Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T04:12:04.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

Published online by Cambridge University Press:  20 August 2015

N. Anders Petersson  and
Björn Sjögreen
N. Anders Petersson*
Affiliation:
Center for Applied Scientific Computing, L-422, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA
Björn Sjögreen*
Affiliation:
Center for Applied Scientific Computing, L-422, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA
*
Corresponding author.Email:andersp@llnl.gov
Email:sjogreen@@llnl.gov
Get access

Abstract

We develop a stable finite difference approximation of the three-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 velocity-stress 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 relies on the summation-by-parts property of the discretization. The new scheme 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.

Keywords

65M0665M1274D0574J0586A1546.16.-x46.35.+z46.40.-f91.30.AbViscoelasticstandard linear solidfinite differencesummation by parts
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 193 - 225
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aki, K. and Richards, P. G., Quantitative Seismology, University Science Books, Sausalito, CA, USA, 2nd edition, 2002.Google Scholar
[2]Apsel, R. J. and Luco, J. E., On the green’s functions for a layered half-space, Bull. Seismol. Soc. Am., 73 (1983), 931–951.Google Scholar
[3]Bielak, J., Karaoglu, H. and Taborda, R., Efficient simulation of anelastic wave propagation by the octree-based finite element method-an improved approach, Seismol. Res. Lett., 81 (2010), 321. Seismological Society of America, 2010.Google Scholar
[4]Carcione, J. M., Wave fields in real media: wave propagation in anisotropic, anelastic and porous media, volume 31 of Handbook of Geophysical Exploration: Seismic Exploration, Pergamon, Elsevier Science, 2001.Google Scholar
[5]Day, S. M., Bielak, J., Dreger, D., Larsen, S., Graves, R., Pitarka, A. and Olsen, K. B., Test of 3D elastodynamic codes: Lifelines program task 1A02, Technical report, Pacific Earthquake Engineering Center, 2003.Google Scholar
[6]Day, S. M. and Bradley, C., Memory-efficient simulation of anelastic wave propagation, Bull. Seismol. Soc. Am., 91 (2001), 520–531.Google Scholar
[7]Day, S. M. and Minister, J. B., Numerical simulation of attenuated wavefields using a Padé approximant method, Geophys. J. R. Astr. Soc., 78 (1984), 105–118.Google Scholar
[8]Emmerich, H. and Korn, M., Incorporation of attenuation into time-dependent computations of seismic wave fields, Geophysics, 52(9) (1987), 1252–1264.CrossRefGoogle Scholar
[9]Glushenkov, V. D., A difference analog of the Korn inequality, J. Soviet Math., 46 (1989), 2176–2182.Google Scholar
[10]Graves, R. W. and Day, S. M., Stability and accuracy analysis of coarse-grain viscoelastic simulations, Bull. Seismol. Soc. Am., 93(1) (2003), 283–300.Google Scholar
[11]Horgan, C. O., Korn’s inequalities and their applications in continuum mechanics, SIAM Rev., 37 (1995), 491–511.Google Scholar
[12]Käser, M., Dumbser, M., de la Puente, J. and Igel, H., An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-III: viscoelastic attenuation, Geophys. J. Int., 168 (2007), 224–242.Google Scholar
[13]Komatitsch, D., Liu, Q., Tromp, J., Süss, P., Stidham, C. and Shaw, J. H., Simulations of ground motion in the Los Angeles basin based upon the spectral-element method, Bull. Seismol. Soc. Am., 94(1) (2004), 187–206.Google Scholar
[14]Komatitsch, D. and Tromp, J., Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophys. J. Int., 139 (1999), 806–822.Google Scholar
[15]Kristek, J. and Moczo, P., Seismic-wave propagation in viscoelastic media with material dis-continuities: a 3D fourth-order staggered-grid finite-difference modeling, Bull. Seismol. Soc. Am., 93(5) (2003), 2273–2280.Google Scholar
[16]Liu, H.-P., Anderson, D. L. and Kanamori, H., Velocity dispersion due to anelasticity, implications for seismology and mantle composition, Geophys. J. R. Astr. Soc., 47 (1976), 41–58.Google Scholar
[17]Liu, P. and Archuleta, R. J., Efficient modeling of Q for 3D numerical simulation of wave propagation, Bull. Seismol. Soc. Am., 96 (2006), 1352–1358.Google Scholar
[18]Moczo, P. and Kristek, J., On the rheological models used for time-domain methods of seismic wave propagation, Geophys. Res. Lett., 32 (2005), L01306.CrossRefGoogle Scholar
[19]Nilsson, S., Petersson, N. A., Sjögreen, B. and Kreiss, H.-O., Stable difference approximations for the elastic wave equation in second order formulation, SIAM J. Numer. Anal., 45 (2007), 1902–1936.Google Scholar
[20]Petersson, N. A. and Sjögreen, B., An energy absorbing far-field boundary condition for the elastic wave equation, Commun. Comput. Phys., 6 (2009), 483–508.CrossRefGoogle Scholar
[21]Petersson, N. A. and Sjögreen, B., User’s guide to WPP version 2.1. Technical report LLNL-SM-4874311, Lawrence Livermore National Laboratory, 2011. (Source code available from https://computation.llnl.gov/casc/serpentine).Google Scholar
[22]Petersson, N. A. and Sjögreen, B., Stable grid refinement and singular source discretization for seismic wave simulations, Commun. Comput. Phys., 8(5) (2010), 1074–1110.Google Scholar
[23]Savage, B., Komatitsch, D. and Tromp, J., Effects of 3D attenuation on seismic wave amplitude and phase measurments, Bull. Seismol. Soc. Am., 100(3) (2010), 1241–1251.Google Scholar
[24]Virieux, J., P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 51(4) (1986), 889–901.CrossRefGoogle Scholar
[25]Zhu, L., The frequency-wavenumber (FK) synthetic seismogram package, available at http://www.eas.slu.edu/People/LZhu/home.html, 1999.Google Scholar
[26]Zhu, L. and Rivera, L. A., A note on the dynamic and static displacements from a point source in multilayered media, Geophys. J. Int., 148 (2002), 619–627.Google Scholar