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

XIAONA MA
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Institute of Geophysics, China Earthquake Administration, Beijing 100081, P.R. China.,
JSE 2021, 30(4), 347–364;
Submitted: 9 June 2025 | Revised: 9 June 2025 | Accepted: 9 June 2025 | Published: 9 June 2025
© 2025 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/ )
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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 electromagnetic
  2. waves. J. Computat. Phys., 114: 185-200.
  3. Bunks, C., Saleck, F.M., Zaleski, S. and Chavent, G., 1995. Multiscale seismic
  4. waveform inversion. Geophysics, 60: 1457-1473.
  5. Bube, K.P. and Nemeth, T., 2007. Fast line searches for the robust solution of linear
  6. systems in the hybrid and Huber norms. Geophysics, 72(2): A13-A17.
  7. Boonyasiriwat, C., Valasek, P., Routh, P., Cao, W., Schuster, G.T. and Macy, B., 2009.
  8. An efficient multiscale method for time-domain waveform tomography. Geophysics,
  9. 74(6): WCC59-WCC68.
  10. Brossier, R., Operto, S. and Virieux, J., 2009. Robust elastic frequency domain full
  11. waveform inversion using the L1 norm. Geophys. Res. Lett., 36: L20310.
  12. Brossier, R., Operto, S. and Virieux, J., 2010. Which data residual norm for robust elastic
  13. frequency-domain full waveform inversion? Geophysics, 75(3): R37-R46.
  14. Crase, E., Pica, A., Noble, M., McDonald, J. and Tarantola, A., 1990. Robust elastic
  15. nonlinear waveform inversion: Application to real data. Geophysics, 55: 527-538.
  16. Djikpéssé, H.A. and Tarantola, A., 1999. Multiparameter L1 norm waveform fitting:
  17. Interpretation of Gulf of Mexico reflection seismograms. Geophysics, 64:
  18. 1023-1035.
  19. Dos Santos, A.W.G. and Pestana. R.C., 2015. Time-domain multiscale full-waveform
  20. inversion using the rapid expansion method and efficient step-length estimation.
  21. Geophysics, 80(4): R203-R216.
  22. Guitton, A. and Symes, W.W., 2003. Robust inversion of seismic data using the Huber
  23. norm. Geophysics, 68: 1310-1319.
  24. Huber, P.J., 1973. Robust regression: asymptotics, conjectures, and Monte Carlo.
  25. Ann. Statist., 1: 799-821.
  26. Hager, W.W. and Zhang, H.C., 2006. A survey of nonlinear conjugate gradient methods.
  27. Pacif. J. Optimiz., 2: 35-58.
  28. Ha, T., Chung, W. and Shin, C., 2009. Waveform inversion using a backpropagation
  29. algorithm and a Huber function norm. Geophysics ,74(3): R15-R24.
  30. Liu, D. and Nocedal, J., 1989. On the limited memory BFGS method for large scale
  31. optimization. Mathemat. Programm., 45: 503-528.
  32. Liu, Y.S., Teng, J.W., Xu, T., Wang, Y.H., Liu, Q.Y. and Jose, B., 2017. Robust
  33. time-domain full waveform inversion with normalized zero-lag cross-correlation
  34. objective function. Geophys. J. Internat., 209: 106-122.
  35. Liu, Y.S., Teng, J.W., Xu, T., Jose, B., Liu, Q.Y. and Zhou, B., 2017. Effects of
  36. conjugate gradient methods and step-length formulas on the multiscale full
  37. waveform inversion in time domain: Numerical Experiments. Pure Appl. Geophys.,
  38. 174: 1983-2206.
  39. Lailly, P., 1983. The seismic inverse problems as a sequence of beforenstack migration,
  40. Proc. Conf. Inverse Scattering, Theory and Application. Soc. Industr. Appl.
  41. Mathemat.: 206-220,
  42. Mora, P., 1987. Nonlinear two-dimensional elastic inversion of multioffset seismic data.
  43. Geophysics, 52: 1211-1228.
  44. Me'tivier, L., Bretaudeau, F., Brossier, R., Virieux, J. and Operto, S., 2014. Full
  45. waveform inversion and the truncated Newton method: quantitative imaging of
  46. complex subsurface structures. Geophys. Prosp. 62: 1353-1375.
  47. Ma, X.N., Li, Z.Y., Xu, S.H., Ke, P. and Liang, G.H., 2017. Comparison of
  48. frequency-band selection strategies for 2D time-domain acoustic waveform
  49. inversion. J. Seismic Explor., 26: 499-519.
  50. Ma, X.N., Li, Z.Y., Ke, P., Xu, S.H., Liang, G.H. and Wu, X.Q., 2019. Research of
  51. step-length estimation methods for full waveform inversion in time domain.
  52. Explor. Geophys., 50: 583-599.
  53. Nocedal, J. and Wright, S.J., 1999. Numerical Optimization. Springer Verlag, Berlin.
  54. Pica, A., Diet, J. and Tarantola, A., 1990. Nonlinear inversion of seismic reflection data
  55. in a laterally invariant medium. Geophysics, 55: 284-292.
  56. Pratt, R.G. and Worthington, M.H., 1990. Inverse theory applied to multisource
  57. cross-hole tomography. Part I: Acoustic wave-equation method. Geophys. Prosp., 38:
  58. 287-310.
  59. Pratt, R.G., Shin, C. and Hicks, G.J., 1998. Gauss-Newton and full Newton methods in
  60. frequency-space seismic waveform inversion. Geophys. J. Internat., 133: 341-362.
  61. Plessix, R.E., 2006. A review of the adjoint-state method for computing the gradient of a
  62. functional with geophysical applications. Geophys. J. Internat., 167: 495-503.
  63. Ravaut, C., Operto, S., Improta, L., Virieux, J., Herrero, A. and Dell’ Aversana, P.,
  64. Multiscale imaging of complex structures from multifold wide-aperture
  65. seismic data by frequency-domain full-waveform tomography: application to a
  66. thrust belt. Geophys. J. Internat., 159: 1032-1056.
  67. Ren, Z.M., Liu, Y. and Zhang, Q.S., 2014. Multiscale viscoacoustic waveform inversion
  68. with the second generation wavelet transform and adaptive time-space domain
  69. finite-difference method. Geophys. J. Internat., 197: 948-974.
  70. Shin, C. and Cha, Y.H., 2008. Waveform inversion in the Laplace domain. Geophys. J.
  71. Internat., 173: 922-931.
  72. Tarantola, A., 1984. Inversion of seismic reflection data in the acoustic approximation.
  73. Geophysics, 49: 1259-1266.
  74. Tarantola, A., 1986. A strategy for nonlinear elastic inversion of seismic reflection data.
  75. Geophysics, 51: 1893-1903.
  76. Tape, C., Liu, Q.Y. and Tromp, J., 2007. Finite-frequency tomography using adjoint
  77. methods - Methodology and examples using membrane surface waves. Geophys. J.
  78. Internat., 168: 1105-1129.
  79. Hu, W.Y., Abubakar, A. and Habashy, T.M., 2009. Simultaneous multifrequency
  80. inversion of full-waveform seismic data. Geophysics, 74(2): R1-R14.
  81. Vigh, D., Starr, E.W. and Kapoor, J., 2009. Developing earth models with full waveform
  82. inversion. The Leading Edge, 28: 432-435.
  83. Virieux, J. and Operto, S., 2009. An overview of full-waveform inversion in exploration
  84. geophysics. Geophysics, 74(6): WCC1-WCC26.
  85. Wang, Y. and Rao, Y., 2006. Crosshole seismic waveform tomography I: Strategy for
  86. real data application. Geophys. J. Internat., 166: 1224-1236.
  87. Warner, M., Ratcliffe, A., Nangoo, T., Morgan, J., Umpleby, A., Shah, N., Vinje, V.,
  88. Tekl, I., Guasch, L., Win, C., Conroy, G. and Bertrand, A., 2013. Anisotropic 3D
  89. full-waveform inversion. Geophysics, 78(2): R59-R80.
  90. Xu, K. and McMechan, G.A., 2014. 2D frequency-domain elastic full-waveform
  91. inversion using time-domain modeling and a multistep-length gradient approach.
  92. Geophysics, 79(2): R41-R53.
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Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
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