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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]

The spatial autocorrelation (SPAC) method has been applied to ambient seismic noise measured by arrays of translational seismometers for inverting phase-velocity dispersion curves of Rayleigh or Love waves for shallow S -wave velocity structure. Recently, it is becoming possible to observe wave spatial gradients such as strain, rotation and tilt owing to the development of dense seismic networks and improving measurement technologies. Therefore, it is desirable to extend the formulation of the SPAC method to strain, rotation and tilt. This study presents analytical expressions of cross-spectra and coherence of the strain, rotation and tilt components that are measured on the free surface. According to the results, both Rayleigh and Love waves contribute to most components of strains. The exceptions are the areal strain and the vertical axial strain (e zz ) on the free surface that are affected by only Rayleigh waves. Only Rayleigh waves contribute to the tilts and rotations around the horizontal axes on the free surface, too. On the other hand, only Love waves contribute to the rotation around the vertical axis. Therefore, different kinds of wave spatial gradients are helpful to separate Rayleigh and Love waves correctly. For practical applications, the analytical expression for the radial axial strain (e rr ) component will be applied directly to distributed acoustic sensing data measured with straight sections of a fibre-optic cable. On the other hand, dense observations of rotation and tilt may still be difficult to carry out at present. However, an application of analytical formulations in this study to arrays of at least several three-component rotational seismometers is attractive because it enables us to separately estimate the phase velocities of Rayleigh and Love waves. [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]