Mathematical Geosciences

, Volume 40, Issue 8, pp 861–873 | Cite as

Transitive Geostatistics for Stepwise Modeling Across Boundaries between Rock Regions

  • J. A. Vargas-GuzmánEmail author


A single intrinsic stationary random field may not account for transitional heterogeneity and abrupt dissimilarity of geological properties across boundaries between rock type regions. This paper proposes the stepwise construction of transitive covariance models for modeling continuous properties correlated across boundaries of multiple disjoint physical domains such as rock type bodies. Modeling in geology is usually simplified by splitting the geological space into rock type geo-domains (e.g., strata, sedimentary facies, soil series, diagenetic regions and alteration zones). Due to the limitations of simultaneous solutions, a simplification is to model each domain independently at the cost of losing the conditioning of properties across domains. This paper proposes to organize the modeling process in a triangular array which follows events in the geological time domain; for example, the younger formations are at the top of the pyramid and the older formation at the base. The estimation may go from top to base by assuming that younger events have perturbed older formations. Geology shows the scars of events that cumulate in rock formations before they are finally eroded. In some cases, older formations may be parent material for younger formations. The continuous property within each geo-domain has a conditional covariance in the main diagonal of the array which may belong to a specific event in the geological time. This sequence leads to transitive estimation and simulations in the physical space. If a simultaneous solution is sought (i.e., the future and past are correlated both ways), the complex covariance functions can be constructed stepwise from conditional spectra.


Petrophysical properties Rock types Diagenesis Multivariate geostatistics Conditional spectra Spatio-temporal Piecewise non-stationarity Large field models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bochner S (1949) Fourier transform. Princeton University Press, London, 219 p Google Scholar
  2. Burrough PA, MacMillan RA, vanDeursen W (1992) Fuzzy classification method for deterministic land suitability from soil profile observations and topography. J Soil Sci 43:193–210 CrossRefGoogle Scholar
  3. Carle S, Fogg GE (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476 CrossRefGoogle Scholar
  4. Deutsch C, Wang L (1996) Hierarchical object-based stochastic modeling of fluvial reservoirs. Math Geol 28(7):857–880 CrossRefGoogle Scholar
  5. Eschard R, Doligez B, Beucher H (2000) Using quantitative outcrop databases as a guide for geological reservoir modeling. In: Armstrong M, Bettini C, Champigny N, Galli A, Remacre A (eds) Geostatistics Rio 2000, vol 12. Kluwer Academic, Dordrech, pp 7–17 Google Scholar
  6. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York, 483 p Google Scholar
  7. Goulard M, Voltz M (1992) Linear coregionalization model: tools for estimation and choice of cross-variogram matrix. Math Geol 24(3):269–286 CrossRefGoogle Scholar
  8. Gutjahr A, Bullard B, Hatch S (1997) General joint conditional simulation using a fast Fourier transform method. Math Geol 29(3):361–389 CrossRefGoogle Scholar
  9. Hatloy A (1994) Numerical facies modeling combining deterministic and stochastic methods. In: Yarus JM, Chambers RL (eds) Stochastic modeling and geostatistics, principles, methods, and case studies. AAPG computer applications in geology, vol 3. Tulsa, Oklahoma, pp 109–120 Google Scholar
  10. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, New York, 600 p Google Scholar
  11. Krumbein WC, Sloss LL (1963) Stratigraphy and sedimentation. Freeman, London, 660 p Google Scholar
  12. Larsen G, Chilingar G (1967) Diagenesis of sediments. Elsevier, Amsterdam, 551 p Google Scholar
  13. Le Ravalec M, Noetinger B, Hu LY (2000) The FFT moving average generator: an efficient numerical method for generating and conditioning Gaussian simulations. Math Geol 32:701–723 CrossRefGoogle Scholar
  14. Matheron G (1973) The intrinsic random functions, and their applications. Adv Appl Probab 5:439–468 CrossRefGoogle Scholar
  15. Myers DE (1982) Matrix formulation of co-kriging. Math Geol 14(3):249–257 CrossRefGoogle Scholar
  16. Pardo-Iguzquiza E, Chica-Olmo M (1994) Spectral simulation of multivariable stationary random fields using covariance Fourier transforms. Math Geol 26(3):277–299 CrossRefGoogle Scholar
  17. Priestley MB (1981) Spectral analysis and time series. Academic Press, New York, 890 p Google Scholar
  18. Vargas-Guzmán JA (2003) Conditional components for simulation of vector random fields. Stoch Environ Res Risk Assess 17(4):260–271 CrossRefGoogle Scholar
  19. Vargas-Guzmán JA (2004) Fast modeling of cross-covariances in the LMC: a tool for data integration. Stoch Environ Res Risk Assess 18(2):91–99 CrossRefGoogle Scholar
  20. Vargas-Guzmán JA, Dimitrakopoulos R (2002) Conditional simulation of random fields by successive residuals. Math Geol 34(5):597–611 CrossRefGoogle Scholar
  21. Vargas-Guzmán JA, Dimitrakopoulos R (2003) Successive estimation of non-parametric distributions. Math Geol 35(1):39–52 CrossRefGoogle Scholar
  22. Vargas-Guzmán JA, Qassab H (2006) Spatial conditional simulation of geo-objects with application to facies modeling. J Pet Sci Eng 54:1–9 CrossRefGoogle Scholar
  23. Vargas-Guzmán JA, Yeh TCJ (1999) Sequential kriging and cokriging: two powerful approaches. Stoch Environ Res Risk Assess 13(6):416–435 CrossRefGoogle Scholar
  24. Vargas-Guzmán JA, Warrick AW, Myers DE (2002) Coregionalization by linear combination of non-orthogonal components. Math Geol 34(4):405–419 CrossRefGoogle Scholar
  25. Wackernagel H (1998) Multivariate geostatistics. Springer, Berlin, 291 p Google Scholar
  26. Webster R (1973) Automatic soil boundary location from transect data. Math Geol 5(1):27–37 CrossRefGoogle Scholar
  27. Yao T, Journel A (1998) Automatic modeling of (cross) covariances tables using fast Fourier transform. Math Geol 30(6):589–615 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geology 2008

Authors and Affiliations

  1. 1.DhahranSaudi Arabia

Personalised recommendations