Three-Dimensional Time-Domain Elastic Wave Modeling using Finite-Difference Method

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2006 Vol.43, Issue 1 Preview Page Next Page

Three-Dimensional Time-Domain Elastic Wave Modeling using Finite-Difference Method 유한차분법을 이용한 시간영역 3차원 탄성파 모델링: 민동주, 김형수, 유해수, 김광희
Dong-Joo Min, Hyoung-Soo Kim, Hai Soo Yoo and Kwang-Hee Kim

28 February 2006. pp. 65-75

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Abstract

We describe a 3-D time-domain, displacement-based, finite-difference elastic wave modeling algorithm that is constructed by defining the material properties within cubes. In the conventional displacement-based finite-difference method, both displacements and material properties are defined at the nodal points, whereas in our finite-difference algorithm, displacements are still assigned to the nodal points but material properties are defined within cubes. In this case, free-surface boundary conditions, which describe stress-free at the free surface, are naturally satisfied by the changes of material properties. Through numerical examples for infinite homogeneous and semi-infinite homogeneous models, we could examine the accuracy of the 3-D finite-difference elastic wave modeling algorithm. Some numerical examples showed that the 3-D finite-differenc elastic wave modeling algorithm satisfies the reciprocity theorem and successfully generates synthetic seismograms and snapshots.

Keywords

3-D

Time-domain

Finite-difference

Elastic wave modeling

Free-surface boundary condition

시간영역에서 변위만을 이용하여 수행되는 3차원 변위근사 유한차분 탄성파 모델링에서 자유면 경계조건을 정확히 묘사하기 위한 방법으로 육면체 내부에 매질의 물성을 정의할 것을 제안한다. 매질의 물성과 변위를 모두 격자점에 정의하는 기존의 방법과는 달리, 매질의 물성을 육면체 내부에 정의할 경우 추가적인 자유면 경계조건을 사용하지 않고도 매질의 물성의 변화만으로 자유면에서 응력이 사라진다는 자유면 경계조건을 만족시킬 수 있다. 무한 균질 매질과 반무한 균질 매질에 대하여 수치적인 해와 해석적인 해를 비교함으로써 육면체 내부에 매질의 물성을 정의하는 3차원 유한차분 탄성파 모델링법의 정확성을 확인하였다. 송신원과 수진기의 위치가 서로 바뀐 경우에 대하여 수치적으로 구한 해들을 비교함으로써 육면체 내부에 매질의 물성을 정의하는 3차원 유한차분 탄성파 모델링 알고리듬이 상반원리 (reciprocity theorem)를 만족시킨다는 것을 알 수 있었다. 또한, 수평층 구조 및 단층구조에 대하여 합성탄성파 단면도 및 스냅단면도를 성공적으로 작성할 수 있었다.

키워드

3차원

시간영역

변위근사

유한차분

탄성파 모델링

자유면 경계조건

References

민동주, 유해수, 2002, “시간영역 가중평균 유한요소법을 이용한 탄성파 모델링:한국자원공학회지,” 제39권 5호, pp. 360-373.
민동주, 유해수, 2003, “시간영역 변위근사 유한차분법의 자유면 경계조건,” 물리탐사, 제6권 2호, pp. 77-86.
Bayliss, A., Jordan, K.E., LeMeusurier, B.J., and Turkel, E., 1986, “A fourth-order accurate finite-difference scheme for the computation of elastic waves,” Bull. Seis. Soc. Am., Vol. 76, pp. 115-1132.
Ewing, W.M., Jardetzky, W.S., and Press, F., 1957, Elastic waves in layered media, McGrow-Hill Book Co.
Fornberg, B., 1987, “The pseudospectral method: Comparisons with finite differences for the elastic wave equation,” Geophysics, Vol. 52, pp. 483-501.
Fornberg, B., 1989, “Pseudospectral approximation of the elastic wave equation on a staggered grid,” Proc. of the 59th annual international meeting, Society of Exploration Geophysics, SEG, pp. 1047-1049.
Jo, C.-H., Shin, C., and Suh, J.H., 1996, “An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator,” Geophysics, Vol. 61, pp. 529-537.
Kelly, K.R., Ward, R.W., Treitel, S., and Alford, R.M., 1976, “Synthetic seismograms: A finite-difference app roach,” Geophysics, Vol. 41, pp. 2-27.
Kosloff, D.D. and Baysal, E., 1982, “Forward modeling by a Fourier method,” Geophysics, Vol. 47, pp. 1402- 1412.
Levander, A.R., 1988, “Fourth-order finite-difference P-SV seismograms,” Geophysics, Vol, 53, pp. 1425- 1436.
Marfurt, K.J., 1984, “Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations,” Geophysics, Vol. 49, pp. 533-549.
Min, D.-J. and Kim, H.-S., 2005, “Feasibility of the Surface-Wave Method for the Assessment of Physical properties of a Dam using Numerical Analysis,” will be published in Journal of Applied Geophysics.
Min, D.-J., Shin, C., Kwon, B.-D., and Chung, S., 2000, “Improved frequency-domain elastic wave modeling using weighted-averaging difference operators,” Geophysics, Vol. 65, pp. 884-895.
Min, D.-J., Shin, C., Pratt, R.G., and Yoo, H.S., 2003, “Weighted-averaging finite-element method for 2-D elastic wave equations in the frequency domain,” Bull. Seism. Soc. Am., Vol. 93, pp. 904-921.
Min, D.-J., Shin, C., and Yoo, H.S., 2004, “Free surface boundary condition in finite-differenc elastic wave modeling,” Bull. Seism. Soc. Am., Vol. 94, pp. 237- 250.
Pilant, W.L., 1979, Elastic waves in the earth, Elsevier Scientific Publishing Co.
Shin, C., 1995, “Sponge boundary condition for frequency- domain modeling,” Geophysics, Vol. 60, pp. 1870-1874.
Shin, C. and Sohn, H., 1998, “A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operators,” Geophysics, Vol. 63, pp. 289-296.
Stekl, I. and Pratt, R.G., 1998, “Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators,” Geophysics, Vol. 63, pp. 1779-1794.
Virieux, J., 1986, “P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,” Geophysics, Vol. 51, pp. 889-901.

Information

Publisher :The Korean Society of Mineral and Energy Resources Engineers
Publisher(Ko) :한국자원공학회
Journal Title :Journal of the Korean Society for Geosystem Engineering
Journal Title(Ko) :한국지구시스템공학회지
Volume : 43
No :1
Pages :65-75

[1] 민동주, 유해수, 2002, “시간영역 가중평균 유한요소법을 이용한 탄성파 모델링:한국자원공학회지,” 제39권 5호, pp. 360-373.

[2] 민동주, 유해수, 2003, “시간영역 변위근사 유한차분법의 자유면 경계조건,” 물리탐사, 제6권 2호, pp. 77-86.

[3] Bayliss, A., Jordan, K.E., LeMeusurier, B.J., and Turkel, E., 1986, “A fourth-order accurate finite-difference scheme for the computation of elastic waves,” Bull. Seis. Soc. Am., Vol. 76, pp. 115-1132.

[4] Ewing, W.M., Jardetzky, W.S., and Press, F., 1957, Elastic waves in layered media, McGrow-Hill Book Co.

[5] Fornberg, B., 1987, “The pseudospectral method: Comparisons with finite differences for the elastic wave equation,” Geophysics, Vol. 52, pp. 483-501.

[6] Fornberg, B., 1989, “Pseudospectral approximation of the elastic wave equation on a staggered grid,” Proc. of the 59th annual international meeting, Society of Exploration Geophysics, SEG, pp. 1047-1049.

[7] Jo, C.-H., Shin, C., and Suh, J.H., 1996, “An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator,” Geophysics, Vol. 61, pp. 529-537.

[8] Kelly, K.R., Ward, R.W., Treitel, S., and Alford, R.M., 1976, “Synthetic seismograms: A finite-difference app roach,” Geophysics, Vol. 41, pp. 2-27.

[9] Kosloff, D.D. and Baysal, E., 1982, “Forward modeling by a Fourier method,” Geophysics, Vol. 47, pp. 1402- 1412.

[10] Levander, A.R., 1988, “Fourth-order finite-difference P-SV seismograms,” Geophysics, Vol, 53, pp. 1425- 1436.

[11] Marfurt, K.J., 1984, “Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations,” Geophysics, Vol. 49, pp. 533-549.

[12] Min, D.-J. and Kim, H.-S., 2005, “Feasibility of the Surface-Wave Method for the Assessment of Physical properties of a Dam using Numerical Analysis,” will be published in Journal of Applied Geophysics.

[13] Min, D.-J., Shin, C., Kwon, B.-D., and Chung, S., 2000, “Improved frequency-domain elastic wave modeling using weighted-averaging difference operators,” Geophysics, Vol. 65, pp. 884-895.

[14] Min, D.-J., Shin, C., Pratt, R.G., and Yoo, H.S., 2003, “Weighted-averaging finite-element method for 2-D elastic wave equations in the frequency domain,” Bull. Seism. Soc. Am., Vol. 93, pp. 904-921.

[15] Min, D.-J., Shin, C., and Yoo, H.S., 2004, “Free surface boundary condition in finite-differenc elastic wave modeling,” Bull. Seism. Soc. Am., Vol. 94, pp. 237- 250.

[16] Pilant, W.L., 1979, Elastic waves in the earth, Elsevier Scientific Publishing Co.

[17] Shin, C., 1995, “Sponge boundary condition for frequency- domain modeling,” Geophysics, Vol. 60, pp. 1870-1874.

[18] Shin, C. and Sohn, H., 1998, “A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operators,” Geophysics, Vol. 63, pp. 289-296.

[19] Stekl, I. and Pratt, R.G., 1998, “Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators,” Geophysics, Vol. 63, pp. 1779-1794.

[20] Virieux, J., 1986, “P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,” Geophysics, Vol. 51, pp. 889-901.

Journal of the Korean Society of Mineral and Energy Resources Engineers ISSN:2288-0291(Print) 2288-2790(Online) 한국자원공학회지

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