Skip to main content
SpringerLink
Log in

Synthesis of Seismic Wave Envelopes Based on the Markov Approximation

  • Conference paper
  • First Online:
Mathematical Analysis of Continuum Mechanics and Industrial Applications

Part of the book series: Mathematics for Industry ((MFI,volume 26))

Abstract

In this study, we use stochastic methods to analyze complex short-period seismograms reflecting small-scale heterogeneities in the Earth. We consider an ensemble of random velocity-fluctuated media and the statistical characteristics of wave propagation. The Markov approximation is a stochastic method and a multiple forward-scattering approximation. In the Markov approximation, we neglect wide-angle and conversion scatterings and directly calculate the statistical average wave envelopes. Even though we cannot model the latter part of the envelope because we neglect wide-angle scattering, we can adequately describe the initial part of the envelope, i.e., from the onset to near the peak arrival time. We can estimate the statistical properties of the small-scale heterogeneities in the Earth by analyzing the envelope broadening effect. The Markov approximation was developed in optics and was introduced to seismology in the late 1980s. Here, on the basis of the Markov approximation, we summarize the development of envelope modeling and describe a method to calculate envelopes on a layered random heterogeneous media.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Google Scholar 

  2. Emoto, K., Sato, H., Nishimura, T.: Sysnthesis of vector wave envelopes on the free surface of a random medium for the vertical incidence of a plane wavelet based on the Markov approximation. J. Geophys. Res. 115, B08,306 (2010)

    Google Scholar 

  3. Emoto, K., Sato, H., Nishimura, T.: Stochastic synthesis of vector wave envelopes in layered random media based on the improved Markov approximation. In: EGU General Assembly (2012)

    Google Scholar 

  4. Emoto, K., Sato, H., Nishimura, T.: Synthesis and applicable condition of vector wave envelopes in layered random elastic media with anisotropic autocorrelation function based on the markov approximation. Geophys. J. Int. 188(1), 325–335 (2012). doi:10.1111/j.1365-246X.2011.05264.x

  5. Emoto, K., Sato, H., Nishimura, T.: Envelope synthesis of a cylindrical outgoing wavelet in layered random elastic media based on the Markov approximation. Geophys. J. Int. 194, 899–910 (2013)

    Article  Google Scholar 

  6. Fehler, M., Sato, H., Huang, L.: Envelope broadening of outgoing waves in 2D random media: a comparison between the Markov approximation and numerical simulations. Bull. Seism. Soc. Am. 90(4), 914–928 (2000)

    Article  Google Scholar 

  7. Fisk, M.D., McCartor, G.D.: The phase screen method for vector elastic waves. J. Geophys. Res. 96, 5985–6010 (1991)

    Article  Google Scholar 

  8. Korn, M., Sato, H.: Synthesis of plane vector-wave envelopes in 2-D random elastic media based on the Markov approximation and comparison with finite difference simulations. Geophys. J. Int. 161(3), 839–848 (2005)

    Article  Google Scholar 

  9. Kubanza, M., Nishimura, T., Sato, H.: Evaluation of strength of heterogeneity in the lithosphere from peak amplitude analyses of teleseismic short-period vector P waves. Geophys. J. Int. 171(1), 390–398 (2007)

    Article  Google Scholar 

  10. Lee, L., Jokipii, J.: Strong scintillations in astrophysics. I. the Markov approximation, its validity and application to angular broadening. Astrophys. J. 196, 695–707 (1975)

    Article  Google Scholar 

  11. Lee, L., Jokipii, J.: Strong scintillations in astrophysics. II. a theory of temporal broadening of pulses. Astrophys. J. 201, 532–543 (1975)

    Article  Google Scholar 

  12. Margerin, L.: Introduction to radiative transfer of seismic waves. In: L.A. Nolet (ed.) Seismic Earth: Array Analysis of Broad-band Seismograms, chap. 14, pp. 229–252. AGU (2005)

    Google Scholar 

  13. Przybilla, J., Korn, M.: Monte Carlo simulation of radiative energy transfer in continuous elastic random media-three-component envelopes and numerical validation. Geophys. J. Int. 173(2), 566–576 (2008)

    Article  Google Scholar 

  14. Przybilla, J., Korn, M., Wegler, U.: Radiative transfer of elastic waves versus finite difference simulations in two-dimensional random media. J. Geophys. Res. 111, B04,305 (2006). doi:10.1029/2005JB003952

  15. Rytov, S.M., Kravtsov, Y.A., Tatarskii, V.I.: Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media. Springer (1989)

    Google Scholar 

  16. Saito, T.: Synthesis of scalar-wave envelopes in two-dimensional weakly anisotropic random media by using the Markov approximation. Geophys. J. Int. 165, 501–515 (2006)

    Article  Google Scholar 

  17. Saito, T., Sato, H., Ohtake, M.: Envelope broadening of spherically outgoing waves in three-dimensional random media having power-law spectra. J. Geophys. Res. 107(10.1029) (2002)

    Google Scholar 

  18. Saito, T., Sato, H., Fehler, M., Ohtake, M.: Simulating the envelope of scalar waves in 2D random media having power-law spectra of velocity fluctuation. Bull. Seismol. Soc. Am. 93(1), 240–252 (2003)

    Article  Google Scholar 

  19. Saito, T., Sato, H., Ohtake, M., Obara, K.: Unified explanation of evnvelope broadening and maximum-amplitude decay of high-frequency seismograms based on the envelope simulation usign the Markov approximation: forearc side of the volcanic front in northeastern Honshu, Japan. J. Geophys. Res. 110, B01,304 (2005)

    Google Scholar 

  20. Saito, T., Sato, H., Takahashi, T.: Direct simulation methods for scalar-save envelopes in two-dimensional layered random media based on the small-angle scattering approximation. Commun. Comput. Phys. 3(1), 63–84 (2008)

    MATH  Google Scholar 

  21. Sato, H.: Broadening of seismogram envelopes in the randomly inhomogeneous lithosphere based on the parabolic approximation: Southeastern Honshu, Japan J. Geophys. Res. 94(B12), 17735–17747 (1989)

    Google Scholar 

  22. Sato, H.: Synthesis of vector wave envelopes in three-dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: plane wave case. J. Geophys. Res. 111(B6), B06,306 (2006)

    Google Scholar 

  23. Sato, H.: Synthesis of vector wave envelopes in three-dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: spherical wave case. J. Geophys. Res. 112(B1), B01,301 (2007)

    Google Scholar 

  24. Sato, H.: Synthesis of vector-wave envelopes in 3-D random media characterized by a nonisotropic Gaussian ACF based on the Markov approximation. J. Geophys. Res. 113, B08,304 (2008)

    Google Scholar 

  25. Sato, H.: Envelope broadening and scattering attenuation of a scalar wavelet in random media having power-law spectra. Geophys. J. Int. 204, 386–398 (2016)

    Article  Google Scholar 

  26. Sato, H., Korn, M.: Envelope syntheses of cylindrical vector-waves in 2-D random elastic media based ont he Markov approximation. Earth Planets Space 59, 209–219 (2007)

    Article  Google Scholar 

  27. Sato, H., Fehler, M., Saito, T.: Hybrid synthesis of scalar wave envelopes in two-dimensional random media having rich short-wavelength spectra. J. Geophys. Res. 109, B06,303 (2004)

    Google Scholar 

  28. Sato, H., Fehler, M., Maeda, T.: Seismic Wave Propagation and Scattering in the Heterogeneous Earth, 2nd edn. Springer (2012)

    Google Scholar 

  29. Sawazaki, K., Sato, H., Nishimura, T.: Envelope synthesis of short-period seismograms in 3-D random media for a point shear dislocation source based on the forward scattering approximation: application to small strike-slip earthquakes in southwestern Japan. J. Geophys. Res. 116, B08,305 (2011)

    Google Scholar 

  30. Shiomi, K., Sato, H., Ohtake, M.: Broad-band power-law spectra of well-log data in Japan. Geophys. J. Int. 130(1), 57–64 (1997)

    Article  Google Scholar 

  31. Shishov, V.: Effect of refraction on scintillation characteristics and average pulse shape of pulsars. Sov. Astron. 17, 598–602 (1974)

    Google Scholar 

  32. Sreenivasiah, I., Ishimaru, A., Hong, S.: Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case. Radio Sci. 11(10), 775–778 (1976)

    Article  MathSciNet  Google Scholar 

  33. Suzaki, A.: Study on vector-wave envelopes in von Karman-type random media. Master’s thesis, Tohoku University, Japan (2007)

    Google Scholar 

  34. Takahashi, T., Sato, H., Nishimura, T.: Recursive formula for the peak delay time with travel distance in von Karman type non-uniform random media on the basis of the Markov approximation. Geophys. J. Int. 173(2), 534–545 (2008)

    Article  Google Scholar 

  35. Takahashi, T., Sato, H., Nishimura, T., Obara, K.: Tomographic inversion of the peak delay times to reveal random velocity fluctuations in the lithosphere: method and application to northeastern Japan. Geophys. J. Int. 178, 1437–1455 (2009)

    Article  Google Scholar 

  36. Williamson, I.: Pulse broadening due to multiple scattering in the interstellar medium. Mon. Not. R. Astr. Soc. 157, 55–71 (1972)

    Article  Google Scholar 

Download references

Acknowledgments

We are very grateful to the staff of the National Research Institute for Earth Science and Disaster Resilience enable Hi-net to continue functioning. The author would like to thank an anonymous reviewer for his valuable comments to improve this manuscript.

Author information

Authors and Affiliations

  1. Graduate School of Science, Tohoku University, Aramaki-Aza Aoba 6-3, Aoba-ku, Sendai, 980-8578, Japan

    Kentaro Emoto

Authors
  1. Kentaro Emoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kentaro Emoto .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Tokyo, Japan

    Hiromichi Itou

  2. Inst of Science and Engineering, Kanazawa University, Kanazawa, Japan

    Masato Kimura

  3. Fujitsu Ltd., Tokyo, Japan

    Vladimír Chalupecký

  4. Hiroshima Kokusai Gakuin University, Hiroshima, Japan

    Kohji Ohtsuka

  5. Kyushu University, Fukuoka, Fukuoka, Japan

    Daisuke Tagami

  6. Asahi Glass Co., Ltd., Tokyo, Tokyo, Japan

    Akira Takada

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Emoto, K. (2017). Synthesis of Seismic Wave Envelopes Based on the Markov Approximation. In: Itou, H., Kimura, M., Chalupecký, V., Ohtsuka, K., Tagami, D., Takada, A. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications. Mathematics for Industry, vol 26. Springer, Singapore. https://doi.org/10.1007/978-981-10-2633-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-2633-1_9

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2632-4

  • Online ISBN: 978-981-10-2633-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation