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Envelope syntheses of cylindrical vector-waves in 2-D random elastic media based on the Markov approximation

Abstract

Seismograms of microearthquakes are complex; however, their envelopes broaden as the travel distance increases. P-waves are recorded in transverse components, S-waves are recorded in the longitudinal component, and waves are observed at sites even in the nodal direction of the source radiation. These phenomena, which are typically found in short-period seismograms, can be interpreted to be the result of scattering due to lithospheric inhomogeneity. We report here our study of a simple statistical model in which the propagation of waves radiated from a point source in two-dimensional (2-D) random elastic media is characterized by a Gaussian autocorrelation function. For the case that the wavelength is shorter than the correlation distance, two methods based on the Markov approximation are introduced for the direct synthesis of vector wave envelopes. One is to analytically solve the stochastic equation for the two-frequency mutual coherence function; the validity of the solution is confirmed by using finite difference simulations. The second is to numerically solve the stochastic equation for the mutual coherence function. The two methods are equivalent, but the latter is applicable to nonisotropic source radiation. For the case of a point shear dislocation source, a peak delay from the onset and a smoothly decaying tail are found to be common to synthesized envelopes in all azimuths. Scattered waves are excited even at receivers in the nodal direction, and amplitudes become independent of the radiation pattern as lapse time increases.

References

  • Aki, K. and B. Chouet, Origin of coda waves: Source, attenuation and scattering effects, J. Geophys. Res., 80, 3322–3342, 1975.

    Article  Google Scholar 

  • Bal, G. and M. Moscoso, Polarization effects of seismic waves on the basis of radiative transport theory, Geophys. J. Int., 142, 571–585, 2000.

    Article  Google Scholar 

  • Fehler, M., H. Sato, and L.-J. Huang, Envelope broadening of outgoing waves in 2-D random media: A comparison between the Markov approximation and numerical simulations, Bull. Seismol. Soc. Am., 90, 914–928, 2000.

    Article  Google Scholar 

  • Frankel, A. and R. W. Clayton, Finite difference 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, 1986.

    Article  Google Scholar 

  • Gusev, A. A. and I. R. Abubakirov, Simulated envelopes of non-isotropically scattered body waves as compared to observed ones: Another manifestation of fractal heterogeneity, Geophys. J. Int., 127, 49–60, 1996.

    Article  Google Scholar 

  • Hoshiba, M., Simulation of coda wave envelope in depth dependent scattering and absorption structure, Geophys. Res. Lett., 21, 2853–2856, 1994.

    Article  Google Scholar 

  • Ishimaru, A., Wave Propagation and Scattering in Random Media, vols. 1 and 2, Academic, San Diego, Calif., 1978.

  • Korn, M. and H. Sato, Synthesis of plane vector wave envelopes in two-dimensional random elastic media based on the Markov approximation and comparison with finite-difference simulations, Geophys. J. Int., 161, 839–848, 2005.

    Article  Google Scholar 

  • Lee, L. C. and J. R. Jokipii, Strong scintillations in astrophysics. I. The Markov approximation, its validity and application to angular broadening, Astrophys. J., 196, 695–707, 1975.

    Article  Google Scholar 

  • Margerin, L., M. Campillo, and B. Van Tiggelen, Monte Carlo simulation of multiple scattering of elastic waves, J. Geophys. Res., 105, 7873–7892, DOI:10.1029/1999JB900359, 2000.

    Article  Google Scholar 

  • Matsumura, S., Three-dimensional expression of seismic particle motions by the trajectory ellipsoid and its application to the seismic data observed in the Kanto district, Japan, J. Phys. Earth, 29, 221–239, 1981.

    Article  Google Scholar 

  • Morse, P. M. and H. Feshbach, Methods in Theoretical Physics, McGraw-Hill, Boston, 1953.

    Google Scholar 

  • Nakahara, H., T. Nishimura, H. Sato, and M. Ohtake, Seismogram envelope inversion for the spatial distribution of high-frequency energy radiation from the earthquake fault: Application to the 1994 far east off Sanriku earthquake, Japan, J. Geophys. Res., 103, 855–867, 1998.

    Article  Google Scholar 

  • Obara, K. and H. Sato, Regional differences of random inhomogeneities around the volcanic front in the Kanto-Tokai area, Japan, revealed from the broadening of S wave seismogram envelopes, J. Geophys. Res., 100, 2103–2121, 1995.

    Article  Google Scholar 

  • Petukhin, A. G. and A. A. Gusev, The Duration-distance relationship and average envelope shapes of small Kamchatka earthquakes, Pure Appl. Geophys., 160, 1717–1743, 2002.

    Google Scholar 

  • Przybilla, J., M. Korn, and U. Wegler, Radiative transfer of elastic waves versus finite difference simulations in 2D random media, J. Geophys. Res., 111, B04305, doi:10.1029/2005JB003952, 2006.

    Google Scholar 

  • Saito, T., H. Sato, and M. Ohtake, Envelope broadening of spherically outgoing waves in three-dimensional random media having power-law spectra, J. Geophys. Res., 107, doi: 10.1029/2001JB000264, 2002.

  • Saito, T., H. Sato, and M. Ohtake, Envelope broadening of spherically outgoing waves in three-dimensional random media having power-law spectra, J. Geophys. Res., 107, doi: 10.1029/2001JB000264, 2002.

  • Sato, H., Energy propagation including scattering effect: Single isotropic scattering approximation, J. Phys. Earth, 25, 27–41, 1977.

    Article  Google Scholar 

  • Sato, H., Attenuation and envelope formation of three-component seis-mograms of small local earthquakes in randomly inhomogeneous litho-sphere, J. Geophys. Res., 89, 1221–1241, 1984.

    Article  Google Scholar 

  • Sato, H., Broadening of seismogram envelopes in the randomly inhomogeneous lithosphere based on the parabolic approximation: Southeastern Honshu, Japan, J. Geophys. Res., 94, 17735–17747, 1989.

    Article  Google Scholar 

  • Sato, H. Synthesis of vector-wave envelopes in 3-D random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Plane wave case, J. Geophys. Res., 111, B06306, doi:10.1029/2005JB004036, 2006.

    Google Scholar 

  • Sato, H., Synthesis of vector-wave envelopes in 3-D random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Spherical wave case, J. Geophys. Res., 112, B01301, doi:10.1029/2006JB004437, 2007.

    Google Scholar 

  • Sato, H. and M. Fehler, “Seismic wave propagation and scattering in the heterogeneous earth”, AIP Press/Springer Verlag New York, 1–308, 1998.

    Book  Google Scholar 

  • Scherbaum, F. and H. Sato, Inversion of full seismogram envelopes based on the parabolic approximation: Estimation of randomness and attenuation in southeast Honshu, Japan, J. Geophys. Res., 96, 2223–2232, 1991.

    Article  Google Scholar 

  • Shishov, V. L., Effect of refraction on scintillation characteristics and average pulsars, Sov. Astron., 17, 598–602, 1974.

    Google Scholar 

  • Sreenivasiah, I., A. Ishimaru, and S. T. Hong, Two-frequency mutual coherence function and pulse propagation in a random medium: An analytic solution to the plane wave case, Radio Sci., 11, 775–778, 1976.

    Article  Google Scholar 

  • Uscinski, J., The Elements of Wave Propagation in Random Media, McGraw-Hill, New York, 1977.

    Google Scholar 

  • Wegler, U., M. Korn, and J. Przybilla, Modeling of full seismogram envelopes using radiative transfer theory with Born scattering coefficients, Pure Appl. Geophys., 163, 503–531, 2006.

    Article  Google Scholar 

  • Williamson, I. P., Pulse broadening due to multiple scattering in the interstellar medium, Mon. Not. R. Astron. Soc., 157, 55–71, 1972.

    Article  Google Scholar 

  • Williamson, I. P., The broadening of pulses due to multi-path propagation of radiation, Proc. R. Soc. London. A., 342, 131–147, 1975.

    Article  Google Scholar 

  • Yomogida, K. and R. Benites, Relation between direct wave and coda: A numerical approach, Geophys. J. Int., 123, 471–483, 1995.

    Article  Google Scholar 

  • Yoshimoto, K., Monte Carlo simulation of seismogram envelopes in scattering media, J. Geophys. Res., 105, 6153–6161, 2000.

    Article  Google Scholar 

  • Zeng, Y., K. Aki, and T. L. Teng, Mapping of the high-frequency source radiation for the Loma Prieta earthquake, California, J. Geophys. Res., 98, 11981–11993, 1993.

    Article  Google Scholar 

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Correspondence to Haruo Sato.

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Sato, H., Korn, M. Envelope syntheses of cylindrical vector-waves in 2-D random elastic media based on the Markov approximation. Earth Planet Sp 59, 209–219 (2007). https://doi.org/10.1186/BF03353097

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Key words

  • Scattering
  • random media
  • envelope
  • simulation