### Stability Analysis for Finite Difference Scheme Used for Seismic Imaging Using Amplitude and Phase Portrait

Received: 22 September 2014    Accepted: 22 October 2014    Published: 14 January 2015
Abstract

A finite difference scheme is produced when partial derivatives in the partial differential equation(s) governing a physical phenomenon like the propagation of seismic waves through real media are replaced by a finite difference approximation. The result is a single algebraic equation which, when solved, provide an approximation to the solution of the original partial differential equation at selected points of a solution grid. Stability of a numerical scheme like that of finite difference scheme in the solution of partial differential equations is crucial for correctness and validity and it means that the error caused by small perturbation in the numerical solution remains bound. This paper considers important concepts like the amplitude and phase portrait used to analyze the stability of finite difference scheme. Applying these concepts produces an amplification factor and celerity for the components of the numerical solution.

 Published in Applied and Computational Mathematics (Volume 4, Issue 1) DOI 10.11648/j.acm.20150401.11 Page(s) 1-4 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2015. Published by Science Publishing Group
Keywords

Finite Difference Scheme, Stability, Seismic Waves, Phase Portrait, Amplitude Portrait, Amplification Factor, Celerity

References
 [1] Coates, R. T., and M. Schoenberg, 1995, Finite-difference modeling of faults and fractures: Geophysics, 60, 1514–1526, http://dx.doi.org/10.1190/1.1443884. [2] Frankel, A., and R. W. Clayton, 1986, Finite difference simulations of seismic scattering: Implications for the propagation of short-period seismic waves in the crust and models of crustal heterogeneity: Journal of Geophysical Research, 91, no. B6, 6465–6489, http://dx.doi.org/10.1029/JB091iB06p06465. [3] Hall, F., and Y. Wang, 2012, Seismic response of fractures by numerical simulation: Geophysical Journal International, 189, no. 1, 591–601, http://dx.doi.org/10.1111/j.1365- 246X.2012.05360.x. [4] Jan Thorbeck and Deyan Draganov.,2011, Finite-difference modeling experiment for seismic interferometry. Geophysics 76(6) H1 - H18 [5] Krebes, E.S and Quiroga-Goode, 1994. A standard finite difference scheme for the time domain computation of anelastic wavefield. Geophysics, 59, 290-296 [6] Peter Manning.,2008, Techniques to enhance accuracy and efficiency of finite-difference modeling for the propagation of elastic waves. PhD Theses. The University of Calgary [7] Saenger E.H.C., Radim, O.S., Kriiger, S.M., Schmalholz., Boris, G & Shapiro, S.A., 2007, Finite-difference modeling of wave propagation on Microscale: A snapshot of the work in progress. [8] Virieux, J., 1984, SH wave propagation in heterogeneous media: Velocity – stress finite-difference method: Geophysics, 49, 1933-1937. [9] Virieux, J., 1986, P – SV wave propagation in heterogeneous media: Velocity – Stress finite-difference method: Geophysics, 51, 889-901. [10] Yan, R., and X. B. Xie, 2010, A new angle -domain imaging condition for elastic reverse time migration: 80th Annual International Meeting, SEG, Expanded Abstracts, 3181–3186. [11] Yan, R., and X. B. Xie, 2012, An angle-domain imaging condition for elastic reverse-time migration and its application to angle gather extraction: Geophysics, 77, no. 5, S105–S115, http://dx.doi.org/10.1190/geo2011-0455.1. [12] Zheng, Y., X. Fang, M. Fehler, and D. Burns, 2011, Double -beam stacking to infer seismic properties of fractured reservoirs: 81st Annual International Meeting, SEG, Expanded Abstracts, 1809–1813.
• APA Style

Olowofela Joseph A., Akinyemi Olukayode D., Ajani Olumide Oyewale. (2015). Stability Analysis for Finite Difference Scheme Used for Seismic Imaging Using Amplitude and Phase Portrait. Applied and Computational Mathematics, 4(1), 1-4. https://doi.org/10.11648/j.acm.20150401.11

ACS Style

Olowofela Joseph A.; Akinyemi Olukayode D.; Ajani Olumide Oyewale. Stability Analysis for Finite Difference Scheme Used for Seismic Imaging Using Amplitude and Phase Portrait. Appl. Comput. Math. 2015, 4(1), 1-4. doi: 10.11648/j.acm.20150401.11

AMA Style

Olowofela Joseph A., Akinyemi Olukayode D., Ajani Olumide Oyewale. Stability Analysis for Finite Difference Scheme Used for Seismic Imaging Using Amplitude and Phase Portrait. Appl Comput Math. 2015;4(1):1-4. doi: 10.11648/j.acm.20150401.11

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journal = {Applied and Computational Mathematics},
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abstract = {A finite difference scheme is produced when partial derivatives in the partial differential equation(s) governing a physical phenomenon like the propagation of seismic waves through real media are replaced by a finite difference approximation. The result is a single algebraic equation which, when solved, provide an approximation to the solution of the original partial differential equation at selected points of a solution grid. Stability of a numerical scheme like that of finite difference scheme in the solution of partial differential equations is crucial for correctness and validity and it means that the error caused by small perturbation in the numerical solution remains bound. This paper considers important concepts like the amplitude and phase portrait used to analyze the stability of finite difference scheme. Applying these concepts produces an amplification factor and celerity for the components of the numerical solution.},
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Author Information
• Department of Physics, Federal University of Agriculture (FUNAAB), Abeokuta, Nigeria

• Department of Physics, Federal University of Agriculture (FUNAAB), Abeokuta, Nigeria

• Department of Physics and Solar Energy, Bowen University, Iwo, Nigeria

• Sections