J. Florian Wellmann and Klaus Regenauer-Lieb.
Uncertainties have a meaning: Information entropy as a quality measure for 3-D geological models.
In Tectonophysics, vol. 526-529, pp. 207-216, 2012.


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Abstract:

Analyzing, visualizing and communicating uncertainties are important issues as geological models can never be fully determined. To date, there exists no general approach to quantify uncertainties in geological modeling. We propose here to use information entropy as an objective measure to compare and evaluate model and observational results. Information entropy was introduced in the 50s and defines a scalar value at every location in the model for predictability. We show that this method not only provides a quantitative insight into model uncertainties but, due to the underlying concept of information entropy, can be related to questions of data integration (i.e. how is the model quality interconnected with the used input data) and model evolution (i.e. does new data - or a changed geological hypothesis - optimize the model). In other words information entropy is a powerful measure to be used for data assimilation and inversion. As a first test of feasibility, we present the application of the new method to the visualization of uncertainties in geological models, here understood as structural representations of the subsurface. Applying the concept of information entropy on a suite of simulated models, we can clearly identify (a) uncertain regions within the model, even for complex geometries; (b) the overall uncertainty of a geological unit, which is, for example, of great relevance in any type of resource estimation; (c) a mean entropy for the whole model, important to track model changes with one overall measure. These results cannot easily be obtained with existing standard methods. The results suggest that information entropy is a powerful method to visualize uncertainties in geological models, and to classify the indefiniteness of single units and the mean entropy of a model quantitatively. Due to the relationship of this measure to the missing information, we expect the method to have a great potential in many types of geoscientific data assimilation problems - beyond pure visualization.

Bibtex:

@Article{        wellman:2012:IEQM,
  author = 	 {J. Florian Wellmann and Klaus Regenauer-Lieb},
  title = 	 {Uncertainties have a meaning: Information entropy as
                  a quality measure for 3-D geological models},
  journal = 	 {Tectonophysics},
  year = 	 {2012},
  volume = 	 {526-529},
  pages = 	 {207-216},
}

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References:

Aug, C., 2004. Mode`lisation ge`ologique 3D et caracte`risation des incertitudes par la me`thode du champ de potentiel: PhD thesis, E.N.S. des Mines de Paris, 198 pp.
Ben-Naim, A., 2008. A Farewell to Entropy. World Scientific, Singapore. 384 pp.
Bond, E.C., Shipton, K.Z., Jones, R.R., Butler, W.R., Gibbs, D.A., 2007. Knowledge transfer in a digital world: field data acquisition, uncertainty, visualization, and data management. Geosphere 3 (6), 568.
Calcagno, P., Chile`s, J.-P., Courrioux, G., Guillen, A., 2008. Geological modelling from field data and geological knowledge: part I. Modelling method coupling 3D potential- field interpolation and geological rules: recent advances in computational geodynamics: theory, numerics and applications. Physics of the Earth and Planetary Interiors 171 (1-4), 147-157.
Caumon, G., Collon-Drouaillet, P., Carlier, Le, de Veslud, C., Viseur, S., Sausse, J., 2009. Surface-based 3D modeling of geological structures. Mathematical Geosciences 41 (8), 927-945.
Cherpeau, N., Caumon, G., Le`vy, B., 2010. Stochastic simulations of fault networks in 3D structural modeling. Comptes Rendus Geoscience 342 (9), 687-694.
Chile`s, J.-P., Delfiner, P., 1999. Geostatistics: modeling spatial uncertainty. A Wiley- Interscience publication. Wiley, New York, NY.
Dewar, 2005. Maximum entropy production and fluctuation theorem. Journal of Physics A: Mathematical and General 28 (21), L371-L381.
Durand-Riard, P., Caumon, G., Muron, P., 2010. Balanced restoration of geological volumes with relaxed meshing constraints. Computers & Geosciences 36 (4), 441-452.
Goodchild, F.M., Chih-Chang, L., Leung, Y., 1994. Visualizing fuzzy maps. In: Hearnshaw, M.H., Unwin, J.D. (Eds.), Visualization in Geographical Information Systems. John Wiley & Sons, New York, pp. 158-167.
Guillen, A., Calcagno, P., Courrioux, G., Joly, A., Ledru, P., 2008. Geological modelling from field data and geological knowledge: part II. Modelling validation using gravity and magnetic data inversion: recent advances in computational geody- namics: theory, numerics and applications. Physics of the Earth and Planetary Interiors 171 (1-4), 158-169.
Jessell, W.M., Ailleres, L., Kemp, A.E., 2010. Towards an integrated inversion of geoscientific data: what price of geology? Tectonophysics 490 (3-4), 294-306. Jones, R.R., McCaffrey, J.K., Wilson, W.R., Holdsworth, E.R., 2004. Digital field data acquisition: towards increased quantification of uncertainty during geological mapping. Geological Society London Special Publications 239 (1), 43.
Klir, J.G., Folger, A.T., Kruse, R., 1988. Fuzzy Sets, Uncertainty, and Information, 159. Prentice Hall, Englewood Cliffs.
Lajaunie, C., Courrioux, G., Manuel, L., 1997. Foliation fields and 3D cartography in geology: principles of a method based on potential interpolation. Mathematical Geology 29 (4), 571-584.
Leung, Y., Goodchild, F.M., Lin, C.C., 1993. Visualization of fuzzy scenes and probability fields. Computing Science and Statistics 416-422.
Lindsay, M., Ailleres, L., Jessell, M., de Kemp, E., Betts, P., 2010. Integrating geological uncertainty into combined geological and potential field inversions. GeoMod 2010 conference proceedings. University of Lisbon, Lisbon, Portugal. 27-29 September 2010.
Luca, A., Termini, S., 1972. A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Information and Control 20 (4), 301-312.
MacEachren, M.A., Robinson, A., Hopper, S., Gardner, S., Murray, R., Gahegan, M., Hetzler, E., 2005. Visualizing geospatial information uncertainty: what we know and what we need to know. Cartography and Geographic Information Science 32 (3), 139-161.
Mann, J.C., 1993. Uncertainty in geology. Computers in Geology-25 Years of Progress. Oxford University Press, Inc, pp. 241-254.
Shannon, E.C., 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379-423.
Suzuki, S., Caumon, G., Caers, J., 2008. Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization. Computational Geosciences 12 (1), 105-119.
Viard, T., Caumon, G., Levy, B., 2007. Uncertainty Visualization in Geological Grids. 27th Gocad Meeting 2007, pp. 1-21.
Wellmann, F.J., Horowitz, G.F., Schill, E., Regenauer-Lieb, K., 2010. Towards incorpo- rating uncertainty of structural data in 3D geological inversion. Tectonophysics 490 (3-4), 141-151.
Yager, R.R., 1995. Measures of entropy and fuzziness related to aggregation operators. Information Sciences 82 (3-4), 147-166.
Zadeh, A.L., 1965. Fuzzy sets*. Information and control 8 (3), 338-353.