Abstract
Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Almeida, A. S., and Journel, A. G., 1994, Joint simulation of multiple variables with a Markov-type coregionalization model: Math. Geol., v. 26, no. 5, p. 565-588.
Anderson, T., 1958, An introduction to multivariate statistical analysis: Wiley, New York, 351 p.
Brieman, L., and Friedman, J. H., 1985a, Estimating optimal transformations for multiple regression and correlation: J. Amer. Stat. Assoc., v. 80, p. 580-598.
Brieman, L., and Friedman, J. H., 1985b, Rejoinder: J. Amer. Stat. Assoc., v. 80, p. 614-619.
Deutsch, C. V., 1992, Annealing techniques applied to reservoir modeling and the integration of geological and engineering (Well Test) data: Stanford University, Stanford, CA, 306 p.
Deutsch, C. V., and Journel, A. G., 1998, GSLIB: Geostatistical software library and users guide, 2nd edn.: Oxford University Press, New York, 369 p.
Froidevaux, R., 1993, Probability field simulation, in Soares, A., ed., Geostatistics Troia 1992: Kluwer Academic, New York, Vol. 1, pp. 73-84.
Gómez-Hernández, J. J., and Srivastava, R. M., 1990, ISIM3D: An ANSI-C Three dimensional multiple indicator conditional simulation program: Comput. Geosci., v. 16, no. 4, pp. 395-410.
Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 483 p.
Isaaks, E. H., 1990, The application of Monte Carlo methods to the analysis of spatially correlated data: PhD Thesis, Stanford University, Stanford, CA, 213 p.
Izenman, A. J., 1991, Recent developments in nonparametric density estimation: J Amer. Stat. Assoc., v. 86, no. 413, p. 205-224.
Johnson, R. A., and Wichern, D. W., 1998, Applied multivariate statistical analysis, 4th edn: Prentice-Hall, NJ, 799 p.
Journel, A. G., 1974, Simulation Conditionnelle—Théorie et Pratique, Thèse de Docteur-Ingénieur: Université de Nancy I, 110 p.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Luster, G. R., 1985, Raw materials for portland cement: Applications of conditional simulation of coregionalization, PhD Thesis, Stanford University, Stanford, CA, 518 p.
Lyall, G., and Deutsch, C. V., 2000, Geostatistical modeling of multiple variables in presence of complex trends and mineralogical constraints: Geostats 2000 Cape Town, South Africa, 10 p.
Rosenblatt, M., 1952, Remarks on a multivariate transformation: Ann. Math. Stat., v. 23, no. 3, p. 470-472.
Scott, D. W., 1992, Multivariate density estimation: Theory, practice, and visualization: Wiley, New York, 299 p.
Xu, W., Tran, T. T., Srivastava, R. M., and Journel, A. G., 1992, Integrating seismic data in reservoir modeling: The collocated cokriging alternative, 67th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Washington, DC, 10p.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Leuangthong, O., Deutsch, C.V. Stepwise Conditional Transformation for Simulation of Multiple Variables. Mathematical Geology 35, 155–173 (2003). https://doi.org/10.1023/A:1023235505120
Issue Date:
DOI: https://doi.org/10.1023/A:1023235505120