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Autoregressive modeling of transfer functions in frequency domain to determine complex travel times


We present a method to determine the complex travel times of impulses in the time domain on the basis of an autoregressive (AR) modeling of superimposed sinusoids in a finite complex series in the frequency domain. We assume that the complex frequency series consists of signals represented by a complex AR equation with additional noise. The AR model in the frequency domain corresponds to a complex Lorentzian in the time domain. In a similar way to the Sompi or extended Prony method, the complex travel times are given by solutions of a characteristic equation of complex AR coefficients, which are obtained as the eigenvector corresponding to a minimum eigenvalue in an eigenvalue problem of non-Toeplitz autocovariance matrix of the complex series. Our method is tested for synthetic frequency series of transfer functions, which show that (1) the complex travel times of closely adjacent pulses in the time domain are clearly resolved, and that (2) the frequency dependence of the complex travel times for physical and structural dispersions is precisely determined by the analysis within a narrow frequency window. These results demonstrate the usefulness of our method with high resolvability and accuracy in the analysis of impulse sequences.


  • Aki, K. and P. G. Richards, Quantitative seismology: Theory and methods, 932 pp., Freeman and Co., 1980.

  • Azimi, Sh. A., A. Y. Kalinin, V. B. Kalinin, and B. L. Pivovarov, Impulse and transient characteristics of media with linear and quadratic absorption laws, Izv. Earth Phys., 2, 88–93, 1968.

    Google Scholar 

  • Bogart, B. P., M.J.R. Healy, and J. W. Tukey, The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum and saphe cracking, in Time Series Analysis, Edited by M. Rosenblatt, Chap. 15, pp. 209–243, New York: Wiley, 1963.

    Google Scholar 

  • Chao, B. F. and F. Gilbert, Autoregressive estimation of complex eigen-frequencies in low frequency seismic spectra, Geophys. J. R. Astr. Soc., 63, 641–657, 1980.

    Article  Google Scholar 

  • Childers, D. G., D. P. Skinner, and R. C. Kemerait, The Cepstrum: A guide to processing, Proc. IEEE, 65, 1428–1442, 1977.

    Article  Google Scholar 

  • Dziewonski, A., S. Bloch, and M. Landisman, A technique for analysis of transient seismic signals, Bull. Seism. Soc. Am., 59, 427–444, 1969.

    Google Scholar 

  • Evans, J. R. and J. J. Zucca, Active high-resolution seismic tomography of compressional wave velocity and attenuation structure at Medicine Lake Volcano, northern California Cascade Range, J. Geophys. Res., 93, 15016–15036, 1988.

    Article  Google Scholar 

  • Evans, J. R. and J. J. Zucca, Active source, high-resolution (NeHT) tomography: velocity and Q, in Seismic Tomography, edited by H. M. Iyer and K. Hirahara, Chap. 25, pp. 695–732, Chapman and Hall, 1993.

  • Haskell, N. A., The dispersion of surface waves in multilayered media, Bull. Seism. Soc. Am., 43, 17–34, 1953.

    Google Scholar 

  • Hildebrand, F. B., Introduction to Numerical Analysis, Chap. 9, 511 pp., New York: McGraw-Hill, 1956.

    Google Scholar 

  • Hori, S., Y. Fukao, M. Kumazawa, M. Furumoto, and A. Yamamoto, A new method of spectral analysis and its application to the Earth’s free oscillations: The“Sompi”method, J. Geophys. Res., 94(B6), 7535–7553, 1989.

    Article  Google Scholar 

  • Kay, S. M. and S. L. Marple, Spectrum analysis—a modern perspective, Proc. IEEE, 69, 1380–1419, 1981.

    Article  Google Scholar 

  • Koketsu, K., The extended reflectivity method for synthetic near-field seismograms, J. Phys. Earth, 33, 121–131, 1985.

    Article  Google Scholar 

  • Kumazawa, M. and Y. Takei, Active method of monitoring underground structures by means of accurately controlled rotary seismic sources. 3. Event detection from a small number of Fourier components observed by ACROSS system, Abstr. Seism. Soc. Jpn., 2, 160, 1994 (in Japanese).

    Google Scholar 

  • Kumazawa, M., Y. Imanishi, Y. Fukao, M. Furumoto, and A. Yamamoto, A theory of spectral analysis based on the characteristic property of a linear dynamic system, Geophys. J. Int., 101, 613–630, 1990.

    Article  Google Scholar 

  • Matsuura, T., Y. Imanishi, M. Imanari, and M. Kumazawa, Application of a new method of high-resolution spectral analysis, “Sompi,” for free induction decay of nuclear magnetic resonance, Appl. Spectrosc, 44, 618–626, 1990.

    Article  Google Scholar 

  • Oppenheim, A. V. and D. W. Schafer, Digital signal processing, pp. 480–531, Prentice-Hall International, London, 1975.

    Google Scholar 

  • Pisarenko, V. F., On the estimation of spectra by means of nonlinear functions of covariance matrix, Geophys. J. R. Astr. Soc., 28, 511–531, 1972.

    Article  Google Scholar 

  • Price, H. J., An improved Prony algorithm for exponential analysis, in Proceedings of the IEEE International Symposium on Electro-magnetic Compatibility, pp. 310–313, Institute of Electrical and Electronics Engineers, New York, 1979.

    Google Scholar 

  • Shankland, T. J., P. A. Johnson, and T. M. Hopson, Elastic wave attenuation and velocity of Berea sandstone measured in the frequency domain, Geophys. Res. Lett., 20(5), 391–394, 1993.

    Article  Google Scholar 

  • Teng, TL., Attenuation of body waves and the Q structure of the mantle, J. Geophys. Res., 73, 2195–2208, 1968.

    Article  Google Scholar 

  • Thomson, W. T, Transmission of elastic waves through a stratified solid, J. Appl. Phys., 21, 89–93, 1950.

    Article  Google Scholar 

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Correspondence to Yoko Hasada.

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Hasada, Y., Kumagai, H. & Kumazawa, M. Autoregressive modeling of transfer functions in frequency domain to determine complex travel times. Earth Planet Sp 53, 3–11 (2001).

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  • Rayleigh Wave
  • Inverse Fourier Transform
  • Impulse Response Function
  • Narrow Frequency Band
  • Wave Element