Acoustic Staggered Grid Modeling in IWAVE |
The IWAVE acoustic staggered grid scheme implements the Perfectly
Matched Layer (PML) approach to absorbing boundary conditions, in one of the
simpler of its many guises (a split field approach -
(Hu et al., 2007)). After some manipulation, the acoustic PML system
for the physical velocity and an artificial vector pressure
takes the form
Many implementations of PML, especially for elasticity, confine the extra PML fields (in this case, the extra pressure variables) to explicitly constructed zones around the boundary, and use the standard physical system in the domain interior. We judged that for acoustics little would be lost in either memory or efficiency, and much code bloat avoided, if we were to solve the system (3) in the entire domain.
Considerable experience and some theory (Moczo et al., 2006; Hu et al., 2007) suggest that the system 3 will effectively absorb waves that impinge on the boundary, emulating free space in the exterior of the domain, if the PML zones outside the physical domain in which are roughly a half-wavelength wide, and .
A simple 2D example illustrates the performance of this type of PML. The physical domain is a 1.8 x 7.6 km; the same domain is used in the experiments reported in the next section. A point source is placed at =40 m, km, with a Gaussian derivative time dependence with peak amplitude at about 5 Hz, and signifcant energy at 3 Hz but little below. The acoustic velocity is 1.5 km/s throughout the domain, so the effective maximum wavelength is roughly 500 m. The density is also constant, at 1 g/. A snapshot of the wavefield at 1.2 s after source onsiet (Figure 1), before the wave has reached the boundary of the domain, shows the expected circular wavefront. At 4.0 s, a simulation with zero-pressure boundary conditions on all sides of the physical domain produces the expected reflections, Figure 2. With PML zones of 250 m on the bottom and sides of the domain, so that only the top is a zero-pressure surface, and , the wave and its free-surfacec ghost both appear to leave the domain (Figure 3, plotted on the same grey scale). The maximum amplitude visible in Figure 2 is roughly , whereas the maximum amplitude in Figure 3 is . The actual reflection coefficient is likely less than , as the 2D free space field does not have a lacuna behind the wavefront, but decays smoothly, so the low end of the wavelet spectrum remains.
It is not possible to decrease the PML layer thickness much beyond the nominal longest half-wavelength and enjoy such small reflections. Figure 4 shows the field at 4.0 s with PML zones of width 100 m on bottom and sides, and an apparently optimal choice of . The maximum amplitude is , and a reflected wave is clearly visible at the same grey scale.
Acoustic Staggered Grid Modeling in IWAVE |