Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Abstract

Abstract. Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

“…After some algebraic manipulations, we obtain the following PML equations (Gao et al . ) The damping coefficients and , in equation , are calculated using the following expression (Collino and Tsogka ) where D indicates the PML thickness and r represents the distance between current position (inside the PML) and PML inner boundary. The value of R for an efficient attenuation is often assumed to be in the range of 10 −3 to 10 −6 .…”

“…After that, we can notice that the system of equations (2) assume the same format as the Maxwell's equations. Furthermore, we can separate the pressure wavefield in the boundary zone into the wavefields P x and P z (Liu and Tao 1997;Gao et al 2017), to obtain the following system of equations:…”

“…Now, the PML equations can be obtained by applying a Fourier transform in time to equation (3) followed by a complex coordinate stretching approach expressed as (Hu et al 2007;Gao et al 2017):…”

“…In addition to that, the efficiency and low computational cost of the PML algorithm contributed to its popularity (Gao et al . ).…”

“…In the literature, the PML has been extensively compared to classical absorbing boundary conditions and showed to be clearly superior to suppress reflections on artificial boundaries. In addition to that, the efficiency and low computational cost of the PML algorithm contributed to its popularity (Gao et al 2017).…”