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Fractional Laplacian

ps2d

According to [Carcione(2010)], the uniform-density pressure formulation is

$\displaystyle \partial_t^2p =\omega_0^{2-2\beta} c^{2\beta} (\partial_x^2+\partial_z^2)^\beta+s$ (12)

where $ s$ is the body force per unit; the order $ \beta$ is $ 1\leq \beta\leq 2$ . When $ \beta\rightarrow 2$ , it introduces stronger attenuation. Regarding the fractional order of the Laplacian operator, we may be able to update the wavefield by

$\displaystyle p^{n+1}=2p^n-p^{n-1}+\Delta t^2 c^2F^{-1}[(-k_x^2-k_z^2)^\beta F p^n]$ (13)

Another choice to perform fractional order wave simulation is

$\displaystyle \rho (\partial_x^\beta \rho^{-1}\partial_x^\beta +\partial_z^\beta\rho^{-1}\partial_z^\beta )$ (14)

which implies the following wavefield extrapolation

$\displaystyle p^{n+1}=2p^n-p^{n-1}+\Delta t^2 c^2F^{-1}[(-1)^\beta(k_x^{2\beta}+k_z^{2\beta}) F p^n]$ (15)

with constant density $ \rho$ .

A snapshot using the code provided in the appendix is shown in Figure 1, in which we use the sponge boundary condition.

snapshotwidth=0.7A snapshot of 2-D acoustic propagation using PSM method


next up previous [pdf]

Next: Conclusion Up: Fourier pseudo spectral method Previous: Computing derivatives using Fourier

2021-08-31