Gaussian beam synthetic seismograms
Article Sidebar
Published:
Dec 19, 1985
Keywords:
Seismic waves,
Synthetic seismograms,
Gaussian beams
Volumes
Vols. 1-18 (1924-1944), ISSN 0044-2801
Main Article Content
Abstract
Numerical modelling of high-frequency seismic wave fields in complex, 2-D and 3-D, laterally varying, layered structures by the summation of elastodynamic Gaussian beams is discussed. The main attention is devoted to the expansion of the wave field into Gaussian beams, to the choice of initial parameters of Gaussian beams in these expansions and to the construction of synthetic seismograms. The Gaussian beam synthetic seismograms are regular even in regions where the ray method fails, such as the caustic region, critical region, etc. Due to the smoothing effects involved in the Gaussian beam procedure, the method is not too sensitive to the approximation of the medium and to minor details of the model. Moreover, the method does not require two-point (source-to-receiver) ray tracing. The eyaluation of Gaussian beam synthetic seismograms requires approximately the same amount of computer time as the evaluation of ray synthetic seismograms. The memory requirements are also approximately the same. Numerical examples of Gaussian beam synthetic seismograms for 2-D and 3-D structures are presented. Various possible applications of Gaussian beams to seismological problem of practical importance are outlined.
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Article Details
How to Cite
Červeny, V. (1985). Gaussian beam synthetic seismograms. Journal of Geophysics, 58(1), 44-72. Retrieved from https://journal.geophysicsjournal.com/JofG/article/view/144
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References
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Arnaud, J.A., Kogelnik, H. (1969) Gaussian light beams with general astigmatism. Applied Optics 8, 1687-1693
Babich, V.M. (1968) Eigenfunctions, concentrated in the vicinity of closed geodesics. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 9, V.M. Babich, ed.: pp. 15-63. Nauka, Leningrad
Babich, V.M., Buldyrev, N.J. (1972) Asymptotic methods in problems of diffraction of short waves (in Russian). Nauka, Moscow
Babich, V.M., Kirpichnikova, N.Y. (1974) The boundary layer method in diffraction problems (in Russian). Leningrad: Leningrad University Press (English translation by Springer-Verlag, 1980)
Babich, V.M., Pankratova, T.F. (1973) On discontinuities of the Green function of mixed problems for wave equation with variable coefficients. In: Problems of mathematical physics (in Russian), Vol. 6, V.M. Babich, ed.: pp. 9-27. Leningrad: Leningrad Univ. Press
Babich, V.M., Popov, M.M. (1981) Propagation of concentrated acoustical beams in three-dimensional inhomogeneous media (in Russian). Akust. Zh. 27, 828-835
Babich, V.M., Ulin, V.V. (1981a) Complex-valued ray solutions and eigenfunctions, concentrated in the vicinity of a closed geodesic. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 11, V.M. Babich, ed.: pp. 6-13. Nauka, Leningrad
Babich, V.M., Ulin, V.V. (1981b) Complex space-time ray method and "quasiphotons". In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 12, V.M. Babich, ed.: pp. 5-12. Nauka, Leningrad
Babich, V.M., Molotkov, I.A., Popov, M.M. (1984) Gaussian beams, concentrated close to lines of solution and their application (in Russian). Preprint No. 24 (396). Moscow: Inst. of Radioengineering and Electronics, Acad. Sci. USSR
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Červeny, V. (1967) The amplitude-distance curves for waves reflected at a plane interface for different frequency ranges. Geophys. J. R. Astron. Soc. 13, 187-196
Červeny, V. (1981) Seismic wave fields in structurally complicated media. Ray and Gaussian beam approaches. Lecture notes. Utrecht: Rijksuniversiteit Utrecht, Vening-Meinesz Laboratory
Červeny, V. (1982) Expansion of a plane wave into Gaussian beams. Stud. Geophys. Geod. 26, 120-131
Červeny, V. (1983) Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method. Geophys. J. R. Astron. Soc. 73, 389-426
Červeny, V. (1985a) Ray synthetic seismograms for complex twodimensional and three-dimensional structures. J. Geophys. 58, 2-26
Červeny, V. (1985b) The application of ray tracing to the numerical modeling of seismic wavefields in complex structures. In: Seismic shear waves, G. Dohr, ed.: pp. 1-124. Handbook of geophysical exploration, Section I: Seismic exploration, K. Helbig and S. Treitel, eds. London: Geophysical Press
Červeny, V., Frangie, A.B. (1980) Elementary seismograms of seismic body waves in dissipative media. Stud. Geophys. Geod. 24, 365-372
Červeny, V., Frangie, A.B. (1982) Effects of causal absorption on seismic body waves. Stud. Geophys. Geod. 26, 238-253
Červeny, V., Jansky, J. (1985a) Fast computation of ray synthetic seismograms in vertically inhomogeneous media. Stud. Geophys. Geod. 29, 49-67
Červeny, V., Jansky, J. (1985b) Fast computation of seismic wave fields in inhomogeneous media. In: Methods of processing of seismic reflection and refraction data (in Czech), M. Fejfar, ed. Brno: Geofyzika N.E. (In press)
Červeny, V., Klimes, L. (1984) Synthetic body wave seismograms for three-dimensional laterally varying media. Geophys. J. R. Astron. Soc. 79, 119-133
Červeny, V., Pšenčik, I. (1983a) Gaussian beams in two-dimensional elastic inhomogeneous media. Geophys. J. R. Astron. Soc. 72, 417-433
Červeny, V., Pšenčik, I. (1983b) Gaussian beams and paraxial ray approximation in three-dimensional elastic inhomogeneous media. J. Geophys. 53, 1-15
Červeny, V., Pšenčik, I. (1984) Gaussian beams in two-dimensional laterally varying layered structures. Geophys. J. R. Astron. Soc. 78, 65-91
Červeny, V., Popov, M.M., Pšenčik, I. (1982) Computation of wave fields in inhomogeneous media - Gaussian beam approach. Geo phys. J .R. Astron. Soc. 70, 109-128
Chapman, C.H. (1978) A new method for computing synthetic seismograms. Geophys. J. R. Astron. Soc. 54, 481-518
Chapman, C.H. (1985) Ray theory and its extensions-W.£5..BJ and Maslov seismograms. J. Geophys. 58, 27-43
Chapman, C.H., Drummond, R. (1982) Body wave seismograms in inhomogeneous media using Maslov asymptotic theory. Bull. Seismal. Soc. Amer. 72, S277-S317
Claerbout, J.F. (1976) Fundamentals of geophysical data processing. McGraw-Hill, New York
Cormier, V.F., Spudich, P. (1984) Amplification of ground motion and wavefront complexity in fault zones; examples from the San Andreas and Calaveras Faults. Geophys. J. R. Astron. Soc. 79, 135-152
Deschamps, G.A. (1971) The Gaussian beams as a bundle of complex rays. Electr. Lett. 7, 684-685
Felsen, L.B. (1976a) Evanescent waves. J. Opt. Soc. Amer. 66, 751-760
Felsen, L.B. (1976b) Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams. Inst. Naz. Alta Matern., Symp. Math. 18, 39-56
Felsen, L.B. (1985) Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams. Geophys. J. R. Astron. Soc. (In press)
Felsen, L.B., Marcuvitz, N. (1973) Radiation and scattering of waves. Englewood Cliffs: Prentice Hall
Fertig, J., Pšenčik, I. (1985) Numerical modeling of P and S waves in explosion seismology. In: Seismic shear waves, G. Dohr, ed. Handbook of geophysical exploration, Section I: Seismic exploration, K. Helbig and S. Treitel, eds. Geophysical Press, London
Fock, V.A. (1965) Electromagnetic diffraction and propagation problems. Pergamon Press, New York
Frazer, L.N. (1983) Feynman path integral synthetic seismograms. Trans. Amer. Geophys. Un. 64, 772
Frazer, L.N. (1985) Reflection seismograms using multi-fold path integrals. Geophys. J. R. Astron. Soc. (In press)
Frazer, L.N., Sen, K.M. (1985) Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium, Part I: Theory. Geophys. J. R. Astron. Soc. 80, 121-147
Grikurov, V.E., Popov, M.M. (1983a) Summation of Gaussian beams in a surface waveguide. Wave Motion 5, 225-233
Grikurov, V.E., Popov, M.M. (1983b) On the computation of waveguides by the method of summation of Gaussian beams (in Russian). Izv. Acad. Sci. USSR, Physics of the Earth No. 8, 101-103
Haines, A.J. (1983) A phase-front method I: Narrow-frequencyband SH waves. Geophys. J. R. Astron. Soc. 72, 783-808
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Haines, A.J. (1984b) A phase-front method III: Acoustic waves, Pand S-waves. Geophys. J. R. Astron. Soc. 77, 65-104
Hanyga, A. (1985a) Gaussian beams in anisotropic elastic media. Geophys. J. R. Astron. Soc. (In press)
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Jobert, N., Jobert, G. (1983) An application of ray theory to the propagation of waves along a laterally heterogeneous spherical surface. Geophys. Res. Lett. 10, 1148-1151
Katchalov, A.P. (1984) Coordinate system for description of the "quasiphoton". In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 14, V.M. Babich, ed.: pp. 73-76. Nauka, Leningrad
Katchalov, AP., Popov, M.M. (1981) The application of the Gaussian beam summation method to the computation of wave fields in the high-frequency approximation. Dokl. Akad. Nauk SSSR 258(5), 1097-1100
Katchalov, A.P., Popov, M.M., Pšenčik, I. (1983) On the application of the method of summation of Gaussian beams to the problems of boundaries with edges. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 13, V.M. Babich, ed.: pp. 65-71. Nauka, Leningrad
Keller, J.B., Streifer, W. (1971) Complex rays with an application to Gaussian beams. J. Opt. Soc. Amer. 61, 40-43
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Klimes, L. (1983) Hermite-Gaussian beams in inhomogeneous elastic media. Stud. Geophys. Geod. 27, 354-365
Klimes, L. (1984a) Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams. Geophys. J.R. Astron. Soc. 79, 105-118
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Arnaud, J.A., Kogelnik, H. (1969) Gaussian light beams with general astigmatism. Applied Optics 8, 1687-1693
Babich, V.M. (1968) Eigenfunctions, concentrated in the vicinity of closed geodesics. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 9, V.M. Babich, ed.: pp. 15-63. Nauka, Leningrad
Babich, V.M., Buldyrev, N.J. (1972) Asymptotic methods in problems of diffraction of short waves (in Russian). Nauka, Moscow
Babich, V.M., Kirpichnikova, N.Y. (1974) The boundary layer method in diffraction problems (in Russian). Leningrad: Leningrad University Press (English translation by Springer-Verlag, 1980)
Babich, V.M., Pankratova, T.F. (1973) On discontinuities of the Green function of mixed problems for wave equation with variable coefficients. In: Problems of mathematical physics (in Russian), Vol. 6, V.M. Babich, ed.: pp. 9-27. Leningrad: Leningrad Univ. Press
Babich, V.M., Popov, M.M. (1981) Propagation of concentrated acoustical beams in three-dimensional inhomogeneous media (in Russian). Akust. Zh. 27, 828-835
Babich, V.M., Ulin, V.V. (1981a) Complex-valued ray solutions and eigenfunctions, concentrated in the vicinity of a closed geodesic. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 11, V.M. Babich, ed.: pp. 6-13. Nauka, Leningrad
Babich, V.M., Ulin, V.V. (1981b) Complex space-time ray method and "quasiphotons". In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 12, V.M. Babich, ed.: pp. 5-12. Nauka, Leningrad
Babich, V.M., Molotkov, I.A., Popov, M.M. (1984) Gaussian beams, concentrated close to lines of solution and their application (in Russian). Preprint No. 24 (396). Moscow: Inst. of Radioengineering and Electronics, Acad. Sci. USSR
Ben Menahem, A., Beydoun, W.B. (1985a) Range of validity of seismic ray and beam methods in general inhomogeneous media. Part I: General theory. Geophys. J. R. Astron. Soc. (In press)
Ben Menahem, A., Beydoun, W.B. (1985b) Range of validity of seismic and beam methods in general inhomogeneous media. Part II: A canonical problem. Geophys. J. R. Astron. Soc. (In press)
Bernard, P., Madariaga, R. (1984) A new asymptotic method for the modeling of near-field accelerograms. Bull. Seismal. Soc. Amer. 74, 539-557
Cernohlavkova, D. (1985) Seismic wave fields in simple structures (in Czech). Diploma Thesis. Prague: Charles University, Faculty of Mathematics and Physics
Červeny, V. (1967) The amplitude-distance curves for waves reflected at a plane interface for different frequency ranges. Geophys. J. R. Astron. Soc. 13, 187-196
Červeny, V. (1981) Seismic wave fields in structurally complicated media. Ray and Gaussian beam approaches. Lecture notes. Utrecht: Rijksuniversiteit Utrecht, Vening-Meinesz Laboratory
Červeny, V. (1982) Expansion of a plane wave into Gaussian beams. Stud. Geophys. Geod. 26, 120-131
Červeny, V. (1983) Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method. Geophys. J. R. Astron. Soc. 73, 389-426
Červeny, V. (1985a) Ray synthetic seismograms for complex twodimensional and three-dimensional structures. J. Geophys. 58, 2-26
Červeny, V. (1985b) The application of ray tracing to the numerical modeling of seismic wavefields in complex structures. In: Seismic shear waves, G. Dohr, ed.: pp. 1-124. Handbook of geophysical exploration, Section I: Seismic exploration, K. Helbig and S. Treitel, eds. London: Geophysical Press
Červeny, V., Frangie, A.B. (1980) Elementary seismograms of seismic body waves in dissipative media. Stud. Geophys. Geod. 24, 365-372
Červeny, V., Frangie, A.B. (1982) Effects of causal absorption on seismic body waves. Stud. Geophys. Geod. 26, 238-253
Červeny, V., Jansky, J. (1985a) Fast computation of ray synthetic seismograms in vertically inhomogeneous media. Stud. Geophys. Geod. 29, 49-67
Červeny, V., Jansky, J. (1985b) Fast computation of seismic wave fields in inhomogeneous media. In: Methods of processing of seismic reflection and refraction data (in Czech), M. Fejfar, ed. Brno: Geofyzika N.E. (In press)
Červeny, V., Klimes, L. (1984) Synthetic body wave seismograms for three-dimensional laterally varying media. Geophys. J. R. Astron. Soc. 79, 119-133
Červeny, V., Pšenčik, I. (1983a) Gaussian beams in two-dimensional elastic inhomogeneous media. Geophys. J. R. Astron. Soc. 72, 417-433
Červeny, V., Pšenčik, I. (1983b) Gaussian beams and paraxial ray approximation in three-dimensional elastic inhomogeneous media. J. Geophys. 53, 1-15
Červeny, V., Pšenčik, I. (1984) Gaussian beams in two-dimensional laterally varying layered structures. Geophys. J. R. Astron. Soc. 78, 65-91
Červeny, V., Popov, M.M., Pšenčik, I. (1982) Computation of wave fields in inhomogeneous media - Gaussian beam approach. Geo phys. J .R. Astron. Soc. 70, 109-128
Chapman, C.H. (1978) A new method for computing synthetic seismograms. Geophys. J. R. Astron. Soc. 54, 481-518
Chapman, C.H. (1985) Ray theory and its extensions-W.£5..BJ and Maslov seismograms. J. Geophys. 58, 27-43
Chapman, C.H., Drummond, R. (1982) Body wave seismograms in inhomogeneous media using Maslov asymptotic theory. Bull. Seismal. Soc. Amer. 72, S277-S317
Claerbout, J.F. (1976) Fundamentals of geophysical data processing. McGraw-Hill, New York
Cormier, V.F., Spudich, P. (1984) Amplification of ground motion and wavefront complexity in fault zones; examples from the San Andreas and Calaveras Faults. Geophys. J. R. Astron. Soc. 79, 135-152
Deschamps, G.A. (1971) The Gaussian beams as a bundle of complex rays. Electr. Lett. 7, 684-685
Felsen, L.B. (1976a) Evanescent waves. J. Opt. Soc. Amer. 66, 751-760
Felsen, L.B. (1976b) Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams. Inst. Naz. Alta Matern., Symp. Math. 18, 39-56
Felsen, L.B. (1985) Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams. Geophys. J. R. Astron. Soc. (In press)
Felsen, L.B., Marcuvitz, N. (1973) Radiation and scattering of waves. Englewood Cliffs: Prentice Hall
Fertig, J., Pšenčik, I. (1985) Numerical modeling of P and S waves in explosion seismology. In: Seismic shear waves, G. Dohr, ed. Handbook of geophysical exploration, Section I: Seismic exploration, K. Helbig and S. Treitel, eds. Geophysical Press, London
Fock, V.A. (1965) Electromagnetic diffraction and propagation problems. Pergamon Press, New York
Frazer, L.N. (1983) Feynman path integral synthetic seismograms. Trans. Amer. Geophys. Un. 64, 772
Frazer, L.N. (1985) Reflection seismograms using multi-fold path integrals. Geophys. J. R. Astron. Soc. (In press)
Frazer, L.N., Sen, K.M. (1985) Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium, Part I: Theory. Geophys. J. R. Astron. Soc. 80, 121-147
Grikurov, V.E., Popov, M.M. (1983a) Summation of Gaussian beams in a surface waveguide. Wave Motion 5, 225-233
Grikurov, V.E., Popov, M.M. (1983b) On the computation of waveguides by the method of summation of Gaussian beams (in Russian). Izv. Acad. Sci. USSR, Physics of the Earth No. 8, 101-103
Haines, A.J. (1983) A phase-front method I: Narrow-frequencyband SH waves. Geophys. J. R. Astron. Soc. 72, 783-808
Haines, A.J. (1984a) A phase-front method II: Broad-frequency-band SH waves. Geophys. J. R. Astron. Soc. 77, 43-64
Haines, A.J. (1984b) A phase-front method III: Acoustic waves, Pand S-waves. Geophys. J. R. Astron. Soc. 77, 65-104
Hanyga, A. (1985a) Gaussian beams in anisotropic elastic media. Geophys. J. R. Astron. Soc. (In press)
Hanyga, A. (1985b) Gaussian beams in anisotropic elastic media and applications. In: Proceedings of the workshop on hybrid formulation of wave propagation and scattering, L.B. Felsen, ed. Amsterdam: Martinus Nijhoff (In press)
Honl, H., Maue, A.W., Westphal, K. (1961) Theorie der Beugung. Berlin-Gottingen-Heidelberg: Springer Verlag
Hudson, J.A. (1980) A parabolic approximation for elastic waves. Wave Motion 2, 207-214
Jobert, N., Jobert, G. (1983) An application of ray theory to the propagation of waves along a laterally heterogeneous spherical surface. Geophys. Res. Lett. 10, 1148-1151
Katchalov, A.P. (1984) Coordinate system for description of the "quasiphoton". In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 14, V.M. Babich, ed.: pp. 73-76. Nauka, Leningrad
Katchalov, AP., Popov, M.M. (1981) The application of the Gaussian beam summation method to the computation of wave fields in the high-frequency approximation. Dokl. Akad. Nauk SSSR 258(5), 1097-1100
Katchalov, A.P., Popov, M.M., Pšenčik, I. (1983) On the application of the method of summation of Gaussian beams to the problems of boundaries with edges. In: Mathematical problems of the theory of propagation of waves (in Russian), Vol. 13, V.M. Babich, ed.: pp. 65-71. Nauka, Leningrad
Keller, J.B., Streifer, W. (1971) Complex rays with an application to Gaussian beams. J. Opt. Soc. Amer. 61, 40-43
Kennett, B.L.N. (1983) Seismic wave propagation in stratified media. Cambridge: Cambridge Univ. Press
Kirpichnikova, N.J. (1971a) Construction of solutions concentrated close to rays for the equations of elasticity in an inhomogeneous isotropic space. In: Mathematical problems of theory of diffraction and propagation of waves (in Russian), Vol. 1, V.M. Babich, ed.: pp. 103-113. Nauka, Leningrad (English translation by American Mathematical Society, 1974)
Kirpichnikova, N.J. (1971b) On the propagation of surface waves concentrated close to rays in an inhomogeneous elastic body of an arbitrary form. In: Mathematical problems of theory of diffraction and propagation of waves (in Russian), Vol. 1, V.M. Babich, ed.: pp. 114-130. Nauka, Leningrad
Klimes, L. (1983) Hermite-Gaussian beams in inhomogeneous elastic media. Stud. Geophys. Geod. 27, 354-365
Klimes, L. (1984a) Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams. Geophys. J.R. Astron. Soc. 79, 105-118
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