Wave Phenomena

  • Matti Lassas
  • Mikko Salo
  • Gunther Uhlmann
Reference work entry


This chapter discusses imaging methods related to wave phenomena, and in particular, inverse problems for the wave equation will be considered. The first part of the chapter explains the boundary control method for determining a wave speed of a medium from the response operator, which models boundary measurements. The second part discusses the scattering relation and travel times, which are different types of boundary data contained in the response operator. The third part gives a brief introduction to curvelets in wave imaging for media with nonsmooth wave speeds. The focus will be on theoretical results and methods.


Wave Equation Riemannian Manifold Wave Speed Sound Speed Gaussian Beam 
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References and Further Reading

  1. 1.
    Anderson M, Katsuda A, Kurylev Y, Lassas M, Taylor M (2004) Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem. Invent Math 158: 261–321MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Andersson F, de Hoop MV, Smith HF, Uhlmann G (2008) A multi-scale approach to hyperbolic evolution equations with limited smoothness. Commun Part Diff Equat 33(4–6):988–1017MATHCrossRefGoogle Scholar
  3. 3.
    Babich VM, Ulin VV (1981) The complex space-time ray method and “quasiphotons” (Russian). Zap Nauchn Sem LOMI 117:5–12MathSciNetMATHGoogle Scholar
  4. 4.
    Belishev M (1987) An approach to multidimensional inverse problems for the wave equation (Russian). Dokl Akad Nauk SSSR 297(3): 524–527MathSciNetGoogle Scholar
  5. 5.
    Belishev M (1997) Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Probl 13:R1–R45MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Belishev M, Kachalov A (1992) Boundary control and quasiphotons in a problem of the reconstruction of a Riemannian manifold from dynamic data (Russian). Zap Nauchn Sem POMI 203: 21–50MATHGoogle Scholar
  7. 7.
    Belishev M, Kurylev Y (1992) To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Commun Part Diff Equat 17: 767–804MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bernstein IN, Gerver ML (1980) Conditions on distinguishability of metrics by hodographs, methods and algorithms of interpretation of seismological information. Computerized seismology, vol 13. Nauka, Moscow, pp 50–73 (in Russian)Google Scholar
  9. 9.
    Besson G, Courtois G, Gallot S (1995) Entropies et rigidités des espaces localement symétriques de courbure strictment négative. Geom Funct Anal 5:731–799MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Beylkin G (1983) Stability and uniqueness of the solution of the inverse kinematic problem in the multidimensional case. J Soviet Math 21: 251–254CrossRefGoogle Scholar
  11. 11.
    Bingham K, Kurylev Y, Lassas M, Siltanen S (2008) Iterative time reversal control for inverse problems. Inverse Probl Imaging 2:63–81MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Blagoveščenskii A (1969) A one-dimensional inverse boundary value problem for a second order hyperbolic equation (Russian). Zap Nauchn Sem LOMI 15:85–90Google Scholar
  13. 13.
    Blagoveščenskii A (1971) Inverse boundary problem for the wave propagation in an anisotropic medium (Russian). Trudy Mat Inst Steklova 65:39–56Google Scholar
  14. 14.
    Brytik V, de Hoop MV, Salo M (2010) Sensitivity analysis of wave-equation tomography: a multi-scale approach. J Fourier Anal Appl 16(4):544–589MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Burago D, Ivanov S (2010) Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann Math 171(2):1183–1211MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Burago D, Ivanov S Area minimizers and boundary rigidity of almost hyperbolic metrics (in preparation)Google Scholar
  17. 17.
    Candès EJ, Demanet L (2003) Curvelets and Fourier integral operators. C R Math Acad Sci Paris 336:395–398MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Candès EJ, Demanet L (2005) The curvelet representation of wave propagators is optimally sparse. Comm Pure Appl Math 58:1472–1528MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Candès EJ, Donoho DL (2000) Curvelets - a surprisingly effective nonadaptive representation for objects with edges. In: Schumaker LL et al (eds) Curves and surfaces. Vanderbilt University Press, Nashville, pp 105–120Google Scholar
  20. 20.
    Candès EJ, Donoho DL (2004) New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities. Commun Pure Appl Math 57:219–266MATHCrossRefGoogle Scholar
  21. 21.
    Candès EJ, Demanet L, Ying L (2007) Fast computation of Fourier integral operators. SIAM J Sci Comput 29:2464–2493MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Chavel I (2006) Riemannian geometry. A modern introduction. Cambridge University Press, Cambridge, xvi+471 ppMATHCrossRefGoogle Scholar
  23. 23.
    Córdoba A, Fefferman C (1978) Wave packets and Fourier integral operators. Commun Part Diff Equat 3:979–1005MATHCrossRefGoogle Scholar
  24. 24.
    Creager KC (1992) Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK. Nature 356:309–314CrossRefGoogle Scholar
  25. 25.
    Croke C (1990) Rigidity for surfaces of non-positive curvature. Comment Math Helv 65: 150–169MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Croke C (1991) Rigidity and the distance between boundary points. J Diff Geom 33(2):445–464MathSciNetMATHGoogle Scholar
  27. 27.
    Dahl M, Kirpichnikova A, Lassas M (2009) Focusing waves in unknown media by modified time reversal iteration. SIAM J Control Optim 48:839–858MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    de Hoop MV (2003) Microlocal analysis of seismic inverse scattering: inside out. In: Uhlmann G (ed) Inverse problems and applications. Cambridge University Press, Cambridge, pp 219–296Google Scholar
  29. 29.
    de Hoop MV, Smith H, Uhlmann G, van der Hilst RD (2009) Seismic imaging with the generalized Radon transform: a curvelet transform perspective. Inverse Probl 25(2):25005–25025CrossRefGoogle Scholar
  30. 30.
    Demanet L, Ying L (2009) Wave atoms and time upscaling of wave equations. Numer Math 113(1):1–71MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Duchkov AA, Andersson F, de Hoop MV (2010) Discrete, almost symmetric wave packets and multiscale geometric representation of (seismic) waves. IEEE Trans Geosc Remote Sens 48(9):3408–3423CrossRefGoogle Scholar
  32. 32.
    Duistermaat JJ (2009) Fourier integral operators, Birkhäuser, BostonGoogle Scholar
  33. 33.
    Greenleaf A, Kurylev Y, Lassas M, Uhlmann G (2009) Invisibility and inverse problems. Bull Amer Math 46:55–97MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Gromov M (1983) Filling Riemannian manifolds. J Diff Geom 18(1):1–148MathSciNetMATHGoogle Scholar
  35. 35.
    Guillemin V (1976) Sojourn times and asymptotic properties of the scattering Matrix. Proceedings of the Oji seminar on algebraic analysis and the RIMS symposium on algebraic analysis (Kyoto University, Kyoto, 1976). Publ Res Inst Math Sci 12(1976/77, Suppl):69–88Google Scholar
  36. 36.
    Hansen S, Uhlmann G (2003) Propagation of polarization for the equations in elastodynamics with residual stress and travel times. Math Annalen 326:536–587MathSciNetGoogle Scholar
  37. 37.
    Herglotz G (1905) Uber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte. Zeitschr fur Math Phys 52:275–299Google Scholar
  38. 38.
    Hörmander L (1985) The analysis of linear partial differential operators III. Pseudodifferential operators. Springer, Berlin, viii+525 ppGoogle Scholar
  39. 39.
    Isozaki H, Kurylev Y, Lassas M (2010) Forward and Inverse scattering on manifolds with asymptotically cylindrical ends. J Funct Anal 258: 2060–2118MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Ivanov S Volume comparison via boundary distances, arXiv:1004–2505Google Scholar
  41. 41.
    Katchalov A, Kurylev Y (1998) Multidimensional inverse problem with incomplete boundary spectral data. Commun Part Diff Equat 23:55–95MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Katchalov A, Kurylev Y, Lassas M (2001) Inverse boundary spectral problems. Chapman & Hall/CRC Press, Boca Raton, xx+290 ppMATHCrossRefGoogle Scholar
  43. 43.
    Katchalov A, Kurylev Y, Lassas M (2004) Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds. Geometric methods in inverse problems and PDE control. In: Croke C, Lasiecka I, Uhlmann G, Vogelius M (eds) IMA volumes in mathematics and applications, vol 137. Springer, New York, pp 183–213Google Scholar
  44. 44.
    Katchalov A, Kurylev Y, Lassas M, Mandache N (2004) Equivalence of time-domain inverse problems and boundary spectral problem. Inverse Probl 20:419–436MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Katsuda A, Kurylev Y, Lassas M (2007) Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Probl Imaging 1:135–157MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Krein MG (1951) Determination of the density of an inhomogeneous string from its spectrum (in Russian). Dokl Akad Nauk SSSR 76(3):345–348MathSciNetGoogle Scholar
  47. 47.
    Kurylev Y (1997) Multidimensional Gel’fand inverse problem and boundary distance map. In: Soga H (ed) Inverse problems related to geometry. Ibaraki University Press, Japan, pp 1–15Google Scholar
  48. 48.
    Kurylev Y, Lassas M (2000) Hyperbolic inverse problem with data on a part of the boundary. Differential equations and mathematical physics (Birmingham, AL, 1999). AMS/IP Stud Adv Math 16:259–272, AMSMathSciNetGoogle Scholar
  49. 49.
    Kurylev Y, Lassas M (2002) Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map. Proc Roy Soc Edinburgh Sect A 132: 931–949MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Kurylev Y, Lassas M (2009) Inverse problems and index formulae for Dirac Operators. Adv Math 221:170–216MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Kurylev Y, Lassas M, Somersalo E (2006) Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems. J Math Pures Appl 86:237–270MathSciNetMATHGoogle Scholar
  52. 52.
    Lasiecka I, Triggiani R (1991) Regularity theory of hyperbolic equations with Nonhomogeneous Neumann boundary conditions. II. General boundary data. J Diff Equat 94:112–164MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Lassas M, Uhlmann G (2001) On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann Sci Ecole Normale Superiéure 34:771–787MathSciNetMATHGoogle Scholar
  54. 54.
    Lassas M, Sharafutdinov V, Uhlmann G (2003) Semiglobal boundary rigidity for Riemannian metrics. Math Annalen 325:767–793MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Michel R (1981) Sur la rigidité imposée par la longueur des géodésiques. Invent Math 65: 71–83MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Mukhometov RG (1977) The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian). Dokl Akad Nauk SSSR 232(1):32–35MathSciNetGoogle Scholar
  57. 57.
    Mukhometov RG (1982) A problem of reconstructing a Riemannian metric. Siberian Math J 22:420–433CrossRefGoogle Scholar
  58. 58.
    Mukhometov RG, Romanov VG (1978) On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian). Dokl Akad Nauk SSSR 243(1):41–44MathSciNetGoogle Scholar
  59. 59.
    Otal JP (1990) Sur les longuer des géodésiques d’une métrique a courbure négative dans le disque. Comment Math Helv 65:334–347MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Pestov L, Uhlmann G (2005) Two dimensional simple compact manifolds with boundary are boundary rigid. Ann Math 161:1089–1106MathSciNetCrossRefGoogle Scholar
  61. 61.
    Rachele L (2000) An inverse problem in elastodynamics: determination of the wave speeds in the interior. J Diff Equat 162:300–325MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Rachele L (2003) Uniqueness of the density in an inverse problem for isotropic Elastodynamics. Trans Amer Math Soc 355(12):4781–4806MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Ralston J (1982) Gaussian beams and propagation of singularities. Studies in partial differential equations. MAA Studies in Mathematics, vol 23. Mathematical Association of America, Washington, pp 206–248Google Scholar
  64. 64.
    Salo M (2007) Stability for solutions of wave equations with C1,1 coefficients. Inverse Probl Imaging 1(3):537–556MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Seeger A, Sogge CD, Stein EM (1991) Regularity properties of Fourier integral operators. Ann Math 134:231–251MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Sharafutdinov V (1994) Integral geometry of tensor fields. VSP, Utrech, The NetherlandsCrossRefGoogle Scholar
  67. 67.
    Smith HF (1998) A parametrix construction for wave equations with C1, 1 coefficients. Ann Inst Fourier Grenoble 48(3):797–835MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Smith HF (2006) Spectral cluster estimates for C1, 1 metrics. Amer J Math 128(5):1069–1103MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Smith HF, Sogge CD (2007) On the Lp norm of spectral clusters for compact manifolds with boundary. Acta Math 198:107–153MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Stefanov P, Uhlmann G (1998) Rigidity for metrics with the same lengths of geodesics. Math Res Lett 5:83–96MathSciNetMATHGoogle Scholar
  71. 71.
    Stefanov P, Uhlmann G (2005) Boundary rigidity and stability for generic simple metrics. J Amer Math Soc 18:975–1003MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Stefanov P, Uhlmann G (2009) Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds. J Diff Geom 82: 383–409MathSciNetMATHGoogle Scholar
  73. 73.
    Stein EM (1993) Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton mathematical series, 43. Monographs in harmonic analysis, III. Princeton University Press, PrincetonGoogle Scholar
  74. 74.
    Sylvester J (1990) An anisotropic inverse boundary value problem. Comm Pure Appl Math 43(2):201–232MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Sylvester J, Uhlmann G (1987) A global uniqueness theorem for an inverse boundary value problem. Ann Math 125:153–169MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Sylvester J, Uhlmann G (1991) Inverse problems in anisotropic media, Contemp Math 122:105–117MathSciNetCrossRefGoogle Scholar
  77. 77.
    D Tataru: Unique continuation for solutions to PDEs, between Hörmander’s theorem and Holmgren’s theorem. Commun Part Diff Equat 20: 855–884Google Scholar
  78. 78.
    Tataru D (1998) On the regularity of boundary traces for the wave equation. Ann Scuola Norm Sup Pisa CL Sci 26:185–206MathSciNetMATHGoogle Scholar
  79. 79.
    Tataru D (1999) Unique continuation for operators with partially analytic coefficients. J Math Pures Appl 78:505–521MathSciNetMATHGoogle Scholar
  80. 80.
    Tataru D (2000) Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Amer J Math 122(2) 349–376MathSciNetMATHGoogle Scholar
  81. 81.
    Tataru D (2001) Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II. Amer J Math 123(3):385–423MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Tataru D (2002) Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J Amer Math Soc 15:419–442MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Uhlmann G (1999) Developments in inverse problems since Calderón’s foundational paper. In: Christ M, Kenig C, Sadosky C (eds) Essays in harmonic analysis and partial differential equations,  Chap. 19. University of Chicago Press, ChicagoGoogle Scholar
  84. 84.
    Wiechert E, Zoeppritz K (2007) Uber Erdbebenwellen. Nachr Koenigl Geselschaft Wiss Goettingen 4:415–549Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Matti Lassas
    • 1
  • Mikko Salo
    • 1
  • Gunther Uhlmann
    • 2
  1. 1.University of HelsinkiHelsinkiFinland
  2. 2.University ofWashingtonSeattle, WAUSA

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