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Successive Nonparametric Estimation of Conditional Distributions

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Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities.

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  1. Bartlett, M. S., 1945, Negative probability: Proc. Cambridge Philos. Soc., v.41, p. 71-73.

  2. Carle, S., and Fogg, G. E., 1996, Transition probability-based indicator geostatistics: Math. Geol., v.28, no.4, p. 453-476.

  3. Carr, J., and Glass, C. E., 1985, Treatment of earthquake ground motion using regionalized variables: Math. Geol., v.17, no.3, p. 221-241.

  4. Chiles, J. P., and Delfiner, P., 1999, Geostatistics: Modeling spatial uncertainty: Wiley, New York, 695p.

  5. Christakos, G., and Hristopulos, D. T., 1998, Stochastic indicator analysis of contaminated sites: J. Appl. Probab., v.34, no.4, p. 988-1008.

  6. Cox, D. R., and Snell, E. J., 1989, Analysis of binary data: Chapman and Hall, London, 236p.

  7. Data-Gupta, A., Xue, G., and Lee, S. H., 1999, Nonparametric transformations for data correlation and integration, theory and practice, in Schatzinger, R. A., and Jordan, J. F., eds., Reservoir characterization recent advances, AAPG Memoir 71, Tulsa, OK, p. 381-396.

  8. Dimitrakopoulos, R., and Dagbert, M., 1993, Sequential modeling of relative indicator variables: Dealing with multiple lithology types, in Soares, ed., Geostatistics Tróia '92: Kluwer, Dordrecht, The Netherlands, p. 413-424.

  9. Fytas, K., Chaouai, N.-E., and Lavigne, M., 1990, Gold deposits estimation using indicator kriging: CIM Bull., v.80, p. 77-83.

  10. Goovaerts, P., 1994, Comparison of CIK, IK and MIK performances for modeling conditional probabilities of categorical variables, in Dimitrakopoulos, R., ed., Geostatistics for the next century: Kluwer, Dordrecht, The Netherlands, p. 18-29.

  11. Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 483p.

  12. Goovaerts, P., Webster, R., and Dubois, J. P., 1997, Assessing the risk of soil contamination in the Swiss Jura using indicator geostatistics: Environ. Ecol. Stat., v.4, p. 31-48.

  13. Hohn, E. M., and McDowell, R. R., 1994, Geostatistical analysis of oil production and potential using indicator kriging, in Yarus, J. M., and Chambers, R. L., eds., Stochastic modeling and geostatistics, AAPG Computer Applications in Geology, No. 3, Tulsa, OK.

  14. Journel, A. G., 1983, Nonparametric estimation of spatial distributions: Math. Geol., v.15, no.3., p. 445-468.

  15. Journel, A. G., and Alabert, F. G., 1990, A new method for reservoir mapping: J. Pet. Technol., Feb., 1990, p. 212-218.

  16. Journel, A. G., and Posa, D., 1990, Characteristic behavior and order relations for indicator variograms: Math. Geol., v.30, no.8, p. 1011-1025.

  17. Lemmer, I. C., 1984, Estimating local recoverable reserves via indicator kriging, in Verly, G., David, M., Journel, A. G., and Marechal, A., eds., Geostatistics for natural resources characterization, Part 1: Reidel, Dordrecht, The Netherlands, p. 365-384.

  18. Soares A., 1992, Geostatistical estimation of multiphase structures: Math. Geol., v.24, no.2, p. 149-160.

  19. Solow, A. R., 1986, Mapping by simple indicator kriging: Math. Geol., v.18, no.3, p. 335-352.

  20. Srivastava, R. M., 1992, Reservoir characterization with probability field simulation, SPE paper #24753.

  21. Sullivan, J., 1984, Conditional recovery estimation through probability kriging–-Theory and practice, in, Verly, G., David, M., Journel, A. G., and Marechal, A., eds., Geostatistics for natural resources characterization, Part 1: Reidel, Dordrecht, The Netherlands, p. 365-384.

  22. Suro Pérez, V., and Journel, A. G., 1991, Indicator principal component kriging: Math. Geol., v.23, no.5, p. 759-788.

  23. Vargas-Guzmán, J. A., and Yeh, T.-C. J., 1999, Sequential kriging and cokriging, two powerful geostatistical approaches: Stochastic Environ. Res. Risk Assess., v.13, no.6, p. 416-435.

  24. Yao, T., and Journel, A. G., 1998, Automatic modeling of (cross) covariance tables using fast Fourier transform: Math. Geol., v.30, no.6, p. 589-616.

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Vargas-Guzmán, J.A., Dimitrakopoulos, R. Successive Nonparametric Estimation of Conditional Distributions. Mathematical Geology 35, 39–52 (2003). https://doi.org/10.1023/A:1022361028297

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  • Non-Gaussian random functions
  • nonparametric estimation
  • conditional covariance
  • cokriging of indicators
  • indicator simulation