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Nonsplit Convolutional-PML (CPML) for acoustics

Another approach to improve the behavior of the discrete PML at grazing incidence consists in modifying the complex coordinate transform used classically in the PML to introduce a frequency-dependent term that implements a Butterworth-type filter in the layer. This approach has been developed for Maxwell’s equations named convolutional PML (CPML) (Roden and Gedney, 2000) or complex frequency shifted-PML (CFS-PML). The key idea is that for waves whose incidence is close to normal, the presence of such a filter changes almost nothing because absorption is already almost perfect. But for waves with grazing incidence, which for geometrical reasons do not penetrate very deep in the PML, but travel there a longer way in the direction parallel to the layer, adding such a filter will strongly attenuate them and will prevent them from leaving the PML with significant energy .

Define $ Ax=\frac{\partial p}{\partial x}$ , $ Az=\frac{\partial p}{\partial z}$ . Then the acoustic wave equation reads

$\displaystyle \frac{\partial^2 p}{\partial t^2}=v^2\left(\frac{\partial Ax}{\partial x}+\frac{\partial Az}{\partial z}\right).

To combine the absorbing effects into the acoustic equation, we merely need to combine two convolution terms into the above equations:

\begin{equation*}\left\{ \begin{split}\frac{\partial^2 p}{\partial t^2}&=v^2\lef...
..._x\\ Az&=\frac{\partial p}{\partial z}+\Phi_z \end{split} \right.\end{equation*} (25)

where $ \Psi_x$ , $ \Psi_z$ are the convolution terms of $ Ax$ and $ Az$ ; $ \Phi_x$ , $ \Phi_z$ are the convolution terms of $ Px$ and $ Pz$ . These convolution terms can be computed via the following relation:

\begin{equation*}\left\{ \begin{split}\Psi_x^n=b_x \Psi_x^{n-1}+(b_x-1) \partial...
..._z \Phi_z^{n-1}+(b_z-1) \partial_z^{n-1/2}P\\ \end{split} \right.\end{equation*} (26)

where $ b_x=e^{-d(x)\Delta t}$ and $ b_z=e^{-d(z)\Delta t}$ . More details about the derivation of C-PML, the interested readers are referred to Collino and Tsogka (2001) and Komatitsch and Martin (2007).

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