Synthesis of a scalar wavelet intensity propagating through von Kármán-type random media: Joint use of the radiative transfer equation with the Born approximation and the Markov approximation

Haruo Sato, Kentaro Emoto

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


In high-frequency seismograms of small earthquakes, we clearly see the excitation of long lasting coda waves and the envelope broadening of an S-wavelet with travel distance increasing. We can interpret those phenomena resulting from scattering by random inhomogeneities distributed in the earth medium. Those phenomena have been theoretically studied by stochastic methods, which deal with velocity inhomogeneities as random media. As a simple mathematical model, we study the propagation of a scalar wavelet for the spherical radiation from a point source in 3-D von Kármán-type random media, of which the power spectral density function (PSDF) decreases according to a power-law higher than the corner wavenumber. Our objective is to propose a method to synthesize the wavelet intensity time trace, the mean square amplitude trace, at a given travel distance by using statistical parameters characterizing the PSDF and the centre wavenumber of the wavelet. When the phase shift is small, we can use the Born approximation to calculate the non-isotropic scattering coefficient representing the scattering power per unit volume. Using the scattering coefficient in the radiative transfer equation (RTE), we are able to synthesize the wavelet intensity time trace. When the centre wavenumber increases in the power-law spectral range, however, we often face the situation of a large phase shift, where the Born approximation is inapplicable, but we are able to use the Markov approximation based on the parabolic approximation. It well explains the intensity time traces showing envelope broadening with peak delay due to multiple scattering around the forward direction and the wandering effect caused by travel time fluctuations; however, it fails to explain rich coda waves composed of scatteredwaves in wide angles. In such a case, here, we newly propose the spectrum division method as follows: at first, taking the centre wavenumber with a tuning parameter as a reference, we divide the random medium spectrum into the low- and high-wavenumber spectral (long- and short-scale) components. The second step is to synthesize the intensity time-trace by using the RTE with the Born scattering coefficient for the short-scale component. The third step is to calculate the envelope broadening and the wandering effects due to the long-scale component. As the fourth step, at each travel distance, we convolve the intensity time trace calculated by the RTE with the envelope broadening and wandering effects and the source function, which gives the intensity time trace reflecting the scattering contribution of all the spectral components. In parallel, realizing random media for given statistical parameters, we conduct finite difference (FD) simulations of waves through them for the spherical radiation of a Ricker wavelet from a point source. We confirm that synthesized intensity time traces well explain averaged FD simulation intensity traces from the onset through the peak to coda for a specific case. Those syntheses will be a theoretical basis for the study of random velocity inhomogeneities in the earth medium from the analysis of high-frequency seismic waves of small earthquakes.

Original languageEnglish
Pages (from-to)512-527
Number of pages16
JournalGeophysical Journal International
Issue number1
Publication statusPublished - 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Geophysics
  • Geochemistry and Petrology


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