A Slowness Matching Eulerian Method for Multivalued Solutions of Eikonal Equations

Journal of Scientific Computing

Volumn 19, Issue 1-3, 2003, Pages 501-526

  Symes, William W ^a   Qian, Jianliang ^b  

^a Rice University   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Hamilton Jacobi; Multivalued eikonal solvers; Slowness matching; Viscosity solution; Weighted essentially nonoscillatory scheme (WENO)

Indexed keywords

CAUSTICS; SLOWNESS MATCHING;

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; COMPUTATIONAL METHODS; FINITE DIFFERENCE METHOD; GEODESY; HAMILTONIANS; PROBLEM SOLVING; SEISMIC PROSPECTING; VISCOSITY;

GEOMETRICAL OPTICS;

EID: 0347900513     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1025380731197     Document Type: Article

References (50)

1. Albert, S., Cockburn, B., French, D., and Peterson, T. (2002). A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case. Math. Camp. 71, 49-76.
2. Benamou, J. D. (1996). Big ray tracing: Multivalued travel time field computation using viscosity solutions of the eikonal equations. J. Comput. Phys. 128, 463-474.
3. Benamou, J.-D. (1999). Direct solution of multi-valued phase-space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52, 1443-1475.
4. Benamou, J.-D., and Solliec, I. (2000). A Eulerian method for capturing caustics. J. Comput. Phys. 162, 132-163.
5. Berger, M., and Oliger, J. (1984). Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484-512.
6. Courant, R., and Hubert, D. (1962). Methods of Mathematical Physics, Vol. II, Wiley.
7. Crandall, M. G., Evans, L. C., and Lions, P. L. (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282, 487-502.
8. Crandall, M. G., and Lions, P. L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277, 1-42.
9. Crandall, M. G., and Lions, P. L. (1984). Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43, 1-19.
10. Dellinger, J., and Symes, W. W. (1997). Anisotropic Finite-Difference Traveltimes Using a Hamilton-Jacobi Solver, 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Soc. Expl. Geophys., pp. 1786-1789.
11. Dennis, Jr., J. E., and Schnabel, R. B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs.
12. Engquist, B., and Runborg, O. (1996). Multi-phase computations in geometrical optics. J. Comput. Appl. Math. 74, 175-192.
13. Engquist, B., Runborg, O., and Tornberg, A.-K. (2002). High frequency wave propagation by the segment projection method. J. Comp. Phys. 178, 373-390.
14. Fomel, S., and Sethian, J. (2002). Fast phase space computation of multiple traveltimes. Proc. Nat. Aca. Sci. 99, 7329-7334.
15. Geoltrain, S., and Brac, J. (1993). Can we image complex structures with first-arrival traveltime? Geophysics 58, 564-575.
16. Gray, S., and May, W. (1994). Kirchhoff migration using eikonal equation traveltimes. Geophysics 59, 810-817.
17. Guillemin, V., and Sternberg, S. (1979). Geometric Asymptotics, American Mathematical Society, Providence.
18. Hu, C., and Shu, C. W. (2000). A discontinous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 666-690.
19. Jiang, G. S., and Peng, D. (2000). Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126-2143.
20. Kim, S., and Cook, R. (1999). 3-D traveltime computation using second-order ENO scheme. Geophysics 64, 1867-1876.
21. Kossioris, G., Makridakis, Ch., and Souganidis, P. E. (1999). Finite volume schemes for Hamilton-Jacobi equations. Numer. Math. 83, 427-442.
22. Kurganov, A., and Tadmor, E. (2000). New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations. J. Comput. Phys. 160, 720-742.
23. Lin, C. T., and Tadmor, E. (2000). High-resolution nonoscillatory central schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2163-2186.
24. Lions, P. L. (1982). Generalized Solutions of Hamilton-Jacobi Equations, Pitman Advanced Publishing Program.
25. Liu, Z., and Bleistein, N. (1995). Migration velocity analysis: Theory and an iterative algorithm. Geophysics 60, 142-153.
26. Milnor, J. (1973)., Morse Theory, Annals of Math., No. 51, Princeton University Press.
27. Operto, S., Xu, S., and Lambare, G. (2000). Can we image quantitatively complex models with rays. Geophysics 65, 1223-1238.
28. Osher, S., Cheng, L.-T., Kang, M., Shim, H., and Tsai, Y.-H. (2002). Geometrical optics in a phase space based level set and Eulerian framework. J. Comput. Phys. 179, 622-648.
29. Osher, S. I., and Sethian, J. A. (1988). Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12-49.
30. Osher, S. J., and Shu, C. W. (1991). High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM J. Num. Anal. 28, 907-922.
31. Podvin, P., and Lecomte, I. (1991). Finite difference computation of travellings in very contrasted velocity models: A massively parallel approach and its associated tools. Geophys. J. Int. 105, 271-284.
32. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterlina, W. T. (1986). Numerical Recipes, Cambridge University Press, New York.
33. Qian, J., Belfi, C. D., and Symes, W. W. (1999). Adaptive Finite Difference Method for Travel-time and Amplitude, 69th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Soc. Expl. Geophys., pp. 1763-1766.
34. Qian, J., Cheng, L.-T., and Osher, S. J. (2001). A level set based Eulerian approach for anisotropic wave propagations. Submitted to Wave Motion.
35. Qian, J., and Symes, W. W. (2001). Paraxial eikonal solvers for anisotropic quasi-P traveltimes. J. Comput. Phys. 173, 1-23.
36. Qian, J., and Symes, W. W. (2002). Adaptive finite difference method for traveltime and amplitude. Geophysics 67, 167-176.
37. Qian, J., and Symes, W. W. (2002). Finite-difference quasi-P traveltimes for anisotropic media. Geophysics 67, 147-155.
38. Qian, J., Symes, W. W., and Dellinger, J. A. (2001). A Full-Aperture Anisotropic Eikonal Solver for Quasi-P Traveltimes, 71st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Soc. Expl. Geophys., pp. 129-132.
39. Rouy, E., and Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM J. Num. Anal. 29, 867-884.
40. Ruuth, S. J., Merriman, B., and Osher, S. J. (1999). A fixed grid method for capturing the motion of self-intersecting interfaces and related PDEs. J. Comput. Phys. 151, 836-861.
41. Schneider, Jr., W. A. (1995). Robust and efficient upwind finite-difference traveltime calculations in three dimensions. Geophysics 60, 1108-1117.
42. Schneider, Jr., W. A., Ranzinger, K. A., Balch, A. H., and Kruse, C. (1992). A dynamic programming approach to first arrival traveltime computation in media with arbitrarily distributed velocities. Geophysics 57, 39-50.
43. Sethian, J. A., and Popovici, A. M. (1999). 3-D traveltime ccmputation using the fast marching method. Geophysics 64, 516-523.
44. Steinhoff, J., Fan, M., and Wang, L. (2000). A new Eulerism method for the computation of propagating short acoustic and electromagnetic pulses. J. Comput. Phys. 157, 683-706.
45. Stoer, J., and Bulirsch, R. (1992). Introduction to Numerical Analysis, 2nd edn., Springer-Verlag, New York.
46. Symes, W. W. (1998). A Slowness Matching Finite Difference Method for Traveltimes Beyond Transmission Caustics, 68th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Soc. Expl. Geophys., pp. 1945-1948.
47. van Trier, J., and Symes, W. W. (1991). Upwind finite-difference calculation of travel-times. Geophysics 56(6), 812-821.
48. Vidale, J. (1988). Finite-difference calculation of travel times. Bull., Seis. Soc. Am. 78, 2062-2076.
49. Vinje, V., Iversen, E., and Gjoystal, H. (1993). Traveltime and amplitude estimation using wavefront construction. Geophysics 58, 1157-1166.
50. White, B. S. (1984). The stochastic caustic. SIAM J. Appl. Math. 44, 127-149.