Abstract
Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.
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References
Ang AHS, Tang W (1990) Probability concepts in engineering planning and design. Volume II: Decision, risk, and reliability. Wiley, New York
Borovkov K (1994) On simulation of random vectors with given densities in regions and on their boundaries. J Appl Probab 31:205–220
Bourgault G (2012) On the likelihood and fluctuations of Gaussian realizations. Math Geosci 44:1005–1037. doi:10.1007/s11004-012-9424-3
Caers J (2011) Modeling uncertainty in the Earth sciences. Wiley, New York
Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York
Dietrich C, Newsam G (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J Sci Comput 18(4):1088–1107
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York
Gutjahr A, Bras R (1993) Spatial variability in subsurface flow and transport: a review. Reliab Eng Syst Saf 42:293–316
Hardin D, Saff E (2004) Discretizing manifolds via minimum energy points. Not Am Math Soc 51:1186–1194
Helton JC, Davis FJ (2002) Illustration of sampling-based methods for uncertainty and sensitivity analysis. Risk Anal 22(3):591–622
Helton JC, Davis FJ (2003) Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab Eng Syst Saf 81(1):23–69
Helton JC, Johnson JD, Salaberry CJ, Storlie CB (2006) Survey of sampling based methods for uncertainty and sensitivity analysis. Reliab Eng Syst Saf 91:1175–1209
Iman RL, Conover WJ (1982) A distribution-free approach to inducing rank correlation among input variables. Commun Stat, Part B Simul Comput 11(3):311–334
Johnson M (1987) Multivariate statistical simulation. Wiley, New York
Johnson R, Wichern D (1998) Applied multivate statistical analysis, 4th edn. Prentice Hall, Upper Saddle River
Kendall M (1945) The advanced theory of statistics, vol I, 2nd edn. Charles Griffin & Company, London
Kroese D, Taimre T, Botev Z (2011) Handbook of Monte Carlo methods. Wiley, Hoboken
Kyriakidis PC (2005) Sequential spatial simulation using Latin hypercube sampling. In: Leuagthong O, Deutsch C (eds) Geostatistics Banff 2004: 7th international geostatistics congress, quantitative geology and geostatistics, vol 14. Kluwer Academic, Dordrecht, pp 65–74
Lemieux C (2009) Monte Carlo and quasi-Monte Carlo sampling. Springer, New York
Li H, Zhang D (2007) Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour Res 43:W09409. doi:10.1029/2006WR005673
McDonald M, Harbaugh A (1988) A modular three-dimensional finite difference ground-water flow model. Tech. Rep. Techniques of Water-Resources Investigations, Book 6: Modeling Techniques, US Geological Survey
McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245
Morgan MG, Henrion M (1990) Uncertainty: a guide to dealing with uncertainty in quantitative risk and policy analysis. Cambridge University Press, Cambridge
Pebesma EJ, Heuvelink GBM (1999) Latin hypercube sampling of Gaussian random fields. Technometrics 41:303–312
Rubin Y (2003) Applied stochastic hydrogeology. Oxford University Press, New York
Scheidt C, Caers J (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41:397–419. doi:10.1007/s11004-008-9186-0
Stein M (1987) Large sample properties of simulations using Latin hypercube sampling. Technometrics 29(2):143–151
Sudicky E, Illman W, Goltz I, Adams J, McLaren R (2010) Heterogeneity in hydraulic conductivity and its role on the macroscale transport of a solute plume: from measurements to a practical application of stochastic flow and transport theory. Water Resour Res 46:W01508. doi:10.1029/2008WR007
Switzer P (2000) Multiple simulation of spatial fields. In: Heuvelink G, Lemmens M (eds) Proceedings of the 4th international symposium on spatial accuracy assessment in natural resources and environmental sciences, pp 629–635. Coronet Books Inc
Xu C, He HS, Hu Y, Chang Y, Li X, Bu R (2005) Latin hypercube sampling and geostatistical modeling of spatial uncertainty in a spatially explicit forest landscape model simulation. Ecol Model 185:255–269
Zhang Y, Pinder G (2003) Latin hypercube lattice sample selection strategy for correlated random hydraulic conductivity fields. Water Resour Res 39(8):1226. doi:10.1029/2002WR001822
Zheng C (1990) MT3D, a modular three-dimensional transport model for simulation of advection, dispersion and chemical reactions of contaminants in groundwater systems. Tech. Rep. Report to the Kerr Environmental Research, Laboratory, US Environmental Protection Agency
Acknowledgements
This work is part of the project “Advances in Geostatistics for Environmental Characterization and Natural Resources Management,” implemented under the “Aristeia” Action of the “Operational Programme Education and Lifelong Learning,” and co-funded by the European Social Fund (ESF) and Greek National Resources. The authors would like to gratefully acknowledge the support of the guest editor Professor Jaime Gómez-Hernández to include this paper in the special issue, as well as the efforts of three anonymous reviewers, whose comments led to significant improvements in the originally submitted manuscript. The support of Stelios Liodakis from the Department of Geography at the University of the Aegean, Greece, in terms of fruitful discussions and administrative assistance for this project is also gratefully acknowledged.
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Kyriakidis, P., Gaganis, P. Efficient Simulation of (Log)Normal Random Fields for Hydrogeological Applications. Math Geosci 45, 531–556 (2013). https://doi.org/10.1007/s11004-013-9470-5
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DOI: https://doi.org/10.1007/s11004-013-9470-5