Abstract
The accuracy of finite-difference schemes of the 2nd and 4th order in 2-D and 3-D regular rectangular grids is studied. The method of designing the schemes and estimating their accuracy is proposed. The paper is devoted to the point schemes, expressed in terms of the discretized (point) values of the wave field and material parameters. Only the common schemes applicable in smooth parts of seismic models, outside structural interfaces, are taken into account. Finite differences at structural interfaces are studied elsewhere.
The inaccuracy of finite-difference schemes is governed, above all, by the error in the phase velocity, caused by discretization. This error is estimated for several finite-difference schemes. It is explicitly dependent on the direction of propagation and on wave polarization. The maximum phase-velocity error over all directions of propagation enables the accuracy of the individual schemes to be appreciated in order to select the best one. The proposed approach is general and applicable to other finite-difference schemes, for example, of the 6th and higher orders.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C. 1964).
Alford, R. M., Kelly, K. R., and Boore, D. M. (1974), Accuracy of Fnite-difference Modeling of the Acoustic Wave Equation, Geophysics 39, 834–842.
Alterman, Z., and Karal, F. C. (1968), Propagation of Elastic Waves in Layered Media by Finite-difference Method, Bull. Seismol. Soc. Am. 58, 367–398.
Bickley, W. G. (1941), Formulae for Numerical Differentiation, Math. Gaz. 25, 19–27.
Dablain, M. A. (1986), The Application of High-order Differencing to the Scalar Wave Equation, Geophysics 51, 54–66.
Fornberg, B. (1988), Generation of Finite Difference Formulas on Arbitrary Spaced Grids, Math. Computation 51, 699–706.
Igel, H., Mora, P., and Riollet, B. (1995), Anisotropic Wave Propagation Through Finite-difference Grids, Geophysics 60, 1203–1216.
Igel, H., Mora, P., and Rodrigues, D. (1991), 3-D Wave Propagation Using Finite Differences, Expanded Abstracts, 61st Annual Int. SEG Meeting, pp. 1577–1579, Soc. Exploration Geophysicists, Tulsa.
Keller, H. B., and Pereyra, V. (1978), Symbolic Generation of Finite Difference Formulas, Math. Computation 32, 955–971.
Levander, A. R. (1988), Fourth-order Finite-difference P-SV Seismograms. Geophysics 53, 1425–1436.
Madariaga, R. (1976), Dynamics of an Expanding Circular Fault, Bull. Seismol. Soc. Am. 66, 639–666.
Marfurt, K. J. (1984), Accuracy of Finite-difference and Finite-element Modeling of the Scalar and Elastic Wave Equations, Geophysics 49, 533–549.
Sei, A. (1993), Computational Cost of Finite-difference Elastic Waves Modeling, Expanded Abstracts, 63rd Annual Int. SEG Meeting, pp. 1065–1068, Soc. Exploration Geophysicists, Tulsa.
Stephen, R. A. (1988), A Review of Finite Difference Methods for Seismo-acoustics Problems at the Seafloor, Rev. Geophys. 26, 445–458.
Virieux, J., and Madariaga, R. (1982), Dynamic Faulting Studied by a Finite Difference Method, Bull. Seismol. Soc. Am. 72, 345–369.
Virieux, J. (1986), P-SV Wave Propagation in Heterogeneous Media: Velocity-stress Finite-difference Method, Geophysics 51, 889–901.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Verlag, Basel
About this chapter
Cite this chapter
Klimeš, L. (1996). Accuracy of Elastic Finite Differences in Smooth Media. In: Pšenčík, I., Červený, V., Klimeš, L. (eds) Seismic Waves in Laterally Inhomogeneous Media. Pageoph Topical Volumes. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9213-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9213-1_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-5648-4
Online ISBN: 978-3-0348-9213-1
eBook Packages: Springer Book Archive