Statistical characteristics of scattered waves in three-dimensional random media: Comparison of the finite difference simulation and statistical methods

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.