Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aktosun, T., Rose, J.H.: Wave focusing on the line. J. Math. Phys. 43(7), 3717–3745 (2002). https://doi.org/10.1063/1.1483894
Belishev, M.I.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Probl. 13(5), R1–R45 (1997). https://doi.org/10.1088/0266-5611/13/5/002
Bingham, K., Kurylev, Y., Lassas, M., Siltanen, S.: Iterative time-reversal control for inverse problems. Inverse Probl. Imaging 2(1), 63–81 (2008). https://doi.org/10.3934/ipi.2008.2.63
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). https://doi.org/10.1016/0165-2125(80)90011-6
Caday, P.: Computing Fourier integral operators with caustics. Inverse Probl. 32(12), 125001 (2016)
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)
Cisternas, A., Betancourt, O., Leiva, A.: Body waves in a "real Earth.". Part I. Bull. Seismol. Soc. Am. 63(1), 145–156 (1973)
Hansen, S.: Singularities of transmission problems. Math. Ann. 268(2), 233–253 (1984). https://doi.org/10.1007/BF01456088
van der Heijden, J.: Propagation of transient elastic waves in stratified anisotropic media. Ph.D. thesis, Technische Universiteit Delft (1987)
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). https://doi.org/10.1137/15M1033010
de Hoop, M.V., Uhlmann, G., Vasy, A.: Diffraction from conormal singularities. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 351–408 (2015)
Kirpichnikova, A., Kurylev, Y.: Inverse boundary spectral problem for Riemannian polyhedra. Math. Ann. 354(3), 1003–1028 (2012). https://doi.org/10.1007/s00208-011-0758-9
Lion, G., Vergne, M.: The Weil representation, Maslov Index and Theta Series, Progress in Mathematics, vol. 6. Birkhäuser, Boston (1980)
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 181
Rose, J.H.: `Single-sided' autofocusing of sound in layered materials. Inverse Probl. 18(6), 1923–1934 (2002). https://doi.org/10.1088/0266-5611/18/6/329. Special section on electromagnetic and ultrasonic nondestructive evaluation
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). https://doi.org/10.1090/conm/630/12670
Stefanov, P., Uhlmann, G.: Thermoacoustic tomography with variable sound speed. Inverse Probl. 25(7), 075011, 16 (2009). https://doi.org/10.1088/0266-5611/25/7/075011
Stefanov, P., Uhlmann, G.: Thermoacoustic tomography arising in brain imaging. Inverse Probl. 27(4), 045004, 26 (2011). https://doi.org/10.1088/0266-5611/27/4/045004
Stolk, C.C.: On the modeling and inversion of seismic data. Ph.D. thesis, University of Utrecht (2001)
Stolk, C.C.: A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous acoustic media. Wave Motion 40(2), 111–121 (2004). https://doi.org/10.1016/j.wavemoti.2003.12.016
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). https://doi.org/10.1002/cpa.10019
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). https://doi.org/10.1080/03605309508821117
Taylor, M.E.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28(4), 457–478 (1975)
Wapenaar, K., Thorbecke, J., van der Neut, J., Broggini, F., Slob, E., Snieder, R.: Marchenko imaging. Geophysics 79(3), WA39–WA57 (2014). https://doi.org/10.1190/geo2013-0302.1
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest regarding this work.
Additional information
Communicated by T.-P. Liu
Rights and permissions
About this article
Cite this article
Caday, P., de Hoop, M.V., Katsnelson, V. et al. Scattering Control for the Wave Equation with Unknown Wave Speed. Arch Rational Mech Anal 231, 409–464 (2019). https://doi.org/10.1007/s00205-018-1283-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1283-8