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Laplace-domain Waveform Inversion using the Pseudo-Hessian of the Logarithmic Objective Function and the Levenberg-Marquardt Algorithm

로그 목적함수의 유사 헤시안을 이용한 라플라스 영역 파형 역산과 레벤버그-마쿼트 알고리듬

  • Ha, Wansoo (Department of Energy Resources Engineering, Pukyong National University)
  • 하완수 (부경대학교 에너지자원공학과)
  • Received : 2019.10.10
  • Accepted : 2019.11.24
  • Published : 2019.11.30

Abstract

The logarithmic objective function used in waveform inversion minimizes the logarithmic differences between the observed and modeled data. Laplace-domain waveform inversions usually adopt the logarithmic objective function and the diagonal elements of the pseudo-Hessian for optimization. In this case, we apply the Levenberg-Marquardt algorithm to prevent the diagonal elements of the pseudo-Hessian from being zero or near-zero values. In this study, we analyzed the diagonal elements of the pseudo-Hessian of the logarithmic objective function and showed that there is no zero or near-zero value in the diagonal elements of the pseudo-Hessian for acoustic waveform inversion in the Laplace domain. Accordingly, we do not need to apply the Levenberg-Marquardt algorithm when we regularize the gradient direction using the pseudo-Hessian of the logarithmic objective function. Numerical examples using synthetic and field datasets demonstrate that we can obtain inversion results without applying the Levenberg-Marquardt method.

파형 역산에 사용하는 로그 목적함수는 관측 자료와 모델링 자료의 로그값의 차이를 최소화하는 목적함수이다. 라플라스 영역 파형 역산에서는 주로 로그 목적함수와 유사 헤시안의 대각 성분을 이용하여 최적화를 수행한다. 이 때 유사 헤시안의 대각 성분이 0 또는 0에 가까운 값이 되는 것을 막기 위해 레벤버그-마쿼트 알고리듬을 적용한다. 본 연구에서는 로그 목적함수의 유사 헤시안의 대각 성분을 분석하여 음향파 라플라스 영역 파형 역산에서는 유사 헤시안의 대각 성분이 0 또는 0에 가까운 값을 가지지 않음을 보였다. 따라서 로그 목적함수의 유사 헤시안을 이용한 경사 방향 정규화시 레벤버그-마쿼트 알고리듬을 적용할 필요가 없다. 수치 예제에서 인공합성 자료와 현장 자료를 이용해 레벤버그-마쿼트 기법 없이도 역산 결과를 얻을 수 있음을 보였다.

Keywords

References

  1. Al-Yahya, K., 1989, Velocity analysis by iterative profile migration, Geophysics, 54(6), 718-729. https://doi.org/10.1190/1.1442699
  2. Aminzadeh, F., Burkhard, N., Nicoletis, L., and Rocca, F., 1994, SEG/EAEG 3-D modeling project: 2nd update, The Leading Edge, 13(9), 949-952. https://doi.org/10.1190/1.1437054
  3. Bae, H. S., Pyun, S., Chung, W., Kang, S. G., and Shin, C., 2012, Frequency‐domain acoustic‐elastic coupled waveform inversion using the Gauss‐Newton conjugate gradient method, Geophys. Prospect., 60(3), 413-432. https://doi.org/10.1111/j.1365-2478.2011.00993.x
  4. Brossier, R., 2011, Two-dimensional frequency-domain viscoelastic full waveform inversion Parallel algorithms, optimization and performance, Comput. Geosci., 37(4), 444-455. https://doi.org/10.1016/j.cageo.2010.09.013
  5. Chapman, C., and Pratt, R., 1992, Traveltime tomography in anisotropic media - I. Theory, Geophys. J. Int., 109(1), 1-19. https://doi.org/10.1111/j.1365-246X.1992.tb00075.x
  6. Chung, W., Shin, J., Bae, H. S., Yang, D., and Shin, C., 2012, Frequency domain elastic waveform inversion using the Gauss-Newton method, J. Seism. Explor., 21(1), 29-48.
  7. Ha, T., Chung, W., and Shin, C., 2009, Waveform inversion using a back-propagation algorithm and a Huber function norm, Geophysics, 74(3), R15-R24. https://doi.org/10.1190/1.3112572
  8. Ha, W., Chung, W., Park, E., and Shin, C., 2012a, 2-D acoustic Laplace-domain waveform inversion of marine field data, Geophys. J. Int., 190(1), 421-428. https://doi.org/10.1111/j.1365-246X.2012.05487.x
  9. Ha, W., Chung, W., and Shin, C., 2012b, Pseudo-Hessian matrix for the logarithmic objective function in full waveform inversion, J. Seism. Explor., 21(3), 201-214.
  10. Kwon, J., Jin, H., Calandra, H., and Shin, C., 2017, Interrelation between Laplace constants and the gradient distortion effect in Laplace-domain waveform inversion, Geophysics, 82(2), R31-R47. https://doi.org/10.1190/geo2015-0670.1
  11. Lee, J., and Ha, W., 2019, Laplace-domain waveform inversion using the l-BFGS method, Geosy. Eng., 22(4), 214-224. https://doi.org/10.1080/12269328.2018.1543032
  12. Levenberg, K., 1944, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2, 164-168. https://doi.org/10.1090/qam/10666
  13. Liu, Z., and Bleistein, N., 1995, Migration velocity analysis: Theory and an iterative algorithm, Geophysics, 60(1), 142-153. https://doi.org/10.1190/1.1443741
  14. Marfurt, K. J., 1984, Accuracy of finite-difference and finiteelement modeling of the scalar and elastic wave equations, Geophysics, 49(5), 533-549. https://doi.org/10.1190/1.1441689
  15. Marquardt, D., 1963, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11(2), 431-441. https://doi.org/10.1137/0111030
  16. Metivier, L., Bretaudeau, F., Brossier, R., Operto, S., and Virieux, J., 2014, Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures, Geophys. Prospect., 62(6), 1353-1375. https://doi.org/10.1111/1365-2478.12136
  17. Metivier, L., and Brossier, R., 2016, The SEISCOPE optimization toolbox: A large-scale nonlinear optimization library based on reverse communication, Geophysics, 81(2), F1-F15. https://doi.org/10.1190/geo2016-0922-TIOgeo.1
  18. Pratt, R., Shin, C., and Hicks, G., 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion, Geophys. J. Int., 133(2), 341-362. https://doi.org/10.1046/j.1365-246X.1998.00498.x
  19. Pujol, J., 2007, The solution of nonlinear inverse problems and the Levenberg-Marquardt method, Geophysics, 72(4), W1-W16. https://doi.org/10.1190/1.2732552
  20. Pyun, S., Son, W., and Shin, C., 2011, Implementation of the Gauss-Newton method for frequency-domain full waveform inversion using a logarithmic objective function, J. Seism. Explor., 20(2), 1-14.
  21. Sava, P., and Biondi, B., 2004, Wave-equation migration velocity analysis. I. Theory, Geophys. Prospect., 52(6), 593-606. https://doi.org/10.1111/j.1365-2478.2004.00447.x
  22. Shin, C., Jang, S., and Min, D.-J., 2001, Improved amplitude preservation for prestack depth migration by inverse scattering theory, Geophys. Prospect., 49(5), 592-606. https://doi.org/10.1046/j.1365-2478.2001.00279.x
  23. Shin, C., and Min, D.-J., 2006, Waveform inversion using a logarithmic wavefield, Geophysics, 71(3), R31-R42. https://doi.org/10.1190/1.2194523
  24. Shin, C., and Ha, W., 2008, A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains, Geophysics, 73(5), VE119-VE133. https://doi.org/10.1190/1.2953978
  25. Shin, C., and Cha, Y., 2009, Waveform inversion in the Laplace-Fourier domain, Geophy. J. Int., 177(3), 1067-1079. https://doi.org/10.1111/j.1365-246X.2009.04102.x
  26. Shin, C., Pyun, S., and Bednar, J. B., 2007, Comparison of waveform inversion, part 1: Conventional wavefield vs logarithmic wavefield, Geophys. Prospect., 55(4), 449-464. https://doi.org/10.1111/j.1365-2478.2007.00617.x
  27. Shin, C., and Cha, Y., 2008, Waveform inversion in the Laplace domain, Geophys. J. Int., 173(3), 922-931. https://doi.org/10.1111/j.1365-246X.2008.03768.x
  28. Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49(8), 1259-1266. https://doi.org/10.1190/1.1441754
  29. Virieux, J., and Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74(6), WCC1-WCC26. https://doi.org/10.1190/1.3238367
  30. White, D. J., 1989, Two‐Dimensional Seismic Refraction Tomography, Geophys. J. Int., 97(2), 223-245. https://doi.org/10.1111/j.1365-246X.1989.tb00498.x
  31. Yilmaz, O., 2001, Seismic data analyis: processing, inversion and interpretation of seismic data, Vol. I, Soc. Expl. Geophys., 288-323.