AccScience Publishing / JSE / Article
Submit to JSE
Cite this article
1
Download
73
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
ARTICLE

Fast line-search for full waveform inversion in the Huber norm

XIAONA MA
Show Less
Institute of Geophysics, China Earthquake Administration, Beijing 100081, P.R. China.,
JSE 2021, 30(4), 347–364;
Submitted: 8 July 2019 | Accepted: 10 February 2021 | Published: 1 August 2021
© 2021 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Download PDF
Cite
XML
Abstract

Ma, X., 2021. Fast line-search for full waveform inversion in the Huber norm. Journal of Seismic Exploration, 30: 347-364. The misfit functions decide the robust performance of full-waveform inversion (FWD) when the data is contaminated by noise. Huber norm combines the L2 norm when residuals are small and L1 norm when residuals are large, which compromise the merits of them. It not only improves the anti-noise ability with the L1 norm, but also keeps smoothness for small residuals with the L2 norm. Line searches are always important for FWI process. Step-length can typically be calculated using an inexact or exact line-search method. Optimal step-length can prevent the over- or under-estimations, and make FWI to reach a global minimum along searching direction with fast convergence rate. Exact line searches for local optimization methods can be performed very efficiently for computing solutions in robust norms, thereby promoting convergence rate of FWI. Therefore, we derive an exact line search method, i.e., the analytical step-length method (ASLM), for the Huber norm. Through numerical tests on noise-free and noisy data of Overthrust model, we demonstrate the efficiency of ASLM for Huber norm. In addition, we also compare Huber with L2 norm on the data contaminated by non-Gaussian noise, such as ground-motion noise. We think that ASLM is an efficient optimal step-length estimation method for the Huber norm in FWI. Meanwhile, the Huber norm makes the FWI more robust than the L2 norm alone.

Keywords
time-domain full waveform inversion
Huber norm
exact line-search method
analytical step-length method
References
  1. Bérenger, J.P., 1994. A perfectly matched layer for the absorption of electromagneticwaves. J. Computat. Phys., 114: 185-200.
  2. Bunks, C., Saleck, F.M., Zaleski, S. and Chavent, G., 1995. Multiscale seismicwaveform inversion. Geophysics, 60: 1457-1473.
  3. Bube, K.P. and Nemeth, T., 2007. Fast line searches for the robust solution of linearsystems in the hybrid and Huber norms. Geophysics, 72(2): A13-A17.
  4. Boonyasiriwat, C., Valasek, P., Routh, P., Cao, W., Schuster, G.T. and Macy, B., 2009.
  5. An efficient multiscale method for time-domain waveform tomography. Geophysics,74(6): WCC59-WCC68.
  6. Brossier, R., Operto, S. and Virieux, J., 2009. Robust elastic frequency domain fullwaveform inversion using the L1 norm. Geophys. Res. Lett., 36: L20310.
  7. Brossier, R., Operto, S. and Virieux, J., 2010. Which data residual norm for robust elasticfrequency-domain full waveform inversion? Geophysics, 75(3): R37-R46.
  8. Crase, E., Pica, A., Noble, M., McDonald, J. and Tarantola, A., 1990. Robust elasticnonlinear waveform inversion: Application to real data. Geophysics, 55: 527-538.
  9. Djikpéssé, H.A. and Tarantola, A., 1999. Multiparameter L1 norm waveform fitting:
  10. Interpretation of Gulf of Mexico reflection seismograms. Geophysics, 64:1023-1035.
  11. Dos Santos, A.W.G. and Pestana. R.C., 2015. Time-domain multiscale full-waveforminversion using the rapid expansion method and efficient step-length estimation.Geophysics, 80(4): R203-R216.
  12. Guitton, A. and Symes, W.W., 2003. Robust inversion of seismic data using the Hubernorm. Geophysics, 68: 1310-1319.
  13. Huber, P.J., 1973. Robust regression: asymptotics, conjectures, and Monte Carlo.Ann. Statist., 1: 799-821.
  14. Hager, W.W. and Zhang, H.C., 2006. A survey of nonlinear conjugate gradient methods.Pacif. J. Optimiz., 2: 35-58.
  15. Ha, T., Chung, W. and Shin, C., 2009. Waveform inversion using a backpropagationalgorithm and a Huber function norm. Geophysics ,74(3): R15-R24.
  16. Liu, D. and Nocedal, J., 1989. On the limited memory BFGS method for large scaleoptimization. Mathemat. Programm., 45: 503-528.
  17. Liu, Y.S., Teng, J.W., Xu, T., Wang, Y.H., Liu, Q.Y. and Jose, B., 2017. Robusttime-domain full waveform inversion with normalized zero-lag cross-correlationobjective function. Geophys. J. Internat., 209: 106-122.
  18. Liu, Y.S., Teng, J.W., Xu, T., Jose, B., Liu, Q.Y. and Zhou, B., 2017. Effects ofconjugate gradient methods and step-length formulas on the multiscale fullwaveform inversion in time domain: Numerical Experiments. Pure Appl. Geophys.,174: 1983-2206.
  19. Lailly, P., 1983. The seismic inverse problems as a sequence of beforenstack migration,
  20. Proc. Conf. Inverse Scattering, Theory and Application. Soc. Industr. Appl.Mathemat.: 206-220,
  21. Mora, P., 1987. Nonlinear two-dimensional elastic inversion of multioffset seismic data.Geophysics, 52: 1211-1228.
  22. Me'tivier, L., Bretaudeau, F., Brossier, R., Virieux, J. and Operto, S., 2014. Fullwaveform inversion and the truncated Newton method: quantitative imaging ofcomplex subsurface structures. Geophys. Prosp. 62: 1353-1375.
  23. Ma, X.N., Li, Z.Y., Xu, S.H., Ke, P. and Liang, G.H., 2017. Comparison offrequency-band selection strategies for 2D time-domain acoustic waveforminversion. J. Seismic Explor., 26: 499-519.
  24. Ma, X.N., Li, Z.Y., Ke, P., Xu, S.H., Liang, G.H. and Wu, X.Q., 2019. Research ofstep-length estimation methods for full waveform inversion in time domain.Explor. Geophys., 50: 583-599.
  25. Nocedal, J. and Wright, S.J., 1999. Numerical Optimization. Springer Verlag, Berlin.
  26. Pica, A., Diet, J. and Tarantola, A., 1990. Nonlinear inversion of seismic reflection datain a laterally invariant medium. Geophysics, 55: 284-292.
  27. Pratt, R.G. and Worthington, M.H., 1990. Inverse theory applied to multisourcecross-hole tomography. Part I: Acoustic wave-equation method. Geophys. Prosp., 38:287-310.
  28. Pratt, R.G., Shin, C. and Hicks, G.J., 1998. Gauss-Newton and full Newton methods infrequency-space seismic waveform inversion. Geophys. J. Internat., 133: 341-362.
  29. Plessix, R.E., 2006. A review of the adjoint-state method for computing the gradient of afunctional with geophysical applications. Geophys. J. Internat., 167: 495-503.
  30. Ravaut, C., Operto, S., Improta, L., Virieux, J., Herrero, A. and Dell’ Aversana, P.,
  31. Multiscale imaging of complex structures from multifold wide-apertureseismic data by frequency-domain full-waveform tomography: application to athrust belt. Geophys. J. Internat., 159: 1032-1056.
  32. Ren, Z.M., Liu, Y. and Zhang, Q.S., 2014. Multiscale viscoacoustic waveform inversionwith the second generation wavelet transform and adaptive time-space domainfinite-difference method. Geophys. J. Internat., 197: 948-974.
  33. Shin, C. and Cha, Y.H., 2008. Waveform inversion in the Laplace domain. Geophys. J.Internat., 173: 922-931.
  34. Tarantola, A., 1984. Inversion of seismic reflection data in the acoustic approximation.Geophysics, 49: 1259-1266.
  35. Tarantola, A., 1986. A strategy for nonlinear elastic inversion of seismic reflection data.Geophysics, 51: 1893-1903.
  36. Tape, C., Liu, Q.Y. and Tromp, J., 2007. Finite-frequency tomography using adjointmethods - Methodology and examples using membrane surface waves. Geophys. J.Internat., 168: 1105-1129.
  37. Hu, W.Y., Abubakar, A. and Habashy, T.M., 2009. Simultaneous multifrequencyinversion of full-waveform seismic data. Geophysics, 74(2): R1-R14.
  38. Vigh, D., Starr, E.W. and Kapoor, J., 2009. Developing earth models with full waveforminversion. The Leading Edge, 28: 432-435.
  39. Virieux, J. and Operto, S., 2009. An overview of full-waveform inversion in explorationgeophysics. Geophysics, 74(6): WCC1-WCC26.
  40. Wang, Y. and Rao, Y., 2006. Crosshole seismic waveform tomography I: Strategy forreal data application. Geophys. J. Internat., 166: 1224-1236.
  41. Warner, M., Ratcliffe, A., Nangoo, T., Morgan, J., Umpleby, A., Shah, N., Vinje, V.,
  42. Tekl, I., Guasch, L., Win, C., Conroy, G. and Bertrand, A., 2013. Anisotropic 3Dfull-waveform inversion. Geophysics, 78(2): R59-R80.
  43. Xu, K. and McMechan, G.A., 2014. 2D frequency-domain elastic full-waveforminversion using time-domain modeling and a multistep-length gradient approach.Geophysics, 79(2): R41-R53.
Share
Back to top
Journal of Seismic Exploration, Print ISSN: 0963-0651, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept