A modified wave equation with diffusion effects and its application as a smoothing scheme for seismic wave propagation simulations

ABSTRACT

We present a new modified wave equation and apply it to develop a smoothing scheme for seismic wave propagation simulations. With mathematical rigour we show that the solution of the new equation, which is derived as an analog of the advection–diffusion equation, can be obtained by the spatial convolution between a solution of the wave equation and the heat kernel and has a finite propagation speed and a diffusion effect. Using numerical experiments we show that the smoothing scheme based on the modified wave equation has the following advantages. Firstly, it preserves the characteristics of the wave equation such as wave propagation speed. Secondly, it selectively removes the short-wavelength components of the solution. Lastly, the energy decreases slowly after the short-wavelength components have been removed. Since our smoothing scheme can be implemented by adding simple correction terms to usual schemes, it can easily be applied to the seismic wave equation.

KEYWORDS:

• Wave equation
• diffusion effects
• smoothing scheme
• seismic wave equation
• long-period ground motion evaluation

2010 AMS Subject Classifications:

• 35K15
• 35L15
• 65M22
• 65Z05

Acknowledgments

We would like to thank Dr Masayuki Yamada and Koji Hada for the constructive discussions. We would like to thank Enago (www.enago.jp) for the English language review. We are grateful to referees for careful reading our manuscript and for giving useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.