Skip to main content

Finite-Difference Methods

  • Chapter
  • First Online:
Full Seismic Waveform Modelling and Inversion

Abstract

The finite-difference method can be considered the classical and most frequently applied method for the numerical simulation of seismic wave propagation. It is based on the approximation of an exact derivative ∂x f (x i) at a grid position x i in terms of the function f evaluated at a finite number of neighbouring grid points.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alford, R. M., Kelly, K. R., Boore, D. M.: Accuracy of finite difference modeling of the acoustic wave equation. Geophysics 39, 834–842 (1974)

    Article  Google Scholar 

  • Alterman, Z., Karal, F. C.: Propagation of elastic waves in layered media by finite-difference methods. Bull. Seism. Soc. Am. 58, 367–398 (1968)

    Google Scholar 

  • Bohlen, T.: Parallel 3-D viscoelastic finite difference modelling. Comput. Geosci. 28, 887–899 (2002)

    Article  Google Scholar 

  • Boore, D. M.: Love waves in nonuniform waveguides: finite difference calculations. J. Geophys. Res. 75, 1512–1527 (1970)

    Article  Google Scholar 

  • Charney, J., Frjørtoft, R., von Neumann, J.: Numerical integration of the barotropic vorticity equation. Tellus 2, 237–254 (1950)

    Article  Google Scholar 

  • Cole, J. B.: A nearly exact second-order finite-difference time-domain wave propagation algorithm on a coarse grid. Comput. Phys. 8, 730–734 (1994)

    Article  Google Scholar 

  • Cole, J. B.: High accuracy solution of Maxwell’s equations using nonstandard finite differences. Comput. Phys. 11, 287–292 (1997)

    Article  Google Scholar 

  • Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen 100, 32–74 (1928)

    Article  Google Scholar 

  • Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of heat conduction type. Proc. Camb. Philos. Soc. 43, 50–67 (1947)

    Article  Google Scholar 

  • Dablain, M. A.: The application of high-order differencing to the scalar wave equation. Geophysics 51, 54–66 (1986)

    Article  Google Scholar 

  • Frankel, A., Clayton, R. W.: A finite-difference simulation of wave propagation in two-dimensional random media. Bull. Seism. Soc. Am. 74, 2167–2186 (1984)

    Google Scholar 

  • Frankel, A., Clayton, R. W.: Finite-difference simulations of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models of crustal heterogeneity. J. Geophys. Res. 91, 6465–6489 (1986)

    Article  Google Scholar 

  • Frankel, A., Vidale, J.: A 3-dimensional simulation of seismic waves in the Santa Clara valley, California, from a Loma Prieta aftershock. Bull. Seism. Soc. Am. 82, 2045–2074 (1992)

    Google Scholar 

  • Furumura, T., Chen, L.: Large-scale parallel simulation and visualisation of 3D seismic wave field using the Earth Simulator. Comput. Model. Eng. Sci. 6, 153–168 (2004)

    Google Scholar 

  • Graves, R. W.: Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bull. Seism. Soc. Am. 86, 1091–1106 (1996)

    Google Scholar 

  • Hicks, G. J.: Arbitrary source and receiver positioning in finite-difference schemes using Kaiser windowed sinc functions. Geophysics 67, 156–166 (2002)

    Article  Google Scholar 

  • Holberg, O.: Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena. Geophys. Prosp. 35, 629–655 (1987)

    Article  Google Scholar 

  • Igel, H., Mora, P., Riollet, B.: Anisotropic wave propagation through finite-difference grids. Geophysics 60, 1203–1216 (1995)

    Article  Google Scholar 

  • Igel, H., Djikpéssé, H., Tarantola, A.: Waveform inversion of marine reflection seismograms for P impedance and Poisson’s ratio. Geophys. J. Int. 124(2), 363–371 (1996)

    Article  Google Scholar 

  • Igel, H., Nissen-Meyer, T., Jahnke, G.: Wave propagation in 3D spherical sections. Effects of subduction zones. Phys. Earth Planet. Int. 132, 219–234 (2002)

    Article  Google Scholar 

  • Jo, C.-H., Shin, C., Suh, J. H.: An optimal 9-point, finite-difference, frequency-space 2-D scalar wave extrapolator. Geophysics 61, 529–537 (1996)

    Article  Google Scholar 

  • Kelly, K. R., Ward, R. W., Treitel, S., Alford, R. M.: Synthetic seismograms: a finite-difference approach. Geophysics 41, 2–27 (1976)

    Article  Google Scholar 

  • Kennett, B. L. N., Furumura, T.: Stochastic waveguide in the lithosphere: Indonesian subduction zone to Australian craton. Geophys. J. Int. 172, 363–382 (2008)

    Article  Google Scholar 

  • Kristek, J., Moczo, P., Archuleta, R. J.: Efficient methods to simulate planar free surface in the 3D 4th-order staggered-grid finite-difference schemes. Stud. Geophys. Geod. 46, 355–381 (2002)

    Article  Google Scholar 

  • Kristekova, M., Kristek, J., Moczo, P., Day, S. M.: Misfit criteria for quantitative comparison of seismograms. Bull. Seism. Soc. Am. 96, 1836–1850 (2006)

    Article  Google Scholar 

  • Kristekova, M., Kristek, J., Moczo, P.: Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals. Geophys. J. Int. 178, 813–825 (2009)

    Article  Google Scholar 

  • Levander, A. R.: Fourth-order finite-difference P-SV seismograms. Geophysics 53, 1425–1436 (1988)

    Article  Google Scholar 

  • Madariaga, R.: Dynamics of an expanding circular fault. Bull. Seism. Soc. Am. 67, 163–182 (1976)

    Google Scholar 

  • Moczo, P., Bystrický, Kristek, J., Carcione, J. M., Bouchon, M.: Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures. Bull. Seism. Soc. Am. 87, 1305–1323 (1997)

    Google Scholar 

  • Moczo, P., Kristek, J., Halada, L.: 3D fourth-order staggered-grid finite-difference schemes: stability and grid dispersion. Bull. Seism. Soc. Am. 90, 587–603 (2000)

    Article  Google Scholar 

  • Moczo, P., Kristek, J., Vavrycuk, V., Archuleta, R. J., Halada, J.: 3D heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli. Bull. Seism. Soc. Am. 92, 3042–3066 (2002)

    Article  Google Scholar 

  • Moczo, P., Kristek, J., Gális, M.: Simulation of the planar free surface with near-surface lateral discontinuities in the finite-difference modeling of seismic motion. Bull. Seism. Soc. Am. 94, 760–768 (2004)

    Article  Google Scholar 

  • Moczo, P., Kristek, J., Galis, M., Pazak, P., Balazovjech, M.: The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. Acta Phys. Slovaca 57(2), 177–406 (2007)

    Google Scholar 

  • Moczo, P., Kristek, J., Galis, M., Pazak, P.: On accuracy of the finite-difference and finite-element schemes with respect to P-wave to S-wave speed ratio. Geophys. J. Int. 182, 493–510 (2010)

    Google Scholar 

  • Mora, P.: Elastic finite differences with convolutional operators. Stanford Explor. Proj. Rep. 48, 277–289 (1986)

    Google Scholar 

  • Muir, F., Dellinger, J., Etgen, J., Nichols, D.: Modeling elastic fields across irregular boundaries. Geophysics 57, 1189–1193 (1992)

    Article  Google Scholar 

  • Ohminato, T., Chouet, B.: A free-surface boundary condition for including 3D topography in the finite-difference method. Bull. Seism. Soc. Am. 87, 494–515 (1997)

    Google Scholar 

  • Pratt, R. G.: Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model. Geophysics 64, 888–901 (1999)

    Article  Google Scholar 

  • Robertsson, J. O. A., Blanch, J. O., Symes, W. W.: Viscoelastic finite-difference modeling. Geophysics 59(9), 1444–1456 (1994)

    Article  Google Scholar 

  • Robertsson, J. O. A.: A numerical free-surface condition for elastic/visco-elastic finite-difference modeling in the presence of topography. Geophysics 61, 1921–1934 (1996)

    Article  Google Scholar 

  • Virieux, J.: SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics 49, 1933–1957 (1984)

    Article  Google Scholar 

  • Virieux, J.: P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics 51, 889–901 (1986)

    Article  Google Scholar 

  • Wang, H. J., Igel, H., Gallovic, F., Cochard, A., Ewald, M.: Source-related variations of ground motions in 3D media: application to the Newport-Inglewood fault, Los Angeles basin. Geophys. J. Int. 175, 202–214 (2008)

    Article  Google Scholar 

  • Zahradník, J., Urban, L.: Effect of a simple mountain range on underground seismic motion. Geophys. J. R. Astron. Soc. 79, 167–183 (1984)

    Google Scholar 

  • Zahradník, J., Moczo, P., Hron, F.: Testing four elastic finite difference schemes for behaviour at discontinuities. Bull. Seism. Soc. Am. 83, 107–129 (1993)

    Google Scholar 

  • Bleibinhaus, F., Hole, J. A., Ryberg, T., Fuis, G. S.: Structure of the California Coast Ranges and San Andreas Fault at SAFOD from seismic waveform inversion and reflection imaging. J. Geophys. Res. 112, doi:10.1029/2006JB004611 (2007)

    Google Scholar 

  • Boore, D. M.: Finite-difference methods for seismic wave propagation in heterogeneous materials. In: Methods in Computational Physics, vol. 11, Academic Press, New York, NY (1972)

    Google Scholar 

  • Dessa, J. X., Operto, S., Kodaira, S., Nakanishi, A., Pascal, G., Virieux, J., Kaneda, Y.: Multiscale seismic imaging of the eastern Nankai trough by full waveform inversion. Geophys. Res. Lett. 31, doi:10.1029/2004GL020453 (2004)

    Google Scholar 

  • Nissen-Meyer, T.: Numerical simulation of 3D seismic wave propagation through subduction zones. Diploma Thesis, Ludwig-Maximilians-Universität München (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andreas Fichtner .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Fichtner, A. (2011). Finite-Difference Methods. In: Full Seismic Waveform Modelling and Inversion. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15807-0_3

Download citation

Publish with us

Policies and ethics