Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Agranovsky, C. Berenstein, and P. Kuchment, Approximation by spherical waves in Lp spaces, J. Geom. Analysis 6(1996), 365–383.
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.
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.
L-E. Andersson, On the determination of a function from spherical averages, SIAM J. Math. Anal. 19(1988), 214–232.
R. Burridge and G. Beylkin, On double integrals over spheres, Inverse Problems 4(1988), 1–10.
J. Boman and E.T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J. 55(1987), 943–948.
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.
R. Courant and D. Hilbert, Methods of Mathematical Physics, II, Wiley-Interscience, New York 1962.
A. Denisjuk, Integral Geometry on the family of semispheres, Fractional Calculus and Applied Analysis, 2(1999), 31–46.
J. A. Fawcett, inversion of N—dimensional Spherical Averages, SIAM J. Appl. Math 42(1985), 336–341.
V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, RI 1977.
S. Helgason, A duality in integral geometry, some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70(1964), 435–446.
L. Hörmander, The analysis of linear partial differential operators I, Springer-Verlag, 1983.
F.B. Jensen, W.A. Kuperman, M.B. Porter, H. Schmidt, Computational Ocean Acoustics, AIP Press, New York.
F. John, Plane waves and spherical means, Interscience, 1955.
M. Lavrent’ev, V. Romanov, and V. Vasiliev, Multidimensional Inverse Problems for Differential Equations, Lecture Notes in Mathematics 167, Springer Verlag, 1970.
S.J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact Solution, J. Acoust. Soc. Am. 64(1980), 1266–1273.
V. Palamodov, Reconstruction from limited data of arc means, preprint, 1998.
B. Petersen, Introduction to the Fourier Transform and Pseudo-Differential Operators, Pittman Boston, 1983.
E.T. Quinto, Pompeiu transforms on geodesic spheres in real analytic manifolds, Israel J. Math. 84(1993), 353–363.
V.G. Romanov, Integral Geometry and inverse Problems for Hyperbolic Equations, Springer Tracts in Natural Philosophy, 26, 1974.
R. Schneider, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26(1969), 381–384.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag/Wien
About this chapter
Cite this chapter
Louis, A.K., Quinto, E.T. (2000). Local Tomographic Methods in Sonar. In: Colton, D., Engl, H.W., Louis, A.K., McLaughlin, J.R., Rundell, W. (eds) Surveys on Solution Methods for Inverse Problems. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6296-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6296-5_8
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83470-1
Online ISBN: 978-3-7091-6296-5
eBook Packages: Springer Book Archive