Modern Techniques in Seismic Tomography

  • Alexander A. Boukhgueim
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 7)


This chapter is focussed on two local inverse kinematic problems of seismology, concerning reflected rays and refracted rays. Both model problems are reduced to a sequence of 2D problems, where theoretical and numerical results are offered. In the case of reflected rays, it is shown how to select a stable problem of recovering a velocity distribution in a layer, by using travel time measurements along rays with one reflection on the boundary. This way, a simple inversion algorithm is obtained for the linearized near a constant velocity case. In the case of refracted rays, a Newton-type algorithm for finding the 3D velocity distribution from 3D travel time measurements is constructed for the local inverse kinematic problem. To this end, a sound velocity that increases linearly with depth is chosen as a first approximation. With this particular choice for the linearization, the underlying problem reduces to a sequence of 2D Radon transforms in discs.


Travel Time Sound Velocity Inversion Formula Seismic Tomography Inversion Algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G.E. Backus (1964) Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths. Bull. Seismol. Soc. Am. 54, 571–610.Google Scholar
  2. 2.
    N.N. Bernshtein and M.L. Gerver (1980) Conditions for distinguishability of metrics by hodographs: methods and algorithms for seismic data interpretation. Computational Seismology, vol. 13, Moscow, Nauka, 50–73.Google Scholar
  3. 3.
    A.A. Boukhgueim (2003) Numerical Algorithms for Attenuated Tomography in Medicine and Industry. Ph.D. dissertation, University of Vienna.Google Scholar
  4. 4.
    A.L. Bukhgeim (1983) On one algorithm of solving the inverse kinematic problem of seismology. Numerical Methods in Seismic Investigations, Nauka, Novosibirsk (in Russian), 152–155.Google Scholar
  5. 5.
    A.L. Bukhgeim, S.M. Zerkal’, and V.V. Pikalov (1983) On one algorithm for solution of a 3D inverse kinematic problem of seismology. Methods for Solution of Inverse Problems. Novosibirsk, 38–47.Google Scholar
  6. 6.
    A.M. Dziewonski (1999) Earth’s mantle in three dimensions. Seismic Modelling of Earth Structure, E.G. Boschi, Ekström and A. Morelli (eds.), Editrice Compositori, Bologna, 507–572.Google Scholar
  7. 7.
    A.M. Dziewonski and D.L. Anderson (1981) Preliminary reference Earth model. Phys. Earth. Planet. Inter. 25, 297–356.CrossRefGoogle Scholar
  8. 8.
    A.M. Dziewonski and D.L. Anderson (1984) Seismic tomography of the Earth’s interior. American Scientist 72, 483–494.Google Scholar
  9. 9.
    A.M Dziewonski, T.A. Chou, and J.H. Woodhouse (1981) Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. J. Geophys. Res. 86, 2825–2852.Google Scholar
  10. 10.
    A.M. Dziewonski, B.H. Hager, and R.J. O’Connel (1977) Large-scale heterogeneities in the lower mantle. J. Geophys. Res. 82, 239–255.Google Scholar
  11. 11.
    A.M. Dziewonski and J.H. Woodhouse (1987) Global images of the Earth’s interior. Science 236, 38–47.Google Scholar
  12. 12.
    P. Funk (1916) Über eine geometrische Anwendung der Abelschen Integralgleichung. Math. Ann. 77, 129–135.MathSciNetCrossRefGoogle Scholar
  13. 13.
    G. Herglotz (1905) Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte. Z. für Math. Phys. 52(3), 275–299.MATHGoogle Scholar
  14. 14.
    H. Kanamori (1970) Velocity and Q of mantle waves. Phys. Earth. Planet. Int. 2, 259–275.CrossRefGoogle Scholar
  15. 15.
    M.M. Lavrentiev, V.G. Romanov, and V.G. Vasiliev (1970) Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics 167. Springer-Verlag.Google Scholar
  16. 16.
    R.G. Mukhometov (1977) A problem of reconstructing 2D Riemannian metric and integral geometry. Dokl. An. SSSR. vol. 296, no. 2, 279–283.MathSciNetGoogle Scholar
  17. 17.
    J. Radon (1917) Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sachs. Akad. Wiss. Leipzig. Math. Nat. Kl. 69, 262–277.MATHGoogle Scholar
  18. 18.
    B.A. Romanowicz (1991) Seismic tomography of the Earth’s mantle. Ann. Rev. Earth Planet. Sci. 19, 77–99.CrossRefGoogle Scholar
  19. 19.
    B.A. Romanowicz (1995) A global tomographic model of shear attenuation in the upper mantle. J. Geophys. Res. 100, 12375–12394.CrossRefGoogle Scholar
  20. 20.
    V.A. Sharafutdinov (1993) Integral Geometry of Tensor Fields. Novosibirsk, Nauka.Google Scholar
  21. 21.
    V.A. Udias (1999) Principles of Seismology. Cambridge University Press.Google Scholar
  22. 22.
    E. Wiechert and K. Zoeppritz (1907) Über Erdbebenwellen. Nachr. Königl. Gesellschaft Wiss. Göttingen 4, 415–549.Google Scholar
  23. 23.
    J.H. Woodhouse and A.M. Dziewonski (1984) Mapping the upper mantle: three-dimensional modeling of Earth structure by inversion of seismic waveforms. J. Geophys. Res. 89, 5953–5986.Google Scholar
  24. 24.
    R.L. Woodward and G. Masters (1991) Upper mantle structure from long-period differential travel times and free oscillation data. Geophys. J. Int. 109, 275–293.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alexander A. Boukhgueim
    • 1
  1. 1.Department of MathematicsUniversity of ViennaAustria

Personalised recommendations