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Analytic and numerical solutions to the seismic wave equation in continuous media. (English) Zbl 1472.86012

Summary: This paper presents two approaches to mathematical modelling of a synthetic seismic pulse, and a comparison between them. First, a new analytical model is developed in two-dimensional Cartesian coordinates. Combined with an initial condition of sufficient symmetry, this provides a valuable check for the validity of the numerical method that follows. A particular initial condition is found which allows for a new closed-form solution. A numerical scheme is then presented which combines a spectral (Fourier) representation for displacement components and wave-speed parameters, a fourth-order Runge-Kutta integration method, and an absorbing boundary layer. The resulting large system of differential equations is solved in parallel on suitable enhanced performance desktop hardware in a new software implementation. This provides an alternative approach to forward modelling of waves within isotropic media which is efficient, and tailored to rapid and flexible developments in modelling seismic structure, for example, shallow depth environmental applications. Visual comparisons of the analytic solution and the numerical scheme are presented.

MSC:

86A15 Seismology (including tsunami modeling), earthquakes
74L05 Geophysical solid mechanics

Software:

OpenSWPC; Wiggle
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Full Text: DOI arXiv Link

References:

[1] Aki K, Richards PG. 2002 Quantitative seismology. San Francisco, CA: Freeman.
[2] Kennett BL. 2001 The seismic wavefield: volume 1, introduction and theoretical development. Cambridge, UK: Cambridge University Press. · Zbl 1048.86009
[3] Lamb H. 1904 On the propagation of tremors over the surface of an elastic solid. Phil. Trans. R. Soc. Lond. A 203, 1-42. (doi:10.1098/rsta.1904.0013) · JFM 34.0859.02 · doi:10.1098/rsta.1904.0013
[4] Garvin WW. 1956 Exact transient solution of the buried line source problem. Proc. R. Soc. Lond. A 234, 528-541. (doi:10.1098/rspa.1956.0055) · Zbl 0071.40101 · doi:10.1098/rspa.1956.0055
[5] Kausel E. 2013 Lamb’s problem at its simplest. Proc. R. Soc. A 469, 20120462. (doi:10.1098/rspa.2012.0462) · Zbl 1371.74030 · doi:10.1098/rspa.2012.0462
[6] Gosselin-Cliche B, Giroux B. 2014 3D frequency-domain finite-difference viscoelastic-wave modeling using weighted average 27-point operators with optimal coefficients. Geophysics 79, T169-T188. (doi:10.1190/geo2013-0368.1) · doi:10.1190/geo2013-0368.1
[7] Carcione JM. 1993 Seismic modeling in viscoelastic media. Geophysics 58, 110-120. (doi:10.1190/1.1443340) · doi:10.1190/1.1443340
[8] Diaz J, Ezziani A. 2010 Analytical solution for waves propagation in heterogeneous acoustic/porous media. Part I: the 2D case. Commun. Comput. Phys. 7, 171-194. (doi:10.4208/cicp.2009.08.148) · Zbl 1364.74032 · doi:10.4208/cicp.2009.08.148
[9] Igel H. 2017 Computational seismology: a practical introduction. Oxford, UK: Oxford University Press. · Zbl 1458.86001
[10] Fichtner A, Igel H, Bunge HP, Kennett BL. 2009 Simulation and inversion of seismic wave propagation on continental scales based on a spectral-element method. JNAIAM 4, 11-22. · Zbl 1189.86003
[11] Komatitsch D, Vilotte JP. 1998 The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seismol. Soc. Am. 88, 368-392. · Zbl 0974.74583
[12] Maeda T, Takemura S, Furumura T. 2017 OpenSWPC: an open-source integrated parallel simulation code for modeling seismic wave propagation in 3D heterogeneous viscoelastic media. Earth Planets Space 69, 102. (doi:10.1186/s40623-017-0687-2) · doi:10.1186/s40623-017-0687-2
[13] Sens-Schönfelder C, Wegler U. 2011 Passive image interferometry for monitoring crustal changes with ambient seismic noise. C. R. Geosci. 343, 639-651. (doi:10.1016/j.crte.2011.02.005) · doi:10.1016/j.crte.2011.02.005
[14] Tsai VC, Minchew B, Lamb MP, Ampuero JP. 2012 A physical model for seismic noise generation from sediment transport in rivers. Geophys. Res. Lett. 39, L02404. (doi:10.1029/2011gl050255) · doi:10.1029/2011gl050255
[15] Paul Winberry J, Anandakrishnan S, Wiens DA, Alley RB. 2013 Nucleation and seismic tremor associated with the glacial earthquakes of Whillans Ice Stream, Antarctica. Geophys. Res. Lett. 40, 312-315. (doi:10.1002/grl.50130) · doi:10.1002/grl.50130
[16] Walters SJ, Forbes LK. 2019 Fully 3D Rayleigh-Taylor instability in a Boussinesq fluid. ANZIAM J. 61, 286-304. (doi:10.21914/anziamj.v61i0.13700) · Zbl 1422.76066 · doi:10.21914/anziamj.v61i0.13700
[17] Boyd JP. 2001 Chebyshev and Fourier spectral methods. New York, NY: Dover. · Zbl 0994.65128
[18] Nelder JA, Mead R. 1965 A simplex method for function minimization. Comput. J. 7, 308-313. (doi:10.1093/comjnl/7.4.308) · Zbl 0229.65053 · doi:10.1093/comjnl/7.4.308
[19] Assi H, Cobbold RS. 2017 Compact second-order time-domain perfectly matched layer formulation for elastic wave propagation in two dimensions. Math. Mech. Solids 22, 20-37. (doi:10.1177/1081286515569266) · Zbl 1371.74135 · doi:10.1177/1081286515569266
[20] Assi H. 2016 Time-domain modeling of elastic and acoustic wave propagation in unbounded media, with application to metamaterials. Doctoral dissertation, University of Toronto, Canada.
[21] Atkinson KE. 1978 An introduction to numerical analysis. New York, NY: John Wiley & Sons. · Zbl 0402.65001
[22] Reeh N, Fisher DA, Koerner RM, Clausen HB. 2005 An empirical firn-densification model comprising ice lenses. Ann. Glaciol. 42, 101-106. (doi:10.3189/172756405781812871) · doi:10.3189/172756405781812871
[23] Schlegel R, Diez A, Löwe H, Mayer C, Lambrecht A, Freitag J, Miller H, Hofstede C, Eisen O. 2019 Comparison of elastic moduli from seismic diving-wave and ice-core microstructure analysis in Antarctic polar firn. Ann. Glaciol. 60, 220-230. (doi:10.1017/aog.2019.10) · doi:10.1017/aog.2019.10
[24] Portugal R. 2019 See https://au.mathworks.com/matlabcentral/fileexchange/38691-wiggle (accessed 18 September 2019).
[25] PGI Community Edition. 2019 See https://www.pgroup.com/products/community.htm (accessed 14 May 2019).
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