Volume 12, Issue 1
Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

N. Anders Petersson & Björn Sjögreen

Commun. Comput. Phys., 12 (2012), pp. 193-225.

Published online: 2012-12

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  • Abstract

Wedevelopastablefinitedifferenceapproximationofthethree-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocitystress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof reliesonthe summation-by-parts propertyof the discretization. The newscheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.


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@Article{CiCP-12-193, author = {N. Anders Petersson and Björn Sjögreen}, title = {Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation}, journal = {Communications in Computational Physics}, year = {2012}, volume = {12}, number = {1}, pages = {193--225}, abstract = {

Wedevelopastablefinitedifferenceapproximationofthethree-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocitystress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof reliesonthe summation-by-parts propertyof the discretization. The newscheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.201010.090611a}, url = {http://global-sci.org/intro/article_detail/cicp/7289.html} }
TY - JOUR T1 - Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation AU - N. Anders Petersson & Björn Sjögreen JO - Communications in Computational Physics VL - 1 SP - 193 EP - 225 PY - 2012 DA - 2012/12 SN - 12 DO - http://dor.org/10.4208/cicp.201010.090611a UR - https://global-sci.org/intro/cicp/7289.html KW - AB -

Wedevelopastablefinitedifferenceapproximationofthethree-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocitystress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof reliesonthe summation-by-parts propertyof the discretization. The newscheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.


N. Anders Petersson & Björn Sjögreen. (1970). Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation. Communications in Computational Physics. 12 (1). 193-225. doi:10.4208/cicp.201010.090611a
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