The waveform inversion problem is inherently ill-posed. Traditionally, regularization terms are used to address this issue. For waveform inversion, where the model is expected to have many details reflecting the physical properties of the Earth, regularization and data fitting can work in opposite directions, slowing down convergence. In this paper, we constrain the velocity model with a model-space preconditioning scheme based on directional Laplacian filters. This preconditioning strategy preserves the details of the velocity model while smoothing the solution along known geological dips. The Laplacian filters have the property to smooth or kill local events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform-inversion scheme to yield geologically meaningful models. We illustrate on a 2-D synthetic example how preconditioning with non-stationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted for. We think that preconditioning could benefit waveform inversion of real data where (for instance) irregular geometry, coherent noise and lack of low frequencies are present.


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