Abstract
Understanding the geological uncertainty of hydrostratigraphic models is important for risk assessment in hydrogeology. An important feature of sedimentary deposits is the directional ordering of hydrostratigraphic units (HSU). Geostatistical simulation methods propose efficient algorithm for assessing HSU uncertainty. Among different geostatistical methods to simulate categorical data, Bayesian maximum entropy method (BME) and its simplified version Markov-type categorical prediction (MCP) present interesting features. In particular, the zero-forcing property of BME and MCP can provide a valuable constrain on directional properties. We illustrate the ability of MCP to simulate vertically ordered units. A regional hydrostratigraphic system with 11 HSU and different abundances is used. The transitional deterministic model of this system presents lateral variations and vertical ordering. The set of 66 (11 × 12/2) bivariate probability functions is directly calculated on the deterministic model with fast Fourier transform. Despite the trends present in the deterministic model, MCP is unbiased for the HSU proportions in the non-conditional case. In the conditional cases, MCP proved robust to datasets over-representing some HSU. The inter-realizations variability is shown to closely follow the amount and quality of data provided. Our results with different conditioning datasets show that MCP replicates adequately the directional units arrangement. Thus, MCP appears to be a practical method for generating stochastic models in a 3D hydrostratigraphic context.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allard D, D’Or D, Froidevaux R (2011) An efficient maximum entropy approach for categorical variable prediction. Eur J Soil Sci 62:381–393. https://doi.org/10.1111/j.1365-2389.2011.01362.x
Armstrong M, Galli A, Beucher H, Loc’h G, Renard D, Doligez B, Eschard R, Geffroy F (2011) Plurigaussian simulations in geosciences. Springer, Berlin
Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geol 39(2):177–203. https://doi.org/10.1007/s11004-006-9075-3
Bajc A, Mulligan R, Rainsford D, Webb J (2015) Results of 2011–13 overburden drilling programs in the southern part of the County of Simcoe, south-central Ontario; Ontario Geological Survey, Miscellaneous Release-Data 324. Techical report
Bogaert P (2002) Spatial prediction of categorical variables: the Bayesian maximum entropy approach. Stoch Environ Res Risk Assess 16(6):425–448. https://doi.org/10.1007/s00477-002-0114-4
Bogaert P, D’Or D (2002) Estimating soil properties from thematic soil maps: the Bayesian Maximum Entropy approach. Soil Sci Soc Am J 66(5):1492–1500. https://doi.org/10.2136/sssaj2002.1492
Boucher A, Costa JF, Rasera LG, Motta E (2014) Simulation of geological contacts from interpreted geological model using multiple-point statistics. Math Geosci 46(5):561–572. https://doi.org/10.1007/s11004-013-9510-1
Chilès J, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New York
Christakos G (1992) Random field models in earth sciences. Academic Press, San Diego. https://doi.org/10.1016/B978-0-12-174230-0.50005-6
D’Or D (2003) Spatial prediction of soil properties, the Bayesian Maximum Entropy approach. Ph.D. thesis, Université catholique de Louvain
D’Or D, Bogaert P (2004) Spatial prediction of categorical variables with the Bayesian Maximum Entropy approach: the Ooypolder case study. Eur J Soil Sci 55(4):763–775. https://doi.org/10.1111/j.1365-2389.2004.00628.x
D’Or D, Bogaert P, Christakos G (2001) Application of the BME approach to soil texture mapping. Stoch Environ Res Risk Assess 15(1):87–100. https://doi.org/10.1007/s004770000057
Jakeman A, Barreteau O, Hunt R, Rinaudo JD, Ross A (2016) Integrated groundwater management—concepts, approaches and challenges, vol 1. Springer, Berlin
Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86. http://www.jstor.org/stable/2236703
Le Blévec TL, Dubrule O, John CM, Hampson GJ (2017) Modelling asymmetrical facies successions using pluri-Gaussian simulations. Geostatistics Valencia 2016, quantitative geology and geostatistics. Springer, Cham, pp 59–75. https://doi.org/10.1007/978-3-319-46819-8_4
Li W (2007) Markov chain random fields for estimation of categorical variables. Math Geol 39(3):321–335. https://doi.org/10.1007/s11004-007-9081-0
Li W, Zhang C (2007) A random-path Markov chain algorithm for simulating categorical soil variables from random point samples. Soil Sci Soc Am J 71(3):656–668. https://doi.org/10.2136/sssaj2006.0173
Li W, Zhang C, Burt JE, Zhu AX, Feyen J (2004) Two-dimensional Markov chain simulation of soil type spatial distribution. Soil Sci Soc Am J 68(5):1479–1490. https://doi.org/10.2136/sssaj2004.1479
Marcotte D (1996) Fast variogram computation with FFT. Comput Geosci 22(10):1175–1186. https://doi.org/10.1016/S0098-3004(96)00026-X
Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):W11536. https://doi.org/10.1029/2008WR007621
Molson JW, Frind EO (2012) On the use of mean groundwater age, life expectancy and capture probability for defining aquifer vulnerability and time-of-travel zones for source water protection. J Contam Hydrol 127(1–4):76–87. https://doi.org/10.1016/j.jconhyd.2011.06.001
Ortiz JM (2003) Characterization of high order correlation for enhanced indicator simulation. Ph.D. thesis, University of Alberta, Edmonton, Alberta, Canada
Orton TG, Lark RM (2007) Accounting for the uncertainty in the local mean in spatial prediction by Bayesian Maximum Entropy. Stoch Environ Res Risk Assess 21(6):773–784. https://doi.org/10.1007/s00477-006-0089-7
Ravalec ML, Noetinger B, Hu LY (2000) The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations. Math Geol 32(6):701–723. https://doi.org/10.1023/A:1007542406333
Refsgaard JC, Christensen S, Sonnenborg TO, Seifert D, Højberg AL, Troldborg L (2012) Review of strategies for handling geological uncertainty in groundwater flow and transport modeling. Adv Water Resour 36:36–50. https://doi.org/10.1016/j.advwatres.2011.04.006
Rezaee H, Asghari O, Koneshloo M, Ortiz JM (2014) Multiple-point geostatistical simulation of dykes: application at Sungun porphyry copper system, Iran. Stoch Environ Res Risk Assess 28(7):1913–1927. https://doi.org/10.1007/s00477-014-0857-8
Rezaee H, Marcotte D, Tahmasebi P, Saucier A (2015) Multiple-point geostatistical simulation using enriched pattern databases. Stoch Environ Res Risk Assess 29(3):893–913. https://doi.org/10.1007/s00477-014-0964-6
Serre ML, Christakos G (1999) Modern geostatistics: computational BME analysis in the light of uncertain physical knowledge—the Equus Beds study. Stoch Environ Res Risk Assess 13(1–2):1–26. https://doi.org/10.1007/s004770050029
Serre ML, Christakos G, Li H, Miller CT (2003) A BME solution of the inverse problem for saturated groundwater flow. Stoch Environ Res Risk Assess 17(6):354–369. https://doi.org/10.1007/s00477-003-0156-2
Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21. https://doi.org/10.1023/A:1014009426274
Tran TT (1994) Improving variogram reproduction on dense simulation grids. Comput Geosci 20(7):1161–1168. https://doi.org/10.1016/0098-3004(94)90069-8
Zhou Y, Li W (2011) A review of regional groundwater flow modeling. Geosci Front 2(2):205–214. https://doi.org/10.1016/j.gsf.2011.03.003
Acknowledgements
Constructive comments from two anonymous reviewers were helpful improving the manuscript. In particular, one reviewer suggested to us the idea of comparing MCP to Gaussian simulator as in Sect. 2.4. The authors thank A. Bolduc, Y. Michaud and H. Russell for their support from the Groundwater Geoscience Program, Geological Survey of Canada, Natural Resources Canada. Research was partly financed by NSERC (RGPIN-2015-06653).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benoit, N., Marcotte, D., Boucher, A. et al. Directional hydrostratigraphic units simulation using MCP algorithm. Stoch Environ Res Risk Assess 32, 1435–1455 (2018). https://doi.org/10.1007/s00477-017-1506-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-017-1506-9