Enhancing Full Waveform Inversion with a deep-learning approximated inverse Hessian: A field data application

In image-domain Least-Squares Migration (LSM), a demigration-migration approach can be employed to identify non-stationary local filters that approximate the inverse Hessian operator. Such filters can be subsequently applied to the original migrated image to compensate for uneven illumination, undo the effect of geometrical spreading, and enhance its resolution. A similar approach can be applied at each iteration of Full Waveform Inversion (FWI) by taking the FWI gradient as input to the demigration-migration process. By doing so, a more balanced update can be constructed whereby both the shallow and deep parts of the model are updated with similar strength. Following our recent line of work, we propose here to approximate the effect of the inverse Hessian with a neural network, which is trained to map the doubly migrated gradient into the FWI gradient. The trained network is later applied to the gradient itself to produce an improved FWI model update. Compared to conventionally used non-stationary local filters, the network can be trained only once (at the first FWI iteration) and cheaply fine-tuned at any subsequent iteration. In this work, we apply the proposed methodology to a challenging field dataset, namely the 2010 ocean-bottom cable Volve dataset. As commonly done in FWI, our approach is naturally embedded into a multi-scale approach where three different frequency bands are subsequently inverted and the network used to approximate the inverse Hessian is re-trained at each outer iteration. When compared to state-of-the-art quasi-Newton methods, the model obtained using our approach is shown to produce images of superior quality and flatter as well as more focused angle gathers.