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
Babenko K (1986) Fundamentals of numerical analysis. Nauka, Moscow
Caers J (2005) Petroleum geostatistics. SPE–Pennwell Books, Houston
Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New York
Chugunova TL, Hu LY (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146
Cressie NA (1993) Statistics for spatial data. Wiley, New York
David M (1977) Geostatistical ore reserve estimation. Elsevier, Amsterdam
David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam
Deutsch C, Journel A (1998) GSLIB: geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York
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
Evans J, Bazilevs Y, Babuška 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
Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, Cambridge
Guardiano J, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. Geosatistics Tróia’92:133–144
Journel AG (1993) Geostatistics: roadblocks and challenges. Stanford Center for Reservoir Forecasting
Journel AG (2003) Multiple-point Geostatistics: a State of the Art. Stanford Center for Reservoir Forecasting
Journel AG, Alabert F (1990) New method for reservoir map**. Pet Technol 42(2):212–218
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic, San Diego
Kitanidis PK (1997) Introduction to geostatistics—applications in hydrogeology. Cambridge Univ Press, Cambridge
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
Mariethoz G, Kelly B (2011) Modeling complex geological structures withelementary training images and transform-invariant distances. Water Resour Res 47:1–2
Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42:245–268
Matheron G (1971) The theory of regionalized variables and its applications. Cahier du Centre de Morphologie Mathematique, No 5
Mustapha H, Dimitrakopoulos R (2010a) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci 36(3):313–334
Mustapha H, Dimitrakopoulos R (2010b) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42:455–473
Pyrcz M, Deutsch C (2014) Geostatistical reservoir modeling, 2nd edn. Oxford University Press, New York
Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, Cambridge
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
Strebelle S (2002) Conditional simulation of complex geological structures using multiple point stastics. Math Geosci 34:1–22
Strebelle S, Cavelius C (2014) Solving speed and memory issues in multiple-point statistics simulation program SNESIM. Math Geosci 46:171–186
Tikhonov AN, Arsenin VY (1977) Solution of Ill-posed problems. Winston & Sons, Washington, DC
Tjelmeland H, Besag J (1998) Markov random fields with higherorder interactions. Scand J Stat 25(3):415–433
Toftaker H, Tjelmeland H (2013) Construction of binary multi-grid Markov random field prior models from training images. Math Geosci 45:383–409
Vargas-Guzmán J (2011) The Kappa model of probability and higher-order rock sequences. Comput Geosci 15:661–671
Vargas-Guzmán J, Qassab H (2006) Spatial conditional simulation of facies objects for modeling complex clastic reservoirs. J Petrol Sci Eng 54:1–9
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
Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New York
Yahya WJ (2011) Image reconstruction from a limited number of samples: a matrix-completion-based approach. Mater Thesis, McGill University, Montreal
Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geosci 38(1):63–80
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).
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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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