Computing and Visualization in Science

, Volume 14, Issue 4, pp 143–156 | Cite as

Distance field computation for geological slab surface data sets

  • Marek Vančo
  • Bernd Hamann
  • Oliver Kreylos
  • Magali I. Billen
  • Margarete A. Jadamec


The three-dimensional shapes of tectonic plates that sink into the Earth’s mantle (slabs) are the starting point for a range of geoscience studies, from determining the forces driving the motion of tectonic plates, to potential seismic and tsunami hazards, to the sources of magmas beneath active volcanos. For many of these applications finite element methods are used to model the deformation or fluid flow, and therefore the input model parameters, such as feature geometries, temperature or viscosity, must be defined with respect to a smooth, continuous distance field around the slab. In this paper we present a framework for processing sparse and noisy seismic data (earthquake locations), defining the shape of the slab and computing a continuous distance function on a mesh with variable node spacing. Due to the inhomogeneous volumetric distribution of earthquakes within the slab and significant inaccuracies in the locations of earthquakes occurring hundreds of kilometers below the Earth’s surface, the seismicity data set is extremely noisy and incomplete. Therefore, the preprocessing is the major part of the framework consisting of several steps including a point based smoothing procedure, a powerful method to use other observational constraints on slab location (e.g., seismic tomography or geologic history) to extend of the slab shape beyond earthquake data set and continuous resampling using moving least squares method. For the preprocessed point data we introduce approaches for finding the three-dimensional boundary of the slab and a subdivision of the slab into quadric implicit polynomials. The resulting distance field is then compiled from distances to the piecewise continuous approximation of the slab and distances to slab boundary.


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  1. 1.
    Amenta, N., Bern, M.: Surface reconstruction by voronoi filtering. In: SCG ’98: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, ACM Press, pp. 39–48 (1998)Google Scholar
  2. 2.
    Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Point set surfaces. In: Conference on Visualization. IEEE Comput. Soc. pp. 21–28 (2001)Google Scholar
  3. 3.
    Amenta, N., Choi, S., Kolluri, R.K.: The power crust. In 6th ACM Symposium on Solid Modeling and Applications. ACM Press, pp. 249–266 (2001)Google Scholar
  4. 4.
    Billen M.I., Gurnis M., Simons M.: Multiscale dynamics of the Tonga–Kermadec subduction zone. Geophys. J. Int. 153, 359–388 (2003)CrossRefGoogle Scholar
  5. 5.
    Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. In: Israel-Korea Bi-National Conference on Geometric Modeling and Computer Graphics. Tel Aviv University, pp. 83–87 (2003)Google Scholar
  6. 6.
    Chiao, L.-Y.: Membrane Deformation rate and geometry of subducting slabs. Master’s thesis, University of Washington (1991)Google Scholar
  7. 7.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH ’99 (1999), ACM Press/Addison-Wesley Publishing Co., pp. 317–324Google Scholar
  8. 8.
    Fleishman S., Drori I., Cohen-Or D.: Bilateral mesh denoising. ACM Trans. Graph. 22(3), 950–953 (2003)CrossRefGoogle Scholar
  9. 9.
    Gärtner, B., Schönherr, S.: Smallest enclosing ellipses—an exact and generic implementation. Tech. Rep. B 98-05, Institut für Theoretische Informatik, ETH Zentrum Zürich / Institut für Informatik Freie Universität Berlin (1998)Google Scholar
  10. 10.
    Gudmundsson Ó., Sambridge M.: A regionalized upper mantle (RUM) seismic model. J. Geophys. Res. 103, 7121–7136 (1998)CrossRefGoogle Scholar
  11. 11.
    Isacks B. L., Barazangi M.: Geometry of benioff zones: lateral segmentation and downwards bending of the subducted lithosphere. In; Island Arcs, Deep Sea Trenches and Back-arc Basins, vol. 1, pp. 99–114 (1977)Google Scholar
  12. 12.
    Jadamec M.A., Billen M.I.: Reconciling surface plate motions with rapid three- dimensional mantle flow around a slab edge. Nature 465, 338–341 (2010)CrossRefGoogle Scholar
  13. 13.
    Jones T.R., Durand F., Desbrun M.: Non-iterative, feature-preserving mesh smoothing. ACM Trans. Graph. 22(3), 943–949 (2003)CrossRefGoogle Scholar
  14. 14.
    Kobbelt, L., Campagna, S., Vorsatz, J., Seidel, H.-P.: Interactive multi-resolution modeling on arbitrary meshes. In: SIGGRAPH’98. ACM Press, pp. 105–114 (1998)Google Scholar
  15. 15.
    Lee, K.-W., Wang, W.-P.: Feature-preserving mesh denoising via bilateral normal filtering. In: 9th International Conference on Computer Aided Design and Computer Graphics. IEEE Comput. Soc., pp. 275–280 (2005)Google Scholar
  16. 16.
    Miller M.S., Gorbatov A., Kennett B.L.N.: Three-dimensional visualization of a near-vertical slab tear beneath the southern Mariana arc. Geochem. Geophys. Geosyst. 7, 6012 (2006)CrossRefGoogle Scholar
  17. 17.
    Nothard S., McKenzie D., Haines J., Jackson J.: Gaussian curvature and the relationship between the shape and the deformation of the Tonga slab. Geophys. J. Int. 127, 311–327 (1996)CrossRefGoogle Scholar
  18. 18.
    Ohtake Y., Belyaev A.G., Bogaevski I.: Mesh regularization and adaptive smoothing. Comput. Aided Des. 33(11), 789–800 (2001)CrossRefGoogle Scholar
  19. 19.
    Ratchkovzki N.A., Hansen R.A.: New evidence for segmentation of the Alaska subduction zone. Bull. Seismol. Soc. Am. 92, 1754–1765 (2002)CrossRefGoogle Scholar
  20. 20.
    Romanowicz B.: Global mantle tomography: progress status in the past 10 years. Ann. Rev. Earth Planet. Sci. 31, 303–328 (2003)CrossRefGoogle Scholar
  21. 21.
    Syracuse, E.M., Abers, G.A.: Global compilation of variations in slab depth beneath arc volcanoes and implications. Geochem. Geophys. Geosyst. 7 (2006)Google Scholar
  22. 22.
    Sipkin, S.A., Person, W.J., Presgrave, B.W.: Earthquake bulletins and catalogs at the usgs national earthquake information center. In: Iris News Letter, vol. 1, pp. 2–4 (2000)Google Scholar
  23. 23.
    Sun X., Rosin P., Martin R., Langbein F.: Fast and effective feature-preserving mesh denoising. IEEE Trans. Vis. Comput. Graph. 13(5), 925–938 (2007)CrossRefGoogle Scholar
  24. 24.
    Sun, X., Rosin, P.L., Martin, R.R., Langbein, F.C.: Random walks for mesh denoising. In: Symposium on Solid and Physical Modeling, ACM, pp. 11–22 (2007)Google Scholar
  25. 25.
    Tackley P.J.: Mantle convection and plate tectonics: toward an integrated physical and chemical theory. Science 288, 2002–2007 (2000)CrossRefGoogle Scholar
  26. 26.
    Taubin, G.: A signal processing approach to fair surface design. In: ACM SIGGRAPH, ACM Press, pp. 351–358 (1995)Google Scholar
  27. 27.
    Vančo, M., Brunnett, G.: Geometric preprocessing of noisy point sets: an experimental study. Computing 79, (2007) [Special issue on Geometric Modeling]Google Scholar
  28. 28.
    Vančo, M., Brunnett, G., Schreiber, T.: A hashing strategy for efficient k-nearest neighbors computation. In: International Conference on Computer Graphics, IEEE Comput. Soc., pp. 120–127 (1999)Google Scholar
  29. 29.
    van der Hist R., Engdahl R., Spakman W., Nolet G.: Tomographic imaging of subducted lithosphere below northwest Pacific island arcs. Nature 353, 37–43 (1991)CrossRefGoogle Scholar
  30. 30.
    Vančo M., Hamann B., Brunnett G.: Surface reconstruction from unorganized point data with quadrics. Comput. Graph. Forum 27(6), 1593–1606 (2008)CrossRefMATHGoogle Scholar
  31. 31.
    Vassiliou M.S.B.H.H., Raefsky A.: The distribution of earthquakes with depth and stress in subducting slabs. J. Geodyn. 1, 11–28 (1984)CrossRefGoogle Scholar
  32. 32.
    Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: New Results and New Trends in Computer Science, LNCS. Springer, Berlin. (1991)Google Scholar
  33. 33.
    wei Zhoua H.: Observations on earthquake stress axes and seismic morphology of deep slabs. Geophys. J. Int. 103(2), 377–401 (1990)CrossRefGoogle Scholar
  34. 34.
    Zhang H., Thurber C., Shelly D., Ide S., Beroza G.C., Hasegawa A.: High-resolution subducting slab structure beneath northern Honshu, Japan, revealed by double-difference tomography. AGU Fall Meet. Abstr. 32(4), 361–364 (2003)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Marek Vančo
    • 1
  • Bernd Hamann
    • 1
  • Oliver Kreylos
    • 1
  • Magali I. Billen
    • 2
  • Margarete A. Jadamec
    • 2
  1. 1.Department of Computer Science, Institute for Data Analysis and VisualizationUniversity of California DavisDavisUSA
  2. 2.Department of GeologyUniversity of California DavisDavisUSA

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