As full-waveform inversion (FWI) is a nonunique and typically ill-posed inversion problem, it needs proper regularization to avoid cycle skipping. To reduce the nonlinearity of FWI, we have developed joint migration inversion (JMI) as an alternative, explaining the reflection data with decoupled velocity and reflectivity parameters. However, the velocity update may also suffer from being trapped in local minima. To optimally include geologic information, we have developed FWI/JMI with directional total variation (TV) as an L1-norm regularization on the velocity. We design the directional TV operator based on the local dip field, instead of ignoring the local structural direction of the subsurface and only using horizontal and vertical gradients in the traditional TV. The local dip field is estimated using plane-wave destruction based on a raw reflectivity model, which is usually calculated from the initial velocity model. With two complex synthetic examples, based on the Marmousi model, we determine that our method is much more effective compared with FWI/JMI without regularization and FWI/JMI with the conventional TV regularization. In the JMI-based example, we also determine that L1 directional TV works better than L2 directional Laplacian smoothing. In addition, by comparing these two examples, it can be seen that the impact of regularization is larger for FWI than for JMI because in JMI the velocity model only explains the propagation effects and, thereby, makes it less sensitive to the details in the velocity model.
Total Variation Regularization Strategies in Full-Waveform Inversion
