Modeling Uncertainty of Complex Earth Systems in Metric Space

  • Jef Caers
  • Kwangwon Park
  • Céline Scheidt
Living reference work entry

Latest version View entry history


Modeling the subsurface of the Earth has many characteristic challenges. Earth models reflect the complexity of the Earth subsurface and contain many complex elements of modeling, such as the subsurface structures, the geological processes of growth and/or deposition, and the placement, movement, or injection/extraction of fluid and gaseous phases contained in rocks or soils. Moreover, due to the limited information provided by measurement data, whether from boreholes or geophysics, and the requirement to make interpretations at each stage of the modeling effort, uncertainty is inherent to any modeling effort. As a result, many alternative (input) models need to be built to reflect the ensemble of sources of uncertainty. On the other hand, the (engineering) purpose (in terms of target response) of these models is often very clear, simple, and straightforward: do we clean up or not, do we drill, where do we drill, what are oil and gas reserves, how far have contaminants traveled, etc. The observation that models are complex but their purpose is simple suggests that input model complexity and dimensionality can be dramatically reduced, not by itself, but by means of the purpose or target response. Reducing dimension by only considering the variability between all possible models may be an impossible task, since the intrinsic variation between all input models is far too complex to be reduced to a few dimensions by simple statistical techniques such as principal component analysis (PCA). In this chapter, we will define a distance between two models created with different (and possibly randomized) input parameters. This distance can be tailored to the application or target output response at hand, but should be chosen such that it correlates with the difference in target response between any two models. A distance defines then a metric space with a broad gamma of theory. Starting from this point of view, we redefine many of the current Cartesian-based Earth modeling problems and methodologies, such as inverse modeling, stochastic simulation and estimation, model selection and screening, model updating, and response uncertainty evaluation in metric space. We demonstrate how such a redefinition greatly simplifies as well as increases effectiveness and efficiency of any modeling effort, particularly those that require addressing the matter of model and response uncertainty.


Euclidean Distance Feature Space Input Model Gaussian Model Inverse Modeling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Alpak F, Barton M, Caers J (2010) A flow-based pattern recognition algorithm for rapid quantification of geologic uncertainty: application to high resolution channelized reservoir models. Comput Geosci, 14:603–621CrossRefGoogle Scholar
  2. Besag J, Green PJ (1993) Spatial statistics and Bayesian computation. J R Stat Soc B 55:3–23MathSciNetGoogle Scholar
  3. Borg I, Groenen P (1997) Modern multidimensional scaling: theory and applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. Caers J, Hoffman T (2006) The probability perturbation method: a new look at Bayesian inverse modeling. Math Geol 38(1):81–100MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dubuisson MP, Jain AK (1994) A modified Hausdorff distance for object matching. In: Proceedings of the 12th international conference on pattern recognition, Jerusalem, vol A, pp 566–568Google Scholar
  6. Evensen G (2003) The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53:343CrossRefGoogle Scholar
  7. Karhunen K (1947) Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann Acad Sci Fennicae Ser A I Math-Phys 37:1–79MathSciNetzbMATHGoogle Scholar
  8. Kwok JT-Y, Tsang IW-H (2004) The pre-image problem in kernel methods. IEEE Trans Neural Netw 15(6):1517–1525CrossRefGoogle Scholar
  9. Lantuejoul C (2002) Geostatistical simulation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  10. Loève M (1978) Probability theory, vol II, 4th edn. Graduate texts in mathematics, vol 46. Springer, BerlinzbMATHGoogle Scholar
  11. Moosegard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res B 100:12431–12447CrossRefGoogle Scholar
  12. Omre H, Tjelmeland H (1996) Petroleum geostatistics. In: Baafi EY, Schofield NA (eds) Proceeding of the fifth international geostatistics congress, Wollongong Australia, vol 1, pp 41–52Google Scholar
  13. Park K, Caers J (2007) History matching in low-dimensional connectivity vector space. In: EAGE petroleum geostatistics conference, Cascais, 10–14 Sept 2007Google Scholar
  14. Ripley BD (2004) Spatial statistics. Wiley series in probability and statistics. Wiley, Hoboken, 251pGoogle Scholar
  15. Roggero F, Hu LY (1998) Gradual deformation of continuous geostatistical models for history matching. In: Proceedings society of petroleum engineers 49004, annual technical conference, New OrleansGoogle Scholar
  16. Scheidt C, Caers J (2009a) Bootstrap confidence intervals for reservoir model selection techniques. Comput Geosci. doi:10.1007/s10596-009-9156-8Google Scholar
  17. Scheidt C, Caers J (2009b) Uncertainty quantification in reservoir performance using distances and kernel methods – application to a West-Africa deepwater turbidite reservoir. SPEJ 118740-PA. Online FirstGoogle Scholar
  18. Scheidt C, Caers J (2009c) Representing spatial uncertainty using distances and kernels. Math Geosci 41(4):397–419. doi:10.1007/s11004-008-9186-0CrossRefzbMATHGoogle Scholar
  19. Scheidt C, Park K, Caers J (2008) Defining a random function from a given set of model realizations. In: Proceedings of the VIII international geostatistics congress, Santiago, 1–5 Dec 2008Google Scholar
  20. Schöelkopf B, Smola A (2002) Learning with kernels. MIT, CambridgeGoogle Scholar
  21. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point geostatistics. Math Geol 34:1–22MathSciNetCrossRefzbMATHGoogle Scholar
  22. Suzuki S, Caers J (2008) A distance-based prior model parameterization for constraining solutions of spatial inverse problems. Math Geosci 40(4):445–469CrossRefzbMATHGoogle Scholar
  23. Suzuki S, Caumon G, Caers J (2008) Dynamic data integration into structural modeling: model screening approach using a distance-based model parameterization. Comput Geosci 12(1):105–119CrossRefzbMATHGoogle Scholar
  24. Tarantola A (1987) Inverse problem theory. Elsevier, Amsterdam, 342pzbMATHGoogle Scholar
  25. Vapnik V (1998) Statistical learning theory. Wiley, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanford, CAUSA

Personalised recommendations