Abstract
A two-dimensional inverse scattering problem in a layered acoustic medium occupying a half-plane is considered. Data is the scattered wavefield from a surface point source measured on the boundary of the half-plane. On the basis of the Radon transform, an algorithm is constructed that recovers the velocity and the acoustic impedance of the medium from the scattering data. An analytical solution is presented for an inverse scattering problem, and several inverse scattering problems are solved numerically.
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References
A. N. Tikhonov, “Uniqueness of the solution to an electrical exploration problem,” Dokl. Akad. Nauk SSSR 69 (6), 797–800 (1949).
A. N. Tikhonov, “Mathematical basis of the theory of electromagnetic soundings,” USSR Comput. Math. Math. Phys. 5 (3), 207–211 (1965).
A. N. Kolmogorov, “Sur l’interpolation et l’extrapolation des suites stationnaires,” C. R. Acad. Sci. Paris 208, 2043–2045 (1939).
J. F. Claerbout and A. G. Jonson, “Extrapolation of time dependent waveforms along their path of propagation,” Geophys. J. R. Astron. Soc. 26 (1–4), 285–295 (1971).
P. S. Schultz and J. F. Claerbout, “Velocity estimation and downward continuation by wave front synthesis,” Geophysics 43, 691–714 (1978).
J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976).
G. A. McMechan, “Migration by extrapolation of time-dependent boundary value,” Geophys. Prosp. 31, 413–420 (1983).
V. G. Romanov, Inverse Problems in Mathematical Physics (Nauka, Moscow, 1984) [in Russian].
M. M. Lavrent’ev, K. G. Reznitskaya, and V. G. Yakhno, One-Dimensional Inverse Problems in Mathematical Physics (Nauka, Novosibirsk, 1982) [in Russian].
E. N. Bessonova, V. M. Fishman, V. Z. Ryaboi, and G. A. Sitnikova, “The tau method for inversion of travel times: 1. Deep seismic sounding data,” Geophys. J. R. Astron. Soc. 36, 377–398 (1974).
C. H. Chapman, “A new method for computing synthetic seismograms,” Geophys. J. R. Astron. Soc. 54, 481–518 (1978).
C. H. Chapman, “Generalized radon transform and slant stacks,” Geophys. J. R. Astron. Soc. 66, 445–453 (1981).
R. W. Clayton and G. A. McMechan, “Inversion of refraction data by wavefield continuation,” Geophysics 46, 860–868 (1981).
T. M. Brocher and R. A. Phinney, “Inversion of slant stacks using finite-length record sections,” J. Geophys. Res 86, 7065–7072 (1981).
G. A. McMechan, “P-x imaging by localized slant stacks of T-x data,” Geophys. J. R. Astron. Soc. 72, 213–221 (1983).
B. Milkereit, W. D. Mooney, and W. M. Kohler, “Inversion of seismic refraction data in planar dipping structure,” Geophys. J. Int. 82 (1), 81–103 (1985).
M. E. Kupps, A. J. Harding, and J. A. Orcutt, “A comparison of tau-p transform methods,” Geophysics 55 (9), 1202–1215 (1990).
B. Zhou and S. A. Greenhalgh, “Linear and parabolic T-p transforms revisited,” Geophysics 59 (7), 1133–1149 (1994).
A. V. Baev, “On t-local solvability of inverse scattering problems in two-dimensional layered media,” Comput. Math. Math. Phys. 55 (6), 1033–1050 (2015).
A. V. Baev, “Imaging of layered media in inverse scattering problems for an acoustic wave equation,” Math. Models. Comput. Simul. 8 (6), 689–702 (2016).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1972; Dover, New York, 2011).
A. V. Baev and N. V. Kutsenko, “Solving the inverse generalized problem of vertical seismic profiling,” Comput. Math. Model. 15 (1), 1–18 (2004).
A. V. Baev and N. V. Kutsenko, “Identification of a dissipation coefficient by a variational method,” Comput. Math. Math. Phys. 46 (10), 1796–1807 (2006).
A. S. Blagoveshchenskii, “Local method of solution of a nonstationary inverse problem for an inhomogeneous string,” Proc. Steklov Inst. Math. 115, 30–41 (1971).
A. V. Baev, “A method of solving the inverse scattering problem for the wave equation,” USSR Comput. Math. Math. Phys. 28 (1), 15–21 (1988).
A. V. Baev, “On local solvability of inverse scattering problems for the Klein–Gordon equation and the Dirac system,” Math. Notes 96 (2), 286–289 (2014).
A. V. Baev, “Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media,” Comput. Math. Math. Phys. 56 (12), 2043–2055 (2016).
A. N. Tikhonov, “On the solution of ill-posed problems and regularization method,” Dokl. Akad. Nauk SSSR 153 (1), 501–504 (1963).
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1986).
A. A. Kaufman, A. L. Levshin, and K. L. Larner, Introduction to the Theory of Geophysical Methods, Part 4: Acoustic and Elastic Wavefields in Geophysics (Nedra, Moscow, 2003) [in Russian].
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Original Russian Text © A.V. Baev, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 4, pp. 550–560.
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Baev, A.V. Radon Transform for Solving an Inverse Scattering Problem in a Planar Layered Acoustic Medium. Comput. Math. and Math. Phys. 58, 537–547 (2018). https://doi.org/10.1134/S0965542518040061
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DOI: https://doi.org/10.1134/S0965542518040061