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  • Article
  • Open Access

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

Earth, Planets and Space200759:BF03353097

  • Received: 11 May 2006
  • Accepted: 4 December 2006
  • Published:


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.

Key words

  • Scattering
  • random media
  • envelope
  • simulation