Synthesis of Seismogram Envelopes in Heterogeneous Media

SUMMARY Wave trains in high-frequency seismograms of local earthquakes are mostly composed of incoherent waves that are scattered by distributed heterogeneities. Their waveforms are very complex and significantly different from those computed for conventional layered structures; however, their envelopes are repeatable, frequency dependent, and vary regionally. Well-logs obtained from deep boreholes show random fluctuations of medium properties superposed over a layered background structure. Recognizing the complexity of seismograms and Earth inhomogeneity, seismologists often focus on understanding envelopes of band-pass filtered traces rather than on unfiltered waveforms. Stochastic approaches are superior to deterministic wave-theoretical approaches for modeling wave envelopes in random media. There are several methods to predict how envelopes vary with travel distance and frequency depending on the power spectra of random media. As the most tractable case, this chapter precisely examines scalar wave propagation in 2-D random media for an impulsive wavelet isotropically radiated from a point source. We mainly discuss three methods: the Markov approximation method, the isotropic scattering model based on the radiative transfer theory, and a hybrid method. By using reference envelopes simulated by stochastic averaging of waveforms calculated using the finite difference method, the three methods for direct envelope simulation are tested. Random medium having Gaussian auto-correlation function with correlation length longer than the seismic wavelength is a simple case to analyze because it is dominated by forward scattering. For this case, we mathematically introduce the Markov 2 approximation method, which directly and reliably predicts wave envelopes based on the parabolic wave equation and an extension of the phase screen method. Then, we examine the validity of the Markov approximation for the case of von Kármán-type random media having realistic power-law spectrum as an asymptote. The envelopes predicted by the Markov approximation satisfactorily explain the reference envelopes for random media having weak short wavelength spectra, which are also dominated by forward scattering. For the case of media with strong short wavelength spectra, however, the coincidence is good around the peak amplitude but becomes poor for the coda portion because of wide-angle scattering. In that case, the isotropic scattering model based on radiative transfer theory well explains the coda portion of the reference envelopes, where the momentum transfer scattering coefficient is used as the effective isotropic scattering coefficient. Introducing a hybrid method using the momentum transfer scattering coefficient and the envelope predicted by the Markov approximation as a propagator in the radiative transfer integral equation for …