A perfectly matched layer for first- and second order time-domain wave equation
Authors: Rezaei Nevisi Arash , Bohlen Thomas 
Published: March 2024

The simulation of wave propagation is an essential part of several cutting-edge geophysical techniques, such as Full-Waveform Inversion (FWI) and Reverse Time Migration (RTM). Due to constraints in computational resources and memory capacity, wave propagation simulations are typically conducted within truncated media. In order to effectively absorb unwanted reflected waves at the boundaries of these simulations, specialized boundary layers are implemented. The Perfectly Matched Layer (PML) is widely acknowledged as a commonly employed technique in the field of seismology for its effectiveness as an absorbing boundary layer.Conventional PML suffers from some well-known drawbacks, including instabilities in long-time simulations and inadequate absorption in cases involving grazing incident and evanescent waves. These limitations can hinder the accuracy and reliability of numerical modeling of seismic wave propagation. CFS-PML (Complex Frequency Shifted Perfectly Matched Layer) addresses the limitations of traditional PML approaches and offers improved absorption and stability in numerical modeling. The CPML technique is widely regarded as highly effective when applied in the context of first-order systems of equations. Nevertheless, this method is not specifically designed for application in second-order displacement formulations. In such cases, alternative numerical techniques, such as finite-element methods and spectral-element methods, have demonstrated greater suitability. Previous studies&#160;have&#160;primarily focused on expanding the first-order formulation to the second-order.Another approach that to incorporate the CFS technique in the wave conventional PML is using ADEs (auxiliary differential equations). The ADE-CFS-PML method incorporates ADEs to drive wave equations equipped with PML in a more simple and straightforward manner than recursive convolution approach.Our contribution is to develop a general scheme that not only satisfies the first-order (velocity-displacement) staggard-grid system, but can easily incorporate in second-order wave equation and address the drawbacks of conventional PML effectively. The proposed scheme demonstrates comparable performance to CPML while avoiding the need for recursive convolution operations. Instead, it introduces the PML into the wave equation through ADEs, which is easily implementable, efficient, and compatible with existing codes and simplifies the implementation process.Our proposed scheme for implementation of the ADE-CFS-PML method has been tested on benchmark models with complex geological structures and has shown excellent performance by demonstrating its effectiveness in absorbing grazing incident waves and maintaining stability in long-term simulations.&#160;It effectively dampens grazing incidence waves and remains stable for long-term simulations. The scheme is suitable for large 3D models due to its on-the-fly computation capabilities, and its memory efficiency, since the coefficients are only varying in the PML area and are constant in the interior media.Overall, it offers an improved method for numerical modeling in various media and PDE orders, while addresses the limitations of traditional PML approaches. The proposed scheme demonstrates enhanced absorption and stability, making it a valuable tool for seismic wave propagation studies and other applications in geophysics and physics.