Abstract
This paper presents a simple, flexible way of introducing stress-free boundary conditions for including cracks and cavities in 2D elastic media by a finite difference method (FDM). The surfaces of cracks and cavities are discretized in a staircase on a rectangular grid scheme. When zero-stress is applied to free surfaces, the resulting finite difference schemes require a set of adjacent fictitious points. These points are classified based on the geometry of the free surface and their displacement is computed as a prior step to later calculation of motion on the crack surface. The use of this extra line of points does not involve a significant drain on computational resources. However, it does provide explicit finite difference schemes and the construction of displacement on the free surfaces by using the correct physical boundary conditions. An accuracy analysis compares the results to an analytical solution. This quantitative analysis uses envelope and phase misfits. It estimates the minimum number of points per wavelength necessary to achieve suitable results. Finally, the method is employed to compute displacement in various models with cavities in the P-SV formulation. The results show suitable construction of the reflected P and S waves from the free surface as well as diffraction produced by these cavities.
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References
Boore D.M., 1972. Finite difference methods for seismic wave propagation in heterogeneous materials. In: B. Alder, S. Fernbach and B.A. Bolt (Ed.), Seismology: Body Waves and Sources. Methods in Computational Physics, Vol. 11, Academic Press, New York, 1–37.
Carcione J.M., 1998. Scattering of elastic waves by a plane crack of finite width in a transversely isotropic medium. Int. J. Numer. Anal. Methods Geomech., 22, 263–275.
Fornberg B., 1988. The pseudospectral method: accurate representation of interfaces in elastic wave calculations. Geophysics, 60, 1514–1526.
Frankel A. and Clayton R., 1986. Finite difference simulation of seismic scattering: Implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity. J. Geophys. Res., 91, 6465–6489.
Hestholm S. and Ruud B., 2002. 3D free-boundary conditions for coordinate-transform finite-difference seismic modelling. Geophys. Prospect., 50, 463–474.
Higdon R.L., 1991. Absorbing boundary conditions for elastic waves. Geophysics, 56, 231–241.
Hong T.K. and Kennet B.L.N., 2004. Scattering of elastic waves in media with a random distribution of fluid-filled cavities: theory and numerical modelling. Geophys. J. Int., 159, 961–977.
Iturrarán-Viveros U., Vai R. and Sánchez-Sesma F.J., 2005. Scattering of elastic waves by a 2-D crack using the Indirect Boundary Element Method (IBEM). Geophys. J. Int., 162, 927–934.
Kawashima K., Omote R., Ito T., Fujita H. and Shima T., 2002. Nonlinear acoustic response through minute surface cracks: FEM simulation and experimentation. Ultrasonics, 40, 611–615.
Kristek J., Moczo P. and Archuleta R.J., 2002. Efficients methods to simulate planar free surface in the 3D 4th-order staggered-grid finite difference schemes. Stud. Geophys. Geod., 46, 355–382.
Kristeková M., Kristek J., Moczo P. and Day S.M., 2006. Misfit criteria for quantitative comparison of seismograms. Bull. Seismol. Soc. Amer. (in print).
Krüger O.S., Saenger E.H. and Shapiro S.A., 2005. Scattering and diffraction by a single crack: an accuracy analysis of the rotated staggered grid. Geophys. J. Int., 162, 25–31.
Levander A.R., 1988. Fourth-Order finite-difference P-SV seismograms. Geophysics, 53, 1425–1436.
Liu E. and Zhang Z., 2001. Numerical study of elastic wave scattering by cracks or inclusions using the boundary integral equation method. Geophysics, 62, 253–265.
Lysmer J. and Drake L.A., 1972. A finite element method for seismology. In: B. Alder, S. Fernbach and B.A. Bolt (Ed.), Seismology: Body Waves and Sources. Methods in Computational Physics, Vol. 11, Academic Press, New York, 181–215.
Madariaga R., 1976. Dynamics of an expanding circular fault. Bull. Seismol. Soc. Amer., 66, 639–666.
Marfurt K.J., 1984. Accuracy of finite-difference and finite-element modelling of the scalar and elastic wave equations. Geophysics, 49, 533–549.
Mittet R., 2002. Free-surface boundary conditions for elastic staggered-grid modelling schemes. Geophysics, 67, 1616–1623.
Moczo P., Bystrický E., Kristek J., Carcione J.M. and Bouchon M., 1997. Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures. Bull. Seismol. Soc. Amer., 87, 1305–1323.
Moczo P., Kristek J., Vavryčuk V., Archuleta R.J. and Halada L., 2002. 3D Heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities. Bull. Seismol. Soc. Amer., 92, 3042–3066.
Moczo P., Kristek J. and Gális M., 2004. Simulation of the planar free surface with near-surface lateral discontinuities in the finite-difference modeling of seismic motion. Bull. Seismol. Soc. Amer., 94, 760–768.
Munasinghe M. and Farnell G.W., 1973. Finite difference analysis of Rayleigh wave scattering at vertical discontinuities. J. Geophys. Res., 78, 2454–2466.
Ohminato T. and Chouet B.A., 1997. A free-surface boundary condition for including 3D topography in the finite-difference method. Bull. Seismol. Soc. Amer., 87, 494–515.
Pérez-Ruiz J.A., Luzón F. and García-Jerez A. 2005. Simulation of an irregular free surface with a displacement finite-difference scheme. Bull. Seismol. Soc. Amer., 95, 2216–2231.
Pointer T., Liu E. and Hudson J.A., 1998. Numerical modelling of seismic waves scattered by hydrofractures: application of the indirect boundary element method. Geophys. J. Int., 135, 289–303.
Robertsson J.O.A., 1996. A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography. Geophysics, 61, 1921–1934.
Rodrigues D., 1993. Large Scale Modelling of Seismic Wave Propagation. PhD Thesis, Ecole Centrale Paris.
Rodríguez-Castellanos A., Luzón F. and Sánchez-Sesma F.J., 2005. Diffraction of seismic waves in an elastic, cracked halfplane using a boundary integral formulation. Soil Dyn. Earthq. Eng., 25, 827–837.
Roth M. and Korn M., 1993. Single scattering theory versus numerical modelling in 2-D random media. Geophys. J. Int., 112, 124–140.
Saenger E.H. and Shapiro S.A., 2002. Effective velocities in fractured media: a numerical study using the rotated staggered finite-difference grid. Geophys. Prospect., 50, 183–194.
Sánchez-Sesma F.J. and Iturrarán-Viveros U., 2001. Scattering and diffraction of SH waves by a finite crack: an analytical solution. Geophys. J. Int., 145, 749–758.
Van Baren G.B., Mulder W.A. and Herman G.C., 2001. Finite-difference modelling of scalar-wave propagation in cracked media. Geophysics, 66, 267–276.
Virieux J., 1984, SH-wave propagation in heterogeneous media: velocity stress finite difference method. Geophysics, 49, 1933–1957.
Vlastos S., Liu E., Main I.G. and Li X.Y., 2003. Numerical simulation of wave propagation in media with discrete distribution of fractures: effects of fractures sizes and spatial distributions. Geophys. J. Int., 152, 649–668.
Zahradník J., O’Leary P. and Sochacki J., 1994. Finite-difference schemes for elastic waves based on the integration approach. Geophysics, 59, 928–937.
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Pérez-Ruiz, J.A., Luzón, F. & García-Jerez, A. Scattering of elastic waves in cracked media using a finite difference method. Stud Geophys Geod 51, 59–88 (2007). https://doi.org/10.1007/s11200-007-0004-9
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DOI: https://doi.org/10.1007/s11200-007-0004-9