Local Tomographic Methods in Sonar

  • Alfred K. Louis
  • Eric Todd Quinto


Tomographic methods are described that will reconstruct object boundaries in shallow water using sonar data. The basic ideas involve microlocal analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model.


Inversion Method Integral Geometry Fourier Integral Operator Sonar Data Hyperplane Tangent 
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  1. 1.
    M. Agranovsky, C. Berenstein, and P. Kuchment, Approximation by spherical waves in Lp spaces, J. Geom. Analysis 6(1996), 365–383.MATHMathSciNetGoogle Scholar
  2. 2.
    M.L. Agranovsky and E.T. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Functional Anal., 139(1996), 383–414.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M.L. Agranovsky and E.T. Quinto, Geometry of Stationary Sets for the Wave Equation in ℝn. The Case of Finitely Supported Initial Data, preprint, 1999.Google Scholar
  4. 4.
    L-E. Andersson, On the determination of a function from spherical averages, SIAM J. Math. Anal. 19(1988), 214–232.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Burridge and G. Beylkin, On double integrals over spheres, Inverse Problems 4(1988), 1–10.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Boman and E.T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J. 55(1987), 943–948.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Cheney and J. Rose, Three-dimensional inverse scattering for the wave equation: weak scattering approximations with error estimates, Inverse Problems 4(1988) 435–447.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Courant and D. Hilbert, Methods of Mathematical Physics, II, Wiley-Interscience, New York 1962.Google Scholar
  9. 9.
    A. Denisjuk, Integral Geometry on the family of semispheres, Fractional Calculus and Applied Analysis, 2(1999), 31–46.MATHMathSciNetGoogle Scholar
  10. 10.
    J. A. Fawcett, inversion of N—dimensional Spherical Averages, SIAM J. Appl. Math 42(1985), 336–341.CrossRefMathSciNetGoogle Scholar
  11. 11.
    V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, RI 1977.Google Scholar
  12. 12.
    S. Helgason, A duality in integral geometry, some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70(1964), 435–446.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. Hörmander, The analysis of linear partial differential operators I, Springer-Verlag, 1983.Google Scholar
  14. 14.
    F.B. Jensen, W.A. Kuperman, M.B. Porter, H. Schmidt, Computational Ocean Acoustics, AIP Press, New York.Google Scholar
  15. 15.
    F. John, Plane waves and spherical means, Interscience, 1955.Google Scholar
  16. 16.
    M. Lavrent’ev, V. Romanov, and V. Vasiliev, Multidimensional Inverse Problems for Differential Equations, Lecture Notes in Mathematics 167, Springer Verlag, 1970.Google Scholar
  17. 17.
    S.J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact Solution, J. Acoust. Soc. Am. 64(1980), 1266–1273.CrossRefMathSciNetGoogle Scholar
  18. 18.
    V. Palamodov, Reconstruction from limited data of arc means, preprint, 1998.Google Scholar
  19. 19.
    B. Petersen, Introduction to the Fourier Transform and Pseudo-Differential Operators, Pittman Boston, 1983.Google Scholar
  20. 20.
    E.T. Quinto, Pompeiu transforms on geodesic spheres in real analytic manifolds, Israel J. Math. 84(1993), 353–363.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    V.G. Romanov, Integral Geometry and inverse Problems for Hyperbolic Equations, Springer Tracts in Natural Philosophy, 26, 1974.Google Scholar
  22. 22.
    R. Schneider, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26(1969), 381–384.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2000

Authors and Affiliations

  • Alfred K. Louis
    • 1
  • Eric Todd Quinto
    • 2
  1. 1.Fachbereich MathematikUniversität des SaarlandesSaarbrückenGermany
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

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