Linearized scattering of surface waves on a spherical Earth
Article Sidebar
Published:
Apr 23, 1987
Keywords:
Seismology,
Normal modes,
Surface waves,
Scattering,
Inversion
Volumes
Vols. 1-18 (1924-1944), ISSN 0044-2801
Main Article Content
Abstract
Recently, a formalism for three-dimensional surface-wave scattering in a plane geometry was derived. Since teleseismic surface-wave data are generally recorded at epicentral distances large enough to be influenced by the sphericity of the Earth, it is necessary to find the effects of a spherical geometry on surface-wave scattering. The theory of surface-wave scattering relies heavily on a dyadic decomposition of the Green's function, and a new derivation is given for the (dyadic) Green's function of a spherically symmetric Earth. This new derivation employs Poisson's sum formula and is more rigorous than previous derivations. Using the dyadic Green's function, a relation is established with the scattering theory in a flat geometry. This finally leads to a linearized formalism for three-dimensional surface-wave scattering on a sphere. Even for shallow surface waves the effects of sphericity are important and necessitate a modification of the propagator terms in the expression for the scattered surface waves.
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Article Details
How to Cite
Snieder, R., & Nolet, G. (1987). Linearized scattering of surface waves on a spherical Earth. Journal of Geophysics, 61(1), 55-63. Retrieved from https://journal.geophysicsjournal.com/JofG/article/view/78
Section
References
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Butkov, E. (1968) Mathematical physics. Addison-Wesley, Reading MA, USA
Dahlen, F.A. (1979a) The spectra of unresolved split normal mode multiplets. Geophys. J. R. Astron. Soc. 58:1-33
Dahlen, F.A. (1979b) Exact and asymptotic synthetic multiplet spectra on an ellipsoidal Earth. Geophys. J. R. Astron. Soc. 59:19-42
Dziewonski, A.M. (1984) Mapping the lower mantle: determination of lateral heterogeneity in P velocity up to degree and order 6. 1. Geophys. Res. 89:5929~5952
Gilbert, F., Dziewonski, A.M. (1975) An application of normal mode theory to the retrieval of structure parameters and source mechanism from seismic spectra. Phil. Trans. Roy. Soc. Lon. A278:187-269
Hudson, J.A. (1977) Scattered waves in the coda of P.J. Geophys. 43:359-374
Hudson, J.A., Heritage, J.R. (1982) The use of the Born approximation in seismic scattering problems. Geophys. J. R. Astron Soc. 66:221-240
Jordan, T.H. (1978) A procedure for estimating lateral variations from low frequency eigenspectra data. Geophys. J. R. Astron. Soc. 52:441-455
Kennett, B.L.N. (1984) Guided wave propagation in laterally varying media - I. Theoretical development. Geophys. J. R. Astron. Soc. 79:235-255
Lerner-Lam, A.L., Jordan, T.H. (1983) Earth structure from fundamental and higher-mode waveform analysis. Geophys. J. R. Astron. Soc. 75:759-797
Levshin, A., Berteussen, K.A. (1979) Anomalous propagation of surface waves in the Barentz Sea as inferred from NORSAR recordings. Geophys. J. R. Astron. Soc. 56:97-118
Malin, P.E., Phinney, R.A. (1985) On the relative scattering of P- and S-waves. Geophys. J. R. Astron. Soc. 80:603-618
Montagner, J.P. (1986) Regional three-dimensional structures using longperiod surface waves. Ann. Geophys. 84:283-294
Nataf, H.C., Nakanishi, 1., Anderson, D.L. (1986) Measurements of mantle wave velocities and inversion for lateral heterogeneities and anisotropy 3. Inversion. J. Geophys. Res. 91:7261-7308
Nercessian, A., Him, A., Taranto Ia, A. (1984) Three dimensional prospecting of the Mont-Dore volcano, France. Geophys. J.R. Astron. Soc. 76:307-315
Nolet, G. (1976) Higher modes and the determination of upper mantle structure. PhD-thesis, University of Utrecht
Nolet, G. (1977) The upper mantle under Western Europe inferred from the dispersion of Rayleigh modes. J. Geophys. 43:265-285
Nolet, G., Romanovicz, B., Kind, R., Wielandt, E. (1985) ORFEUS, Observatories and Research Facilities for European Seismology
Nolet, G., Trier, J. van, Huisman, R. (1986) A formalism for nonlinear inversion of seismic surface waves. Geophys. Res. Lett. 13:26-29
IRIS (1984) PASSCAL, Program for Array of the Continental Lithosphere. Issued by the Incorporated Research Institute for Seismology
Snieder, R. (1986a) 3-D Linearized scattering of surface waves and a formalism for surface wave holography. Geophys. J. R. Astron. Soc. 84:581-605
Snieder, R. (1986b) The influence of topography on the propagation and scattering of surface waves. Phys. Earth Planet. Inter. 44:226-241
Spakman, W. (1986) Subduction beneath Eurasia in connection with the Mesozoic Thethys. Geol. Mijnbouw 65:145-154
Vlaar, N. (1976) On the excitation of the Earth's seismic normal modes. Pure Appl. Geophys. 114:863-875
Woodhouse, J.H., Dziewonski, A.M. (1984) Mapping the upper mantle: three-dimensional modeling of the Earth structure by inversion of seismic waveforms. J. Geophys. Res. 89:5953-5986
Woodhouse, J.H., Girnius, T.P. (1982) Surface waves and free oscillations in a regionalized Earth model. Geophys. J. R. Astron. Soc. 68:653-673
Wu, R., Aki, K. (1985) Scattering characteristics of elastic waves by an heterogeneity. Geophysics 50:582-595
Yomogida, K., Aki, K. (1985) Total waveform synthesis of surface waves in laterally heterogeneous Earth by Gaussian beam method. J. Geophys. Res. 90:7665-7688
Babich, V.M., Chikhachev, B.A., Yanovskaya, T.B. (1976) Surface waves in a vertically inhomogeneous elastic halfspace with a weak horizontal inhomogeneity. Izv. Phys. Earth 12:242~245
Backus, G.E. (1964) Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi circular paths. Bull. Seism. Soc. Am. 54:571~610
Ben-Menahem, A., Sing, S.J. (1968) Eigenvector dyads with applications to geophysical theory. Geophys. J. R. Astron. Soc. 16:417-452
Bungum, H., Capon, J. (1974) Coda pattern and multipathing propagation of Rayleigh waves at NORSAR. Phys Earth Planet. Inter. 9:111-127
Butkov, E. (1968) Mathematical physics. Addison-Wesley, Reading MA, USA
Dahlen, F.A. (1979a) The spectra of unresolved split normal mode multiplets. Geophys. J. R. Astron. Soc. 58:1-33
Dahlen, F.A. (1979b) Exact and asymptotic synthetic multiplet spectra on an ellipsoidal Earth. Geophys. J. R. Astron. Soc. 59:19-42
Dziewonski, A.M. (1984) Mapping the lower mantle: determination of lateral heterogeneity in P velocity up to degree and order 6. 1. Geophys. Res. 89:5929~5952
Gilbert, F., Dziewonski, A.M. (1975) An application of normal mode theory to the retrieval of structure parameters and source mechanism from seismic spectra. Phil. Trans. Roy. Soc. Lon. A278:187-269
Hudson, J.A. (1977) Scattered waves in the coda of P.J. Geophys. 43:359-374
Hudson, J.A., Heritage, J.R. (1982) The use of the Born approximation in seismic scattering problems. Geophys. J. R. Astron Soc. 66:221-240
Jordan, T.H. (1978) A procedure for estimating lateral variations from low frequency eigenspectra data. Geophys. J. R. Astron. Soc. 52:441-455
Kennett, B.L.N. (1984) Guided wave propagation in laterally varying media - I. Theoretical development. Geophys. J. R. Astron. Soc. 79:235-255
Lerner-Lam, A.L., Jordan, T.H. (1983) Earth structure from fundamental and higher-mode waveform analysis. Geophys. J. R. Astron. Soc. 75:759-797
Levshin, A., Berteussen, K.A. (1979) Anomalous propagation of surface waves in the Barentz Sea as inferred from NORSAR recordings. Geophys. J. R. Astron. Soc. 56:97-118
Malin, P.E., Phinney, R.A. (1985) On the relative scattering of P- and S-waves. Geophys. J. R. Astron. Soc. 80:603-618
Montagner, J.P. (1986) Regional three-dimensional structures using longperiod surface waves. Ann. Geophys. 84:283-294
Nataf, H.C., Nakanishi, 1., Anderson, D.L. (1986) Measurements of mantle wave velocities and inversion for lateral heterogeneities and anisotropy 3. Inversion. J. Geophys. Res. 91:7261-7308
Nercessian, A., Him, A., Taranto Ia, A. (1984) Three dimensional prospecting of the Mont-Dore volcano, France. Geophys. J.R. Astron. Soc. 76:307-315
Nolet, G. (1976) Higher modes and the determination of upper mantle structure. PhD-thesis, University of Utrecht
Nolet, G. (1977) The upper mantle under Western Europe inferred from the dispersion of Rayleigh modes. J. Geophys. 43:265-285
Nolet, G., Romanovicz, B., Kind, R., Wielandt, E. (1985) ORFEUS, Observatories and Research Facilities for European Seismology
Nolet, G., Trier, J. van, Huisman, R. (1986) A formalism for nonlinear inversion of seismic surface waves. Geophys. Res. Lett. 13:26-29
IRIS (1984) PASSCAL, Program for Array of the Continental Lithosphere. Issued by the Incorporated Research Institute for Seismology
Snieder, R. (1986a) 3-D Linearized scattering of surface waves and a formalism for surface wave holography. Geophys. J. R. Astron. Soc. 84:581-605
Snieder, R. (1986b) The influence of topography on the propagation and scattering of surface waves. Phys. Earth Planet. Inter. 44:226-241
Spakman, W. (1986) Subduction beneath Eurasia in connection with the Mesozoic Thethys. Geol. Mijnbouw 65:145-154
Vlaar, N. (1976) On the excitation of the Earth's seismic normal modes. Pure Appl. Geophys. 114:863-875
Woodhouse, J.H., Dziewonski, A.M. (1984) Mapping the upper mantle: three-dimensional modeling of the Earth structure by inversion of seismic waveforms. J. Geophys. Res. 89:5953-5986
Woodhouse, J.H., Girnius, T.P. (1982) Surface waves and free oscillations in a regionalized Earth model. Geophys. J. R. Astron. Soc. 68:653-673
Wu, R., Aki, K. (1985) Scattering characteristics of elastic waves by an heterogeneity. Geophysics 50:582-595
Yomogida, K., Aki, K. (1985) Total waveform synthesis of surface waves in laterally heterogeneous Earth by Gaussian beam method. J. Geophys. Res. 90:7665-7688