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    Gilbert, R. P. Xu, Y. and Thejll, P. 1992. An Approximation Scheme for the Three-dimensional Scattered Wave and Its Propagating Far-field Pattern in a Finite Depth Ocean. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 72, Issue. 10, p. 459.
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An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

  • Yongzhi Xu (a1)
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The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

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Copyright
COPYRIGHT: © Edinburgh Mathematical Society 1991
References
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1.Ahluwalia, D. and Keller, J., Exact and asymptotic representations of the sound field in a stratified ocean, in Wave Propagation and Underwater Acoustics (Lecture Notes in Physics 70, Springer, Berlin 1977).
2.Angell, T. S., Colton, D. and Kirsch, A., The three dimensional inverse scattering problem for acoustic waves, J. Differential Equations 46 (1982), 46–58.
3.Angell, T. S., Kleiman, R. E. and Roach, G. F., An inverse transmission problem for the Helmholtz equation, Inverse Problems 3 (1987), 149–180.
4.Colton, D., The inverse scattering problem for time-harmonic acoustic waves, SIAM Rev. 26 (1984), 323–350.
5.Colton, D., and Monk, P., A novel method of solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math. 45 (1985), 1039–1053.
6.Colton, D. and Monk, P., A novel method of solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region: II, SIAM J. Appl. Math. 46 (1986), 506–523.
7.Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory (John Wiley, New York, 1983).
8.Gilbert, R. P. and Xu, Yongzhi, Starting fields and far fields in ocean acoustics, Wave Motion 11 (1989), 507–524.
9.Gilbert, R. G. and Xu, Yongzhi, Dense sets and the projection theorem for acoustic harmonic waves in homogeneous finite depth ocean, Math. Methods Appl. Sci. 12 (1989), 69–76.
10.Kirsch, A., and Kress, R., An optimization method in inverse acoustic scattering, in Boundary Elements, Vol. 3. Fluid Flow and Potential Applications (1988), to appear.
11.Millar, R. F., The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers, Radio Science 8 (1973), 785–796.
12.Sleeman, B. D., The inverse problem of acoustic scattering, IMA J. Appl. Math. 29 (1982), 13–142.
13.Van Den Berg, P. M. and Fokkema, J. T., The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle, IEEE Trans. Antennas and Propagation AP-27 (1979), 577–583.
14.Xu, Yongzhi, The propagating solution and far field patterns for acoustic harmonic waves in a finite depth ocean, Appl. Anal. 35 (1990), 129–151.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
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