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An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme



In finite-difference (FD) acoustic forward modelling, the parameter settings for the perfectly matched layer (PML) are case-dependent. There is no explicit PML formula that can be applied for most acoustic models without tuning, especially, for the fourth-order FD scheme. In this paper, we propose an explicit PML formula for the acoustic frequency-domain FD with second-order and fourth-order accuracies, respectively. The fourth-order FD scheme uses a special treatment for the boundary. The number of points in the PML is fixed to be 10 and 15 for the second-order and fourth-order FD schemes, respectively. The maximum artificial attenuation parameter associated with the PML formula is automatically calculated based on the FD grid size and the value of the compressional velocity of the boundary cells of the interior domain. Numerical tests confirm that this empirical formula achieves the desired accuracy for 2-D and 3-D media for grid sizes varying from 1 to 200 m. For the fourth-order FD scheme, the proposed PML formula works effectively up to 25 points per wavelength for both 2-D and 3-D media. Beyond that, the error of the PML discretization becomes larger than the discretization error in the interior domain. For such cases and to keep a fourth-order accuracy, a larger number of points in the PML (thicker PML region) needs to be employed.