Reverse-time continuation of elastic waves
Tóm tắt
Methods based on downward continuation of an oscillation field have received wide recognition in the processing of multichannel seismic prospecting data. Physically, the idea behind this approach is clear: a wavefield observed on some surface is continued into the medium and backward in time. Mathematically, all the continuation algorithms used are based on scalar wave equation modeling, which takes proper account of the wave nature of oscillations of some types, but defies consideration of their vector nature. It is well known that a system of equations of dynamic elasticity theory (Lame equations) is a more adequate model for describing seismic oscillations. In this paper, a continuation of a field of elastic oscillations in an inhomogeneous isotropic medium is considered.
Tài liệu tham khảo
Petrashen’, G.I., Osnovy matematicheskoi teorii rasprostraneniya uprugikh voln (Foundations of Mathematical Theory of Elastic Wave Propagation), Leningrad: Nauka, 1978.
Aki, K. and Richards, P., Quantitative Seismology: Theory and Methods San Francisco: Freeman, 1980.
Morse, P.M. and Feshbach, H., Methods of Teoretical Physics, Parts I and II, New York: McGraw Hill, 1953.
Mikhailov, V.P., Differentsial’nye uravneniya v chastnykh proizvodnykh (Partial Differential Equations), Moscow: Nauka, 1976.
Tsibulchik, G.M., On the Formation of a Seismic Image Based on the Holographic Principle, Geol. Geofiz., 1975, no. 11, pp. 97–106.
Tsibulchik, G.M., Analyzing Solution to a Boundary Value Problem Modeling the Image Formation Process in Seismic Holography, Geol. Geofiz., 1975, no. 12, pp. 22–31.
Krein, S.G., Funktsional’nyi analiz (Functional Analysis), “SMB” Ser., Moscow: Fizmatgiz, 1972.
Babich, V.M., Sobolev-Kirchhoff Method in the Dynamics of an Inhomogeneous Elastic Medium, Vestnik LGU, 1957, vol. 13, no. 3, pp. 146–160.
Babich, V.M., Fundamental Solutions of Dynamic Equations of Elasticity Theory for an Inhomogeneous Medium, Prikl. Math. Mekh., 1961, vol. 25, no. 1, pp. 38–45.
Kupradze, V.D.,Metody potentsiala v teorii uprugosti (Potential Methods in Elasticity Theory), Moscow: Fizmatlit, 1963.
Novatskii, V., Teoriya uprugosti (Elasticity Theory), Moscow: Mir, 1975.
Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1971.
Bleistein, N. and Cohen, J., Nonuniqueness in the Inverse Source Problem in Acoustics and Electromagnetics, J.Math. Phys., 1977, vol. 18, no. 2, pp. 194–201.
Alekseev, A.S. and Tsibulchik, G.M., On a Relationship between Inverse Problems of Wave Propagation Theory and Problems of Wave Field Visualization, Dokl. Akad. Nauk SSSR, 1978, vol. 242, no. 5, pp. 1030–1033.
Bukhgeim, A.L., Uravneniya Volterra i obratnye zadachi (Volterra Equations and Inverse Problems), Novosibirsk: Nauka, 1983.