Two-layer model

The purpose of our first example is to investigate the accuracy of the solution of the constant-$Q$ wave equation using the proposed low-rank scheme, in the presence of a sharp contrast in both velocity and $Q$. We use an isotropic two-layer model with $v=1800\;m/s$ in the top layer and $v=3600\;m/s$ in the bottom layer. The model is discretized on a $200 \times 200$ grid, with a spatial sampling of $8\;m$ along both $X$ and $Z$ directions. An explosive source with a peak frequency of $50\;Hz$ is located at the center. The reference frequency is $\omega_0=1500\;Hz$. Wavefield snapshots are taken at $t=330\;ms$. Figure 1a shows the acoustic case, in which the model has the velocity discontinuity but no attenuation ($Q=\infty$). For comparison, Figure 1b demonstrates the effect of homogeneous attenuation where $Q=30$. Both velocity dispersion and amplitude loss can be observed. In Figure 1c, we set $Q=30$ in the top layer and $Q=100$ in the bottom layer. The transmitted arrival exhibits less attenuation compared with that in Figure 1b. In Figure 1d, both velocity and $Q$ remain the same as those in Figure 1c; however, the fractional power of Laplacians, $\gamma$, is taken to be the averaged value, which corresponds to the original implementation by Zhu and Harris (2014). To compare the results modeled by the two strategies, a middle trace at $x=800\;m$ is extracted from both wavefield snapshots (Figures 1c and 1d). Figure 2 shows the two traces, along with their difference. Errors caused by using a constant $\gamma$ instead of a spatially varying $\gamma$ can be easily observed.

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Figure 1. Viscoacoustic wave propagation in a two-layer model: (a) acoustic modeling with $v=1800\;m/s$ in top layer and $v=3600\;m/s$ in bottom layer; (b) same velocity as (a), homogeneous $Q=30$; (c) same velocity as (a), $Q=30$ in top layer and $Q=100$ in bottom layer; (d) wavefield propagated using a constant averaged fractional power $\gamma$ using same model as (c).

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Figure 2. Traces at $x=800\;m$ extracted from wavefield snapshots and their difference. Red, long-dashed line corresponds to averaged $\gamma$; blue, solid line corresponds to variable $\gamma$; black, shot-dashed line is their difference.