Full waveform inversion of surface seismic data using damped least-squares method

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2009 Vol.46, Issue 3 Preview Page Next Page

Full waveform inversion of surface seismic data using damped least-squares method 감쇠 최소자승법을 이용한 지표 탄성파 탐사자료의 전파형 역산: 주용환, 설순지, 변중무
Yonghwan Joo, Soon Jee Seol and Joongmoo Byun

30 June 2009. pp. 322-331

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Abstract

The gradient method has been used for full waveform inversion more often than the least-squares method that requires the huge computational memory in solving matrix equation. However, since misfits of the gradient method show an oscillating tendency and updated velocities converge to true values very slowly, the gradient method requires many iterations. In this study, we have presented an inversion algorithm to find the subsurface velocity structure using the damped least-squares method in the frequency domain. To verify our waveform inversion scheme, we applied the algorithm to the synthetic data, and found that comparatively correct solutions were acquired even with a few iterations. To reduce computer memory requirements and computing time, we utilized inversion blocks consisting of 2×2 elements, which are larger than that of forward modelling, and parallel processing technique based on MPI (Message Passing Interface). For investigating the effects of the initial model on inversion results, we compared and analyzed inversion results for two initial velocity models. One is a gradually increasing velocity model and the other is a smoothed horizontally layered velocity model. The latter was constructed based on the assumption that we have horizontally layered velocity information from zero-offset VSP or well logging data. The inversion result obtained by using the smoothed horizontally layered velocity model shows interfaces of subsurface velocity structures more clearly and provides more accurate velocity values than those using the gradually increasing velocity model. In addition, since the damped least-squares method calculates the sensitivity matrix unlike the steepest-descent method, we can identify the influence of frequency range on inversion results by comparing the sensitivity matrices for a couple of frequencies.

Keywords

Full waveform inversion

Frequency domain

Damped least-squares method

Zero-offset VSP data

전파형을 이용한 역산에서는 행렬식의 계산을 위해 대용량의 메모리를 요구하는 최소자승법보다는 대부분 경사법이 사용되어왔다. 하지만 경사법은 잔차가 진동하는 경향을 보이며 수렴 속도가 느려 상대적으로 많은 연산횟수를 필요로 하는 단점을 가진다. 본 연구에서는 주파수 영역에서 감쇠 최소자승법을 사용하여 지하의 속도모델을 효과적으로 찾아가는 역산 알고리듬을 구현하고, 이 알고리듬의 타당성 및 적용성 검토를 위해 수치모형 실험을 수행하여 적은 반복 연산 횟수에도 비교적 정확한 해를 찾아가는 것을 확인하였다. 감쇠 최소자승법의 메모리 문제를 해결하기 위하여 모델링에 이용된 셀의 2 × 2 크기로 역산블록을 설정하고 계산시간의 단축을 위하여 병렬처리 프로그램을 이용하였다. 또한 초기모델의 선정이 역산에 미치는 영향을 확인하기 위해 심도에 따라 속도가 증가하는 초기모델과 제로-오프셋 VSP나 음파검층 자료가 있을 때 이로부터 획득한 구간속도를 평활화한 초기모델을 사용하여 역산한 결과를 비교･분석하였다. 그 결과, VSP 자료나 음파검층 등의 자료를 초기속도모델로 사용한 경우 지하 속도구조의 층 경계들을 잘 영상화하고 보다 정확한 속도 정보를 제공함을 확인할 수 있었다. 또한 감쇠 최소자승법의 경우 최대경사법과 달리 민감도 행렬을 계산하기 때문에 이러한 정보를 이용하여 주파수에 따른 영향범위를 확인할 수 있다.

키워드

전파형 역산

주파수 영역

감쇠 최소자승법

VSP 자료

References

설순지, 주용환, 변중무, 2008, “탄성파 주파수영역 모델링을 위한 개선된 스펀지경계조건에 관한 연구,” 제91회 한국지구시스템공학회 추계학술발표논문집, 한국지구시스템공학회, 한양대학교, 서울, 10월 16일, pp. 171-173
Cho, H., Min, D. -J., Cha, Y. H., Shin, C. and Ko, S., 2008, “A comparative study of cascaded frequency‐selection strategies for 2D frequency‐domain acoustic waveform inversion,” SEG Expanded Abstracts of the 78th Annual meeting, Las Vegas, USA, pp. 1930-1934.
Choi, Y., Shin, C. and Min, D. -J., 2007, “Frequency domain elastic full waveform inversion using the new pseudo- Hessian matrix: elastic Marmousi-2 synthetic test,” SEG Expanded Abstracts of the 77th Annual meeting, San Antonio, USA, pp. 1908-1912.
Clayton, R. and Engquist, B., 1977, “Absorbing boundary conditions for acoustic and elastic wave equations,” Bull. Seis. Soc. Am., Vol. 67, No. 6, pp. 1529-1540.
Gauthier, O., Virieux, J. and Tarantola, A., 1986, “Twodimensional non-linear inversion of seismic waveforms: numerical results,” Geophysics, Vol. 51, pp. 1387-1403.
Hu, W., Abubakar, A. and Habashy, T. M., 2009, “Simultaneous multifrequency inversion of full-waveform seismic data,” Geophysics, Vol. 74, pp. R1-R14.
Kim, H. J., Song, Y. and Lee, K. H., 1999, “Inequality constraint in least-squares inversion of geophysical data,” Earth Planets Space, Vol. 51, No. 6, pp. 255-259.
Kolb, P., Collino, F. and Lailly, P., 1986, “Pre-stack inversion of a 1-D medium,” Proc. IEEE, Vol. 74., pp. 498-508.
Lailly, P., 1983, “The seismic inverse problem as a sequence of before stack migrations,” Conference on Inverse Scattering: Theory and Application, eds Bednar, J.B., Redner, R., Robinson, E. and Weglein, A., Soc. Industr. appl. Math., Philadelphia, PA.
Levenberg, K., 1944, “A Method for the Solution of Certain Non-Linear Problems in Least Squares,” Quart. Appl. Math., Vol. 2, pp. 164-168.
Lines, L. R. and Treitel, S., 1984, “Tutorial: A review of least-squares inversion and its application to geophysical problems,” Geophys. Prospect., Vol. 32, pp. 159-186.
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.
Menke, W., 1984, Geophysical data analysis: Discrete inverse theory, Academic Press Inc.
Mora, P. R., 1987, “Nonlinear two-dimensional elastic inversion of multioffset seismic data,” Geophysics, Vol. 52, pp. 1211-1228.
Oristaglio, M. L. and Worthington, M. H., 1980, “Inversion of surface and borehole electromagnetic data for twodimensional electric conductivity models,” Geophys. Prospect., Vol. 28, pp. 633-657.
Pratt, R. G., 1990, “Inverse theory applied to multi-source cross -hole tomography II: Elastic wave-equation method,” Geophys. Prospect., Vol. 38, pp. 311-330.
Pratt, R. G. and Worthington, M. H., 1990, “Inverse theory applied to multi-source cross-hole tomography Ⅰ: Acoustic wave-equation method,” Geophys. Prospect., Vol. 38, pp. 287-310.
Pratt, R. G., Shin, C. S. and Hicks, G. J., 1998, “Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion,” Geophys. J. Int, Vol. 133, pp. 341-362.
Shin, C., 1995, “Sponge boundary condition for frequencydomain modeling,” Geophysics, Vol. 60, No. 6, pp. 1870-1874.
Shin, C. S. and Cha, Y. H., 2008, “Waveform inversion in the Laplace domain,” Geophys. J. Int, Vol. 173, pp. 922-931.
Shin, C. and Min, D. -J., 2006, “Waveform inversion using a logarithmic wavefield,” Geophysics, Vol. 71 , No. 3, pp. R31-R42.
Sirgue, L. and Pratt, R. G., 2004, “Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies,” Geophysics, Vol. 69, pp. 231-248.
Tarantola, A., 1984, “Inversion of seismic reflection data in the acoustic approximation,” Geophysics, Vol. 49, pp. 1259-1266.

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 : 46
No :3
Pages :322-331

[1] 설순지, 주용환, 변중무, 2008, “탄성파 주파수영역 모델링을 위한 개선된 스펀지경계조건에 관한 연구,” 제91회 한국지구시스템공학회 추계학술발표논문집, 한국지구시스템공학회, 한양대학교, 서울, 10월 16일, pp. 171-173

[2] Cho, H., Min, D. -J., Cha, Y. H., Shin, C. and Ko, S., 2008, “A comparative study of cascaded frequency‐selection strategies for 2D frequency‐domain acoustic waveform inversion,” SEG Expanded Abstracts of the 78th Annual meeting, Las Vegas, USA, pp. 1930-1934.

[3] Choi, Y., Shin, C. and Min, D. -J., 2007, “Frequency domain elastic full waveform inversion using the new pseudo- Hessian matrix: elastic Marmousi-2 synthetic test,” SEG Expanded Abstracts of the 77th Annual meeting, San Antonio, USA, pp. 1908-1912.

[4] Clayton, R. and Engquist, B., 1977, “Absorbing boundary conditions for acoustic and elastic wave equations,” Bull. Seis. Soc. Am., Vol. 67, No. 6, pp. 1529-1540.

[5] Gauthier, O., Virieux, J. and Tarantola, A., 1986, “Twodimensional non-linear inversion of seismic waveforms: numerical results,” Geophysics, Vol. 51, pp. 1387-1403.

[6] Hu, W., Abubakar, A. and Habashy, T. M., 2009, “Simultaneous multifrequency inversion of full-waveform seismic data,” Geophysics, Vol. 74, pp. R1-R14.

[7] Kim, H. J., Song, Y. and Lee, K. H., 1999, “Inequality constraint in least-squares inversion of geophysical data,” Earth Planets Space, Vol. 51, No. 6, pp. 255-259.

[8] Kolb, P., Collino, F. and Lailly, P., 1986, “Pre-stack inversion of a 1-D medium,” Proc. IEEE, Vol. 74., pp. 498-508.

[9] Lailly, P., 1983, “The seismic inverse problem as a sequence of before stack migrations,” Conference on Inverse Scattering: Theory and Application, eds Bednar, J.B., Redner, R., Robinson, E. and Weglein, A., Soc. Industr. appl. Math., Philadelphia, PA.

[10] Levenberg, K., 1944, “A Method for the Solution of Certain Non-Linear Problems in Least Squares,” Quart. Appl. Math., Vol. 2, pp. 164-168.

[11] Lines, L. R. and Treitel, S., 1984, “Tutorial: A review of least-squares inversion and its application to geophysical problems,” Geophys. Prospect., Vol. 32, pp. 159-186.

[12] 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.

[13] Menke, W., 1984, Geophysical data analysis: Discrete inverse theory, Academic Press Inc.

[14] Mora, P. R., 1987, “Nonlinear two-dimensional elastic inversion of multioffset seismic data,” Geophysics, Vol. 52, pp. 1211-1228.

[15] Oristaglio, M. L. and Worthington, M. H., 1980, “Inversion of surface and borehole electromagnetic data for twodimensional electric conductivity models,” Geophys. Prospect., Vol. 28, pp. 633-657.

[16] Pratt, R. G., 1990, “Inverse theory applied to multi-source cross -hole tomography II: Elastic wave-equation method,” Geophys. Prospect., Vol. 38, pp. 311-330.

[17] Pratt, R. G. and Worthington, M. H., 1990, “Inverse theory applied to multi-source cross-hole tomography Ⅰ: Acoustic wave-equation method,” Geophys. Prospect., Vol. 38, pp. 287-310.

[18] Pratt, R. G., Shin, C. S. and Hicks, G. J., 1998, “Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion,” Geophys. J. Int, Vol. 133, pp. 341-362.

[19] Shin, C., 1995, “Sponge boundary condition for frequencydomain modeling,” Geophysics, Vol. 60, No. 6, pp. 1870-1874.

[20] Shin, C. S. and Cha, Y. H., 2008, “Waveform inversion in the Laplace domain,” Geophys. J. Int, Vol. 173, pp. 922-931.

[21] Shin, C. and Min, D. -J., 2006, “Waveform inversion using a logarithmic wavefield,” Geophysics, Vol. 71 , No. 3, pp. R31-R42.

[22] Sirgue, L. and Pratt, R. G., 2004, “Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies,” Geophysics, Vol. 69, pp. 231-248.

[23] Tarantola, A., 1984, “Inversion of seismic reflection data in the acoustic approximation,” Geophysics, Vol. 49, pp. 1259-1266.

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