Mathematical Geology

, Volume 35, Issue 2, pp 155–173

# Stepwise Conditional Transformation for Simulation of Multiple Variables

• Oy Leuangthong
• Clayton V. Deutsch
Article

## 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.

multivariate transformation multivariate geostatistics model of coregionalization

## REFERENCES

1. 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.Google Scholar
2. Anderson, T., 1958, An introduction to multivariate statistical analysis: Wiley, New York, 351 p.Google Scholar
3. Brieman, L., and Friedman, J. H., 1985a, Estimating optimal transformations for multiple regression and correlation: J. Amer. Stat. Assoc., v. 80, p. 580-598.Google Scholar
4. Brieman, L., and Friedman, J. H., 1985b, Rejoinder: J. Amer. Stat. Assoc., v. 80, p. 614-619.Google Scholar
5. 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.Google Scholar
6. Deutsch, C. V., and Journel, A. G., 1998, GSLIB: Geostatistical software library and users guide, 2nd edn.: Oxford University Press, New York, 369 p.Google Scholar
7. Froidevaux, R., 1993, Probability field simulation, in Soares, A., ed., Geostatistics Troia 1992: Kluwer Academic, New York, Vol. 1, pp. 73-84.Google Scholar
8. 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.Google Scholar
9. Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 483 p.Google Scholar
10. 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.Google Scholar
11. Izenman, A. J., 1991, Recent developments in nonparametric density estimation: J Amer. Stat. Assoc., v. 86, no. 413, p. 205-224.Google Scholar
12. Johnson, R. A., and Wichern, D. W., 1998, Applied multivariate statistical analysis, 4th edn: Prentice-Hall, NJ, 799 p.Google Scholar
13. Journel, A. G., 1974, Simulation Conditionnelle—Théorie et Pratique, Thèse de Docteur-Ingénieur: Université de Nancy I, 110 p.Google Scholar
14. Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.Google Scholar
15. Luster, G. R., 1985, Raw materials for portland cement: Applications of conditional simulation of coregionalization, PhD Thesis, Stanford University, Stanford, CA, 518 p.Google Scholar
16. 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.Google Scholar
17. Rosenblatt, M., 1952, Remarks on a multivariate transformation: Ann. Math. Stat., v. 23, no. 3, p. 470-472.Google Scholar
18. Scott, D. W., 1992, Multivariate density estimation: Theory, practice, and visualization: Wiley, New York, 299 p.Google Scholar
19. 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.Google Scholar

© International Association for Mathematical Geology 2003

## Authors and Affiliations

• Oy Leuangthong
• 1
• Clayton V. Deutsch
• 1
1. 1.Department of Civil & Environmental EngineeringUniversity of AlbertaEdmontonCanada