Multi-level elastic full waveform inversion in isotropic media via quantitative Lipschitz stability estimates

Abstract : We study the seismic inverse problem for the (complex) frequency-domain elastic isotropic wave equation; in particular the recovery of the Lamé parameters and density. We consider the case of land acquisition with interior sources. We employ a Full Waveform Inversion where the iterative minimization is based on a gradient descent. The elastic inverse problem shows a Lipschitz-type stability where the Fréchet derivative has a strictly positive `lower bound'. This bound is connected to the stability constant and can be approximated using the Gauss-Newton Hessian. Then we effectively estimate the stability by computing the smallest singular value and condition number of the Gauss-Newton Hessian. The successive stability estimates provides a control of the convergence of our algorithm. It helps us decide on the parametrization (quantities to inverse) and the model representation (partitioning). We develop a multi-level approach with a hierarchical compressed reconstruction. The compression is based on a structured domain partitioning of the sub-surface, while the hierarchy is established through successive refinement of the partitioning. The coefficients (Lamé parameters and density) are assumed to be piecewise constant functions following the domain partitioning.The partitioning is naturally defined with the successive stability estimates to maintain the radius of convergence, while refinement provides resolution. It allows us to start with minimal prior information for the coefficients. The multi-parameters FWI follows two stages driven by the stability and convergence analysis. The Lamé parameters are reconstructed jointly while assuming an unknown fixed density model. We can also reconstruct the density assuming the knowledge of the Lamé parameters. The algorithm is perfectly suitable for complex frequency and we carry out numerical experiments in two and three dimensions.
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Contributor : Florian Faucher <>
Submitted on : Wednesday, December 17, 2014 - 10:22:18 AM
Last modification on : Friday, April 12, 2019 - 10:42:02 AM


  • HAL Id : hal-01096272, version 1


Florian Faucher, Jia Shi, Maarten V. de Hoop, Henri Calandra. Multi-level elastic full waveform inversion in isotropic media via quantitative Lipschitz stability estimates. MATHIAS – TOTAL Symposium on Mathematics, TOTAL, Oct 2014, Paris, France. ⟨hal-01096272⟩



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