Advertisement

Computational Geosciences

, Volume 16, Issue 4, pp 933–952 | Cite as

Accuracy and efficiency of stencils for the eikonal equation in earth modelling

  • Tor Gillberg
  • Øyvind Hjelle
  • Are Magnus Bruaset
Original Paper

Abstract

Motivated by the needs for creating fast and accurate models of complex geological scenarios, accuracy and efficiency of three stencils for the isotropic eikonal equation on rectangular grids are evaluated using a fast marching implementation. The stencils are derived by direct modelling of the wave front, resulting in new and valuable insight in terms of improved upwind and causality conditions. After introducing a method for generalising first-order upwind stencils to higher order, a new second-order diagonal stencil is presented. Similarly to the multistencil fast marching approach, the diagonal stencil makes use of nodes in the diagonal directions, whereas the traditional Godunov stencil uses solely edge-connected neighbours. The diagonal stencil uses nodes close to each other, reaching upwind, to get a more accurate estimate of the angle of incidence of the arriving wave front. Although the stencils are evaluated in a fast marching setting, they can be adapted to other efficient eikonal solvers. All first- and second-order stencils are evaluated in a range of tests. The first test case models a folded structure from the Zagros fold belt in Iran. The other test cases are constructed to investigate specific properties of the examined stencils. The numerical investigation considers convergence rates and CPU times for non-constant and constant speed first-arrival computations. In conclusion, the diagonal stencil is the most efficient and accurate of the three alternatives.

Keywords

Numerical stencils Stencil accuracy and efficiency Eikonal equation Fast marching method Front propagation Earth modelling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alaei, B., Petersen, S.A.: Geological modelling and finite difference forward realization of a regional section from the Zagros fold-and-thrust belt. Pet. Geosci. 13(3), 241–251 (2007)CrossRefGoogle Scholar
  2. 2.
    Alkhalifah, T.: Implementing the fast marching eikonal solver: spherical versus cartesian coordinates. Geophys. Prospect. 70(2), 245–178 (2001)Google Scholar
  3. 3.
    Antiga, L., Steinman, D.A.: VMTK: the Vascular Modeling Toolkit. http://www.vmtk.org (2010). Accessed 1 Mar 2012
  4. 4.
    Appia, V., Yezzi, A.: Fully isotropic fast marching methods on cartesian grids. In: K. Daniilidis, P. Maragos, N. Paragios (eds.) Computer Vision—ECCV 2010, Lecture Notes in Computer Science, pp. 73–85. Springer, Berlin (2010)Google Scholar
  5. 5.
    Bak, S., McLaughlin, J., Renzi, D.: Some improvements for the fast sweeping method. SIAM J. Sci. Comput. 32, 2853–2874 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bancroft, J., Du, X.: Circular wavefront assumptions for gridded traveltime computations. CREWES Report 17, 1–14 (2005)Google Scholar
  7. 7.
    Berre, I., Karlsen, K.H., Lie, K.A., Natvig, J.R.: Fast computation of arrival times in heterogeneous media. Comput. Geosci. 9(4), 179–201 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Danielsson, P.E., Lin, Q.: A modified fast marching method. In: SCIA’03: Proceedings of the 13th Scandinavian Conference on Image Analysis, pp. 1154–1161. Springer, Berlin (2003)Google Scholar
  9. 9.
    Gillberg, T.: A semi-ordered fast iterative method (SOFI) for monotone front propagation in simulations of geological folding. In: MODSIM2011, 19th International Congress on Modelling and Simulation, pp. 631–647 (2011)Google Scholar
  10. 10.
    Gremaud, P.A., Christopher, Kuster, M.: Computational study of fast methods for the eikonal equation. SIAM J. Sci. Comput. 27, 1803–1816 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hassouna, M.S., Farag, A.A.: Multistencils fast marching methods: a highly accurate solution to the eikonal equation on cartesian domains. IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1563–1574 (2007)CrossRefGoogle Scholar
  12. 12.
    Hjelle, Ø., Petersen, S.A.: A Hamilton–Jacobi framework for modeling folds in structural geology. Math. Geosci. 43(7), 741–761 (2011)zbMATHCrossRefGoogle Scholar
  13. 13.
    Huang, J.W., Bellefleur, G.: Joint transmission and reflection traveltime tomography using the fast sweeping method and the adjoint-state technique. Geophys. J. Int. 188(2), 570–582 (2012)CrossRefGoogle Scholar
  14. 14.
    Hysing, S.R., Turek, S.: The eikonal equation: numerical efficiency vs. algorithmic complexity on quadrilateral grids. In: Proceedings of ALGORITMY (2005)Google Scholar
  15. 15.
    Jeong, W.K., Whitaker, R.T.: A fast iterative method for a class of Hamilton–Jacobi equations on parallel systems. Tech. rep., University of Utah (2007)Google Scholar
  16. 16.
    Jeong, W.K., Whitaker, R.T.: A fast iterative method for eikonal equations. SIAM J. Sci. Comput. 30(5), 2512–2534 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kadlec, B., Dorn, G.: Leveraging graphics processing units (GPUs) for real-time seismic interpretation. Lead. Edge 29(1), 60–66 (2010)CrossRefGoogle Scholar
  18. 18.
    Kim, S.: An O(N) level set method for eikonal equations. SIAM J. Sci. Comput. 22(6), 2178–2193 (2001)zbMATHCrossRefGoogle Scholar
  19. 19.
    Kim, S.: 3-D eikonal solvers: first-arrival traveltimes. Geophysics 67(4), 1225–1231 (2002)Google Scholar
  20. 20.
    Kimmel, R., Sethian, J.A.: Optimal algorithm for shape from shading and path planning. J. Math. Imaging Vis. 14, 237–244 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lambaré, G., Lucio, P.S., Hanyga, A.: Two-dimensional multivalued traveltime and amplitude maps by uniform sampling of a ray field. Geophys. J. Int. 125(2), 584–598 (1996)CrossRefGoogle Scholar
  22. 22.
    Li, H., Elmoataz, A., Fadili, J., Ruan, S.: Dual front evolution model and its application in medical imaging. In: MICCAI 2004. Saint Malo, France (2004)Google Scholar
  23. 23.
    Lin, Q.: Enhancement, extraction, and visualization of 3D volume data. Ph.D. thesis, Linköping University, Institute of Technology (2003)Google Scholar
  24. 24.
    Mallet, J.L.: Space–time mathematical framework for sedimentary geology. Math. Geol. 36(1), 1–32 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Perrin, M., Zhu, B., Rainaud, J.F., Schneider, S.: Knowledge-driven applications for geological modeling. J. Pet. Sci. Eng. 47(1–2), 89–104 (2005, Intelligent Computing in Petroleum Engineering)CrossRefGoogle Scholar
  26. 26.
    Petersen, S.A.: Compound modelling, a geological approach to the construction of shared earth models. In: EAGE 61th Conference & Exhibition, Extended Abstracts (1999)Google Scholar
  27. 27.
    Petersen, S.A.: Optimization strategy for shared earth modeling. In: EAGE 66th Conference & Exhibition, Extended Abstracts (2004)Google Scholar
  28. 28.
    Petersen, S.A., Hjelle, Ø.: Earth recursion, an important component in shared Earth model builders. In: EAGE 70th Conference & Exhibition, Extended Abstracts (2008)Google Scholar
  29. 29.
    Petersen, S.A., Hjelle, Ø., Jensen, S.L.: Earth modelling using distance fields derived by fast marching. In: EAGE 69th Conference & Exhibition, Extended Abstracts (2007)Google Scholar
  30. 30.
    Podvin, P., Lecomte, I.: Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys. J. Int. 105, 271–284 (1991)CrossRefGoogle Scholar
  31. 31.
    Popovici, A.M., Sethian, J.A.: 3-D traveltime computation using the fast marching method. Geophysics 64(2), 516–523 (1999)Google Scholar
  32. 32.
    Popovici, A.M., Sethian, J.A.: 3-D imaging using higher order fast marching traveltimes. Geophysics 67(604, Issue 2), 604–609 (2002)Google Scholar
  33. 33.
    Qin, F., Luo, Y., Olsen, K.B., Cai, W., Schuster, G.T.: Finite-difference solution of the eikonal equation along expanding wavefronts. Geophysics 57(3), 478–487 (1992)Google Scholar
  34. 34.
    Rawlinson, N., Hauser, J., Sambridge, M.: Seismic ray tracing and wavefront tracking in laterally heterogeneous media. Adv. Geophys. 49, 203–267 (2009)CrossRefGoogle Scholar
  35. 35.
    Rawlinson, N., Sambridge, M.: Multiple reflection and transmission phases in complex layered media using a multistage fast marching method. Geophysics 69(5), 1338–1350 (2004)Google Scholar
  36. 36.
    Rickett, J., Fomel, S.: A second-order fast marching eikonal solver. Stanford Exploration Project Report, vol. 100, pp. 287–292 (1999)Google Scholar
  37. 37.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  38. 38.
    Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton-Jacobi equations. Proc. Natl. Acad. Sci. U.S.A. 98(20), 11,069–11,074 (2001)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Contr. 40(9), 1528–1538 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Tveito, A., Bruaset, A.M., Lysne, O. (eds.): Simula Research Laboratory—by Thinking Constantly About It, chap. Turning Rocks into Knowledge. Springer, Berlin (2010)Google Scholar
  41. 41.
    Vidale, J.: Finite-difference calculation of travel times. Bull. Seismol. Soc. Am. 78(6), 2062–2076 (1988)Google Scholar
  42. 42.
    Weber, O., Devir, Y.S., Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Parallel algorithms for approximation of distance maps on parametric surfaces. ACM Trans. Graph. 27(4), 1–16 (2008)CrossRefGoogle Scholar
  43. 43.
    Xu, S.G., Zhang, Y.X., Yong, J.H.: A fast sweeping method for computing geodesics on triangular manifolds. IEEE Trans. Pattern Anal. Mach. Intell. 32(2), 231–241 (2010)CrossRefGoogle Scholar
  44. 44.
    Zhang, Y.T., Chen, S., Li, F., Zhao, H., Shu, C.W.: Uniformly accurate discontinuous Galerkin fast sweeping methods for eikonal equations. SIAM J. Sci. Comput. 33(4), 1873–1896 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Zhang, Y.T., Zhao, H.K., Qian, J.: High order fast sweeping methods for static Hamilton–Jacobi equations. SIAM J. Sci. Comput. 29(1), 25–56 (2006)MathSciNetGoogle Scholar
  46. 46.
    Zhao, H.K.: A fast sweeping method for eikonal equations. Math. Comput. 74(250), 603–627Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Tor Gillberg
    • 1
  • Øyvind Hjelle
    • 2
  • Are Magnus Bruaset
    • 3
    • 4
  1. 1.Computational GeosciencesSimula Research LaboratoryLysakerNorway
  2. 2.Kalkulo ASLysakerNorway
  3. 3.Simula Research LaboratoryLysakerNorway
  4. 4.Department of InformaticsUniversity of OsloOsloNorway

Personalised recommendations