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ps2d
According to [Carcione(2010)], the uniform-density pressure formulation is
![$\displaystyle \partial_t^2p =\omega_0^{2-2\beta} c^{2\beta} (\partial_x^2+\partial_z^2)^\beta+s$](img26.png) |
(12) |
where
is the body force per unit; the order
is
. When
, it introduces stronger attenuation. Regarding the fractional order of the Laplacian operator,
we may be able to update the wavefield by
![$\displaystyle p^{n+1}=2p^n-p^{n-1}+\Delta t^2 c^2F^{-1}[(-k_x^2-k_z^2)^\beta F p^n]$](img31.png) |
(13) |
Another choice to perform fractional order wave simulation is
![$\displaystyle \rho (\partial_x^\beta \rho^{-1}\partial_x^\beta +\partial_z^\beta\rho^{-1}\partial_z^\beta )$](img32.png) |
(14) |
which implies the following wavefield extrapolation
![$\displaystyle p^{n+1}=2p^n-p^{n-1}+\Delta t^2 c^2F^{-1}[(-1)^\beta(k_x^{2\beta}+k_z^{2\beta}) F p^n]$](img33.png) |
(15) |
with constant density
.
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
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2021-08-31