Relation between direct wave Q and coda Q: a numerical approach

SUMMARY Using the boundary integral method to simulate SH waves numerically in 2‐D homogeneous full‐ or half‐space media with randomly distributed cavities, we compare the amplitude attenuation of direct waves with the temporal decay of the coda. The boundary integral method includes the effect of any degree of multiply scattered waves for a wide frequency range, up to wavelengths smaller than the size of the cavities. We consider seismograms on the free surface so that heterogeneities exist only on one side of the receivers, a situation that resembles actual seismic observations. Seismograms are computed for a vertically incident plane wave and for an isotropic line source. In both cases, the value of Q −1 as a function of kd , where k is the wavenumber and d is the cavity diameter, peaks around kd = 2 for the direct wave, which is consistent with some single‐scattering models. Coda Q −1 determined by the temporal decay of the coda envelope agrees well with Q −1 for the direct wave for models with a root‐mean‐square fluctuation of velocity, Q , of about 10 per cent in a half‐space. On the other hand, the coda Q −1 is systematically larger than the direct wave Q −1 in full‐space models, that is, without the inclusion of the reflection at the free surface. When the cavity density is doubled (σ > 20 per cent), the coda energy increases rapidly and its temporal decay decreases, so that coda Q −1 becomes smaller than the direct wave Q −1 , even for full‐space models. With a smaller value of cr (about 5 per cent), the coda decays rapidly and the relation between the two types of Q −1 is reversed: the coda Q −1 becomes larger than the direct wave Q −1 . By comparing results from seismograms composed only of singly scattered waves with those that include multiply scattered waves, we can compare the relative contribution of each singly and multiply scattered wavefield to the two measures of Q . Single scattering mainly determines both the direct wave Q −1 and the coda Q −1 for the smallest value of a , while the values of both kinds of attenuation, particularly the direct wave Q −1 , are strongly affected by multiple scattering when σ is large. Our results imply that a reasonable estimate of scattering attenuation can be obtained by measuring the temporal decay of the coda, if the scattering character of the Earth is similar to our models with a σ of around 10 per cent, where the single scattering is found to be dominant.