza, E., and Chica-Olmo, M., 1994, Spectral simulation of multivariable stationary random functions using covariance Fourier transforms: Math. Geology, v. 26, no. 3, p. 277–299.Google Scholar
  20. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. A., 1986, Numerical recipes: Cambridge University Press, Cambridge, 963 p.Google Scholar
  21. Priestly, M. B., 1981, Spectral analysis of time series, Vol. 1: Academic Press, London, 890 p.Google Scholar
  22. Sen, M., and Stoffa, P. L., 1995, Global optimization methods in geophysical inversion: Elsevier, Amsterdam, 281 p.Google Scholar
  23. Tatarski, V. I., 1961, Wave propagation in a turbulent medium: McGraw-Hill, Inc., New York, 285 p.Google Scholar
  24. von Kármán, T., 1948, Progress in the statistical theory of turbulence: J. Mar. Res., v. 7, p. 252–264.Google Scholar
  25. Wu, R.-S., and Aki, K., 1985, The fractal nature of the inhomogeneities in the lithosphere evidenced from seismic wave scattering: Pure Appl. Geophys., v. 123, no. 6, p. 805–818.Google Scholar

Copyright information

© International Association for Mathematical Geology 1999

Authors and Affiliations

  • John A. Goff
    • 1
  • James W. JenningsJr.
    • 2
  1. 1.University of Texas Institute for GeophysicsAustin
  2. 2.University of Texas Bureau of Economic GeologyAustin

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