Abstract
A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.
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References
E. V. Nikol’skii, “On solution of direct and inverse seismic problems for a one-dimensional inhomogeneous medium in the case of a normally incident plane wave,” in Seismic Exploration Methods (Nauka, Moscow, 1965), pp. 190–205 [in Russian].
A. S. Blagoveshchenskii, “An inverse problem in seismic wave propagation theory,” in Problems in Mathematical Physics (Leningr. Gos. Univ., Leningrad, 1966), pp. 68–81 [in Russian].
B. S. Pariiskii, “An inverse problem for the wave equation with a depth effect,” in Direct and Inverse Problems in Seismology (Nauka, Moscow, 1968), pp. 25–40 [in Russian].
V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984; VNU Science, Utrecht, 1987).
M. I. Belishev and A. S. Blagoveshchenskii, Dynamic Inverse Problems in Wave Theory (S.-Peterburg. Univ., St. Petersburg, 1999) [in Russian].
S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sibirskoe Nauchoe, Novosibirsk, 2009) [in Russian].
A. L. Bukhgeim, Volterra Equations and Inverse Problems (Nauka, Novosibirsk, 1983) [in Russian].
A. V. Baev, “On the solution of an inverse problem for the wave equation with the help of a regularizing algorithm,” USSR Comput. Math. Math. Phys. 25 (1), 93–97 (1985).
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 dissipative scattering problems,” J. Inverse Ill-Posed Probl. 9 (4), 227–247 (2001).
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. L. Bukhgeim, Introduction to the Theory of Inverse Problems (Nauka, Novosibirsk, 1988) [in Russian].
A. V. Baev and S. N. Bushkov, “Numerical solution of an inverse problem for the wave equation by regularized inversion of a difference scheme,” Vestn. Mosk. Univ. Ser. 15: Vychisl. Mat. Kibern., No. 4, 52–54 (1086).
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. V. Baev and G. Yu. Mel’nikov, “Inverse dissipative problems in vertical seismic profiling,” J. Inverse Ill-Posed Probl. 7 (3), 201–220 (1999).
M. I. Belishev and A. P. Kachalov, “The methods of boundary control theory in the inverse spectral problem for an inhomogeneous string,” J. Sov. Math. 57 (3), 3072–3077 (1991).
M. I. Belishev and T. L. Sheronova, “Methods of boundary control theory in the nonstationary inverse problem for an inhomogeneous string,” J. Math. Sci. 73 (3), 320–329 (1995).
S. I. Kabanikhin, Projection-Difference Methods for Determining Coefficients of Hyperbolic Equations (Nauka, Novosibirsk, 1988) [in Russian].
S. I. Kabanikhin, “On linear regularization of multidimensional inverse problems for hyperbolic equations,” Dokl. Akad. Nauk SSSR 309 (4), 791–795 (1989).
S. I. Kabanikhin and M. A. Shishlenin, “Boundary control and Gel’fand–Levitan–Krein methods in inverse acoustic problem,” J. Ill-Posed Probl. 12 (2), 125–144 (2004).
S. I. Kabanikhin and M. A. Shishlenin, “Numerical algorithm for two-dimensional inverse acoustic problem based on Gel’fand–Levitan–Krein equation,” J. Inverse Ill-Posed Probl. 18 (9), 979–995 (2011).
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, “Mathematical modeling of waves in layered media near a caustic,” Math. Models Comput. Simul. 6 (4), 364–377 (2014).
V. L. Trofimov, V. A. Milashin, F. F. Khaziev, et al., “Prediction of geological features on seismic data of high resolution,” Seism. Technol. 4, 49–60 (2009).
V. L. Trofimov, V. A. Milashin, F. F. Khaziev, et al., “Special processing and interpretation of seismic data in complex geological conditions by high-resolution seismic,” Seism. Technol. 3, 36–50 (2009).
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Original Russian Text © A.V. Baev, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 12, pp. 2073–2085.
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Baev, A.V. Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media. Comput. Math. and Math. Phys. 56, 2043–2055 (2016). https://doi.org/10.1134/S0965542516120034
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DOI: https://doi.org/10.1134/S0965542516120034