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For the study of the random velocity fluctuation of the solid Earth medium, it is useful to measure the collapse of a seismic wavelet with increasing travel distance and the excitation of coda waves. Radiative transfer theory (RTT) is a powerful tool for synthesizing the propagation of a seismic wavelet in random media statistically characterized by the power spectral density function (PSDF) of the fractional velocity fluctuation. The Born scattering coefficient is a key building block of RTT. As the centre wavenumber of a wavelet increases, the phase shift across the correlation length increases and the Born approximation leads to an extremely large forward scattering exceeding the applicable range of the perturbation method. In such a case, the Eikonal approximation is able to explain the envelope broadening with increasing travel distance; however, it can not explain the coda excitation. To overcome the difficulty, we have proposed a hybrid Monte Carlo (MC) simulation for scalar waves. In the case of von Kármán-type random media, when the centre wavenumber is higher than the corner wavenumber, taking the centre wavenumber as a reference, we divide the PSDF into two spectral components. Applying the Born and Eikonal approximations to the high- and low-wavenumber spectral components, we statistically evaluate the wide-angle scattering and the narrow-angle ray bending, respectively. The proposed MC simulation serially using two kinds of scattering processes successfully synthesizes the time trace of the wave energy density from the onset to the late coda. The travel-distance fluctuation derived from the one-way propagation of the Eikonal approximation is also important. This paper extends this method for the propagation of a vector wavelet in random elastic media. We suppose that random fractional fluctuations of the P - and S -wave velocities and the mass density are linearly proportional to each other, which maintains the linear polarization of an S wave throughout the scattering process. Using the hybrid MC simulation with the spectrum division, we synthesize three-component energy density time traces for the anisotropic radiation from a moment tensor source, from which we derive three-component root mean square (RMS) velocity amplitude time traces at different azimuths. In parallel, we synthesize the propagation of a vector wavelet in many realized random elastic media by the finite-difference simulation, then we calculate three-component RMS velocity amplitude time traces. Using them as a benchmark, we confirm the validity of the proposed MC simulation for specific cases. [ABSTRACT FROM AUTHOR]

In high-frequency seismograms of a small earthquake, coda waves continue for a long time and the S -wavelet is broadened as travel distance increases. Those phenomena can be interpreted as results of scattering by random inhomogeneities distributed in the solid earth. Recent measurements show that the power spectrum of the random medium heterogeneities decreases according to some power of wavenumber. As a mathematical model, we study the propagation of a scalar wavelet through von Kármán-type random media. Our objective is to propose a stochastic method to synthesize intensity time traces, the mean square amplitude traces, by using a few number of parameters statistically characterizing the random media. The most conventional method is known to use the Born approximation in the radiative transfer equation. However, as the centre wavenumber of a wavelet increases higher than the corner wavenumber, the Born approximation becomes inapplicable because the phase shift increases. We proposed a spectrum division method to avoid the difficulty in our previous papers. Taking the centre wavenumber as a reference, we divide the power spectrum into two components. The short-scale component is chosen to satisfy the applicable condition of the Born approximation, which leads to wide-angle scattering per volume. Here, we newly propose to use the phase screen approximation for the long-scale component, which leads to narrow-angle ray bending per length. We simultaneously introduce both the wide-angle scattering due to the short-scale component and the narrow-angle ray bending due to the long-scale component into the Monte Carlo simulation code in the framework of the radiative transfer theory. Synthesized wavelet intensity time traces show envelope broadening and peak delay with increasing travel distance and long lasting coda waves at a short distance. If the power spectrum is properly divided, synthesized intensity time traces well explain the averaged intensity time traces calculated by the finite difference simulation of wavelets in random media from the onset through the peak to coda for a wide dynamic range. The proposed method has a potential to adopt easily the spatial variation of randomness and intrinsic absorption. [ABSTRACT FROM AUTHOR]

Random velocity fluctuations distributed in the solid Earth function as sources of seismic wave scattering. Scattering effects are often observed in high-frequency seismograms of earthquakes as the broadening of the apparent duration of an S -wavelet and the emergence of coda waves. We conduct large-scale 3-D finite difference (FD) simulations of the scalar wave equation to analyse the intensities of scattered waves propagating through random small-scale heterogeneous media. First, we compare ensemble averaged intensities (mean square amplitudes) derived by the FD simulation with those synthesized based on statistical methods such as the radiative transfer equation with the Born approximation and the newly developed spectrum division methods. We consider several types of random media characterized by von Kármán type autocorrelation functions with different characteristic distances and mean square fractional fluctuations. In the case of a large characteristic distance, the forward scattering is dominant and the fluctuation of the traveltime is large. Even in that case, the newly developed spectrum division method can reproduce the average intensity derived by FD simulations in the entire lapse time range. We further investigate the characteristics of scattered waves. To know the property of the fluctuation of intensities due to the small-scale heterogeneity is important for the ground motion prediction. We reveal the gradual shift of intensity fluctuations from the log-normal distribution to the exponential one with the increase of lapse time. The timing of the shift varies depending on the random medium parameters. This can be explained as the shift from the multiple forward scattering regime to the incoherent wide-angle scattering one. The decay rate of the intensity changes from r −2 to r −4 with the increase of the propagation distance r, which corresponds to the empirical relationship of the observed intensity. This timing of the change also depends on the random medium parameters. [ABSTRACT FROM AUTHOR]

In the heterogeneous earth medium, short period seismograms of an earthquake are well characterized by their smooth envelopes with random phases. The Markov approximation has often been used for the practical synthesis of their envelopes for a given frequency band. It is a stochastic extension of the phase screen method to synthesize wave envelopes in media with random fluctuations under the condition that the wavelength is shorter than the correlation distance of the fluctuation. We propose an extension of the Markov approximation for the envelope synthesis to the case that an isotropically outgoing wavelet is radiated from a point source in horizontal layered random elastic media, where different layers have different randomness and different background velocities. In each layer, we solve the master equation for the two frequency mutual coherence function (TFMCF) which contains the information of the intensity in the frequency domain. Just below each layer boundary, we calculate the angular spectrum which is the expression of the TFMCF in the transverse wavenumber domain for up-going wavelets. The angular spectrum shows the ray angle distribution of intensities of scattered waves. Multiplying it by the square of transmission coefficients calculated from the background velocity contrast at the boundary, we evaluate the angular spectrum just above it. We neglect P to S (S to P) conversion scattering inside of each layer; however, we take into account the mode conversion at the layer boundary. Different from the vertical incidence of a plane wavelet, the wavefront expands with time and its curvature is modified at the layer boundary due to the Snell's law. Approximating the wavefront in the second layer by a circle for a small incidence angle, we may shift the real origin to the pseudo-origin of the wavefront circle, which leads to the change in geometrical spreading factor. Finally, we calculate the mean square envelope from the TFMCF by using an FFT. By multiplying the angular spectrum by conversion or reflection coefficients and calculate the TFMCF for the converted or reflected wavelets at layer boundaries, we can calculate any phase generated due to velocity discontinuities. 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 envelopes in 2-D two-layered random elastic media and compare them with the averaged envelopes calculated by finite difference (FD) simulations of the wave propagation in random elastic media. We find that the Markov envelopes well agree with the FD envelopes not only for a transmitted wavelet but also for a P to S converted wavelet and a reflected wavelet at a layer boundary. [ABSTRACT FROM AUTHOR]