Envelope synthesis of a cylindrical outgoing wavelet in layered random elastic media based on the markov approximation

Kentaro Emoto, Haruo Sato, Takeshi Nishimura

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1 Citation (Scopus)


In the heterogeneous earth medium, short period seismograms of an earthquake are wellcharacterized by their smooth envelopes with random phases. The Markov approximation hasoften been used for the practical synthesis of their envelopes for a given frequency band. Itis a stochastic extension of the phase screen method to synthesize wave envelopes in mediawith random fluctuations under the condition that the wavelength is shorter than the correlationdistance of the fluctuation. We propose an extension of the Markov approximation forthe envelope synthesis to the case that an isotropically outgoing wavelet is radiated from apoint source in horizontal layered random elastic media, where different layers have differentrandomness and different background velocities. In each layer, we solve the master equationfor the two frequency mutual coherence function (TFMCF) which contains the informationof the intensity in the frequency domain. Just below each layer boundary, we calculate theangular spectrumwhich is the expression of the TFMCF in the transverse wavenumber domainfor up-going wavelets. The angular spectrum shows the ray angle distribution of intensitiesof scattered waves. Multiplying it by the square of transmission coefficients calculated fromthe background velocity contrast at the boundary, we evaluate the angular spectrum just aboveit. We neglect P to S (S to P) conversion scattering inside of each layer; however, we takeinto account the mode conversion at the layer boundary. Different from the vertical incidenceof a plane wavelet, the wavefront expands with time and its curvature is modified at thelayer boundary due to the Snell's law. Approximating the wavefront in the second layer bya circle for a small incidence angle, we may shift the real origin to the pseudo-origin of thewavefront circle, which leads to the change in geometrical spreading factor. Finally, we calculatethe mean square envelope from the TFMCF by using an FFT. By multiplying the angularspectrum by conversion or reflection coefficients and calculate the TFMCF for the convertedor reflected wavelets at layer boundaries, we can calculate any phase generated due to velocitydiscontinuities. For the reflected wavelets, we solve the master equation of the TFMCF downward. To confirm the validity of the method, we directly synthesize mean square envelopesin 2-D two-layered random elastic media and compare them with the averaged envelopescalculated by finite difference (FD) simulations of the wave propagation in random elasticmedia. We find that the Markov envelopes well agree with the FD envelopes not only for atransmitted wavelet but also for a P to S converted wavelet and a reflected wavelet at a layerboundary.

Original languageEnglish
Pages (from-to)899-910
Number of pages12
JournalGeophysical Journal International
Issue number2
Publication statusPublished - 2013 Jul


  • Body waves
  • Computational seismology
  • Wave scattering and diffraction


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