Stochastic Modeling of Continuous Geospatial or Temporal Properties

  • Y. Z. Ma


This chapter presents geostatistical methods for stochastically simulating continuous geospatial properties. For facilitating the presentations, it uses many temporal data in stochastic simulations benefiting from the 1D simplification. The commonly used simulation methods for spatial data include sequential Gaussian simulation and spectral simulation. Unlike estimation methods (e.g., regression and kriging), one main goal of stochastic simulation is to model the heterogeneities of physical properties.

Stochastic simulations are often mathematically extended from estimation methods. Therefore, the kriging methods presented in Chap.  16 are used as a methodological basis for stochastic simulations. Readers should be familiar with kriging, especially simple kriging, before reading this chapter. The main texts in this chapter focus on basic methodologies and three appendices cover more advanced topics.


  1. Bracewell, R. (1986). The Fourier transform and its application. New York: McGraw-Hill.zbMATHGoogle Scholar
  2. Daly, C., Quental, S., & Novak, D. (2010). A faster, more accurate Gaussian simulation, AAPG Article 90172. CSPG/CSEG/CWLS GeoConvention.Google Scholar
  3. Deutsch, C. V., & Journel, A. G. (1992). Geostatistical software library and user’s guide. Oxford: Oxford University Press, 340p.Google Scholar
  4. Fournier, F., & Ma, Y. Z. (1988). Spectral analysis by maximum entropy: Application to short window seismic data. Research Report, 66 pages, Elf-Aquitaine.Google Scholar
  5. Gray, R. M. (2009). Probability, random processes, and ergodic properties (2nd ed.). Berlin: Springer. A revised edition is available online: Last accessed 27 Nov 2017.CrossRefGoogle Scholar
  6. Hardy, H. H., & Beier, R. A. (1994). Integration of large- and small-scale data using fourier transforms. In paper presented at ECMORIV Conference, Roros, Norway, June 7–10, 1994.Google Scholar
  7. Helstrom, C. W. (1991). Probability and stochastic processes for engineers (2nd ed.). New York: Macmillan.Google Scholar
  8. Huang, X., & Kelkar, M. (1996). Integration of dynamic data for reservoir characterization in the frequency domain. Society of Petroleum Engineers.
  9. Journel, A. G. (2000). Correcting the smoothing effect of estimators: A spectral postprocessor. Mathematical Geology, 32(7), 787–813.MathSciNetCrossRefGoogle Scholar
  10. Journel, A. G., & Deutsch, C. V. (1993). Entropy and spatial disorder. Mathematical Geology, 25(3), 329–356.CrossRefGoogle Scholar
  11. Journel, A. G., & Huijbregts, C. J. (1978). Mining geostatistics. New York: Academic, 600p.Google Scholar
  12. Journel, A. G., & Zhang, T. (2006). The necessity of a multiple-point prior model. Mathematical Geology, 38(5), 591–610.CrossRefGoogle Scholar
  13. Lantuejoul, C. (2002). Geostatistical simulation: Models and algorithms. Berlin: Springer.CrossRefGoogle Scholar
  14. Lebowitz, J. L., & Penrose, O. (1973, February). Modern ergodicity theory. Physics Today, pp. 23–29.Google Scholar
  15. Lee, Y. W. (1967). Statistical theory of communication (6th ed.). New York: Wiley, 509p.Google Scholar
  16. Ma, Y. Z. (1992). Spectral estimation by simple kriging in one dimension. In P. A. Dowd & J. J. Royer (Eds.), 2nd international codata conference on geomathematics and geostatistics (Sciences de la terre, Vol. 31, pp. 35–42).Google Scholar
  17. Ma, Y. Z., Seto, A., & Gomez, E. (2008). Frequentist meets spatialist: A marriage made in reservoir characterization and modeling. SPE 115836, SPE ATCE, Denver, CO, USA.Google Scholar
  18. Marple, L. (1982). Frequency resolution of Fourier and maximum entropy spectral estimates. Geophysics, 47(9), 1303–1307.CrossRefGoogle Scholar
  19. Matern, B. (1960). Spatial variation. Meddelanden Fran Statens Skogsforskningsinstitut, Stockholm, 49(5), 144p.Google Scholar
  20. Matheron, G. (1988). Suffit-il, pour une covariance, d’etre de type positif? Sciences de la Terre Informatiques (Vol. 86, pp. 51–66).Google Scholar
  21. Matheron, G. (1989). Estimating and choosing – An essay on probability in practice. Berlin: Springer.CrossRefGoogle Scholar
  22. Mega Millions. (2018). Last accessed 16 Oct 2018.
  23. Papoulis, A. (1965). Probability, random variables and stochastic processes. New York: McGraw-Hill, 583p.Google Scholar
  24. Pardo-Iguzquiza, E., & Chica-Olmo, M. (1993). The Fourier integral method: An efficient spectral method for simulation of random fields. Mathematical Geology, 25(2), 177–217.CrossRefGoogle Scholar
  25. Pettitt, A. N., & McBratney, A. B. (1993). Sampling designs for estimating spatial variance components. Applied Statistics, 42, 185–209.CrossRefGoogle Scholar
  26. Soares, A. (2001). Direct sequential simulation and cosimulation. Mathematical Geology, 33, 911–926.CrossRefGoogle Scholar
  27. Stein, M. L. (1999). Interpolation of spatial data: Some theory for kriging. New York: Springer, 247p.Google Scholar
  28. Yao, T. (1998). Conditional spectral simulation with phase identification. Mathematical Geology, 30(3), 285–308.CrossRefGoogle Scholar
  29. Yao, T., Calvert, C., Jones, T., Ma, Y. Z., & Foreman, L. (2005). Spectral component geologic modeling: A new technology for integrating seismic information at the correct scale. In O. Leuangthong & C. V. Deutsch (Eds.), Quantitative geology and geostatistics (pp. 23–33). Dordrecht: Springer.Google Scholar
  30. Yao, T., Calvert, C., Jones, T., Bishop, G., Ma, Y. Z., & Foreman, L. (2006). Spectral simulation and its advanced capability of conditioning to local continuity trends in geologic modeling. Mathematical Geology, 38(1), 51–62.CrossRefGoogle Scholar
  31. Zhan, H. (1999). On the ergodicity hypothesis in heterogeneous formations. Mathematical Geology, 31, 113–134.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Y. Z. Ma
    • 1
  1. 1.SchlumbergerDenverUSA

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