Abstract
This chapter deals with a method of computing wave fields in inhomogeneous media. It is based on a hybrid formulation — in the frequency domain — in terms of finite elements and a boundary integral representation. Numerical aspects of the method are discussed, and computational results for a simple model configuration are compared with analytical results. Finally the method is applied in a seismic modeling experiment. In this experiment the effect of a low velocity sediment fill at the surface of a halfspace on an incident teleseismic wave is modelled. The results clearly illustrate the strong amplification effect of the sediments on the surface displacements.
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References
Abramowitz, M., and Stegun, L, 1964. Handbook of mathematical functions, New York, Dover.
Aki, K. and Lamer, K. L., 1970. Surface Motion of a Layered Medium Having an Irregular Interface due to Incident Plane SH Waves, J. Geophys. Res., 75, 933–954.
Aki, K., and Richards, P. G., 1980. Quantitative Seismology, San Francisco, W.H. Freeman.
Alford, R.M., Kelly, K.R., and Boore, D.M., 1974. Accuracy of finite-difference modelling of the acoustic wave equation, Geophysics, 39, 834–842.
Alterman, Z.S., and Karal, F.C., 1968. Propagation of elastic waves in layered media by finite difference methods, Bull. Seism. Soc. Am., 58, 367–398.
Banaugh, R. P., and Goldsmith, N., 1963. Diffraction of steady acoustic waves by surfaces of arbitrary shape, J. Ac. Soc. Am., 35.
Bard, P.Y. and Bouchon, M., 1980. The seismic response of sediment-filled valleys. Part 1. The case of incident SH waves, Bull. Seism. Soc. Am., vol.70, 1263–1286.
Belytschko, T., and Mullen, R., 1978, On dispersive properties of finite element solutions. In, Modern Problems in Elastic Wave Propagation, eds Julius Miklowitz, and Jan D. Achenbach. Wiley.
Ben-Menahem, A., and Singh, S. J., 1981. Seismic waves and sources, Springer Verlag.
Berkhoff, J.C.W., 1974. Linear wave propagation problems and the finite element method. In Finite elements in fluids 1, Viscous flow and hydrodynamics, proceedings “International Symp. on the f.e.m. in flow problems”, Univ. College of Wales, eds. Oden, J.T., Taylor, C., Gallagher, R.H. and Zienkiewicz, O.C.
Berkhoff, J.C.W., 1976. Mathematical models for simple harmonic linear water waves wave diffraction and refraction, PhD Thesis TH-Delft.
Bolomey, J.-C. and Tabbara, W., 1973. Numerical aspects on coupling between complementary boundary value problems., IEEE Transactions on Antennas and propagation, AP-21, 356–363.
Boore, D. M., 1972. Finite difference methods for seismic wave propagation in heterogeneous materials. In Seismology: Surface waves and earth oscillations, ed B.A. Bolt, 11, Academic press.
Boore, D. M., Lamer, K., and Aki, K., 1971. Comparison of two independent methods for the solution of wave scattering problems: Respons of a sedimentary basin to vertically incident SH waves, J. Geophys. Res., 76, 558–569.
Bouchon, M. and Aki, K., 1977. Near-field of a seismic source in a layered medium with irregular interfaces, Geophys. J. R. Astr. Soc, 50, 669–684.
Burton, A.J., and Miller, G.F., 1971. The application of integral equation methods to the numerical solution of some exterior boundary value problems., Proc. Roy. Soc. London, Ser. A, 323, 201–210.
Chin, R.C.Y., Hedstrom, G. and Thigpen, L., 1984. Numerical Methods in Seismology, J. Comp. Phys., 54, 18–56.
Cooley, J.W., and Tukey, S.W., 1965. An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19, 297–301.
Copley, L.G., 1968. Fundamental results concerning integral representations in acoustic radiation, J. Ac. Soc. Am., 44, 28–32.
Courant, R., and Hilbert, D., 1968. Methoden der Mathematischen Physik, Berlin, Springer-Verlag.
Crichlow, J. M., Beckeis, D., and Aspinall, W.P., 1984. Two-dimensional analysis of the effect of subsurface anomalies on the free surface respons to incident SH-waves, Geophys. J. R. Astr. Soc, 79.
Colton, D. and Kress, R., 1983. Integral Equation Methods in Scattering Theory, New York, John Wiley. 669–684.
Colton, D. and Kress, R., 1983. The unique solvability of the null field equations, Q. Jl Mech. Appl. Math., 36, 87–95.
Day, S.M., 1977. Finite element analysis of seismic scattering problems, PhD Thesis Univ. Cal., San Diego. Desai, CS., and Abel, J. F., 1972. Introduction to the Finite Element Method, New York, van Nostrand Reinhold.
Dongarra, J.J., Moler, C.B., Bunch, J.R., and Stewart, G.W., 1979. LINPACK User’s Guide, Philadelphia, SIAM.
Eringen, A. C, and Suhubi, E. S., 1975. Elastodynamics, New York, Academic press.
Hoenl, H., Maue, A.W., and Westfahl, K., 1961. Theorie der Beugung. In Handbuch der Physik, ed S. Fluegge, 25, 218–583, Springer.
Hong, M., and Bond, L.J., 1986. Application of the finite difference method in seismic source and wave diffraction simulation, Geophys. J. R. Astr. Soc, 87.
De Hoop. A.T., 1958. Representation theorems for the displacement in an elastic solid and their application to elastodynamic diffraction theory, PhD Thesis Techn. Univ. Delft.
Kelly, K. R., 1983. Numerical study of Love wave propagation, Geophysics, 48, 833–853.
Kelly, K. R., Ward, R. W., Treitel, S., and Alford, R. M., 1976. Synthetic seismograms: a finite difference approach, Geophysics, 41, 2–27.
Kennett, B.L.N., 1983. Seismic wave propagation in stratified media, Cambridge university press.
Kuepper, F.J., 1958. Theoretische Untersuchungen ueber die Mehrfachaufstellung von Geophonen, Geophys. Prosp., 6, 194–256.
Kupradze, V. D., 1956. Randwertaufgaben der Schwingungstheorie und Integralgleichungen, Berlin, VEB Deutscher Verlag der Wissenschaften.
Kupradze, V. D., 1963. Dynamical problems in elasticity. In Progress in solid mechanics, eds I.N. Sneddon and R. Hill, Vol. 3.
Levander, A.R., and Hill, N.R., 1985. P-SV resonances in irregular low velocity surfaces, Bull. Seism. Soc. Am., 75, 847–864.
Lysmer, J., and Drake, L. A., 1971. The propagation of Love waves across nonhorizontally layered structures, Bull. Seism. Soc. Am., 61, 1233–1251.
Marfurt, K. J., 1984. Accuracy of finite difference and finite element modeling of the scalar and elastic wave equations, Geophysics, 49, 533–549.
Mitchell, A. R., 1969. Computational methods in partial differential equations, New York, Wiley.
Olsen, K., and Hwang, L. S., 1971. Oscillations in a bay of arbritrary shape and variable depth, J. Geophys.Res., 76, 5048–5064.
Ralston, A., 1965. A first course in numerical analysis, New York, McGraw-Hill.
Reynolds, A. C, 1978. Boundary conditions for the numerical solution of wave propagation problems, Geophysics, 43, 1099–1111.
Sanchez-Sesma, F. J., 1983. Diffraction of elastic waves by three dimensional surface irregularities, Bull. Seism. Soc. Am., 73, 1621–1636.
Sanchez-Sesma, F. J., Herrera, I., and Aviles, J., 1982. A boundary method for elastic wave diffraction: application to scattering of SH waves by surface irregularities, Bull. Seism. Soc. Am., 72, 473–490.
Schenck, H. A., 1968. Improved integral formulation for acoustic radiation problems, J. Ac. Soc. Am., 44, 41–58.
Schuster, G.Th., 1984. Some boundary integral equation methods and their application to seismic exploration, PhD Thesis, Univ. Columbia.
Segal, A., 1980. AFEP user manual,, Dept. of Math. Techn. University Delft.
Sharma, D. L., 1967. Scattering of steady elastic waves by surfaces of arbitrary shape, Bull. Seism. Soc. Am., 57, 795–812.
Smith, W. D., 1974. A non reflecting plane boundary for wave propagation problems, J. Comp. Phys., 15, 492–503.
Smith, W. D., 1975. The application of finite element analysis to body wave propagation problems, Geophys. J. R. Astr. Soc, 42, 747–768.
Stoer, J., 1972. Einfuehrung in die numerische Mathematik I, Berlin, Springer-Verlag.
Strang, G., and Fix, G.J., 1973. An analysis of the finite element method, Prentice-Hall.
Tan, T. H., 1975. Scattering of elastic waves by elastically transparant obstacles, Appl. Sci. Res., 31, 29–51.
Tan, T. H., 1975. Diffraction theory for time harmonic elastic waves, PhD Thesis, Techn. Univ. Delft.
Tobocman, W., 1984. Calculation of acoustic wave scattering by means of the Helmholtz integral equation, J. Ac. Soc. Am., 76, 599–607.
Van den Berg, A. P., 1984. A hybrid solution of wave propagation problems in regular media with bounded irregular inclusions, Geophys. J. R. Astr. Soc, 79, 3–10.
Van den Berg, A. P., 1987. A hybrid method for the solution of seismic wave propagation problems, PhD Thesis Univ. Utrecht.
Wilkinson, J. H., 1965. The algebraic eigenvalue problem, Oxford University Press.
Wilton, D.T., 1978. Acoustic radiation and scattering from elastic structures., Int. J. Num. Meth. Eng., 13, 123–138.
Wong, H. L., and Jennings, P. C, 1975. Effects of canyon topography on strong ground motion, Bull. Seism. Soc. Am., 65, 1239–1257.
Zahradnik, J., and Urban, L., 1984. Effect of a simple mountain range on underground seismic motion, Geophys. J. R. Astr. Soc, 79, 167–183.
Zienkiewicz, O. C, 1977. The finite element method, 3rd edition, McGraw-Hill.
Zienkiewicz, O. C, Kelly, D. W., and Bettes, P., 1977. The coupling of the finite element method and boundary solution procedures, Int J. Num. Meth. Engineering, 11, 355–375.
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van den Berg, A. (1988). A hybrid solution for wave propagation problems in inhomogeneous media. In: Vlaar, N.J., Nolet, G., Wortel, M.J.R., Cloetingh, S.A.P.L. (eds) Mathematical Geophysics. Modern Approaches in Geophysics, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2857-2_5
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DOI: https://doi.org/10.1007/978-94-009-2857-2_5
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