The Modification of the Matrix Method for the Modelling of Propagation of the Body Waves
Earth Sciences
Volume 3, Issue 1, February 2014, Pages: 1-8
Received: Dec. 12, 2013; Published: Jan. 10, 2014
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Author
Anastasiia Pavlova, Carpatian Branch of the Institute of geophysics named after S.I. Subbotin NAS of Ukraine, Lviv, Ukraine
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Abstract
The modification of the matrix method of construction of wavefield on the free surface of an anisotropic medium is presented. The earthquake source represented by a randomly oriented force or a seismic moment tensor is placed on an arbitrary boundary of a layered anisotropic medium. The theory of the matrix propagator in a homogeneous anisotropic medium by introducing a "wave propagator" is presented. It is shown that for anisotropic layered medium the matrix propagator can be represented by a "wave propagator" in each layer. The matrix propagator P(z,z0=0) acts on the free surface of the layered medium and generates stress-displacement vector at depth z. The displacement field on the free surface of an anisotropic medium is obtained from the received system of equations considering the radiation condition and that the free surface is stressless. The approbation of the modification of the matrix method for isotropic and anisotropic media with TI symmetry is done. A comparative analysis of our results with the synthetic seismic records obtained by other methods and published in foreign papers is executed.
Keywords
Matrix Propagator, Seismic Moment Tensor, Anisotropic Medium
To cite this article
Anastasiia Pavlova, The Modification of the Matrix Method for the Modelling of Propagation of the Body Waves, Earth Sciences. Vol. 3, No. 1, 2014, pp. 1-8. doi: 10.11648/j.earth.20140301.11
References
[1]
A1fоrd R. M., Kelly K.R., Вооге D.М. Accuracy of finite difference modeling of the acoustic wave equation. Geophysics, V. 39, 1974, p. 834–842.
[2]
Alterman Z., Loewenthal D. Computer generated seismograms – In: Methods in computational physics. New York, V. 12, 1972, p. 35–164
[3]
Auld, B. A. Acoustic Fields and Waves in Solids, John Wiley & Sons, Vol. 1, New York. 1973
[4]
Babuska V. Anisotropy of Vp and Vs in rock-forming minerals. Geophys. J. R. astr. Soc, NO. 50, 1981, р. 1-6.
[5]
Bachman R.T. Acoustic anisotropy in marine sediments and sedimentary rocks. J. Geophys. Res., V. 84, 1979, p. 7661-7663.
[6]
Backus G.E. Long-wave elastic anisotropy produced by horizontal layering. J.Geophys.Res., V. 67, NO. 11, 1962.
[7]
Behrens E. Sound propagation in lamellar composite materials and averaged elastic constants. J.Acoust.Soc.Amer., V. 42, NO. 2, 1967, p. 168-191.
[8]
Buchen P.W., Ben-Hador R. Free-mode surface-wave computa¬tions. Geophys. J. Int., V. 124, 1996, p. 869-887.
[9]
Cerveny V. Seismic ray theory. Cambridge University Press, 2001.
[10]
Chapman, C.H. The turning point of elastodynamic waves, Geophys. J.R. astr. Soc., V. 39, 1974, p. 613-621.
[11]
Chen, M. Numerical simulations of seismic wave propagation in anisotropic and heterogeneous Earth models—the Japan subduction zone, California Institute of Technology, 143, 2008.
[12]
Crampin, S. A review of the effects of anisotropic layering on the propagation of seismic waves, Geophys. J. R. astr, Soc., 49, 9-21, 1977
[13]
Crampin, S. A review of wave motion in anisotropic and cracked elastic media, Wave Motion, 3, 343-391, 1981.
[14]
Dziewonski, A.M., Andersin, D.L. Preliminary reference Earth model, Physics of the Earth and Planetary Interiors, 25, 297-356, 1981.
[15]
Fryer, G.J., Frazer, L.N. Seismic waves in stratified anisotropic media. Geophys J. R. and Soc. 78, 691-710, 1984
[16]
Fryer, G.J., Frazer, L.N. Seismic waves in stratified anisotropic media.-II. Elastodynamic eigensolutions for some anisotropic systems. Geophys J. R. and Soc. 73-101, 1987
[17]
Fuchs K. Explosion seismology and the continental crust-mantle boundary. Journal of the Geological Society, - 1977. - V. 134. - p. 139-151.
[18]
Haskell N.A. The dispersion of waves in multilayered media. Bull. Seism. Soc. Am., V. 43, NO. 1, 1953, P. 17–34.
[19]
Helbig K., Treitel S. Wave fields in real media: wave proragation in anisotropic, anelastic and porous media. Oxford, 2001.
[20]
I1an A., Ungar A., Alterman Z.S. An improved representation of boundary conditions in finite difference schemes for seismological problems. Geophys. J. Roy. Astron. Soc., V. 43, 1975, p. 727–745.
[21]
Karchevsky A.L. Finite-difference coefficient inverse problem and properties of the misfit functional. Journal of Inverse and Ill-Posed Problems, V. 6, NO. 5, 1998, p. 431-452.
[22]
Kawai, K, Takeuchi, N, Geller, R. J. Complete synthetic seismograms up to 2 Hz for transversely isotropic spherically symmetric media. Geophys. J. Int. 164, 411-424, 2006
[23]
Keith, C. M., Crampin, S., Seismic body waves in anisotropic media: propagation through a layer, Geophys. J. R. astr. Soc., 49, 200-223, 1977a.
[24]
Kennet B.L.N. Seismic waves in laterally inhomogeneous media. Geophys. J.R. astr. Soc., V. 27, NO. 3, 1972, p. 301–325.
[25]
Kennett, B.L.N. Seismic wave propagation in stratified media, Cambridge Univ. Pres., Vol. 1, Cambridge. 1983.
[26]
Kennett, B.L.N., Engdahl E. R., Traveltimes for global earthquake location and phase identification, Geophysical Journal International, 105 (2), 429-465, 1991.
[27]
Langer R.E., On the connection formulas and the solution of the wave equation. Phys. Rev., Ser., V. 2, NO. 51, 1937, p. 669-676.
[28]
Sacks P., Symes W.W. Recovery of the elastic parameters of a layered half-space. Geophysical Journal of Royal Astronomical Society, V. 88, 1987, p. 593-620.
[29]
Stephen R. A. Seismic anisotropy observed in upper oceanic crust. Geophysical Research Letters., V. 8, NO. 8, 1981, p. 865–868.
[30]
Strang G. Linear algebra and its applications. Massachusetts Institute of Technology, Academic Press, New York - San Francisco - London, 1976.
[31]
Thomson W.T. Transmission of elastic waves through a stratified solid medium. J. appl. Phys., V. 21, 1950, p. 89-93.
[32]
Ursin B. Review of elastic and electromagnetic wave propagation in horizontally layered media. Geophysics, V. 48, 1983, p. 1063-1081.
[33]
Wang R.A. simple orthonormalization method for stable and efficient computation of Green’s functions. Bull, seism. Soc. Am., V. 89, 1999, p. 733-741.
[34]
Woodhouse J.H. Asymptotic results for elastodynamic propagator matrices in plane stratified and spherically stratified Earth models. Geo¬phys. J. R. astr. Soc., V. 54, 1978. p. 263-280.
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