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On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

Full waveform inversion is a PDE-constrained nonlinear least-squares problem dedicated to the estimation of the mechanical subsurface properties with high resolution. Since its introduction in the early 80s, a limitation of this method is related to the non-convexity of the misfit function which is minimized when the method is applied to the estimation of the subsurface wave velocities. Recently, the definition of an alternative misfit function based on an optimal transport distance has been proposed to mitigate this difficulty. In this study, we review the difficulties for exploiting standard optimal transport techniques for the comparison of seismic data. The main difficulty is related to the oscillatory nature of the seismic data, which requires to extend optimal transport to the transport of signed measures. We review three standard possible extensions relying on a decomposition of the data into its positive and negative part. We show how the two first might not be adapted for full waveform inversion and focus on the third one. We present a numerical strategy based on the dual formulation of a particular optimal transport distance yielding an efficient implementation. The interest of this approach is illustrated on the 2D benchmark Marmousi model.

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Acknowledgements

This study was partially funded by the SEISCOPE consortium ( http://seiscope2.osug.fr ), sponsored by CGG, CHEVRON, EXXON-MOBIL, JGI, SHELL, SINOPEC, STATOIL, TOTAL, and WOODSIDE. This study was granted access to the HPC resources of the Froggy platform of the CIMENT infrastructure (https://ciment.ujf-grenoble.fr), which is supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA), the OSUG@2020 labex (reference ANR10 LABX56), and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche, and the HPC resources of CINES/IDRIS/TGCC under the allocation 046091 made by GENCI.

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Authors and Affiliations

  1. ISTerre/LJK, CNRS, Univ. Grenoble Alpes, Saint-Martin-d’Hères, France

    L. Métivier

  2. LJK, Univ. Grenoble Alpes, Saint-Martin-d’Hères, France

    A. Allain & E. Oudet

  3. Univ. Grenoble Alpes, ISTerre, Grenoble, France

    R. Brossier & J. Virieux

  4. LMO, Univ. Paris Sud, Orsay, France

    Q. Mérigot

Authors
  1. L. Métivier
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  2. A. Allain
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  3. R. Brossier
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  4. Q. Mérigot
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  5. E. Oudet
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  6. J. Virieux
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Corresponding author

Correspondence to L. Métivier .

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Editors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

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Métivier, L., Allain, A., Brossier, R., Mérigot, Q., Oudet, E., Virieux, J. (2018). On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_11

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