Skip to main content


We’d like to understand how you use our websites in order to improve them. Register your interest.

Differentiation operation in the wave equation for the pseudospectral method with a staggered mesh


In the present analysis we introduced a calculation strategy of the staggered grid differentiation by using the real value FFT and real inverted FFT for the pseudospectral method and applied the technique to seismic wave simulation. The calculation method introduced here is one third faster on average than the traditional differentiation method by using the complex FFT. The introduced differentiation strategy is very efficient in economy. For example we apply the staggered grid differentiation by real valued FFT to the simulation of seismic wave propagation in inhomogeneous medium. The results show the validity of the present method.


  1. Aki, K. and P. G. Richards, Quantitative seismology theory and methods, pp. 224–243, W. H. Freeman and Company, San Francisco, U.S.A., 1980.

  2. Cerjan, C., D. Kosloff, R. Kosloff, and M. Reshef, A nonreflecting boundary condition for discrete acoustic and elastic wave equation, Geophysics, 50, 705–708, 1985.

  3. Chen, H. W., Staggered grid pseudospectral simulation in viscoacoustic wavefield simulation, J. Acoust. Soc. Am., 100, 120–131, 1996.

  4. Fornberg, B., The pseudospectral method: Comparisons with finite difference for the elastic wave equation, Geophysics, 52, 483–501, 1987.

  5. Furumura, T. and H. Takenaka, A stable method for numerical differentiation of data with discontinuities at end-points by means of fourier transform-symmetric differentiation, Geophys. Expl., 45, 303–309, 1992 (in Japanese).

  6. Furumura, T., B. L. N. Kennett, and H. Takenaka, Parallel 3-D pseudospectral simulation of seismic wave propagation, Geophysics, 63, 279–288, 1998.

  7. Herrmann, R. B., SH-wave generation by dislocation sources—A numerical study, BSSA, 69, 1–15, 1979.

  8. Irikura, K., K. Kudo, H. Okada, and T. Sasatani, The effect of surface geology on seismic motion, in Chap. 3, simultaneous simulation of Kobe, A.A. Balkema, 1594 pp., Rotterdam Press, Netherlands, 1999.

  9. Kawase, H., S. Matsushima, R. W. Graves, and P. G. Somervelle, Three-dimensional wave propagation—The cause of the damage belt during the 1995 Hyogo-ken Nanbu earthquake, Zishin, 50, 431–449, 1998.

  10. Kosloff, R. and E. Baysal, Forward modeling by a Fourier method, Geophysics, 42, 1402–1412, 1982.

  11. Orszag, S. A., Comparison of pseudospectral and spectral approximation, Stud. Appl. Math., 51, 253–259, 1972.

  12. Ozdenvar, T. and G. A. McMechan, Causes and reduction of numerical artefactsinpseudo-spectral wavefield extrapolation, Geophys. J. Int., 126, 819–828, 1996.

  13. Pitarka, A., K. Irikura, T. Iwata, and H. Sekikuchi, Three-dimensional simulation of the near-fault ground motion for the 1995 Hyogo-ken Nanbu (Kobe), Japan, earthquake, BSSA, 88, 428–440, 1998.

  14. Reshef, M., D. Kosloff, M. Edwards, and C. Hsiung, Three-dimensional elastic modeling by the Fourier method, Geophysics, 53, 1184–1193, 1988.

  15. Saatcilar, R. and S. Ergintav, Solving elastic wave equations with the Hartley method, Geophysics, 56, 247–278, 1991.

  16. Virieux, J., SH wave propagation in heterogeneous media: velocity-stress finite difference method, Geophysics, 49, 1933–1957, 1984.

  17. Virieux, J., p-sv wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 51, 889–901, 1986.

  18. Witte, D. C., The pseudospectral method for simulating wave propagation, Doctor Thesis, Columbia University, 29 pp., 1989.

  19. Zhao, Z.-X. and R. Kubota, The distribution of strong ground motion from uppermost crustal structure-comparison with disaster from the Hyogoken nanbu earthquake, Proceeding of the 97th SEGJ Conference 71–74, The Society of Exploration Geophysicists of Japan, October, 1997, Sapporo, Japan, 1997.

  20. Zhao, Z.-X. and R. Kubota, The empirical Green’s function method by using simulated waveforms, in The Effect of Surface geology on Seismic Motion-Recent Progress and New Horizon on ESG Study, Volume 3, edited by K. Irikura, K. Kudo, H. Okata, and T. Sasatani, pp. 1435–1442, A.A. Balkema, Rotterdam Press, Netherlands, 1999.

  21. Zhou, B., On the use of the Hartley transform in geophysical applications, Geophysics, 57, 196–197, 1992.

Download references

Author information



Corresponding author

Correspondence to Zhixin Zhao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, Z., Xu, J. & Horiuchi, S. Differentiation operation in the wave equation for the pseudospectral method with a staggered mesh. Earth Planet Sp 53, 327–332 (2001).

Download citation


  • Ground Motion
  • Stagger Grid
  • Seismic Wave Propagation
  • Elastic Wave Equation
  • Traveltime Curve