Constrained full-waveform inversion by model reparameterization

The waveform inversion problem is inherently ill-posed. Traditionally, regularization schemes 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: the former smoothing and the latter adding details to the model. We propose constraining estimated velocity fields by reparameterizing the model. This technique, also called model-space preconditioning, is based on directional Laplacian filters: It preserves most of the details of the velocity model while smoothing the solution along known geological dips. Preconditioning also yields faster convergence at early iterations. The Laplacian filters have the property to smooth or kill local planar events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models. We illustrate with 2D synthetic and field data examples how preconditioning with nonstationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted and when sharp velocity contrasts are present. Adding geological information with preconditioning could benefit full-waveform inversion of real data whenever irregular geometry, coherent noise and lack of low frequencies are present.