We study the seismic inverse problem for elastic isotropic medium associated with the time-harmonic wave equation, in particular the recovery of the Lamé parameters. We employ full waveform inversion (FWI) where the multi parameters reconstruction is based on iterative minimization techniques. This inverse problem shows a Lipschitz stability where the stability constant is related to the (conditional) lower bound of the Frchet derivative, when assuming a piecewise constant representation of the parameters. We successively estimate the stability constant for different model partition to control the convergence of our scheme. Hence we define a multi-level (multi-scale, multi-frequency) algorithm where the natural progression of frequency is paired with the model partition. We implement our new scheme and present numerical experiments for elastic parameters reconstruction assuming no-prior information in the initial models.


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