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A High-Order, Data-Driven Framework for Joint Simulation of Categorical Variables

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Geostatistics Valencia 2016

Part of the book series: Quantitative Geology and Geostatistics ((QGAG,volume 19))

Abstract

Relatively recent techniques for categorical simulations are based on multipoint statistical approaches where a training image (TI) is used to derive complex spatial relationships using patterns. In these cases, simulated realizations are driven by the TI utilized, while the spatial statistics of the hard data is not used. This paper presents a data-driven, high-order simulation approach based upon the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances similar to variogram models for two-point methods. It is shown that the higher-order statistics are connected with lower orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from hard data. Finally, conditional distribution is calculated using Bayes rule and random values are simulated sequentially for all unsampled grid nodes. The main advantages of the proposed technique are its ability to simulate without a training image, which reproduces the high-order statistics of hard data, and to adopt the complexity of the model to the information available in the hard data. The approach is tested with a synthetic dataset and compared to a conventional second-order method, sisim, in terms of cross-correlations and high-order spatial statistics.

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Bibliography

  • Arndt C (2004) Information measures: information and its description in science and engineering. Springer, Amsterdam

    Google Scholar 

  • Babenko K (1986) Fundamentals of numerical analysis. Nauka, Moscow

    Google Scholar 

  • Caers J (2005) Petroleum geostatistics. SPE–Pennwell Books, Houston

    Google Scholar 

  • Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New York

    Book  Google Scholar 

  • Chugunova TL, Hu LY (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146

    Article  Google Scholar 

  • Cressie NA (1993) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • David M (1977) Geostatistical ore reserve estimation. Elsevier, Amsterdam

    Google Scholar 

  • David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam

    Google Scholar 

  • Deutsch C, Journel A (1998) GSLIB: geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York

    Google Scholar 

  • Dimitrakopoulos R, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42:65–99

    Article  Google Scholar 

  • Evans J, Bazilevs Y, BabuÅ¡ka I, Hughes T (2009) N-widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Engrg 198:1726–1741

    Article  Google Scholar 

  • Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, Cambridge

    Google Scholar 

  • Guardiano J, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. Geosatistics Tróia’92:133–144

    Google Scholar 

  • Journel AG (1993) Geostatistics: roadblocks and challenges. Stanford Center for Reservoir Forecasting

    Google Scholar 

  • Journel AG (2003) Multiple-point Geostatistics: a State of the Art. Stanford Center for Reservoir Forecasting

    Google Scholar 

  • Journel AG, Alabert F (1990) New method for reservoir mapping. Pet Technol 42(2):212–218

    Article  Google Scholar 

  • Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic, San Diego

    Google Scholar 

  • Kitanidis PK (1997) Introduction to geostatistics—applications in hydrogeology. Cambridge Univ Press, Cambridge

    Book  Google Scholar 

  • Mao S, Journel A (1999) Generation of a reference petrophysical/seismic data set: the Stanford V reservoir. 12th Annual Report. Stanford Center for Reservoir Forecasting, Stanford

    Google Scholar 

  • Mariethoz G, Kelly B (2011) Modeling complex geological structures withelementary training images and transform-invariant distances. Water Resour Res 47:1–2

    Article  Google Scholar 

  • Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42:245–268

    Article  Google Scholar 

  • Matheron G (1971) The theory of regionalized variables and its applications. Cahier du Centre de Morphologie Mathematique, No 5

    Google Scholar 

  • Mustapha H, Dimitrakopoulos R (2010a) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci 36(3):313–334

    Article  Google Scholar 

  • Mustapha H, Dimitrakopoulos R (2010b) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42:455–473

    Article  Google Scholar 

  • Pyrcz M, Deutsch C (2014) Geostatistical reservoir modeling, 2nd edn. Oxford University Press, New York

    Google Scholar 

  • Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Straubhaar JRP, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43:305–328

    Article  Google Scholar 

  • Strebelle S (2002) Conditional simulation of complex geological structures using multiple point stastics. Math Geosci 34:1–22

    Google Scholar 

  • Strebelle S, Cavelius C (2014) Solving speed and memory issues in multiple-point statistics simulation program SNESIM. Math Geosci 46:171–186

    Article  Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solution of Ill-posed problems. Winston & Sons, Washington, DC

    Google Scholar 

  • Tjelmeland H, Besag J (1998) Markov random fields with higherorder interactions. Scand J Stat 25(3):415–433

    Google Scholar 

  • Toftaker H, Tjelmeland H (2013) Construction of binary multi-grid Markov random field prior models from training images. Math Geosci 45:383–409

    Article  Google Scholar 

  • Vargas-Guzmán J (2011) The Kappa model of probability and higher-order rock sequences. Comput Geosci 15:661–671

    Article  Google Scholar 

  • Vargas-Guzmán J, Qassab H (2006) Spatial conditional simulation of facies objects for modeling complex clastic reservoirs. J Petrol Sci Eng 54:1–9

    Article  Google Scholar 

  • Wang W, Pottmann H, Liu Y (2006) Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Transactions on Graphics 25(2):214–238

    Article  Google Scholar 

  • Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New York

    Book  Google Scholar 

  • Yahya WJ (2011) Image reconstruction from a limited number of samples: a matrix-completion-based approach. Mater Thesis, McGill University, Montreal

    Google Scholar 

  • Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geosci 38(1):63–80

    Google Scholar 

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Acknowledgments

Funding was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 239019 and mining industry partners of the COSMO Lab (AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers Canada, Kinross Gold, Newmont Mining, and Vale).

Author information

Authors and Affiliations

  1. COSMO, Stochastic Mine Planning Laboratory, McGill University, Montreal, QC, Canada, H3A 0E8

    Ilnur Minniakhmetov & Roussos Dimitrakopoulos

Authors
  1. Ilnur Minniakhmetov
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  2. Roussos Dimitrakopoulos
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Corresponding author

Correspondence to Ilnur Minniakhmetov .

Editor information

Editors and Affiliations

  1. Institute for Water and Environmental Engineering, Universitat Politècnica de València, Valencia, Spain

    J. Jaime Gómez-Hernández

  2. Institute for Water and Environmental Engineering, Universitat Politècnica de València, Valencia, Spain

    Javier Rodrigo-Ilarri

  3. Institute for Water and Environmental Engineering, Universitat Politècnica de València, Valencia, Spain

    María Elena Rodrigo-Clavero

  4. Institute for Water and Environmental Engineering, Universitat Politècnica de València, Valencia, Spain

    Eduardo Cassiraga

  5. Saudi Aramco, Dhahran, Saudi Arabia

    José Antonio Vargas-Guzmán

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Minniakhmetov, I., Dimitrakopoulos, R. (2017). A High-Order, Data-Driven Framework for Joint Simulation of Categorical Variables. In: Gómez-Hernández, J., Rodrigo-Ilarri, J., Rodrigo-Clavero, M., Cassiraga, E., Vargas-Guzmán, J. (eds) Geostatistics Valencia 2016. Quantitative Geology and Geostatistics, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-46819-8_19

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