Abstract
A method for combining the asymptotic operator designed by Beylkin (Born migration operator) for the solution of linearized inverse problems with full waveform inversion is presented. This operator is used to modify the standard L2 norm that measures the distance between synthetic and observed data. The modified misfit function measures the discrepancy of the synthetic and observed data after they have been migrated using the Beylkin operator. The gradient of this new misfit function is equal to the cross-correlation of the single scattering data with migrated/demigrated residuals. The modified misfit function possesses a Hessian operator that tends asymptotically towards the identity operator. The trade-offs between discrete parameters are thus reduced in this inversion scheme. Results on 2-D synthetic case studies demonstrate the fast convergence of this inversion method in a migration regime. From an accurate estimation of the initial velocity, three and five iterations only are required to generate high-resolution P-wave velocity estimation models on the Marmousi 2 and synthetic Valhall case studies.
Highlights
Full Waveform Inversion (FWI) has reached today sufficient maturity for being routinely included in the industrial seismic imaging workflow
While in standard FWI the estimation first focusses on the shallow part and progressively reconstructs the deeper parts of the model, the modification introduced by the Beylkin operator allows to reconstruct deep structures in the very first iterations
The reconstructed models obtained after 10 l-BFGS iterations using the modified misfit function g(m) and 20 l-BFGS iterations using standard FWI with a diagonal pre-conditioning are compared to the exact and the initial models
Summary
Full Waveform Inversion (FWI) has reached today sufficient maturity for being routinely included in the industrial seismic imaging workflow. The two key ingredients involved in the FWI process are firstand second-order derivatives of the misfit function, namely the gradient and the Hessian operators These ingredients are used at each iteration to construct the update that corrects the current subsurface model estimation. This study suggests that the method presented here may be well suited in a quantitative migration context, for the reconstruction of high-resolution estimates of the subsurface parameters, as it is the case for instance in Metivier (2011) and Metivier et al (2011) It shows that far from the high-frequency regime, the asymptotic approximation has still a significant effect on the conditioning of the FWI problem
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