5:15 PM - 6:30 PM
[SSS09-P09] Numerical simulations of weak localization of scalar waves based on the finite difference method (2)
Keywords:weak localization, scalar waves, finite difference method, discrete scatterers
Short-wavelength random heterogeneities in the Earth’s lithosphere scatter seismic waves propagating therein. In most cases, we can consider scattered waves to be incoherent. Nevertheless, there exists an interference phenomenon due to multiple scattering called weak localization (WL), if a wave source and a receiver are located at the same position. WL is caused because a pair of waves, which propagate along reciprocal paths via identical scatterers, have the same phase and hence constructively interfere. Margerin et al. (2001) theoretically investigated WL of scalar waves and concluded that the intensity at around the source is enhanced twice as large as that in a distant position, the diameter of the volume of intensity enhancement corresponds to the wavelength of the input wave, and the emergence time of this volume is governed by the scattering mean-free time (SMFT).
In our previous study (2019, SSJ fall meeting), we simulated WL of scalar waves in 2-D von Karman-type random media, using the finite difference method (FDM). We confirmed that the obtained intensities at the sources were nearly doubled as time passed. The peak widths of intensity were about half the dominant wavelength of the input waves. However, we found no systematic relationship between the peak emergence times and the SMFTs predicted based on the Born approximation (Sato et al., 2012). A possible reason may be the breakdown of the approximation under the present model setting.
In this study, we simulated WL of scalar waves in media with randomly distributed discrete scatterers, like cracks or inclusions. In contrast to random media, the total scattering cross section (TSCS) of a discrete scatterer can be defined without approximations. If scatterers with TSCS σ0 are distributed with a number density n, then the SMFT is given as (nσ0V0)-1, where V0 is the background wave velocity.
We adopted again a velocity-stress scheme FDM (Virieux, 1984). We modeled scatterers as small low-velocity squares with the side length of several grid intervals. These scatterers were distributed randomly, but without overlapping, in an area on the grid. We repeated simulations for many such areas with the same number density, and then calculated the ensemble average of intensity distributions from synthesized wavefields. We also estimated the TSCS of each scatterer after Suzuki et al. (2006). We repeatedly simulated plane waves propagating in areas with scatterers, and then evaluated scattering Q-1 as a function of wavenumber k from the spectra of the ensemble-averaged waves. We then obtained the TSCS σ0(k) via the relation Q-1=nσ0/k. Here we assumed the sparse distributions of scatterers and the wavelength much longer than the scatter size, which means nearly isotropic scattering.
Our simulations denoted again the intensity enhancement at the sources, with the relative peak heights of intensity being about two. The peak widths of the intensity were close to the dominant wavelength of the input waves. The peak emergence times were fairly smaller than the SMFT, but we found positive correlation between them. The peaks tended to emerge earlier for denser distribution of scatterers, that is, stronger scattering.
In our previous study (2019, SSJ fall meeting), we simulated WL of scalar waves in 2-D von Karman-type random media, using the finite difference method (FDM). We confirmed that the obtained intensities at the sources were nearly doubled as time passed. The peak widths of intensity were about half the dominant wavelength of the input waves. However, we found no systematic relationship between the peak emergence times and the SMFTs predicted based on the Born approximation (Sato et al., 2012). A possible reason may be the breakdown of the approximation under the present model setting.
In this study, we simulated WL of scalar waves in media with randomly distributed discrete scatterers, like cracks or inclusions. In contrast to random media, the total scattering cross section (TSCS) of a discrete scatterer can be defined without approximations. If scatterers with TSCS σ0 are distributed with a number density n, then the SMFT is given as (nσ0V0)-1, where V0 is the background wave velocity.
We adopted again a velocity-stress scheme FDM (Virieux, 1984). We modeled scatterers as small low-velocity squares with the side length of several grid intervals. These scatterers were distributed randomly, but without overlapping, in an area on the grid. We repeated simulations for many such areas with the same number density, and then calculated the ensemble average of intensity distributions from synthesized wavefields. We also estimated the TSCS of each scatterer after Suzuki et al. (2006). We repeatedly simulated plane waves propagating in areas with scatterers, and then evaluated scattering Q-1 as a function of wavenumber k from the spectra of the ensemble-averaged waves. We then obtained the TSCS σ0(k) via the relation Q-1=nσ0/k. Here we assumed the sparse distributions of scatterers and the wavelength much longer than the scatter size, which means nearly isotropic scattering.
Our simulations denoted again the intensity enhancement at the sources, with the relative peak heights of intensity being about two. The peak widths of the intensity were close to the dominant wavelength of the input waves. The peak emergence times were fairly smaller than the SMFT, but we found positive correlation between them. The peaks tended to emerge earlier for denser distribution of scatterers, that is, stronger scattering.