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.
Two related formulations are proposed for target-oriented joint least-squares migration/inversion of time-lapse seismic data sets. Time-lapse seismic images can be degraded when reservoir overburden is complex or when acquisition geometries significantly differ, because the migration operator does not compensate for the resulting amplitude and phase distortions. Under these circumstances, time-lapse amplitudes are poor indicators of production-related changes in reservoir properties. To correct for such image degradation, time-lapse imaging is posed as joint inverse problems that utilize concatenations of target-oriented approximations to the linear least-squares imaging Hessian. In both formulations, spatial and temporal constraints ensure inversion stability and geologically consistent time-lapse images. Using two numerical time-lapse data sets, we confirmed that these formulations can attenuate illumination artifacts caused by complex overburden or geometry differences, and that they yield better-quality images than obtainable with migration.
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