Importance of double-pole CFS-PML for broad-band seismic wave simulation and optimal parameters selection

The perfectly matched layer (PML) is an efficient absorbing technique for numerical wave simulation. The complex frequency-shifted PML (CFS-PML) introduces two additional parameters in the stretching function to make the absorption frequency dependent. This can help to suppress converted evanescent waves from near grazing incident waves, but does not efficiently absorb low-frequency waves below the cut-off frequency. To absorb both the evanescent wave and the low-frequency wave, the double-pole CFS-PML having two poles in the coordinate stretching function was developed in computational electromagnetism. Several studies have investigated the performance of the double-pole CFS-PML for seismic wave simulations in the case of a narrowband seismic wavelet and did not find significant difference comparing to the CFS-PML. Another difficulty to apply the double-pole CFS-PML for real problems is that a practical strategy to set optimal parameter values has not been established. In this work, we study the performance of the double-pole CFS-PML for broad-band seismic wave simulation. We find that when the maximum to minimum frequency ratio is larger than 16, the CFS-PML will either fail to suppress the converted evanescent waves for grazing incident waves, or produce visible low-frequency reflection, depending on the value of a. In contrast, the double-pole CFS-PML can simultaneously suppress the converted evanescent waves and avoid low-frequency reflections with proper parameter values. We analyse the different roles of the double-pole CFS-PML parameters and propose optimal selections of these parameters. Numerical tests show that the double-pole CFS-PML with the optimal parameters can generate satisfactory results for broad-band seismic wave simulations.