Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 409–464 | Cite as

Scattering Control for the Wave Equation with Unknown Wave Speed

  • Peter Caday
  • Maarten V. de Hoop
  • Vitaly Katsnelson
  • Gunther UhlmannEmail author


Consider the acoustic wave equation with unknown wave speed c, not necessarily smooth. We propose and study an iterative control procedure that erases the history of a wave field up to a given depth in a medium, without any knowledge of c. In the context of seismic or ultrasound imaging, this can be viewed as removing multiple reflections from normal-directed wavefronts.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



P.C. and V.K. were supported by the Simons Foundation under the MATH + X program. M.V.dH. was partially supported by the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-1559587, and by the members of the Geo-Mathematical Group at Rice University. G.U. is Walker Family Endowed Professor of Mathematics at the University of Washington, and was partially supported by the National Science Foundation, a Si-Yuan Professorship at Hong Kong University of Science and Technology, and a FiDiPro Professorship at the Academy of Finland.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest regarding this work.


  1. 1.
    Aktosun, T., Rose, J.H.: Wave focusing on the line. J. Math. Phys. 43(7), 3717–3745 (2002). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Belishev, M.I.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Probl. 13(5), R1–R45 (1997). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bingham, K., Kurylev, Y., Lassas, M., Siltanen, S.: Iterative time-reversal control for inverse problems. Inverse Probl. Imaging 2(1), 63–81 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burridge, R.: The Gel' fand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems. Wave Motion 2(4), 305–323 (1980). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caday, P.: Computing Fourier integral operators with caustics. Inverse Probl. 32(12), 125001 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chazarain, J.: Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques. In: Journées Équations aux Dérivées Partielles de Rennes (1975), pp. 165–181. Astérisque, No. 34–35. Society Mathematical France, Paris (1976)Google Scholar
  7. 7.
    Cisternas, A., Betancourt, O., Leiva, A.: Body waves in a "real Earth.". Part I. Bull. Seismol. Soc. Am. 63(1), 145–156 (1973)Google Scholar
  8. 8.
    Hansen, S.: Singularities of transmission problems. Math. Ann. 268(2), 233–253 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    van der Heijden, J.: Propagation of transient elastic waves in stratified anisotropic media. Ph.D. thesis, Technische Universiteit Delft (1987)Google Scholar
  10. 10.
    de Hoop, M.V., Kepley, P., Oksanen, L.: On the construction of virtual interior point source travel time distances from the hyperbolic Neumann-to-Dirichlet map. SIAM J. Appl. Math. 76(2), 805–825 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de Hoop, M.V., Uhlmann, G., Vasy, A.: Diffraction from conormal singularities. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 351–408 (2015)Google Scholar
  12. 12.
    Kirpichnikova, A., Kurylev, Y.: Inverse boundary spectral problem for Riemannian polyhedra. Math. Ann. 354(3), 1003–1028 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lion, G., Vergne, M.: The Weil representation, Maslov Index and Theta Series, Progress in Mathematics, vol. 6. Birkhäuser, Boston (1980)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972). Translated fromthe French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181Google Scholar
  15. 15.
    Rose, J.H.: `Single-sided' autofocusing of sound in layered materials. Inverse Probl. 18(6), 1923–1934 (2002). Special section on electromagnetic and ultrasonic nondestructive evaluation
  16. 16.
    Safarov, Y.: A symbolic calculus for Fourier integral operators. In: Geometric and spectral analysis, Contemp. Math., vol. 630, pp. 275–290. American Mathematical Society, Providence, RI (2014).
  17. 17.
    Stefanov, P., Uhlmann, G.: Thermoacoustic tomography with variable sound speed. Inverse Probl. 25(7), 075011, 16 (2009).
  18. 18.
    Stefanov, P., Uhlmann, G.: Thermoacoustic tomography arising in brain imaging. Inverse Probl. 27(4), 045004, 26 (2011).
  19. 19.
    Stolk, C.C.: On the modeling and inversion of seismic data. Ph.D. thesis, University of Utrecht (2001)Google Scholar
  20. 20.
    Stolk, C.C.: A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous acoustic media. Wave Motion 40(2), 111–121 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stolk, C.C., de Hoop, M.V.: Microlocal analysis of seismic inverse scattering in anisotropic elastic media. Comm. Pure Appl. Math. 55(3), 261–301 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tataru, D.: Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem. Commun. Part. Differ. Equ. 20(5–6), 855–884 (1995). zbMATHGoogle Scholar
  23. 23.
    Taylor, M.E.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28(4), 457–478 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wapenaar, K., Thorbecke, J., van der Neut, J., Broggini, F., Slob, E., Snieder, R.: Marchenko imaging. Geophysics 79(3), WA39–WA57 (2014).

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Peter Caday
    • 1
  • Maarten V. de Hoop
    • 1
    • 2
  • Vitaly Katsnelson
    • 1
  • Gunther Uhlmann
    • 3
    • 4
    Email author
  1. 1.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  2. 2.Department of Earth, Environmental, and Planetary SciencesRice UniversityHoustonUSA
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA
  4. 4.Institute for Advanced StudyHong Kong University of Science and TechnologyClear Water BayHong Kong

Personalised recommendations