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Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images

Mathematical Geosciences volume 46, pages 625–645 (2014)Cite this article

Abstract

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.

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References

  • Alharthi A, Lange J (1987) Soil water saturation: dielectric determination. Water Resour Res 23(4):591–595

    Article  Google Scholar 

  • Alumbaugh D, Chang P, Paprocki L, Brainard J, Glass R, Rautman C (2002) Estimating moisture contents in the vadose zone using cross-borehole ground penetrating radar: a study of accuracy and repeatability. Water Resour Res 38(12):1309

    Google Scholar 

  • Bayer P (2000) Aquifer-Analog-Studie in grobklastischen ‘braided river’ Ablagerungen: Sedimentäre/hydrogeologische Wandkartierung und Kalibrierung von Georadarmessungen—Diplomkartierung. Master’s thesis, University of Tübingen, Tübingen, Germany (in German)

  • Bayer P, Huggenberger P, Renard P, Comunian A (2011) Three-dimensional high resolution fluvio-glacial aquifer analog: part 1: Field study. J Hydrol 405:1–9

    Article  Google Scholar 

  • Caers J, Ma X (2002) Modeling conditional distributions of facies from seismic using neural nets. Math Geol 34(2):143–167

    Article  Google Scholar 

  • Cardiff M, Kitanidis P (2009) Bayesian inversion for facies detection: an extensible level set framework. Water Resour Res 45(10):W10416

    Google Scholar 

  • Coléou T, Poupon M, Azbel K (2003) Unsupervised seismic facies classification. Lead Edge 22(10):942–953

    Article  Google Scholar 

  • Comunian A, Renard P, Straubhaar J, Bayer P (2011) Three-dimensional high resolution fluvio-glacial aquifer analog: part 2: Geostatistical modeling. J Hydrol 405:10–23

    Article  Google Scholar 

  • Constable S, Parker R, Constable C (1987) Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 52(3):289–300

    Article  Google Scholar 

  • Cordua K, Hansen T, Mosegaard K (2012) Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information. Geophysics 77(2):H19–H31

    Article  Google Scholar 

  • Dafflon B, Barrash W (2012) Three-dimensional stochastic estimation of porosity distribution: benefits of using ground-penetrating radar velocity tomograms in simulated-annealing-based or Bayesian sequential simulation approaches. Water Resour Res 48(5):W05553

    Google Scholar 

  • Davis J, Annan A (1989) Ground-penetrating radar for high-resolution mapping of soil and rock stratigraphy. Geophys Prospect 37(5):531–551

    Article  Google Scholar 

  • Day-Lewis F, Lane J Jr (2004) Assessing the resolution-dependent utility of tomograms for geostatistics. Geophys Res Lett 31(7):L07503

    Google Scholar 

  • Day-Lewis F, Singha K, Binley A (2005) Applying petrophysical models to radar travel time and electrical resistivity tomograms: resolution-dependent limitations. J Geophys Res 110(B8):B08206

    Google Scholar 

  • de Matos M, Osorio P, Johann P (2007) Unsupervised seismic facies analysis using wavelet transform and self-organizing maps. Geophysics 72(1):P9–P21

    Article  Google Scholar 

  • Doser D, Crain K, Baker M, Kreinovich V, Gerstenberger M (1998) Estimating uncertainties for geophysical tomography. Reliab Comput 4(3):241–268

    Article  Google Scholar 

  • Eppstein M, Dougherty D (1998) Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site. Water Resour Res 34(8):1889–1900

    Article  Google Scholar 

  • Gómez-Hernández J, Wen X (1998) To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology. Adv Water Resour 21(1):47–61

    Article  Google Scholar 

  • Heinz J, Kleineidam S, Teutsch G, Aigner T (2003) Heterogeneity patterns of quaternary glaciofluvial gravel bodies (SW-Germany): application to hydrogeology. Sediment Geol 158(1–2):1–23

    Article  Google Scholar 

  • Hu L, Chugunova T (2008) Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour Res 44(11):W11413

    Google Scholar 

  • Hubbard S, Chen J, Peterson J, Majer E, Williams K, Swift D, Mailloux B, Rubin Y (2001) Hydrogeological characterization of the South Oyster Bacterial Transport Site using geophysical data. Water Resour Res 37(10):2431–2456

    Article  Google Scholar 

  • Hubbard S, Rubin Y (2000) Hydrogeological parameter estimation using geophysical data: a review of selected techniques. J Contam Hydrol 45(1–2):3–34

    Article  Google Scholar 

  • Journel A, Posa D (1990) Characteristic behavior and order relations for indicator variograms. Math Geol 22(8):1011–1025

    Article  Google Scholar 

  • Kitanidis P (1997) Introduction to geostatistics: applications in hydrogeology. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Klotzsche A, van der Kruk J, Meles G, Vereecken H (2012) Crosshole gpr full-waveform inversion of waveguides acting as preferential flow paths within aquifer systems. Geophysics 77(4):H57–H62

    Article  Google Scholar 

  • Kostic B, Becht A, Aigner T (2005) 3-D sedimentary architecture of a quaternary gravel delta (SW-Germany): implications for hydrostratigraphy. Sediment Geol 181(3):147–171

    Article  Google Scholar 

  • Krishnan S, Journel A (2003) Spatial connectivity: from variograms to multiple-point measures. Math Geol 35(8):915–925

    Article  Google Scholar 

  • Künze R, Lunati I (2012) An adaptive multiscale method for density-driven instabilities. J Comput Phys 231:5557–5570

    Article  Google Scholar 

  • Linde N, Binley A, Tryggvason A, Pedersen L, Revil A (2006) Improved hydrogeophysical characterization using joint inversion of cross-hole electrical resistance and ground-penetrating radar traveltime data. Water Resour Res 42(12):W04410

    Google Scholar 

  • Mariethoz G, Renard P, Caers J (2010a) Bayesian inverse problem and optimization with iterative spatial resampling. Water Resour Res 46(11):W11530

    Google Scholar 

  • Mariethoz G, Renard P, Straubhaar J (2010b) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):W11536

    Google Scholar 

  • Meerschman E, Pirot G, Mariethoz G, Straubhaar J, Meirvenne M, Renard P (2013) A practical guide to performing multiple-point statistical simulations with the direct sampling algorithm. Comput Geosci 52:307–324

    Article  Google Scholar 

  • Meles G, Greenhalgh S, der Kruk J, Green A, Maurer H (2011) Taming the non-linearity problem in GPR full-waveform inversion for high contrast media. J Appl Geophys 73:174–186

    Article  Google Scholar 

  • Menke W (1989) Geophysical data analysis: discrete inverse theory, vol 45. Academic Press, New York

    Google Scholar 

  • Moysey S, Caers J, Knight R, Allen-King R (2003) Stochastic estimation of facies using ground penetrating radar data. Stoch Environ Res Risk Assess 17(5):306–318

    Article  Google Scholar 

  • Moysey S, Knight R, Jol H (2006) Texture-based classification of ground-penetrating radar images. Geophysics 71(6):K111–K118

    Article  Google Scholar 

  • Moysey S, Singha K, Knight R (2005) A framework for inferring field-scale rock physics relationships through numerical simulation. Geophys Res Lett 32:L08304

    Google Scholar 

  • Ory J, Pratt R (1995) Are our parameter estimators biased? The significance of finite-difference regularization operators. Inverse Probl 11:397–424

    Article  Google Scholar 

  • Paige C, Saunders M (1982) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans Math Softw 8(1):43–71

    Article  Google Scholar 

  • Peterson J (2001) Pre-inversion corrections and analysis of radar tomographic data. J Environ Eng Geophys 6:1–18

    Article  Google Scholar 

  • Podvin P, Lecomte I (1991) Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys J Int 105(1):271–284

    Article  Google Scholar 

  • Rankey E, Mitchell J (2003) That’s why it’s called interpretation: impact of horizon uncertainty on seismic attribute analysis. Lead Edge 22(9):820–828

    Article  Google Scholar 

  • Rauber M, Stauffer F, Huggenberger P, Dracos T (1998) A numerical three-dimensional conditioned/unconditioned stochastic facies type model applied to a remediation well system. Water Resour Res 34(9):2225–2233

    Article  Google Scholar 

  • Richardson J, Sangree J, Sneider R (1987) Applications of geophysics to geologic models and to reservoir description. J Pet Technol 39(7):753–755

    Article  Google Scholar 

  • Roth K, Schulin R, Flühler H, Attinger W (1990) Calibration of time domain reflectometry for water content measurement using a composite dielectric approach. Water Resour Res 26(10):2267–2273

    Google Scholar 

  • Sen P, Scala C, Cohen M (1981) A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46(5):781–795

    Article  Google Scholar 

  • Singha K, Day-Lewis F, Moysey S (2007) Accounting for tomographic resolution in estimating hydrologic properties from geophysical data. In: Hyndman DW, Day-Lewis FD, Singha K (eds) Subsurface hydrology: data integration for properties and processes. Geophysical monograph series, vol 171. AGU, Washington, pp 227–242

    Google Scholar 

  • Singha K, Moysey S (2006) Accounting for spatially variable resolution in electrical resistivity tomography through field-scale rock-physics relations. Geophysics 71(4):A25

    Article  Google Scholar 

  • Straubhaar J, 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 

  • Straubhaar J, Walgenwitz A, Renard P (2013) Parallel multiple-point statistics algorithm based on list and tree structures. Math Geosci 45(2):1–17

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Tinga W, Voss W, Blossey D (1973) Generalized approach to multiphase dielectric mixture theory. J Appl Phys 44(9):3897–3902

    Article  Google Scholar 

  • Tran T (1994) Improving variogram reproduction on dense simulation grids. Comput Geosci 20(7):1161–1168

    Article  Google Scholar 

  • Vidale J (1988) Finite-difference calculation of travel times. Bull Seismol Soc Am 78(6):2062–2076

    Google Scholar 

  • Western A, Blöschl G, Grayson R (2001) Toward capturing hydrologically significant connectivity in spatial patterns. Water Resour Res 37(1):83–97

    Article  Google Scholar 

  • Williamson P (1991) A guide to the limits of resolution imposed by scattering in ray tomography. Geophysics 56(2):202–207

    Article  Google Scholar 

  • Zhou H, Gómez-Hernández J, Li L (2012) A pattern-search-based inverse method. Water Resour Res 48(3):W03505

    Google Scholar 

  • Zinn B, Harvey C (2003) When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields. Water Resour Res 39(3):1051

    Google Scholar 

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Acknowledgements

This research was funded by the Swiss National Science Foundation (SNF) and is a contribution to the ENSEMBLE project (grant no. CRSI22_132249). We would like to thank Philippe Renard for many helpful suggestions during the course of this study. We thank the three anonymous reviewers for their constructive comments that helped to improve the manuscript.

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Authors and Affiliations

  1. Applied and Environmental Geophysics Group, Faculty of Geosciences and Environment, University of Lausanne, Lausanne, Switzerland

    Tobias Lochbühler & Niklas Linde

  2. Center for Hydrogeology and Geothermics, University of Neuchâtel, Neuchâtel, Switzerland

    Guillaume Pirot & Julien Straubhaar

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  1. Tobias Lochbühler
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Correspondence to Tobias Lochbühler.

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Lochbühler, T., Pirot, G., Straubhaar, J. et al. Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images. Math Geosci 46, 625–645 (2014). https://doi.org/10.1007/s11004-013-9484-z

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  • DOI: https://doi.org/10.1007/s11004-013-9484-z

Keywords

  • Multiple-point statistics
  • Multiple-point direct sampling
  • Geophysical tomography
  • Conditioning