Skip to main content
SpringerLink
Log in

Improving the Pattern Reproducibility of Multiple-Point-Based Prior Models Using Frequency Matching

  • Published:
Mathematical Geosciences Aims and scope Submit manuscript

Abstract

Some multiple-point-based sampling algorithms, such as the snesim algorithm, rely on sequential simulation. The conditional probability distributions that are used for the simulation are based on statistics of multiple-point data events obtained from a training image. During the simulation, data events with zero probability in the training image statistics may occur. This is handled by pruning the set of conditioning data until an event with non-zero probability is found. The resulting probability distribution sampled by such algorithms is a pruned mixture model. The pruning strategy leads to a probability distribution that lacks some of the information provided by the multiple-point statistics from the training image, which reduces the reproducibility of the training image patterns in the outcome realizations. When pruned mixture models are used as prior models for inverse problems, local re-simulations are performed to obtain perturbed realizations. Consequently, these local re-simulations lead to additional pruning in the set of conditioning data, which further deteriorates the pattern reproduction. To mitigate this problem, it is here suggested to combine the pruned mixture model with a frequency matching model. The multiple-point statistics of outcome realizations from this combined model has improved degree of match with the statistics from the training image. An efficient algorithm that samples this combined model is suggested. Finally, a tomographic cross-borehole inverse problem with prior information expressed by the combined (prior) model is used to demonstrate the effect of pattern reproducibility on the resolution of an inverse problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Alabert F (1987) The practice of fast conditional simulations through LU decomposition of the covariance matrix. Math Geol 19(5):369–386

    Article  Google Scholar 

  • Binley A, Winship P, Middleton R, Pokar M, West J (2001) High resolution characterization of vadose zone dynamics using cross-borehole radar. Water Resour Res 37:2639–2652

    Article  Google Scholar 

  • Bishop CM (2006) Pattern recognition and machine learning. Springer, New York

    Google Scholar 

  • Boucher A, Kyriakidis PC, Cronkite-Ratcliff C (2008) Geostatistical solutions for super-resolution land cover mapping. IEEE Trans Geosci Remote Sens 46(1):272–283

    Article  Google Scholar 

  • Caers J, Hoffman T (2006) The probability perturbation method: a new look at Bayesian inverse modeling. Math Geol 38(1):81–100

    Article  Google Scholar 

  • Castillo E, Gutiérrez JM, Hadi AS (1997) Expert systems and probabilistic network models. Springer-Verlag New York, Inc, New York

    Book  Google Scholar 

  • Cordua KS, Looms MC, Nielsen L (2008) Accounting for correlated data errors during inversion of cross-borehole ground penetrating radar data. Vadose Zone J 7(1):263–271

    Google Scholar 

  • Cordua KS, Nielsen L, Looms MC, Hansen TM, Binley A (2009) Quantifying the influence of static-like errors in least-squares-based inversion and sequential simulation of cross-borehole ground penetrating radar data. J Appl Geophys 68:71–84

    Article  Google Scholar 

  • Cordua KS, Hansen TM, Mosegaard K (2012) Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information. Geophysics 77:H19–H31

    Article  Google Scholar 

  • Cressie N, Davidson J (1998) Image analysis with partially order Markov models. Comput Stat Data Anal 29:1–26

    Article  Google Scholar 

  • Daly C (2005) Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong O, Deutsch CV (eds) Quantitative geology and geostatistics, geostatistics Banff 2004, part 1. Springer Netherlands, pp 215–224

  • Deutsch CV, Journel AG (1998) GSLIB: Geostatistical software library. Oxford University Press, Oxford

    Google Scholar 

  • Emery X, Lantuéjoul C (2013) Can a training image be a substitute for a random field model? Geosci Math. doi:10.1007/s11004-013-9492-z

    Google Scholar 

  • Faucher C, Saucier A, Marcotte D (2013) A new patchwork simulation method with control of the local-mean histogram. Stoch Environ Res Risk Assess 27:253–273

    Article  Google Scholar 

  • Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741

    Article  Google Scholar 

  • Geyer CJ, Thompson EA (1995) Annealing Markov chain Monte Carlo with applications to ancestral inference. J Am Stat Assoc 90:909–920

    Article  Google Scholar 

  • Gómez-Hernández J, Cassiraga EF (2000) Sequential conditional simulations with linear Constrains. In: Monestiez P, Allard D, Froideveaux R (eds) Geostatistics’2000 Cape Town. Geostatistical Association of Southern Africa, Cape Town

    Google Scholar 

  • Guardiano FB, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics-Troia, vol 1. Kluwer Academic Publishers, Dordrecht, p 133–144

  • Hansen TM, Mosegaard K, Cordua KS (2008) Using geostatistics to describe complex a priori information for inverse problems. In: Ortiz JM, Emery X (eds) GEOSTATS 2008, Proceedings of the Eighth International Geostatistics Congress, Chile, vol 1. p 329–338

  • Hansen TM, Mosegaard K, Cordua KS (2009) Reducing complexity of inverse problems using geostatistical priors. IAMG 2009, Stanford, CA

  • Hansen TM, Cordua KS, Mosegaard K (2012) Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling. Comput Geosci 16:593–611

    Article  Google Scholar 

  • Hansen TM, Cordua KS, Looms MC, Mosegaard K (2013) SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information Part 2–application to crosshole GPR tomography. Comput Geosci 52:481–492

    Article  Google Scholar 

  • Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109

    Article  Google Scholar 

  • Irving J, Singha K (2010) Stochastic inversion of tracer test and electrical geophysical data to estimate hydraulic conductivities. Water Resour Res 46:W11514

    Article  Google Scholar 

  • Jensen CS, Kjærulff U (1995) Blocking Gibbs sampling in very large probabilistic expert systems. Int J Hum Comput Stud 42:647–666

    Article  Google Scholar 

  • Journel AG (1974) Geostatistics for conditional simulation of ore bodies. Econ Geol 69:673–687

    Article  Google Scholar 

  • Journel A, Zhang T (2006) The necessity of a multiple-point prior model. Math Geol 38:591–610

    Article  Google Scholar 

  • Kjønsberg H, Stien M, Kolbjørnsen O, Fjellvoll B, Abrahamsen P (2012) Using multiple grids in Markov mesh facies modeling. In: Ninth international geostatistics congress, Oslo, Norway, 11–15 June

  • Lange K, Frydendall J, Cordua KS, Hansen TM, Melnikova Y, Mosegaard K (2012) A frequency matching method: solving inverse problems by use of geologically realistic prior information. Math Geosci 44:783–803

    Article  Google Scholar 

  • Mariethoz G, Kelly BFJ (2011) Modeling complex geological structures with elementary training images and transform-invariant distances. Water Resour Res 47:W07527

    Article  Google Scholar 

  • Mariethoz G, Renard P, Caers J (2010a) Bayesian inverse problem and optimization with iterative spatial resampling. Water Resour Res 46:W11530

    Google Scholar 

  • Mariethoz G, Renard P, Straubhaar (2010b) The Direct Sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46:W11536

  • McCullagh P, Nelder J (1989) Generalized linear models, 2nd edn. Chapman and Hall, Boca Raton

    Book  Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 1(6):1087–1092

    Article  Google Scholar 

  • Mosegaard K (2006) Monte Carlo analysis of inverse problems. Doctoral Thesis, Faculty of Science, University of Copenhagen

  • Mosegaard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res 100(B7):431–447

    Article  Google Scholar 

  • Newell GF, Montroll EW (1953) On the theory of the Ising model of ferromagnetism. Rev Mod Phys 25:353–389

    Article  Google Scholar 

  • Stien M, Kolbjørnsen O (2011) Facies modeling using a Markov mesh model specification. Math Geosci 43:611–624

    Article  Google Scholar 

  • Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34:1–21

    Article  Google Scholar 

  • Strebelle S, Zhang T (2004) Non-stationary multiple-point geostatistical models. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff 2004. Springer, New York, pp 235–244

    Google Scholar 

  • Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. Society of Industrial and Applies Mathematics, Philadelphia

    Book  Google Scholar 

  • Tarantola A, Valette B (1982) Inverse problems = quest for information. J Geophys 50:159–170

    Google Scholar 

  • Tjelmeland H (1996) Stochastic models in reservoir characterization and Markov random fields for compact objects. PhD thesis, Norwegian University of Science and Technology. Thesis Number 44:1996

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

    Article  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 

  • Tuanfeng Z, Stein IP, McCormick D (2008) Patched path and recursive servo system in multi-point geostatistical simulation. In: Proceeding of Geostats 2008. Santiago, Chile

  • Zelt C, Barton P (1998) Three-dimensional seismic refraction seismic refraction tomography—a comparison of two methods applied to data from the Faeroe Basin. J Geophys Res 103(B4):7187–7210

    Article  Google Scholar 

Download references

Acknowledgments

We thank DONG E&P for financial support. Moreover, we are very grateful for the comments and suggestions that were provided by associated editor Odd Kolbjørnsen and two anonymous reviewers, which we believe improved this paper considerably.

Author information

Authors and Affiliations

  1. Center for Energy Resources Engineering, National Space Institute, Technical University of Denmark, Electrovej, Building 327, 2800 , Kongens Lyngby, Denmark

    Knud Skou Cordua, Thomas Mejer Hansen & Klaus Mosegaard

Authors
  1. Knud Skou Cordua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Mejer Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Mosegaard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Knud Skou Cordua.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cordua, K.S., Hansen, T.M. & Mosegaard, K. Improving the Pattern Reproducibility of Multiple-Point-Based Prior Models Using Frequency Matching. Math Geosci 47, 317–343 (2015). https://doi.org/10.1007/s11004-014-9531-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11004-014-9531-4

Keywords

Search

Navigation