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A review of numerical experiments on seismic wave scattering

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Abstract

This paper reviews applications of the finite-difference and finite-element methods to the study of seismic wave scattering in both simple and complex velocity models. These numerical simulations have improved our understanding of seismic scattering in portions of the earth where there is significant lateral heterogeneity, such as the crust. The methods propagate complete seismic wavefields through highly complex media and include multiply scattered waves and converted phases (e.g.,P toSV, SV toP, body wave to surface wave). The numerical methods have been especially useful in cases of moderate and strong scattering in complex media where multiple scattering becomes important. Progress has been made with numerical methods in understanding how near-surface, low-velocity basin structures scatter surface waves and vertically-incident body waves. The numerical methods have proven useful in evaluating scattering of surface waves and body waves from topography of both the free surface and interfaces buried at depth. Numerical studies have demonstrated the importance of conversions from body waves to surface waves (andvice versa) when lateral heterogeneities and topographic relief are present in the uppermost crust. Recently, several investigations have applied numerical methods to study seismic wave propagation in velocity models which vary randomly in space. This stochastic approach seeks to understand the effects of small-scale complexity in the earth which cannot be resolved deterministically. These experiments have quantified the relationships between the statistical properties of the random heterogeneity and the measurable properties of high-frequency (≥1 Hz) seismograms. These simulations have been applied to the study of many features observed in actual high-frequency seismic waves, including: the amplitude and time decay of seismic coda, the apparent attenuation from scattering, the dispersion of waveforms, and the travel time and waveform variations across arrays of receivers.

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References

  • Aki, K. (1969),Analysis of Seismic Coda of Local Earthquakes as Scattered Waves, J. Geophys. Res.74, 615–631.

    Google Scholar 

  • Aki, K. (1973),Scattering of P Waves under the Montana LASA, J. Geophys. Res.78, 1334–1346.

    Google Scholar 

  • Aki, K. (1980),Scattering and Attenuation of Shear Waves in the Lithosphere, J. Geophys. Res.85, 6496–6504.

    Google Scholar 

  • Aki, K., andChouet, B. (1975),Origin of Coda Waves: Source, Attenuation and Scattering Effects, J. Geophys. Res.80, 3322–3342.

    Google Scholar 

  • Aki, K., andLarner, 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.

    Google Scholar 

  • Alford, R. M., Kelly, K. R., andBoore, D. M. (1974),Accuracy of Finite-difference Modeling of the Acoustic Wave Equation, Geophysics39, 834–842.

    Google Scholar 

  • Alterman, Z. S., andKaral, F. C., Jr. (1968),Propagation of Elastic Waves in Layered Media by Finite-difference Methods, Bull. Seismol. Soc. Am.58, 367–398.

    Google Scholar 

  • Alterman, Z. S., andRotenberg, A. (1969),Seismic Waves in a Quarter Plane, Bull. Seismol. Soc. Am.59, 347–368.

    Google Scholar 

  • Alterman, Z. S., andLoewenthal, D. (1970),Seismic Waves in a Quarter and Three Quarter Plane, Geophys. J.,20, 101–126.

    Google Scholar 

  • Alterman, Z. S., andLoewenthal, D. (1971),Corner Generated Surface Pulses, Geophys. J.25, 285–298.

    Google Scholar 

  • Alterman, Z. S., andNathaniel, R. (1975),Seismic Waves in a Wedge, Bull. Seismol. Soc. Am.65, 1697–1719.

    Google Scholar 

  • Andrews, D. J. (1989),Can Earthquake Source Spectra be Inferred from Coda Spectra?, submitted to Bull. Seismol. Soc. Am.

  • Bache, T. C., Marshall, P. D., andBache, L. B. (1985),Q for Teleseismic P-waves from Central Asia, J. Geophys. Res.90, 3575–3587.

    Google Scholar 

  • Bayliss, A., Jordan, K. E., LeMesurier, B. J., andTurkel, E. (1986),A Fourth Order Accurate Finite-difference Scheme for the Computation of Elastic Waves, Bull. Seismol. Soc. Am.76, 1115–1132.

    Google Scholar 

  • Bolt, B. A., andSmith, W. D. (1976),Finite Element Computation of Seismic Anomalies for Bodies of Arbitrary Shape, Geophysics,41, 145–150.

    Google Scholar 

  • Bond, L. J. (1979),A Computer Model of the Interaction of Acoustic Surface Waves with Discontinuities, Ultrasonics17, 71–77.

    Google Scholar 

  • Boore, D. M. (1970),Love Waves in Nonuniform Wave Guides: Finite Difference Calculations, J. Geophys. Res.75, 1512–1527.

    Google Scholar 

  • Boore, D. M., Larner, K. L., andAki, K. (1971),Comparison of Two Independent Methods for the Solution of Wave-scattering Problems: Response of a Sedimentary Basin to Vertically Incident SH Waves. J. Geophys. Res.76, 558–569.

    Google Scholar 

  • Boore, D. M.,Finite-difference methods for seismic wave propagation in heterogeneous materials, In vol. 11,methods in Computational Physics (ed. Bolt, B. A.) (Academic Press, New York 1972a).

    Google Scholar 

  • Boore, D. M. (1972b),A Note on the Effect of Simple Topography on Seismic SH Waves, Bull. Seismol. Soc. Am.62, 275–284.

    Google Scholar 

  • Boore, D. M. (1973),The Effect of Simple Topography on Seismic Waves: Implications for the Accelerations Recorded at Pacoima Dam, San Fernando Valley, California, Bull. Seismol. Soc. Am.63, 1603–1609.

    Google Scholar 

  • Boore, D. M., Harmsen, S. C., andHarding, S. T. (1981),Wave Scattering from a Step Change in Surface Topography, Bull. Seismol. Soc. Am.71, 117–125.

    Google Scholar 

  • Červený, V., Popov, M. M., andPšenčik, I. (1982),Computation of Wave Fields in Inhomogeneous Media—Gaussian Beam Approach, Geophys. J. R. Astr. Soc.70, 109–128.

    Google Scholar 

  • Chernov, L. A.,Wave Propagation in a Random Medium, (McGraw-Hill, New York 1960).

    Google Scholar 

  • Claerbout, J. F., andJohnson, A. G. (1971),Extrapolation of Time-dependent Waveforms along their Path of Propagation, Geophys. J. R. Astr. Soc.26, 285–293.

    Google Scholar 

  • Clayton, R. W., andEngquist, B. (1977),Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations, Bull. Seismol. Soc. Am.67, 1529–1541.

    Google Scholar 

  • Clifton, R. J. (1967),A Difference Method for Plane Problems in Elasticity, Qu. Appl. Math.25, 97–116.

    Google Scholar 

  • Cray Research, Inc. (1986),Multitasking Strategies for Large-scale Problems, Cray Channels, Spring 1986, 2–5.

  • Day, S. M., andMinster, J. B. (1984),Numerical Simulation of Attenuated Wavefields Using a Padé Approximant Method, Geophys. J. R. Astr. Soc.78, 105–118.

    Google Scholar 

  • Der, Z. A., Marshall, M. E., O'Donnell, A., andMcElfresh, T. W. (1984),Spatial Coherence Structure and Attenuation of the L g Coda, Bull. Seismol. Soc. Am.74, 1125–1147.

    Google Scholar 

  • Drake, L. A. (1972a),Love and Rayleigh Waves in Nonhorizontally Layered Media, Bull. Seismol. Soc. Am.62, 1241–1258.

    Google Scholar 

  • Drake, L. A. (1972b),Rayleigh Waves at a Continental Boundary by the Finite-element Method, Bull. Seismol. Soc. Am.62, 1259–1268.

    Google Scholar 

  • Drake, L. A. (1980),Love and Rayleigh Waves in an Irregular Soil Layer, Bull. Seismol. Soc. Am.70, 571–582.

    Google Scholar 

  • Drake, L. A., andMal, A. K. (1972),Love and Rayleigh Waves in the San Fernando Valley, Bull. Seismol. Soc. Am.62, 1673–1690.

    Google Scholar 

  • Drake, L. A., andBolt, B. A. (1980),Love Waves Normally Incident at a Continental Boundary, Bull. Seismol. Soc. Am.70, 1103–1123.

    Google Scholar 

  • Emerman, S., andStephen, R. (1983),Comment on “Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations” by R. Clayton and B. Engquist, Bull. Seismol. Soc. Am.73, 661–665.

    Google Scholar 

  • Emerman, S. H., Schmidt, W., andStephen, R. A. (1982),An Implicit Finite-difference Formulation of the Elastic Wave Equation, Geophysics,47, 1521–1526.

    Google Scholar 

  • Fehler, M., andAki, K. (1978),Numerical Study of Diffraction of Plane Elastic Waves by a Finite Crack with Application to Location of a Magma Lens, Bull. Seismol. Soc. Am.68, 573–598.

    Google Scholar 

  • Flatté, S. M., andTappert, F. D. (1975),Calculation of the Effect of Internal Waves on Oceanic Sound Transmission, J. Acoust. Soc. Am.58, 1151–1159.

    Google Scholar 

  • Fornberg, B. (1987),The Pseudospectral Method: Comparison with Finite Differences for the Elastic Wave Equation, Geophysics52, 483–501.

    Google Scholar 

  • Frankel, A., andClayton, R. W. (1984),A Finite-difference Simulation of Wave Propagation Through Random Media, Bull. Seismol. Soc. Am.74, 2167–2186.

    Google Scholar 

  • Frankel, A., andClayton, R. W. (1986),Finite-difference Simulations of Seismic Scattering: Implications for Propagation of Short-period Seismic Waves in the Crust and Models of Crustal Heterogeneity, J. Geophys. Res.91, 6465–6489.

    Google Scholar 

  • Frankel, A., andWennerberg, L. (1987),Energy-flux Model of Seismic Coda: Separation of Scattering and Intrinsic Attenuation, Bull. Seismol. Soc. Am.77, 1223–1251.

    Google Scholar 

  • Fuyuki, M., andMatsumoto, Y. (1980),Finite-difference Analysis of Rayleigh Wave Scattering at a Trench, Bull. Seismol. Soc. Am.70, 2051–2070.

    Google Scholar 

  • Fuyuki, M., andNakano, M. (1984),Finite-difference Analysis of Rayleigh-wave Transmission Past an Upward Step Change, Bull. Seismol. Soc. Am.74, 893–911.

    Google Scholar 

  • Gao, L. S., Lee, L. C., Biswas, N. N., andAki, K. (1983),Comparison of the Effects Between Single and Multiple Scattering on Coda Waves for Local Earthquakes, Bull. Seismol. Soc. Am.73, 377–390.

    Google Scholar 

  • Gazdag, J. (1981),Modeling of the Acoustic Wave Equation with Transform Methods, Geophysics46, 854–859.

    Google Scholar 

  • Gauthier, O., Virieux, J., andTarantola, A. (1986),Two-dimensional Nonlinear Inversions of Seismic Waveforms: Numerical Results, Geophysics51, 1387–1403.

    Google Scholar 

  • Gibson, B. S., andLevander, A. R. (1988),Modeling and Processing of Scattered Waves in Seismic Reflection Surveys, Geophysics53, 466–478.

    Google Scholar 

  • Gilbert, F., andKnopoff, L. (1960),Seismic Scattering from Topographic Irregularities, J. Geophys. Res.65, 3437–3444.

    Google Scholar 

  • Harmsen, S., andHarding, S. (1981),Surface Motion over a Sedimentary Valley for Incident Plane P and SV Waves, Bull. Seismol. Soc. Am.71, 655–670.

    Google Scholar 

  • Haskell, N. A. (1960),Crustal Reflection of Plane SH Waves, J. Geophys. Res.65, 4147–4150.

    Google Scholar 

  • Helmberger, D. V., andVidale, J. E. (1988),Modeling Strong Motions Produced by Earthquakes with Two-dimensional Numerical Codes, Bull. Seismol. Soc. Am.78, 109–121.

    Google Scholar 

  • Hill, N. R., andLevander, A. R. (1984),Resonances of Low-velocity Layers with Lateral Variations, Bull. Seismol. Soc. Am.74, 521–537.

    Google Scholar 

  • Ilan, A., Ungar, andAlterman, Z. S. (1975),An Improved Representation of Boundary Conditions in Finite-difference Schemes for Seismological Problems, Geophys. J.43, 727–742.

    Google Scholar 

  • Ilan, A., andLoewenthal, D. (1976),Instability of Finite-difference Schemes due to Boundary Conditions in Elastic Media, Geophys. Prospecting24, 431–453.

    Google Scholar 

  • Ilan, A. (1977),Finite-difference Modeling for P-pulse Propagation in Elastic Media with Arbitrary Polygonal Surface, J. Geophys.43, 41–58.

    Google Scholar 

  • Ilan, A., Bond, L. J., andSpivack, M. (1979),Interaction of a Compressional Impulse with a Slot Normal to the Surface of an Elastic Half Space, Geophys. J.57, 463–477.

    Google Scholar 

  • Joyner, W. B., Warrick, R. E., andFumal, T. E. (1981),The Effect of Quaternary Alluvium on Strong Ground Motion in the Coyote Lake, California Earthquake of 1979, Bull. Seismol. Soc. Am.71, 1333–1349.

    Google Scholar 

  • Kafka, A. L., andDollin, M. F. (1985),Constraints on Lateral Variation in Upper Crustal Structure beneath Southern New England from Dispersion of R g Waves, Geophys. Res. Lett.12, 235–238.

    Google Scholar 

  • Karal, F. C., andKeller, J. B. (1959),Elastic Wave Propagation in Homogeneous and Inhomogeneous Media, J. Acoust. Soc. Am.31, 694–705.

    Google Scholar 

  • Kelly, F. C., Ward, R. W., Treitel, S., andAlford, R. M. (1976),Synthetic Seismograms: A Finite-difference Approach, Geophysics,41, 2–27.

    Google Scholar 

  • Kennett, B. L. N. (1972),Seismic Waves in Laterally Inhomogeneous Media, Geophys. J. R. Astr. Soc.27, 301–325.

    Google Scholar 

  • Key, F. A. (1968),Some Observations and Analyses of Signal Generated Noise, Geophys. J. R. Astr. Soc.15, 377–392.

    Google Scholar 

  • Knepp, D. (1983),Multiple Phase-screen Calculation of the Temporal Behavior of Stochastic Waves, Proc. IEEE71, 722–737.

    Google Scholar 

  • Kosloff, D. D., andBaysal, E. (1982),Forward Modeling by a Fourier Method, Geophysics47, 1402–1412.

    Google Scholar 

  • Kosloff, D. D., Reshef, M., andLoewenthal, D. (1984),Elastic Wave Calculations by the Fourier Method, Bull. Seismol. Soc. Am.74, 875–891.

    Google Scholar 

  • Kummer, B., andBehle, A. (1982),Second-order Finite-difference Modeling of SH-propagation in Laterally Inhomogeneous Media, Bull. Seismol. Soc. Am.72, 793–808.

    Google Scholar 

  • Levander, A. R. (1985a),Use of the Telegraphy Equation to Improve Absorbing Boundary Efficiency for Fourth-order Acoustic Wave Finite-difference Schemes, Bull. Seismol. Soc. Am.75, 1847–1852.

    Google Scholar 

  • Levander, A. R. (1985b),Finite-difference Calculations of Dispersive Rayleigh Wave Propagation, Tectonophysics113, 1–30.

    Google Scholar 

  • Levander, A. R., andHill, N. R. (1985),P-SV Resonances in Irregular Low-velocity Surface Layers, Bull. Seismol. Soc. Am.75, 847–864.

    Google Scholar 

  • Lysmer, J., andDrake, L. A. (1971),The Propagation of Love Waves Across Nonhorizontally Layered Structures, Bull. Seismol. Soc. Am.61, 1233–1252.

    Google Scholar 

  • Lysmer, J., andDrake, L. A.,A finite-element method for seismology, In vol. 11,Methods in Computational Physics (ed. Bolt, B. A.) (Academic Press, New York 1972).

    Google Scholar 

  • Lysmer, J., andKuhlemeyer, R. L. (1969),Finite Dynamic Model for Infinite Media, J. of the Engineering Div., ASCE101, 859–877.

    Google Scholar 

  • Macaskill, C., andEwart, T. E. (1984),Computer Simulation of Two-dimensional Random Wave Propagation, IMA J. Appl. Math.33, 1–15.

    Google Scholar 

  • Madariaga, R. (1976),Dynamics of an Expanding Circular Fault, Bull. Seismol. Soc. Am.66, 639–666.

    Google Scholar 

  • Mal, A. K., andKnopoff, L. (1965),Transmission of Rayleigh Waves Past a Step Change in Elevation, Bull. Seismol. Soc. Am.55, 319–334.

    Google Scholar 

  • Malin, P. E., andPhinney, R. A. (1985),On the Relative Scattering of P and S Waves, Geophys. J. R. Astr. Soc.80, 603–618.

    Google Scholar 

  • Marfurt, K. J. (1984),Accuracy of Finite-difference and Finite-element Modeling of the Scalar and Elastic Equations, Geophysics,49, 533–549.

    Google Scholar 

  • Martel, L. (1980),Love Wave Propagation Across a Step by Finite Elements and Spatial Filtering, Geophys. J. R. Astr. Soc.61, 659–677.

    Google Scholar 

  • May, T. W., andBolt, B. A. (1982),The Effectiveness of Trenches on Reducing Seismic Motion, Earthquake Eng. and Struct. Dyn.10, 195–210.

    Google Scholar 

  • McCowan, D. W., Glover, P., andAlexander, S. S. (1977),A Static and Dynamic Finite-element Analysis of the 1971 San Fernando, California, Earthquake, Geophys. J. R. Astr. Soc.48, 163–185.

    Google Scholar 

  • McLaughlin, K. L., Anderson, L. M., andDer, Z. A.,Investigation of seismic waves using 2-dimensional finite difference calculations, InMultiple Scattering of Waves in Random Media and Random Surfaces (The Pennsylavania State University 1985) pp. 795–821.

  • McLaughlin, K. L., andAnderson, L. M. (1987),Stochastic Dispersion of Short-period P-Waves due to Scattering and Multipathing, Geophys. J. R. Astr. Soc.89, 933–964.

    Google Scholar 

  • McLaughlin, K. L., Anderson, L. M., andLees, L. C. (1987),Effects of Local Geologic Structure on Yucca Flats, NTS, Explosion Waveforms: 2-dimensional Linear Finite-differnce Simulations, Bull. Seismol. Soc. Am.77, 1211–1222.

    Google Scholar 

  • McLaughlin, K. L., andJih, R. S. (1988),Scattering from Near-source Topography: Teleseismic Observations and Numerical Simulations, Bull. Seismol. Soc. Am.78, 1399–1414.

    Google Scholar 

  • McLaughlin, K. L., andJih, R. S. (1989),Finite-difference Simulations of Rayleigh Wave Scattering by 2-D Rough Topography, submitted to Bull. Seismol. Soc. Am.

  • Menke, W. (1984),Asymptotic Formulas for the Apparent Q of Weakly Scattering Three-dimensional Media, Bull. Seismol. Soc. Am.74, 1079–1082.

    Google Scholar 

  • Mereu, R. F., andOjo, S. B. (1981),The Scattering of Seismic Waves Through a Crust and Upper Mantle with Random Lateral and Vertical Inhomogeneities, Phys. Earth Planet. Inter.26, 233–240.

    Google Scholar 

  • Mitchell, A. R.,Computational Methods in Partial Differential Equations (John Wiley and Sons, New York 1969).

    Google Scholar 

  • Munasinghe, M., andFarnell, G. W. (1973),Finite-difference Analysis of Rayleigh Wave Scattering at Vertical Discontinuities, J. Geophys. Res.78, 2454–2466.

    Google Scholar 

  • Ohtsuki, A., andHarumi, K. (1983),Effect of Topographies and Subsurface Inhomogeneities on Seismic SV Waves, Earthquake Eng. Struct. Dyn.11, 444–461.

    Google Scholar 

  • Ohtsuki, A., Yamahara, H., andHarumi, K. (1984a),Effect of Topography and Subsurface Inhomogeneity on Seismic Rayleigh Waves, Earthquake Eng. Struct. Dyn.12, 37–58.

    Google Scholar 

  • Ohtsuki, A., Yamahara, H., andTazoh, T. (1984),Effect of Lateral Inhomogeneity on Seismic Waves, II. Observations and Analyses, Earthquake Eng. and Struct. Dyn.12, 795–816.

    Google Scholar 

  • Ottaviani, M. (1971),Elastic Wave Propagation in Two Evenly-welded Quarter Spaces, Bull. Seismol. Soc. Am.61, 1119–1152.

    Google Scholar 

  • Phillips, W. S., andAki, K. (1986),Site Amplifications of Coda Waves from Local Earthquakes, Bull. Seismol. Soc. Am.76, 627–648.

    Google Scholar 

  • Richards, P. G., andMenke, W. (1983),The Apparent Attenuation of a Scattering Medium, Bull. Seismol. Soc. Am.73, 1005–1022.

    Google Scholar 

  • Sato, H. (1977),Energy Propagation Including Scattering Effects, Single Isotropic Scattering Approximation, J. Phys. Earth25, 27–41.

    Google Scholar 

  • Sato, H. (1982),Attenuation of S Waves in the Lithosphere due to Scattering by its Random Velocity Structure, J. Geophys. Res.87, 7779–7785.

    Google Scholar 

  • Sato, H. (1988),Temporal Change in Scattering and Attenuation Associated with the Earthquake Occurrence — A Review of Recent Studies of Coda Waves, Pure Appl. Geophys.126, 465–498.

    Google Scholar 

  • Satô, Y. (1972),A Numerical Experiment on Wave Propagation in an Elastic Quarter Space, J. Phys. Earth20, 287–299.

    Google Scholar 

  • Schlue, J. W. (1979),Finite-element Matrices for Seismic Surface Waves in Three-dimensional Structures, Bull. Seismol. Soc. Am.69, 1425–1437.

    Google Scholar 

  • Schlue, J. W. (1979),Love-wave Propagation in Three-dimensional Structures Using Finite-element Techniques, Bull. Seismol. Soc. Am.69, 319–328.

    Google Scholar 

  • Schlue, J. W. (1981),Seismic Surface Wave Propagation in Three-dimensional Finite-element Structures, Bull. Seismol. Soc. Am.71, 1003–1010.

    Google Scholar 

  • Schlue, J. W., andHostettler, K. K. (1987),Rayleigh Wave Phase Velocity and Amplitude Values in the Presence of Lateral Heterogeneities, Bull. Seismol. Soc. Am.77, 244–255.

    Google Scholar 

  • Scott, P., andHelmberger, D. V. (1983),Applications of the Kirchhoff-Helmholtz Integral to Problems in Seismology, Geophys. J. R. Astr. Soc.72, 237–254.

    Google Scholar 

  • Smith, W. D. (1975),The Application of Finite-element Analysis to Body Wave Propagation Problems, Geophys. J. R. Astr. Soc.42, 747–768.

    Google Scholar 

  • Stead, R. J., andHelmberger, D. V. (1988),Numerical-analytical Interfacing in Two Dimensions with Applications in Modeling NTS Seismograms, Pure Appl. Geophys.128, 157–193.

    Google Scholar 

  • Stephen, R. A. (1983),A Comparison of Finite-difference and Reflectivity Seismograms for Marine Models, Geophys. J. R. Astr. Soc.72, 39–58.

    Google Scholar 

  • Stephen, R. A. (1984),Finite-difference Seismograms for Laterally Varying Marine Models, Geophys. J. R. Astr. Soc.79, 185–198.

    Google Scholar 

  • Stephen, R. A., Cardo-Casas, F., andCheng, C. H. (1985),Finite-difference Synthetic Acoustic Logs, Geophysics50, 1588–1609.

    Google Scholar 

  • Suteau, A., andMartel, L. (1980),Surface Waves in Structures with Locally Irregular Boundaries, Bull. Seismol. Soc. Am.70, 791–808.

    Google Scholar 

  • Szelwis, R. (1983),A Hybrid Approach to Mode Conversion, Geophys. J. R. Astr. Soc.74, 887–904.

    Google Scholar 

  • Szelwis, R. (1984),Rayleigh Wave Scattering at a Basin Type Heterogeneity, J. Geophys. Res.89, 11,476–11,494.

    Google Scholar 

  • Tarantola, A. (1984),Linearized Inversion of Seismic Data, Geophys. Prospecting32, 998–1015.

    Google Scholar 

  • Thurber, C. H. (1983),Earthquake Locations and Three-dimensional Structure in the Coyote Lake Area, Central California, J. Geophys. Res.88, 8226–8236.

    Google Scholar 

  • Trifunac, M. D. (1971),Surface Motion of a Semi-cylindrical Alluvial Valley for Incident Plane SH Waves, Bull. Seismol. Soc. Am.61, 1755–1770.

    Google Scholar 

  • Vidale, J., Helmberger, D. V., andClayton, R. W. (1985),Finite-difference Seismograms for SH Waves, Bull. Seismol. Soc. Am.75, 1765–1782.

    Google Scholar 

  • Vidale, J. E., andClayton, R. W. (1986),A Stable Free-surface Boundary Condition for 2-D Elastic Finite-difference Wave Simulation, Geophysics51, 2247–2249.

    Google Scholar 

  • Vidale, J. E., andHelmberger, D. V. (1988),Elastic Finite-difference Modeling of the 1971 San Fernando, California Earthquake, Bull. Seismol. Soc. Am.78, 122–141.

    Google Scholar 

  • Virieux, J. (1984),SH-propagation in Heterogeneous Media, Velocity-stress Finite-difference Method, Geophysics49, 1933–1955.

    Google Scholar 

  • Virieux, J. (1986),PSV-wave Propagation in Heterogeneous Media, Velocity-stress Finite-difference Method, Geophysics51, 889–901.

    Google Scholar 

  • Wu, R.-S. (1982),Attenuation of Short-period Seismic Waves due to Scattering, Geophys. Res. Lett.9, 9–12.

    Google Scholar 

  • Wu, R.-S. (1985),Multiple Scattering and Energy Transfer of Seismic Waves—Separation of Scattering Effect from Intrinsic Attenuation—I. Theoretical Modeling, Geophys. J. R. Astr. Soc.82, 57–80.

    Google Scholar 

  • Wu, R.-S., andAki, K. (1985a),Scattering Characteristics of Elastic Waves by an Elastic Heterogeneity, Geophysics50, 582–595.

    Google Scholar 

  • Wu, R.-S., andAki, K. (1985b),Elastic Wave Scattering by a Random Medium and the Small-scale Inhomogeneities in the Lithosphere, J. Geophys. Res.90, 10,261–10,273.

    Google Scholar 

  • Zienkiewicz, O. C.,The Finite Element Method in Engineering Science (McGraw-Hill, London 1971.

    Google Scholar 

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    Arthur Frankel

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Frankel, A. A review of numerical experiments on seismic wave scattering. PAGEOPH 131, 639–685 (1989). https://doi.org/10.1007/BF00876267

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