Abstract
This chapter deals with a multidimensional inverse dynamic problem for the acoustic equation. This problem is reduced to inverse kinematic, integral geometry problems, a Dirichlet problem for a quasilinear elliptic equation, and for some special case to standard tomography problems. The structure and some properties of the fundamental solution of the Cauchy problem for the acoustic equation are used for this reduction. The structure and properties are described in detail.
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References
A.S. Alekseev. A numerical method for solving the three-dimensional inverse kinematic problem of seismology. Mat. Problemy Geofiz., 1:176–201, 1969 (Russian).
A.S. Alekseev. A numerical method for determining the structure of the Earth’s upper mantle. Mat. Problemy Geofiz., 2:141–165, 1971 (Russian).
Yu.E. Anikonov, N.B. Pivovarova, and L.V. Slavina. The three-dimensional velocity field of the Kamchatka focal zone. Mat. Problemy Geofiz., 5:92–117, 1974 (Russian).
Yu.E. Anikonov. Some Methods for the Study of Multidimensional Inverse Problems for Differential Equations. Nauka, Novosibirsk, 1978 (Russian).
G.Ya. Beil’kin. Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions. Soviet Math. Dokl., 21:3–6, 1983.
I.N. Bernshtein and M.L. Gerver. A problem of integral geometry for a family of geodesic and an inverse kinematic problem of seismology. Dokl. Earth Sc. Sections, 243:302–305, 1978.
A.L. Bukhgeim. Some problems of integral geometry. Siberian Math. J., 13:34–42, 1972.
L.E. El’sgol’tz. Differential Equations and Calculus of Variations. Mir, Moscow, 1970.
I.M. Gel’fand and M.I. Graev. Geometry of Homogeneous Spaces, Representations of Groups in Homogeneous Spaces and Related Questions of Integral Geometry. American Mathematical Society Translations, Providence, RI, 1964.
I.M. Gel’fand, M.I. Graev, and N.Ya. Vilenkin. Generalized Functions, Integral Geometry and Representation Theory. Academic Press, London, 1966.
M.L. Gerver and V.M. Markushevich. Determining seismic-wave velocities from travel-time curves. Comput. Seism., pp. 3–51, Plenum Press, New York, 1972.
G. Herglotz. Uber die elastizitat the erde bei berucksichtigung ihrer variablen dichte. Z. Math. Phys., 52:275–299, 1905.
F. John. Bestimmung einer funktion aus ihren integralen uber gewisse manningfaltigkeiten. Math. Ann., 109:488–520, 1933.
F. John. Abhangigkeiten zwishen den flachen integralen einer stetingen funktionen. Math. Ann., 111:541–559, 1935.
A.A. Khachaturov. Determination of the value of the measure of region of n-dimensional Euclidean space from its values for all half-spaces. Uspekhi Mat. Nauk, 9:205–212, 1954 (Russian).
R.O. Kostelyanetz and Yu.G. Reshetnyak. Determination of a completely additive function by its values on half-spaces. Uspekhi Mat. Nauk, 9:131–140, 1954 (Russian).
M.M. Lavrent’ev and A.L. Bukhgeim. On a class of problems of integral geometry. Soviet Math. Dokl., 14:38–39, 1973.
M.M. Lavrent’ev and A.L. Bukhgeim. On a class of operator equations of the first kind. Functional Anal. Appl., 7:44–53, 1973.
M.M. Lavrent’ev, V.G. Romanov, and S.R Shishatskii. Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society Translations, Providence, RI, 1986.
R.G. Mukhometov. The problem of reconstructing a two-dimensional Riemannian metric and integral geometry. Soviet Math. Dokl., 18:32–35, 1977.
R.G. Mukhametov and V.G. Romanov. On the problem of finding an isotropic Riemannian metric in n-dimensional space. Soviet Math. Dokl., 19:1279–1281, 1978.
F. Natterer. The Mathematics of Computerized Tomography. Wiley, Stuttgart, and B.G. Tenbner, Leipziz, 1986.
J. Radon. Uber die bestimmungen von funktionen durch ihre integralwerte langs gewisser manningfaltigkeiten. Ben Verh. Sachs. Ges. Wiss. Leipzig Math.-Phys. Kl., 262–277, 1917.
V.G. Romanov. Inverse Problems of Mathematical Physics. VNU Science Press, Utrecht, The Netherlands, 1987.
V.G. Romanov. Integral geometry on the geodesics on isotropic Riemannian metric. Soviet Math. Dokl, 19:290–293, 1978.
V.G. Romanov. Structure of the fundamental solution of the Cauchy problem for Maxwell’s systems of equations. Differents. Uravn., 22:1577–1587, 1986 (Russian).
V.G. Romanov and S.I. Kabanikhin. Inverse Problems for Maxwell’s Equations. VNU Science Press, Utrecht, The Netherlands, 1994.
VS. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, New York, 1971.
V.G. Yakhno. Inverse Problems for Differential Equations of Elasticity. Nauka, Novosibirsk, 1990 (Russian).
V.G. Yakhno. Multidimensional inverse problems in the ray formulation for hyperbolic equations. J. Inverse Ill-Posed Problems, 6:373–386, 1998.
E. Wiechert, K. Zoeppritz. Uber erdbebenwellen. Nachr. Konigl Ges. Wiss. Göttingen Math.-Phys. Kl., 4:415–549, 1907.
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Yakhno, V.G. (2001). Multidimensional Inverse Problem for the Acoustic Equation in the Ray Statement. In: Taroudakis, M.I., Makrakis, G.N. (eds) Inverse Problems in Underwater Acoustics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3520-8_10
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DOI: https://doi.org/10.1007/978-1-4757-3520-8_10
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