Advertisement

Multiple imputation framework for data assignment in truncated pluri-Gaussian simulation

  • Diogo S. F. Silva
  • Clayton V. Deutsch
Article
  • 171 Downloads

Abstract

Truncated pluri-Gaussian simulation (TPGS) is suitable for the simulation of categorical variables that show natural ordering as the TPGS technique can consider transition probabilities. The TPGS assumes that categorical variables are the result of the truncation of underlying latent variables. In practice, only the categorical variables are observed. This translates the practical application of TPGS into a missing data problem in which all latent variables are missing. Latent variables are required at data locations in order to condition categorical realizations to observed categorical data. The imputation of missing latent variables at data locations is often achieved by either assigning constant values or spatially simulating latent variables subject to categorical observations. Realizations of latent variables can be used to condition all model realizations. Using a single realization or a constant value to condition all realizations is the same as assuming that latent variables are known at the data locations and this assumption affects uncertainty near data locations. The techniques for imputation of latent variables in TPGS framework are investigated in this article and their impact on uncertainty of simulated categorical models and possible effects on factors affecting decision making are explored. It is shown that the use of single realization of latent variables leads to underestimation of uncertainty and overestimation of measured resources while the use constant values for latent variables may lead to considerable over or underestimation of measured resources. The results highlight the importance of multiple data imputation in the context of TPGS.

Keywords

Geostatistics Geomodeling Missing data analysis Gibbs sampler 

References

  1. Agresti A (2002) Categorical Data Analysis, 2nd edn. Wiley Series in Probability and Statistics, Wiley, New YorkCrossRefGoogle Scholar
  2. Armstrong M, Galli A, Beucher H, Loc’h G, Renard D, Doligez B, Eschard R, Geffroy F (2011) Plurigaussian simulations in geosciences, 2nd edn. Springer-Verlag, Berlin HeidelbergCrossRefGoogle Scholar
  3. Arroyo D, Emery X, Peláez M (2012) An enhanced gibbs sampler algorithm for non-conditional simulation of gaussian random vectors. Computers & Geosciences 46:138–148CrossRefGoogle Scholar
  4. Astrakova A, Oliver DS, Lantuéjoul C (2015) Truncation map estimation based on bivariate probabilities and validation for the truncated plurigaussian model. ArXiv e-prints https://arxiv.org/pdf/1508.01090.pdf. Accessed 29 June 2016
  5. Barnett RM, Deutsch CV (2015) Multivariate imputation of unequally sampled geological variables. Mathematical Geosciences 47(7):791–817CrossRefGoogle Scholar
  6. Christakos G (1990) A bayesian/maximum-entropy view to the spatial estimation problem. Mathematical Geology 22(7):763–777CrossRefGoogle Scholar
  7. Davis MW (1987) Production of conditional simulations via the lu triangular decomposition of the covariance matrix. Mathematical Geology 19(2):91–98Google Scholar
  8. Deutsch CV, Cockerham PW (1994) Practical considerations in the application of simulated annealing to stochastic simulation. Mathematical Geology 26(1):67–82CrossRefGoogle Scholar
  9. Deutsch JL, Deutsch CV (2014) A multidimensional scaling approach to enforce reproduction of transition probabilities in truncated plurigaussian simulation. Stochastic environmental research and risk assessment 28(3):707–716CrossRefGoogle Scholar
  10. Emery X, Arroyo D, Peláez M (2014) Simulating large gaussian random vectors subject to inequality constraints by gibbs sampling. Mathematical Geosciences 46(3):265–283CrossRefGoogle Scholar
  11. Galli A, Gao H (2001) Rate of convergence of the gibbs sampler in the gaussian case. Mathematical Geology 33(6):653–677CrossRefGoogle Scholar
  12. Galli A, Beucher H, Le Loc’h G, Doligez B, Heresim Group (1994) The pros and cons of the truncated gaussian method. Geostatistical simulations, vol 7, 1st edn. Springer, Netherlands, pp 217–233CrossRefGoogle Scholar
  13. Geman S, Geman D (1984) Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6(6):721–741CrossRefGoogle Scholar
  14. Journel AG (1983) Nonparametric estimation of spatial distributions. Journal of the International Association for Mathematical Geology 15(3):445–468CrossRefGoogle Scholar
  15. Kyriakidis PC, Deutsch CV, Grant ML (1999) Calculation of the normal scores variogram used for truncated gaussian lithofacies simulation: theory and fortran code. Computers & Geosciences 25(2):161–169CrossRefGoogle Scholar
  16. Lantuéjoul C, Desassis N (2012) Simulation of a gaussian random vector: a propagative version of the gibbs sampler. In: The 9th International Geostatistics Congress, Oslo, http://geostats2012.nr.no/pdfs/1747181.pdf. Accessed 29 June 2016
  17. Little RJ, Rubin DB (2002) Statistical analysis with missing data, 2nd edn. John Wiley & SonsCrossRefGoogle Scholar
  18. Matheron G, Beucher H, De Fouquet C, Galli A, Guerillot D, Ravenne C (1987) Conditional simulation of the geometry of fluvio-deltaic reservoirs. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Texas, pp 591–599Google Scholar
  19. Pyrcz MJ, Deutsch CV (2014) Geostatistical reservoir modeling, 2nd edn. Oxford University Press, New YorkGoogle Scholar
  20. Rossi ME, Deutsch CV (2014) Mineral resource estimation, 1st edn. Springer, NetherlandsCrossRefGoogle Scholar
  21. Shannon CE (1948) A mathematical theory of communication. The Bell System Technical Journal 27(379–423):623–656CrossRefGoogle Scholar
  22. Silva D, Boisvert J (2014) Mineral resource classification: a comparison of new and existing techniques. Journal of the Southern African Institute of Mining and Metallurgy 114(3):265–273Google Scholar
  23. Snowden DV, Glacken I, Noppé MA (2002) Dealing with demands of technical variability and uncertainty along the mine value chain. Value Tracking Symposium, vol. 69. Australasian Institute of Mining and Metallurgy, Brisbane, pp 93–100Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Centre for Computational Geostatistics, 6-247 Donadeo Innovation Centre For EngineeringUniversity of AlbertaEdmontonCanada

Personalised recommendations