A numerical study of multi-parameter full waveform inversion with iterative regularization using multi-frequency vibroseis data

  • Jia ShiEmail author
  • Elena Beretta
  • Maarten V. de Hoop
  • Elisa Francini
  • Sergio Vessella
Original Paper


We study the inverse boundary value problem for time-harmonic elastic waves, for the recovery of P- and S-wave speeds from vibroseis data or the Neumann-to-Dirichlet map. Our study is based on our recent result pertaining to the uniqueness and a conditional Lipschitz stability estimate for parametrizations on unstructured tetrahedral meshes of this inverse boundary value problem. With the conditional Lipschitz stability estimate, we design a procedure for full waveform inversion (FWI) with iterative regularization. The iterative regularization is implemented by projecting gradients, after scaling, onto subspaces associated with the mentioned parametrizations yielding Lipschitz stability. The procedure is illustrated in computational experiments using the continuous Galerkin finite element method of recovering the rough shapes and wave speeds of geological bodies from simple starting models, near and far from the boundary, that is, the free surface.


Full waveform inversion Finite element method Stability and convergence 


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J.S. would like to thank Jean Virieux, Peter Caday, Ruichao Ye, and Florian Faucher for useful discussions. The authors would also like to thank two anonymous referees for their helpful comments.

Funding information

MVdH and JS were financially supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and by the members of the Geo-Mathematical Group at Rice University.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jia Shi
    • 1
    Email author
  • Elena Beretta
    • 2
  • Maarten V. de Hoop
    • 3
  • Elisa Francini
    • 4
  • Sergio Vessella
    • 4
  1. 1.Department of Earth, Environmental and Planetary SciencesRice UniversityHoustonUSA
  2. 2.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanItaly
  3. 3.Simons Chair in Computational and Applied Mathematics and Earth ScienceRice UniversityHoustonUSA
  4. 4.Dipartimento di Matematica e Informatica “U. Dini”Universita di FirenzeMilanItaly

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