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Unified approach to amplitude attenuation and coda excitation in the randomly inhomogeneous lithosphere

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Abstract

A unified model is proposed for explaining the frequency dependent amplitude attenuation and the coda wave excitation on the basis of the single scattering process in the randomly inhomogeneous lithosphere. Adopting Birch's law and a direct proportion between density and wave velocity, we statistically describe the inhomogeneous medium by one random function characterized by the von Karman autocorrelation function. We calculate the amplitude attenuation from the solid angle integral of scattered wave energy on the basis of the Born approxiimation after subtracting the travel-time fluctuation effect caused by slowly varying velocity inhomogeneities. This subtraction is equivalent to neglect energy loss by scattering within a cone around the forward direction. The random inhomogeneity of the von Karman autocorrelation function of order 0.35 with the mean square fractional fluctuation of 7.2×10−3 ≈1.3×10−2 and the correlation distance of 2.1≈5.1 km well explains observed backward scattering coefficientg π and the ratioQ −1P /Q −1S , and observed and partially conjecturedQ −1S for frequencies between 0.5 Hz and 30 Hz.

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References

  • Aki, K. (1969),Analysis of the 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. (1980a),Attenuation of Shear Waves in the Lithosphere for Frequencies from 0.05 to 25 Hz, Phys. Earth Planet. Inter.21, 50–60.

    Google Scholar 

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

    Google Scholar 

  • Aki, K. (1982a),Scattering and Attenuation, Bull. Seismol. Soc. Am.72, S319-S330.

    Google Scholar 

  • Aki, K. (1982b),Three-dimensional Seismic Inhomogeneities in the Lithosphere and Asthenosphere: Evidence for Decoupling in the Lithosphere and Flow in the Asthenosphere, Rev. Geophys. Space. Phys.20, 161–170.

    Google Scholar 

  • Aki, K. (1985),Theory of Earthquake Prediction with Special Reference to Monitoring of Quality Factor of Lithosphere by the Coda Method, Earthq. Predict. Res.3, 219–230.

    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., Christoffersson, A., andHusebye, E. S. (1976),Three-dimensional Seismic Structures of the Lithosphere under Montana LASA, Bull. Seismol. Soc. Am.66, 501–524.

    Google Scholar 

  • Aki, K., Christoffersson, A., andHusebye, E. S. (1977),Determination of the Three-dimensional Seismic Structure of the Lithosphere, J. Geophys. Res.82, 277–296.

    Google Scholar 

  • Aki, K., andRichards, P. G.:Quantitative Seismology (W. H. Freeman Co., San Francisco 1980) pp. 1–932.

    Google Scholar 

  • Anderson, D. L. andHart, R. S. (1978),Attenuation Model of the Earth, Phys. Earth Planet. Inter.16, 289–306.

    Google Scholar 

  • 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 

  • Bakun, W. H., Bufe, C. G., andStewart, R. M. (1976),Body Wave Spectra of Central California Earthquake, Bull. Seismol. Soc. Am.61, 55–64.

    Google Scholar 

  • Batchelor, G. K.,The Theory of Homogeneous Turbulence (Cambridge Univ., London 1953), pp. 14–33.

    Google Scholar 

  • Beaudet, P. R. (1970),Elastic Wave Propagation in Heterogeneous Media, Bull. Seismol. Soc. Am.60, 769–784.

    Google Scholar 

  • Berteussen, K. A., Christoffersson, A., Husebye, E. S., andDahle, A. (1975),Wave Scattering Theory in Analysis of P Wave Anomalies at NORSAR and LASA, Geophys. J. R. Astr. Soc.42, 403–417.

    Google Scholar 

  • Birch, F. (1961),The Velocity of Compressional Waves in Rocks to 10 kb: Part 2 J. Geophys. Res.66, 2199–2224.

    Google Scholar 

  • Biswas, N. N., andAki, K. (1984),Characteristics of Coda Waves: Central and Southcentral Alaska, Bull. Seismol. Soc. Am.74, 493–507.

    Google Scholar 

  • Butler, R., McCreery, C. S., Frazer, L. N. andWaker, D. A. (1987),High-frequency Seismic Attenuation of Oceanic P and S Waves in the Western Pacific, J. Geophys. Res.92, 1383–1396.

    Google Scholar 

  • Capon, J. (1974),Characterization of Crust and Upper Mantle Structure under LASA as a Random Medium, Bull. Seismol. Soc. Am.64, 235–266.

    Google Scholar 

  • Cheng, C. C., andMitchell, B. J. (1981).Crustal Q Structure in the United States from Multi-mode Surface Waves, Bull. Seismol. Soc. Am.71, 161–181.

    Google Scholar 

  • Chernov, L. A.,Wave Propagation in a Random Medium (McGraw-Hill, New York 1960) pp. 35–57 and 84–124.

    Google Scholar 

  • Christensen, N. I. (1968),Chemical Changes Associated with Upper Mantle Structure, Tectonophys.6, 331–342.

    Google Scholar 

  • Console, R., andRovelli, R. (1981),Attenuation Parameters for Friuli Region from Strong Motion Accelerogram Spectra, Bull. Seismol. Soc. Am.71, 1981–1991.

    Google Scholar 

  • Dainty, A. M. (1981),A Scattering Model to Explain Seismic Q Observations in the Lithosphere between 1 and 30 Hz, Geophys. Res. Lett.8, 1126–1128.

    Google Scholar 

  • Dainty, A. M. (1984),High Frequency Acoustic Backscattering and Seismic Attenuation, J. Geophys. Res.89, 3172–3176.

    Google Scholar 

  • Dainty, A. M., andToksöz, M. N. (1981), Seismic Codas on the Earth and the Moon: A Comparison, Phys. Earth Planet. Inter.,26, 250–260.

    Google Scholar 

  • Del Pezzo, E., andZollo, A. (1984),Attenuation of Coda Waves and Turbidity Coefficient in Central Italy, Bull. Seismol. Soc. Am.74, 2655–2659.

    Google Scholar 

  • Fedotov, S. A., andBoldyrev, S. A. (1969),Frequency Dependence of the Body Wave Absorption in the Crust and the Upper Mantle of the Kuril-Island Chain (English Trans.), Izv. Akad. Nauk SSSR, Fiz. Zemli9, 17–33.

    Google Scholar 

  • Frankel, A. (1982),The Effects of Attenuation and Site Response on the Spectra of Microearthquakes in the Northeastern Caribbean, Bull. Seismol. Soc. Am.72, 1379–1402.

    Google Scholar 

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

    Google Scholar 

  • Frisch, U.,Wave Propagation in random media, InProbabilistic Method in Applied Mathematics 1 (ed. Bharucha-Reid, A. T.), (Academic, New York 1968) pp. 76–198.

    Google Scholar 

  • Gao, L. S., Biswas, N. N., Lee, L. C., andAki, K. (1983),Effects of Multiple Scattering on Coda Waves in Three-dimensional Medium, Pure Appl. Geophys.121, 3–15.

    Google Scholar 

  • Herraiz, M., andEspinosa, A. F. (1987),Coda Waves: A Review, Pure Appl. Geophys.125, 499–577.

    Google Scholar 

  • Herrmann, R. B. (1980),Q Estimates Using the Coda of Local Earthquakes, Bull. Seismol. Soc. Am.70, 447–468.

    Google Scholar 

  • Hoang-Trong, P. (1983),Some Medium Properties of the Hohenzollern Graben (Swabian Jura, W. Germany) Inferred from QP/Qs Analysis, Phys. Earth Planet. Inter.31, 119–131.

    Google Scholar 

  • Horie, A., andAki, K. (1982),Three-dimensional Velocity Structure Beneath the Kanto District, Japan, J. Phys. Earth30, 255–281.

    Google Scholar 

  • Howe, M. S. (1971),Wave Propagation in Random Media, J. Fluid Mech.45, 769–783.

    Google Scholar 

  • Hudson, J. A., andHeritage, J. R. (1981),The Use of the Born Approximation in Seismic Scattering Problems, Geophys. J. R. Astr. Soc.66, 221–240.

    Google Scholar 

  • Husebye, E. S., Christoffersson, A., Aki, K., andPowell, C. (1976),Preliminary Results on the Three-dimensional Seismic Structure under the USGS Central California Seismic Array, Geophys. J. R. Astr. Soc.46, 319–340.

    Google Scholar 

  • Jackson, D. D., andAnderson, D. L. (1970),Physical Mechanisms of Seismic Wave Attenuation, Rev. Geophys. Space Phys.8, 1–63.

    Google Scholar 

  • Jin, A., Cao, T., andAki, K. (1985),Regional Change of Coda Q in the Oceanic Lithosphere, Geophys. Res.90, 8651–8659.

    Google Scholar 

  • Kanamori, H., andMizutani, H. (1965),Ultrasonic Measurements of Elastic Constants of Rocks under High Pressures, Bull. Earthq. Res. Inst. Univ. Tokyo43, 173–194.

    Google Scholar 

  • Karal, F. C., andKeller, J. B. (1964),Elastic, Electromagnetic and Other Waves in a Random Medium, J. Math. Phys.5, 537–547.

    Google Scholar 

  • Kikuchi, M. (1981),Dispersion and Attenuation of Elastic Waves due to Multiple Scattering from Cracks, Phys. Earth Planet. Inter.27, 100–105.

    Google Scholar 

  • Knopoff, L., andHudson, J. A. (1964),Scattering of Elastic Waves by Small Inhomogeneities, J. Acoust. Soc. Am.36, 338–343.

    Google Scholar 

  • Kopnichev, Y. F. (1975),A Model of Generation of the Tail of Seismogram (English Trans.), Dokl. Akad. Nauk SSSR222, 333–335.

    Google Scholar 

  • Kurita, T. (1975),Attenuation of Shear Waves Along the San Andreas Fault Zone in Central California, Bull. Seismol. Soc. Am.65, 277–292.

    Google Scholar 

  • Liu, H., Anderson, D. L., andKanamori, H. (1976),Velocity Dispersion Due to Anelasticity: Implication for Seismology and Mantle Composition, Geophys. J. R. Astr. Soc.47, 41–58.

    Google Scholar 

  • Lundquist, G. M., andCormier, V. C. (1980),Constraints on the Absorption Band Model of Q, J. Geophys. Res.85, 5244–5256.

    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 

  • Mandelbrot, B. B.,The Fractal Geometry of Nature (W. H. Freeman Co., San Francisco 1982) pp. 84–96.

    Google Scholar 

  • Manghnani, M. H., andRamananantoandro, R. (1974),Compressional and Shear Wave Velocities in Granulite Facies Rocks and Eclogites to 10 kb, J. Geophys. Res.79, 5427–5446.

    Google Scholar 

  • Mason, W. P., Marfurt, K. J., Beshers, D. N., andKuo, J. T. (1978),Internal Friction in Rocks, J. Acoust. Soc. Am.63, 1596–1603.

    Google Scholar 

  • Mavco, G. M. (1980),Velocity and Attenuation in Partially Molten Rocks, J. Geophys. Res.85, 5173–5189.

    Google Scholar 

  • Mavco, G. M., andNur, A. (1979),Wave Attenuation in Partially Saturated Rocks, Geophys.44, 161–178.

    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 

  • Menke, W. (1984),Asymptotic Formulas for the Apparent Q of Weakly Scattering Three-dimensional Media, Bull. Seismol. Soc. Am.74, 1079–1081.

    Google Scholar 

  • Miles, J. W. (1960),Scattering of Elastic Waves by Small Inhomogeneities, Geophys.15, 642–648.

    Google Scholar 

  • Minster, J. B., andAnderson, D. L. (1980),Dislocations and Nonelastic Processes in the Mantle, J. Geophys. Res.85, 6347–6352.

    Google Scholar 

  • Modiano, T., andHatzfeld, D. (1982),Experimental Study of the Spectral Content for Shallow Earthquakes, Bull. Seismol. Soc. Am.72, 1739–1758.

    Google Scholar 

  • Nevskiy, M. V., Nikolaev, A. V., andRiznichenko, O. Yu (1982) (English Trans.) Dokl. Akad. Nauk SSSR265, 53–57.

    Google Scholar 

  • Nikolaev, A. V.,The Seismics of Heterogeneous and Turbid Media (English Trans.) (Akad. Nauk SSSR, Moscow 1975) pp. 1–140.

    Google Scholar 

  • O'Connell, R. J., andBudiansky, B. (1977),Viscoelastic Properties of Fluid Saturated Cracked Solids, J. Geophys. Res.82, 5719–5735.

    Google Scholar 

  • Pulli, J. J. (1984),Attenuation of Coda Waves in New England, Bull. Seismol. Soc. Am.74, 1149–1166.

    Google Scholar 

  • Rautian, T. G., andKhalturin, V. I. (1978),The Use of Coda for Determination of the Earthquake Source Spectrum, Bull. Seismol. Soc. Am.68, 923–948.

    Google Scholar 

  • Rautian, T. G., Khalturin, V. I., Martinov, V. G., andMolnar, P. (1978),Preliminary Analysis of the Spectral Content of P and S Waves from Local Earthquakes in the Garm, Tadjikistan Region, Bull. Seismol. Soc. Am.68, 949–971.

    Google Scholar 

  • Richards, P. G., andMenke, W. (1983),The Apparent Attenuation of a Scattering Medium, Bull. Seismol. Soc. Am.,73, 1005–1021.

    Google Scholar 

  • Rodriguez, M., Havskov, J., andSingh, S. K. (1983),Q from Coda Waves near Petatlan, Guerrero, Mexico, Bull. Seismol. Soc. Am.73, 321–326.

    Google Scholar 

  • Roecker, S. W. (1982),Velocity Structure of Pamir-Hidu Kush Region: Possible Evidence of Subducted Crust, J. Geophys. Res.87, 945–959.

    Google Scholar 

  • Roecker, S. W., Tucker, B., King, J., andHatzfeld, D. (1982),Estimation of Q in Central Asia as a Function of Frequency and Depth using the Coda of Locally Recorded Earthquakes, Bull. Seismol. Soc. Am.72, 129–149.

    Google Scholar 

  • Rovelli, A. (1982),On the Frequency Dependence of Q in Friuli from Short Period Digital Records, Bull. Seismol. Soc. Am.72, 2369–2372.

    Google Scholar 

  • Rovelli, A. (1983),Frequency Relationship for Seismic Q β of Central Southern Italy from Accelerograms for the Irpinia Earthquake (1980), Phys. Earth Planet. Inter.,32, 209–217.

    Google Scholar 

  • Rovelli, A. (1983b),Time-frequency Analysis of Seismic Excitation and Estimates of Attenuation Parameters for the Friuli (Italy), Local Earthquakes, Phys. Earth Planet. Inter.33, 94–110.

    Google Scholar 

  • Rovelli, A. (1984),Seismic Q of the Lithosphere for Montenegro Region (Yugoslavia): Frequency, Depth and Time Windowing Effects, Phys. Earth Planet. Inter.34, 159–172.

    Google Scholar 

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

    Google Scholar 

  • Sato, H. (1978),Mean Free Path of S Waves Under the Kanto District of Japan, J. Phys. Earth,26, 185–198.

    Google Scholar 

  • Sato, H. (1979),Wave Propagation in One-dimensional Inhomogeneous Elastic Media, J. Phys. Earth27, 455–466.

    Google Scholar 

  • Sato, H. (1982a),Amplitude Attenuation of Impulse Waves in Random Media Based on Travel Time Corrected Mean Wave Formalism, J. Acoust. Soc. Am.71, 559–564

    Google Scholar 

  • Sato, H. (1982b),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. (1984a),Attenuation and Envelope Formation of Three Component Seismograms of Small Local Earthquakes in Randomly Inhomogeneous Lithosphere, J. Geophys. Res.89, 1221–1241.

    Google Scholar 

  • Sato, H. (1984b),Scattering and Attenuation of Seismic Waves in the Lithosphere: Single Scattering Theory in a Randomly Inhomogeneous Medium (in Japanase), Rep. National. Res. Ctr. Disast. Prev.33, 1–186. [Ph.D. thesis, Dept. Geophys., Univ. of Tokyo, Japan].

    Google Scholar 

  • Sato, H. (1988a),Temporal Change in Scattering and Attenuation Associated with the Earthquake Occurrence: A Review of Recent Studies on Coda Waves, Pure Appl. Geophys.126, 465–497.

    Google Scholar 

  • Sato, H. (1988b),Is the Single Scattering Model Invalid for the Coda Excitation at Long Lapse Time? Pure Appl. Geophys.128, 43–47.

    Google Scholar 

  • Sato, H., andMatsumura, S. (1980),Q −1 Values for S Waves (2∼32z) under the Kanto District (in Japanese), Zisin33, 541–543.

    Google Scholar 

  • Savage, J. C. (1966),Thermoelastic Attenuation of Elastic Waves by Cracks, J. Geophys. Res.71, 3929–3938.

    Google Scholar 

  • Schilt, S., Oliver, J., Brown, L., Kaufman, S., Albaugh, D., Brewer, J., Cook, F., Jensen, L., Krumhansl, P., Long, G., andSteiner, D. (1979),The Heterogeneity of the Continental Crust: Results from Deep Crustal Reflection Profiling Using the VIBROSEIS Technique. Rev. Geophys. Space Phys.17, 354–368.

    Google Scholar 

  • Singh, S. K., Apsel, R. J., Fried, J., andBrune, J. N. (1982),Spectral Attenuation of SH Waves Along the Imperial Fault, Bull. Seismol. Soc. Am.72, 2003–2016.

    Google Scholar 

  • Singh, S., andHerrmann, R. B. (1983),Regionalization of Crustal Coda Q in the Continental United States, J. Geophys. Res.88, 527–538.

    Google Scholar 

  • Suzuki, H., Ikeada, R., Mikoshiba, T., Kinoshita, S., Sato, H., andTakahashi, H. (1981),Deep Well Logs in the Kanto-Tokai Area (in Japanese), Rev. National Res. Ctr. Disast. Prev.65, 1–162.

    Google Scholar 

  • Tatarsky, V. I.,Wave Propagation in a Turbulent Medium (R. A. Silverman (trans.)) (McGraw-Hill, New York 1961) p. 18.

    Google Scholar 

  • Taylor, S. R., Bonner, B. P., andVon Zandt, G. (1986),Attenuation and Scattering of Broadband P and S Waves Across North America, J. Geophys. Res.91, 7309–7325.

    Google Scholar 

  • Tittmann, B. R. (1977),Internal friction measurements and their implications in seismic Q structure model of the crust, InThe Earth's Crust (AGU monograph 1977) pp. 197–213.

  • Tittmann, B. R., Clark, V. A., andRichardson, J. M. (1980),Possible Mechanism for Seismic Attenuation in Rocks Containing Small Amount of Volatiles, J. Geophys. Res.85, 5199–5208.

    Google Scholar 

  • Tsai, Y. B., andAki, K. (1969),Simultaneous Determination of the Seismic Moment and Attenuation of Seismic Waves, Bull. Seismol. Soc. Am.59, 275–287.

    Google Scholar 

  • Tsang, L., andKong, J. A. (1980),Thermal Microwave Emission from a Three-layer Random Medium with Three-dimensional Variations, IEEE Trans. Geoscience Remote Sensing GE-18, 212–216.

    Google Scholar 

  • Tsujiura, M. (1978),Spectral Analysis of the Coda Waves from Local Earthquakes, Bull. Earthq. Res. Inst. Univ. Tokyo53, 1–48.

    Google Scholar 

  • Ukawa, M., andFukao, Y. (1981),Poisson's Ratio of the Upper and Lower Crust and the Sub-Moho Mantle Beneath Central Honshu, Japan, Tectonophys.77, 233–256.

    Google Scholar 

  • Walsh, J. B. (1966),Seismic Wave Attenuation in Rock Due to Friction, J. Geophys. Res.71, 2591–2599.

    Google Scholar 

  • Walsh, J. B. (1969),New Analysis of Attenuation in Partially Melted Rock, J. Geophys. Res.74, 4333–4337.

    Google Scholar 

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

    Google Scholar 

  • Wu, R.-S. (1982b),Mean Field Attenuation and Amplitude Attenuation due to Wave Scattering, Wave Motion4, 305–316.

    Google Scholar 

  • Wu, R.-S., andAki, K. (1985a),Scattering of Elastic Waves by a Random Medium and the Small-scale Inhomogeneities in the Lithosphere, J. Geophys. Res.90, 10261–10276.

    Google Scholar 

  • Wu, R.-S. andAki, K. (1985b),The Fractal Nature of the Inhomogeneous Lithosphere Evidenced from Seismic Wave Scattering, Pure Appl. Geophys.123, 805–818.

    Google Scholar 

  • Zener, C. M. (1948),Elasticity and Anelasticity of Metals (Univ. Chicago, Chicago 1948), p. 170.

    Google Scholar 

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  1. National Research Center for Disaster Prevention, Tennodai 3-1, 305, Tsukuba-shi, Ibaraki-ken, Japan

    Haruo Sato

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Sato, H. Unified approach to amplitude attenuation and coda excitation in the randomly inhomogeneous lithosphere. PAGEOPH 132, 93–121 (1990). https://doi.org/10.1007/BF00874359

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