Elastic isotropic full waveform inversion via quantitative stability estimates
Résumé
We study the seismic inverse problem in the complex frequency-domain. Seismic inversion is realised for
depth imaging and standardly uses the Full Waveform Inversion (FWI) method. FWI utilizes surface
observation of waves phenomena, and minimize their difference with a simulated propagation, which
makes use of an approximate model. The following iterative minimization of those residuals leads to the
update of the approximated model and eventually the reconstruction of the targeted parameter (velocity,
density).
We focus on the inverse problem for the elastic isotropic case and recovery of the Lamé parameters and
density. Our FWI algorithm is based on the projected descent method. Recent results on the conditional
well-posedness [1] show that the problem follows 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 of the Gauss-Newton Hessian. The successive stability estimates provide a
control of the convergence of our algorithm, it decides on the parametrization (quantities to inverse) and
the model representation (partitioning). Hence 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 refinements 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 in order to
maintain the radius of convergence, while refinement provides resolution. It allows us to start with minimal
prior information for the coefficients. Moreover 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 for elastic reconstruction in two and three dimensions.