Advertisement

Journal of Mathematical Sciences

, Volume 86, Issue 3, pp 2773–2786 | Cite as

Diffraction tomography: Construction and interpretation of tomographic functionals

  • V. N. Troyan
  • G. A. Ryzhikov
Article
  • 18 Downloads

Abstract

On the basis of a linearized model of propagation of seismic wave fields the notion of tomographic functionals is introduced. The physical interpretation of tomographic functionals is that their integral kernels are spatial functions of the influence of variations of the medium parameters in question upon particular measurements of the wave field of the sounding signal. The norm of a tomographic functional is determined by the intensity of the influence function related to the interaction operator. The field is generated by a “source” with the dependence on time determined by the apparatus function of the seismic channel. The analysis of tomographic functionals makes it possible to mathematically design tomographic experiments for monitoring active seismic zones by controlling the parameters of tomographic functionals. The richness in content of a tomographic experiments is determined not only by the norms of tomographic functionals, but also by the region where their supports overlap. Tomographic functionals for the wave and Lamé equations are analyzed. Bibliography:14 titles.

Keywords

Seismic Wave Physical Interpretation Wave Field Seismic Zone Spatial Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    M. I. Belishev, “The Gel'fand-Levitan equations in the multidimensional problem for the wave equation,”Zap. Nauchn. Semin. LOMI,156, 15–20 (1987).Google Scholar
  2. 2.
    A. S. Blagoveshenskii, “On a local method for solving a nonstationary inverse problem for an inhomogeneous medium,”Tr. Mat. Inst. Akad. Nauk SSSR,115, 28–38 (1971).Google Scholar
  3. 3.
    A. S. Blagoveshenskii, “Inverse problems of the theory of propagation of elastic waves,”Izv. Akad. Nauk SSSR, Ser. Fiz. Zemli, No. 12, 16–29 (1978).Google Scholar
  4. 4.
    A. S. Blagoveshenskii, “On an inverse problem of the theory of propagation of seismic waves,” in:Problems of Mathematical Physics [in Russian], Leningrad (1966), pp. 18–31.Google Scholar
  5. 5.
    V. G. Romanov,Inverse Problems of Mathematical Physics [in Russian], Moscow (1984).Google Scholar
  6. 6.
    V. G. Romanov,Some Inverse Problems for the Hyperbolic-Type Equations [in Russian], Novosibirsk, (1972).Google Scholar
  7. 7.
    L. D. Faddeev, “Inverse problems in the quantum theory of scattering,” in:Modern Problems of Mathematics (Itogi Nauki Techniki) [in Russian], Vol. 3, Moscow, (1974), pp. 93–181.Google Scholar
  8. 8.
    A. L. Buchgeim,Introduction to the Theory of Inverse Problems [in Russian], Novosibirsk (1988).Google Scholar
  9. 9.
    A. N. Tichnov and V. Ya. Arsenin,Methods of Solving Incorrect Problems [in Russian], Moscow (1986).Google Scholar
  10. 10.
    V. N. Troyan and Yu. M. Sokolov,Methods for Approximating Geophysical Data on Computers [in Russian] Leningrad (1989).Google Scholar
  11. 11.
    G. A. Ryzhikov and V. N. Troyan,Tomography and Inverse Problems of Remote Sensing [in Russian], St. Petersburg (1994).Google Scholar
  12. 12.
    G. A. Ryzhikov and V. N. Troyan, “Tomographic functionals in interpretation problems of the elastic waves sounding,”Vopr. Dinam. Teor. Raspr. Seismich. Voln,28, 87–90 (1989).Google Scholar
  13. 13.
    G. A. Ryzhikov and V. N. Troyan, “Diffraction tomography and backprojection,”Proc. of the 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, Vieweg, 47–52 (1991).Google Scholar
  14. 14.
    G. A. Ryzhikov and V. N. Troyan, “On regularization methods in 3-D tomography,”Proc. of the 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, Vieweg, 53–61 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • V. N. Troyan
  • G. A. Ryzhikov

There are no affiliations available

Personalised recommendations