This website uses cookies to improve your experience. If you continue without changing your settings, you consent to our use of cookies in accordance with our cookie policy. You can disable cookies at any time.

×
Advanced search
Skip main navigation

Previous
Next
No AccessGEOPHYSICSVolume 83, Issue 4

Source estimation for wavefield-reconstruction inversion

Authors:
https://doi.org/10.1190/geo2017-0700.1

ABSTRACT

Source estimation is essential for all wave-equation-based seismic inversions, including full-waveform inversion (FWI) and the recently proposed wavefield-reconstruction inversion (WRI). When the source estimation is inaccurate, errors will propagate into the predicted data and introduce additional data misfit. As a consequence, inversion results that minimize this data misfit may become erroneous. To mitigate the errors introduced by the incorrect and preestimated sources, an embedded procedure that updates sources along with medium parameters is necessary for the inversion. So far, such a procedure is still missing in the context of WRI, a method that is, in many situations, less prone to local minima related to so-called cycle skipping, compared with FWI through exact data fitting. Although WRI indeed helps to mitigate issues related to cycle skipping by extending the search space with wavefields as auxiliary variables, it relies on having access to the correct source functions. To remove the requirement of having the accurate source functions, we have developed a source-estimation technique specifically designed for WRI. To achieve this task, we consider the source functions as unknown variables and arrive at an objective function that depends on the medium parameters, wavefields, and source functions. During each iteration, we apply the so-called variable projection method to simultaneously project out the source functions and wavefields. After the projection, we obtain a reduced objective function that only depends on the medium parameters and invert for the unknown medium parameters by minimizing this reduced objective. Numerical experiments illustrate that this approach can produce accurate estimates of the unknown medium parameters without any prior information of the source functions.

REFERENCES

  • Akcelik, V., 2002, Multiscale Newton-Krylov methods for inverse acoustic wave propagation: Ph.D. thesis, Carnegie Mellon University.
    Google Scholar
  • Amestoy, P., R. Brossier, A. Buttari, J.-Y. L’Excellent, T. Mary, L. Metivier, A. Miniussi, and S. Operto, 2016, Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea: Geophysics, 81, no. 6, R363–R383, doi: 10.1190/geo2016-0052.1.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Aravkin, A. Y., and T. van Leeuwen, 2012, Estimating nuisance parameters in inverse problems: Inverse Problems, 28, 115016, doi: 10.1088/0266-5611/28/11/115016.INPEEY0266-5611
    Web of ScienceGoogle Scholar
  • Askan, A., V. Akcelik, J. Bielak, and O. Ghattas, 2007, Full waveform inversion for seismic velocity and anelastic losses in heterogeneous structures: Bulletin of the Seismological Society of America, 97, 1990–2008, doi: 10.1785/0120070079.BSSAAP0037-1106
    Web of ScienceGoogle Scholar
  • Bernstein, D. S., 2005, Matrix mathematics: Theory, facts, and formulas with application to linear systems theory: Princeton University Press.
    Google Scholar
  • Brossier, R., S. Operto, and J. Virieux, 2009, Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion: Geophysics, 74, no. 6, WCC105–WCC118, doi: 10.1190/1.3215771.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Chen, Z., D. Cheng, W. Feng, and T. Wu, 2013, An optimal 9-point finite difference scheme for the Helmholtz equation with PML: International Journal of Numerical Analysis and Modeling, 10, 389–410.
    Web of ScienceGoogle Scholar
  • Golub, G., and V. Pereyra, 2003, Separable nonlinear least squares: The variable projection method and its applications: Inverse Problems, 19, R1–R26, doi: 10.1088/0266-5611/19/2/201.INPEEY0266-5611
    Web of ScienceGoogle Scholar
  • Huang, G., R. Nammour, and W. Symes, 2017, Full-waveform inversion via source-receiver extension: Geophysics, 82, no. 3, R153–R171, doi: 10.1190/geo2016-0301.1.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Li, M., J. Rickett, and A. Abubakar, 2013, Application of the variable projection scheme for frequency-domain full-waveform inversion: Geophysics, 78, no. 6, R249–R257, doi: 10.1190/geo2012-0351.1.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Li, X., A. Y. Aravkin, T. van Leeuwen, and F. J. Herrmann, 2012, Fast randomized full-waveform inversion with compressive sensing: Geophysics, 77, no. 3, A13–A17, doi: 10.1190/geo2011-0410.1.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Liu, J., A. Abubakar, T. Habashy, D. Alumbaugh, E. Nichols, and G. Gao, 2008, Nonlinear inversion approaches for cross-well electromagnetic data collected in cased-wells: 78th Annual International Meeting, SEG, Expanded Abstracts, 304–308.
    Google Scholar
  • Nocedal, J., and S. J. Wright, 2006, Numerical optimization: Springer-Verlag.
    Google Scholar
  • Peters, B., C. Greif, and F. J. Herrmann, 2015, An algorithm for solving least-squares problems with a Helmholtz block and multiple right-hand-sides: Presented at the International Conference on Preconditioning Techniques for Scientific and Industrial Applications.
    Google Scholar
  • Peters, B., and F. J. Herrmann, 2017, Constraints versus penalties for edge-preserving full-waveform inversion: The Leading Edge, 36, 94–100, doi: 10.1190/tle36010094.1.
    Google Scholar
  • Peters, B., F. J. Herrmann, and T. van Leeuwen, 2014, Wave-equation based inversion with the penalty method: Adjoint-state versus wavefield-reconstruction inversion: 76th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: 10.3997/2214-4609.20140704.
    Google Scholar
  • Plessix, R.-E., 2006, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications: Geophysical Journal International, 167, 495–503, doi: 10.1111/j.1365-246X.2006.02978.x.GJINEA0956-540X
    Web of ScienceGoogle Scholar
  • Pratt, R. G., 1999, Seismic waveform inversion in the frequency domain. Part 1: Theory and verification in a physical scale model: Geophysics, 64, 888–901, doi: 10.1190/1.1444597.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Symes, W. W., D. Sun, and M. Enriquez, 2011, From modeling to inversion: Designing a well-adapted simulator: Geophysical Prospecting, 59, 814–833, doi: 10.1111/j.1365-2478.2011.00977.x.GPPRAR0016-8025
    Web of ScienceGoogle Scholar
  • Tarantola, A., and B. Valette, 1982, Generalized nonlinear inverse problems solved using the least squares criterion: Reviews of Geophysics, 20, 219–232, doi: 10.1029/RG020i002p00219.REGEEP8755-1209
    Web of ScienceGoogle Scholar
  • van Leeuwen, T., A. Y. Aravkin, and F. J. Herrmann, 2014, Comment on: “Application of the variable projection scheme for frequency-domain full-waveform inversion” (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249–R257): Geophysics, 79, no. 3, X11–X17, doi: 10.1190/geo2013-0466.1.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2013a, Fast waveform inversion without source encoding: Geophysical Prospecting, 61, 10–19, doi: 10.1111/j.1365-2478.2012.01096.x.GPPRAR0016-8025
    Web of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2013b, Mitigating local minima in full-waveform inversion by expanding the search space: Geophysical Journal International, 195, 661–667, doi: 10.1093/gji/ggt258.GJINEA0956-540X
    Web of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2015, A penalty method for PDE-constrained optimization in inverse problems: Inverse Problems, 32, 015007, doi: 10.1088/0266-5611/32/1/015007.INPEEY0266-5611
    Web of ScienceGoogle Scholar
  • Virieux, J., and S. Operto, 2009, An overview of full-waveform inversion in exploration geophysics: Geophysics, 74, no. 6, WCC1–WCC26, doi: 10.1190/1.3238367.GPYSA70016-8033
    Web of ScienceGoogle Scholar
  • Warner, M., T. Nangoo, N. Shah, A. Umpleby, and J. Morgan, 2013, Full-waveform inversion of cycle-skipped seismic data by frequency down-shifting: 83rd Annual International Meeting, SEG, Expanded Abstracts, 903–907.
    Google Scholar
  • Zhou, B., and S. A. Greenhalgh, 2003, Crosshole seismic inversion with normalized full-waveform amplitude data: Geophysics, 68, 1320–1330, doi: 10.1190/1.1598125.GPYSA70016-8033
    Web of ScienceGoogle Scholar